The Theory, Construction, and Field Use of a Direct Current Potentiometer for Measuring Earth Resistivity Citation Doolittle, Russell Carter (1940) The Theory, Construction, and Field Use of a Direct Current Potentiometer for Measuring Earth Resistivity. Master's thesis, California Institute of Technology. doi:10.7907/D941-WY06. https://resolver.caltech.edu/CaltechTHESIS:03122010-083823737 Abstract The fundamental theory underlying direct current methods of measuring "apparent" earth resistivity and of interpreting these field measurements to obtain the actual resistivities of the parts of a composite earth is extensively reviewed. The three main types of apparatus used in making the field measurements, the Gish-Rooney, "Megger", and "porous pot" instruments, are briefly outlined. Considerations involved in making a general design for a "porous pot" outfit are briefly discussed, and a detailed description of the apparatus used by the writer is given. Field data obtained with this apparatus is interpreted by the several methods discussed in the literature. Item Type: Thesis (Master's thesis) Subject Keywords: Geology Degree Grantor: California Institute of Technology Division: Geological and Planetary Sciences Major Option: Geology Thesis Availability: Public (worldwide access) Research Advisor(s): Peterson, R. A. Thesis Committee: Unknown, Unknown Defense Date: 1 June 1940 Record Number: CaltechTHESIS:03122010-083823737 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:03122010-083823737 DOI: 10.7907/D941-WY06 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5594 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 13 Apr 2010 17:58 Last Modified: 16 Aug 2023 23:35 Thesis Files Preview PDF - Final Version See Usage Policy. 180MB Repository Staff Only: item control page

The THEORY, CONSTAUCLION, AND FIELD USE of a DIRECT CURREN? POTENTIOMETER for MEASURING EARTH RESISTIVITY by Russell Carter Doolittle Thesis Submitted in Partisl FPulfilwent of the Requirements for the Mester of Science Degree Division of Geological Seiences California Institute of Technology Pasadena, California June 1, 1940 Chapter I II Tit iv vil CONTENTS Abstract Acknowledgements Introduction Theory of direct eurrent methods of measuring earth resistivit Definitions and songepts Derivation of resistivity formlae Infinite homogeneous medium Semi-infinite medium Potential and current distribution in & Bemi-infinite medium Errors in basic assumptions Methed of images in & semi-infinite . nedium Method of images in the cape of two horizontal layers Methed of integrals in the case of swo horizontal layers Depth of penetration and resolving power Design and construction of the appe- retus General considerations Detailed design and construction Primary cirouit Potential cireuit Porous pote Potentiometer cirevuit Conditioning cireuit Hull instrument Field work Introduction Hastings Rench survey Delano survey Victerville survey interpretation of apparent resistivity~- electrode seperation eurves introduction General outline of methods Preparation of two-layer master Tage ' igina) meth a 8 or ad metho Tegg's modified method Pirgon's successive a: proximation nethod (i) Page ? 12 14 17a 18 21 26 23 CONTENTS continued Chepter Page yit Roman's superposition method 93 continued Use of three-layer theoret. ical curves Paletea superposition method 36 Bibliography 104 TABLES I Resistivities of common roeks and rock materials 3a Iz Data from Hastings Ranch near Sierra Madre, California 74a Tit Dats obtained near Delanc, California 74h IV Data obtained near Vietorville, California 742 Vv Roman's table for twoeluyer master gurves 788 VI Interpolations in Table V 782 VII Values of %</7 derived from Table Vo 80a VIII Wetsel and MeMurry tables fer three. layer master curves 948 (34) I Abstract the fundamental theory underlying direct current methods of measuring “apparent" earth resistivity and of interpreting these field meugurements to obtain the actual resistivities of the parts of a composite earth is extensively reviewed. The three main types of apparatus used in miking the field measurements, the Cish-Rooney, "Megger", and "porous pot" instruments arg briefly outlined. Considerations iuvolved in meking a general design for a “porous pot" outfit are briefly discussed, and « detailed description of the apparatus used by the writer is given. Field data obtuined with this sppsratus is interpreted by the severul methods discussed in the liter..ture. (1) il Acknowledgements The subject of this thesis was suggested by Dr. Re ‘ Peterson of the Culifornio Institute Division of Geological Sciences, whose supervision and helpful suggestions have been of great value. Funds for buying parts for the apparatus and for paying field expenses were kindly furnished by the Division of Ge.logi¢cal selences together with the use of its 1931 Yord station wagone The Inatitute Physice Depurtment allowed the use of electrical inatruments borrowed from its stock room whiel afforded a considernble saving of expense. the writer is yvrateful thut liis problem and that of Martin Nichelberger required in purt the use of the vame equipment ‘9 the eneryy ond skill of the latter were invaluable in ussembling the battery ond potentio- meter units, constructing an amplifier, and cooperating in some of the field tests. Mr. RK. We. Lohman of South Magadena generously sugwested the locution of the test near Sierra Madre and uyon its completion allowed the writer to compure results with some he had previously obtained. ‘ermission to run the profile ot the voint sugested on the Hustings Ranch was kindly given by Mr, Christy, and the carrying out of field operations was made possible through the assistance of Nessre. Wm. Bonell, Robert Hull, and om. Keorris. The second test made neur Delano, Cale was suggested by Dr, Peterson (2) and carried out with the assistance of Mr. Bichelberger. The United Geophysical Company kindly allowed the use of a truck and cable in the field. Dr. Maxson of the Insti- tute Geology Department spent a day in the field outlining the work at Victorville which was financed by the department and carried out with the help of Dr. Robert Dreyer and Mr. Ellis Roberts. Finally, the writer is greatly indebted to Dr. Potapenko for valuable advice and eriticism, ¢s- pecially on interpreting field curves with the Paletoa | method. (3) TABLE I Some Resistivities of Common Kocks und Rock Materials This table indieates the wide range of resistivities commonly found for different rock typea, and for the game rock type under different conditions by different observers, It is by no means comolete ana in most cuses is of value only in suggesting the order of ma;nitude to be expected. ‘the vulues ure taken from the following sources: (a) we and Keys, Applied Geophysics, 2nd. edition, p. 94, Table I. Taken re AmbronnLehrbuch der .nwvewandten Geophysik pe 110; Sundberg, Lundberg, and “klund.- | ‘lectrical ae in Sweden ppe 11-123 Gutenberg~Lehrbuch der Geophysik p. 571 (vo) Swe and Keys, Loe. Cit. Yootnote-aTaken from Koenigeaberger, AIME Tech. ?uper #125, 1928 {c) Sherwin Kelley~.A4 Uniform ™xpression for Resistivity AIMEE Geophysical Prospecting, 1932, pe 142. (4) Karl Sundberg-.ffect of Impregnating Waters on Electrical Conductivity of toils nnd Kooks, ALMis Geoohysical Prospecting, 1935, De GSB. (@) “otapenko--C,I.T., Lecture in Course on “‘lectrical Methods of Geophysical Prospecting, 1940. (f£) Bdge and faby~.I.G. 1.5. pe 11, Table II. Ae Ulectrolytes Material Resistivity (ohm-om.} Source Surface waters: Arens of voleanie rocks end crystalline schists 3% 109, -5 x1io* 4 \reas of younger sediments L0”..~10 d Seu water About 25 8 Salt water 25-10 @ Ground waters? 105...104 € Normal (potable) 103..195 f Areas of volcanics and achists 3 x 10%-.1.5 x 104 4 Areas of younger sediments 100 and greuter d Suline (1% Nacl 75.0 f 5% NaCl 15.0 : 10% =~ 8.25 f 20% Nec 5.10 f Connate waters: Areas of younger sediments 10 and greater a B. Metallic Ore Minerals Zinc blende, hematite, limonite, stibgite, wolframite 10°_.106 r Zine blende 1.5 x 108 a Moly bdenite LawSQ ft (3a) TABLN I (Cont'd) Vaterial Reaistivity (ohm-~om.) Source Chalcopyrite, bornite, chulcocitea, pyrite, pyrrhotite, galens, magnetite, svecular hematite 1074.21 f Chaleopyrite 1,0 | a Pyrrhotite 0.01 & Galena 0. 00340540 e Nagnetite 0.6 a Cuprite Oe Rw ols O @ Copper 1.7 x 1078 a Lead 21 x 1075 a Mercury 0.96 x 1074 a Silver 1.6 x 10°6 a C. Dielectrics Coal i t 109 e Gypsum tote @ Vein quartz | 1029, 1942 @ Caleites (chalk, marble) 108 .-10 4 @ Mica (biotite muscovite) 1029. 1015 e Rocksalt (mineral) 10> e Sulphur 1027 _ 21018 e Crude oil 1040. 1948 @ D Ores and Rocks Formation with about 5% sulphides 108 e 5S to 20% 10, e 20% to BO’ 105 e 50% or more 10 @ Blende-galens ore (Illinois, UsSs Ac, 4 dry } 166 x 10, a Illinois, U.SeAe, moist) 1.5 x 10, a Sweden, dry) lo 3 x 10% a (Sweden, moist) 3.0 x 104 a Pyrite ore, dry 1.0 x 10, & Pyrite ore, moist 6.0 x 1g° a Reeksalt (in ground) 10° w=, 6 b Igneous rocks, gneiss, schist 2 % 10 gr 0 f Teneous rocks 105-1054 8 Granite 109~-~10 a porphyry 10 5 a Diabase, dry 3.0 x 104 & Diebagse, moist 2.0 x 10 6 a Loaves Greater than 10 c Voleanic rocks and erystalline schists 5 Much greaier Rhan LO 6 a Serpentine SQ x 1g =e x 107 a sedimentary 10 °- #10 e Compact sandstuone, quartzite, recrystulsdized limestone About 10 rf (3b) TABLE I (Cont'd) Material Resistivity (ohm-cm.) Source Sedimentary (Cont'd) Shale, mudstone, sandstone, limeatque, slates 10%.5 x 104 5 f Conglomerates up to 1.3 x 10° b Shales up to ie x,40% c Bandatone 5 x 10°%.~.10 5 a Sandstone 5.1 x 10%..4 x 10 b Sundatone (dry) up to 40 4 ec Sandstone (moist) 3x 10°24 x 10 5° Limestone aeS x 10%.6.3.5 x 10”b Limestone (dry) 6.8 x 104 a Limestone (moist) 4.0 x 104 a. Normal limestone and sundstone saturated with surface waters 10°..108 3 d guturated with connate watera 5 x 10°..10 a Marls, clays, sands, elluvium 4 (non arid 50 5-10 f Yost e41 sands 107.109 e Yellow river sand 86% water 830 a Yellow river sand 1,52% water 380 a Yellow river sand 9,5%¢ water 95 2 & Glacial drift 5 x 10"..4 x 105 c Clay 464% water 1450 a Clay 16.8% water 50 a Cluy 28,0% water 16 a Clays snd sands saturated with 5 surface waters 4x 10%..4 x 10 d Porous sands and clove saturated 5 with surface woters 6 x 105-2 x 10 d Clays and sands saturated with connate waters 2x 10*..4 x 10° d Porous sunds and clays saturated with connate waters 50~=-200 d Limestone, dolomite, morl, loess saturated with surface 4 waters 3x 10%.2 x 10 d with connate waters 15...40 d Garden g0i1, 3.3% water 1670 & Garden soll, 17.3% water 60 a (3e) III Introduction Geophysics or “physics of the earth" is a brunch of seience which investigates the physical properties of the earth. In its broudest sense it is concerned with the constitution und physical environment of the whole eurth including the utmosphere, In Applied Jeophysiecs, « knowledyve of local snoma- lies in the physical properties of the exarth is used a8 o basis for solving geological wroblems, Tus vyarinotions in the esrth's magnetic and eravitational fields, local differences in the sneed of propagation of seismic wives, ound anomalous accumulations of rodio- active substances have all been meusured in the seurch for subterranean structures geologically favoruble for the accumulation of oil, Variations in the electrical resistivity of the earth have been used to detect potuble water in urid regions, metullic ore-hodies, to locate faults; ad to determine geological structure where conditions muke surface geolowic studies useless. he great roenge of resistivity in rocks is illustrvated in table I, A method for measuring the resistivity of an electrically homogeneous terrain was suggested by Wenner ap early as 1915, und various methods of interpreting variations of "apparent rosistivity"” with electrode spacing or location huve been described. ‘The porous (4) pot, direct-current method of measuring appurent re. sistivities requires relatively simple a rpuratus and wag chosen us most suitable for iillustrceting the prin- ciples of resistivity investigations to studenta. This pauper is divided into five parte, as follows: le ithe theory underlying direct current methods of earth resiativity investiga. tion. Chopter IV.. ae Direct current methods of measuring earth resistivity wnd the principle instruments used. Chapter V. 3 Details of construction of the uppuratus used. Chapter V, 44 Tield use of the uppuratuse. Chapter VI. 56 Interpretation of the field uuta obtained with it. Chapter VII. Of these parts, lo and 2 are common knowledge, and based on the literature, Part 3 is originel insofar as small changes dictated by expediency are concerned, and in part 5, methods of interpretation in common use ore applied to data obtained by the writer in the field, 4 large and varied literature hus arisen on earth. resiletivity prospecting due to its commercial apolica- tion. It consists in part of wu few books combining theoretical discussion with the results of experimental investigation, but mainly of short articles on apeciul phases of the subject in several languages scuttered through « great number of diverse periodicals, on extensive bibliography is included at the end of the Papers (5) IV The Theory of Direct Current Methods of Kexsuring Warth Hesistivity The methods of measuring resistivity, and of interpreting meusurements obtained in the field are bused on the tieor, of flow of direct currents in homegeneous media, and in geneh special cases of ordered heterogeneity as herigontal conducting layers and regions separated by vertical faults. The follow. ivig pages present a review of this theory. iefinitions and fundsmental concepts involved in this review are given in single-spaced type for the benefit of readers unfamiliar with the subject. 1. An eleetric current is now commonly accepted to be » flow of amull units of matter, galled electrons. These have masp,of about 9.00 x 10° grams, «a radius of about 2x 195 5 centimeters, and «a nexative charge of 4.774 x 107 0 slectrostatie unite. \ gram of eleotrons in close cubie piling would occupy about 7 x 10° cuble centimeters, and two grams ut a distance of «a mg her would repel each other with a force of 4 x 10°* tons, a force greatey, than faye due to wravita- tion in the ratio of 4.2 x 10°* to 19) mectrically neutral matter is cumposed of positively charged neuclei surrounded by negatively charged electrons, and vositive charges characterize jarticles of matter from which part of the normel number of electrons has been removed. me The foree of repulsion batween any two point charges is oxpressed by Coulomb's Law r® where Gj, dp are the charges, r is the distance between them, and k is a constant copending on the unite used. When k is unity for Ff in dynes and r in centimeters, the equation defines the GaGeSe unit of electrostatic charge. (a) Jeans - Blectricity and Magnetism. p. 20 (6) If the two charges ure of opposite sign, the force is ttructive. 3, The force ucting on un electric charge is the veotor gum of forces acting due to the presence of all other charges distributed throughout space, and ut every voint of scace a vector may be drawn indicating the direction :nd magnitude of the electric force which a unit »vositive charge would experience if it were placed at that noint without changing the pre- existing charge distribution. This force is called the electric intensity and is represented by the vector T 4, A quantity useful in solving problema of electrostatics and of ateady current flow is the poten. tial which is defined as the work required to bring a unit positive charge to 2 nooint in space by any path, against the forces exerted on it by the other charges in tha field. ‘Yor a given charge distribution the potential varies only with the location of the ooint and each soint in Buclidian space is churacterized by a pingle value of potential. The potential is defined ns zero at un infinite distunce from a charge. Foints having the same potential Lie on equipotential surfaces and no work is required to move « charge from one point on an equipotential surface to another. The gradient of this scalur vcotential field is a vector field giving the magnitude and direction of fastest Increase of potential and this direction is perpendicular to the equipotential surfaces, 5. <A charge at a voint of given potential is acted upon by 2 force directed tovurd points of lower potential, and if the charge is free to move it will take a puth normal to the eyuipotential surfaces so that the direc. tion of the lines of force (or line of flow if motion takes place) is ovposite to that of the grudient. Thus the electric intensity at an, point is the negative of the gersdient of the petential at the point, or in vector notation _ _ im .VV {2) Blectromotive force is » force which produces an electric current and.is mesaured in terme of the potential difference of points between which the current flows. The volt is that potential difference tigainst which one joule of work is done in the transfer of one coulomb. B® ia commonly expressed in volts per centimeter, 6. The quantity of the Cegs3. unit of electric charge is derived from eyuution (1), but the practical unit of electrie charge is the coulomb, and is defined ag the quantity of electricit; required to Liberate 2001118 groms ot silver from a solution of silver nitrate uccording to certain apecifications. A coulomb is equal to 3,0 x 109 evgeus and each is made up of an integral multiole of the churge on «4 single electron The practical unit of current, the umoere, is « flow of (7) one coulomb per second, und cerresponds to the motion of 6.3 x 1015 electrons per second. % ‘The direction of the electric current is tuken by convention to be onposite to the motion of electrons and corresponds to the direction of motion of equivalent positive charges. (it any voint in ssace the motion of electrons will be in » given direction and the current will be in the opposite direction. The current is a vector quantity huving direction and mugoitude, and if at anv voint « coordinate s stem is set up it con be resolved into components purallel to the coordinate axese The direction of the current le normul tu the glane throuwh which the wreatest nwaber of electrons pass at a ,iven point, and the current density is u veotor auentity in the direction of the current with magnitude equal to the number of aomoeres which pss through unit cross section areay @.@. the number of coulombsa which pass through a square centimeter in a second. The relution between current and eurrent dengity is exoressed b the equation Is fitas = fi cos(n,i) de (3) where I ig the current, i the current density of magnitude i, and (n,i) the angle between the direction of the current density and the normsl to the surface through which the current flow is txcken. 8 Electric currents will flow enly through electric circuits, Hlectrons truvel in cloged paths, from n0ints of one votentiul to these of lower jotentinl, und back to the original point, the electrical energy required to reach the originel points of higher potential being farnished by chemical or other means, «An electron will be driven by an applied electromotive force to a point in » solid conductor onl, when other electrons formerly at that point have been driven to yet another point by the same force, . 9 <A conductor is « body through which motion of electrons can take place. .. dieleotric is « substance through which no motion of electrons is possible. In nature, all materials will permit some flow of electrons if under uw sufficiently large electric intensity so that in practice, distinction is between good con- ductors and poor, rather than between conductors and dielectrics, 10. <A line of flow is «© Line drawn in « conductor such that ut every point it is tun;ent to the direction of the current ot that pointe <A tube of flow is a tubular region of infinitely smell cross-section pounded by lines of flow. No current cun cross the bounduries of uw tube of flow, ond the apgreeute current flowing scross 11 sections of the tube is constant, ang a measure of the strength of the tube of flow. (8) ll. Resistance is the property of evonductors, dgevending upon their dimensions, materials, and tempera ture, which determines the current produced by a given difference of potentinl. The practical unit of resis. tance, the ohm, is that resistance through which | difference of nootential of one volt will produce a eurrent of one ampere. It is defined by the Ohm's Luw equations R 2 WI (4) R is in ohma when “ is in volts, I in amperes. Con. duetance is the reciprocal of resistance and is ex. pressed by the reciprocal of the same equation and is measured in mhos for & in volts and I in amperes, In linear conductors the resistance is preportionnl direct. ly to the length and inversely to the cross section of the ecunductor,. RseC@L/A | \5) and this equation defines the resistivity s8 resistagee of a body of unit cross section and unit length at o°C, then R is in ohms, . in square centimeters, and L in centimeters, € is expressed in ohm.centimeters. Other units in use ure the ohm-meter eyucl to 100 ohm-centi- meters, anu the ohm-foot eyual to 30.48 ohm-centinmeters, Conductivity, , is the reciprocal of repistivitye A three dimensional body may be considered as made up of linear conductors or tubes of flow und Unm's wow tukes the form Sd {sO H, or Be ei (6) Derivation of the Potential Functions und Jxpressions for “arth Resistivity : It is not vossible to measure earth reaistivity directly, and it is necessury to express the resistivity in terms of measurable quantities, so that it can be commuted from measurements feusible in the field. infinite Homogeneous Medium We will first consider the c.se of on infinite, homogeneous medium of uniform resistivity © into which (9) a current I is introduced at a point source and from which the same current flows by o »coint sink. (Figure 1) The potential at any v.oint other than the source or sink may be derived from the following conditions: (1) The electricity must flow in continuous paths. This may be expressed divie od or Vif zo (7) or since I =7R and 0 = 1/e is constant, T VeR es Q (8) and since Be ~VV | (2) “TR VV = 0 ox Vev = 0 (9) which ig Laplace's equation. (2) The current diverges raudislly from the point source and converges radially toward the point sink. The actual current density $ is aubject to the condition V*f = 0 and may be considered as composed of two current densities i, and dp due to current issuing from the source, und that Leaving ot the sink respectively provided that Ve(fy + fp) = V-iy+V-T, = 0 (10) This will be true provided V«T, = 0 and Vets 0. The actual potential V is the sum of potentials V, and V 1 2 iy 2 -T VV)» i. (3) The potential must vinish .t infinity ut least which may be defined by the equations as fast ne 1/r where r is the distance from the source or sink to the point where the potential is measured. (10) Diagram explaining Sym bols used in deriving potential in qa infinite homogeneous medium Figure / If we consider the source shown at C, in Figure 1 a8 an origin, then at a point ” on a sphericol surface of radius r from C, the current density of entering current will have a magnitude I/4ur* and the direction of the unit vector é@y,. I, = (1/4nr®) 6, (a1) If the sink shown as Cy is tuken »s origin, the current density at « similar point 4 due to leaving current will have magnitude 1/4aur*, and direction «8p. fo = -(I/4rr") oy (12) Substituting f,/e for 7, in (11), and «VV, for &, we have ~WVife = (1/4m®) 8, (13) or in soherical coordinutes after multiplying through by (8) L av ha JV ge, + ~ £65 ay a + - Se, + Fsw eo yea a 4rr* “(14) ak From this it follows thot Bs = 4G = O and equating coefficients of éy Vv. . SE (15) oF Ae from which Vy = (CI/4nr4 c _ (16) Since the potentiel is defined to be gero as Yr becomes infinitely great, C = Oe By the sume process, Vo & =@I/4nr (17) | Differentiation shows that y“v, 2 W"Vp > 0 sutisfying condition (1) so that at a point Pyy ryz distant from Oy and ryo diatant from Cy Vp, * C1/anr74 ~€I/anrjg s€ 1/4 1/eq, - rz) (18) (12) That this is the reyuired potenti.l ia guaranteed (since it satisfies Laplace's equation and the boundary conditions) by the Uniqueness Theorem which sys that two solutions of Laplace's equation thit satisfy the boundury conditions can differ only by an additive constant and this can be get to zero, Similarly ut Pp Vp, * e1/4am\(1/rer - W/rog) (19) Subtracting (19) from (18) we have Pa ‘pe 2 (¢1/4m) (1/rir = W/ryp - 1/tgy + 1/Po9) (20) Vpy This is an expression reluting the resistivity to the current, potential difference, and various distance. Semi Infinite Nedium However, in order to gain acceas to points Cy, Cp» Py and P,. for meugurement it is necessary to consider the cage of u homogeneous medium extending infinitely in all directions below o bounding plane. Above this plane we postulate a medium of infinite resistivity which is well epproximated in practice by the air. If the point source and sink are located in the bounding plane, since all current enters the semi infinite medium through hemi«sphericel surfaces, i, = (1/em®) é,, i, = «(1/enr*) 6, and equation (20) becomes *™ Voy - Vpn 2 C1/2I/rir - VWrig = rei + W/reg) (21) (a) For another derivation see Wenner, F. - U.S. Bureau of Standards Bulletin #258, 1916 ~ A Method of Measuring Barth Resistivity ~ Vol, 12g, No. 4, (12) If we consider the pointa Cy, Py, Po, Cy to lie on a straight line separsted by equal intervals of length a, as shown in Fige (2a), f]1 ® 4, Yyp = Ray Fp, = Bay Foo % &, and (21) becomes > b, = (€1/2m1/a ~ 1/2a - 1/Za + 1/a) = (€r/en(i/a) = C1/27a (22) This is Wenner'ts Formula, and when solved for the resistivity takes the form C= 21a (Vp, * Vp.) = SvaB/I = 27aR (22a) where @, Vpy - Voor and I are all measurable and in practice Cy is the current anode, Gy the current cathode, and points Py, Pg are occupied by potential electrodes. In the Lee-Partition method, (Figs 2b) a third potential electrode Py is inserted half way between Py und Pp so that YD, ” Yb, =(eI/2m{ l/a - 1/2a ~ 2/3a 4+ 2/3a) = CI/4ra (23) and @ = 47a (Vp, “ p,)/t (23a) and gimilarl = 47a (V,. = V,_)/1 (23b) and similarly e ( pz > “pp f The equation for the resistivity when one of the current electrodes is moved to a wreat distance (single electrode prove method » Fig. 2¢) may be obtained from eyuation (21). When ry1 7 a Ty. FO, To, F Ds Pgp FO% ge (eI/ev)(1/a - 1/2b) = C1(b-u)/2tab (24) C= 2nbaR/(b - a) | (24a) It should be noted thut in the ubove derivation ¢ is assumed constant in all directions throughout the medium, (13) YUL, MM THT® [Vézoind 77 ULE W//11111 Werner's Configuration — 1 S gt SF E, Ez e--——_@ — see —-nle— Ws —- = @— — TITIT TTITTTIT ITT AT ITTTATT TITS WITT ITS TTTTTT, Lee Configuration Ea RT ee —- @ —— Je _-b-@ — 5 MUM LLL ALT MLM TLL TT Single electrode Configuration E ad r<— QD iar Kee-—-— @ —— — se — — A — — oe — — —- UH -— —- => MMM WALL TY TTT lnverted Werner Contigura ation P Figure a fhe current and potential electrodes ure ussumed to be infinitely esmull and on the olane boundary surfiuce between the two medis, and the wir ig .saumed to have infinite resistivity, Furthermore, it is useumed that the medium acts as u metallic conductor being unchunged by the passage of current, that no current is drawn from the petentiul electrodes in measuring their potential difference, and finally that no cther currents or poten- tials than those vostulated ure present. Chard terietios of the Potential and Merce Fields in & Bemi-infinite Medium Before considering some of the effects of variations from these asaumptions which occur in field oractice, it is inetructive to consider certain churncteristics of the votential in a semi-infinite medium | (1) If the current and notential electrodes in the Wenner configuration ore interchanged (Figs 2d), the difference of potential for a given current between the inner electrodes is the aume us would be set up between the inner electrodes by the same current in the normal configuration, This follows from dvean's recip. roc.tion theorem(®) und holde for any distribution of resistivities in space, It may be seen for homogeneous media by substituting the new values fq 7 a, Fyp = Rihy re, * 2a, foo # @ in equation (21) Vp, - Vp * ( €1/er)(1/a - 1/24 - 1/fa - lfa) = C1/27a which is equotion (22), se thet, the resistivity for both the Yenner and inverted Wenner configurations is given (a) Jeans, tlectricity and Magnetiam - p» 926 Schlumberger, C. and Schlumberger, Me, As lal, Geophysical ee 1932, o. 144, footnote 1, (14) by equation (22a). (2) The potential can heve a moximum value only at a current source, and « minimum only at a current sink, Laplace's equation prohibiting the condition for an extremum--thut the second purtiuls be all néegutive, or nll positive. This will hold for any distribution of the resietivity in the ground as it followa from Vei = 0 whieh holds for « steady current in any type of medium. (3) It ie instructive to inveatigute the equipoten. tiol surfnces and lines of flow in the cuse of 1 semi- infinite medium given wbove. The equipotential surfaces may be found by putting the expression for sotential equal to a constunt Vv w(¢I/er)(1/r,- 1/r,) > K (25) or since ¢€I/2t is « constant, — I/ry - lf/tg 2 ¢ (26) From Figure (3) it is evident that the potential is symmetricel about the X oxis, os the point P ean be rotated about the sxis without changing the values of ry or rg, and va equipotentiul surfaces sre therefore surfaces of revolution about the X axis, they may be visualized from their trace on the X.2 plene, The slope of the tangent to the equipotentiaf lines is given by | dz xix - 4)® 4 pe) 3/2 . x. ~ d)(x* dx Z(H” 4 2) 37a [(x = 4 “ , Th obtained by differentiating (262) ee with ett to x, umd for 2 = 0, and finite x # 0 or d, the lines (15) (27) * TOTTRRRLITITIT TTT TTT Po OGraund < ee Figure 3. Diagram of symbols ysed in discussing potential and current distribution ina semi-infinite homogeneous medlum x approach the x axis perpendicularly. The slope of the tangent to the lines of force or lines of flow at a point dn the X*4 plane nay be found by substituting the coordinates of the point into the equation Hh - © By gg af [Ux = al? 4 aA) 5/2 » (x? 4 2P)87F (96) ere Be 3k 9 x[tx = a)? 