AMERICAN INSTITUTE OF MINING AND METALLURGICAL ENGINEERS Technical Publication No. 1743

(CLASS G. PETROLBUM DIVISION, NO. ax?; CLASS L. GEOPEYSICS. NO. IOI)

DISCUSSION OF THIS PAPER IS INVITED. Discussion in writing (a copies) may be sent to the Secretary. American Institute of Mining and.Metallurgica1 Engineers. a9 West 39th Street. New York 18. N. Y. Unless spec181 arrangement is made. dscusston of thls paper ml1 close Oct. 30. 1944. Any discussion offered thereafter should preferably be in the form of a new paper.

An Empirical Method of Interpretation of Earth-resistivity Measurements

BY R. WOODWARD MOORE* (New York Meeting. February 1944)

A B S T ~ C T A GRAPHICAL method of analyzing the data

obtained from shallow earth-resistivity depth tests is presented. The method is based upon empirical results and has no theoretical basis. The usual apparent resistivity-electrode spac- ing curve is used together with a cumulative resistivity-electrode spacing curve plotted on the same sheet. The greatly reduced scale re- quired for plotting the cumulative values of resistivity together with the &ect of the sum- mation of the individual resistivity values serves to minimize the effect of purely local surface anomalies and inadvertent errors of measurement. The point of intersection of tangents or straight lines drawn to intersect a t zones of maximum curvature in the cumulative curve indicates the depth to the underlying material. Numerous figures are presented in which data from published reports and from recent field studies are analyzed and the re- sults compared with actual depths established by borings or with depth values obtained by the use of theoretical methods of analysis. Smoothly rounded curves of apparent resis- tivity such as are often obtained in the field, and which have been a serious drawback to attempts to analyze the data empirically heretofore, appear to be susceptible to rather accurate analysis by the method described.

The method is best suited to analyses in- volving shallow two-layer formations. I t has been applied successfully, however, in analyz- ing the data obtained from tests made over shallow three-layer formations. As with most empirical methods, its chief advantage is its simplicity and ease of application.

There have been published many papers that discuss the interpretation of data obtained from earth-resistivity tests when using the four-terminal method of electrode spacing developed by Wenner.' The majority of these have dealt with theo- retical analyses for two-layer and three- layer formations. Some sets of "master curves" have been presented for use in analyzing field data to determine the depth to the first and possibly the second horizon below the earth's surface. Although practically all of these theoretical methods of analysis have appeared to have par- ticular merit and some have been used successfully in practice, they have been found to be of little value where the local conditions surrounding the test failed to conform to those assumed in the theory.

. I n certain fields, particularly in civil engineering, relatively shallow explorations are often involved and geophysical methods of test must compete with the direct methods of exploration ordinarily used. Only when i t can be demonstrated that geophysical methods of test can materially reduce the time and cost of a given explora- tion project will the civil engineer abandon direct' methods in favor of the interpre- tations of geophysical exploration data.

Empirical methods of analyzing earth- resistivity data have been used in many instances in the past. Such methods have

Published by permission of the Public Roads been employed for a number of Years by Administration. Manuscript received at the the public ~ d ~ i ~ i ~ t ~ ~ t i ~ ~ in re-

I office of the Institute Jan. 28, 1944. Associate Civil Engineer. Division of Tests, search work relative to the application of

Public Roads Administration. Washington. 1 D. C. 1 References are at the end of the paper. - -

Copyright. 1944. by the American Institute of Mining and Metallurgical Engineers, Inc. PETROLBUM TBCHNOLOCY. July 1944. and separately for the Geophysics Committee. Printed in U. S. A.

INTERPRETATXON OF EARTH-RRSISTMTY hfEASUBEaNTS

TABLE I.-Resistivity Depth Test Data Used in Figure I

Remarks

Test pit shows clay to depth of 14.0 ft. under- ltun by hard ruck

Electrude Spacing. Pt.

3 6 9 I a 15 18 a1 a4 a7 30

Apparent Resistivity, Ohm-cm.

--- 8,000 6,800 7.050 7.510 8.600 9.800

11.550 1 a . 4 ~ 0 13.500 14.650

Cumulative Resistivity. Ohm-om.

