An experimental study of electromagnetic frequency soundings

OF ELECTROMAGNETIC FREQUENCY SOUNDINGS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECH-NISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. IR. F.J. KIEVITS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET

COLLEGE VAN DEKANEN TE VERDEDIGEN OP WOENSDAG 21 FEBRUARI 1979 TE 14.00 UUR

DOOR

DIRK TJASSE BIEWINGA

DOCTORANDUS IN DE GEOFYSICA GEBOREN TE VELSEN

^-^Ci I c£ ö ~ ~ < / ^ ^

1979

DRUKKERIJ J.H. PASMANS, 'S-GRAVENHAGE

BIBLIOTHEEK TU Delft P 1156 5446

1. INTRODUCTION 5 2. THE FUNDAMENTAL RELATION BETWEEN THE MUTUAL

IMPE-DANCE RATIO AND THE LAYER DISTRIBUTION IN THE EARTH 9

2.1 Introduction 9 2.2 The mutual impedance ratio of the horizontal coils system 10

2.3 The mutual impedance ratio for the perpendicular and the vertical coaxial

coils system 11 2.4 The calculation of electromagnetic frequency sounding curves 12

3. MASTER CURVES AND THE INTERPRETATION OF FIELDCURVES 15

3.1 Introduction 15 3.2 The theoretical basis of mastercurves 15

3.3 The inference of layer parameters with the aid of master curves 17 3.4 The influence of variation of resistivity and coil spacing on the response

curve of a homogeneous earth 18 3.5 Master curves with two layers 19 3.6 Master curves with three layers 21

3.7 The interpretation 22 4. INTERPRET ATIONAL PROBLEMS 28

4.1 Introduction 28 4.2 The restrictions of the method itself 28

4.3 Restrictions caused by instrumental facts 36 4.4 The influence of measuring errors and noise 37 5. JOINT INTERPRETATION OF MEASUREMENTS WITH DIFFERENT

COEL SPACINGS AT THE SAME STATIONS 44

5.1 Introduction 44 5.2 Parameter estimation 47 5.3 Improvement of the interpretation accuracy by combined interpretations 48

6. THE FIELD WORK 51 6.1 Introduction 51 6.2 Description of the measurements in the field 57

6.3 The calculation of the mutual impedance ratio from the quantities

measured in the field 58 6.4 The results of the fieldwork 59

7.1 Introduction 63 7.2 The transmitter and receiver 63

8. ALTERNATIVE ELECTROMAGNETIC METHODS, AND

APPLICA-TIONS 70

8.1 Introduction 70 8.2 Comparison of resolution in the interpretation of different

electro-magnetic parameters, for two layer models with P2 > Pi 71

8.3 The transient response 74 8.4 Electromagnetic profiling 75

APPENDK A 77 SUMMARY AND CONCLUSIONS 80

SAMENVATTING EN CONCLUSIES 82 REFERENCES

CHAPTER 1

INTRODUCTION

The electrical resistivity of rocks in the earth's crust depends mainly on four proper-ties: the mineral content, the texture, the porosity, the amount and resistivity of the pore filling material (water, oil, gas). Knowledge of the resistivity in the earth can be of great value in the prospecting for minerals, geothermal reservoirs, structu-res etc. In groundwater surveys, for example, the structu-resistivity of the water bearing rock is controlled principally by its water content, if the resistivity is known it is possible to estimate roughly the water content.

Geothermal reservoirs consist of very porus rocks fully saturated with water at a high temperature because of local heat conduction. The hot water usually dissolves much salt. As a consequence, the resistivity of these geothermal reservoirs is charac-teristically low.

The development of methods to infer the resistivity in the earth from measurements on the surface has greatly aided the study of these phenomena. The Schlumberger method, named after the pioneer Conrad Schlumberger, is most commonly used. This method allows the determination of the resistivity of horizontal layers. Usually the restriction to horizontal layering is of little practical significance because in many geological situations the resistivity layering is approximately horizontal. In the Schlumberger method a DC current is driven through a pair of electrodes A and B, see fig. 1.1.1.

The potential gradient is measured with the closely spaced electrodes M and N. The current I creates a potential difference AV between M and N; a quantity called the apparent resistivity, can then be calculated:

where AB and MN are the distances between the electrodes. For a homogeneous earth the apparent resistivity is equal to the intrisic resistivity. In the multilayer case, the layer resistivities can be derived from the apparent resistivity measured as a function of the electrode distance AB. The penetration depth of the current and hence the influence on the apparent resistivity of the deeper layers, increases with increasing AB (fig. 1.1.1). When plotting apparent resistivity on log/log paper as a function of AB/2, the layer resistivities and thicknesses may be inferred with the aid of indirect interpretation techniques. For accurate measurements, with a good signal to noise ratio, the potential difference between M and N caused by the current 1 should be much greater than the potential difference caused by natural currents.

A M^N B _{P 2 > PT}

• ' ..:'f^W^)-':^ .•^-^•: : M

;•. : - . • • : : :.'••:'. -. - . - . " • > ; ; A,-/-- :•:•':• " -' • ' - * • ' - , ' - • • ' - • . - •

Fig. 1.1.1. With increasing distance between A and B the depth penetration of the current also

increases.

This may create problems for very great values of AB and in areas with a very resis-tant top layer. Because of this resistivity soundings in arid and permafrost areas become doubtful or even impossible. To allow meaningful resistivity soundings to be undertaken in such areas, the use of electromagnetic methods has been considered. The theory of the induction methods was already derived in the sixties - Wait (1954/55/58) - Kozulin (1960/63) - Vanyan (1967) - Keller and Frischknecht (1966) - Dey and Ward (1970) - Ryu, Morrison and Ward (1970), but only few field experiments have been done so far — Plouff (1961) — Ryu, Morrison and Ward (1972).

Plouff, Keller, Frischknecht and Wahl, (1961) successfully measured the thickness of thick ice floating on the Arctic Ocean. This experiment showed clearly the useful-ness of electromagnetic frequency soundings for the investigation of simple problems.

TRANSMITTER COIL RECEIVER COIL

Fig. 1.1.2. Diagram of a coil configuration used for electromagnetic frequency soundings.

The coil configuration "horizontal coplanar" used for electromagnetic frequency soundings is shown in fig. 1.1.2. The electromagnetic field is generated by a trans-mitter coil, carrying an AC current (I e^'^'). The electromagnetic field will be influ-enced by the conducting earth layers, so that the field measured with the receiver coil (H receiver) can be seen as consisting of two components: one component Ho that represents the field that would have been measured in free space, the primary field, and a component H, that represents the influence of the eddy currents, induced in the earth.

The depth penetration of the electromagnetic field depends on the coil spacing and the frequency; because of the skin effect the depth penetration increases with increasing period. For this reason these electromagnetic measurements are named, electromagnetic frequency soundings.

An electromagnetic frequency sounding consists of the measurement of the magnetic field strength at a number of discrete frequencies at the same coil spacing.

The ratio of the measured field strength and the value that would have been measured in free space is the mutual impedance ratio Z.

Z= iHo +H, 1/ IHol (1.1.3)

The value of HQ can be calculted from the dipole moment and the coU spacing. The mutual impedance ratio plotted versus the logarithm of the period is the electromag-netic responsecurve. An example of a curve measured in the field is given in fig. 1.1.3; the field curve is indicated with crosses and the full line denotes the response curve of a well fitting model.

Field curves may be interpreted by comparing them to a response curve that is calcul-ated for a starting model; usually this model is then improved by trial and error methods. This interpretation procedure is called the indirect method.

The purpose of the investigation described in this thesis, was to test the usefulness of electromagnetic frequency soundings in the investigation of multi layer problems in the field; for this purpose measurement equipment and interpretational techniques have been developed.

STATION 106 HOR. COILS COILDISTANCE = 300 m d, = 21 m P, = i.3ohrafn P, = 21.5 + + FIELD CURVE INTERPRETATION 0." O < UJ o z < n LU z <

• /

/ / PERIOD 10" 10 10"' Fig. 1.1.3. The field curve and the response curve of a well fitting model.

In chapter 2 the fundamental relation between the mutual impedance ratio and the layer distribution in the earth is considered, and a description of a computer program for the calculation of response curves is given.