4 ey F Tx = a) (ae + 22)9/2 This shows for 2s 0, and x any finite value other than 0 or Gd that one line of flow 1s alonr the X axis, It - will be soon from a comparison of oecuations (27) and (28) that the lines of flow, and of equal votential are perpendicular as their slopes are novative roeciprocals of each other. Figure (“:) showing the nature of the equipotential lines in the “#% plane for the case when the distance d botween the source and sink vas. taken as 10 units is sketched from the values of C computed by substitutin,, values for x and g in formula (26a). C 2 U/(x2 + 22)*= 1/[(x - a)2 + 22] * (26a) The lines of force were then drawn perpendicular to the equipotential lines. From the closoness of the spacing or the oquinotential contour Lines one can et an impres« Sion of the spacial variation of the field strength and of the current density which is proportional to it. The dintribution of current in the «round is of fundomental importance in determining depth of penetration and resolr= ing power of resistivity methods, These problems are discussed in sone detail at the end of the chapter, but a fow conclusions which may be reached after an exanination (16) — —3)Ga -100 -50-Yo 30 1 + T =F T 77 T ee eee a J ie : Figure 4. “arrent and eayipotential /ines ina sem/-infinite homogeneous medium. ; , Solid line: /ines of constant Potential Numbers af margins give potential V= FE ((X74Z4) 4 ~[ho-x)- +z] =} In volts when X,Z andl aretnmeters, P=1x/0* ohm-centimeters, and I= 0.2 amperes. Brokentine: lines with direction of current flow. Numbers along centra/ line of zero potential give current density through median plane X= 35 ia microamperes persquare meter for same Case. of Ficuve 4 may be mentioned here, Tirst, the equipeten« tial lines are crowded tocether noar the current source and sink, This is one reason for not using the twoeelece trode method, which measures the potential and resulting current between a sinvle cair of eloctrodes in the eround as the createst part of the voltae drop for a miven current takes place very noar the electrodes so that the resistivity measured is that of the small region at the surface tirvetly surrounding the current stalces, The depth of penetration is therefore nerlicible whatever the ancunt of separation of the stakes, Seconlly, the field varies almoct linearly along the nart of the Line betwoen the current electrodes from 3/10 to 7/10 of the cAstanee fror one to the other. Placing the poten~ tial electrodes in the central 2/5 of the current sproad ag in the Wenner confircuration is therefore desirable as it reiuees the error introduced into the calevlation of the vesintivity when they ere shifted somewhat fron thelr normal position, It they vere nearer to the current oloetroves a shift in position would result in a lareox difference between the measured potontial difference and that which would be measired with the alectrode vosition assumed in the resistivity formulae. Tt is also anvorent that the field becomes rapidly uni-+ form bu. very woak with increasing distance from the source and sink. (17) To return to the problem of experimentally determin« im: olec rical resistivity of the round from surface moamurenents, we have Pound that F aTa E/L (22af for the Wenner confiruration when it 4s assumed that 1. The electrodes are on a plane surface bounding the cemlinfinite medium. &, Current and potential electrodes are infinitely 3. The resistivity of the air is infinite. 4, Ff is constant throughout a semi-infinite medium. 5, The conduction through the medimm is metallic, 6, No current is drawn from the potential electrodes in uweasurin: their potential difference. 7. There are no potentials cet up by other char-e accunuletiens or currents. Ltfect oF Lyrror in Assumptions In practice it usually happens that none of the seven assumptions is strictly true, (1) As it is aiffieult to make proper correction for toporraphy, lines should be made es level as possible in rouch country. ‘The effect of small toparraphic ivro;~nlerities in even country is not serious, (8) Finite size of the eloetroles is not a verios cauce of error if thoy are small with veapect to the alacirode separation as In earlier nebols of interpreting (1%a) field data it was nocossary to use very small electrode separations to determine the surtace resistivity and much difvierlty was experienced in retting reliable values, but with the now universally used superposition methods the use of small spreads is not necessary, It is coumon practice to water the electrodes in planting them to expedite the digging of holes, and to cut down contact -esistance if the surface material is hard or dry, This inercases the effective diameter of the electrodes to a foot or more, and for that reason It is desirable to hove the electrode interval "a" at least ten feet. (3) As no appreciable lonization of the air takes place, its resistivity is so preat that no appreciable error is found from the fact that it is finite. It eceurs in the formulae in the form 5 air ™ 5 garth Sair + 5 earth which is assumed unity. If Fae ew LOOO Seexth? the ratio is 999/1001 and the error is only one nart in 1000, (4) The ground is not homogeneous, and even in the simplest eeolo-leal case of two rock layers such as erystalline bei roek and overburden there are commonly (17) irregularities in the resistivity which may take the form of several parallel layers, or be so irreguler that no theoretical interoretation is possible. In non- homogeneous ground, the measured value 27a B/I no longer is a true resistivity but a complexly and irregulerly weighted average of the uctual resiativities. Tt is therefore termed the upparent resistivity and denoted by Cae The mathematical oroblem then arises of expressing the actual resistivities in terms of the measurable apparent resistivity. This has been worked out for various cuses including the cause of any number of perallel layers und the case of a vertical fault. The cases of two and three rock lnyers will be treated here ws they heve been quite largely used in interpretation of field data and have received much attention in the literuture. Method of Images in a Semi-~Infinite Nedium Before turning to the ease of two rock layere (which because of the air is u three layer ohysicul problem) it is instructive to rework the case of the semi-infinite medium to illustrate the general method of attack, Conaider all apace to be divided into. two regions of resistivity Cy and @g respectively by # horizontal plune boundary surface (Figure 5). Take X and Y uxes in the plene and 4 4 uxis positive downward. The potential at any point in the first region is denoted by Vy = V(x, Yl» 21<0), that ot a point in the second region by Vg = Valxes Yor Zo 70)- Postulate a point current source of strength I at C,(0,0,49) and « corresvonding sink at C (xg,0,29)+ The potential Vz must now satisfy three (18) FP (%,¥,,2)) _ oe Nt) -2, (-*.T) De, (0,0,25) Ca (Ke, 0,25) / aa fo Lad a 7 / SL, Va \ ur / , Oey (Xa, O,-Ze) 4 (- 2) laa P, (ir. ¥, Zh) Pa Vv Figure 5, Diagram explaining symbols used in der/ving expressions for potential and resistivity ‘pn asemi-infin/te hamageneous medium by the method of images, ‘poundary conditions aga well as Luclace's equetion VeVi = 0. Theee ore le At two infinitely close points on oppoxite sides of the interfxce the potential must be the enme ns no finite work is done in moving electricity on infinitely small aistence from one voint to the other, Therefore, at g = 0 Vy # Vo 2. All the current which posses through one side of unit crea of the interface must omerge trom the other, or at xy = Xe, ¥Y2 ® Yor @ ® Oy Aq, * ipng where iy, denotes the component of current density in the direction of the positive % axis in the first medium, and deg that in the second medium(®? ox - dV, Vv —&, — ~ 6 2 Va. ‘ j2 * eZ MA or Ce Ve when z = 0 3. The potential must vanish at infinity at least ag fast as 1/r. The problem of Yinding the potential whieh will satisfy Loplace's equation and tha two boundsry eonditions From these¢ two lows may be derived the luws of refraction for lines of force at the interface, From (1) OV 2 OMe dS 5S where se is any direction in the buundary planes Bisine, « Resinge (29) From (2), Coy co 86, = C5 Mecos@y (30) Dividing, tan@;/lo = teun@e/@, or tund,/tundo = Po/Py (3i) (19) is solved by the method of images. (4) The materLul of ressrviry f) is replaced by material Of ney > und w current source of unknown strength k)I is placed at Cg, ag far below the interface on a line CyCz perpendicular to it us Cy ia above it, and a sink image of Cg is placed similarly at C4. The potential Vz at any point ?y(x1, yi, 21) in the first region is then thought of as due to the combined currents of the real and image sources and sinks in a single infinite homogeneous medium of resiativity G, Vo at any point Polxe, yes 42) in the second region is thought of as due to w single source at Cy of strength kel and sink at Cy of strength ~koI in the same infinite medium, The unknown constants ky and kg are then deter. mined so as to satisfy the two po undaxy conditions, while the form of the sotential is that known to satisfy Laplace's equation in a homogeneous infinite medium Vy = (1/47) (1/ry - U/tyo) + (Gk, 1/47) (1/ry% = 1/ry4) (32 Vo = (CykoI/4r) (1/rg, ~ 1/reg) (33) or in cartesian coordinates » f ! ! 2 PEL! aoe) = =— = “ f ; — — SSeS * 7 IMIR > 12-20)" APE) "Sor Wiseman) Warn) e 1G (32a) MS aEaSey Fre) (33a) To satisfy condition 1, »%t we O, Vi # Vp (€,1/47) (1 4+) = Qket/4y or kos 14k, = (34) (a) See Jeans, Blectricity and Magnetism p. 208, Maxwell, Slectricity mad Wasnatt metiem, 2nd ed. Vol. i, PPe £28, 404. er rd ed. Vole I, po 435. BEhrenburg and Watson, AeIe Melle Geophysical Pros. | pecting, 1932, pe 420. (20) To satisfy the second condition differentiating (_V 1 and Cie with respect to z and equating gives Wa (21a) Ge aa) +h, (21+20)( dos “73 )] = £6 (2-2) ( z3) - Putting x1 = Xp, ¥1 = Yor 21 * 2g = Oy we have, einee riz * Fag * Tey Tig = Tia = Fue Q(B) =-@ tog (Ade )-R-6 Simce Wan IK, (36) so that ky = §-9- (36a) Ke = ote (37) For u point A in the interfuce, . 3 If in particular we teke two points, Py(a,0,0,) and Po(2a,0,0,} and put G S°, 29 70, xX > 3a Yo. = Ye = (Cgt/2 1) (1/a ~ 1/2a . 1fga . Ifa) = Pol /ema (22) ag found before. METHOD OF maces Iw THE CASE OF TWo foRIzewral LAYERS In the case of two rock Layers(*) the process of finding the potential is complicated by the presence of two bounding planes, and two sets of boundury conditions, go thut the imoge whieh will sutisfy one set disturbs the potentin] «t the other boundary In the physic.1 cuse shown in Bigure 6 it is con. venient to take ss origin of coordinates the real current source, a Z axis positive downward, and an X.Y axis parallel to the beds. The whole epace is aguin regarded (a) This case his been worked out using images by Hd ge and Laby Te Ga Re Ge Pa 255, 1931 — Hummeli - sei tschritt fur Geoohysik, Vol. 5, os 89, 1929. or Aslewa se Geornysical Prospecting, 1922, pe 400. Lancas ter.Jones~-«#The Vol, 43, July, 1930, De 19 ning Magazine, (21) mw Pp (Heal, KgsL) { ed | H VM, bya) | 5, Vv, 2(h-d) ey 7° | : Air * ida Keil) Dp | |¢: ta, 0, a) P R. Cz (¥s,0, d) B ed qd K--— a--%—-—a-—->ke—-4--> i oF a | f 6) L, Kost) } hal Upper ground layer h | FM | uA 7 T — AT h-d | F; Lower ground layer * Ot, tai) J ee 5 Vy a O OKgt, Ky, T) | | | 2. (haf | | | | | i" $ Figure 6, Diagram erplaining symbo/s used in deriving expressions for potential and resis -— tivity in a two-layer ground by the method of ‘mages, as of resistivity (1, .nd the potential is modified by &@ geries of images to Batisfy the boundery conditions, If an image of strength Ka, T is placed us shown to satisfy conditions at the a boundary), and an image of kp, to satisfy the B boundary conditions, other images kp. and ka, must be used to nullify the undesired dis- tortion of the A boundary by KB) and ef the B boundary by Kai These in turn disturb the conditions «t the boundaries, and aso ons The potentionl, however, is approximated more end more closely by successive sets of images, and is accurately given by an infinite series. The SOSERTEES in the three regions is expressed as V.. = £ a8 oes \ neni a: = aire] eeqir ce * Teeter ” Dep te pat by (39) erie. ‘aeat Ce 2 ht? a. rece (Z,-2h)~ Fo (x7742 +2, y2a)™ ETF, he CZi4thpe (40) s AL Lf Now. + __ New aa Vi iT PS Merfetze | TXSeye Grady yet e(Eet2h)* (42) where Xs Yor fg are the coordinates of any point in the region of resistivity ( und the constunts sre chosen to satisfy the boundary conditions le Vy, # VY: : 0 ' 5 when 2] # 29 = dy, X, * Xo; Ve . # BV, e« Sah Sz, Y. = Yo iv when 2) = @, =h.d, yi = Yoo GE eee + Was 1. Ve = op LETTE * Gerseaaap * eae -] (39a) Vi= Ge lattice * es —t~ J ET IKE IR KS Ye Cd-2hye VEY? ye Cde2h)” (408) (22) Equating coefficients of like terms, Koy = 1 + k"B, (41a) Ka, ® ka, + k'p, (42b) Kip * Kap + k's, (41¢) . Bef Cd), ty Cd-2h) Kan (od POD, ge as 2, fibe te thr [erie OP ta aay + aa ae) l (42a) £ ay REE td ~-ra) "Ed. “7h Ki Hed ~ (4-2 wha — bs) 4--~| 92, OF oI, dy ee a dh)” Fee +9, H+ (dezary™ * (43a) Again equating coefficients, ko, =F(1 - k'y,) (44a) 2 Ky, ® sale - k'g,) (44b) Kas =? (k', 27 k'g,) (44¢) 3 fF It pHa Kor 5 _ Ning Fee yg ob * Me NIS+ h-d)* hea ated)” REA +8 Dh) J (40) 27° => = ; YW LIT +5 a "hee vlpe. 47+ (hed) ~ er + 09-5h Equating coefficients, Hyp = Lek, (45a) kp # Kk! k! 45 Be Ag tag (45¢) , D ms (Aa = * Saja pr [Beri Te? Ce hea) + rd ah m9 #488) mM pry » lbs td + alah) Kewl Wien Gel Racer aye try te he) Dee sy “fl4ara) Rquating Goettderentes kyo = “ (1 = kM, a) (48a) Kgs ® han - k",,) (48¢) From 41a end 44a we have 2h Koy FR+PR = ky 41 (23) So-P, =. By * Rts, * ky From 41b and 44b k! k' Bp » kyko Kay ke(1 + key) From 4le and 44¢ 1 2 k'g, # ky "Kp ky @ Kyk,(1 +k,) From 45a and 49a we heve oe Pe Se “9 From 45b and 48b H kta, = kykp kg, = ky (1 + kp) From 45¢ and 48e « » & k', = k, ky Kyo ky ko(2 + ky) The oxrevasions for the petensies then become " : ie - Verde ON irae * heer * The Gam to pa 7 ie (39>) 3 — ——— —— wn, i. _ A, a 7 (RSS = * ES ee + fete T Lea Fe, Um 4+ G-eheta” "Ry leek)” TRG lene)” Mf 2 es + Hs : — "te VX," Y, 4 (2,4 2d) wae - 4 (2,42h)~ ~ Vx, *+4, *4 (2; mat? (40¢) fe ( Wx - V2 {TT 77 OM) ee + WK ae ee Re YI, = (z.42N ~~ | (41d) These serice can sted iaaiiaial in the form dip rd § i Ha itty al Ys *4+ C20-EM h # Zed) (39¢) v= Hef & [- a ret tt bs kvx,* soar jn2enp)” ” ea “4 @,+2nnezd)” + 2 le “+ Se yo + pol 5 (404) of ae 2 I+ ‘% to ee ; LE +h) e_ Dresti "4 (2,4emh)~ + Rte. ranks (41¢) If we now include the effect due to a sink at Co, we must add a term of the form: (24) mow ' 3 rc Ki,” ne ae 7 Fla (AeA ea ee ee Ay sia 2 ae <—— +E pe * Vix,~ ape +(Zt2Ah 5 (49) If we now put Pp =-o, d = 0, then k, 7 41, und the origin moves to Cy, ko > k 5 fog » ound at a point P fa105 0) in the B tntantans distunt a from Cy, Vy: = a | Sea 24 (amhy™ ae Taz — al Lan ay” ; { 50) i “LE [LaF eae — ak — 25 ete} = Ji ef pm ' ‘ira f 1+ 4E 0" peer ~ WERE ame) | (50a) (50b) and at point Pp(fa,0,0) _ SES Pet J nn a ial —¥ rises $ Ven ~ 2a LS may TL fap (amh)& Lae a Laren _ FE 5, an a: dal _ a “25 jaa —k + *2_ ey Va7+ ae] | fx _)- _ ! qrA { 4 [a Ee ~ rae \ a Vi-Ve 2 2S {eth Cae eee | “p> Ys F fyT/eta (1 + 4F) (514) 84 = Py (1 + 4F) (Sle) The serles - By 5 42a TE) where k = i converges, and has been evaluated for assumed values of h/a and ky The varintion of */p with h/u vor different values of k is used in the interpreta- tion of field data discussed in w luter seetion of this paper. The convergence is slow, and the evaluation (@) is tedious, 30 that it is instructive to consider an (a) Roman.-U.% Bureau of Mines, Tech, Paper #502, 1932 (25) Air Ve stank van A) RAS | | Upper ground (ayer A KM | 1 Lower ground /ayer 5, B) V, a v Frgure 7 Diagram explaining symbols used in deriving expressrons for Potential and resistivity tna two-layer ground by the integral method, alternative type of solution '#) which makes mechanical evaluation possible. This is especially imnortant for three und more rock liyers for whieh tie serios solution ia more complicated, ‘Ye will rework the case of two layers with thia method for the sake of compurisone INTEGRAL METHOD IN THE CASE OF TWo HomzonrAs LAYERS Cylindrical coordinates are set wp (Figure 7 wi th. origin.at the source, » 4 axis positive downward, and an r (=(x* 4 y2)z) axig in the boundury plunee As the potential is s;mmetrical about the “% axis, Laplace's equation in cylindrical coordinates takea the form Sv ydoyv, gv Lg We Kap? 377 =P (9a) A solution is sought of the form Ve U(r) e% z <0 » U(r) en"? 2 >0o (52) where A is a positive real number as the potential must vunish for 2 infinitely large. If the value of V in equation(S52)is substituted in equition (9a), there results vu, Lt iv *U =O Set Pap t? (53) the Bessel'sa equation of zero order, whose solutions ure two kinds of Bessel function series denoted by % far) and Yo (xr) « In this oase we are eoneerned only with the series J (xr) = (54) We now have particular golutions of the type Vs voir) 7A Z2o az Versace #40 (52a) A general solution is found by summing particular solu. tions multiplied by arbitrary functions of » and the (a) This case ia Menated Sy Chiandores Mrdstrome, p» 69 26 infinite sum of solutions with different values of , may be replaced with the integral forme (Hankel'ts or Bessel. Fourier integrals) Va <2 5, 8 (a) Je Gr) eda Ve Cf piyrurne**da Z<o (55) zZ>o where C, ©()) » da) are to be evaluated to fit the boundary conditions, - veber() hugs shown that Se ub (ar) CAAA= ene > (2 > 0) Sar) er? aa = ieyp = R (2 0) so that in the case of «a semi-infinite medium where Ve C= fe Awo Ola) =) In the two layer case, the potential is regarded as made up of a primary part (that set up in un infinite medium) modified by a secondary potentixl due te the boundaries, \V = Gx - oto , “> Se LEtar er24a + (°@ Gd dolar) CM ZdA + Se Fol Uotar)e™dayy 56) V, = SE Su, ee | Az are A +(e &, (x) Jotare™ AD +f “b,(s) Jo (ar) € da (57) 2 fe ar 15a wire 240 + elsune 24a + So Ga (Muefar) ewanK 58 ) Singe the potentiil must upproach zero as 2 becomes infinitely large, Vg cannot contain a term of the form Sa, On) oane**d z. as @ is negative. Therefore Op= 0. Similarly ¢,2 0 When 2 = 0, Vz = Vo or f= ¢+46, (a) Fou fs 2M and since 32) dor) @* 9 35 Fede) = Oo for z= QO AR R=-ABO+ARS, or §- & (4,-6,) (») (a) See Webster, Partial Differential Kguations of Ma thy~ matical P Physica - ps 120. { ory) 3 =)4 When 2 = hy Vi = V2, 2, e ah + 4, @ah =O, 2 3 Bg ah eopoy % One St he 0.077 (6 faz hot) eet ee” 4a) = PLE te) a # (ire, +ger) —I (a) , F + £0 Bguating (a) end (b) gives ¢,=% a7 (e) _p j-2ah Rquating (¢) and (d) gives = ers $e “My Bt e (f) Nquating (e) and (f) gives F.-f, =2rh F+8. hi-fi e~ f.-F, ~ FF, eta If we now put fL-2 4 ©& becomes jo-4re* PP, WHERE FT = % Rf f= by (e) A=26, wy (a) e.- 6,cire7*") by (ce) - ae ae n* iM, = Be {fate ar +f cA lAY) cae Ce ) 44y for the potential due to the sink, « term must be added of the form v= ~ SENG, (Aus-r]) e = Ah + Jo(Ate-0)) the ae (e** 4, e7) 4ag where rg is the distance from source to sink Thus at P,(r = a, 2 2 0), with rg= 3a, 0o fo =] SEA V, = Le Lf [trap Coragildd +2.) Lde lag) de 2 >01] apeg am dA! The series form may be gotten from the integral form -ZAh =2Ph 2 o-dah | B3p-Shh ie _ we 477 + by expanding 7oye- ~ ve r ety Bah, - 3 . Vp = f Ea 3s 5 a do(na) Le te Ny. .] AX =a fu, (anal +r ey Jaa (28) vy, = DES Ek +2 limaae amt --]-2 2 (pies + ape od [7 2Tr Q 24 ; 2 —_ a Vis tea [3 r7J ream eee) § (52) The case for n layers haus been worked out by both the image and integral methoda\®) mnd in. particular, the integr.1 form worked out by Sehlichter for the cuse of three rock layers hos been evaluated. by vetgel and Moxurray (?) on the differentiul nnalyzer(®), Rquations _ for the apparent resistivity huve vlso been worked out for the case of anisotropic ayers(4) , a verticul gunitl@), and numerous others which will not be considered here. These expressions for the »otentinl are alao fundsmentul to other surface sotentiul methods such us the potentisl. droperatitdna potential gradient methoda(®), (5) The fifth assumption made in deriving | emer's formula that ¢conduetion is metallic is not generally true 48 in most cuses the current is carried through the ground we ome i (a) Stefanesco and are pet a erie De Physiyue et Le Sehiiciter, Le Be--Physics 4, p. 307 (1931) Ghrenburg and satgon~-AyIeM. 2s Geophysical Prospecting, 1932, pe 423 b) Geophysics, Vol. 2, 1937, pe 329. V. Bush--Journul, Franklin Inatitute, 212 (1931) p. 447. Piraoh, Bulle jAsds Pe (1935) Vole 19, #1, 37-57 Schlumberger, Cade, Leonarden, Ae Le Me lle y Geophysical Prospecting, 1934, p. 159. (e) Rdge & laby, 1.6.3.5.) po 252 Tagg~-Mining Mug., Sept. 1930, p. 155, (f) Lundberg & Zuschlag, AsIekeie, Geophysical Proapecting | (1932) p. 4%. Hedstrom--Mining Magazine, April, 1932, Vol. XiVI p. 201. {g) Heiland.-N.R.C. Pt. Il, 1937, ps» 581. (29) by salts dissulved in the ground water, and obeys the rules of electrolytic conduction, The conductivity of the rocks is therefore a function of their porosity and permeability, and of the presence or absence of suluble calts within the rocks which may increase locally the number of ions in the ground water and consequently (a) the conductivity. “lectrochewicsl studies have ghowe that when metal stake electrodes ire used with epplied voltage avove wu certain value (of the order of mugnitude of «a volt) further increase in voltage will result in proportionate current increase, so thut Ohm's law hulde. The failure of Ohm's law for voltuges less than the eritical value, is due to the development of e back @emef, due to accumulation of iona at the electrodes. There is also an increase in contact resistance ut the electrode due to formation of gas films, etc. This pheno- menon of 4 back or polurization voltage has negligible effect on the current and potential distribution set up by the applied eam fs whieh is large compared to the critics1 voltage. It has o lurge effect on the difference of potential mensured between the potential electrodes, ap the polarization voltege often amounts to several hundred millivolts, and is of the order of mayguitude of n the Potencial efectpodes the potentiel difference set uprby the applied current. (a) Sundberg--A.1.M.%,, Geophysical! Prospecting, 1932~ Fritsche«flectrotechnische Zeitschrift, Vol. 58, (These theoretical and experimental studies show thot the problem ia quite complicated. } (30) To eliminate thie effect, us well as thet of the yalvanie potential comnonl, set up between the two potentilal stakes becnuge of difference of the churweter of the electrolyte at the two pointa, it ia customury to commutate both the applied and resulting voltages, or to replace the metal potential stakes with non-poluriging potential electrodes, Carl Barus (4) encountered this trouble in 1882, in the Comstock Lode, and remedied it by usine gine electrodes in » solutivn of 2ine aulphate in a semi permeable goutskin bage vost workera have used co over rods and & suturated golution of copper sulvhate in poroue eluy pots xe originated by Schlumberger, The saturated solution ensures both potential sets being at the same concentration after « current hus passed. ‘This is the method used by the writer. According to Dr. Potapenko(), the use of gine und zine sulphute in place of copper and copper sulph te may be more soatiafactory in some types of soil. He also hag found that if a ‘igh input resistance is put in the potential circuit so that current flow between the potentinl electrodea is very amall, it ie vossible to use amore diiute solution of about 1/7 aauturation concentra~ tion of sine sulphates This low eoncentration avoids the clowging of the oorous walls of the clay pot with sulphate erystals which would plock the interchunge of fons between the pot and the ground and prevent equslizing of the (s} U.5.0.5. Monograph #3, Chapter X b C.Tit.--Lecture course in electrical methods of ueophysical Prospecting. (32) potentiuls of the pots in the ground. Unfortunately the writer .ag not aware of thig until after the field work had been completed. (6) If much current ig drawn from the votential electrodes, the flow will reduce their potentiul difference and « low reading will result. This effect is more serious when the potential electrodes have a large contact resistance us the potential difference between the two contacts in the instrument where it is messured will be suvaller than that in the actuul ground by an IR drop, most of which is in the contact resistance. (7) The final souree of error is the presence of potentials not considered in the theory. These may be clussified for our purvose ae due to (1) spontaneous polarization, (B) instrumental leakage between circuits, end (C) other couses, both natural ond artificial, (\} Sponteneous polarisetion is known to oceur in sulphide ores of a pyritiec nature, in anthracitic conls and graphite bearing ores, pyrrothite deposite, and in peat boga!