8.ooo 14.800 a1.850 ag.360 37.960 47.760 59.310 71.740 85.240 99.890

R. WOODWARD MOORE 3

the resistivity test in the field of highway surface layer is encountered, an inflection construction. Prior to 1940 the empirical will presumably be found in the apparent relation proposed by Gish and Rooney2 resistivity-electrode spacing curve. Gish was generally used for the study of the and Rooney found that wherc such an

ELECTRODE SPACING-FEET FIG. 2.-REPRODUCTZON OF TAGG'S CURVE FOR STATION A.'

field data. In this method of analysis, the apparent resistivities measured for a suc- cession of gradually increased electrode spacings are plotted as ordinates and the corresponding electrode spacings as ab- scissas. An empirical relation appears to exist, in which the so-called "effectiGe depth" of current flow approximates the value of the electrode spacing used. If, within this "effective depth," an under- lying formation with a specific resistance materially different from that of the

inflection appeared the value of the elec- trode spacing at the point of inflection could be interpreted as the approximate thickness of the surface layer. Where a low- resistivity material is underlain by one of a higher resistivity a U-shaped curve is often obtained and in such cases the electrode spacing for the low point of the curve is usually interpreted as indicating the depth to the underlying material.

In many instances, in connection with shallow work abrupt breaks in the curves

4 INTERPRETATION OF EARTH-RESISTIVITY MEASUREMENTS

for apparent resistivity vs. electrode The following discussion relates to an spacing have been found, which, when empirical method of analysis that has interpreted in this manner, give a reason- been developed in connection with the ably accurate indication of the position resistivity research of the Public Roads

FIG. ~.-REP~ODUC~ION OF TAGG'S CURVE FOR STATION B.'

of the interface between the surface materials and the underlying material in simple two-layer formations. There are other instances, however, in which resis- tivity depth tests in two-layer or multilayer formations yield apparent resistivity-depth curves that are smoothly rounded, with no marked inflections or other indications of the position of the various strata. For such curves no satisfactory empirical method of analysis is available and such theoretical methods as have been proposed are both uncertain and time consuming.

Administration. When applied to data obtained both by this agency and by others, over a wide range of field conditions, it seems to offer definitely better corre- lations than other methods where the test data are of the type described above.

In the proposed method of analysis the data are obtained in the field using the Wennerl four-electrode configuration and a conventional Gish-Rooney curve (appar- ent resistivity vs. electrode spacing) is

R. WOODWARD MOORE 5

prepared. The Gish-Rooney curve is used shallow work, is chosen arbitrarily and the for whatever indication it may give of electrode spacing is then increased regu- subsurface conditions a t the point of test. larly by increments of 3 ft. for each The data in the curve are replotted on the successive determination. The initial value

CURVE BY ASSUMING A VALUE OFp = 10000 OHM- CMS. IN THE RAT I 0 fi/p USED IN HIS FIG. 8

= =, h

- H ~ Y E L ' S CURVE OF APPARENT RESISTMTY FOR CASE OF hl = 3h AND RATIO PO: = m: 1:o.g: m (HIS FIG. 8).4

same sheet in the form of a cumulative resistivity-electrode spacing curve. By reason of a greatly reduced ordinate scale and the effect of successive summations of individual resistivity values, this plot- ting of the data tends to minimize the effect of any single resistivity value and thus eliminate purely local effects caused by surface anomalies or any peculiarity of a particular setting of the electrodes.

In obtaining the field data for plotting I in this manner, an initial electrode spacing

of some convenient value, say 3 ft. for

of apparent resistivity is plotted as the initial ordinate of the cumulative curve. Each subsequent value of apparent resis- tivity is added to the sum of all preceding resistivity values and each total thus obtained is plotted as the ordinate of another point in the cumulative curve. By using regularly increased electrode spacings (for example, 3 ft., 6 ft., 9 ft., etc.) it would appear that a substantially straight line with a given slope should be obtained as long as the "effective depth" of current flow remains primarily within the surface

6 INTERPRETATION OF EARTH-RESISTIVITY MEASUREMENTS

layer and this layer consists of a relatively corresponding approximately to the depth homogeneous material. I n practice, how- of the surface layer the plotted cumulative ever, where the surface material is rela- curve usually tends to change direction, the tively shallow and the soil layer is not new slope depending upon the relative

FIG. 5.-HUHMEL'S CURVE OF APPARENT RESISTIVITY FOR TEE CASE OF hl = 3h AND RATIO p o : p : p ~ : pe = 0 : 1 : 3 : 0 (HIS FIG. 9).4

perfectly homogeneous the plotted data frequently are in the form of a generally smooth curve of gentle curvature rather than in a straight line. Probably this is due to gradual changes of resistivity with depth and with soil variation.