The interpretation of electromagnetic frequency sounding curves is treated in the chapters 3,4, and 5.

In the chapters 6 and 7 the fieldwork and the equipment are described, while in chapter 8 alternative electromagnetic methods and apphcations are discussed. The application of the method is of particular importance in the search for ground-water in arid areas. Because of this the fieldwork was undertaken in southern Tune-sia and was supported by NUFFIC. (Netheriands Universities Foundation For Interna-tional Co-operation).

CHAPTER 2

THE FUNDAMENTAL RELATION BETWEEN THE MUTUAL IMPEDANCE RATIO AND THE LAYER DISTRIBUTION IN THE EARTH

2.1. Introduction

In theoretical work dealing with electromagnetic fields, the transmitter and receiver

coil are usually treated as a dipole. If the separation between the source loop and receiver loop is at least five times the diameter of either loop, both loops may be treated mathematically as dipoles (Wait, 1954). For practical reasons a horizontal transmitter loop was used, while the orientation of the receiver coil could be changed to measure the vertical or the radial component of the magnetic field. These are the horizontal and the perpendicular coils systems respectively.

The fundamental equation for the magnetic field generated by a vertical oscillating magnetic dipole at the surface of a horizontally stratified earth, with homogeneous layers, has been given by Kozuhn (1963). His equations are used, but with the nota-tion of Keller and Frischknecht (1966).

Notation. r = coilspacing X = variable of integrafion dj = thickness of layer i Pi = resistivity of layer i CTj = conductivity of layer i R = Kernel function. 1^0 = Jo = f = CO = T = J =

magnetic permeability in free space Bessel function of order zero frequency

27rf period. imaginary unit. The layering notation is given in fig. (2.1.1.).

surface z-o z - d i Z.d,-Hl2 p„_». air <i2 P> z - d i + d 2 - - - d r j - i " PN-2 PN-1 PN

2.2. The mutal impedance ratio of the horizontal coils system

According to Kozulin (1963) the mutual impedance ratio for the horizontal coils system is given by:

Z i = l - ƒ r^X^ R(X,di,ai,f) Jo(Ar)dX (2.2.1)

x=o

The Kernel function R depends on the layer parameters and the frequency; this function can be obtained by means of a recurrence relation. To this end a double

suffix is attached to R:

R ( > ^ ) = R O , N O ) (2.2.2)

The first suffix relates to the field in free space just above the ground surface, the

second suffix N is equal to the number of subsurface layers. The recurrence equation is:

Ri_ 1 N(X) = -^-^ -'^^-^—TT— (2.2.3) ' l+Vi-,,iRi,N(A)e-''''^*

and the Kernel function of the deepest layer is zero:

RN,N(X) = 0 (2.2.4)

where Vj =\/^^+yf,y^ = j 2nfioaj{ and vj ^ = (vi — Vk)/(vj -i- v^). Evaluating the Kernel function, the mutual impedance ratio can be calculated with the aid of formula (2.2.1). Due to the oscillating character and the slow decay of the Bessel function numerical evaluation of this integral is a rather time consuming task. For this reason alternative methods have been developed. Wait (1955) derived a simple formula for the mutual impedance ratio of a homogeneous earth:

Zi = - T X [ 9 - ( 9 + 9 7 r + 47'r2 + 7 ' r 3 ) e - ^ ^ ] (2.2.5)

y r

where y^ r^ = J/JQ o i wr^ .

Koefoed, Ghosh and Polman (1972) presented a method for the calculation of the mutual impedance ratio of a multilayered earth, by means of a linear filter analogy, details of the method are given in section 2.4.

2.3. The mutual impedance ratio for the perpendicular and the vertical coaxial coils

system

There are four ways in which the transmitter and receiver coils may be oriented with respect to each other, which are commonly used in electromagnetic prospecting (fig. 2.3.1).

Horizontal coplanar

_± ±_

Vertical coplanar

O O

JAW.VVV AW.VW ."W/^AyJ > A^V m

Vertical coaxial Perpendicular

zt ±_ _±

^

Fig. 2.3.1. Coil configurations used for soundings.

The formula for the mutual impedance ratio for the perpendicular coils system is given by Kozuhn (1963).

Z 2 = - ƒ r3X^R(X,di,ai,f) J,(Xr)dX (2.3.1)

where Ji is a Bessel function of the first order, and the other symbols are defined in section 2.1.

The mutual impedance ratio for the vertical coaxial coils system

The formula for the mutual impedance ratio for the vertical coaxial coils system, will be given only for a homogeneous earth (Wait, 1955)

Z4 = [12 +127r-i-57^r^ +7'r^] e l l i + 2 - 12 (2.3.2)

^ 2 ^ 2

T^r^ for the notation reference is made to equation (2.2.5).

2.4. The calculation of electromagnetic frequency sounding curves

In the present work type curves have been obtained numerically, following the

method of Koefoed et al. (1972) and Verma and Koefoed (1973). An outline of this method wül be given in this section. It forms the theoretical basis of a com-puter program for type curves in the horizontal coils system; a listing is given in appendix A.

The method is based on the observation that a linear relation exists between the Kernel function (as a function of the variable of integration), and the magnetic field strength (as a function of the separation between transmitter and receiver coil). This relation is turned into a convolution form by substituting x = ln(r) and y = ln(l/X), in the equations (2.2.1) and (2.3.1),

Z, = 1 - T^ e3(^->')R(y,di,ai,0Jo(e''-'')dy (2.4.1)

Z 2 = - Te^(''-^>R(y,di.ai,f)Ji(e''->')dy (2.4.2)

and the convolution may be interpreted as the action of a hnear filter. The symbol * is used to signify convolution. Let us assume an input g(x) is being convolved in the filter with response h(x). This yields:

f(x) = g(x)*h(x)

f(x) = T g(y) h(x - y) dy

In the Fourtier transformed domain, convolution becomes simple algebraic multi-plication, so that

F(k) = G(k)H(k)

where F(k), G(k) and H(k) are the Fourier transforms of f(x), g(x) and h(x) respec-tively. H(k) can be obtained by a simple devision.

H(k) = F(k)/G(k)

Finally h(x) is calculated from H(k), with the inverse Fourier transform, this is an example of deconvolution.

In this way the mutual impedance ratio can be obtained by subjecting the Kernel function to a linear transformation.

Koefoed et al, (1972) calculated the digital filter coefficients for a sampling distance of ln(10)/10, and the mutual impedance ratio can be calculated by a simple summa-tion, instead of a very time consuming numerical integration.

Z = S Ck f(yk) (2.4.3)

k=0

where Ck denotes the filter coefficients and f(yk) the sampled input function. The sample values yk can be obtained with the formula:

yk=ln(r)-T?o+kln(10)/10. (2.4.4)

where r?o determines the value of the first sample point, k = 0.

In Verma and Koefoed (1973), possibilities for shortening the number of filter coefficients n, are considered. The filter coefficients are considered in conjunction with the values of the input function, or in the range of values which the input func-tion may assume. This filter is then cut off at such a place that the sum of the products of input function and filter coefficients incurs a tolerable error, i.e. at most a few units in the fourth decimal point.

The first filter coefficient suffix that must be used, called KS, depends on the layer parameters and coilspacing. For the horizontal and perpendicular coils system emperical formulas have been derived to calculate KS.

Improvement in shortening the filter in the horizontal coils system was obtained by using the following input function:

Z, = l - /°° e^(''-^^R(y,di,ai,0 e(''~^^Jo(e''~*'^dy (2.4.5)

— oo . .

input function filter function

The value used for TJQ in equation (2.4.4) was 8.75198, for KS the following emperical formula was derived:

KS = entier{4.1hi(1000 d,/r) + ln(10"pminT/r^)/l.lVl + ln(1000 d,/r'} (2.4.6) where p^,,, denotes the minimum resistivity in the layer sequence and di/r should satisfy the condition d, /r > 0.001.

If d, /r> 0.01, equation (2.4.6) may be simpHfied to:

KS=entier{41n(1000d|/r)} (2.4.7) For both formulas there is the condition KS < 22.

Also for the perpendicular coils system improvement in shortening the filter was ob-tained.