#), In the ease of the sulphide ore bodies, the upper end undergoes oxidation because of the .round water and becomea the ne,utive end of the body, while the lower end ig reduced becoming poattives current is thus set up from bottom to top of the ore hody through the ground, and sets up « potential field @imilar to that artifleislly produced by un upplied current. (a) For a discussion of spontaneous tate tameatia gee Bull, 5 ry ee 6 polsini=-deovhjaieal eploration By ape nt wy arigation Methods, Mining “Hagazine, 1938-1939 LIX pe 278ff, and p. 347ff und LX, ped2Q und p. 90 (32) Although this has o disturbing effect on resistivity mengurements which may be serious unless the applied current is lurge, it hus also the advantuge that if no current is applied, the potential grudient due to the polurization (which muy be as large us 25 millivolts per 100 feet) con be measured, and a "nexative center" of potentiul will be found over the ore body. (B) Leakuge between the applied voltuges ond the gepurate circuit which meesures the e.mf, between the potentiol electrodes will be serious according to tie magnitude of the leakuge, but is necsligible with proper construction of the appuratua, | (c) Other potentials (4) will have an effect depending upon their magnitude. To eliminate the effect of uni- directionsl direct currents, the upplied current may be gent through the ground in both directions. In any ease it is well to use lurge 4polied voltages so that the effect of undesired potentials will be relatively small, Lbepth of Penetration and Resulving Power In concludinw a discussion of the theory underlying earth resistivity measurements it 1s well to c.nsider the yuestion of the depth of penetrution and te resolving pover of restivity-deoth profiles. In general, the greater the aleectrode aepuration is midge, the deeper the current penetration and the deeper are the regions (a) Gish and Rooney have made extended studies of natural earth potentials, and interested readers ure referred to the bibliogrephy ot the end of the paper. (33) which contribute to the value of the apnarent resistivity, Cish and Rooney) have found empirically that for a Wenner configuration the depth at which a discontinuity occurs is equal to tho cleotrode separation at which a turning point oceurs on the apparent resiativity~eloctrode separation curve. This hac been froquently substantiated in the cowrse of resistivity prospecting, but very often is ereatly in error, ‘Vagg and others have developed the-«# oretical methods of determining the depth to an interface in the case of two, three, or more layers in terns of the electrode separation, and it may be seen from equation S1e abova to depend uron the values of the resistivities, The Sehlumborgers(?) have found that for many practical apolicationa it is suffiectently aceurabe to consider the apparent resistivity to represent a layer of earth to a depth about equal to + the separation between the current electrodes, or 3/4 the Wenner configuration separation "a", Lancaster Jones '*) estimates the depth to a horizontal discontinuity as 2/3 of the electrode separation at which an inflection point between points of maximum curvature oceurs on the electrode separation-apparent rosistivity curve, An explanation for the validity in many cases of the approximation that the depth to a discontinuity is roughly 1/3 the distance between the current electrodes may be iy A.i,u.E,, Géophysieal Prospecting, 1985, p. 53. b) AIM, Gevphysical frospecting, 1938, p. 134. (e) Mining Magazine, Vol. 43, p. 22. (34) seen from the following analysis, (2) fhe marnitude of the field at a point r, distant from a current source of strength I on the surface of a sevieinfinite medim 1s B= ¢q7* and thet from a corres- ponding sink © distant from the point is a « The magnitude of the resultant field is Wa Err 62-2 Ey Er EG Since i L is the distance between ource and sink, E£ = 5 Ve ry THE and the corresponding ~“AGee , The current current density is 1-35 eee density in the median plane perpendicular to the line join« ing source ond sink ab its midpoint m is found by putting r, <%= fiayth where h is the distance in the median plane from the midnoint m to the point at which the current density is desired, The equation then reduces to the fom (ETT RE The total anount of current Plowing through a semiecirele of radius h in the median plane with mas a center is then ;iven by the integral Ly=f tah dh whieh gives the expression [= E(i- Tae). This current (S about 1/10 the total current when h «# L/4 and increases to more than $ the total cusrent when h » L and to about 5/4 the total current for h ew 2L. The ratio inc'oases fastest for an increase in h/L at the point of inflection found by putting 32(F)=7 heft) yeh (iri) *=ofrom whieh h/l. is found to be 0,554, If now a diseontinuity at a certain depth h is considered it follows that the spread length L at which a chance in L will cause the createst change in (a) Dr. Potapenko, Leeture Course in electrical methods of prospecting. (34a) the amount of the cvrrent down to the depth of the dise» continuity ia h/0.354 or about Sh, and the break in the SaL curve is to be oxpected at this ratio of L/h, This is only approximate as the prosence of a rasiativity discontinuity was not postulated in ceriving the exoression for the current density and also because the formula holds exactly only in the median plane. lowevor it is found to be of sreat value in determining the best of several possible interpretations in ease of ambiguous data as discussed in Chapter VII, When the ratio of resistivi-~ ties at a discontinuity is very much different from mity, the broal: in the resistivity-electrode spread curve tends to come at values of L corresponding to larger values of L/h, so that the values of h siven by the approximation are too Larrea. zvjen‘®) has considered the case of n horizontal layers, defining the depth of penetration as the denth to or below which exactly 4 of the total current penetrates. In the case of a somieinfinite homoreneous earth, this {a} Geowhysics, Vol. 3, 78, March, 19%6,, p. 78, (34) depth ig equal to + the distance between the current electrodes, lle undertokes to find a theoraticul value for the depth factor (ratio of depth to « discontinuity to the electrode separation) for various types of electrode configuration assuming a horizontally layered earth, and a point source of current (Figure 8). jie considers the function oa Fr (Q-f) = Weep er Via ~ PE =2r bee where V(r) is the potential on the surface of the ground due to the point source, and hy is the depth of the image of atrength Cys The series is then transformed into an integral by teuns of the definitions Image density g@(z) = O for a # hy lim, g(a) = oO for 2 = hy hata fala)az = 2Cy ba-Q go that Wr) LS ae + Be and by defining the sum of image densities to a depth a by 2 G(a) = fela)ae 0 he obtains W(r) is now regarded as « weighted avernge of the sum of image densities G(z), greatest weight being given to the depth given by making ae (2427) a maximum. This is found by setting 2 eae ) = 0 to be zy, = r//2 and since images appear at twiee the depth of the inter. face to whieh they are Ae a pe depth fuctor is taken as "aA . no (Z) r a Top ground /ayer D §, q g d q f, | S. 4 Oompa Se | ee Ty UY VO €y (£5 % I) Ye 4 eiki ao SYM Figure 8&8. Diagram explaining symbols used in review oF Fufen discussion of depth of penetration of adic. resis- tryity method in aground of many hori- zontal layers, a first approximation to be 1/2/82. By replacing the weight function . 24 Gs ~ (r#4z*)@ with more sharply pecked functions formed from higher derivatives of {gq with respect to r it ip possible to increase the resolving power. In particular for u Wenner configuration he finda a depth factor of about 1/3, ise. “when a layer is present in the ground euch that the surface log of apparent resistivity has o maximum or minimum, this maximum or minimum usgusily will come at a total eleetrode spread which is about nine times the depth to the upper interfuce," In general the vertical resolving power will be greater, the greater the discontinuity in resistivites, and the shallower its depths The depth fuctor will de- pend both uvon the magnitude of the imuge density sums and the weight functions, which in turn will depend upon the magnitude of the resistivity discontinuities and their depths. The resistivity method hus a much greater resolving power for horizontal changes in resistivity which muy be found by moving the set-up along the surface, and is espeoelally useful in engineering foundxtion problems, ond in seeking the surface trace of buried faultse (36) The resulving power will also depond on the location of the electrodes, If the potential electrodes ave too far from the current electrodes their potential difference will be very small, and chances in potential difference due to chance in their location will be very difficult to measure accurately. For example, in the Wenner configura« tion, the potential difference for constant current is inversely proportional to the separation "a", If 4 single euvrent clectrode is used and the potential electrodes are kept a constant distance apart and moved farther end farther away from it, the voltage between them is invorsely pro- portional to the square of the distance to the current electrode, If the potential electrodes are too far from each other, the volume of ground whose resistivity is measured (the volume botween the equipotential surfaces on which the potential electrodes are located to a depth of about the distance between the ourrent electrodes) will be so larce that the measured avnorent resistivity will be insensitive to a stiall increase of electrode separation to ineluie a small volume of material of different resistivity. (36a) Vv Design and Construction of the .pparatus Ae shown in Part IV, the “appurent resistivity" from which the true distribution of resistivity in the ground is obtained requires the measurement of (1) the strength of «a direct current, I, applied to the ground, (2) the resulting potential difference, 4, which the applied current sets up between two potentianl electrodes, and (3) various distances between electrodese Two separate eireuitse are thus necessary, of which one applies « voltage of desired maynitude to the current electrodes, and meagures the strength of current resulting, and the other measures the potential difference, % Bem qause of the neceselty of eliminating the adverse effects of polariantion, galvanic, and other potentials present in the zround, and of taking as amsulil » eurrent aos possible from the potential electrodes, several modifi.~ cationa of each circuit are in use. ‘®) ‘the Gish» {a} In addition to the ingtruments mentioned in the text, there is the new more complicuted equipment de- signed by Jukosky for continuous profiling. nly one electrode is moved und thie, one of the two current electrodes, is u radially projecting blade on the rim of the reor wheel of « truck which is driven at speeds up to severel miles an hour while o continuous reeord of the resistivity is made. (dJskosky,J.J+« Geophysics III, #2, Somewhat different types of apparatus used in two other surface.potential methods ure the “racom" which measures potentiul-drop-ratiog from which resistivity ratios are obt.ined, (Lundberg and Zuschlig, sIMB "Geophys- ical Prospecting" 1932 p47, Hedetrom, The Mining Nagazine, April 1932), and the 'resistanceecridiometer' by which a "higher derivative of the surface-potential function is measured in such se manner thut the depths to formation boundaries are resd directly from the instrument without conversion." (Heiland, Trans. Am. Geophe Union, NeRe Cy 1937, Pt. II, p» 581.) (37) Rooney ‘#) apparatus uses "B" batteries us a current source, and to reduce the effects of polurization, reverses the direetion of the soplied current through the ground at on desired frequency of the order of 30 times per second. This is accomplished by putting one half of » double commutator in the cireuit between the battery and current meter, snd the current electrodes. In the secondary cireuit,. the potentiel difference to be mensured is opposed by the @emef. of a potentiometer through ao galvanometer used as a null instrument. The potentiometer is adjusted until the meter shows thot no current is flowing in the cireuit and the potentiometer reading then gives the desired voltage. The measured potential difference must be applied to the cireuilt always in the same direction, and for this reason, since it reverses direction with the apvlied current, its polarity is kept correct.ewith the other half of the double commutators A condenser is inserted in the potentiol eircuit to eliminate stray direct currenta. In the Megeer(?), un instruntent commonly used to determine insulation resistance in electrical machinery, the source of ayplied current is a hand driven direct current generator und the current is reversed with a double commututor as beforts (a) Further detsils of the design of this spparatua are given in US Patent a Wap ed ae accounts of ite field uge are -iven in papers by Cish and Rooney and others listed in the Bibliography (b) A more detailed deseription of the 'megger' is Elias by Drysdale and Jolley in their “Mleetrical Mi ¢ Instruments” Part Il, pp. 385.388 ond 398, 399. Accounts of field use of the instrument may be found among the papers liated in the bibliography. (38) The current and voltage are not mezeured separately but ore passed through the current and voltage coils of a galvanometer.which records the ratio directly in ohms. Originally no potentiometer was used, and potential electrode stakes sa much as 12 feet long were sometimes necessary to reduce the contact-registiunce voltage drop between the «round and the instrument. When non polarizing electrodes (described below) are used, it is no longer necessary to commutnte the eurrent, and the eleatrical appuratue is simplified by replacing the double commutator with reversing switches, General Considerations In designing apparatus for measuring resistivity, the following factors are to be considered, le Range of values which will be encountered, 2 \ccurscy deeiruble, _ 3 Convenience for field use.~-ruggedness 4, Ixpenses In this case, low expense and ruggedness were con- sidered most important, beecsnuse, as the apparatus is to be used only occasionally and then for student inatruction there ig no demand for extreme accuracy or great range of values. 4p it worked out, stock instruments borrowed from the Physics department were used to uvoid unnecessary duplication by the inatitute, and the runge, uccuracy, and field convenience of the apparatus were controlled by what was available. The quantities whose runge should be considered are (39) the electrode separation a, the upplied current TI, and the potential difference i, The value of a is governed by the depth of penetra- tion desired, and the only requirement placed on the apparatus by larger values is the necessity for longer cables connecting the electrodes to the instrument. The Culifornia Institute of Technology Division of Geological Seiences kindly made available the use of its 1931 Model \ Ford stution wagon (Figet5) originally outfitted for Selsmograph use, end this wae equipped with four cable reels making poasible Wenner separations of 350 feet. In the field tests made it was not deemed worth while to increase this separation, and the con- siderable expense of buying cable wos saveds The values of & and I oceur in the ratio "/T = Re Most values of R found from the grent amount of resistivity prospecting thut has been done fall in « range of from 0 to 300 ohms. The instrumental value of EB was limited above, by the use of a Leeda and Northrup student potentiometer, to 1.6 volts, and belew, to a large enough value (from about .010 to «200 volts depending or local conditions) so that polarization and other diaturbing potentials would not cause too large ou relative error. The value of current should be limited to 200 mi Lidampal®) to avoid excessive drain on B batteries, but in the limited amount of field work done, no appreciable wear on the batteries was noticed with oceasional currents of «7? amps. The lower limit of current is flexible, and (a) Hawkine--A.1eM.l Geophysical Prospecting, 1934, pe 117. (40) should be large enough to overshadow the effects of naturul potentials. In the variety of places at which measurements were made, the extreme values encountered were ae 200'; R= 007-3 Ks .046 volts; I = .6464"Near Deluno a= 2"; Rs 1500 ng Bt Opin, I 2 .0006ame Near Victorville on grunite No determination of the xccuraey of measurement wis made, but the error was probably not greater than ten percent, nnd because of the inacuracies inherent in making theoretical interpretations of duta obtained from a heterogeneous terrain it is likely that incrensed accuracy in measurements would be of no real values Detailed Design and Construction The design used is tuken in outline from an urticle vy BRy Sheperd(S), put the details hive been dictated by expedience, »nd the ideas of other writers have been used o¢casionally, The reader will be aided in following the ensuing deseription by consulting Plate I at the end of the paper. The Srimary Circuit Yor a current source, 8 22.5445 volt Burgess heavy duty B batteries (B1, Plate I) were connected in series. The negative terminal wis connected to the moving contact of 4 17 point non-shorting type yaxley selector switch, (S;) and a wire from each of the positive terminals was soldered to one of the stationary contucts making possible o range of applied voltage from 224 to 360 volts ut 22) volt stepu. (a) Trans. im. Geoph. Union--N.R.0,, 1935, °te 1, De 78s (42) Figure 9. Primary cireult unit: battery box with current meter (M;, Plate I) and controls on top, switeh S, on side above handle, and reversing switeh 8% to which current electrodes are con~ nected when in use below and to the Left. Figure 10. Open battery box showing heavy duty 5 batteries, voltage selector switeh 81 at right, and rheostat RH at left, on topes In nddition the negative terwninul wus connected to the off position of the selector switch so thit natural earth currents could be measured on the milliommeter(M,) in the circuit. However it was found that with the high stake contact resistance in the circuit the current resulting from natural sotentials was toc smull to measure on a meter with 1 milliampere full runges As a precaution against too great a current at 224 volts in case contact and ground resistance should fall to a value of « few ohms, « 500 ohm rhecatnt(RH) was placed in series with the batteries, This proved convenient for making small adjuatments of current when the resulting 2.D, had such o value that the bulance point came at one end of the potentiometer slide wire. "or measuring the vulue of applied current it was originally intended to use » multirainge instrument with ranjes of 10, 50, 100, 500, and 1000 milliamperes, ‘The extension of range by « series of external shunt resistors of accuracy as high as called for by that of the meter ie not expensive, and the cost of meter and shunts with 2% guaranteed accuracy should run about $15,00. Tlowever, it was found expedient to use two borrowed meters with 2 L50 and » 750 amperes full seale deflection, These were -ived to » switeh (Sq) so that either one could be put into the circuit at will, When small currents were to be measured, one or the other meter was replaced by the 10 milliampere meter used in the conditioning circuit described below. (42) The two ends of the part of the circuit desoribed (battery, selector switch, rheostat, meter) were then connected to the two center poles of a double pole, double throw porcelain base copper knife ewitch(Ss). Crosea connections were made, so that the polarity of the two end poles connected to the current electrode cables could be reversed. No. 18 Tirex portable 5.7. cord two conductor rubber insuluted cuble was used. ‘The two conductors were shorted toyxether where they wer: connmeeted to the reversing switeh, and at the opposite end were attached to the female part of an ordinary domestic attachment olugs The male counterpart was then attached to one end of « few feet of eable whose opposite end was fustened to « current electrode stake (C1,C2p). | The gtakes were of two types, one of 1} x 1) x 3/16 ir sentian iron, the other of 14 x 14 x 3/16 angle iron. The two types were made because of the desire to see which type would have the least contact resistance. Although, unfortunately, no comparative teats were made, whatever small difference may have existed waa Of no practical importance in the course of field work. Two aStukes of each type were made enabling two stakes to be connected in parallel at each electrode position, The stakes were about 18 inehes long pointed at one end, The other ends were cut to the proper size so that the three end points of the angle or tee would lie on the (43) cireuuference of @ circle a little larger thun the inner edge of a 2 inch steel cvp. The cups were then hammered on, the edges of the stake cutting through the soft lining materlul containing the thresds, The cups formed © solid surfuce lurge enough to hit solidly with the 6 pound sledge used for driving the stakes. In the course of field operations several of the cap tops split where the top of the stuke cut into them, und one was replaced. However, it wes found that the iron ring left when the c.ip top broke off «served sufficiently to keep the stuke tops from spreading, und no difficulty arose fro pounding on the stoke tops themselves, A hole wag cut in each side of the angle or in the stem of the T below the cap, and a 3-foot piece of Glothes line was passed through the holesand tied. The rope wos looped ut the other end to form u handle, and oroved a convenient means of pulling out the stokes once they had been loosened by tapping their sides with the sledge. Some trouble waa at firet experienced with the manner in wnich the short length of cuble was connected to the stakes. a brass bolt wus placed through u hole in the stuke below the cup, and the cuble wire wus wound wround the bolt between two washers and tightened with a nut. This connection cee loose every time a stake was driven. This was remedied by applying u thick cout of solder with a hot «as flame. This kept the washers, nut, and cable firm indefinitely, but trouble now uwrose from snupping of the cable where it wae (44) Figure 11. Angle iron current electrode stake with steel cap for driving, and rope for removing from the ground. Figure 12. Potential electrode; asphault coated porous clay pot with paper cap to keep the cable from twisting and to keep dust from the copper electrode and sulphate solution. weakened just outside the connection by removal of the insulation, heuting from the so lden’’ and the continual twisting. This wus fixed by soldering the cuble ends to a thin steel lug which kept the cable from twisting. The lug waa then soluered firmly to the stake, and strengthened with a bridge of solder sbout the size of the cable. These etakes worked well, but had « high contuct resistance in dry ground. ‘This was reduced by souking the ground uround the stakes with a solution of salt watere In one Case, where the set up was on solid yprunite, & gatisfuetory contact was obtained b, Laying the stakes on cloths souked with salt water and weighting them down with slubs of rock. Potential Circuit The The potential circuit consisted of non polurising electrodes, switches for selecting (wuen a Lee Configura- tion was used) the electrodes whose potential difference was to be measured, 2 switch for reversing the direction in which 1t was upplied to the cireuit, » potentiometer cireult to produce an opposing voltage of known value, a null indicator to show when the two e.mf.'s were in balance, and a key to muke and break the circuit so that current would only flow instuntuneously when the system wae out of balance The vnon-polsrizing potentinl electrodes (PysPoePs) consisted of porous »ots containing pure copper pipe in @ saturated solution of C.?, cupric sulphute with an excess of crystals. The pots were of ungluzed por. celain 76 mm. in dismetoer, 3/16 of an inch thick in the (45) walls, and 177 mm. in height. To avoid excessive loss of solution by evaporation the outside walls were suturated with « solution of usphalt in xylene so that the outside as dry except for the untreated base which waa wetted ' by the copp@r sulphate solution inaide, (@) The coat ef ssphalt oenetrated perhaps only «a fifth of the way through the walle, and several pots avlit when excess pressure was placed on the solution because of hydraulie aation by the fluid in the pores of the walls. The pots. were greatly atrengthened by coating the inner walls with the ssphilt, thus keeping the solution out of their poreg. The presence of the inner counting of sephalt did not noticeably increase the size of the polarizetion potential. It was found in working in dry arenas that contact resistance wea too high with only the bottom of the cup wet. Wew pots were ussembled with only the upper half of the pet coated with asphalt so that when the pots were planted in holes liberally sprinkled with a solution of technical powdered copper sulphate the inereased contuct srea greutly reduced the resistance. This method of waterproofing the walls proved ver, convenient. The asphalt dissolved rapidly in the xylone, and when applied to the pots with a puint brush dried a0 rapidly that o thick coating could be applied and dry within an hours The copper electrodes were cut from a single piece of ¢ inch i.pes. electrolytic copper vipe, the commercial (a) Partly glazed pots for electrodes ure availuble- commercially from Geophysical Instrument Company in Washington, D.C. and trom Coors in Boulder, Colorado. (46) grade being fortunately more than 99% pure, go that no special treatment was needed to avoid “locul action" po- tentials between the copper und metallio impurities such ug Zinc. \LL electrodes were cut from the same plece of vipe to insure uniformity. The) were sbout 9 inches long, with holes cut near the end immersed in the solu. tion to obtain the increused electrochemical stability of having an almost doubly large contact surface of solution and metal with the svulution filling the interior of the pipe too, The bottom of the pipe waa raised above the vottom of the pot by oluging it with a cork. This was to prevent the possibility of having the copper in direct contact with the solutions of dissolved salts in the ground. The solution was mude up by dissolving C.?. copper sulphate crystals in distilled water until it wos saturated and in excess of crystals remoined. This ensured the pots being at the sume concentration of solution und aveided the votentials set up by a concentrution cell. The pipe was kept vertical in the middle of the pot by passing its top through a tight hole bored in the flat cork which fitted into the top of the pot ond seuled in the golution. To further prevent evaporation and leakage, the cork was dipped in a bath of melted paruffin to aeal up its pores, and a thick coating of “air-tite" auto-top sesler was placed over the cork, especially at its junctions with the pot walls, and with the outside of the pipe. This inexpensive subatance (47) has the desirable properties of staying fluid in u tube indefinitely, and of drying to a smooth-surfuced, rubbery consistency after about 24 hours exposure to the alr. It will form a water and air-tight seal indefinitely and yet the cark was easily removed by a gentle pull when the pot. was to be recouted with asphalt. The upper end of the sipe was sealed with « cork when it was not used for fllling the pot with solution. The upper end of the pipe ubove the pot was groovad and a hole bored through it. The two strunds of copper cable were pussed through the hole und wrapped in the groove to form the contuct. In later pots « thin-walled copper tubing was used, and the cable was attached to a brass bolt passed through s hole in the tubing. The contact potential between the brass and copper was negligibly email. To keep out dust and to keep the cable from twisting, a tin can was placed over the upper end of the pipe bottom end up, so that the cork in the upper end of the pipe was fustened to the bottom of the can with a small woodserew, and the top border of the can was pressed aguwinet the top of the clay pot with a flexible collar of 20 guuge copper drawn together and tightened with « braes bolt. The cable was paseaed through a rubber grommet in a hole in the top of the can wall. This proved unsatisfactory both because the solution of copper sulphate in which the pots were kept would splash against the tin cover, with danger of galvunic action disturbing the potential eyuilibrium of the pots, and because the copper collar put too great a pressure (48) on the clay pote The can wags then revluced with « large inverted tapered paper cup which was held in pluce sufficiently well by the cable passing through a grommet near lis wottom, and by the friction where it slid over the top of the porous pote To the other end of the cable attnuched to the pipe was attached the male part of an uttachment plugs The pote when not in use were kept in vo saturated solution of copper sulphate with excess of eoryatals in a 2 gallon stone jar, and the terminals were shorted together with » double socket to keep the pots ut the sume potential. The pots were kept tight in the jar by wedging in pieces of paper and eloth so that they would not jostle and break in trenait. To prevent splashing, a plece of 16 ounce canvas folded to four thicknesses wis placed over the jar and secured by a piece of wire wound sround the jaws upper circumference beneath u small projecting rim The pots proved very durable, none breaking in the field work, even though two were knocked over by inquisi- tive calvea. It was originally planned to have double chumbered pots of the I1.G.8.5. type used by itdge and taby '*), in which the chamber in contact with the ground contains gelatin goaked with the copper sulphate solution so that there is less lose of copper sulphate into the ground, and less chance of leaknge. However, no double chambered pots were available, and the oroblem of cutting (49) a piece of clay for a partition und cementing it was given up after a few attempts as of more bother than us@e FYortunutely, it was found that the pots as constructed could be used several days without refilling so that the copper sulphate expense was not excessives In some locnlities, the wround was sufficiently moist so that the pots could be set directly upon the ground surfaces In sandy soil, a hole several inches deep was scooped out with w trowel and filled with an impure solution of copper sulphate. In a set up on eranite, the pote were eet on pieces of cloth sorked in copper sulphates The other end of the cables, to which the electrodes were connected in use were counected to two single pole double throw switches(34,55) in such 4 manner that the potential difference between any two of the potentinl electrodes in a Lee Configuration could be spplied to two end poles of s double pole, double throw switch(dg). This switch was cross wired so thut the potential could be thrown seross the two middle poles in either direction. This was necessary in order to huve the ground potential applied to the null instrument always in the sume direc. tion (opposed to the potentiometer emf.) whatever the direction of apvlied current, or pair of electrodes uged. The two center poles were then connected te two end poles of « second double pole, double throw switch(s,) so that when the switeh blades were thrown to thia end ef the switch, the porous pots were thrown into the circuit. (50) When the switch was thrown in the other direction, the potential of a standard cell or “culibration circuit" (see below) was applied to the circuit and used for adjusting the variable resistance (RB) in the potentiometer circuit. The potentiometer circuit (Plate II ) sonsisted of a Leeds and Northrup student potentiometer,(2) two dry cells(Bo) hooked in series to pass a current through the slide wire, a 4 dial 999.9 ohm decade resistance box (RB) in series with the batteries to udjust the current through the potentiometer to .01 emps, and co aingle-pole double throw switeh(Sg) to select either of the two potentiometer ranges. When the "1" range wus used, -0O1 amperes flowed through the slide wire, giving a total RI drop range of 1.6 volts through the 15 10-ohm resistance coils and the 10-ohm slide wire in series with them When the switch was in the ".01" position, the current entered the potentiometer through a different terminul und was shunted so that only «0001 amperes flowed through the resistances and slide wire reducing the totel range to 16 millivolts. énother single pole double throw switeh(Sg) was inserted into the cireuit to make sure that thia circuit was completely open when not in use. This instrument proved satisfactory except that the control knob which ran the moving contacts along the slide wire worked unevenly and it was hard to reach the exact bulance point without jumping past it when the sensitivity of the null instrument (Mp) was high. (51) Plate ID. Diagram showing Four dral decade resistance Sox > 0O-97949.9 ofms details of potential circ uit. L.