I t has been found that as the electrode spacing approaches and passes a value

resistivities of the two layers of material. From the electrical principles involved, this slope should increase if the lower formation possesses a higher resistivity than the surface layer, and, conversely, should decrease if the underlying formation is the more conductive. I t has been found that lines drawn tangent to the cumulative

R. WOODWARD MOORE 1

TWO-LAYER MODEL EXPERIMENT Ifl

H OBTAINED BY TAGG'S METHOD

ELECTRODE SPACING-INCHES

100- z e W

a 7 5 -

5 0 -

2 5

FIG. 7.-CUMULATIVE-BESISTMTY-CURVE METHOD OF ANALYSIS APPIdED TO MODEL EXPERIMENT DATA OBTAINED BY R. J. WATSON (HIS FIGS. 2 0 AND Z I ) . ~

O- OO 100 2 0 0 0 100 2 0 0 3 0 0

CLECTRODE SPACING- FEET

FIG. 6.-CURVES PLOTTED FROM FIELD DATA OF HUBBERT'S TABLE 1.)

8

4 -

- 2

"I

P-pPr

0

,.*. I('

I SHALE

STATION 3

6---. NOTE- TAGG'S METHOD

NOT APPLICABLE

i...*----X .... ,..*.-.*...

. . E L E C T R O D E SPACING - F E E T

FIG. 8.-REPRODUCTION OF CROSBY AND LEONARDON'S CURVES (FIGS. 6-13).'

R . WOODWARD MOORE 9

MUDDY WATER I SATURATED SAND

ELECTRODE SPACING-INCHES

ELECTRODE SPACING- FEET FIG. IO.-C~ULATIVE-BES~STM~-CURVE METHOD OP ANALYSIS APPLIED TO ROMAN'S FIGURE 9.'

I 0 INTERPRETATION OF EARTH-RESISTIVITY MEASUREMENTS

curve and intersecting in the region where method of presenting the data is employed the change in slope occurs will give a good in each of the subsequent figures and for approximation of the depth to the interface simplicity the identification of the curves between the two materials if the point of is not repeated.

OF TEST LOCATION

ELECTRODE SPACING - FEET FIG. 11.-RESISTIVITY TEST NEAR WEST ABUTXENT OF ARLINGTON MEMORIAL BRIDGE, WASH-

INGTON, D. C.

intersection of the tangents is projected to the horizontal or depth axis. Fig. I

illustrates the method applied to a typical case; the data used in plotting the curves are given in Table I. The values of electrode spacing and. of apparent resistivity in columns I and 2, respectively, were ob- tained in field tests in the vicinity of

Referring to Fig. I , the presence of the high-resistance rock formation a t the rela- tively shallow depth of 14 ft. affects strongly the measured apparent resistivity beyond an electrode spacing of about 10 ft., and for this reason the plotted values of cumulative resistivity continue to show a rather marked degree of curva-

Washington, D. C., where there existed a ture beyond what might be termed the simple two-layer formation consisting of "critical point" in the curve. The trend clay underlain by rock. of the Gish-Rooney or individual test-value

In Fig. I the Gish-Rooney or individual curve is used to indicate the probable test-value curve is shown by crosses con- "critical point," which in this curve nected by a dashed-line curve and the appears to be a t an electrode spacing of cumulative resistivity curve, by plotted ~ b - t o 12 ft. Guided by the indications of circles. For clarity the curve connecting this curve and such correlating data as the circles has been omitted. This same may be available from test pits or drill

holes in the general area, the additional of analysis, together with the points that tangent intersections beyond the "critical result when the data are replotted in the point" may or may not be disregarded. manner previously described. Fig. 3 shows

Admittedly, this is empirical in every a similar treatment of Tagg's station B.