In this coils system, the input function was:

Z j = - e 2 ' ' ƒ" e-2>'R(y,di,(7i,0 e<^->'>J,(e'<->')dy (2.4.8)

input function filter function

The value of TJQ in equation (2.4.4) is 7.48268, for KS the following formula was derived:

KS = entier{2.5 ln(10' p^m T/r^)} (2.4.9) where Pn, in is the minimum resistivity in the layer sequence, and for KS there is

the condition KS < 17.

Hence for the mutual impedance ratio in the horizontal coils system:

n

Z , = S Ck f ( y k ) K = Ks

where n = 50 and 0 < KS < 22, Ck denotes the filter coefficients for the horizontal coils system.

For the perpendicular coils system it has been found that

n

Z j = 2 Ck f(yk) K = KS

where n = 40 and 0 < KS < 17, Ck denotes the filter coefficients in the perpendicular coUs system.

A description of a computer program based on the linear filter method is given in Appendix A.

CHAPTER 3

MASTER CURVES AND THE INTERPRETATION OF FIELDCURVES

3.1. Introduction

Response curves measured in the field are often interpreted by comparing them to curves calculated from a layered model. This is a so called indirect method of interpretation.

When a well fitting curve is found, the layer parameters of this model are said to be a solution to the fieldcurve. Fortunately it is not necessary to calculate the curves separately for every fieldcurve, because it is possible to use mastercurves, this is discussed in section 3.2.

3.2. The theoretical basis of mastercurves

In this section it is shown that the mutual impedance ratio depends on the

following parameters: r^ fa^, aja^, dj/r. The response curves of different models will coincide if the values, of the parameters mentioned above, are the same. To handle this problem the formulas for the mutual impedance ratio given in section 2.2, may be applied.

The following substitutions are made in equation (2.2.3)

a[ = ailai d,'= dj/r Vi' = rvi

This yields:

Vi' = r Vi = rVX^ +j27rMoaif'= V(^r)'' +fl-niXoo[ a, fr^' = Vi'(Xr, a[, a, fr^) (3.2.1) r(vi + Vk)

Vi,k = — r = Vi,k r(vi - Vk)

e-2diVi=,-^lL"i^^-2divi

From the last three equations it is concluded that the Kernel function Ro,N =Ro,N(^r,CTÏ'di,a|fr^)

(3.2.2)

and substitution in (2.2.1) gives:

Z, = 1 - / (Xr)^Ro,N(Xr,a;,d;,a,fr^)Jo(Xr)d(Xr)

\ r = 0

where Xr is a variable of integration, which leads to the statement in the beginning of this section, namely that the mutual impedance ratio depends on dJ. o[, a, fr^, and not on the separate values of resistivity, layer thickness and coilspacing. In the special case of a homogeneous earth (where d i = °° and aja-i =1) the mutual impedance ratio depends only on a, fr^. This is in accord with equation (2.2.5) for the mutual impedance ratio of a homogeneous earth derived by Wait (1955).

Plotting response and master curves

For the purpose of comparison to mastercurves, the mastercurves were plotted on semi-log paper, the mutual impedance ratio on linear scale and Tpi /r^ on log scale. The standard values used for the resistivity and coil spacing were 1000 Ohm-m and 1000 m, respectively, so that the mutual impedance ratio was plotted against T.IO"'' The mutual impedance ratio of the field curves was plotted against the logarithm of the period.

The advantage of this representation lies in the fact that response curves for models with different values of d|, Pj and coil spacing, but with the same values for dj/r and Pi/Pi, will have exactly the same shape. The only difference is that the curves are shifted on the period axis, as illustrated in fig. 3.2.1.

I MUTUAL IMPEDANCE RATIO

HOR. COILS , - ' ' " ' . < ^ X , COILSPACING-iOOm / / ^^s., \ ^ 1. - ,1-® / -0) CURVE NO. 1 2 d, 50 . d2 w w d3 • » Pi 50 200 P? 5 20 P3 SO 200 10"' 10"' 10"'

Fig. 3.2.1. The response curves of different models with the same values for dj/r and P{IPt have the same shape, but are shifted on the period axis,d{m), p(ohm-m).

For example the mutual impedance ratio in both points A and B is 0.7, while the periodes are Ti and T i , respectively.

To verify this statement it may be noted that the mutual impedance ratio depends only on P\lp\, dj/r and T/a, r^. However dj{,p;lpi and the mutual impedance

ratio are the same for both curves, so it may be concluded that: TiPi _ TiPf

or log T, = log T2 + log(pi*r|/p, rj) = log Tj + constant.

where pf is the resistivity of the first layer of model two. Accordingly the horizontal distance between any two points of the curves, with the same value for the mutual impedance ratio, is constant. Of course, it must be remembered that a value of the mutual impedance ratio greater than one, will occur at two periods, and in that case only the corresponding periods should be compared. The fact that curves with the same values for dj/r and Pi/Pi have the same shape is a very important aspect of the interpretation with mastercurves, as will be seen in section 3.3.

3.3. The inference of layer parameters with the aid of master curves

The procedure may be illustrated best by the example in fig. 3.3.1. This represents a curve measured in the field, while fig. 3.3.2 gives a set of master curves of two layer models with di /r = 0.07 and a resistivity contrast varying from one to infinity.

MUTUAL IMPEDANCE RATIO

Fig. 33.1. The field curve. Fig. 3.3.2. A set of master curves, the field

The field curve is shifted over the master curves until a well fitting master curve is found. The master curve with P2/P1 =5 has a good fit with the field curve (indicated with crosses in fig. 3.3.2). It may be observed that at the period of 10"^ of the field curve the value of Tpi /r^ of the master curve is 4.8 x 10''. From the similarity of the field and master curves and the results of section 3.2, it is inferred that dj/r, Pi/Pi, and Tpi/r^ are the same.

Hence, from the field curve:

T p , 10"'p, 300^ = 4.8 10"' so pi = 4.3 Ohm-m and pj = 5 p, = 21.5 Ohm-m.

The thickness of the first layer can be found from the ratio djr-curve; d,/300 = 0.07, sod, =21 m.

0.07 of the master

3.4. The influence of variation of resistivity and coil spacing on the response curve

of a homogeneous earth

In section 3.2 it has been shown that the mutual impedance ratio of a homogeneous

earth can be written as a function of Tpi /r^; variation of the resistivity or coil spacing has no influence on the shape of the response curve.

Increasing the resistivity while the coil spacing is constant means that one measures the same value for Z, at a smaller period, as illustrated in fig. 3.4.1.

Similarly increasing the coil spacing will result in a shift of the curve to the greater periods (fig. 3.4.2). Variation of the coil spacing can be used to "bring" the rising flank of a response curve to the frequency range of the instrument. The use of this relation between resistivity, coil spacing, frequency and mutual impedance ratio is referred to later.

MUTUAL IMPEDANCE RATIO MUTUAL IMPEDANCE RATIO

_ HOR. COILS HOMOCENEOUS E A R T H COILSPACING 1000 m

R E S I S T I V I T Y 1000 AND / 100 o h m - m /

1000 ohm. m—Y / 100 ohtT

-^ \ v ^ PERIOD HOR. COILS HOMOGENEOUS EARTH COILSPACING - 300 AND 600 rr RESISTIVITY - 1000 ohm-m

Fig. 3.4.1. Variation of the resistivity causes only a shift of the curve on the period axis.

Fig. 3.4.2. Variation of the coil spacing causes only a shift of the curve on the period a.xis.

3.5. Master curves with two layers

The response curve of any model will have the same general shape: the mutual impe-dance ratio is zero for very short periods; going to the greater periods the curve rises to a maximum and approaches unity for very long periods.

For most models, the maximum of the mutual impedance ratio is between 1.29 and 1.33, low values like 1.22 - 1.29 occur only when there are great resistivity contrasts at a shallow depth. In case of two layers there are two types of curves depending on the resistivity contrast P2/P1, that can be greater or smaller than one.