& NW student potentiqneter ze no 7651 0~16 volts 7 On’ mitivolts to conditioning circuit R L | 17080 -4. /) T J Mz fo switches 4,5,6 and potential electrodes If more field work were done it would be desirable to gear down the control knob so that the contact could be moved siong the slide wire in smaller increments. ‘A telegraph key (K) was used to make and break the current through the circuit and cause s deflection of the null instrument when the opposed potentials were out of balunces 4A 10,000 ohm resistor (R) was wired across the two end poles of u single pole, double throw switch (S49) which was placed in series in the circuit by hooking in the center and one end pole, This resistor was left in the cireuit to protect the potentiometer and null inatrument until approximate balance wag obtained. It wus then shorted out by reveraing the throw of the switch so that « more delicate balance could be obtuined,. The potentiometer current was adjusted by moving the resistance box diala in 1 ohm ateps until the null indicator did not deflect for 4% setting on the potentio. meter dials eyual to that of the known opposed voltage thrown into the circuit in place of the porous pot potential. <‘s field use is too hard on a stundard cell, a conditioning cireuit similar to that used by tee *) Wie set up. This consists of passing a current whose vulue is measured by u meter in the cireuit through two precision resistors of the proper vulue. The potential drop across one of the resistors is than placed acrosa the terminuls of the double pole switch This voltage is then given (a) Lee, ™W.--Information Cireular, 6899, ug 1936--_ U. GS Bure of Mines (52) Figure 13, Potentiometer unit; Potentiometer (P), resistance box (RB), dry cells (Bz), switches 6, 9, and 10, protective resistor (R), and key (K). Figure 14. High gain amplifier with meter in plate cireuit of third stage used by Bicshelberger as null indicator, (Mo, Plates I and IT) by the product of the resistance in ohme and the current: in amperes. The value of the resiatunee, which is suificiently constant with good resistors and soldered connections, can be determined initdully by comparison with a stundard cell. If good resistors ure used, ond a fresh dry cell, the current in the circuit would not change in the time required to adjust the potentiometer eireuilt, snd chan,ves in current over. period of a duy or so caused by fluctuations in the dry cell @eme?. cun be nllowed for by taking the product of the resistance and new current values as 4 new setting for the potentio- meters In the present case, a 1.5 volt dry cell (Bs) passed current through ao 10 milliampers meter (iia) in sérica with a 300 ohm Shaicross and « 100 ohm I.R.C. precision re- sistor. The current was .0039 emperes, and the voltage across the 300 ohm resistor was found to be 1.1406 volts by Balaveing it against the potentiometer Voltuve pre. viously cet with a eston standard cell. This wave a resiatence vulue of 292.4 ohms. Fs} it happened, the current value remained et .0039 oumoseres through the four duys of field use, and the potentiometer setting for calibration was kept at 1.1406 volts. The choice of an instrument to indicate the dezree of balance of the opposed potentials is of great importance, for on it devend the speed, ease, and accuracy of the measurements. If the sensitivity is too low, the system will appear to be in balance for a considerable range of (83) potentiometer settings and the measurement will net be uceurate.e If the sensitivity is too high, the system will appesr to be out of bulance for potentiometer settings sufficiently close to the exact value, and there will be trouble in finding this value, The instrument should be slightly underdemped. hen insufficiently dumped, the oscillations of the indicutor cause » delay in reaching the balance point, and when overdamped the hesitation of the needle in reaching the full value of deflection will be . source of deluy und inuccuracye If the instrument does not indicate which of the two out of balance voltages is lorger, time ia required to determine whether the estentionater setting should be increased or decreuged to reach the balance point, The instrument commonly used ts u portable moving coil gulvonometer with deflection in both directions from a central zero. The instruments used by Hawkins‘) and Bage and Laty'?) had 4 sensitivity of one fourth microampere per acale division of 1 millimeter width, and an internal resistance of about 150 ohme. In this cuge other instruments were used ns described below, but vere not very satisfuctory, and in any future field work with the apparatus it would be advisuble to select a gnivonomater of vroper sensitivity and required external damping resistance to fit the electrical charsecteristiaes of the measuring circuit including the potentiometer and the path through the potentinl electrodes, on (2) ‘eIsMs EB. "Geoohysical Prospecting" 1934, Pave 115. €:9) Ie Ga Be ae Page O66 (54) instrument with high externsl damping resistance is proferable to avoid overdamoing when the contact resistance at the potential electrodes is high, In ureas of low contact resistance a resistance could be thrown in series with the meter to avoid underdamping. To avoid the expense of a gadvonoweter which would be of the order of $25,00 it was decided to use the amplifier-valvonometer unit (Plate ) already assembled in the station wagon for seiemie recording. In this unit, the seismometer output was passed through a input transformer to « three stage amplifier and the amplifier output passed through a second transformer whose secondary wag in series with the moving coil of the gulvonometer which had optical recording. The light from an automobile headlight bulb was reflected from a rotating mirror mounted on the galvonometer coil and thrown onto a slit behind which the film for photoyrsuvhic recording was located. The magnitude of the seismometer voltuge ia then indicated by the size of the displucement of the light beam along the slit. The total voltage #sin of the system is about 1000, For use as u null instrument in the porous pot apparatus, the voltuge of unbulance in the potentiometer circuit was applied through the input trunsformer to the amplifiex. The photographie equipment was removed, and the motion of the light spot -on the slit waa observed visually, The optical system wos 80 Unstendy that it was a laborious task to adjust the light apot to full on the slit, and us the sdjuatment had to be made every time the truck was jarred the attempt (55) Plate IT. Diagram showing circult of seismograph amplifier ~ga/va- hometer unit used as null instrument. (Mz, Plates I and IL) 2uF ___| | I II Ground 8 rt/8o0 V. B +90. B+e7v. *B8 +67. to use the galvonometer was abandoned. However, it was noticed that a 1.5 milliampere direct current meter, connected to a selector awiteh so that it could be thrown in series with the plate current of either the second or third gtuges of the amplifier, was deflected when the votentiometer circuit was broken und closed with the keys ‘This made a satistarkory null inetrument for reaching a balance when checking the potentiometer voltage urainst the standard cell or conditioning circuit, where the cireuit resistance was low ind the balsnce point known. However, in the field work, it worked satisfactorily only in a few cases. Usually it was too insensitive on the second stage of am lificution, and too sensitive on the third stage. It was a laborious task to adjust the sensitivity properly with the rheostat in geries with the meter, and often proper sensitivity could not be obtained. A direct current 1 milliampere full scale meter on the plate current of a three stage amplifier built by Martin “ichelberger (PlateIv) worked much more satise factorily, but also hud the drawback of not indicuting by the direction of deflection whether the potentiometer setting was too lurge or too small. A Weston Model 301 DC millisameter with full scale range of 1 milliumpere, used in the field work near Victorville, California, indieated the direction and degree of unbalance well enough in most causes, but was not sensitive enough in ease of high contact resistance (56) Plate IZ. Circuit diagram of Eichelberger’s high gain amplifier used wrth input transformer as null indicator (Mz, Plates ZL and). 0.025 ME 9.025K8F | | h 54,000 4. GUT | : GUS olMFE | Maariv T = >} a aatuaied d ; — -— == J 305M eee posm Soman. Fs olNF | | sind f_ a | fnpoak fons —— 2.0 aa ie 3 etn. _— Man. 0.1 NE! “ey ; % i 4.0ne 2. Mo i 5M * 4 ra ala - |0-5¥F —=T25u 3 Soca 6.5AF to fransformer BS — J = 3 = [ $5000. _ S20 : : 3 50,0007, Frta. 1 B+40 v. BtrI70 18+ 90 Vv. even when the vorogus pots were buried in ground wet with cooper sulphate gulution. It also had the disxdvantuge of defleeting in one direction only so that when wired for positive deflection by the component of current from the potentiometer voltage, the current in the cireuit when the ovotentiometer setting was below the balunce noint forced the indicating needle against tie stopa in the negative directions (57) VI Bield Work Introduction J@ have seen above that the resistivity of « ground homogeneous to an unlimited depth is given by the expres~ sion Ss aya B/I (22a) where a is the separstion be- tween any two adjacent electrodes in the enner con- figuration, und & is the potential difference between the porous potas when uw current I is made to pase between the current stakes. In such a homogeneous body the computed value of resistivity would be the eame for any choice of electrode separation, and for epreads of any one separation located anywhere on the wround surruce. Actually the ground is never completely homogeneous, and the computed value is not the uctual resistivity. As it is useful in determining the actual resistivities, the vulue is computed from the formula, but fe culled the appurent resistivity and designated by the symbol f,,. Two main types of field test are made. One method is to take a series of measurements with increasing eleetrode sepurutions about « single fixed spread center. this will be called 2 depth profile and the data is usunlly plotted in the form of a graph with apparent resistivity as ordinate and electrode separation aus abscissa The curve obtained in practice is usually not the straight horizontal line which a truly homogeneous ground would give, but is either smooth or irregular. If the curve (58) is smooth, and if similar curves tiuken ut the same location with the line of electrodes in different direc. tions are also smooth and of the aame type, « theoretical treatment is possible as the symmetry of the problem indicstes a regular distribution of resistivities in apace. A full deseription of methods of interpreting such smooth curves is ,iven in Chapter VII, end illustrated with data obtained by the writer in the field. If the curve is jagged, or the curves for different azimuths of electrode lines do not indicate symmetry, wn empirical interpretation is necessary, Type curves may be compured with the known geology at a point, and irregularities found to correspond with definite ceologiec units. If such correlation is found to exist at several points, and a definite relationship to exist between the magnitude of the electrode sepsration and such physical dimensions of the problem os known de ths to formations, it is then possible to estimate such dimensions from the electrode separations at intermediate pointe of unknown geology, provided the curves obtained ure correlatable with the type curves. “ven with smooth ourves subject to theoretical treatment it is very important to get empirical check on the result if one is to feel much confidence in the interpretotion, In the other field method, the spread is moved from point to point in an ares while the electrode separation is kent constant. The computed resistivity as ordinate ig then plotted uzsinst the location of the apread center (59) as sbacissa. Here, again, the curve obtained is seldom the horizontal straight line which should be found for a gingle homogeneous ground, but may either be jagged or approximate a smooth curve. The first cause requires empirical interpretation, and the other, although subject to theoretic .l trestment has much more value if subject to frequent check with known geologye The methods of theoretical interpretation of curves obtpined in such a traverse are not specifically treated in this paper, but the general method of sttack ie the same us for the depth profiles in thst the field curves ure compared with master curves in which the apparent resistivity was derived in terms of o postulated distribution of actual resistivities by the use of the surfoce potential function. Depth profiles are commonly used in simple cuses of fairly shallow depth such as to determine the cepth of overburden covering bed rock, knowledge useful to the civil engineer in planning highway, bridge and dam construction, or to indicate the location of highly conductive ore bodies. Deeper profiles obtained with large electrode spacing have been used to determine the position of aquifers at depths of more than 1000 feet to help in locating witer welle, and to locate the presence of possible oil structures at depths of a few thousand feet. Horizontal traverse profiles ure useful in reveecling high engle contacts such as foult contacts or those (60) between vertical beds where the surfuce trices ure obscured by a veneer of surface material, and in indicate ing the presence of shallow ore bodies. “hey have the advantage over deoth orofiles of having greater resolving power as the chunges in resistivity take place nearer to the electrodes. They are also of use in regions of anooth apparent resaistivity~electrodé separation curves in conjunction with the more time ocunsuming depth pro. ~ files. For exomple the depth to a horizontal contact may be found at severnl points with depth profiles. The depth to the contact may then be estimated at intermediate nointa by taking a traverse with a critical value of electrode separation and seeing whether the value of npparent resistivity for that separation increases or decreasess The field work carried out by the writer (See Map I) consists of three parte. The first was a depth profile run at Hastings Raneh near Sierra Madre, to see whether the values obtained would be the same us those obtained with commercial apvarntus., The second was a depth profile made near Delano to see how the resistivities and curve interpretations would compare with the informa. tion obtained by lowering electrodes into » selamograph shot hole, The third waa a series of depth profiles made near Victorville to see whether such profiling was a avactivakis wey of determining the depth of alluvium over the bed rock, The rest of this chapter is devoted to a discussion of these three teatas (61) San Francisce CALIFORNIA NEVADA © cetano © Bawerstield PACIFIC .- OCEAN O Victorville Los Angeles O Sierra Madre ARIZONA 2 San Diego = " MEXICO 2 25 50 'o9 (go ° 200 nm A A 4 L i Poe ok SCALE OF PAILES Map I. Index map of part of the state of Californata showing general location of places at which Field worn was carried out atcenters of double circles. Field Work at Hastings Ranch To learn whether the appsratus would work satis~ factorily in the field, the first test was the running of » depth profile at a location where Mr. Lohman, a consulting geophysicist of South Pasadena, had previous- ly run such a profile with Gish-Rooney ee The work went slowly becuuse of a lack of a sufficient number of field assistants, ound some difficulty in cetting the potentiometer circuit to bnlance properly. ‘The days of Webruary 11, 12, and 19 ond Moreh 4 and 5, 1939 were required for the work. After the profile had been com. pleted and an apparent resistivity-electrode separation curve plotted, Mr. Iohman gave the writer a copy of the curve which he hod obtained two years before, These curves will be discussed Inter. The profile was laid out in a nearly north.~south direction about a center point 1500 feet north and 500 feet east of the fence corner at the northeast corner of the intersection of Sierra Madre Villa Road and Foot. hill Boulevard, near Sierra Madre, Direetions were taken from a Brunton compass, and distances measured with a 200 foot steel chain. The pointa to be occupied by porous pots and current stokes were marked out with short pieces of lath. Yenner spacings of from 5 to 260 feet were used, and in some of the apacings a third potential electrode was used with the Lee configuration us a check, For every value of electrode spacing, a, valuea of the potential (62) Figure 15. Institute Model A station wagon and apparatus set up for field work at Hasting's Ranch near Sierra Madre, California. Figure 16. Line of profile at Hasting's Ranch. San Gabriel Mountains in the background. difference i between the pots was taken for two values of applied current I in each direction of current between the stakes. The four ratios of B/I = R are then nveraged to obtain the apparent resistivity, Pa * 2raR, Re. verging the direction of current and averaging eliminates the effect of constant potentials from other sources which may be present in the ground, Using two magnitudes of applied current «ffords an indtcation of how well Ohm's law is holding, and by inoreasing the number of observations used for each value of resistivity cuts down the effect of non syatematic errors in the individual measurements. Readinzs were also taken of the potentiometer setting Eygg at which balance with the standard cell was attained. From this the value of current through the potentiometer slide wire can be obtained, and if it is different from .01 amperes, corrections in the value of E can be made accordingly. It was found that by adjusting the 4 d#n1 resistance box in series with the slide wire every half hour or so, the velues of ! were sufficiently accurate without correction. Headings of the resistence box setting Rpy st which the potentiometer setting agreed with the standard cell at balance were taken ag a check on the voltage stability of the dry cells supplying the potentiometer current. Values of the applied voltage Hg necessary to obtain the applied current I were also taken, and the ratio computed to give an indication of the size of the stuke contact resistance, The large and irregular effect of this contact resistance on the vulue of B,/I (Figure 18) (63) (onms) ee ; ee l i] 16 oo} Lh =f 1 /4000 iF l y 1 13 000 = i J2 ooat t/ 090 10 ogo}— Jaoo te f 7000 |-— t——t a 60001 eI aT 40900 74 Fr tT ot 3009} oo 2000!}— [| a 1000 E—+- { has prevented the measuring of variation of ground resietance with two electrodes. A few measurements of the potential difference existing between the porous pots with no eurrent spplied were made when it waa noticed that the values of i differed gmatly in the two directiona. Thie potential difference ran as high as 50 millivolts with the pots about 250 feet aparts The identities of the porous pot were kept by patting a different number of pieces of udhesive tape ground the terminal of each, Polarization potentials batween the pots were checked from time to time by inter- changing the pote, and found to be of the order of one or two millivolts, not large enough to make any appreciable error in the values of resistivity. The data obtained in thia work sre listed in Table II at the end of the chapters tr, Lohmanta curve and the writer's sre compared in PLATE The resistivity values of the latter are lower, und much less regular, and their general trend is a atraight line about tangent to Mr. Lohman's curve for electrode spacings of about 200 feet. The higher Gish-Rooney values might seem to contradict the test of Te and Keys at Pigsher maf), New York, where the porous pot valuea were higher, but it is not improbable thet the heavy rains just before the porous pot survey at Hastings Raneh sufficiently watered the ground to reduce the resistivity well below the value ut the time two years @arlier when lr. Lohman's teat was mad@s (64) 2) ApPIED GEOFAYSICS, 2wv. EP, 7933, AUS The much smoother curve obtained by Nr, Lohman may be expluined by three differences in the field work. Mx. Lohman's work was all carried out in one day, while the vriter's required five daya over u three week periods Tt ia possible that the ground resistivity veried aome. what from day to day. Algo there was difficulty in reaching the correct balance point and this delay in setting the potentiometer balenced affects the value of reaistivity determined, because due to polarization around the current electrodes and perhaps other causes, the values of M and I fluetuate. The fluctuations are more noticeable in = than in I, and fluctuatione in the measured registivity result, which would be eliminated by rapid commutation of the current, and reaching the balanee point ropidly. Perhaps the largest cause of irregularity is the difficulty in finding the correct balance point when there (3s insufficient sensitivity. In some places the value taken for B was estimated as the center of a renge of potentiometer settings of ag much as 10 milli- volts over which there was vo deflection of the null instrument. A discontinuity is to be noticed in Vr. Lohman's curve at an electrode separstion of 520 feet. This is due to the fact that one of the current electrodes passed over the water main at that separation, but does not mean that the water main wag at anywhere near that depth. The effeet of such a main at so great a depth on the appsrent resistivity of the large mase of sround (65) Bess GTS fils iy / i 2 t i GE ; ZY te Re CALE! ZT Goo Goo | é Wie NN A MW / — A iE | wen = 2000 FEET Sj a g ‘ ; Dae | 8) mezzo 7240 NL Pea A\EIIS ( e Map II. Part of the United States Geological Survey topographic map of the Sierra Madre Quadrangle, California, edition of 1928. The position of the profile at Hasting's Ranch on an &@lluvial fan at the foot of the San Gabriel Mountains ig shown. penetrated with » 520 foot apacing or 1560 feet between the current eleotrodes would be nagligibles Two layer interpretstiona of the two curves by Roman's guperposition method (Chanter VII} do not show close agreement of either curve with . muster curve, but a rough approximation gave values of depth to a gecond lower resistivity layer as about 120 feet for ir. Lohman's curve, end 90 feet for the writer's, Thie is mentioned, not in supnert of the theais that the difference corresponds with the rising of the water tuble during heavy rains that lewered also the resistivity of the upper layers, but ae an example of the type of aneculative interpretation to be avoided, Not only is there no empiries1 check nvailuble, but the location of the profile on an alluvial fun at the base of the Gan Gabriel Mountnings (Mopiar) suggests that there may be a considerable number of lenses of sand and @ravel of varying resistivity down to the depth of 100 feet, so that even if the greatest change of realativity occurs nt the water table, use of master curves based on the nesumption of two and only two lavera would be prohibited. Field Work Near Delano In April 1939, the writer was priviledged to accompany ir. Hichelberger and Dr. Peteraou to Bukersfield, Sen Joaquin Valley heudyquerters of the 'mited cecphysical Compuny, The portable battery, potentiometer, and amplifier units were taken along as well as two sete of current and potential electrodes, one to be lowered into (66) a seismograph shot hole, ond the other to be used in a surface depth profile. The United | eophysical Company allowed the use of 2 shot hole several miles southeast of Delano, California and furnished the use of o truck for field uses ‘4 complete deseription of the whole survey is given in Martin Mchelberger's Masters Thesis (Institute Division of Geological “elencesa, 1939), For purposes of this paper it is sufficient to sa) that the four electrodes were placed at » constant spacing of about 3 feet along es hard rubber tube which was lowered into the shot hole on a cable in such a way that the depth of the center of the apread below the ground agurfnee was knewn. This t-pe of profile is identical with s eurfsce traverge with Wenner configuration of a three foot sevaration with the exception that the electrodes become completely surrounded by ground (except for the negligible volume of the eylindrical hole) po that the opparent resistivity is ¢iven by the formula f, = 47a B/I, the expression for an infinite rather thsn semi-infinite homogeneous medium, The apparent resistivity in this cuse is approxim:tely the actual resistivity if the ground is Romogeneous in layers thick compared to the spread lenuth, and at any rate reflects much more sharnly the variation of true resistivity with depth. 