ELECTRODE SPACING- FEET FIG. 12.-RESISTIVITY TEST AT LOCATION OF PROPOSED RAILWAY UNDEWASS NEAR PETERSBURG,

VIRGINIA.

sense but the results that have been The cumulative values in Figs. 2 and 3 obtained in the analysis of data from many indicate depths that closely approximate tests under various field conditions have those obtained by Tagg with a rather been encouraging. I n addition to the use laborio'us method of analysis. In connection of this method of analysis for data obtained with these curves and others that are to in field tests in connection with the work be presented, it should be pointed out that of the Public Roads Administration, the frequently it is rather difficult to read literature has been reviewed and resistivity accurately 'values of the coordinates from curves presented by various authors have the necessarily small figures found in the been replotted and analyzed by the method published papers when taking off data for wherever the necessary depth data were replotting. given. The results obtained appear to Figs. 4 and 5 show Hummel's4 curves of substantiate the ..conclusions reached in apparent resistivity for two theoretical analyzing the data obtained in the field. cases in which two surface layers of differ-

Fig. 2 shows Tagg'sS "classical" curve ing resistivity and thickness are underlain for his station A, representing typical by, in one case, a layer having an infinitely field data to which he applied his method high resistivity and in the other a layer

I 2 INTERPRETATION OF EARTH-RESISTNITY MEASUREMENTS

having an infinitely high conductivity, as three-layer condition was involved in both shown in his Figs. 8 and g. The ordinate v instances. of Hummel's Figs. 8 and g is given as the The curves shown in Fig. 6 were plotted ratio p./p in which p is the true resistivity ,from data presented in Table I of a paper

ELECTRODE SPACING-FEET FIG. 13.-~SISTMTY DEPTH &ST I N SALT MARSH NEAR HAYPTON, NEW HAMPSHIRE.

of the upper layer of the ground and p. is the average or apparent resistivity resulting from the influence of the under- lying layers on the values of resistivity as measured at the ground surface. In plotting the dotted-line curves in Figs. 4 and 5, a value of 10,ooo ohm-cm. was assumed for p and applied to the ratios p./p taken from Hummel's curves. This resulted in conventional curves for appar- ent resistivity vs. electrode spacing, from which the data for plotting the cumulative curves were obtained. The position of the interfaces as determined from the cumu- lative curve appears to agree with that assumed in the theoretical treatment rather closely despite the fact that a

by Hubbert.6 The data for his stations I ,

2 and 4 check the known conditions reasonably well when plotted using the cumulative value of resistivity. As stated by the author, Tagg's method of analysis, although giving good intersection zones for the three sets of curves plotted for stations I , 2 and 4 gives depth indications that are in error by 50, 54 and 70 per cent for the respective stations. The individual test value or Gish-Rooney curves for these stations were of little value in deter- mining the depths involved. The data for station 3 apparently cannot be interpreted by any known method.

In Fig. 7 are data from a paper by W a t ~ o n , ~ in which curves obtained from

R. WOODWARD MOORE I 3

model experiments are described. In the checked the actual conditions existing experiments he performed a carefully beneath the surface within about 3 per prepared sandy shale was screened and cent. The average error for all curves was mixed with different percentages of mois- 5.8 per cent.

ELECTRODE SPACING-FEET FIG. ~~.-RESISTMTY DEPTH TEST ON SAND AND GRAVEL FILL APPROXIMATELY I S FEET ABOVE

SEA LEVEL NEAR HAYPTON, NEW HAMPSHIRE.

ture to form surface layers having different resistivities. A rubber sheet was placed under the surface layer to simulate a bottom layer having infinite resistivity. In both cases compared the cumulative curve indicates rather accurately the location of the buried rubber sheet as well as the interface between layers of soil of different resistivity.

Fig. 8 contains reproductions of several curves published by Crosby and Leo- n a r d ~ n . ~ The depth to rock as predicted by these authors is noted on each of the graphs in Fig. 8 and the depth to rock as obtained by drilling is shown along .

the lower edge. With three exceptions, the cumulative-curve method of analysis

No data were included in Crosby and Leonardon's paper relative to variations in the material classed as glacial till, which overlay the rock. I t is possible that the presence of a water table is responsible for the breaks in some of the cumulative resistivity curves a t depths less than those given for the rock surface.