Two layer master curves with P2 <Pi

In the figs. 3.5.1 and 3.5.2 an example is shown of two layer curves with d, /r = 0.07, respectively 0.45 while the rafio of the resistivities P2/P1 varies from 1 to 0.01. The comparison of the curves of fig. 3.5.1 with the response curve of a homogeneous earth (P2/P1 = 1) reveals that the two layer curves rise a little above the curve of the homongeneous earth at PCI, and at PC2 the curves fall below the homogeneous earth curve. It may be observed that the curvature of the top is less than for the homogeneous earth curve.

MUTUAL IMPEDANCE RATIO I MUTUAL IMPEDANCE RATIO

HOP. COILS TV^O LAYER CURVES d r . .(.5

1. .2 .05 .01

P,T/,;

Fig. 3.5.1. Two layer master curves with d,/r = 0.07 and p, >

p^-Fig. 3.5.2. Two layer master curves with d,/r = 0.45 and p , > p j .

In fig. 3.5.2 the same facts are observed, but the periods PCI and PC2 are much greater. Because of the much larger thickness d, the response curve is now sensitive to the second layer at relatively long periods (due to the skin effect).

Since the depth penetration of the electromagnetic field decreases with decreasing period, for sufficient short periods the mutual impedance ratio is only effected by the first layer. Therefore, the beginning of every master curve must be that of a homogeneous earth and when the period increases the influence of the second layer on the response curve will appear. In the field however, the beginning of the response curve is often influenced already by more than one layer, because of the limit of the highest

fre-quency. One may expect the beginning of the curve to be fitted by a homogeneous earth curve only when the top layer is thick enough. The two intersection points of the two layer curve and the homogeneous earth curve, are called the first critical point PCI, and the second critical point PC2. Because of the inaccuracy of the measurements the first critical point is often undetectable, and one deals only with PC2.

With increasing thickness of the first layer the critical points will move to greater periods.

This was verified in an investigation of the relation between the thickness of the first layer and the critical period. If the resistivity of the second layer was smaller than that of the first layer, the following empirical formula was obtained:

d, = 14.5 • 10' • TPC2 (3.5.1)

where TPC2 denotes the period at the critical point PC2. The accuracy of this formula is 14% when 0.1 < d, /r < 0.7. A similar formula was found for measure-ments in the perpendicular coUs system:

d. 18 . 10^ • TPC2 a, . r when 0.1 < d , / r < 0 . 9 .

It was impossible to find a relation between dj and TPC2 if the resistivity of the second layer was greater than that of the first layer.

Two layer master curves with P2 > P i

In the figs. 3.5.3 and 3.5.4, examples of two layer curves are shown with d, /r • 0.07 and 0.45 respectively, while the resistivity ratio varies from 1 to 1000. MUTUAL IMPEDANCE RATIO MUTUAL IMPEDANCE RATIO

F^. 3.5.3. Two layer master curves with d,/r = 0.07 and Pj > p, .

Fig. 3.5.4. Two layer master curves with d, /r = 0.45 and p , > p, .

In fig. 3.5.3 it may be observed that the beginning of the two layer curve (P2/P1 = 5) falls below the homogeneous earth curve and then rises with a sharp turn above it. The curvature of the top is stronger and the maximum is narrower compared to the homogeneous earth curve; in fig. 3.5.3 one may also observe a decreasing of the maximum value with increasing resistivity contrast.

Comparing the deviations from a homogeneous earth curve for two layer curves with P2/P1 > 1 and withp2/p 1 < 1, there are much larger deviations for the case P2/P1 < I (e.g. the curves with P2/P1 = 0.1 and 10, in figs. 3.5.2 -3.5.4).

This leads to a conclusion that it is easier to detect a sub-surface layer with a relatively low resistivity, than one with a relatively high resistivity.

3.6. Master curves with three layers

Descending or Q type Bowl or H type Bell or K type Ascending or A type

When dealing with three layer models, four types may be distinguished: Pi > P2 > P3 fig- 3.6.1 curve no. 1

Pi >P2 <P3 fig. 3.6.1 curve no. 2 Pi <P2 >P3 fig- 3.6.2 curve no. 1 Pi <P2 <P3 fig. 3.6.2 curve no. 2

The shapes of these curves are characteristically different, and this feature enables us to make inferences on the type of resistivity contrast in the earth to be made. The descending type (fig. 3.6.1, curve no. 1) falls below the homogeneous earth curve (3) so that P2 < Pi; and because the top is very flat there is an indication of a third layer with Pa < P2 . However this possibility should'be considered only when it appears impossible to find a two layer solufion with a good fit.

The Bowl type (fig. 3.6.1, curve 2) also falls below the homogeneous earth curve, so P2 < Pi. However the evidence of a third layer with p-j > p2 is now clear, because the sharp curvature of the top is not possible for a two layer model with P2 < P i .

I MUTUAL IMPEDANCE RATIO 1 MUTUAL IMPEDANCE RATIO

_ HOR. COILS THREE LAYER CURVES COILSPACING- 1000 m ^

«k

, / / /

v//

/ CUfiVE © d CURVE ® d CURVE Q)

'S

@ - d, - tM - a , - 1 5 0 J3)

^S>^

ft-WOO Pj-500 P, - 1000 Pi - 500 P, - « 0 0 P,yr2 Oj P3 _ HOR. COILS THREE LAYER CURVES COILSPACING = 1000 m

-c

<z

X /

f

H

1

1

Ir'

l/h>

1 CURVE 0 a CURVE CD d CUfiVE (3) i 1 ^ . \ - : -d? -150 - a , -150 ^ ^ ' ' = ^ -. = 1000 P;-2000 , -1000 p.yr^ i HX» ,-^

Fig. 3.6.1. Three layer models characteristic for Fig. 3.6.2. Three layer models characteristic for the Q and H type; d(m), p(ohm-m). the K and A type; d(m), p(ohm-m).

The Bell type, (fig. 3.6.2, curve no. 1) rises above the homogeneous earth curve (3), so P2 > Pi. A third layer with p^ < P2 is required because a two layer model with P2 > Pi has a sharp or normal curvature of the top rather than the flat top of curve no. 1. The Ascending type (fig. 3.6.2, curve 2) rises with a sharp turn in the beginning of the rising flank above the homogeneous earth curve, so P2 > Pi. There is no clear indication of a third layer, so that as with de descending type, it should be introduced only when a fit of two layer curves fails.

3.7. The interpretation

Insection 3.5 and 3.6 it has been shown that characteristics of the shape of a response curve are discriminative with respect to the resistivity contrasts to be expected in the earth. This fact is very important in the interpretation procedure which may be summarized as follows:

1 — compare the curve with a homogeneous earth curve; if the curve rises above the homogeneous earth curve there must be a second layer with a higher resistivity and in the opposite case the second layer has a lower resistivity.

2 — try to find a two layer model that fits the whole curve; if this is not possible it must be decided which type of three layer model (Q, A, K, or H) is appropriate to the whole curve. As discussed in section 3.6, the curvature of the top of the response curve can be an indication of the resistivity contrast between the second and third layer.

3 — try to find a well fitting curve of the selected three layer model.

Of course it is possible to continue with a fourth layer, but resolution of more than three layers was not warranted from the measurements taken, which had an accuracy of about 3% and a frequency range of two decades.

The interpretation of field curves was done in two ways.

(i) By comparing the field curves with master curves, that were plotted as a function of Tp, /r^ (with Pi =1000 Ohm-m and r = 1000 m ) The estimation of the layer parameters from the parameters of the fitting master curve, has already been dis-cussed in section 3.3.

(ii) By comparing the field curves with curves calculated for the coil spacing used, with the aid of a Raytheon computer and a graphical display. The procedure used was as follows:

the computer was fed with the data of a measurement, to make a plot of the field curve on the screen of the graphical display. The computer was then fed with the coil spacing and the parameters of a trial model. The calculated mutual impedance ratio could be compared to the data on the screen. By trial and error the model was im-proved until a good fit to the field data was obtained.

Totally two curves could be drawn on the screen, a dashed line that indicates the former model and the full line indicates the last model.

With the aid of a hardcopy device, copies of the curves on the screen could be made. An advantage of the method described is that it is not restricted by the number of master curves.

The interpretation of field curves in practice.

Several practical limitations require some modifications to the procedure, discussed above: The frequency range of the instrument used was two decades, while a response curve from the beginning of the rising flank up to the asymptotical value covered at least three decades.