4A surface depth profile was algo made with vwenner configuration and electrode sepurationa of from 5 to 200 feet. A miniature traverse profile was innde with 10 feot (67) electrode sepurition, and spreud centers at 30 foot intervals from 270 feet north of the center paint to 270 feet south of the center point of the north south line of electrodes of the depth profile. The purpose of this traverse to see whether horizontul variations in the gurface rasistivity might be causing breake in the apparent resistivity values of the depth profile. The surface resistivity as meusured with the 10 foot epacing was found to be quite uniform along moat of the length of the profile and no correlation was found between surf.es variations and apparent resistivity in the depth profile. The duta are tabulated in Table ITI ‘at the end of the chuoter, and shown yruphicelly in the curves of Figure 19, Curve 1 is the apparent resistivity- depth curve obtained with electrodes in the hole. Curve 2 is the apparent realativity-eleetrode separation curve for the surface depth profile. Curve 3 is the uppurent resistivity-spread location curve for the truverse profile, and the set of curves 4, ig derived from Curve 4 and showa the surfsxee resistivity value for each 6lectrode corres. ponding to each electrode separation in the depth profile. Curve 1 shows us the depth to the level of water in the hole, found by lowerin the electrodes from the surface until there was » sudden sharp inerease in the applied current with the eoplied battery voltuge kept consatunt, to be fifty feet. It also shows o repid variation of resistivity with depth below the water level, sugyesting the oresence of numerous liyers of grouud with different resistivities. It shows also, un average resistivity of (68) 00a &000 5000 Ni fateh al iS urface ca Line pf elestrodes a —}- ! aul | poor 1 T as Ee | eae | | ’ any METE a Se 66-Wetier lyst ri at { I coglrayals =", mee hele! ‘oe: a (Cri ba f a bp Be, Sati uae So geo tS Ee =! Fxrplanation - it 4 a 1 “Ve gs ttre negro Se ee of Apparent ‘resistivity — “dental ciclo, “Eleiivedies eke aE lowered into sho? hole: Electrode inferval 9) Po ara —eonstant-at!—2 -F- Feet —-~1-—- Ph gt ote 7 ma Hinged nintletey L depth re te. |ESoc rade (Se pt eb | Separation vairteol elects on Surface.: : sis | : 3. Apporent resist Wty against lotion efisptead) . |i Pa [- center a” ad oF ef ea tts es or 2. Electrode are ie L——+--$€ fore consitan# a 40 Feet: --— 1 sy Appareatiressstivity. af gra a aroha |_| oe ee de ee ee ae eek electrode Far e@ Specie oa PED Sem a Sei et i Bditem afaob eh oa] eee me ee aE Pa Gaara LC af. i t ee Rigel t r ! I 1 i I { = ; : 1 } y cert i t i t ta rp Q f | Es a 2 ¥ bare | ee | es ioe ee ‘ |: Pay pee ee AS: au ee aA iF be, S: % (ieee 8 2 esl tenn = Pee See a7 ie = = ae | Satara ietecaer-a ayes i ae a i a moor a rH iam aie | ee ie 242 fi. mel ea E 4 ie seared aE is qj} eS Seas = oa er =| Taal base i i. a = ren = ay | i a See en | a ME SCA Getler oe ie bot ee Ce A bedas feces 10 to 15 ohm-weters for ground below the water level. Compariaon with curve 2 slhowa the litter to be much siioother illustrating how small the offect of uw change of resistivity ut depth is on the upparent resistivity of the whole ground down to and including that depth, It may aleo be seen that the value of apparent reaistivity at the longer electrode sepurations, corresponding to. .teauter deptha of penetration are ubout twice us greut — as for the bole agurvey, This is due in part to the fact that the water in the hole added during drilling may raise the water level there above the regienul water table go thut the hole elactrodes were surrounded by abnormally wet ground, in part to the fuet that the hole is lined with a low resistivity layer of drilling mud, and finally to the fact that the electrodes in tue hole no .oner are greatly influenced by the piasence ar the higher resistivity surface layer while the apparent reaistivity of the surface profile ise The shupe of the surface depth ourve would sugwest a three Layer interpretation with a saceond layer at shallow depth of resistivity wreater than the surfuce layer overlying the more conductive deeper ground. In. sufficient data at the critical electrode sepuration of the turning point are available for making u three luyer interpretation, and it is likel; that the resiativity value at the shortest spuoing is too low, us the ualt water and copper sulphute solutions used tu cul down contact resistance muy have lowered appreciably the (69) resiativity of tie volume of ground we..sured vith the five foot spacing. Two layer interpretutions mude in Chapter VII for purpose of illustrating vurlous methoda show u depth to the interface of 10 to 15 feet, but in the absence of confirmatory dutu the presence of u lithologvic boundury ot that depth is problematical, Field Work Near Victorville, California The final field tasts wera made aarly in June 19359 at the suggeation of Dr, Maxson, The geologic problen vas to determine whether the buried portions of padiment aurfaces towarda the edge away from the higher portiona of the bed rock on which they ure cut are convex ar coneuve wowardse The elevation of the surface of the ulluvium ovseuriug the bedrock surface is known from surface mapping. If the depth to bedrock below the surface of uwlluvium cun be determined, the elevation ou top of the bedrock is found by subtricting the thickness af uwlluviura trom the elevation of the alluvium surface, und contours of the bedrock surface from this information would show tha shape of the pediment surface, Dr. Mumxson chese an urea for this inveastig.tion 3 or 4 miles north of Victorville, southeast of Oro Grande und eust of the Mojave River (liug TIT). We and Mr. le... Lewis sccompunied the writer on .. trip to the urea in whieh a pro. ran of fields work ws outlined, This field work wus cerried out in the four cays June 4 through June 7, 1939, as «& prelim. inary investigation to see whether sutisfuctory results could be expected from the resistivity method. The work (70) t Ad elanito 12885 BV Z R.4 W.. . 3 4 qn BERNARDINO 40 Mi.” ar mat 4 oles z S Seale TREO SAN BERNARDINO 35 Mi, oe ASS t q ‘ Fs eel he 4 4 is op IG ——_— = — = = SS F Z "Se, 2 fi 2. By ok 2 3 & ". 3 Tae eres : - ' % | OS ole j a Q EN elle © Contour interval 100 feet & Datum ts mean sec level Map III. Part of the United States Geological Survey popoyrephiec map of the Barstow Quadrangle, California, edition of 1934. Crosses indicate the points at which resistivity measurements* were made. consisted of five depth profiles, the first at a point where the depth to the granite bedrock was known, to give & check on values reached by theoretical interpretation. The second profile was made on an exposure free of alluvial covering to determine directly the resistivity of the granite basement. The other three tests were made at typical points in the areca to see whether smooth curves susceptible of theoretical treatment would be obtained. The data from these profiles are given in Table IV at the end of the chapter, and the apparent resistivity» electrode separation curves are plotted to a log log scale in Figurea 32, 34, 35, and 36 below, Test Point no. 1 (see Map III) is located some distance from the other points and acrosa the Mojave River from them. The profile was laid out in a roughly north-south direction parallel to an adjacent road cut in which the section below the elea- trodes was exposed. This section showed the bed rock con- tact to be 16 feet below the ground surface on which the electrodes were placed. Unfortunately, no similar exposure was available near the other points. The curve obtained at Test Point no. 1 shows the problem to be a three layer one electrically, with a lower resistivity layer intermediate between the surface layer and the bed rock. This renders interpretation of field curves more complicated and less certain than in the two layer case. The known depth to bed rock of 16 feet checks well with the 15 foot electrode separation at the turning point in agreement with Rooney's (71) empirical generalization, and with the depth found by superposition (Chapter VII) of 16 feet. However, it is likely that with similar curves at other points where the minimum of the curve occurs at larger separations, the greater depth to bedrock may change the depth factor or ratio of bedrock depth to electrode separation from unity to some smaller value. In this case, if no further empirical check is available, it would seem safer to trust the theoretical solution if this differed from the electrode separation of the curve minimum. Test Point no. 2, located about 100 feet from Test Point no. 1, was not profiled, ewen though it would give. a desirable check on the first curve, because of the limited time available. With euch slow progress (a whole day was required for the profile at Test Point no. 1) it seemed wiser to see whether curves obtained at points across the river, where it was most important to determine bed rock depth, would be of the same type or sufficiently smooth for theoretical. treabment. Test Point nc. 3% was next occupied, and here the resistivity of the unweathered granite basement was found to be of the order of 500,000 ohm-centimeters, something like fifty times that of the overburden. it is interesting to note that the value determined for the bed rock resistivity in Chapter VII is between 25,000 and 50,000 ohm-centimetera, or only 1/10 to 1/20 as great. This suggeste that where the bedrock 45 (#2) Figure i7. ‘The apparatus set at Test Point no. 3 on granite uear Victorville, Galifornia,. Boulder dam power line in middle distance, not oxposed it may be decayed to some extont by weathering. A ratio of something like 5 to 1 in the resistivities of bedrock and surface gravels should be sufficient to sive @ pronounced bend in the field curves, even if the depth | to the discontinuity should be at considerable depth, Curve 3 was tyken about £200 feet southeast of the noarest granite outcrop, It is a three layer curve like that at #1. Unfortunately measurements were made at too few diaznostic electrode separations, and the value of 68 fect found for the bedrock depth may bo more in error than the values at the other points, Gurve & was taken at a point about 150 feet south of the center of the spread for Curve 3 and about 300 feot south east of the nearest eranite outerop., It is a 4 layer curve, and indicates a bed rock depth of about 60 feot. Curve 5 was taken with a center point about 65 feet south of the center of that of Curve 4, This 3 layer curve indicates the bedrock to lie at a depth of about $0 feet, The data from these tests are given in Table IV at the end of the chapter, and the results. of interpretation of the curves are shown in Plate VI, The individual inter- pretations are discussed in Chapter VII below, It is the opinion of the writer that this preliminary work indicates the probable success of the method in fiving a qualitative answer to the question of the shape (73) Plafe YZ. {5.8 interpretation of Fre/d curves From near Vicrorville , California, Numbers in columns are resistivities m chm-centime ters. Broken Jines between columns are +wo of the possible pasitrans of the overburden —"granite’ contact within the zone of probable error in determining the depth of the contact. ~~ CURVE 3 — " CURVE Y cuaves 19,200 wing S re 18,900 4 ag aes corr if overburden ; ae } 30 S a apn ee at ee | ever burden P —— — |__| 50 | Re Aterb ae — se L___1 5’ 35/00 parr { 58". bt --—- og ‘i oF ee 4% 500 — _ 47 600 granite” F en bea Ee = a . a aoe cage henge SE ERM Rw BE ‘ — } ——— gain PRLS eae ; 7 aa | fig a a an } “ Gn ‘ t ” rr sy a rel ‘granite a a © Cig? ee x wy a* ries ae got NX ee : : hy Xj at ~ ie ee Ts | . — 4 aXe Oo ny 100 | . ie _. I} NORIZOWTAL ANO VERTICAL SCALES : { (WON EQUALS 33.3 FEET Ley i of the bad rook surface, Measurements made with sufficiently good apparatus, and with sufficient care should cive the depth to bedrock within 10%, From this it should be possible to deter ine whether the surface is considerably colcave upward, considerably concave downwards or about plane, or whathor the surface is step like or irregular, Any future work should be carried out with a better potentio~ meter and null instrument as sugested in Chapter VV, Field curves should be plotted on loraritimic paper in the field sO that a avufficiont number of accurate values will be had at diagnostic electrode spacings, All the »~reliminary curves were taven with the electrodes in a roughly ecastewesat line about parallel to the topographic strike, In future work 1t would be deslrabloe to take aproads in soveral agZimuths at each of a few points to see how much symmetry thore ig in the resistivity distribution. Fotnts should be accurately profiled at intervals of about 4 mile on a erid with sides parallel to the expected din and strike of the bedrock, These would be far enouch apart so that if there is any pronounced divergence between the slopes of the alluvium and bedrogk surfaces 1t should show up in variations in depth to bed roek laree compared to the probable error in determining the depth. To cheek on the regularity at intermediate points profiles could be run fairly rapidly using only three or four valucs of eleetrode separation, (74) 50 45 60 13 9@ 105 Data from Hastings Ranch near Sierra Medre, California Bh se 10 PA) 25 ad Potential measured between potential electrodes in volts TABLE If Part i. Field observations Explanation, of symbols; Dietance between current electrodes in feet Distance between potential electrodes in feet Voltage applied between current electrodes Current between current electrodes in amperes Direction from. positive to negative electrodes Setting of rheostat in potentiometer cireult in ohms Measured potential of standard cell in volte 270 560 360 270 yes 30" 560 157% 5 oSooc eeu Fo ww zs = = 8 *wst # aoe es 2 oSooce coco jw PPPS PT SF RR in Dia OD ~ ranstnsan ss. 8 + HF win Ot ooco cooec ceoceeg So SHS > Ww oes & Aas Sn * ££ @ & - 0.802 14425 1643 ? 0.815 12434 0.9499 0.9407 i. 5627 1. 5407 1+ 524 0.916 1.04 ? 165157 . Self ts 521 12464 _ On947 te 264 in BA? i232 0.921 0.904 1133 0.889 0.0003 0.420 0+ 939: 0.910 U.4U5 ‘Dir NS Sail Nad Bell Had Sail WS WS S=il Ned Sa Ned Sad Rap 150,40 149 «0 148.8 148.7 146.5 148.44 Bsg 1,0189 1.0215 160193 1.0187 1.0191 1.0187 1.0190 1.0 187 1.0190 120187 1.0 189 1.0187 TeUizi Above measurements made on rebruary 11, 173Y (744) 120 135 150 165 180 iP) 210 225 240 49 45 be) 60 65 70 10 80 TABLE IL Part 16 Continued Measurements meade on February 12, 1939 Ba 202, 5 ie 360 202.5 r 60 180 r 360 180 186 8 360 360 180 360 180 360 360 180 135 315 515 135 135 270 270 135 180 360 360 180 i 0.0265 0.0470 0.0445 0.0250 0.0143 0.0283 0.0287 0.0143 0.0207 0,040 5 0.0425 0.0207 0,0205 0.0405 0.079 0.0775 0.0395 0,0387 0.0703 0.0945 6.0391 0.0473 0.09 1; 0.091 0,04790 OG > f Po Bay = + ss * Sococ coco oooce ocec = = & * 0.0360 0.0680 0.0722 0.0378 E 0.38 0.67 0.642 0463.56 0, 188 0.366 06357 0.175 0,232 0.447 0.421 0.216 0. 364 0696 0.694 0.653 0.330 ss ¢ +. 8 —— Sauer Secs 5 PSA, eqcoo ocesc cece PO Opa Wo *¢* ¢ * 6 oh Bre aatintnws <3 (74) Dir Nes Sai NaS S =i Ned Sali Had Sali N-5S Sail NS Sal NaS Sali NaS Sli Nad Sali Rup 146.6 146.5 146.4 146.4 146.2 146.5 146.2 146.1 Es ¢ 1.0187 1.0187 1.0187 1.0190 1.0187 140 190 1.0 187 1.0187 1.0189 1.0192 12.6187 1.0187 1.0189 1,0194 1,0 189 1.0187 1.0187 1.0191 ba a 8 90 a5 95 ™ Part te He 180 60 60 180 TABLE {1 Measuremenmtea made 90 45 100 110 247. 5 360 247. 5 2472 5 dl os 30 360 Continued i E Dir 0.0655 0.354 = Sal 0.02 Os 15 N«S 0,0465 0.264 G.0490 0.254 S.-H 0.0245 0.116 0.0235 0. 108 NaS 0.40305 O» 146 0.0320 0. 139 8M 0.0228 0.112 0.0 180 0,092 oS O.0315 o: | 0.0310 0. 146 Sai 0.0155 0.055 on February 19, 0.0365 0, 1814 NaS 0.0 53 0,292 0.0483 0, 249 Sal 0 «0327 O. 183 0.0330 0.089 N«S 0.0310 0.08: S-N 0.046 0. 13 O»0 523 0. 146 eS 0.052! 0, 123 O.049 0.114 Sal 0.0300 0. 133 Nas 0.0480 04.217 0.0447 0. 180 Sil 0.0283 0.116 0.0485 0.115 WS 0.0483 0.110 Sai $ +048) 0,088 0.0485 0.097 Nad 0.0353 0.155 Nad 0.0835 03353 0.0925 0, 3: Sali O 0353 i 14569 14.5.6 14.5.3 1939 145.9 145.6 145.3 14561 144.9 Esg 1.0187 1.0189 1.0187 1.0187 1.0187 1.0187 1.0190 1.0187 140196 140189 1.0195 1.0187 1.0192 1.0189 1.0187 1.0187 and Sin last column indicate whether north or south (74¢) Di Nalt of spread is tested when Lee partition method is used. 330 360 390 360 390 420 4 50 480 510 oo 120 60 130 65 120 130 140 150 160 1970 Part 1. Continued Ea I ¥ Dir 22! 0.083 0.175 NS rey: ooBes 0.170 Sal 225 0.0190 0,126 HS 225 0.0865 0.148 Sd 90 0.0425 0.1457 Ned 90 06,0423 0.1247 SaM 180 9100s aie | 180 06,0845 0.276 HS 180 0.0835 0.1 No 13 9°e040 O19 SH 180 0.0818 QO. 1487 | 180 0,0800 O. 151 NaS 112.5 0.0462 0.157 4S 225 eet ree 225 0.0895 0,353 Sell 112.5 00,0465 04 124 225 0.0888 0,194 NaS 225 0.0839 0.135 225 0.0898 0.134 Sail Measurements made on March 4, 1939 90 0.0425 04139 eS 180 0,9 ly 0.290 190 0.0805 0.274 SN 90 0.0410 0.138 157.5 0.0796 a i N-~S 13}69 0:0) os eet Sai 360 0.1630 0.6007 NaS 157.5 0,0470 0. 384 NaS 13113 Oloa7d = Oltge ae Me VOT, QO. 176 157:5 0.0500 0,086 Sail’ 180 0.0885 0.267 NS 180 0.0685 0,200? SW 180 0.0895 0,286 Has 180 0.0905 0.189 Si 135 ety 0 264 NewS 135 0.0975 6, 227 Sli 135 0.0925 0. 227 Ned 135 0.0930 0.167 Sal 90 0.0640 0.119 90 0.0640 0.144 Hag (744) TABLE Il Rup 144.9 144.6 144.4 144.3 144,2 146.7 146.5 14569 146.0 145.9 145.7 14552 14.5.3 %g¢ 1.0198 140189 1.0190 1.0187 1.0189 1.0187 1.9189 1.0189 1.0188 1.0187 1.0195 1.0187 TABLE II Part i. Goneluded L e Eg Bi E wir Rep Esg 540, 180 90 0.0445 0,083 NS 145.2 1.0187 , 180 0.08 é o190 : : 180 0,0880 0,1 SH 90 0.0450 6.060% 1.0187 570 «6190 ~—s 180 0.02235 0.058 Ne 144,797 1.0189 360 0.0450 0.1007 180. 0.0223 ee S-N 369 0.0440 0.04 1.0183 Measurements made en Mareh 5, 1939 519-190 0 | 0,035 Nd 360 0.0467 ars 144.9 1.0187 360 0.0465 0.057 Sal 360 0.090 0.109 Nes 95 360 0.0465 0,053 360 0.0495 ©,0047 Hed (1.0192 600 200 0 0.04 7 Hes 292.5 0.0615 0,082 Sel 143.8 1.0187 292.5 0.0613 0,081 Had 180 0.0390 0.057 100 )=—s_« 180 0.0390 0,050 180 0,0390 0,032 12.0187 660 220 360 0.0575 0.053 HS 143.6 1.0187 360 0.0575 0,052 Sai 1.0189 720 240 360 0.0215 0.02? ae aber oes xs 14.8 8 202350 © sO8 le ry Q 360 0.0225 O.064¢ Sail J 113065 780 260 360 6.0482 06,0677# SN 15.6 1.5980 | 360 0.0502 ee ee si id 180 60 270 0.0965 200% Ne 316.0 1. 5980 270 0.0965 0,768 144.0 1.0187 270 0,0973 0,945 Swi 300-270 0.0967 9.4372 270 0.0975 0,393 Had 270 0.0965 0.370 Sl 270 0.0978 0.375 Had # Potentiometer reading must be ee by 1.0187/ 2 sc to get correot value of E because slide wire current was decreased in order to eon bigger velues easier to measure, (74) i & TABLE IT Part 2. Apparent resistivities. Symbols: he Distance between current electrodes in feet. a. Distance between conseoutive potential akootretes in feet, Ray. Average value of R (equals E/I) from values in Part 1 for euch configuration expressed in ohms, S.. apparent resistivity in ohm-centimeters., Computed from the formulae * = = be JnLR,y for Wenner configura. tion (L= 3a) and = x 473 )rLReY for Lee configuration: (L=6a). ik equals 30.48 to convert feet to centimeters. Gon. Configuration. W. Wenner Le Lee 8. South part of apread BH. Hoxrth part of spread AY) a Ray Fa anem. Con 15 5 114,8 109,900 W 30 10 52.4 100 ,400 W 45 15 33,0 94,700 W 60 20 6.9 103,100 W 15 25 23.4 112,000 W 90 30 1992 110,200 W 105 35 1509 106 , 500 W 2/11/39 120 40 14,4 110,200 W @/ 12/39 135 45 12.9 109 ,400 W Fe *) i940 100 ,400 W 165 55 . 90, 300 We 180 60 8.0 91,900 W 95 6 [2 85 *600 W 210 4 :) 91,200 W 225 bs 6.5 93,500 W 240 0 6.0 91,900 W 255 85 5,8 94,400 W e70 90 i 91,300 W 300 =: 100 4.5 86 , 200 W a/ 12/39 270 90 53 91,300 W af 19/39 45 28 oe "dap z 6 , 45 a3 19,200 L WB 300 §=6100 4.2 0,400 W R 2.4 91,900 L 8 3300-110 4.28 8,400 W 55 2.0 84,300 L 5 2 1.6 67,400 L W 360s 3.2 3,500 W ff) 1.8 2,700 L 8 390 = «130 3e 4,600 W 65 2» 2? 109, 5007 L 8 65 15 82,900 L WM 2¢ 19/39 (742) & 120 140 1é0 i 170 red 200 100 100 220 240 260 60 30 oF TABLE IL Part 2, Concluded Ray Fa accom 354 76,100 2.8 75,100 24 ie 600 2 251 [ps6 oes yt Oo? 1,87 6 OF 0,6? es, aut 1.3 49, 0,8? 647300? cer gee 0,87 +8007 2,077 92,0007? tee 59 ,800 28 89,600 "2 89,600 " 87 ,300 (746) Gon Peet £2h hehe Bee a ee it 3/4/39 Bt 5f39 Symbolsy : i a L E Dir Fa. L a 15 5 30 10 60 20 70 30 120 40 150 pi!) TABLE III pata obtained near Delano, California Part is. Surface depth profile Distance between current electrodes in feet. Distance between potential electrodes in feet. Current through current electrodes in aps, rt no aha onic between potential electrodes VOLES. | Direction from positive to negative electrodes. Average value of ratio E/I for each velue of Ly (in-), Apparent resistivity in ohm-centimeters = /7/5 @Rgye i 0. 188 0.135 9,20 0.18 0,148 O,11 0.24 0.188 0 04.135 0,23 0. 24 Q 0.128 0..128 0.2.19 0.197 0 04.12 0417 04.195 0 170 0.118 Ow1i2 0.423 0.412 EK 1.21 1.081 1,308 1. 206 04 925 0,095 0,003 0,050 0.034 0. 143 0.13 0,004 # not included in Ryy. Dir Nod S-H N83 5H N-S Si NaS doll Sil TS Sli NaS Sli SN Ray ¥ 6.45 tt 56 ae 1*,26 £3 © 0+35 ang Polarizacton ## natural potential (74) Sa (a- cm) 6,860 4,025 3,605 3302 180 240 270 300 360 480 60 70 80 90 jou 120 140 160 I 0. 142 0. 138 0.365 0» p18 0. 122 0. 123 0. 364 6.286 are eee t 0: 384 0.2? 0.29 0» 31, O45 0. 298 0,30: O51 0.498 0 0.22 0.222 0. 398 O» 547 Q 0, 23 0.22 0+ D7 0.492 “ 228 ae 0: 366 TABLE {II Part E 0,046 0.036 0,111 Os 104 0,004 ococeoe oocoo ss 2, 2 8 . cocoe ro Sassy (ea “6 PON! Bo ooooe +e 2 8 ow © O Ow 0.048 0.048 0,082 0.077 0.00 14 0.026 8088 0.068 0,069 0.0007 0,024 0.024 0,052 0.0.51 Po 4 Qococ 7-2 & & ooace CS Pino St Gi OD Continued Dir N-S S-ii i-S Sai N-8 i-5 i~5 Sli iS i-~3 8-H HS 5-H i-5 Nod Sli HS Sali i-S S-i Sail iS SH Hos Sali N-S 5 =i Se a i~5 Sali Sal Hed Si NS S=il R-S (744) Ray 0.429 0,20 0.18 0. 16 0.12 0. 10 0.09 ae (-2-em.) 31333 3064 3104 3064 2) 58 2683 27 58 TABLE IIT Part 1. Concluded iL. & Li E Dir Ray Faq (a-em.) JAG 180 0» 548 0,028 NS 0, 348 0.025 Si 0.563 0,057 NaS | 0,08 29.58 600 200 0.328 0.024 Hef 0.646 0,04 ie-S 0.618 0,041 Sali 0.04 2682 Above measurements made April, 20, 1939 Part 2 Surface traverse profile, April 21, 1939 Wenner configuration, L constant at 30 geet. Locus is distance of center of set up in-feet north or south of center point of measurements in Part 1. Loous i E Dir Ryy Pa (a-em.) 0 0,246 0,932 Wad 04245 0.825 Sell Oe462 14534 Gell 06435 145452 lied 06425 16422 Sa 0.UG34 Self 5.35 6420 30 8 0,282 0.989 Wad 0.266 1,014 Sal 06377 126516 3=Soll 0 0 53.51 6730 60 & 0,22 0.683 0S aed! 1.172 Hed 0. 10101 Sell 0 0.003 Sail 339 6500 90 8 0.189 0.639 NaS 6.192 0.652 Sail 0.402 1.365 Wad O47 1.409 Sell 0 0,002 Sell 3.40 6515 (749) 0 CUS eu we jou wv 24 2 30 NN as Us,ca] Ust3'] Us t S| Ue 240 Ue a4'/ U2416 ue4I1d Ue iPS 0. 19'/ Us. 25u Us 176 U.cVe Us 515 u U, 258 U,. 256 U.4U0 Ve ced 0.226 0.471 0.498 Us BU Ue4t4 i4 0.415 U, 228 0, 22h 0, 53¢ U Tabi sil Lart ce. is wir ve fo| Wed in%0c Nem i,#0 1 Foren ve i i al 1.366 Neo 1.305 sell VeUyUD Ned U,661 Sat V.t5i Ne le 54 pH 1.250 N=b ¥.003 dD =i 0.066 Neo 0,06 D =i 1.3/4 Neo 1. 3'/b nell V.003 AN mes U.01'/ N= Uv 10 nell 1. c¥u Neo je 2’ju Paes 0.65" N-S U.6 pel Ie 343 Neo in4c¥ pli i.uuy N= Ue¥¥O = Dall i. 366 N= 1.362 Sal Ue'/4y Del 1.074 Ly md 1,086 Pore 0.004 IN mid (74s) Gountinued Bay 53? Be 9% 4,34 4.232 Zeit] 2.87 D+ 5D Fa {4 -<m.) 6 duu o4 10 8330 Oo 26u 6USU 55 63 40 Locus 60 90 120 130 180 210 240 270 i 0, 282 0.457 0.452 0.202 0,202 0.3562 0. 358 Part 25 z 0.746 O« 753 te 301 1.416 0.002 Oe 869 TABLE Til Concluded Dir H~5 5—ii NeS Sli 5 Swill Had S alii NaS 5—N Nod S—Hi Hed So . Boll Yas 5 =i Rad SN Ned NaS 5= il NaS 5-H Swill HS wil li-S Sell h-S BS Sul aid S= Mf Maid HS Sli NS Sai (741) Roy 3213 3.40 5671 3.64 3a DA # exanginded Be OY 2.92 4.35 SS (a -cm,) 6010 6520 W115 6980 6786 6880 II 8330 RABLE IIT Part 3, Summary of data from verticel traverse profile Meagurements made April 19, 1939 with constent Wenner spacing of 2.9 feet and electrodes lowered down seismograph shot hole at center of spread of resistivity depth profile in Part i. % depth of center of electrodes below ground surfece in TRCG.« ys Ray, 2-m & Rav ¥ (-a.-cm.) 181 (bottom) 1,25. 1388 155 1,19 1322 105 1.38 1489 115 1.61 1782 125 1642. 1718 135 Me 16009 145 0. 2 988 155 Ie 1 1312 165 1.38 | 1553 WF 0.92 1022 180 (bottom) 0695 10 57 155 1,20 1333 150 i. 10 1222 145 0.79. 878 140 1.36 1511 135 1.44 1600 130 1.34 1489 125 1.62 1800 115 1. 59 1768 10 1626 1400 be 14 53 1700 3 hep 08 ] Oe? 65 1467 18 57 50 (water table) 2.8 3175 Observations made in the order listed. (74m) PABLE IV Data obtained near Victorville, California Part 1. Resistivity-depth profile at fest Point 1. Symbols: L Distance between current electrodes in feet. a Distance between potential electrodes in feet. I GQurrent through current electrodes in amperes. E Voltage measured between potential electrodes. Ray Average of the ratio Evi gor each value of I in ohms. g Apparent resistivity (equals 191.5 4 R,,) ohm-centi. meters. i & i E Dir Ray G, Ca-em.) 6 2 0,070 1.562 NaS 0,070 1.547 Sadi 0.053 16130 Nad 02051 1,132 Sad 22.3 8 550 18 6 0,066 0.443 Nai 0.055 0.4373 Sul 0. 1353 0.410 Pe 6.78 7800 30 10 0,084 0,272 NaS 0.075 0.242 3H 0.129 0.422 Sufi 0.141 0.463 Bag 3.255 6240 39 13 0.242 0,560 Nef 0.233 0.522 Sa 2.2975 5670 6.5 04255 0.258 HS u 0.254 0.229 SN 0.955 4750 6.5 0.262 0.344 Had S 0.257 05343 Gell 1,320 6570 45 15 0,282 0.575 Hd 0.248 0.4 | 0.120 QO.251 Nad 0.127 0,234 Sal 1.95 5600 51 1] 0.252 0.428 G8 0.328 0,574 Bad 1.725 5620 8.5 0.203 0.212 Sail a 0.325 0.265 WaS 0.782 5090 8.5 o.872 0.262 Swi S 0.282 0,269 Hes 0.958 6230 60 20 0,288 0,511 Nad 0.300 0. 528 Sai 0.416 0.211 Sui (74m) TABLE IV Part 1. Continued XL a i E Dir Boy ve Ca-~cm.) 15 25 0.266 0.470 Ned 0.261 0.4253 Sell O.132 0,220 Ned 6.130 0,213 SW 1664 78 50 90 30 04350 0.542 Nad 06.327 0.517 Sail 0,129 0.210 HS 0.135 0.208 Si 16575 9046 120 40 0» 290 0.402 Nad 0.282 0.385 SoH 0.135 0,183 Sal 04133 0,200 Und 0.130 0,193 Nad 04129 04172 Suii, 1.40 107 20 150 50 0.4312 0.405 Was 0.328 0,400 Sal 0.124 04162 Sul 0.115 0.153 Nd 0.112 04145 Ned 129 12340 Above measurements made on June 4, 1939 Part 2. Measurements at Test Point 3. June 5, 1939 6 2 0,0006 0.93 aa 1500 57] 5000 15 5 0.0028 1,45 a 0.0027 1.25 by 505 474000 60 20 0.006 0,8 a 0.009 1.3 a 1385 551000 Part 3. Measurements for Curve 3, ewe 7, 1737 6 2 0,042 1,302 ry 0.039 1.195 b 30.8 11800 15 5 Ov 1435 1.165 a 0,12 1,045 b 8.14 71790 30 10 860.174 0,874 a 0.162 0,814 b 5,03 y640 60 20 0.162 06.540 a 0.164 0,546 v 3.325 12730 # Bearing of spread and direction of current not recorded, & indieates current through the ground in one direction and b in the opposite direction, (740) TABLE iV Part 3, Continued L a I E bir Ray Fa. (-n.-m.) 120 40 0.192 0.265 a 7 0,187 0.260 db, «143585 10620 180 60 0.175 0.109 w& 0.112 0.106 b. 04947 10890 300 «100 «420,142 0.109 « | 0.136 0,105 »b O.770 14760 600 200 0.168 @,.090 a Qe 158 Oe 105 b ; Oe 149 0. 100 Bb, , 0.143 0.075 @ 0,601 23000 Part 4, Resistivity depth profile of Curve 4, June 6, 1939 6 2 0.0125 1.438 a. | 0.0135 1,555 » 11,51 4410 15 5 0.052 4323 a | 50 10 0,014 0.1730 a, 0.014 0,129 b 9,25 17720 45 15 0.066 0,356 2 | 0,067 5 Sé9 b 5,465 15720 60 20 0.117 0.431 a. O14 0.430 b, 3.680 14.100 15 25 0.042 O.113 @ 0.040 0.113 db. 2.955 13200 10 5 0.092 0, 168 & 1 a 7 0,088 0,148 db «14955 11780 150-04 1BR 0413, ; 0.136 9 Us 149 db 0.975 9340 225 75 0.14@ O.085 db | 6.149 0.065 a. 0.126 0,065 @. 0.56 7760 300 «1000s, 132 = GB a. | 0. 1352 0.07 b 0.625 11980 375 125 0.046 0.035 a. 0.049 0.020 »b | 0.049 0.036 a 0.027 0.010 b 0.57 136 50 (749) TABLE IV Fert 4, Continued 450 150 OsttS O.075 9 a. 0,111 rad b. 0,58 16680 600 200 0.129 0.098 a 0.423 0.052 b. 0,114 06072 a. Gaii2 6.044 b 0.51 19550 950 250 00133 04069 a O.127 0.055 db §= G46 5 22250 1050 350) «66,073 04019 v. 0.044 0,04 a. Osi 5 0604 a 0.134 04043 bd, 0.134 0,040 b. 0,415 27820 Part 5. Resistivity depth profile of Curve 5, Jume 7, 1939 6 2 0.011 1,483 a OoO1i 16410 Bb 131.5 50350 15 5 9.070 1,469 a . 0,062 te 3i@ B) 27.05. 20400 30 10 04122 0.882 a . 