Fig. 9 reproduces curves 3 and 4 of Fig. 7 of a paper by Manhart.8 He reported on model tank experiments in which he made numerous tests under laboratory- controlled conditions. The two curves shown in the figure were obtained when using a surface layer of muddy water underlain by saturated sand. The typically smooth curves obtained from Manhart's

14 INTERPRETATION OF EARTH-RESISTIVITY MEASUREMENTS

experiments lend themselves well to a solution by the cumulative-curve method of analysis. Of the many curves presented by Manhart, 13 were analyzed by the

curve of apparent resistivity vs. electrode spacing and also in the form of the corre- sponding cumulative curve. pa, the appar- ent resistivity, was assumed to be given in

120

I00

80

60

p 40 W

W z : 20- 0

; Y 6 ; b w

ELECTRODE SPACING-FEET

FIG. I 5.-RESISTIVITY DEPTH TESTS OVER PEAT-BOG FORMATIONS

-

-

-

-

-

O-

proposed method, four two-layer and nine three-layer curves comprising 22 possible depth determinations, and in only two or three cases were the conditions indicated by the analysis materially different from those established in the laboratory.

In Fig. 10 a curve given by Romang in his Fig. 9 has been replotted as a normal

ohm-cm. Roman suggests that further theoretical studies are required to make possible the interpretation of such curves. Although use of the original apparent-resis- tivity and electrode-spacing data would have given a more accurate curve for use in the empirical analysis that is proposed, the data shown in Fig. 10 indicatd:that

m t 0 h-

g g '20 m 3 0'3 8 0

; I00 t L V) -

e o -

60

40

20

-

'

.

-

-

0 -

R. WOODWARD MOORE

the cumulative-curve method of analysis stated previously, both the Gish-Rooney has given a reasonably accurate depth and the cumulative curves are used in determination. The depth of 80 ft. to interpreting the data from a particular bedrock was given in the discussion that test. The general shape of the Gish-Rooney

ELECTRODE SPACING -FEET

FIG. I~.-EMPI~CAL METHOD OF ANALYSIS APPLIED TO RESISTIVIn CURVES FOR A CLAY STRATUM UNDERLAIN BY ROCK IN THE VICINITY OF WASHINGTON, D. C.

followed the presentation of Roman's paper. Mention also was made in the dis- cussion of the presence of swampy mate- rials underlain by hardpan which overlay the rock a t the test location. The change indicated in the cumulative curve a t a depth of approximately 2 2 ft. may be a t the boundary between these two surface materials.

Many other published data have been replotted and analyzed by the cumulative- curve method, with about the same general degree of success. There are border-line conditions of test which do not lend them- selves to an analysis by this method. As

curve serves to indicate the probable ground conditions, whether two-layer or more, and it may also give some clue to the approximate depth of the surface layers.

FIELD TESTS CONDUCTED BY THE PUBLIC ROADS ADYINISTRATION

The results of tests conducted by the Public Roads Administration that involve a variety of surface and subsurface con- ditions encountered in the vicinity of Washington, D. C., and in New Hampshire are shown in Figs. 11 to 14. The curve in Fig. 11 .was obtained near the west abut-

x6 INTERPRETATION OF EARTH-RESISTIVITY MEASUREMENTS

ment of the Arlington Memorial Bridge spacing of 48 ft. The cumulative curve across the Potomac River a t Washington, checked the known depth and the depth D. C. At this location a few feet of earth indicated by refraction seismic tests made fill overlay soft mud, which in turn was near by within about 1.7 and 6.9 per cent,

FIG. I 7.-IRREGULAR SURFACE OF ROCK FORMATION THAT WAS INVOLVED I N THE RESISTIVITY TESTS.

underlain by a rock formation. The depth of 43 ft. to rock indicated by the cumu- lative curve is in reasonable agreement with the depth of 40 ft. found bj. drilling a t the abutment location about 150 ft. to the south. The change shown by the cumu- lative curve a t a depth of about 15 ft. was probably where the fill material merged with the-underlying mud.

Fig. 12 shows data for a test where heavy clay was underlain by coarse, water- bearing sand a t a depth of 14.0 f t ; The relatively smooth individual value or Gish- Rooney curve shows'no indication of the sand stratum a t that depth but it is located a t 14.2 ft. on the cumulative resistivity curve.

Fig. 13 contains the resistivity curve obtained a t a location in the salt marshes near Hampton, N. H., where rock was known to be a t a depth of approximately 60 ft. The Gish-Rooney curve shows an increased upward trend a t an electrode

respectively. In the absence of previous knowledge of the general nature of the sub- surface forniation, it may be necessary to put down one or two drill holes to establish the significance of data that show several changes in direction or "breaks" in the cumulative curve.