To solve this problem two soundings with different coil spacings at a station were always made. The greatest coil spacing was able to measure the beginning of the rising flank of the response curve, while the smallest coil spacing was able to measure the other part of the curve (figs. 3.7.1 and 3.7.2).

STATION 62 HOR. COILS COILDISTANCE - 3CW m d , - U m p, - 52 o h m . P. - 15.6 „ / / - /'/ / PERIOD HOR. COILS CQILDPSTANCE - 2 0 0 m - J i m ft- S l o h m - f i . P . - 15.6 FIELD CURVE INTERPRETATION /

1

1 PERIOD

Fig. 3.7.1. The field curve of station 62, with

a coilspacing of 300 meter. Fig. 3.7.2. The field curve of station 62, with a couspacing of 200 meter.

Increasing the coil spacing has a three fold influence:

a — the response curve shifts to the greater periods, because when r increases the period T must increase too in order to keep Tp ,/r^ constant, reference is made to fig. 3.4.2. This variation of coil spacing was used to bring the beginning of the rising flank in the frequency range of the instrument.

b — the ratio of layer thickness to coil spacing, dj/r, becomes smaller; which may obscure the presence of thin layers.

c — the depth penetration of the electromagnetic field increases, so it may be pos-sible that the curve measured at the short coU spacing needs less layers for a well fitting solution than the curve measured at the great coil spacing.

Apart from the number of layers, solutions for the different spacings may show differences in the layer parameter values due to other reasons, such as lateral varia-tions in the earth (and there is also the general problem of non-uniqueness). For a good interpretation a model must be found that fits both the field curves. This way of joint interpretation improves the reliability of the model, in chapter five this subject wül be treated in more detail.

Examples of interpretation with two layers.

These examples are the results of measurements made during the field work in southern Tunesia.

Station 62; reference is made to the figs. 3.7.1 and 3.7.2, the field curves are indi-cated with crosses. If the field curve of fig. 3.7.1 is compared with the dashed homo-geneous earth curve it may be concluded that there must be a second layer with a lower resistivity. With the aid of master curves a good fitting solution to both the coil spacings could be fotind.

Station 30.

The field curves of station 30 are shown in the figs. 3.7.3 and 3.7.4; a two layer model was able to fit both measurements.

IMUTLIAL IMPEDANCE RATIO HOR. COILS STATION 30 COILSPACING-300m 1.

-10-' -10-' -10-' "• t o - ' 10"= 10"'

Fig. 3.7.3. The field curve of station 30, with Fig. 3.7.4. The field curve of station 30, with

a coil spacing of 300 meter. a coil spacing of 200 meter.

In fig. 3.7.3 the field curve rises above the homogeneous earth curve, so it may be concluded that the second layer has a greater resistivity. The parameters of the model, that gave response curves with a good fit, are indicated in the figures.

Examples of interpretation with three layers

The Bowl or H type (station 36)

The field curves of station 36 are shown in the figs. 3.7.5 and 3.7.6.

In fig. 3.7.6 it is observed that the first part of the rising flank falls below the homo-geneous earth curve (dashed line), so that Pi > P2. The sharp curvature of the top is a clear indication of a third layer with pj > P2 . With the aid of master curves and trial and error methods a good fit was found for both the coilspacings:

d| = 2 0 m p, = 28 Ohm-m d 2 = 4 0 m P2 = 14 Ohm-m dj = =0 m P3 = 140 Ohm-m

MUTUAL IMPEDANCE RATIO

Fig. 3.7.5. The field curve of station 36, with

a coil spacing of 200 meter.

[MUTUAL IMPEDANCE RATIO HOR, COILS STATION 36 COILSPACING-300 d (ml p (ohm-m) 20 28 W U oo uo 10-' Fig. 3.7.6. The field curve of station 36, with

a COÜ spacing of 300 meter.

The descending or Q type (station 27)

The field curves of station 27 are shown in the figs. 3.7.7 and 3.7.8.

Comparing the field curve of fig. 3.7.7 with the homogeneous earth curve it can be seen that the resistivity of the second layer must be smaller than that of the first layer.

MUTUAL IMPEDANCE RATIO MUTUAL IMPEDANCE RATIO

HOR. COILS STATION 27 COILSPACING-300 m / 1 1 / 1 „ ' -/ -/ / PERIOC

'?

dim 21 57 op r — • p lohm-m) 19 I i 3LS HOR. COILS STATION 27 COILSPACING-200r -- " " " ^ 1 / 1 If 1

$1

^,

/ V

f

" > ^ - i - - < ^^^ d i m l p 21 57 00 PERIOD 1 ~~"^ (ohm-ml 19 14 34

Fig. 3.7.7. The field curve of station 27, with

a coil spacing of 300 m.

Fig. 3.7.8. The field curve of station 27, with

a coil spacing of 200 m.

It was not possible to find a well fitting two layer solution, so a third layer was introduced. A model that gave response curves, which fitted the field curves, was the following: d, = 21m d2 = 57m d3 = <»m Pi = 19 Ohm-m P2 = 14 Ohm-m P3 = 3.8 Ohm-m

The Ascending or A type (station 26)

The field curves of station 26 are shown in the figs. 3.7.9 and 3.7.10.

MUTUAL IMPEDANCE RATIO [MUTUAL IMPEDANCE RATIO

HOR. COILS STATION 26 COILSPACING-300m ,HOM. EARTH d (ml p (ohm-m 1 30 15 45 30 oo 75

Fig. 3.7.9. The field curve of station 26, with a Fig. 3.7.10. The field curve of station 26, coil spacing of 200 m. with a coil spacing of 300 m.

In fig. 3.7.10 it can be seen that the field curve rises with a sharp turn above the homogeneous earth curve, so it is concluded that P2 > P i . It was impossible to find a well fitting two layer curve, so a third layer was introduced; the sharp curva-ture of the top of the field curve indicated that the resistivity of the third layer should be greater than that of the second layer.

The model that gave well fitting response curves for both the coil spacings was the following: di = 30 m d2 = 45 m ds = °°m Pi = 15 Ohm-m P2 = 30 Ohm-m P3 = 75 Ohm-m The Bell or K type (station 3 1)

The field curves of station 31 are shown in the figs. 3.7.11 and3.7.12.

[MUTUAL IMPEDANCE RATIO MUTUAL IMPEDANCE RATIO

HOR. COILS STATION 31 COILSPACING ^ 300 m /

1

1

/ " " " " ^ / d ( m l 5 60 oo PERIOD < P " (ohm- m 1 16 130 24 HOR. COILS STATION 31 COILSPACING = 150 rr

Fig. 3.7.11. The field curve of station 3 1 , with a coil spacing of 300 meter.

Fig. 3.7.12. The field curve of station 31, with a coil spacing of 150 meter.

Figure 3.7.11 shows that even the coU spacing of 300 meter was not great enough to bring the beginning of the rising flank in the frequency range of the instrument. For this reason a comparison with a homogeneous earth curve was not very helpful. Comparing the field curves with each other, shows that the slope of the rising flank is steep for the 150 m curve, but much less steep for the measurement of 300 m. For the steep flank of the 150 m curve a resistivity contrast with P2/P1 > 1 is deduced while the flank of the 300 m curve indicates a resistivity contrast smaller than one.

This is not a contradiction because the depth penetration at 300 m. coil spacing is twice that of 150 meter, so different layers are involved.

Because of this a model with pi <P2> P3 was tried; by trial and error methods two, nearly the same, solutions were found. Only the resistivity of the first layer was different, which may have been caused by lateral inhomogeneities.

The layer parameters for The layer parameters for a coil spacing of 150 m. a coil spacing of 300 m.

d| = 5 m Pi = 10 Ohm-m dj = 5 m pi = 16 Ohm-m d2 =60m P2 =130OhiTi-m dz = 60 m p2 =130 Ohm-m da = °° m P3 = 24 Ohm-m da = o°m P3 = 24 Ohm-m

CHAPTER 4

E^ERPRETATIONAL PROBLEMS

4.1. Introduction

The interpretation of electromagnetic frequency sounding curves is sometimes troublesome and the reasons for these difficulties are of different natures.