0.119 0.861 b 7,23 13840 60 20 0,12) 0.217 a O.116 Os 8ey d. 1.88 7210 120 40 0.102 0.071 a 0.1015 0091 d. 0.107 0.068 a. 0.1005 0,091 [:) 0,792 6060 180 60 0.10) 0,051 #8 | 0.094 0.05 Od. 0.092 0.053 a | | 0.088 0.049 b. 0+ 564 6480 240 80 046165 0.061 «@ 0.133 0.090 »b. 0.128 0,0 a. O. T2e 0,084 b Q, 507 7170 500 100 0,082 0.033 & . 0.078 0.045 » 0.076 0.030 a2 (74@) ZABLE IV Part 5. Concluded . * é . bir Ray §. (a-em.) & 0.138 0,055 b | 0.1368 0.064 a 0.136 6.052 »b 0.426 12220 600 200 0,105 0.032 a 0.101 O,U52 b . 0, 101 0.051 b 0.406 15550 (742) Vit Interpretation of Apparent Resistivity-Eleetrode Separation Curves Introduction It is the purpose of thie Chapter to discuss in some detail the various methods that have been proposed for determining the depths to horizontal discontinuities in resistivity, and the magnitudes of resistivity from field measurements of apparent resistivity at different - electrode separations using Wenner's configuration, of three equal intervals between the four electrodes, To avoid the confusion which would reevlt from plunging into detailed deseriptions at the outset, the first part of the chapter is devoted to a general outline of the methods, and of their rolative importance in practice, Most of them are of historical importance only. The balance of the chapter is «iven to detailed examples of the various methods ~ as applied to data obtained in the field by the writer, General Outline of Methods The following outline will he used as a guide Por the following discussion, and the detailed examples at the end of the chapter will be considered in the order in which they fit into this schone, I, mnpirical Methods, It, ‘Theoretical Methods, B. Graphical Ll. Tage's method and variations, a. Two layer case (1) Tagets orlginal method (Examples I and II-Delano Data) (75) (2) Tage'ts modified method (iixample IiI~Delano Data) b. Three and more layers (1) Pirsonts approximation method (Example IV-Victorville test point #1) | (2) Taegts modified method £2, Supervosition Methods as Two layer cases (1) Roman's method, (2) Paletea method. (Example V- Delano data) b. Three and more layers Fy Three-Llayer master curvos. 2) Paletea method (Ixemples VI, VII, VIIT, IXy-Victorville eurves 1, 3, 4, and 5) Yt. A number of onpirical rules for interpreting the rosistivity-separation curves were mentioned in Chapter IV. Of these the most valuable is the one which states that the depth to an interface causing a bend in the field curve is about 1/3 the distance between the current eleotrodes at which the effect of the chance in resistivity appears in the apparent resistivity. This rule is not wiversally applicable and is much less reliable than theoretical methods. it is, however, a useful cheel: on the order of magnitude of theoretically obtained valucs, and to choose the most probable value if the other methods sive several apparently equally likely values, It's use will be illustrated in connestion with the examples of the faletca method. ITA. Wor the sale of completeness it da well to point out that there are no analytical m-thois of determining the depths to discontinuities, and resistivites from the field data, by use of the resistivity formulas. Jiven in the simplcat ease of two layers the expression for apparent Pp alo ot i re resistivity J. =F it A (RETR ia Hank) )| (52) (76) is so complicates that there is no possibility of solving for hand & » Jor that reason mraphical methods must be UEGOE » IIs, There are two main types of method for graphical interpretation of ficld data. The first of these plots the field data in curves with the apparent resistivity as ordinate and the distance between the potential oleetrodes "a" a9 abscissa, It further requires mumerical enlcula- | tions from a cet of theoretical master eurves deseribed below and the drawing of other sets of craphs. The other method plots the field data on double lorarithmic senle, and determines the depths and resistivities directly from the master euvves without any calenuletion or further curve trawing, This latter method 1s ao superior to the other that for practical work it may be considered the only method. It te an interesting fact that although this superposition method (Faletca method) wes developed in Russia as early or earlier than the first method, it was not Introduced in the United States for a numbor of yoars after its inception, probably because of the languarre barrier, Meanwhile the Varc method had become entrenched in the Literature of this eountry. It is illustrated here because of ite historical Interest and for comparison to cdenonstrate the great superiority of the superposition method, The procedures, and diffieultios which are encountered are broveht out in the examples and will not be discussed further hove. | (77) Preparation of the heoretical Twoehayor Laster Curves The twosLayer methods of interpretation to be deserlbed below make use of sets of master curves derived from the eee \™ J gs | 1+ 42 BY Teste varesee (58) derived in Chapter IV. This equation gives the value of the ratio of apparent resistivity to surface resistivity in terms of the resistivities of the two layers, the depth "ht to their horigontal plane intorface, and the Vvonner . confi;uration clectrode separation "a", It is to be noted that the rasintivities in the risht hand member oecur only in the reflection factor k = Sp and that the depth and electrode separation occur only in the form of their ratio, The right menbor of equation (52) has been evaluated by various workers , but the only set of tabulated values to whieh the .riter has access are those civen by Luvin Roman, Vhese are included in his article Some Interpretae tions of | Herth Resistivity peta‘®! and are also appended to his paper How to Compute Tables for Jetermining Bleotri- gal Resistivity of Underlying Beds ani thelr Au iication to Ueophysical Lvoblems, 'P) Because many interpretations given below are based on these tables, they ave copied herve as able yls), In working, with the curves made up from these tables (Fipwre 21, Plates VITI and Ix) it was found necessary in some cases to intorpolate for valucs of "EK" Antormediate to those siven in Table V, This te} Asleleu. “Goophysical Prospecting" (1954 Ye» 18S U,3,Bureau of hiines, Teehnical Paper #502 (1931) (c) The notation used in this paper is subst’ tuted for that of Roman who uses "Q" for the reflection factor "k", ra" for the cleetrode soparation “a" and "e" for the interface depth “"h", (78) TaBLE body of table gives ratio ‘“/ys eorresponding to values of k listed at top and @/h at the siée. a/h & 0.1 0.2 0.3 0.4 0.5 Or 1.900} 1.0002 1,0002 1.0003 1,0004 0.2 1.0006 1.0012 1.0018 1.0024 1.003% 0.3 1,0019 1.0038 1.0058 i,0079 1.0101 0.4 1,0042 1.0085 1.0131 1.0177 1.0226 0.5 1.0076 tevi 1eVe30 1,035 inwale 0.6 i,0122 1.0249 1,0380 1.0516 1.0658 0.7 1.0178 1.0361 1.0552 1.0751 1.0959 0.8 1.0241 1.0490 1.0750 1.17022 1.1307 0.9 71,0310 1.0632 1.0968 1.1320 1.1691 1.0 1.0383 1.0782 1.1200 1,1639 1.2104 461 1.0456 1.0937 1.1441 1.1972 1.4535 1.2 1.0534 1.1095 1.1686 1.2312 1.2977 103 1.0610 1.1253 1.1932 1.2654 1.3424 1.4 1.0685 1.1408 1.2176 1.2995 1.3872 1.5 1.0758 161560 1.2415 1.3331 1.4316 16 1.0828 11,1708 142649 1.3660 1.4753 ie 1.0959 te $987 163094 1.4292 1.5597 1.9 1.1020 1.2718 1.3304 1,4593 1.6002 2.0 1.1078 1.2243 163505 1.4683 71,6395 2.1 1.1134 1.2362 1.3698 1.5162 1.6776 2.2 1.1186 1.2475 1.3882 1.5429 1.7143 203 1.1235 1.2502 1.4058 1.5686 1.749 2e4 1.1261 1.2683 1.4225 1, 5932 1.76) 2.5 1.1325 1.27860 1.4384 1,6168 1.8168 2.6 161367 1.2871 164536 1.6395 1,8486 ie | 1.1406 1.2957 1.46081 1.6612 1.8792 2.5 1.1442 1.3039 1.4620 1.6620 1.9087 29 1.1476 163417 1.4952 1.97019 1.9372 3.40 1.1508 1.3190 1.5077 1.7210 1.9646 3,1 1,1538 1.3259 1.5196 1.7392 1.9910 3.2 161567 163325 1.5309 147567 €.0164 3 121595 1463388 165417 is daez 0409 $74 1,1621 1.3448 1, 5521 ae 96 2.0646 3,5 1.1645 163504 1.5620 1.8051 2.0875 346 1.1668 1.3558 11.5715 71,6200 2.1096 30] 1.1690 1.3609 145806 71,8343 2.1309 3. 1.1711 1.3657 1.5892 1.8479 2.1514 3.9 1.1730 1.3705 1.5974 1.8610 2.1712 4.0 141748 1439747 1.6053 1.8737 2.1905 Qe faken from Roman, AIME Geophysical Prospecting 1934 p» 183, with change of notation. (78a) TABLE V continued afh k 0.6 0.7 0.8 0.9 1.0 1.0005 1.0006 71,0007 1.0008 1.0009 1.0038 1.0045 71,0053 1.0061 1.0070 1.0125 1.0146 11,0171 1.0198 1.0227 160276 1.0329 1.0385 1.0445 1,051e 1.0 504 1.0662 1.0705 1,081 120939 1,0807 1.0964 141131 t,131t 147513 161178 181409 141656 121924 1,2225 1.1607 151926 1.2268 1.2640 1.3062 1.20864 11,2502 1.2952 1.3445 1.4008 162596 1451235 143094 1.4322 1, 5045 ie i 163134 145999 1.4480 1s 3294 1.6157 é 143689 144458 1.5298 1,625 169389 te3 1.4253 165151 146138 1s 724s 1.8550 16827 > +. 8 © soot coeooc _2 * &- Be 2 Cw Mean OG in Bin fw = 1.4 164819 1.5851 1.6991 1.9810 ws 14 5385 166555 147851 169327 2. 1099 1.5942 1.7251 18713 2.0385 2.2410 12,6492 1s 1982 W572 2,1448 2.3740 1.9031 146624 2.0424 2.2510 2, 508 Hy Ft 1.9295 2.1268 2.3571 2.643 1.8072 149954 2.2103 2.4627 2.97799 168573 2.0599 222926 255678 2.9167 14,9059 2.1230 2.39737 2.6721 3.0540 1.9531 21847 2.4535 e+ 11 36 3,19 16 1.9989 2.2450 2.5321 2.8782 3.3294 2.0434 2.3039 2.6093 2.9600 3.4675 2.0865 243613 2.6852 3.0808 3.6057 2.1283 2.4173 2.7598 3. 1806 341439 2.1688 2.4719 268330 3.2795 5.8823 2.2081 2.5252 2.9050 3.3774 4.0206 2.2462 2.5772 2.9757 36497435 4.1593 5 enh 2.6280 3.0451 3.5703 4.2978 2.3189 2.6776 3.1133 3.6653 4.4364 2.3536 257 a1 5e7593 4.5750 263073 2.9730 53,2463 3.0525 4.1133 ‘ Y, de 4. 4 4 4 4 es es © © @ OS 2 a oe. 8 o & 28 © & @ *2 se 8 Ow cen32S tnkis the Ow O26 in Pin po 4200 2.0197 bee 2 264518 2.8640 3.5946 2e 5124 i 8 BAS 6 5, 2699 205415 2.9928 3.5590 03045 $4065 @» 5696 3,03536 3,61 4 4, 5922 je 54 5G 60 4.9907 263 5.1924 159 Fiwaetela tilstotw Sm mh POM PP Ne es we s es = & @ (780) a/h 4,1 4.¢ 4.4 4.) 4.6 ref. 4.9 ped o/h, > +; © = & =©Ocoo oocec OW OA Weipa - > eo 2 % 8 ooh ee ae tet ook *® © & & & PO es ot st es. ee; © + & ON Pea As Sis foo Q.1 1.1765 1. 1781 1.1997 1, 1813 1. 1827 1. 1840 t» 1853 1, 1865 1 1808 ~O. 1 0.9999 0.9994 0.9982 049759 0.9925 0. 9861 0.9828 0.9768 0.9702 0.9633 0.9562 0.9490 0.9419 0.93.50 0.9284 0.9220 049159 0.9101 0. 7044 0.899 0.6949 0.9904 0.8862 0.6823 0.8787 948755 0.8721 0.869 1 0.8664 0.8639 TABLE WV continued O.2 163189 is 38z9 1, 38649 16 3905 163938 1.3972 1,4004 1640354 1,4062 124091 #02 O29799 0.9989 0.9964 049919 0. 98 53 0.9767 0. 7662 0. 7543 0.9414 0.9279 0.9141 . COND ec o fo * a - O = foto * 2: *= @ @ aueed bceee E ws inka & GS WS BPHeotws BYMSE FVS 7> = * & & wee oS rs Ar a * es. &£= 2 & g k 0.3 1.6128 1466200 ~0.3 0.7948 0.9983 0.9946 G. 7880 0.97782 G. 96 54 0.9500 016936 0.8736 0.8534 0.8335 0.8142 067958 o es 3 =i Dm ce | oe cocoo eoce ce 8 * 2 Ow.) Gx © 04 1.8859 1.8977 1.9090 1.9199 149304 1.9405 1.9502 13eb4 1699775 rs “Ia =) oo Latin @ OS 3 fe & Hs OM pr, oa zs = &® & # =» = = *® *@ =I a Zosé= & WAR GAARA SCOCcMo coocoe scooco cooceo BS Sat SRS Mn b4 SOSH DQ WO an (y eeae1e # R Co s ©: &t « & BUSEY. POwse Sin 0.5 2.2042 2.2271 ey 244d 2 2613 2.2776 &. 2933 2, 3086 2.3254 203577 263517 “0.5 0.9973 0.7912 0.9804 0. 9646 * «+ + ® & > ees PS two SOW Ono PSN GH co.oks CO -* 2 2 * 8 LEP HE .<. op wm Op Beatin fos Pings © fe wOAntn sr ff 8 & tn & = WO OO a/h ses @#¢ 8 * PPR PSL ESE Swoon in is ho = a/h wsOOooo ococo es 68 8 8 2» @¢@ &© 8 # aed ck aed aed 2 es eee Ono Mea SF {yp Sin P= PS =o et ot _— 2 e*@ se = @ Ow Ano ihe 0.6 0.999 3°39 +4 0.9768 0.9 0493555 0.9041 077967 01603 0:87 0.6522 0.6192 0, 5881 0. 5592 0. S34 0.5077 0.48.50 0.4642 044452 0.4280 0.4104 0. 3982 0.38 54 02738 04 503 0.553 0.5452 TABLE V continued On] 3.0758 May 3; 1886 3, 2251 3» 260 32295 by 5302 3 2037 3.5965 0.641 0.6024 O« 56 52 Oe 5304 0.4980 0.4682 0.4408 0.4157 0.3929 0.3723 k 0.8 3.6767 5.6464 3.9010 3.954 4 0, 2009 0.1918 (784) Oe? 4,4991 4, 5051 we NO > *e es Saree PERS : ar ewes ees oan ooo ooooo ocooce =) HS ses ££ @ ws K ree GS COlN a hy: o ce = « - 8 ww oS 0.4222 0.3488 0: 2880 0, 2620 os el 0.8178 0. 1991 0. 1625 0.167 0, 1546 0» 1430 0. 7132; 0. 123 0.0693 0.0604 6 54 z Z #1,0 #O. 5 7 t-Ft ££ = 0.0597 ) OS 4 0,36 7 03607 0968 0.0345 8 08 0.0109 9Q 0,0095 o74 0.0082 660 06,0070 648 0.0061 0637 QO. 0.0 0.0 0.0 0.0 0,0 354 O43 46 de 4 4652 0.3671 1 0 1156 0.0526 1085 0.0457 102 0919 O,0299 0728 6.0126 g “0.4 “069 Ssocce ” * » ” * zs e2 ££ + SO000 1638 0 1766 0 1702 0 1645 0 1594 0 ~0.3 “0,8 * 2 ©£ e® = €§ & &< &€ & SF F&F F&F SF SF = & Fe LS Ue |B *2s # * & } 9 13 90,2022 - TABLE V soneluded “0,2 “007 042202 042167 2136 0.2905 0.2108 66 0.2955 (0. “0.6 8 062794 «0,1 0.2879 0,2083 0,1 0, 3083 , 04 2041 0.30 0.29 0.2855 ‘0, 2061 0.2 a/vh 0052 6 0.004 0.0039 0,00354 e14 610 2 8 6 a 3 0.2004 0.129% 0, 1988 0.12 a ‘QO. 12 J,1962 0.125 0,1950 0,1 QO, 6 ‘Os 0.2777 0.2761 0.294 0.2733 re: a 4.9 oe0 (78¢) a/h Pl et et et »-.?e 2s & Wo tere ho ‘ 2 © # & tn > Blin es 2. 282 &© 2 * a/h soe *» Soe GMn se 2 @ @ CONOw rst CHOMRan Ow And & Bit =» =» @® #2 8&8 & *@ = . tS tf > — > Se AN wk nk wet tt es @ eal = =) Cd La Wet : ee ed ee eed eed ed ee ass. 2*# # © # 2, 248 Ge 306 (782) =e WHI~2-2 THO SO Sin hl OWS AG & ro & 4e 653 TABLE VWI continued a/h k 1 15 “225 baal 55 m~r45 “. ) 0.8 0.966 0.943 0,922 0.901 0.881 0.9 0.956 O.92 es: 0,874 06.848 1.0 0.946 0.911 0.877 0.646 0,014 142 0.925 0.877 0.831 oT 0.744 13 0.914 0.8 Q 0.8086 0.45 —« Oe 10 16 0.885 0.613 asa} 0.680 0,616 1. 0,868 0,786 Go409 0.636 0,566 1.9 0.860 0.774 0.692 0.616 0.543 2.0 0.853 0.762 0.677 0.597 0.522 2.1 0.646 0.751 0.663 0.580 0.502 2.2 0.840 0.741 0.650 0.564 0,484 2.3 0.834 0.732 0.636 0.550 0.468 . 0.828 O.f24 O4627 0.537 0.453 2.6 0.818 0.708 0.607 0.514 0,42 i 0.814 06.702 0.5979 0.504 0,41 2. 0.810 0.695 re 0.495 0.40 269 0.806 0.690 0,584 0.486 +384 3.0 0.802 0.684 0.577 6.479 OO. 3.1 0.799 0.680 0.571 0.472 0.381 568 009796 0.675 0.565 0.465 0.394 3.3 0.793 OnG 7! 0,560 0.460 0,368 344 0.790 06607 0.555 0.454 0,362 345 0.788 0.663 0.551 0 449 O 357 Ze] 0.784 0,657 0.543 0.441 0,348 3 0.762 6,654 0,540 0,437 0,344 549 0.9780 0.651 0.537 0.434 0.341 4.0 0.778 0.649 0.554 0.4351 0,338 42 0.775 06644 0,529 0.6425 0.332 4,3 e712 0.642 0,526 0.423 0.330 4,4 0.772 O.641 O4524 0.420 0.3527 405 0.771 0.639 0,522 0.418 0.325 4.70 0.768 0,636 0.516 6.414 0,322 4, 0.767 0,634 0,519 0.413 0.3520 4.9 0.766 06633 0.515 0.411 0.318 5.0 0.765 0.632 0.514 0.410 0.317 (786) ash oe OG OO * @ 2 28 8 =e, 8 & TABLE VI vontinued k “0,75 0.65 0,842 oi 54 0.925 0,708 0.675 0.663 0.625 0.629 0.577 0,539 O.4 ; or2eh 0.449 Os4i2 03436 orga 0.409 QO. 34 0,363 0.51 0.359 0.294 b ose 04315 osa8e 0,286 9.216 0.272 0.202 Ou 259 G.189 0.243 04177 + 00856 0:10) 0,221 0.150 0.207 0.13 0.207) G2 131 0. 196 0.126 ot ge hi9 1G] a it? a4 O. 114 0,180 @.171 0.177 0.108 02175 04106 28 Soe, o: 0.166 rat 0, 165 609 O, 163 00958 0 0.162 0.0946 0.167 0.0956 0.160 0.0926 “095° 0080s 02839 O» 642 0. 336 0.446 0,321 51268 sad rie : 0. i¢ 3] O. 151 0.136 Ow 123 Os iti On 01 G49920 0,0841 Os “213 10 a a 56 0.0609 0.0568 0.0532 0 40,500 0,047. 0.044 0.042 0,0406 040392 010505 040554 38 G.0 53 0,03 ae 0.0322 6 18 09 0.558 0,510 558 196 0.770 O.772 0.744 0.5 4 0.52 6 0, 0.750 0.720 615 0.571 0.620 0.798 +650 0.604 0 +293 0.547 a» BVS 0. 583 0.2 16 0. Qe 2 0.628 0.585 0 Oo 02 0,552 0,02 640 0 830 0.768 0.708 0,652 0.597 0.545 0.4 =a 385 842 6.621 0.600 Oo. 780 28 0,664 90.642 9.600 18 0.6 08 0.6 e275 0 ofBaa 0.829 E36 *) 690 0. 4 O49 p Of O. 3 44 0 49 0. 42 Qs é 888 0.66 8 7 1 156 TABLE VI continurd 88° 0.8 0.877 0.6 e801 Oe “175 U.912 0. ‘0,900 QO. 0 6.822 0. 0,008 0. 5 0.8 1 O.7 ‘ 0.920 0.8 b) 0.711 g O93 0.9% 0.8 0.8 oy 0.85 ayh 6 28 0.539 O° abs 1.586 0.534 0.484 0,582 0.528 0.478 0.578 0.526 0.474 00 0,642 oO, 96 0,638 o 104 0,646 0 164 0. 3 | 0.755 0.692 0,634 25 0.761 0.74 0,8 0.8 0.8 0,8 0.8 ae 21 we yg e¢* # #6 @ PAY QP yy 436 0.440 0,438 0.43 548 0.492 z° 542 01488 ) +546 0.4 544 (0.4 »668 0,606 0. 2667 0.605 0 603 0, 4 0,602 06, 0.66 (784) 0 0 a/h so @ es oe 8 es 2 6 @ eoaes i Pf COMI BE Pie Po PSPs PS s + > s a e*eese ose 8 8 * eo © & LAPR PR SPREE Sow UA eo es & & COOH ture 475 0.776 0.746 0,718 0.690 0,664 0,640 0.618 ae 0.57 0. 560 O. 52g 04515 0. 505 0.492 bets i 0.472 0.463 0,456 0,448 0.442 0.436 0.450 0.425 0.420 0.416 O+412 04410 0.406 0.404 0,400 0.398 geeee 0.394 0.392 0. 590 0.388 0.38 0438 SCa0co coscooo coos s @2s8 ¢@¢ a. ¥ = » =~ J GG PS = = = war Aa wh eS tH NSP ses &© += & 3 ona : Ww ah ow Cc CAA TABLE VI continued “0915 ~.625 04734 0.71 01698 OLaTe (0,665 0.640 (0.633 0,606 0.603 0.574 0.575 0+ 544 0.526 0. 491 O46 504 0,468 0.483 0.446 04464 0.427 0.448 (0.410 0.432 0.394 0.418 (0.379 0.406 0,366 0.595 0,355 01396 92354 0, 366 0°325 £.33 (0.317 (785) SOODC SOfSOO CooSes 0.693 0.653 0.614 0.578 com > > es s+ ¢ & & Fner- Ve ©0 soEeSO coOSoGScO esos =z 2s &=& 8 # es & & Pp, BODO w= Antti S_ hs £ a — ro Ww = ses = * ‘Ov Role Min Oe zs +t = © Owls CG POs SK Sista es 2s, # 2 @ a = = & & EERIE miro cs * ts 0, 240 0,232 0,226 0,220 0.214 0.210 0.206 0.202 0.198 0.196 0. 193 0.190 0,186 0. 186 0. 182 0.184 0. 180 0.178 0,178 C2eooo oacoc sooo i © ZF =~ An s + Ra Hoes es - Ss = & POX\AAAAN Lh SSP Son mon ee PPM. PR Gin ps Pro 0.268 & 8 POPS RS fo PO fo PO fo PS at et oe tt ed ss. ¢t © 8 s «#8 #8 © es 2e?0 ® (2 = & fo ® 320 sc: ¢ 2 @ WPPPHP PNPPRPHA Ptshoe i OO M2 init fo = zs es = @& @ Oven i Bt WIA Ay Pwr — ese 2 @ OND OO _ eocooce a eee st 0.654 0. 538 Oe Sod 0.462 0.426 0.39 0.362 0.334 0.3510 0,288 0, 268 0» 249 0, 2353 0,218 0» 206 0, 174 0, 184 0.175 0, 167 « 160 0.15 0, 14 0, 142 0, 138 0, 134 0, 130 0, 128 0, 124 0, 122 0.118 0.116 0,114 ee ed ed DB ak mek os ed oO = fein Soooco ceocecoe so = 8 coecoe coocoe ec LaRES S sos: es#s 2 > «ss &= (78) VI concluded k e925 Ow 59 0, 54 0.497 0,452 04410 0,243 ' 0,249 0,224 0,202 O» 104 OU» 166 0.152 O» 139 0.127 0.11 0,100 red + 0.0686 0.0812 0.0764 =e 15 0.579 0. 526 0.47 0.42 0. 384 0. 344 0.307 0,274 0,244 0.217 0, 192 0.172 0.152 0.135 0. 120 0. 107 14 wes & s+. 8s 2 ¥ or 8 sv ee *# # @ # ss??? = 8 interpolation was linear and the valucs computed by the writer are listed in Table VI, A few characteristics of these eurves may be seen from an exemination of equation (62), Whe value of & anproaches Sf, for infinitely small values of the eleatrode separation "a" as the term a/h then becomes infinitely great and the oxprassion in the brackets approaches gzoro, Similarly, for infinitely large valuce of "a" where a is large conpared to nh for oven very larce vlucs of n, the term in the brackots approaches the value 4 ana 4 anproaches | + 2p (ge) - + r= if Seis plotted asainet af/h, curves for all values of k £-% y where Poe K+] will start at (1,0) and avproach & 2 + for very large valucs of a/hy The curvee approach the values Ss, sx 1 and Vw Sve norizontalzy(@) so that for very small finite valuos of a, Ss 4a still very nearly:.S , and for laree finite valuos of a, Fy. ie nearly §. fhe .alotea superrogition method makes use of master curves in which Fp as plotted aceinet, L/h on double logarithmic paper where L equals 3a, and is the distance between the current clectrades, The data for ‘these Feletca curves was conputed from the interral form of the expression for the apparent resistivity, Tables of data a-e hot senerally availeblie, but the curves are shovwm in Plate VII at the end of the toxt through the courtesy of Dr. Potapenko, Differences in the manner of plotting the data for the master curves are lliustrated in the oxam les (a) Hummel A.I.M.E. "Goophysical /rospacting” 1932 p.404_ (79) The use of '‘lave's modified mothod requires curves show- ing the value of tha ratio of two apparent reaiotivities gotten with a choven ratio of eleectroie separations in ‘terms of the ratio of the smaller separation to the intere face depth h for various values of k, That is, values of Sra ls _ +E Gree 7 ee a TE EA 14 OW Gee oiaxcactae. are computed for a chosen value of n in terms of a/h and for a uerios of different values of k, Table VII shows the values coumputed with the aid of Table V for the case ne 2 ‘The values of # are listed in Table VY opnosite the value of af/h corresponding to the electrode separation a. The value corresponding to an electrode separation Ba is then found in Table V epposite a value of a/h twice as large. The division of + ba by © a) was carried out on oa slide tule. ‘he curves plotted from this data are shown in ileure 24, below, It may be noted here that whether the ratio teen as abscisda is expressed as a/h or the recipro= cal h/a makes no difference in the evaluation of the curves, or in their use, other than In the matter of convenicnce of the numerical units in which the abscissa is oxpressed, The same is true of the use of the resistivity ratio % or the conductivity ratio + as, ordinate, Tange preferred the use of the conductivities where the lower layer has the ereater resis ativity as this limits to unity the rane of ordinate values, but the writer has used the resistivity form because that is the form used in Roman's tables, I+ is true that use of reciprocal values of coordinates will (80) a 4 #0. 5 651 661 # the top of the =140 «0,4 0, 99 0.98 0.98 0,944 v.92 8 8 1 ] 9 0.804 0.93: 0,720 0.644 k Oo. B34 0, 0.727 0. 0.735 0 2 0.744 OO. 9 0.755 0. VIL 1 0,816 0. -0.8 9G 0.996 0.996 0.996 0.996 a Lista values of SasK% 0.971 0,884 0,84 TABLE The body of the Gable 888 0 28 8 «2 62 0.970 O.¥42 0.913 0,88 2904 0,857 é Ue Zo 92} wo» I ® GO i) 0 Q 0 61 ce 38 34 NOAA 0.999 0.999 0.998 0.998 0.994 0.990 “Oe 1 On.7 pe 0.909 0.9 “026 eorresponding to the values of k in table and of a/sh on the aide. a/nh 0,1 O.@ 0.43 04 0.5 0.9675 +207 26 ro 019170. 06757 04733 0.843 0.8: 05 0.671 BG 2 B2 3 1 oe a ® 87 804 * * 80 O.977 0.974 0 4 0 5 0 7 0 43 0.93 0 ) oS. OA 0 § Q 4 74 0.6405 0.535 hate: 46 17 06457 00659 0.519 0, 4 0,590 0. 4 e171 o 145 266 0 O.3e9 0,840 2B2 0 0.363 0,284 0,302 0.203 O. 5 O.415 7 0.30% 0.470 0,374 OO (80a) 48 0.400 0.3 0.557 0.472 1 04586 0.513 0.439 291 Or 570 O« Ge 0. S eeoee Ore co. Fo Po fo Po fo TO wo ee ss © & 2 An tn Poe & *> 2 &® @# @ TRONTRONON OL oowyinti fo ONO SOW SO Arse ooceoo eocece *- ¢ = * ®& (80b) 0 0.418 0.436 66 0.456 0,298 0.476 0.498 TABLE VII coneluded k “0.7 +068 “0.9 0472 06372 0.257 0.478 0.3 0.252 0.488 0,3 0.251 0.500 0,390 0,254 0.514 403 0,260 530 0.270 548 0,282 0.315 «334 change the shane of the curves if a ¢c nventional numboring of coordinate units is maintained, but in the final result, it is merely a matter of whether one wishes to think of the derth as twice the electrode separation, or of the electrodes separation as ¢ the depth, 1. Zare's Methods; as Interpretation of Two-Layer Field Curves (1) Toge's original method, The following deseription of the procedure is quoted from Tags:'s vapor LartheResistivity 1, Determine the surface resistivity, 5 , by a series of careful measuvencnts at small electradée intervals, 2, betermine the apparent resistivity for a number of values of eleetroie interval. % Fiot a curve of apparent resistivity arcainst olectrode separation. 4, Prom this curve read off the values of apparent resistivity © for various values of electrode separation "a", and for cach value of "a" deter# mine the value of *4i?7 ® is less than & , or the value of “/eAift sie greater than f . 5. il’rom the master curves read off for each value of "a" a sories of corresponding valuce of h/a and k, and from these calculate a series of corres« ponding values of "h" and "kk", 6G. Plot curves, for each value of cleetrode interval, of "h® arainst "k", These curves should all intersect in a voint (iving the true values of Li ht and Wyte # Ixanple I, To illustrate this method we will use the data from the profile taken near Delano. (Table IIT Part I end of Chapter VI) 1. Let us assime, for purvoses of illustration, that the apparent resistivity value obtained for the five foot separation is the average of a series of careful measure- mentea with several short separations, ff =1éen.~«m. &e The values of apparent resistivity obtained for ®& number of values of electrode separation are shown in the following table, (a) The Mining Macazine, Vol, LITI #3 (Sept, 1935) p 150 (81) a et _ t | j T 1 i | ‘ —— re am cs ! | “i | | as 1 i ! a 5 pe. eee ee Oe a 7 aE a b } | {3 _Apbarent peolebh ur jity~ ae ae seporatven leurye ae (aed ar tae =o aoe fro Delano data. FP TT, Curve; ; ee “YF, ee ae t ! 1 1 a a re a ca ne OO an ae ae ee Ea pore rp Hol ee 4 v4 ts bl _| A doe fe kk : i oi : a 5 a ot: a en Fae eae es eed ee [ Tt — Teed Leb be |b 4 eee ee a | ee ee - pooh Jf ebe pe pe Pp Pe ba pe | - po ns feb sp a fp Ga aa Ee iG er ae t J i ifs i a (Feet) S, (a.-¢m.) a (Feet) 9, (a-em,) ag 4830 90 3100 30 4.020 LOO 3060 40 3600 120 2760 50 3350 140 2680 — 60 3330 160 2760 70 3220 180 2760 8O 3060 BOO e680 5. The curve of apparent resistivity against olectrode senaration is shown in il:-ure 20, 4, ‘the folilowlng values of % and corresponding values of "a" were read off the curve of Picure 20, As these values are less than f, , the values of *%% are comouted (using the value §» 6180 «-em, from 1) rather than the valuesof “« . As the curve deviates considers ably from some of the observed values, points were chosen whero the curve eoincides with the experimental data, The value for the 100 foot senaration talen from the curve is lower than the experimental value and ineluded to see how consistent the values of h and It obtained for this ,olnt of the curve are with the values obtained at points where the curve is closer to the observed values, a (Pt.) — Rala-m) Kis a (Pty) — &(a-n) Fas 20 4B, 5 781 80 30.6 495 40 56,0 582 . 160 2746 oA4& 50 33.5 +542 200 26.8 +433 5, The corresponding; values of a/h and k were then read from the master curves (Figure 21), These may be checked by laying a straight edse across the fieure perallel to the afh axis for the values of *%141%ted above, By dividing the values of a/h into the corresponding values of ‘a, &@ series of corresponding valuos of h and It are ob« tained. These values are listed in the following table. (a2) i ced Phe a Il io rs eis |e oe 4— st i. 4 i | oe ee sy | cia hae fy = 2 jihad Por tree] pi re mee | | | | a-—4 | | 1 i Tread from data” i ee ie L | ad L a8 Peer Fe turvas La T _ aste ; | ie Tw | { g | FF | | —t - , ~{= ’ 1 4 t | bias Tice - ot) p+ . vide a aay as ~ farm ey | | 4 | | | | | a3 fessahege of Keto | wF0.9 | Ko0.8 | rF0.7 | wto.g | THOS | HOY | WHOS | Ke.2 Ss, reer reer Freer reer| reer eeeT Freer fear Freer seer! "|| | 1 Val bh Lal h | Yal h Yl A lY| hlValh | Awl ar Yala ZO |.78!| |0,83|24.) |0.87)/23.0 |©.9)|22.0|0,97/ 20,6) /.04|)9.2 | 1.15 [77.y | 1-30 | 15-4 | 1.57|/2-7 |2.30 B70 3© | 650) |),05/28-6] 1.1) [27,0 L)8 [25.4] },27|23.6|440| 224 [1-58 | /9.0[1.97 [18.7 |2.65 [7.3 HO |,582]|).y¢ |34.5| /.25|32.@ | /.33 |30.) 1.Y5|27.6| 4. 63 |24.6| 1.87 |27.4 |2.38 lees 939| Vi FO |, $42 |1.25 | 4.0 1.33 |37.6|!-93 |35. 0] £.56|32.0| 1.76 (28.4 [2-09 123.912.78\/8.0 BO |.4495 11 1.34 |59.7 | 1-43 |56.01 5515/6 |1.70 97.7 1.94 |9/.2 |2.39 |34.2 [3.53 |22.7 10 0} 474) 11.38 [72.4 | £97 68.0] /.60 |62.5|1.77 |56. 5|2.05| 8 g|2. 55 |39.2/4.23 |23.6 160). 996|| ,g3)siz [434/109 | 1.68/75.3 |1.87|85.6| 2.19/73, 1| 2.83) 56-6 ZOO} 493] pyel/37 | 57/127 [470 |77 [292/704] 2,25] 88.92.96 [E76 Se Curves of h arcailnst k vere plotted for cach value of ‘a" in Figure 22, Those for separations of 20, 30, and 40 feet intersect at h # 14,5t, k « +,375, The curves for larger separations cannot be extended below h » 15', kt w «.4 Prom the data in Ficure 21, because for large values of ‘a, the sieller values of *h correspond to values of afh pr renter than five, By interpolating k it is possible to axtend some of the curves to values of a/h closer to the upver limit of five to which Roman! s tables extend, The dotted portions of the curves in Figure 22, were oxen tended by use of Piate VIII in which master curves are shown for values of k at ,025 intervals. ts use ig no different from that of FPicure 21 althourh a double logarith- mie scale is used. The fact that no extension of the hel curve for the 100! spacing obtained indicates only that h is léss than £0 feet for values of k equal to and creater than «,575, so that .s far as our data goes, it may be const stent with the curves for the other spacings . Let us saume that the correct value of h and k is h x 15 feet, "i # 375. 