In Fig. 14 are shown the results of tests made on a sand and gravel fill near the sea and about 15 ft. above sea level a t Hampton, N. H. The cumulative curve indicates a low-resistivity medium such as salt water a t about 15.0 feet.

Curves representative of tests made with the resistivity apparatus over peat bogs are shown in Fig. IS. In this series of tests the cumulative curve proved to be a satis- factory means of analyzing the data obtained. These tests are of interest because i t seemed probable that the peat and marl formations overlying the sand or hardpan usually comprising the bottom formations of the bogs would approach the uniform

R . WOODWARD MOORE I 7

layers of homogeneous materials assumed tivity traverse" covering an area of about in theory, thus presenting conditions that 150 acres, a map was prepared showing would be particularly suitable for a satis- subsurface rock elevations by means of factory analysis by the resistivity method. rock-surface contours. Numerous check

FIG. 18.-RESULTS OF TESTS IN TILED POOL.

Furthermore, it was relatively simple to check directly the indications of the resis-

* tivity tests by means of rod soundings with equipment capable of taking samples a t any desired depth.

Fig. 16 shows data from an extensive resistivity survey project completed in the spring of 1942 m the vicinity of Washing- ton, D. C. Based upon some 475 'to 500 depth tests and about 10.5 miles of "resis-

borings, with post-hole auger equipment, and direct observations in excavations made subsequent to the preparation of this map, indicated an average accuracy of I to 2 ft. for the rock contours as estab- lished. The depth of the surface clay varied from about 4 ft. to 30 ft. The underlying rock was a granite formation of quite irregular contour, as shown m Fig. 17. I t was found that a difference of 6 ft.

I 8 INTERPRETATION OF EARTH-RESISTIVITY MEASUREMENTS

vertically could be encountered by a post- hole auger in as little as 6 ft. horizontally in places where such irregularities were the more pronounced. Under such con- ditions the resistivity test should offer more dependable over-all depth values than a limited number of individual borings.

The results of tests conducted in the tiled pool in the plaza of the National Academy of Science, Washington, D. C., are given in Fig. 18. The Gish-Rooney method of plotting the data showed no indication of the depth of the water ia the pool, whereas an analysis with the cumu- lative curve shows a remarkable correlation with the measured depth of water, 17 inches.

The cumulative-resistivity-curve method of analysis offers a simple and rapid means of determining the depth t o an underlying formation in cases that show no definite indication of its presence in the apparent- resistivity-electrode-spacing curve.

Although best suited to shallow two- layer formations, the cumulative curve has been applied with good results to data from relatively deep tests and for both two-layer and three-layer formations. It

has proved to be particularly useful where rapid reconnaissance surveys are being made over relatively large areas and where more detailed and time-consuming methods - of analysis are not justified economically. This method of analysis is intended to augment and supplement other methods in current use rather than to displace them.

REPERENCES I . P. C. Wenner: A Method of Measuring

Earth Resistivity. Nat. Bur. Stds. Sci. Paper 258 (1915).

2. 0. H. Gish and W. T. Rooney: Measure- ment of Resistivity of Large Masses of Undisturbed Earth. Terrestrial Mag- netism and Atmosbheric Electricity (1925) . ~-

30. 161. 3. G. F. Tagg:.Interpretation of Earth-resis-

tivity Measurements. Trans. A.I.M.E. (1934) 110, 135. C"

4. J. N. Hummel: A Theoretical Study of Apparent Resistivity in Surface Potential Methods. Trans. A.I.M.E. (1932) 97, 392.

5. M. K. Hubbert: Results of Earth-resis- tivity Survey. on Various Geologic Struc- tures in Illinois. Trans. A.I.M.E. (1934) 1 II0 ,9 . .

6. R. J. Watson: A Contribution to the Theory of the Interpretation of Resistivity Measurements Obtained from Surface Potential Observations. Trans. A.I.M.E. 1 (1934) 110, 201.

7. I. B. Crosby and E. G. Leonardon: Elec- trical Prospecting Applied to Foundation Problems. Trans. A.I.M.E. (1929) 81, 199.

8. T. A. Manhart: Model Tank Experiments and Methods for Interpretation of Resis- tivitv Curves. Colorado School of Mines ' ~ u a i t e r l ~ (Jan. 1937) 32.

9. I. Roman: Some Interpretations of Earth- resistivity Data. Trans. A.I.M.E. (1934) 1x0, 183. I