The method itself has its restrictions, because the interpretation is based on horizon-tal and homogeneous layers, and the depth penetration is limited.

Moreover, models with different layer parameters may have nearly the same response curves; this problem of non-uniqueness is also known in DC resistivity soundings where two types, equivalence and suppression are distinguished.

The limitations of the instrument, the dipole moment and the frequency range, may prevent the measurement of the important part of the response curve at the most suitable coil spacing.

Measurement errors and noise will influence the results; sometimes the response curve can be distorted in such a way that interpretation becomes impossible. These problems will be investigated in this chapter.

4.2. The restrictions of the method itself

The depth penetration of electromagnetic frequency soundings. An important

ques-tion is how deep a layer may be situated if it is still to have a measurable effect on the shape of the electromagnetic response curve. This is not an easy question since a number of different factors like the resistivity contrast and measurement accuracy are involved. For reasons of simplification the investigation was restricted to two layer model.^with a resistivity contrast of a factor one hundred.

To find the depth penetration, the values of di /r, for which there was still a clear difference between the curves of the two layer models and the homogeneous earth curve, were investigated.

As expected, fig.4.2.1 (P2/P1 =0.01) shows that the difference between the two layer curves and the homogeneous earth decreases with increasing value of d, /r, but even for dJr = 0.7 the difference with the homogeneous earth curve is still visible. It may be seen that the rising flanks of both curves coincide, but before their maxi-mum, they start to deviate from each other. Curves for models with di /r values greater than 0.7 will show only small differences with the homogeneous earth curve, so that they are not detectable with equipment that measures with an accuracy of a few percent. The conclusion is that the depth penetration for two layer curves with P2/P1 = 0.01 is a little better than 0.7 of the coil spacing.

[MUTUAL IMPEDANCE RATIO HOR. COILS COILSPACING -1000m /

~ 1

1

/ " " ^

^—'T~^^^'^^~-~-^ ^ "" CIWVE no d, p, d j D, O » «00 -(7) 7x woo " w (3) ur WD » o [ PERIOD 1

Fig. 4.2.1. Up to d, /r = 0.7 there is a clear

difference between the two layer curve, with

Pi/p, =0.01 and the homogeneous earth

curve; d(m), p(ohm-m).

MUTUAL IMPEDANCE RATIO

COILSPACING - 1000 m CURVE "o d, p, d , p, 0 im n x <B naooo (?) un m» <B max) 10''

Fig. 4.2.2. For d, /r = 0.7 there is only a small

difference between the two layer curve, with /3,/p, = 100 and the homogeneous earth curve; d(m), p(ohm-m).

In the field however, conditions are less favourable; here several layers are dealt with, so the difference between three and two layer response curves has to be exam-ined. Also the limited frequency range and the measurement errors (reference is made to section 4.3) have a negative influence on the resolution or, in other words the possibility of distinguishing layers from each other.

Experience from the survey in Tunesia indicated that layers with a resistivity contrast ( p j / p i ) smaller than one at a depth greater than d | /r = 0.6 could not be resolved. The depth penetration for a resistivity contrast (P2/P1) that is greater than one is less favourable, fig. 4.2.2 shows that the difference between the homogeneous earth curve and the two layer curves is much smaller than in fig. 4.2.1.

The maximum value of d, /r at which there is stUl a clear difference between the two layer curve and the homogeneous earth curve is about 0.6. In the field the maximum depth t o find a resistant layer was about dj /r = 0.4.

Equivalence and suppression

These two phenomena are known from the normal Schlumberger resistivity soun-dings; sometimes it is very difficult to determine the characteristics of layers whose thicknesses are small compared to their depths.

The principle of equivalence concerns a layer whose resistivity is either greater than, or less than, those of the beds above and below itself (Bell and Bowl type). It is found that a resistant bed between two more conductive beds, manifests itself mostly by its "transverse resistance" (resistivity x thickness); on the other hand, a conductive bed between two more resistant beds, shows essentially its "horizontal conductance" (thickness/resistivity). In other words it will be difficult, if not im-possible, to distinguish between two more resistant beds of different thickness and resistivity if the product of thickness and resistivity is the same.

APPARENT RESISTIVITY MUTUAL IMPEDANCE RATIO

Fig. 4.2.3. The Schlumberger curve for the equivalence of the "transverse resistance" of the second layer.

Curve no. 1 2 3 4 d,(m) 10 10 10 10 res, (ohm-m) 100 100 100 100 d,(m) 5 10 20 40 resj (ohn 4000 2000 1000 500

Fig. 4.2.4. The mutal impedance ratio curve for the equivalence of the "transverse resis-tance" of the second layer.

djtm) ^es3(ohm^^l) 1 1 1 1 The product of dj x resj = 20000. for each model.

resistivity soundings, may also produce electromagnetic response curves that nearly coincide is examined. Consider first the case of "transverse resistance". For this pur-pose, both the apparent resistivity and the mutual impedance ratio are calculated for three layer models, which were examples of equivalence for the transverse resistance (^2 X P2) of the second layer, at Schlumberger soundings.

It appears that, while the apparent resistivity curves nearly coincide (fig. 4.2.3) the electromagnetic response curves show great differences see fig. 4.2.4. With layer parameters of the Bell type there is no clear equivalence for the "transverse resis-tance" in electromagnetic frequency soundings.

The second case of equivalence, the "horizontal conductivity", may be investigated in the same way (figs. 4.2.5 and 4.2.6).

Here also the electromagnetic curves show only small differences; the greatest differ-ences occur in the beginning of the rising flank. So the equivalence of the "horizontal conductivity" also occurs with electromagnetic frequency soundings.

APPARENT RESISTIVITY

MUTUAL IMPEDANCE RATIO

CURVE no 0) (0 (=) _® d , | m ) 10 10 10 10 5 10 20 U p , ( o h m - f r WO too 100 100 - i ^ p , 25 1 0 ' 5 i o ' 10 1 0 * 20 1 0 '

Fig. 4.2.5. The Schlumberger curve for the equivalence of the "horizontal conductivity" of the second layer.

Curve no. 1 2 3 4 d,(m) 10 10 10 10 res, (ohm-m) 100 100 100 100 '2 On) 5 10 20 40

Fig. 4.2.6. The mutual impedance ratio curve, for the equivalence of the "horizontal con-ductivity", of the second layer.

resj (ohm i n ) 2.5 5 10 20

datm) res, (ohm-m) 10'

10' 10' 10»

The product of d, /reSj = 2 for each model.

Suppression

The principle of suppression relates to those beds whose resistivities are intermediate between the resistivities of the enclosing beds; descending and ascending types. Such beds will cause hardly any differnce with the corresponding two layer case; when the thickness of the intermediate bed begins to grow, the bed will begin to effect the response curve, but before the bed itself can be identified the effect remains at first undistinguishable from that due to a change in thickness of the enclosing beds. For the descending type this problem was investigated with the following models; curve no. 1 2 d i ( m ) 10 10 res I (ohm-m) 1000 1000 d2(m) OO 10 res2 (ohm-m) ds (m) 1 50 ress (ohm-m) 1 An example of suppression is given in fig. 4.2.7, where the second layer causes only a small difference between the curves two and one. The electromagnetic response curves of the same models are shown in fig. 4.2.8, in which very great differences between the curves in the beginning of the rising may be seen. So there is no strong suppression for these models.

I APPARENT RESISTIVITY 1000

MUTUAl IMPEDANCE RATIO

Fig. 4.2.7. The Schlumberger curves to show the Fig. 4.2.8. The electromagnetic response curves to suppression for descending type; d(m), p(ohm-m). investigate suppression for the descending type;

d(m), p(ohm-m).

Suppression for ascending type curves, is investigated with the following models, curve n o . d i ( m ) r e s , ( o h m - m ) d 2 ( m ) res2(ohm-m) d 3 ( m ) res3(ohm-m)

1 10 10 <=" 10*

2 10 10 10 200 <» 10*

In figs. 4.2.9 and 4.2.10 it may be seen that the curves nearly coincide for b o t h m e t h o d s . It thus appears that suppression for ascending type curves also exists for electromagnetic frequency soundings and that suppression is very weak for descending t y p e curves.