8 is then equal to 9, (ERI/I-H) $6180 x B5575 OT about 2O00 obm contimeters, Exemple II. Ll. As an illustration of the effect of an error in the determination of § on the value of h and k obtalned, let us assume that the apparent resistivity for the ten foot spacing (6860 ohm=centimeters) is correet, and that the surface resistivity is still larger, say 8000 obn- centimeters. We will discard the low apparent resistivity (65) SS! SE ; - Figure 22. Fxampbe TL: Tag's orrginal mie tho d=: hak: curiv e.. - a ae ij -_ VF, = od eee _ tf. }: | foe noun inna eee : 4 | i } nif mentee ee | | | | s.] tu in | ret t : Resmeye Sie © Senpe eeees os ee ULL: 1 wan whe of the five foot spacing which is now inconsistent with a twoelayer interpretation, &, The apparent resistivity for the various spacings was shown in the table at the tor of page 82. 3. The apparont vreslstivity~sloctrode separation is shown in Figure 20. It coincides with that of Mxample I exeopt for small electrode separations where 1t is shown by the broken line. 4, The same values of apparent resistivity and elec~ trode separation are usad, but the use of a different value rs of §, sives the following different values of *%, a (Ft.) Slam) la (Fe) Rlrm) Fre 20 4050 «605 80 3060 +382 aQ 4020 «508 LOO 2930 6 566 40 3600 0449 160 2760 0345 30 900 +418 BOC BG 20a5 § The following values of a/h and k were read from computed, 9 K=-f.0 he= 0.9 Kr>-0.8 W=~+o.9 Hs O0€ W=-0.8 K2~o.9 Kr OF Cy a Q Feer feerT Feer FEET Feer Freer FEET PGE ! Ala Wnt bh 19) AW th Vel A Yel A Ml A Wale FEET 20 | 603/74 12.6 WAz0 | (6-728 ISS ).33 14.0|| 1,55 | 12,9 || 1.76 | 2 |iz.20 19,05 3.531567 3O |, 502 ||7.33| 22.610 Yo [204 53 1/76 | 1.67 |18.0 1. 9) [15.2 || 2,33 [22.4 |19.37/8.70 FO |v49|\ 1.43 [28.07.53 |262 || 1.67/24 1] 1 86 [20-5 far? 78.9 [|276 [74.4 SO |.418 | 7.49/33, 6||4 $/ [37.0] 475 |28. 6} /.79 125.1 112.37 |20. 1 ||F20 | 15.6 BO |.382 |1 7.59 |57.0] 1.70/47.) |[688 |42,6/|2. 16 |37.0]]2.63|F° F114. 02 119.9 100 | 366 ||. 62|67.7|| 1.75 |57.2||" 74 [57.6 ||2.24|9%9 2-78 36-01] 470/273 760 | 345 ||) ¢7 195.8 £60 (96-9 liz, 03 (78.7 || 2.36 [67.613 00 | 53.3 200 | 335||Le9 |4& ||/.83|/27 |12.06|97. | ||2. 43 [82.6 ||3 /7 [643 (84) Tag a's ‘original method: AIK bury Se 8000 ohm -cm ft ~O194 6. The values of kh and k are plotted in Figure 23, Tha gurves for the forty and fifty foot cpacings were extended (broken lines) using Plate VIII as in Uxample I, The data from blate VIII is shown in the following table, Ew «0,5 kom ~0,475 k# w ~0,45 | k& eo «0,425 a/h oh efh oh | afh oh | a/b oh 40 ,.449 || 2.78 14,4/ 5.09 15,0/ 3,51 113.4/ 4.26 9.4 e (Ft.) 50 8.416 ||5.18 15.7| 3,65 1347 /4.50 L211 The eenter of the points of intersection is in thie case at about h = 10 fect, k = +44, The assumption of a high wv surface rosistivity led in this case to a value of interface depth about a third less than that obtained in Exaaplo Is . (2) knowing the surface resistivity in finding h and k, Tage tage's modified uethod. To avoid the necessity of BY ected the following mothoa '#) which Ils illustrated step by step with the same data used in itxamples I and II, Exauijple IIL, . 1. Determine the apparent resistivities for a number or different electrode intervals, ‘These valucs are tabulated on page ? . &. Jllot an apparent resistivity-electrode separation curve, This has been done in Figure 20, o iead from the curve the apparent resistivities corresponding to a number of different electrode intervals, Also read the apparent resistivities corresponding to a second set of eleetroda intervals bearing a set ratio to tio first. for example, let us take for the first set of intervals 20, $0, 40, and 50 feet, Our second sot is then 46, GO, SO, and 100 feet If we select the ratio 2, (We heve chosen the ratio two, because the master curves in fi, ure 24 were derived for such a ratio. There is no reason for not using pairs of resistivities for a series of ratios of olectrode spacings such as 20 and 40 feet, n m= 2; 20 and 30 fect, nw 1.5 ete, As will be shown below the pre- paration of a set of master curves for cach ratio is not necessary. The only Limit toe the choice of ratios is thet they be large enough so that the difference between the intervals represents a significant portion of the apparent resistivity curve and that they are not so larre that the larger interval ia more than five times the depth. (a) The Wining Marazine, Vol. LIil, #3 (September 1935) p.148 (85) al AeEL Lp te at Vou oa eee 4 a pe =! reed Td | mee Opa eae The second restriction is imposed because of the limited range of Roman's tables, wor purposes of illustrating the method as first civen by Tags, we will assume master curves of tho type in Figure 24 are necescary, and to avold the labor of computing several sets of such curves, use only the ratio 8.) We will designate the first set of resis- tivities & , tho sseond seth. . The ratio of the resis~ tivities for sach value of a is shown in the following table» adie) LL (a-em) Ba (rr) —— F.al-a-cm.) Frag 20 4880 40 3600 0746 50 4,080 60 3250 808 40 3600 80 3060 «849 50 3350 100 2950 875 4+, Fron Ligure &4 we read the following corresponding values of a/h and k in the same manner as from Firure 21 in lixample I, Frow these values of a/h and the correspond- ing values of ‘a, the corresponding values of h are computed Pea fs-ho K=-6.9 2-9-8 T= ~ 0.7 | We-O6 &= -0.5 W=~0.y W=49.3 ” K=-0.2 a 7 Tad ¥ nae eetr| Se || Val [nl “h [te | | al h hl ap |b | Mal | Yael h | Yel A Bi e.4 O58 at pe 4.2 38.8 36.5 33.8 or lo787|_le.seo| Zo Bt 10.5 -74 bllo.4ge) os Jo. SYR o.S92|%" |o.6s6 2 AY) U4 67. 7 64.4 60.6 55.3 47.6 92.2 ah pes 36 @8°Bllo.y20 0.442, 0.466 lo. 506 o.S4¥3 0.6.06 2716 1,83 107 rer 96.7 Well 1643 wae 67-2 lerae|o”” 46-8 o 4 899 O374 2.396 0.414 10. 44O. 0.479 0,$20 ©.596 2.38 197 7190 132 124 Wwe 107 9.2 29.2 TSE 10, 50 . | ZZ - 875 |lo.syo 0.358 2.378 Yo 0, 24 0.468 0, 52¢ 6.63 1.8Y ; So The curves of h arainst i for each value of ‘a’ are . shown in Figure 25 as solid lines, The data are insuffici- ent to show any of tho points of Intersection. The broken lines in Figure 25 extend the curves so as to intersect. The values of h and k for the extended portions of the eurves were obtained from Plate VIII by the following method, <A piece of tracing paper was placed over l’late Vil\ and three dots marked on it. One was over the origin of the graph, a second along the log(a/h) axis at a dise tance from the origin equal to the ratio of electrode inter- vals (in this case 2), and the third with the same abscissa, and an ordinate eaual to the ratio ™“/,, The tracing paper was then slid over the master curves until the first and third of these dots both fell on the same thong For the value of k corresponding to this cur a, va fe) could then be read from the abscissa. Opposite the Pest ae. (86) pH hp - — ~$e-- poy 1 { As the curves are plotted on a leg log scale, what we have done is to find a value of a/h on a curve with a civen valuc of k, for which the antilog of Lag (**/9, ) = log (#/4,) (or %a/%) has the desired value, vas done in etep 4, above, except that the use of a Lo; arithmic a table and set of craphs such as these in Pirure 24, Care must be taken to keep the axes of the master curves amd of the trneinge 1 oy ar allel by keeping the first This is in effect what -cele obviated the intermediate step of making and second of the three dots parallel to the abscissa of the master evivea, values of a ‘his ;rephical method has another adventaze in the use of tables which are only extended to As the curves approach hor» a/h as creat as 5, izontal assymntotes for large values of a/h, and are smooth, it is por gible to approximote fairly closely by eve the extension of the curves to larcor values of a/h, and this rst coma os cet wt & critical smallor values of h, cucvos for varlous values of a and n obtained rraphisally, these curves converge at a value of h of about 15 feet, imilenting that the lower resistivity used in iixanple I was the better. The cata For these curves is :‘iven in the ible the extension of the hel curves to the Fieure 26 shows several f-Llowing tables 7 k= wo Ww | a " Ea kK | OF K ba alin $a, ||O-$_|-2-9F|-0 4280.4 [- 0.34 5|-0.95| -cats|—0. ¥ |=0.275 |-0.25|-0,225|- O27 3 — Pr, [AT Frle, |er hla, |\F Fla, |\Fr ar Er er FY, AY eur! | Se TAL AI%Ih [VIA LMA Ile |ZIA |Uln (Mla wR IBA lA el a | 8.0 7 4.5 10.3) Wy —) = 2. 235) at b “7s. 7 = : zo!/|2|- el os ee 2.5 we 6.5 | [746 Io. 8 aso) 297 \.06 | | oO ~ fle, 10.7 ee Ue say 73.3 v4.3 15.6 18.5 ! 2:75 oT ___|24 2.2.5) Ret 1.92) Ly. Jo | 2. iso.0] 45.41 74) faa [395 aa] a) 30.0 14.9 806 lloce| la ace 70| (276! fea tel 2a Was | 7 en i ew) 38 4.3 3 S71 | CS 18,2] Ze, 23.6 FP. 25° RAT z 2. h70 255 2.¥ 2 195 { | 9o\2 77. 67-6 e783 678 as Wa eal SB 0.0 125 car) 897 »- 52. 9,58 lo. lo, 59) ocy 2 e-70 O72 lo. Bo | a.e3 1.0 | (4.3 74.9 15.6 76.7 76.4) rr. 17 18.9 ce 2a. 8 23. 27, 3.5. ut 3.05 on EW) 45 |2 ae, 2B 50)|2 404 PY i ls az 710 TY 64) seo] 8.11 875 | 0,53 0.60 6) 9 P| ‘ 2. 2.89 oy, 73 89 8.7 TET a4 as 2.78 225 Lrkas 2s 260 1/.5 22, 2. Cy TH) 6.2 sy ‘834 logs | —|n00| _|y, aon |e a7| i278 [ea [esl lag 230 2.0: (Ph £3 L7F VA 26 /2.5 o:4 27, Era 23.8 20.8 699 lose 72| __ |o.7 CH 78 10.8 "8 er} BS 15.8 2.05 VA 1551 lt, 22. 20|¥ 5.5 You) pes] [3n0 28.6 639 |p.¥y Crd 0.s2 OS logs] [70] ' 27 10,5, 10,9 1 & ry 3 17.3 3 3.4/0 2 2S 2.5. 24 Qe LTS 0 133 e495 33. B12, URnq 2B6 j2s.0 22,2 3.8 : 0.97a! 0.76 1,02. fos Az 43S nto ae 70.7 AC a Be as gl (or 20:7 278 2.6 2.44 12,30] 2.10! th 19. 30 n Fo. 37.5 5.2 a 30.3 an 25-8 "671.933 llozs| fo, lo 2 0.961 1/03 16 (87) _ — i a a eee rr t | | {ear are: F = hig re 9's es eee ier a ee eee ee t aa thod a 30" n ea lb. Interpretation of fiold curves involving more than two layors (lL) Pirson's successive approximation method The extension of two Layer mathods to the case of three layers is based on the assumption which exporience has shown to be correct that the two upper layers may be troated as a sinsle layer of their combined thickness, and of an effective resistivity intermediate to their Individual rosictivities, ‘The third layer is then the lower layer of @ second two layer problem, ‘To use Tage's original method 4t is neeessary to mow the value of this effective resis tivity of the combined upper layers, As this value de» pends on the depth to the third layer which is not known | from the solution of the curve for the upper two layers, 8 value is cnessed by an empirics] approximation and a4 tentative value of the otfective resistivity is computed, Usine this value as an effeetive first layer resistivity the depth to the third layer is ealeulated by Tagg's origi- nal method and if this value differs much from the assumed value, it is used to caleulate an Improved value of the effective resistivity and the method is repeated, Example IV To illustrate this method, the data (Table IV) obtained at Tost Foint #1 near Victorville are used bocause the depths to the interfaces are shallow and the depth to the second interface is known from observation of an adjacent road cut to be about 16 feet, The apparent rosistivity~ eleetrode separation cwrve is shown in Ficure 27, We will estimate the value of § to be 12,000 ohm-centimeters. (This value is that ¢iven by the superposition method lixample VI below, The lower value of J, of about 8600 ohm=centimoters sugeested by the values of apparent resis« tivity for small spacim-s does not give a good intersection of h, k curves and may be considerably in error bocause of the effect on the resistivity of the ground for a small distance around the electrodes which were watered to cut down contact rosistaree, (88) +f trehore terete pot te tres! (tt. fan att +I i fi rte hes fe t attr 15 + toa aa } Bal TEE are tt SESE HEH eee +} if rE eh prt I ass | ae gt LS x WB EEE TT t +4 t it core Son tele ie : = pt a Ke Sy ea 3-017 sla 6 =-0.7 | "=-6,¢6 | =-0.5 t=-0,ye5|ie-0,Y |ta— 0.375] Soo 35 ns-ams| fhe- 0.3 ; ‘lay JFT T, a ‘yer FT. Fri. lF Pr fF F Fr g; a Gq, g re |. x nl T : eegr|o-cm| 7 |\Ialh |\Ml hl Alb | lh Vl hb Mlb [Mala lA (Mal b [MA [Vals 7 57 ExT [5.08 4.69 429 78 3.33 7g) 12.96 rs ey) 6 Exe 63). hos Aut [4/8 £28 4¥o £59. 46d __\|297 2 2.20 zs9|___|ae «78 eyo) PF 598 4,9) yu Er 3,22] |3.03/ lan 2.35 482 8 7000| 58 lhe $25 134 246) Wes £90 224 a 2.64 PRL fj ¥, 7,87 730 6.76 6.4 SF 4.07| p.cq [a26 [2.79 ag ie (0 \e2yo|$2/||.29| |va7|_ [ave | _|nes| _egs| [22a zs] Bol __|s, 58 a 9.43 8.80 AS 732z| |€35 ell PL avo] | y | 3 |se7o |.yz3\lese| |eva [eée| |nra| os] bss) (3s an _| 3 _ 10.7 2.0 9.26 8.33 7.18) 16.78) 410 3.30} i fo ‘3 | 600] yer|lius ec ae 80 La 2.61] [4 The above table ives the data obtained from Plate VIII for the h, k curves in Vicure 88, The intersection of the curves gives values for h,and k,of 3 feat and -~0,.4, The next problem is to find the depth h, to the second intertace. To do this, the two upper layers are considered replaced by an equivalent single layer of thickness he and resistivity f', The first step is to find this effective yepistivity so that the ratios &/p may be computed and values for the h ,k ecyryos obtained from the two-layer master curves, Liummel'®’ has shown that ror eloctrode separations large with respect to the second interface depth h.,the effective resistivity of the two upper layors is given by Kirchof?'s law for resistances in parallel. ‘The value is therefore calculated from the formula i i lie or SRG ken v a of the quantities necessary for determining 9’, hoy, and ®% are mown trom the interpretation of the TMxrst acvroxie nation curve. It is atill necassary to get a value of 4. The LoneastoreJones approximation of two-thirds the depth bo the infleetion point in the second part of the field curve is. used to cet a preliminary value of }., In this case we have the values h, » St, f= 12000 2-m , f. @ 12000 x 6/164 =~ 5000 .-cm,., and hi » 8/3 x S5' ~*~ 23t, ' 3 We caleulate the value 7 i t= ~ 5400 +-~and use this 12.0! 660 ag an eaquivalent surface resistivity. The following table rives the data for the h,k curves of liswre 29, (a) AWE Geophysical Prospecting 1932 p 418 (89) a . * Figure 28. i = Bink TZ, Firson's Suceessive approximation method: | ; i 1 tt so _— pe es or : a } 4 RF cirves fori iwpper tive ayers at Test Pernt ao./ Victorville, - e, 2 a 5,= 42000 n- om.) a 6 a, ¥F,2 a aa me = 5000. foo. : beh “OHS og Ste melee 1 Be t : | ! ; | J I 1 a i : ae an | ea oo eo fe t 4 ° Fos ps ee ae “TT f | | | I ! W 7 r way rv — " YT f 8 Lo 6.9 °.8 0.7 O¢ o.'5 2.4 2.3 o.2 a |va y4 ¥ eT, PR Fr. Fr. Fr. Fr. FT. Fr: Fr 4 . lq aP ; : : , peer |acm.| S114 |b |MIA lh (AIA IY la |Yele [Mla |VAlA ~ 27.4 26.6 23, $ 22,0 20,6 18.2 16.0 42.9 94 20 | 6750 | 1.25 lloz3 028 6.85 o.9 46a 116 A2S 155 220 26.3 24.5 22.7 20.7 1B.5 6.) B08 WE 25 7900 | 1-46 lla. Lor hd Ady 435 h§S 490) 12.6) 266 24,2 a2. Z 20.2 (2.7 14.9 as 64 30 [87001 1.65 Ilia | |aay Aas AY9 1.20 Zoz| |2.6 #6 28.6 25.8 Z3.0) Fo. v7.8 13.) 8.0 40 10700) 1.78 ll 4 ISS A249 £98 2.35 13. O€ 5.0 5 [36.6 26.6 23,6 20,2 | 15.8 2.0 a 12 450\2.3) || 7 29 4a8 212 a8 EA 4.55 From Fisure 89 the value of h,z may be taken as 20 feet, the value et the interseetion of the eurves for the thirty, forty, and fifty foot spacings, It is to be noted that the values for the opacinss nearer the minimum in avvarent resistivity eurve-the 20 and ®5 foot snacings-are not consistent with the others, This is an illustration of the fact that the two layer aprroximation sourves deviate moat from the theoretical three layer curves at electrode separations in the remwion where they intersect, We may now apply Lummel's: formula again to cet a new and better value of F usin the better value of h.. pis . a 342200 + 17/5000 We nay now reapply Vage'a original method to the ‘econd pproximation eurvve using the improved value of the ef~ foctive resistivity of the upper layers, The data for the curves in Firuwre 30 are ~iven in the following table. 54aoo0 a-em (90) ia Se 4 Vary seen a g 5 ho oF °.8 0.7 o¢ | os oY 0.3 O12 | BF Wray |r | gy [Fr Fr. Fr, Fr er. Fr. Ale “% [FEE Flac hl hiVUlhiwyia (% hIAjhA [MAL AIVl A Mia 1% 1h le ~ | | feat faze | a oo ee oe er Ans 9.31 ao? | n23llen,| lo» 982] _loeel Jogel |rog 419 48} jz04| | 26.6 24, 8 22.4 he 3.9 \€.3 34 0.6 25 [Pool 1.94 logy be __|4o9| ine A32| ass £82 25a | ; 27.3 So 12.2.9 20,2 48,2. Sy 42.) Rd [3° |ego0] 162 {yo fae “3s 45 Les Les x mi ; 29.2, 25.8 23.7 AO? cv A | 43.2 &.?7 | Yo fopoe | 1.95 43> 485 Ae 49 ZIe 2.72] |%€ = | bo 30.8 27.6 24y 20.8 16.7 we) | a 124950| 2.L5 || 4 Ls 2.05 vo 3.0 #.3 | From #icure 30 the Intersection of, the h,k curves for the 25, 50, 40, and 50 foot spacines «ives the value of about ai feet for hz,» This value is so near that used in calcue latinn tho Improved value of effective resistivity that there is no reason to repeat the method in this cane, If the improved value of h had been amaller, perhans of the order of fifteen feet, a atill closer apvroximation nicht have been expected by repeating the method. If a fourth layer had been present and represented by a third bend in the field curve to which a third approxi} mation curve might be fitted, the method conld be applied using an effective resistivity calevlated from the formula fy Loh, -hy ~h i ia ice a whore h. is the depth to the top of the fourth layor and estimated as before by the Lancaster«Jones approxinetion, In the case of n Layers the formmuia hones = h, h -~ h = hime es = — ga TS, i * — is used to determine the effective resistivity of the top nwl layers, It is apparent that the method ig more and more laborious as more and more layers are involved, and that errors in determining tho resistivitios and thiolce nesses of the usper layers will produce errors in the values computed for the lower Layers, (91) (2) Page's modified method This method, described under interpretation of two Layer field curves, may be apylied une changed to any seement of an nelayer field curve, and elim inates the necessity of predetermining the effective resis~ tivity of the upper layers, It has the inherent disadvantage of requiring each bend of the field eurve approximated by a two layer curve to be sufficiently long that values of "a" ean be found on it which are sufficiently far from maxima and minima of apparent resistivity and which are yet far enough apart to give values of f/f appreciably different from unity. 2, Superposition methods a. Interpretation of two layer. field curves The superposition methods differ only as to the mechani- cal details used in applying the common principle on which they are all based. This is the fact that if the lorarithn of apparent resistivity is plotted against the locarithm of the distance between the electrodes the shape and size of the eurve is independent of the units in which the field data are expressed, and of the absolute size of the values since the distance and direction between any two points on the curve is determined only by the ratios of the two values of ordin- ate and of abscissa. Further these curves will be identical with those two layer master curves (when plotted on the same double logarithmic scale) which are derived for the same ratio of resistivities (or k value) regardless of the ab- solute value of the resistivities, When a field curve ia plotted on transparent paper and slid over a set of master curves until it coineides with a particular eurve, the value of k which represents the ratio of resistivities in the ground is given direetly by the master curve with which the fit was made, In making the fit, tre axes of the field and master (92) eurves must be kept parallel so that in shifting the field eurve the only change made is in the relative positions of the two ordinutes, From the intercepts of the axes of the master curves on the axes of the field curve, when a mateh has been made,are obtained the depth to the interface between the two layers, and the size of the rosistivities in the eround. The zero abscissa of the master set is the line along which the log of the ratio of apparent resistivity to the surface resistivity is zero, or the ratio itself is unity. This will intersect the zero ordinate of the field curve at a value of anvarent resistivity which is ecual to the resis- tivity of the surface layer. Similarly the zero ordinate of the master set will intersect the zero abseissa of the field curve at a value of electrode separation equal to the depth of the intorface,. (1) Roman’ method Irwin Roman was the first in this country Bp pu the suggestion of using a superposition method, (@ He Jotermined the Logarithms of the data for the fiold and master curves from a set of log tables and plotted them on ordinary craph paper. Tt was then necessary to com- pute antilosarithms to find the surface resistivity and interface depth. (2) Paletea method This method was developed in Russie several years before Roman deseribed his method, and although it has been Uppoussed in Russian has not yet found its way into lnglish. It differs in two respects from Roman's method, The most important chance is that the curves are plotted on logarithmic paper in which the coordinates are expressed in the field units and ratios involved, but are spaced according to the losarithms of these values. This eliminates the use of log tables. The other chance is in using for electrode separation the distance between the (a) Some Interpretations of Earth lesistivity wate “Aslelin Es Geophysical Frospecting, 1954, p. 183. (b) The discussion of the Faletea method «iven here is based on lectures given by Dr. Potapenko in a course on elete trical methods of prospecting. (93) ee ae ser (Broven Takes show. ar LATEaE BCI) POLIS R An —Tpestrror < a fret d cureeot best a of , e serene Por eel ep ee ee mrt ‘ ' i i 4 current electrodes rather than that between the potential electrodes. This has been found convenient where field con- ditions necessitate the shifting of potential electrodes from the oxact locations required by the Wenner configuration, Rxample V. Figure 31 illustrates the «reat simplicity of the superposition method, The data used in Examples I, IT, and III is plotted on logarithmic graph paper and the fit with the master curves of Plate VITI is shown, The values h,w 14 feet and 5 # 6300 ohm-centimoters are in close agrea- ment with those obtained in Examples I and TIT and hours of computing were saved, Plate VIII is a hybrid botween Roman and Paletea becauce it uses a logarithmic graph paper but uses the potential electrode separation instead of the cure rent electrode separation, It is also unique in that to’ fit the master curves on the sheet of Lapety hh. payee it was necessary to plot decreasing values of log (%/7) in the dir» ection of increasing ordinate. It shows however the great advantage of the superposition principle. 2b. Interpretation of field curves of more than two layers. In general field curves will not be simple two-layer curves but will be considerably more complioated. The super position method can be used for these field curves age well as for the two layer case, Theoretical master curves have been prepared for the three layer case and are discussed below, However these are clumsy to use and impractical. ‘No date for theoretical curves for cases of more than three layers is available, and interpretation of more complicated eurves are ¢arried out by repeated use of the two-layer master curves, As this extended use of the superposition mebhod is by far the best method to use in routine inter« pretations of depth profile field data it ie deseribed in some detall in connection with the three layer field curves obtained at Victorville. ‘(1) Use of three~layor theoretical curves, Theoretically it is more accurate to compare three-layer field curves with master curves computed especially for that case. If the field curves are approximated by two two-layer curves, thesa will (94) a TABLE VIII The table lists the value of fe or % since f +1. Part 1, h,-6, h,=8 bs c ¥ : A §a/h,. 2 4 8 10 12401841 0.979 0,869 0.693 0,518 Bs 66 21/3 01949 0.869 0,69 0.52 0376 1 04979 9,870 0,706 04 53 on5 8 +10 0,979 0,871 0,709 0. 544 0 1396 13913401 0-970 0,848 0,674 0,510 0,380 2 143 0,983 0.898 0.761 0.634 0,529 31 0,986 0.978 0.808 0.691 0,621 13 0098] 0.927 0.8351 04120 0.719 3 10 0,98 0-93? 0,845 OF ; 0.745 18 1/33,01 0.970 0,865 0,713 0.550 0.416 ait 0° 980 0° 89a 01163 0.623 0,495 : 1 0.989 0.949 0.898 0.855 0.832 t 3 0.997 0.994 0,998 1,030 1,080 310 0.999 1,000 1.020 1,092 1,223 3 100 1,000 1.023 1.060 1,148 1.284 1:33.01 0.990 0.980 0.970 0,900 0.815 to] 1,000 0.995 0,980 0,945 0-873 31 1.010 1,045 1,100 1,150 1,180 +10 1,025 1.155 14345 1.590 1,850 2100 1,04 1,170 1.42 1,73 2,09 13 102,01 1.01 1,05 1,14 1,250 1,280 731 1.01 1,06 1,16 1,265 1,390 1 1,02 411 1,24 1,575 1,480 23 1,04 1.150 1,30 1,495 a i : 100 1,04 1,19 1.49 1,885 2,260 2100%,1 1,02 113 14540 1,700 2,010 11/3 1,02 1.13 1441 1,705 2,02 1 1.02 1,14 1,43 1.918 2.05 33 1,02 14 1.45 12730 2,0 : 10 1,02 1415 1.47 1.78 2,14 a, Wetzel and MeMurry, Geophysics, Yol. 2, #4, 1937 Table Il, P+984, wanie III, p.335, fable IV, De 336 (948) PABLE VIII Part Te Goneluded h,=6,% h,-8 83. BE Sash, 15 20 25 30 40 13,012.17 0.736 0.0785 0,0 0763 0.0810 0.0861 g1/3 0. 160 0,1075 0.1150 O12 § 0, 1490 1 0.190 0,144 0.15 0.17 0, ae 3 0, 194 0.14 hah Dy 20! 0.266 210 0.205 0,166 0,1 0.21 0, 285 1g01t.01 0.148 0,062 0. 032 0.9196 00,0116 te Ae id: 04336 0.345 0.341 0,340 0,605 0,625 0, AS 0.665 Oy 700 3 ee He 0.665 1,00 1613 143.50 310 Hd 1.025 142 1.425 te920 ¢ 100 4 .8 Oo Wilt 1.3 1,620 ee 150 1s 1/3g—01 0,182 bet 0,035 0.0196 0,01 tel 0.278 On 16 0, 144 O04 129 QO, 112 31 0.650 0,850 0.673 0.895 0.930 $3 1, 26 1,44 is 1474 109 $10 Te 61 a4 00 2y5 ow 2 3.28 1233.01 ee 0.383 04182 0,106 . taf eu 064: 0, 294 0,214 0, 128 1/3 066 0 0, 540 0,460 0.3582 1 le 40 te t75 1413 1410 1,08 310 2,49 5.05 5455 4,02 4,82 3100 5.01 ae 99 Aa? 598 W495 13103401 1,21 1,01 Os‘ 5 0, 558 Uy 230 21/3 1. 1,21 1,04 0.895 0,653 3 e095" 2,32 ay 5 2467 2,86 : 100 e355 = 4 AB 5452 6.60 8.95 1874053) 5176402 (946) M4BLE VIII h, sie Fart 2. 10 WOO MANw me Fong A ++ OnNo+toO FARRAR * #8 = # &€ @ oocecse AO Ko A om POO O<t~ e@ @2¢@ 58 88 oocooo LUN © UNO oN ee ata tata’ DAAANN oy 2 aN oo gre Ee 20 40 60 KD D8 oe -_ ° =e ~ espete_ss8 @ 2 S EO OMEN ooceoe eve @ 8 @ @ ee ee ee ee =~ wy o onrm o +e ere eye ae SP eb SS Ce ae oS = ae — (94¢) TABLE VIII Part 2. h,-4, h.=8, Coneluded at 5s? yo co = ~ = r 15 20 25 30 40 1t.01t,1 0,056 0.0496 0.0540 06,0588 6,0665 21/3 0.0665 0.06 0.077 00,0875 0,105 1 0.0720 0.0775 0.090 0.103" 0,130 33 0.0742 06,0810 0.0960 0,112 0,14 : 10 0.0748 06,0823 0.0978 0,114 0,149 18.1%.01 sort 0.0345 0,0213 eet 0.0111 11/3 On 2% 042359 0,261 0.27 0.301 21 0.342 0.400. 0,458 0.510 0.600 33 04415 0.510 0,618 0,720 0.910 210 0.455 6,57 0.720 0,858 1.12 : 100 0,475 0,61 0.765 0.920 1,24 131/33,01 0,111 06,0480 0.0242 0,0162 0.0108 e491 0,210 Oe 154 0, 134 0.122 0.111 31 0.700 0.745 0.795 0,830 0.890 : 1402 1e21 Bd ‘ 1+ 51 Ne 210 1420 1. 53 1484 2,13 2.68 100 1.32 Te 74 2.16 2,60 3.45 1332.01 0.820 0,455 rhe 4! 0, 148 te 0.905 0.595 0.385 0.262 0,140 3143 1,03 0.800 - 0,610 0.495 0,390 31 1,42 14351 teal 1.15 1.10 2 10 2,97 Be 2 4,20 4.70: 56 50 : 100 3.80 4.95 6.15 7430 9.60 12103,01 14 94 1.91 1442 1.11 0.565 t.1 1.99 1938 ie, a Or iee 3173 2,06 te g - 1464 103 0» 60 ¢1 ria ¥ eee] Bel2- 1,92 1643 23 2.84 341 Be35 Behl 5430 : 100 4, 50 5e95 7440 280 1146 4:100%.1 4.14 5400 5662 6.00 6440 81/3 4,14 5,00 5.62 6.05 64 50 1 4,19 5,08 5,72 6,20 6,60 £3 4425 5.25 6,00 6.55 Ace : 10 4,56 be JO 6,70 7055 8470 (744) 10 088 0.040 0,0295 8a/h 0,088 0 h-2, h,- 8 TABLE VIII Part Be 40 0365 0380 0. 0.265 0.106 0,055 0,0385 0047 0,03 «052 OO, 62 0,102 0 -105 0,0 0.252 O. 0,259 0,264 0 0.2 5 1 Ov2e9. 0 e@5i. 25 0.166 0, >? 9.3 0.350 0,206. 0, 61 16 0 36 AO 0.3 é 4 0 ~~ f= CoS oe 0 0.4255 00 605 0.661 a 0.5 0. On 533 et 0. 545 é 0.233 0.166 —Oed 2379 0,490 0.4 5 6 581 560. 0.610 0,703 0. 99 00 610 a 5 0,508 0.3 0.820 9,540 6,3 0,840 0 53 OO, * co 0,80 0,848 0 0.85 0.855 0. 1 2 Qo Ooo 2 eye tye eo OF 4 £5 6% ce 131/43 20125 2.72 2,13 2,1 2,% wn, Ayo WO Gs wm ON sey so e,mctlUstmUhOthUMS MY ATY YP Py Ey 2.4 ry, 0 2-94 ood a. ee @a12 2.354 1.36 1°36 F 1734 1,42 1 3 00 ors a ger eye se ¢e e ge ee 13 103 (94e) TABLE VIII Part 3, &h,<-2, h.=8 Concluded 15 20 25 30 40 120.01t41 0.0311. 0.0362 0.0410 0.0459 06,054 21/3 0.0368. 0,0442 06,0529 06,0610 sere 21 0.0388. eet ( 0.0575 0.0670 0.0 1301801 0.0348 06,0220 0.0160 06,0130 0,010 1173 0,176 . 06202 0,225 0,242 0,267 1 0.250 . 04314. 0,370 0.422 0,505 23 0.296 . 04380. 0.462 0,540 0,685 :10 0.322 0,421. 0,525 0,630 0,825 1100 04329 0.432 0,540 0.655 0.880 121/3:,01 0,068 . 0,030, 06,0185 0.0140 0.010 t,1 0.166 0.138. 0,124 0.11 0,106 21 0.565 0,640, 0.9705 0.75 0,820 +3 0.780 . 0.960. 1412. 1,26 1.51 $10 0.920 1419 1.44. 14.69 2.14 1100 1.03 1637 1,70 2.03 2,647 1: 33,01 0.970 04555, 0.325 0,205 0,100 tal 1,04 0.650. 0.420 0, 283 0,152 ¢1 149 1055 . 193 1427 1,14 110 Be . 4446 +10 5.60 6.39 : 100 5,10 6460 015 9,63 12.6 1:10:.01 2,97 2. 53 9 1.32 0.608 21/3 3.19. 875 2,18 1.64 0420 $ 3, 54 3,25 2674 2,23 1.4 t 4,10 4,09 4,00 3,87 gvet : 100 175 10,0 12,2 14,2 18.0 1210041 a 92.00 . 9,90. 10,4 10.8 11/3 Pyit) 9,10 . 10,0 10,5 11.1 +1 7,85 . 9643 © 10,5 11402 11.8 $10 8.80 1143 1345 15.2 1949 (942) SSoouln MS ~— deviate somewhat from the corresponding three~layer curve, and this deviation will be greatest where the two approxi~ mating; curves intersect, as at a maximum or minimum in the field curve. However, the preparation of such curves, besides being a laborious task,involves the problem of representing graphically curves with a greater number of variables. In the two layer case the curves contain three unknowns, £,8, and h, and can all be plotted on a single sheet with 4/: and a/h or L/h as coordinates and k » $54 as a parameter, In the three layer case there are five unknowns,*,%,%h,, and hi, so that the curves must be plotted by groups for special cases in which the number of variables 4s reduced by apecify- ing ratios between some of thems This is exemplified in Table VIII where valucs of Yzare liated for various retios ot the three resistivities (with?S=) and various values of 8 a/h, for a single value of the ratio h./h.. This data is taken from Wetzel and MeMurry's article A oat of | | aver Assis Inte patios the Three | broblen CCU les IT, and TT end LV» This is the only systematic set of tables published so far as.the vriter is aware, Curves from this data plot the log. of *,azainst that of 8 a/h,for a civen ratio of h,/h, and of BEN/ R19 » It le suggested in the article that these curves be grouped in families with a single value of f/r/3,for values of h/h. of g, g, and 3/4 on a single sheet of graph paper together with the two limiting two layer curves for the cases h,/h, = 0/4 and 4/4, This is illustrated by Figure 35 which shows the field curve obtained at Test Point #1 near Victor~ ville superimposed on the, family which two layer approximation shows to be the closest fit. The awkwardness of using 3=layer curves becomes apparent when it is noted that 14 sets of these eurve families ave needed to include only those field curves whieh contain a minimum of apparent resistivity. The field curve is compared by superposition to families of master curves until one is found which includes a curve of the size and shape of the field curve, Then the field curve is fitted into its place in this family by eye until its shape and size are intermediate to those of the master curves between which it falls, The position relative to the other master curves gives the ratio h/h.; the value of abscissa of the field curve opposite the master curve ordinate Sa/h. = 8 gives the value of hpj and the Intersect on the fleld ordinate determines §% asa in the two-layer case, * and % are then determined from the resistivity ratio of the family of master curves, This method at present is not strictly one of supere position because there are master curves for so few values of hy/hg that most field curves fall between them rather than on them and the matching involves extensive visual inter-~ | polation, Further, the field conditions do not often exactly correspond with any of the scte of resistivity ratios aseumed in the master curves, and in the absence of a close fit with ono of the master curves there ia often no way to tell te whieh of various families the field curve is most closely related, or to determine the value of hj/he, (a) Geophysics, Vol. II, #4, p,. 329. October 1937, (95) (2) Tho Paletea method ‘The Paletca curves for all values of k are assymptotic at very small values of L/h to the L/h axis where the apparent resistivity is equal to that of the upper layer. At very large values of L/h they are assymptotic to the abselissa along which the epparent resis« tivity oquals that of the lower layer. Between those two assymptotic portions of each curve is a connecting segment in which the apparent resistivity varies continuously from the resistivity of the upper layer to that of the lower. In the following diseuesion this connecting segment is called a "Tink", The link contains at each end a point of inflection whieh will be called a "turning point", The values of L/h at which the: turning points ocour are detormined mainly by the depth to the interface between the two layers, and some~« what by the ratio of the resistivities, The ¢reater the difference in the resistivity ratio or the larcer the absolute value of the reflection factor k, the sooner the effect of the second layer on tho apparent resistivity ls noticed and the smaller is the value of L/h at which the first turning point occurs, and the larger the value of L/h for the second turing point, The steepness of the link inereases with the abs lute value of k in spite of the increased L/h interval between the turning points because of a greater increase of the 1? intercept between apparent resistivity values of % and &, . Field curves of more than two layers are inter~ preted by assuming them to be made up of a number of two layer curves, Ifa third layer is present at sufficient depth, (96) the apparent resistivity may very nearly reach its assymptotic value before the effect of the third layer is noticeable, Yurther inorease in L/h values will cause a third turning point in the field curve followed by a second link turning through a fourth turning point into a third assymptotic por- tion where the apparent resistivity 4s very nearly that of the third layer. If the second layer is thin, the apparent resistivity may not apnroach its resistivity asaymptotically before the effect of the third layer becomes pronounced, and if the second layer ls too thin, its effect on the apparent resistivity may be negligible go that its presence isn't noted. <A field curve with n distinet Links represents the presence of n +1 layers of distinetiy different resistivities and of thicknesses shown by experience to be from &) to 10% at least ot their depth, In outline the procedure of interpreting the field curves is very simple, Sach suceessive link of the curve is fitted with one of the Paletea curves, From the fit of the first link io determined the surface resistivity f, tho depth h, to the top of the second layer, and from the values of fF and the ¢ of the Faletea curve, the resistivity of the second layer §, as in the two layer cage. Trom the second fit are deter- mined the effective resistivity 7’ of the first two layers, the depth h, to the top of the third layer, and the resistivity & of the third layer. ‘he third fit gives the effective resistivity fof the first three layers, hg the depth to the (97) top of the fourth layer, and 5 the resistivity of the fourth layer, and so on, If each link of the curve wags sufficiently long to inelude both turning points there would be no ambiguity in the fit, but in practise the links are usually short se that they may be fitted to parts of links of Paleteca curves for several valuos of k, and there le doubt as to whether it belongs to a certein portion of one curve, or to a less steep portion of a steeper curve. The different fite will five different values of the epths an’ resistivities, The following arbitrary mile is of assistance in determing the most likely of the possible fits. If the slope of the link 4s steep (ikl> °3) the mo t lilsely depth is that closest to 3/8 the value of L at the bend where the presence of the layer below the interface starts to make itself felt. By "bend" is meant the small portion of the field curve which contains two turning points at the meeting of two successive links, If the slope is small, (ie.5) the most likely interface depth 4a that closest to 3/8 the value of L at the middle of the link which is being interpreted. Joints on the field curve Links close to the turning points should not be trustod be- cauce they may deviate considerably from theoretical two layer curves, It may happen that Links are represented by only two points so that they will fit the Paletea curves in many places, In this case only an approximate sulution is possible. A depth to the interface can be approxineted by the rule given above, and the field curve moved over the Faletca curves, with the value of L equal to th approximate depth (98) ee = ont ee Bon ee - fortes aes ce Kee Be Bw Ea ae Te. — oo he ne ee ee ey fi ale ane | . \ ‘ f + wll tg cee bad \ | fe et eee res m ' i ee ae 1 kept on the L/h =1 ordinate, until the slopes of the curves are the same and thus determine an spproximate value of Kk and of the resistivities, 4 valuable check on the depths and resistivities deter. mined is furnished by the application of Hummei'ts fermula; ee ee oe here will be ni of these equations in the case of n links or n+i layers, Although they are approximate, they hold quite closely. IZ the two members of an equation differ by a two to one retio after numerical values have been substituted, the nature of the inequality is helpful in deciding which depths or resistivities are probably too large or too small. If the members differ by ratios of five or ten to one, it may be udvisable to change the interpretation. The procedure is illustrated in the following exemples, Example Vi. the curve obtuined at fest Point #1 near Victor. Wilie is show in Figure 32. the first link is unreliable because the point for which L= 18 feet mey be affected by the finite size of the electrodes, and because the points at L yalues of 39, 49s end 51 feet are close to the turning point 4% appreciably influenced by the third isyer. Only one good fit was obtained using the first three points, and this gives the yalues k.= «0,35, '.= 9600 ohm-centimeters, b,- 5.1 teet and \ is then computed to be 4630 ohm-centimeters. The second link is more reliable. Ali five points fit quite closely Paletca curves for twp values of K. 2. k.=0. » F'= 2200 Ohm=CGies Fy he = q feet Since k is large, the best value of h.may be teken as that nearest 3/9 the velue of L at the bend, or 3/8 of 45 feet which is 16,9 feet. the first fit is therefore the more prob- able and from this & is computed to be 24,400 ohm-centimeters, A break in the slope of the second Link ean be seen at L=90 feet. If this is interpreted as indicating the presen e (99) ' —- + 25 : {-.—+- 3 5 at who = Sse ee ta, : = — --+* $= = ak Sere ou Tee 1 ies ge oe - ee FE Ed [apes :, ee ae re FS, “fF xomple HZ Paletca method: Dashed tine ts “Fa, curve — =Eee poe ERE TS Fpom Test Pott no) near Victor ville fitted isto the appropriate - | ~ 5 valaes of \f and: hy determined from the Paleteca Curve si | ane ae Se a ee eg eee eee ee BP = af ee pp Setoh Wetzel MeMurry 3-layer. master, curves by the. | SPE t — | ja Er 3 = PRESSE pe — =e —-— of a fourth layer, each sublink ean be interpreted sepa- rately but this is poor practise. In the first place, a ehenge in the apparent resistivity at the 99 foot spacing of in, whieh 18 well within the range of error of field measurements would remove the break entirely. Secondly, 4f£ the break is reel, it is so slight that the ratio of resistivities above end below the interface is nearly unity end the diseontinuity is not important. thirdly, in this case it replaces one reliable 5 point link with two less reliable 3 point Links. However, if the link is subdivided and the best £1% chosen by the three-cighths rule, the depth h. ia founi te be about 17 feet, and a third layer 17 feet thick of 37000 ohm-centimeter resis. tivity to overlie a basement of 39000 ohm-centimeters. fhe difference in the resistivity determined for the basement in the two interpretations illustrates the aiffi- culty of accurately determining the resistivity of the lower layer when k is large. When |k! is near /.o, an error in determining the effective upper resistivity is multiplied many times in computing thet of the Lye: deyer, and algo an error in determining kris greatest*when k is nearly plus or minus one, Application of Hummel's formule hi lk Bach 68 ~ St 4» 10.7 er = F e % > y OR Y3e0 F600 F630 results in the slight inequality 0.00368 = 0.00285 which indicates that the three layer interpretation is sufficiently good, Taking the values h,- 5,1 feet, h.- 15.8 feet, i.= 9600 ohm-cm., % -4630 ohm-com., and ’,=24,400 ohm-cm., we find %./¢,/¢,-1/0.482/2, 54 and can compere the curve with the theoretical three-layer family for i/(1/3)/3. Figure 33 shows the field curve fitted into the family, oriented by the values of h. and §f, determined by the Paletea method. It shows an excellent agreement in shape and size with the master curves, and in the indicated ratio of b, /d (a= h.+«h) whieh agrees well with the Paletea figure of 1.3/2.7, The a deviation from the theoretical curves is seen at the pend. The value for h. of 15,4 feet agrees very well with the 16 feet measured with a steel tape in an adjacent road cut, Example Vil. Curve 3 shown in Figure 34 contains too few points to be reliable. The first three points, correspondim to L values of 30, 60, and 120 fect , may indicate the pres. ence of three layers ebove the basement, It might olso be that the apparent resistivity at the 30 foot spacing is too smell from the efrect of watering the electrodes, or that the 60 foot value is lerge because of irregulabities in the top gurface of the granite. For simplicity we will assume a three leyer solution, ignoring the value at the 40 foot spread. The slope of the link between the 60 and 120 foot points is small so the depth to the second layer is about (3/8 x 60) 30 feet. Moving the field curve until the link fits one of . the Paletea curves with the L-30 feet ordinate placed over that for L/n=i1, we find en approximate value k= 0.35. (100) F 5. is 13,600 and f& 6559 ohm-centimetera., The second link is more satisfactory, but still not very reliable because 4t has only three points of which one is near the bend, hia link fits best at k.-0.7, °'= 84002-49547, 6002", B= 67, 5b» fhis value of h is 3/8 of 180 and thet value of L is near- the bend in the field curve so that h. is of the right order of magnitude, Application of Hummel's rule gives 0.0683-0.007 whieh shows the fit is all right. For further cheek the curve was compared to the closest corresponding three-layer . family whieh again is 1/(1/3)/3. In this case the fit determined by the Paletca values didn't cheek so well, but this is not surprising when it is remembered that the resis. tivity ratios are somewhat different, that the values for § and h, cannot be considered reliable, and that h. may easily be in error by 10%. When an attempt was made to fit the curve to the three-layer family independently, it was found that the master ourves are so far apart that there is @ considerable range of values of h. ands’, for which o fit can be made. Certainly the failure to agree with the three layer curves is not sufficient reason to change the interpretation, Exemple Viil. Curve 4 is illustrated in Figure 2 fhe first link fite very nicely for the five points from i= 30 to i05 feet, giving the values k,= 0.3.97) = 19, 200.-,.h= 10,330 a-cnm, and b,=12 feet. Interpretation of the middle part of the curve is hagardeus, There is apparently a third leyer of Low resistivity above the granite, but there are too few Foints on this link. Of the three, thowe at L=105 treet and i= 225 feet are on turning points, and the former is also included in the first link. The slope of the link between the points L= 225 and L =300 feet is steeper than is possible if the layers are herizontel. To get a tentative solution, the point at the 225 foot spacing wae ignored, and h. was teken ag 40 feet (3/8 x 105). Using this value of h. k. was found to be ~0.7, und Ss’ to be 15000 ohmesentimeters, The last part of the curve appears to break in siope at the 450 foot specing, but the break is too small to bother with. Within the Limits of error im determining apparent resistivity, the third link fits nisely for k,-0,.8, %"= 5500, h,-58 feet. This value of h corresponds to L= 155 feet at the bend , which is not unreasoneble when the point at the 225 foot spacing is cant out, Substituting the values in Hummel's formula gives for the first two layers 0.00267= 0.00333, amd for the first three layers 0,01053=0.01012 which is surprisi good, agreement, However, the interpretation should not be consid. ered reliable. Exemple Ix. Curve 5 taken near Vietorville is shown in Figure 36. Here the first link is unreliable because there ure only two points sufficiently far from the bend, K appears to be large, but since the vulue of L ut the first turning point is not known, the three-eighths rule cannot be applied. However the second link is more reliable. It fits the Paletoa curve (101) eee eee oo) | eae $ rat pest oop he po ed ee of er pee ee ste a pt is a ee 1 n t thee 4 ee) ee are | a a i ae | ae |e a | rit a tbe op ao - po 4 ppp — gaat ve =e =" : =it 4 = eg. ee ee ae Ee ee eees ie ser ' + Mi) seis a. acre meen ‘cea eae _— on) Eee + yr { t ony Sai a a ‘ i : * dts — _ . _ . snk are 1. i pesmscaeg, gilmore gepoagety | i = f i mR outs a ie ss gk ees = De eA: ee - | be — e e e i 492 | geen rant Sse mw hy Sig ep aero na ee ie ot T i . ’ T _ i 4: z ra 4 {E-series al vo eae a a tpt aS 5 ea be a I eA © See es ee ae aed Bt Se6 50.4 cm 7 af 1 T 6 i a I —_ es —— | oe = --iit, — I —< ot 7“ < ——_} 19, hoa SEE °C RE ARS 2 iy ee ye for k-0,8 quite well tor values of h. trom 50 to 73 fect. Of these the lower limit is closest to the as feet given by the three-eighths rule, 4s a cheek, seven possible fits of the first two points of the first Link were tried with three possible tite of the second link in different combi-~ nations to see which agreed best with Hummel's formula. The fits giving the decidedly closest gheek (0,01282-0,01328) f= 3y00 ohm=om,, h,- 50 fect, K.=0.8,7,= 35,700 ohm-cm, Two comments may be made on these examples. Firat, they show the supreme importanee of having reliable field eurves, fo ensure good values, the data should be plotted on log. arithmie paper in the field, and any ambiguities removed by repeating observations or taking intermediate spacings. . Second, it should be noted that the depths and resistivities ure eXpressed in more precise numbers than is justizied by the accuracy of the method, Even with good data, and careful use of the Paletca curves, an error of 10% may be cxpected in the depth determinations, The exact values were given to enable the reader to see the fit whieh appeared best to the writer, It 18 usual for interpretations te vary a few percent according to the individual, The methods of interpreting the apparent resistivity- electrode separation curves discussed in this chapter all assume that the layers are horizontal. If links steeper than the Paletea curves for k equals plus or minus ome are found, and if there is no error in the field measurements, some other distribution of resistivities is indicated. Some idea of the nature of the irregularity may be hed by running depth profiles in several azimuths about ea common spread center, (102) For layers with Gips of less than thirty degrees, satisfactory results can be obtained if the line of electrodes is kept parallel to the strike of the beds, For steeper dips it is advisable to Giseadéd the use of depth profiles und use surface traverse profiles with « constant electrode separation and a moving spread center, (103) Bibliography The following list of references is intended to enable the reader more casily to find some of the more important papers on direct current measurement of earth resistivity. As only & smell part of the total literature could be in- cluded, preference was given to sources most apt to be available, and to articles in Mnglish, Contributions to interpretation of resistivity data, and descriptions of the main types of apparatus are quite completely covered, and discussions of applications to particular areas, while only a small part of the whole, are representative. For @ more exhaustive list the reader is advised to consult the Geophysical Abstracts published monthly by the U.S. Bureau of Mines through June 1936, and indexes of leading mining, engineering, petroleum, and geological periodicals, Barus, Carl, On the electrical activity of ore bodies: U. 5. Geol, Survey Mon. III (Becker, G F., Geology of the Comstock lode) Chapter X, pp. 309 + 367, 1882, Broughton Kige, A B, and Laby, 2. He, Principles and puather of geophysical prospecting (report of imperial Geophysical Experimental Survey), Cambridge University Presse, 1931. Bruckshaw, Je M. and Dixey, F, Ground water investigations by geophysical methods: Mining Megezine, Vol. 5, Ha. 2 ppe 13 cd 84, and He, Bay PPR 147] cal 154, Febs and. Maz, 1934. Crosby, i. B, and Leonardion, ‘Ee G., Hlesvtriceal prospec ting applied to foundation problems: A. I. M. E Geophysi Prospecting 1929, pPe 199 = 211, Ehrenburg, De O. and Watson, R, J., Mathematical theory of electrical flow in stratified media with horizontal, homogencous, and isotropic layers: A. I. M. EB. Geophysical Prospecting 1932, pp. 423 = 4580 Eve, 4. 5, and Keys, D. A,, Applied geophysics, 34. ed,, Chapter 3, Cambridge University Press, 19386 Evjen, H. M, Depth factors and resolving power of electrical measurements: Geophysies, Vol. 3, No. 2, ppe 78 = 95, Mereh 1938. Fritseh, Volker, Der Einfluss des Wassergehaltes auf den eee mae ae ee ak a wires Sel tu HG. Py ‘ee eft. i2 OL» PRe 3IF » 320 Berlin, March 25, 1937s . ‘ tHe FAY » 320s Gilchrist, le, Geophysical investigations made in 1 30% Canada-Geol, Survey, Mem, 170 Pt. Il, pde 65 « 99, 19326 Gish, O. H. and Rooney, We. J», Measurement of resistivity of large masses of undisturbed earth: Terr, Mag, and Atmosph. Electricity, Vol. SEX, PP. 161 = 188, 19256 ( 104) Hewkins, Re He, <Application of resdativity methods to Hiorthern Ontario lignite deposits: A. I. M. E. Geophysi- eal, Prospecting 1934, DP. 76 + 1206 Hedstrom, Helmar, llectrical prospecting for eauriferous quartz veins and reefs: Mining Magazine, Vol, 46, No. 4, pPe 201 = 213, ApFAil 1932s Hopper, KR. H, and Jakosky, J. J., the effects of moisture on the direct current resistivities of oi] sands and. vooke;s Geophysics, Vol. 2, Hoe 1, pps 33 = 55, Jame 1937. Hotchkiss, W. 0», Rooney, We Je, and Fisher, James, Barth resistivity measurements in the Lake Superior copper eountry; A. I, lM. BE, Geophysical-Prospecting 1929, pre 51 ~ 65, Hubbert, M. K,, Results of resistivity survey on various geologic structures in Illinois: a, I. Me E. Geophysical Prospecting 1934, pps 9 = 29. Hubbert, M. Ke and Weller, J. M., Location of faults in Herdin County, Illinois, by the earth-resistivity method; Ae I, Me Ey Geophysical Prospecting 1934, Dds 40 + 44s Hummel, J. He, <A theoretical atudy of apparent resistivity in surface potential methods: A, I. M. BE, Geophysical Prospecting 1932, Dpe392 ~ 4226 dakosky, J. Js, Continuous electrical profiling: Geophysics Vole 3, Hoe 25 DP 130 » 153, March 19386 Karcher, J. C. and McDermott, E., | Deep electrical prospecting; Kurtemacher, K. So, Some practical applications of resis. tivity-measurements to highway prebleme: A. I, Ms Ey Geophysical Prospecting 1934, pDe 49 « Se Lancaster J ones, Be » the eartheresistivity method of electrical prospecting; Mining Megegine, Vol. 42, Ho. 6 BPs rn 355, and Vol. 43, Now 1, PD» 19 +» 29, June and ‘ e Lee, 7. We, Measuring the variation of ground resistivity with @ megger:; U. S. Bur. Mines Weeh. Paper 440, 1928, Lee, Fe We, Geophysical prospecting for underground waters in desert areas; U. S. Bur. Mines, Info, Cire, 6599, august 1936, Lee, Fe We and Swarts, J, Hy, Resistivity measurements of 012 bearing beda; U. & Bur. Mines Tech, Paper 488, 1930. Leonardon, HE. G and Kelly, S.. Fe, Some applications of potential methods to structural studies: A. I. Me. Ee Geophysical Prospecting 1929, PP» 180 + 1996 ( 105) lew, Bela, Kelly, S. F., and Greagmile, W. Be, Apptying the megger ground tester in electrical exploration; A I, M. E, Geophysical Prospecting 1932, DP» 114 » 125, Lugeon, M. and Schlumberger, G., ‘the electrical study of dam formations, Mining Megazine, Vol. 48, Now6, PP. 340 » 345, June 19336 Muskat, Morris, Potential distribution about an electrode on the surface of the earth; Physies, 4, ppe iz) = 149; 19330 Qliendorf, Franag, Evistrome, Berlin, Springer, 1928. Pirson, 5. Je, Interpretation of three-layer resistivity vie As t M. E, Geophysical Prospecting 1934, Ppp» 145 » 1505 Pirson, 8, ds, iifect of anisotropy on apparent resiativity Pls t73 50 Poldini, E., Geophysical, exploration at spontaneous polari. gation methods: Mining Magazine, Vol. 59, Ho. (ior, ) Ol. 60, Nos 1, DPy 22 + 27, No. 2, PD. 90 ~ 94, Jan. end Feds 19396 Potter, E. Ve, Results of electrical resistivity and eles. trical induction measurements eat Abana Mine, Quebec, Roman, Irwin, How to compute tables for determining electrical resistivity of underlying beds end their application to 44 DDe, 19316 Komen, Irwin, Some interpretations of earth resistivity data: Ae I, Me Ky Geophysical Prospecting 1934, pp» 183 « 197. Rust, W. Moy Jere, A historical review of electrical pros- peoting methods; Geophysics, Vols 3, No. i, DDe 1 = 6, anuary 19366 Sayre, A. li, and Stephenson, HE. L., ‘The use of resistivity- methods in the location of salt-water bodies in the EL Paso, Texas, areas flat. Research Counell, am Geophys. Union trans. 1937, Pte Ii, pp» 393 = 398, Slichter, L. B,, the interpretation of the resistivity prospecting method for horigontal structures; Physics, 4, Pye RUT ~ 32a, 9336 Schlumberger, G. and M,, The method of the ground resistivity nap and its practical applications; Canadian Min. and het. Bull. 226, pDs 271 = 294, Feds 19346 ( 106) Schlumberger, G. and M., and Leonardon, EB. Ge, Eleotrical exploration of water covered areas: A. I. M. E. Geophy &- ical Prospecting 1934, pp» 122 » 134. Schlumberger, C. and M., and Leonardon, E, Ge, Some observations concerning eleetrical measurements in an. jgotrople media and their interpretation: A. I. M, E. Geophysical Prospecting 1934, ppe 159 «= 181, Schlumberger, G. and M., aud Leonardon, BE, G, Electrical ecoting; a method of determining bottom-hole data by elec. trical measurementa: A. I. M. E, Geophysical Prospect ng 1934, DDe 237 ~ 272. Shepard, Ee Re, Subsurface exploration by earth-resistivi ty and aseismic methoda; Nat. Kesearch Council, am. Geophy 8. Stetanesco, &, Schlumberger, M. and Co, Sur la distribution electrique potentielle autour d'une prise de terre ponctuelle dans un terrain a couches horizontales, homogenes, et isotropes; Jour. de Physique et Redium, Serle VII, fare I, Sundberg, Karl, Effect of impregnating waters on electrical conductivity of soils and roeke: A, I. M. BH. Geophysical Prospecting 1932, Dp. 367 + 391, Swartz, Je He, Artificial measurements upon artificial beds; U. S&S» Bur. Mines Info. Circ. 6445, 9 pp., Febs 1931. Swarte, J. He, O11 prospecting in Kentusky by resistivity methods: U. S. Bur. Mines Tech. Paper 521, 23 pRer 1932. Tags, Ge Be, The earth resistivity method of geophysical Prenpeetizg. Some theoretical considerations: Mining tege, Ge Bs, Interpretation of resistivity measurements; A. I. Me E. Geophysical Prospecting 1934, pp» 1235 = 145, Tage, Ge F., Zerth-resistivity surveying: Mining Magazine, Wateon, Re de, A contribution to the theory of the interpre- tation of resistivity measurements obtained from surface potential observations; A. I. M. E. Geophysical Prospecting 1934, DP» 2Oi ~ 232. Watson, Re J», und Johnson, J. Fe, Qn the extension of two~ layer methods of interpretation of carth resistivity data to three and more layers: Geophysics, Vol. 3, No» 1, PP» 7 = 2i, Januery 1936. Wenner, Prank, A method of measuring earth resistivity: U. 5. Bur. Standards Bull, 258 (Vol. i2, Nos 4), 1916. hipaa We nd ne ere a Ve» A set of curves to assist in e interpretation of the three layer resistivity. problem; Geopayaian. 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