APPARENT RESISTIVITY MUTUAL IMPEDANCE RATIO

SCHLUM6ERGER SOJNOIN'

A S / ;

Fig. 4.2.9. The Schlumt)erger curves, to show suppression for the ascending type; d(m), p(ohm- m). _ HOR. COILS COILSPACING - 2 0 0 m

, y

0-6 10"5 1

s y ^

/ CUBVE / / "' \ (') © 0-3 d, a, dj 10 o " e -PERIOD 1 10-P, 10 to I P J xc to* PJ 10 ' 10-'

Fig. 4.2.10. The electromagnetic response curves, to show suppression for the ascending type; d(m),p(ohm-m).

The influence of a very conductive surface layer

To investigate this problem a homogeneous earth curve may be compared with that of the same earth with a thin surface layer. The homogeneousearth, (fig. 4.2.11 curve l),has a resistivity of 1000 Ohm-m and the curves 2 and 3 are the response curves of two layer models with a good conducting surface layer of 1 and 10 Ohm-m respectively.

I MUTUAL IMPEDANCE RATIO [MUTUAL IMPEDANCE RATIO

HOR. COILS COILSPACING - 1000 m

Fig. 4.2.11. The effect of agood conducting

surface layer; d(m), p(ohm-m).

Fig. 4.2.12. Equivalence of two layer models

with the same "horizontal conductivity" for the surface layer; d(m), p(ohm-m).

The effect of the surface layer is a shift of the whole curve by about one decade to the greater periods and also the slope of the rising flank is a httle steeper.

Curves 2 and 3 clearly coindice; this is described as the equivalence of good con-ducting surface layers: when the ratio of dj /p, is the same for both models the curves will coincide. Another example of this electromagnetic equivalence of good conducting surface layers is given in fig. 4.2.12, where di/pj =3/1 = 30/10. It is obvious that this electromagnetic equivalence of good conducting surface layers may cause problems in the interpretation.

IMUTUAL IMPEDANCE RATIO HOR. COILS COILSPACING - 1 0 0 0 m -1

i

®/k

Jm-— - ^ i : ^ ,

1 CURVE 0 d, - « m / CURVE ( T ) di - n m r d) - » CUBVE 0 d, -50 m D 1 PERIOD 1 P, . » ' ohm-m p , - 1 0 * ohBi-m P , - I 0 ' p , - ) 0 ' oNn-m P , - « '

The influence of a resistant surface layer

To investigate the influence of a very resistant surface layer, a homogeneous earth curve may be compared with that of the same earth with a very resistant surface layer.

Fig. 4.2.13 shows that.a surface layer of 10 meter (di /r = 0.01) and with a resistivity contrast of a factor 10^, hardly causes any difference with the homogeneous earth curve.

When the thickness of the surface layer increases greater differences appear with the homogeneous earth curve, as shown in fig. 4.2.13 curve 3.

The conclusion is that a resistant surface layer, with a thickness of a few percent of the coilspacing, will not be resolved. The resistivity of the first layer that will be inferred from interpretation corresponds to the resistivity of the second layer of the real model.

Electromagnetic frequency soundings are not sensitive to thin layers

This problem was manifest when interpreting electromagnetic response curves of known earth models.

It was possible to find good fitting solutions with two or three layers while the real model had many more layers. For example, fig. 4.2.14 gives the result of the inter-pretation as a two layer model.

The resistivity of the interpreted layer is an average of the resistivities of the four layers of the real model and the thickness is about the sum of the four thicknesses. Fig. 4.2.15 shows the Schlumberger curves of the five layer model and the two layer interpretation model the great differences between these two curves is caused by the resistant layers of 135 and 65 Ohm-m, which were not "seen" in the electromagnetic method.

An other example is a four layer model which is interpreted as a three layer solution, see fig. 4.2.16.

Fig. 4.2.17 shows the Schlumberger curves for the three and four layer model, again great differences between the apparent resistivity curves may be seen.

The reason that the interpretation of electromagnetic soundings is limited to about three layers is rather complicated; the following comments relate to this problem. 1 — the accuracy of field observations is about three percent, so small differences between electromagnetic response curves are not significant.

2 — the ratio of the thickness to the coil spacing dj/r should be greater than a certain minimum for the response curves to be different. To cause a clear difference between a homogeneous earth curve (p = 1000 ohm-m) and the response curve of a two layer model (pj/pi = 2 or 0.5, while P2 = 1000 ohm-m), the value of di must be greater than 5% of the coil spacing. For models with more layers this ratio will increase for the deeper layers, depending also on the resistivity contrasts. Bearing in mind that the depth penetration is about half the coil spacing, one may not expect to find many more than four layers. A possibUity to find more layers with electromagnetic frequency soundings is to perform measurements with short and greater coil spacings.

I MUTUAL IMPEDANCE RATIO HOR. COILS COILSPACING - 2 5 0 m 1

r

ORIGINU. MOOtL Loyw d p I 2.5 m r la as 3 a t 13 u n H S • 9 PERIOD 1 RESULT OF INTEHPnETATlDN Loy»r d p 1 u j g 7 - as I APPARENT RESISTIVITY « 0 0 - : ® . , . u . S C H L U H b £ R C € R SOUtlDING A B / 2

Fig. 4.2.14. The result of the interpretation

of a five layer model is a two layer model; d(m),p(ohm-m).

Fig. 4.2.15. The Schlumberger curves of the

five and two layer models of figure 4.2.14.

[MUTUAL IMPEDANCE RATIO HOR. COILS COILSPACING-250 m b R I O I N U . MOOEL L u , * d p 1 7.J SS Ï U ) 13 3 U . U t - 18 fttSULTOF iWTHWETATTW i j y w a p 1. 11 50 55 « IS I APPARENT RESISTIVITY 1000 -- <^ 1 CUBVE 0 a , - * m d l - 5 0 CURVE @ d , - 7,2 m d i - t d i - S Ï S C H L U H K R O E R SOUNDING 1 1 " / ^ P , - 1 1 o H m - m P , - 5 5 P , - « P , - W o h m - m P , - 1 J P,-« 1 10-'

Fig. 4.2.16 The result of the interpretation of Fig. 4.2.17. The Schlumberger curves of the

a four layer model is that of a three layer model; four and three layer models of figure 4.2.16.

d(m),p(ohm4Ti).

A short coil spacing implies a small depth penetration and as a consequence less layers that have influence on the response curve. So a possibility to find solufions with more than three layers is to start the interpretation with the response curve of the shortest coil spacing. To the response curves of the greater coil spacings one may try to fit models by adding new layers to the solution of the shortest coil spacing. Another possibility to improve the interpretation is to use both the electro-magnetic and the Schlumberger soundings. The improvement is not only caused by the more sensitive Schlumberger method,but also by the fact that the phenomena of equivalence and suppression are rather different for the two methods, see table 4.2.1.

PROBLEM

Equivalence Bell type Equivalence Bowl type Suppression descending type Suppression ascending type Equivalence for a thin good

conduc-ting surface layer with dj /pi = constant

Not sensitive to a resistant surface layer Resolving power Table 4.2.1 EM. not clear yes not clear yes yes yes moderate METHOD Schlumberger. yes yes yes yes no no fair

This procedure will cause great improvement and should be important in areas where no borehole data is available.

4.3. Restrictions caused by instrumental facts

The interpretation of field curves is sometimes made more difficult by measurement errors and limitations of the instrument, such as the maximum possible dipole moment and the frequency range. Most instruments range through a frequency range of only two or three decades, where as four decades is needed to measure a field curve from the beginning of the rising flank up to the point where the mutual impe-dance ratio asymptotically approaches unity. Another restriction is imposed by the highest frequency of the instrument, because the smallest coil spacing at which a field curve can be measured, from the beginning of the rising flank (* Zi = 0.03), is dependent on the highest frequency of the instrument and the parameters of the earth layers.

In case of a homogeneous earth the shortest coil spacing can be calculated by means of a simple formula (equation 2.2.5).

The mutual impedance has the value Z, = 0.03 when 1/(CTI r^ f) = 1.35 • 10"*, so

the coil spacing can be calculated by substitution of the conductivity and the highest frequency.

This equation can also be used to estimate the shortest coil spacing for a multi layer problem, but then a resistivity has to be substituted that is an average of the resistiv-ities of the shallow layers.

The other restriction on the coil spacing is the desired depth penetration, which has been estimated to be about half the coil spacing. In order to obtain a field curve with the desired depth penetration a coil spacing about twice the penetration depth

should be chosen, but the spacing has also to be selected to bring the important part of the response curve inside the frequency range of the instrument.

In order to avoid a conflict between these two recommended coil spacings a very broad frequency range is desirable, but may not be realized because of electronical restrictions.

The disadvantage of a frequency range of two decades has already been mentioned in chapter three; it is absolutely necessary to extend the frequency range to both higher and lower frequencies.

Deviations of the dipole moment, and the amplification factors of the receiver system

Deviations from the desired dipole moment of the transmitter and deviations from the amplification factors of the receiver system, will cause systematical measurement errors in the mutual impedance ratio. These effects may be corrected by calibration at every frequency. The correction factors can be obtained by performing measure-ments above a resistant earth and with a short coU spacing, because in that case the part of the curve where it approaches unity is measured. Correcting factors are thus obtained from measured deviations from unity. This is done for every frequency of the instrument.

4.4. The influence of measuring errors and noise

Errors in the coil spacing, the orientation of the coils and the slope of the terrain can distort the field curve in such a way that interpretation based on master curves for horizontal and homogeneous layers, becomes impossible.

To investigate this problem the deviations in the mutual impedance of a homogeneous earth caused by misorientation of the transmitter coil, a small terrain slope and an error in the coil spacing have been calculated.

Errors caused by misorientation of the coils and slope of the terrain

These errors have been investigated by Verma (1973). If the assumption is made that the receiver coil has no misorientation error then the following formula is valid for the error in Z] (notation see fig. 4.4.1 and section 2.3)

AZ,/Z, = — 1/2 sin^0r - 1/2 sin^0y + sin a(sin 0r — sin a) +

Z2/Z1 sin 0r + 2.Z4/Z1 sin a(sin 0r - sin a) (4.4.1)

The values of Z2/Z, and Z4/Z, depend on the layers of the model and the coilspacing. These values have been estimated, in the case of a homogeneous half space, as a function o f Z, (table 4.4.1).

TRANSMITTERCOIL

RECEIVERCOIL

a TERRAIN SLOPE ( Z - R PLANEl

0r a -f Or

P TERRAIN SLOPE l Y - Z PLANEl

Oy SAME DEFINITION AS ^ r . BUT IN [ Y - Z PLANEl

0y 3 t »,

Fig. 4.4.1. Transmitter-receiver orientation along sloping ground for the horizontal coils system.

TABLE 4.4.1. An estimation of the values Z4/Z1 and Z^/Zi.

z,

Z4/Z, Z2/Z, 0.0 0 0 -0.05 40 6 0.1 20 5 0.2 10 4 0.4 5 3 0.6 3.5 2 0.8 2.5 1.5 1.0 1.7 1.0 1.2 1.2 1.0 1.33 0.8 0.7 1.2 0.8 0.4 1.1 0.9 0.3 1.05 0.9 0.1 1.00 1. 0.0

The following errors in the orientation angles have been investigated. case 1 Vr = —3 a= 3 0^ = 0y = 0 case 2 case 3 case 4 ^ r = VT = Vt = 3 0 - 2 a = ot = a = 0 3 " 3 0r = 0y = 0r = 0y = 0r = 0y = 3 3 - 5

Corresponding errors in Z, are given in table 4.4.2, for reference the following nota-tion has been used:

Error part A = — 1/2 sin^ 0r — 1/2 sin^ 0y -1- sin a(sin 0^ - sin a) Error part B = Z2/Z, sin 0^

Error part C = 2 Z4/Z1 sin a(sin 0r — sin a).

From table 4.4.2 it follows that parts B and C have more influence than part A. The influence of the terrain slope (case 1) is less important than a misorientation of a coil (case 2).

The errors, caused by misorientation of a coil, in the mutual impedance ratio are troublesome for the interpretation. In the interpretation by curve fitting errors of a

z,

0.0 0.05 0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.32 1.2 1.1 1.05 1.00 Error part % A B 0 3 -„ — -„ — „ -„ _ -„ — _ „ — „ -„ — -C - 2 3 - 1 1 - 6 - 3 - 1 . 5 - 1 3 - 1 - 0 . 6 - 0 . 5 - 0 . 5 - 0 . 5 - 0 . 5 - 0 . 5 - 0 . 5 0y=O° AZi% - 2 3 3 - 1 1 3 - 6 3 - 3 3 - 1 . 8 - 1 . 6 - 1 3 - 0 . 9 - 0 . 8 - 0 . 8 - 0 . 8 - 0 . 8 - 0 . 8 - 0 . 8 AZ, _ -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 Error part % A B C 0 3 -+31 26 20 15 10 „ 8 5.2 5.2 3.6 2.1 1.5 0.5 0.0 -0 y = 3 AZ,% _ 3 0 7 2 5 7 197 14.7 9 7 7.7 4.9 4.9 3 3 1.8 1.2 0 3 - 0 3 0 AZ, _ .02 .03 .04 .06 .06 .06 .05 .05 .04 .02 .01 .00 .00 Error part % A B C 0 3 31% -26 20 15 10 „ 8 5.2 5.2 3.6 -2.1 1.5 -0.5 0.0 -* . = 3 ° AZ,% AZ, _ _ 3 0 7 0.02 2 5 7 0.03 19.7 0.04 14.7 0.06 9.7 0.06 7 7 0.06 4.9 0.05 4.9 0.06 3 3 0.04 1.8 0.02 1.2 0.01 0 3 0.00 - 0 3 - 0 . 0 0 Error part % A B 0 . 6 -- 5 2 - 4 3 , - 3 6 - 2 6 - 1 7 - 1 3 - 9 - 9 - 6 - 3 . 4 - 2 . 6 - 0 . 1 0.0 C -, +14.5 7.2 3.6 1.8 1.2 1 0.6 0.4 0 3 0 3 0 3 0 3 0 . 3 l * y = - 5 AZi% _ - 3 8 . 1 - 3 6 . 4 - 3 2 . 0 - 2 4 . 8 - 1 6 . 4 - 1 2 . 6 - 9 - 9 . 2 - 6 3 - 3 . 7 - 2 . 9 - 0 . 4 - 0 . 3 O AZ, _ - . 0 2 - . 0 4 - . 0 6 - . 1 - . 1 - . 1 - . 0 9 - . 1 1 - . 0 8 - . 0 4 - . 0 3 - . 0 0 - . 0 0

few % wül cause problems, especially for greater values of the mutual impedance ratio (that is for the great values of the absolute error). In fig. 4.4.2 the value of the mutual impedance ratio is plotted with the errors of the cases 3 and 4. The main effects are 1) a shift of the rising flank without much change in shape and 2) a devia-tion of the maximum of the curve from the true value 1.33.

I MUTUAL IMPEDANCE RATIO

10'' lO"'- 10"' 10'' 10"'

Fig. 4.4.2. The influence of the errors of the cases 3 and 4 on the mutual inpedance ratio of a

homogeneous earth.

In case 3 interpretation by curve fitting, based only on the rising flank, is possible and will cause a small error in the resistivity of the halfspace; fitting the whole curve is impossible, because there is no maximum greater than 1.33 on any curve. In case 4 curve fitting, based only on the rising flank, will give a misleading solution with two layers and it is impossible to find a solution that fits the whole curve. The effect of procentual errors, that are about the same for every value of the mutual impedance ratio, can be compensated by plotting the mutual impedance ratio on log scale. Because of the great differences between the errors, this is not very helpfuU in cases 3 and 4. These facts lead to the conclusion that interpretation by curve fitting of response curves with errors greater than 1% becomes doubtful. Also when the inter-pretation is based on the minimalisation of the average procentual error, the measur-ing errors cause severe problems for the interpretation.

Errors in the coil spacing

The mutual impedance ratio is obtained by dividing the measured magnetic field strength by the value of the field strength in free space, HQ . the latter is calculated by: