384

• Physics of the Earth and Planetary Inferiors, 53 (1989) 384-404

Elsevier SciencePublishers B.V.,Amsterdam - Printed in The Netherlands

A rapid inversion technique for transient electromagnetic soundings

Perry A. Eaton and Gerald W. Hohmann

Department of Geology and Geophysics, University 0/Utah, Salt Lake City, UT 84112 (U.S.A.)

(Received January 5, 1987; revision accepted August 20, 1987)

Eaton, P.A. and Hohmann, G.W., 1989. A rapid inversion technique for transient electromagnetic soundings. Phys, Earth Planet. Inter., 53: 384-404.

Interpreting electromagnetic (EM) data is difficult because responses of three-dimensional (3-D) models of the Earth are complicated and expensive to calculate. To simplify interpretation we developed an approximate method based on the behavior of transient currents in a layered-Earth model to estimate the variation of resistivity with depth beneath a controlled-source EM survey. Responses measured in the vicinity of each source are approximately matched at each delay time by the magnetic field of a current filament in free space. This filament, which moves downward with time, is an image of the loop or grounded-wire transmitter used in the survey. A continuous resistivity profile with depth is estimated from the velocity of the filament.

Our inverse solution is simple and rapid, and provides meaningful interpretations even when the measured responses are contaminated by random or geological noise. Although overshoots in the estimated resistivity profile and a spatial smearing of the zones of anomalous conductivity can occur, our technique seems to be less biased by 3-D effects than is computer-intensive, layered-Earth model fitting using constrained nonlinear optimization. This is because we do not artificially parameterize the Earth into a finite number of layers. In a direct comparison using the two techniques to interpret a set of field data from a geothermal site, we find that certain features of the interpreted models are similar, and thus probably essential, while others are not. Given the inherent ambiguities and ubiquitous bias in EM inversion, the smoothed resistivity profile produced by the approximate scheme often is adequate.

1. Introduction lustrated electric field patterns in layered-Earth models and showed that rather than individual

Our understanding of the behavior of transient smoke rings moving in each layer there is only one currents diffusing into a layered Earth is based in smoke ring which diffuses with time through all of part on the pioneering work of Nabighian (1979) the layers. This smoke ring is distorted from the and Hoversten and Morrison (1982). For step­ half-space pattern by the different velocities and function excitation due to a transmitting loop on a attenuation rates associated with the conductivity homogeneous half space, Nabighian showed that and thickness of each layer. the transient response can be approximately repre­ This work has provided insight into interpre­sented by a downward moving and outward ex­ ting transient electromagnetic (TEM) soundings panding current filament, of diminishing ampli­ over more complicated structures. Using this in­tude, having the same shape as the transmitting sight, workers are now devising methods for trans­loop. Because responses are measured as a discrete forming responses measured as a function of time function of time, the current filament resembles a into resistivity values as a function of depth. One system of 'smoke rings' blown by the transmitting such scheme was recently described by Macnae loop into the Earth. Hoversten and Morrison il- and Lamontagne (1987). Both their scheme and

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.. the one we discuss in this paper are based on equating the response measured at the sur face of the Earth at each delay time to the magnetic field of images of the source. These images are simp ly current filaments in free space and their fields can be computed quickly according to the Bicr-Savart Law.

\ The fact that the system of images does not necessarily bear resemblance to the real current distribution in the ground is exemplified by an image solution for transient loop-source excitation posed by Smythe (1968) for an infinite thin-sheet model in free space. Clearly the real currents are confined to the sheet, but the measured response at any point can be calculated exactly from a single image of the source which recedes away from the sheet with a velocity inversely propor­tional to the sheet's .conductance. If the position of the image at several delay times were de­termined by comparing measured and image fields, the velocity, and therefore the sheet's conduc­tance, could be estimated.

.... For a half-space model the current filament

moves downward with a velocity inversely propor­tional to the square root of time and conductivity (Nabighian, 1982). In both our scheme and the one discussed by Macnae and Lamontagne (1987), the resistivity of the Earth is estimated from the variation of the image's position with time. Our .. scheme differs from theirs in that we use only a single image and we extract resistivity estimates directly from its velocity. A related technique ap­plied to central-loop data over a layered-earth using approximateI: image representation was de­scribed by Nekut ](1987). Barnett (1984) used a current filament approach in the design of an inversion procedure for plate-like conductor re­sponses.

In this paper we discuss a new technique for estimating resistivity as a function of depth from TEM sounding data measured at the surface of the Earth using a square transmitting loop or a grounded-wire source. Data must consist of or be converted to the step response, i.e., magnetic field

~ measurements owing to a step-function current waveform , measured at one or several positions near the transmitter. From our program one ob­tains a continuously varying, estimated resistivity

curve, together with an indication of the continu­ity of resistivity laterally along the survey profile. We plot the curve directly beneath the center of the source , although the influence of nearby con­ductors can produce distortions in the curve which are unrelated to the resistivity structure directly beneath the source . For 1-D (layered-Earth) mod­els the estimated curve is a smoothed approxima­tion to the true resistivity structure.

After discussing how the scheme works, we show examples of interpreting synthetic layered­Earth and 3-D responses, as well as a set of field data. We believe that techniques such as the one we describe in this paper will soon become a popular means of obtaining a first guess at the 3-D distribution of resistivity in the Earth.

2. Apparent resistivity-a function of time

Often the first step in interpreting a transient sounding is to compute an app arent resistivity curve as a func tion of time. This quantity is easy to calculate, is a convenien t means of normalizing the data, and may provide some interpretational insigh t. Raiche (1983), Spies and Eggers (1986), and Newman et al. (1987) advocated the use of an apparent resistivity function based on the mag­netic field, rather than its time derivative. This approach elimin ates some problems associated with apparent resistivity overshoots and multi-val­ued functions.

We compute apparent resis tivity from the verti­cal magnetic field at each time by successively refining a guess of the resist ivity of the half space which would yield the measured field at that time. A bisect ion technique is used and refin em en t pro­ceeds until numerical convergence of the resisti v ­ity value is obtained. As long as the apparent resistivity function is single valued , the technique may be used for any survey configura tion . To our knowledge, this iterative approach applied to arbi ­trarily shaped sources has not been reported i n the literature.

The vertical magnetic field of a loop or grounded source on a h alf space of conductivity, (1 , and permeability, M, may be computed by integrating an an alytic express ion for a horizontal

386

LOOP SOURCE

_-.:i__ x/ __...... .".. ""'--1- - - - ­

dol i..-"-"- p Z

GROUN DED SOURCE

--------- y-:-;-/~1------

dol y",.",-p Z

Fig. 1. Graphical illustration of the quantities related to com­puting the magnetic field owing to current flowing in a source wire on a homogeneous half space-see eqn (1).

electric dipole along the length of the source as shown in Fig. 1. For this dipole

Hz(x, y) = Idry {3 exp - U / ( 'JT 1(2 u)

+ [1- 3j(2u 2 ) ] erf(u)}/(4?1i)

(1)

where

u= [a.u p2/ (4t) ]1/2

P = (x 2 + y2)1/2

I is the current in the source prior to turn-off, and t is the delay time. For large u (early times) eqn. (1) becomes

Hz(x, y) - I dly[l- 3/(2u 2 ))j (4'71'p3)

and for small u (late times)

Hz(x, y) - I dly(8u3/ '17" l/2)

X (1/15 - u 2j 35 + u4/ 126)/ (41Tp3 )

These equations were derived from Kaufman and Keller (1983).

For loop-source calculations the central-loop response is utilized. At this location the apparent resistivity function is single valued, and symmetry may be applied to reduce the amount of numerical

integration in the case of a square transmitting loop. For grounded-source calculations, the responses at equal distances on either side of the wire are averaged to reduce the effect of noise in the measurements. In this case the function is again single valued. For the vertical component there are no grounding terms associated with the grounded-source fields.

We compute apparent resistivities from the measured response by finding the unique resistiv­ity of the half space which would yield the same response based on integrating eqn. (1) along the length of the source.

3. From sounding data to image depths

In a TEM sounding one typically measures the voltage induced in a receiving coil at one or more stations for several transmitter positions. In Fig. 2 we illustrate the loop and grounded-source survey configurations which provide convenient data sets for computing the inverse solution. We assume that the sources are laid out on the surface of the Earth along a profile. Neither the size of the source nor the number of receivers (crosses along the profile) is critical in computing the solution. The notion of a smoke ring diffusing downward away from the source with time suggests that we compute an image solution, i.e., estimated resistiv­ities as a function of depth, for each source posi­tion using only those responses measured in the vicinity of that source. Short-offset measurements are less likely to be influenced by lateral variations in resistivity than are long-offset measurements. Our solution technique would have little meaning if applied to long-offset measurements, such as those described by Strack (1984).

Also shown in the figure are images of the i th loop source and the i th grounded source whose magnetic fields approximate that measured at the receivers owing to the real Earth at the jth delay time. Fitting free-space magnetic fields of an image at a given time requires that we know the mag­netic field response of the Earth owing to a step tum-off of current in the transmitter, i.e., the step response. Some systems measure this directly, for example, see West et al. (1984). However, most

----

LOOP 80UAC~

I I I II I I I I , I I

:dj: I : I I I I I I I I

I I I I !!LtV'·"·, l-----R------{

QROUNDED IOURCE

dJ

\ X ............ ::>- ImOIlBIJ

~--~ y'"

Fig. 2. Transient sounding surveys and the images of the ith source at the jth delay time for a loop and grounded wire. The crosses represent the receivers used in conjunction with the it11 source. The images have the same dimension (R) and current (I) as the source, and are at a depth denoted by dj .

systems measure the time derivative of the mag­netic field due to a current waveform terminated with a linear ramp. Levy (1984), Nekut (1987) and Eaton and Hohmann (1987) described techniques for converting this type of measurement into the step response. If harmonic measurements have been made over a broad range of frequencies they could be converted directly to the step response using the digital filter transformation technique described by Newman et al. (1986).

The magnetic field of the current filaments shown in Fig. 2 are computed by breaking each

387

into linear elements and applying the following expression from Telford et aI. (1976)

HI/> = I(sin 132 - sin.B1)/4'1Tr (2)

to the situation illustrated in Fig. 3. This image field is <t>-directed with respect to a cylindrical co-ordinate system whose axis coincides with the element. The vertical component of the field at the position of the receiver can be determined geomet­rically and the total field is simply the sum of the fields due to each elemen t.

A more complex current filament than that illustrated in Fig. 2 or multiple images would be needed to precisely fit sounding data from layered-Earth (I-D) and 3-D models. Our image solution was designed on the principle that an exact fit to even noise-free data is not necessary. The position of the image is established by simply minimizing the difference between the measured and image fields. This optimization procedure is more efficient and more likely to be successful if the number of parameters describing the image is small. Furthermore, fixing the current and dimen­sions of the image at that possessed by the source preserves the magnetic moment, i.e., the product of the current and area (length) of the loop (grounded) source. In general, allowing the cur­rent filament to expand and its current to decrease with time does not improve the image solution;

x

_....... --­_.---­

Receiver

\ /32

\ \

\ r\

z \ \

\ \\ i' \ \ c,~ \ «.~ \ .... \ 1lJ~

......\ \ c.l

Fig. 3. Graphical illustration of the quantities related to com­puting the magnetic field owing to a linear current filament in free space-see eqn. (2).

388

.1

therefore the depth of the filament is typically the only unknown parameter to be estimated at each delay time. Although we do not use multiple images in our solution, there may be merit in doing so, e.g., see Macnae and Lamontagne (1987).

When interpreting loop-source responses we use only the vertical magnetic field. However, in the case of a grounded source, it is not possible to obtain an approximate fit to 3-D vertical magnetic field responses at intermediate and late times using the image shown in Fig. 2. Furthermore, it is not possible to fit the horizontal response of most layered-Earth and 3-D models at early time. We even tried a more complex image forming a con­tinuous vertical loop, intended to simulate more closely the flow of current in typical models at early time, e.g., see Gunderson et al. (1986). These problems were solved by using the vertical field at early time and the horizontal field at intermediate and late times. At any particular delay time a single component of the measured field is used in the solution. Alternatively, the sum of the squares of the vertical and horizontal components at each time could be used.

The procedure of approximately fitting image and measured fields at the jth delay time (/ j )

consists of searching for a minimum of

N

\f;j= L IHk'(tJ -H£(dJ\JN (3) k~l

in the vicinity of the point in parameter space associated with the ~j-l solution. H!:'(tj) is the step response measured at the kth receiver and Hk(dj ) is the magnetic field at the position of the kth receiver owing to the image at a. depth dj

beneath the source. N is the total number of receivers. We use the nonlinear simplex method for function minimization described by Nelder and Mead (1965) and constrain the jth solution so that dj > dj _ 1 > O.

The procedure is repeated, beginning with the earliest delay time and ending with the latest delay time. At this point the data, measured as a func­tion of time at one or more receivers owing to a single source, have been converted into a set of image depths.

4. Estimated resistivity-a function of depth

Having determined the position of the image at discrete times d , = d(t), the next step is to esti­mate resistivities and the depth which each resis­tivity value corresponds to in the interpretation. Our procedure is slightly different for loop- and grounded-source interpretations, but in each case we begin by scaling the image depths to corre­spond to the position in the Earth of the maxi­mum of a particular electromagnetic field compo­nent. The next step is to determine velocity by computing the derivative of this depth function with respect to time. The rate of diffusion of the 'smoke ring' of currents in the Earth directly depends on resistivity. We estimate resistivity by comparing discrete velocity estimates to those computed analytically for that field component in a homogeneous half space. Finally, we rescale the depth values so that a change in the estimated resistivity values occurs approximately at the de­pth where the actual resistivity changes. We dis­cuss the treatment of loop- and grounded-source data separately.

4.1. Loop source

The velocity function is estimated from

v(tj ) = ag(t)/atj t~1 /

(4)

where g( t) corresponds to the depth of the maxi­mum of 3h / at in the Earth. We found that thisz

depth is essentially proportional to the image de­pth

g(tj ) = ad(tj ) (5)

The derivative in eqn. (4) is estimated numerically using a cubic spline interpolant. According to :1

Raiche and Gallagher (1985), the vertical diffusion 11

velocity, V, of this maximum at a time, t, in a homogeneous half space of conductivity, c, and permeability, u; for a circular loop of radius, a, is

V= )'1/2{ C1 + (c,f + 2f/2

+[1 + C1/ ( C{ + 2)1/2] )'C2} /( C1J.La ) (6)

389

•

where

3'lT112(1- y/4 ­C= ~ 1 k=2

x {(2k - 3)!!/[k!( k+ l)!J)( r/2)' );4

C2=3?T1/ 2(1/2+ r {(2k-l)!!

k=l

/[k!(k + 2)!]} (Y/2)k )/4 and

y = aJLQ2/4t

We estimate resistivity from the velocity estimate for each t} by finding the unique resistivity of the half space which would yield the same velocity at that time based on eqn. (6). A bisection technique is used to refine the resistivity computed at the previous time until convergence of the resistivity value is obtained at the current time.

Lastly, the depths Zj associated with the resis­tivity values are estimated by scaling the depth function

Zj = /3 g( ti) = af3d ( tj ) (7)

In our solution the values of ex and /3 are 1.10 and 0.40, respectively. These values are model and time independent, and were determined em­pirically using the responses of 1-D models with layer resistivities ranging from 1 to 10 000 Q m.

4.2. Grounded source

In this case, g( t) in eqn. (5) corresponds to the depth of the maximum of the component of the electric field parallel to the wire and a was esti­mated to be 0.67 for vertical magnetic fields and 0.50 for horizontal magnetic fields. Very little work has been done on interpreting grounded­source data, and to our knowledge there are no analytic forms available in the literature which describe electromagnetic field behavior in the Earth. The parallel electric field component was chosen because its behavior is compatible with our image solution. In most conductive Earth models

104 , ;;> ------~---­1

103

'0' I 10. 6 10. 5 10-4 10-3 10"2 10"

TIME (ad Fig. 4. Depth of the maximum of the electric field component parallel to a grounded-wire source, directly beneath the wire in a homogeneous half space, as a function of time.

there exists a maximum in this component which diffuses downward beneath the source with time. The rate of diffusion is directly related to the resistivity.

We computed electric fields in a half space using a numerical modeling technique described by Gunderson et al. (1986). At any depth beneath the wire the parallel component is initially nega­tive, which implies that current flows in the Earth under the influence of this electric field in a direction opposite to that of current flowing in the source before it is turned off. With time the field increases and becomes positive, reaches a maxi­mum value, and then decreases. In Fig. 4 the depth of the positive maximum is plotted as a function of time for three half-space resistivities and a 10 m source with solid lines. These calcula­tions suggested that as a first-order approximation

g ( t) = 1000 ( 'ITt / (J ) 1/2 (8)

The rate at which this maximum recedes into the Earth is therefore

V= ag(t)/8t = 500 (1T/at/12 (9)

These expressions are not dependent on R, the length of the source, and would probably be ade­quate for interpreting measurements from a grounded-dipole source. However, the use of eqn. (9) yielded resistivity estimates that were biased at early times for long sources. In Fig. 4 we compare computed get}) values from the image solution for several source lengths and the 100 a m model

.. 390

(dashed curves). This graphical discrepancy sug­gested that we add a term to eqn . (8)

2get ) = 1000 (7Tt/a / / [1 - exp( - w)/ 2] (10)

where

2 ) 1 /~ IV = (100r.t /a/lR

Then the vertical diffusion velocity becomes

v = 500 ( IT/ at )1/ 2[1 - exp( - w)/ 2

+wexp( -w)/ 4] (11)

which is used to compute the resistivity estimates. In order to calculate dep ths zJ corresponding to

these estimates, we scale the depth function in eqn. (7) with {3 = 0.66. As in the case of the loop source, the scale factors CI. and {3 were determined empirically from layered-Earth model results. Surprisingly, the product of CI and {3 is the same for both loop- and grounded-source solutions using the vertical-field.

There is an important point to be noted regard­ing eqn. (8) (or (10» an d the diffusion process. This expression does not predict the depth at which the maximum in the electric field, con­toured in cross-section beneath the source, is ob­served at a given time, e.g., see Gunderson et al. (1986). The field at that position is only large there compared with other position s at that time. At an earlier time the field is actually larger at this position, but the observed maximum occurs at a shallower depth then because there the fields are even larger. To put this in perspective, at 1000-m depth in a 300 Q m half space the electric field for a 1000-m source reaches a (positive) maximum value of - 10 - 5 V m - 1 at - 1 ms. However, it takes - 30 ms for the observe d maximum (2.5 X 10- 7 V m - 1) to be positioned at this depth.

From eqn. (8) it is a simple matter to deduce that

C1 = (7T 2/ 5/l)a2t/agl (12)

so that a may be estimated from the 'slowness' of the image, e.g., see Macnae and Lamontagne (1987). For some models this conductivity esti­mate is more accurate than is the estimate derived from the velocity of the image. However, for other-

mode ls it is difficult to approximate accurately the second derivative in eqn, (12) based on using a single image scheme to compute the dep th func­tion .

5. Interpretation of synthetic data

The expressions presented in the previous sec­tion yield meaningful image solutions for a wide range of model and survey parameters. However, for values of y in excess of two the series in eqn. (6) for a loop source are numerically unwieldy. For grounded-source interpretations, the velocity expression in eqn . (11) for large values of ( ap.R 2/ 4t ) also is inadequate. At very early times, we compute apparent resistivities and estimate image depths by assuming tha t the position of the current filament associated with a dipole trans­mitter on a homogeneous half space can be lin­early scaled to match the computed depth of the image at the earliest time at which the image solutio n can be obtained. Despite this difficulty, recognized by Nekut (personal communication, 1987) in his formulation as well, both the loop­and grounded-source solutions yield good ap­proximate resistivity profiles over a wide range of time and hence dep ths. To resolve shallow struc­tures in a cond uctive environ ment one should use a small enough source to keep y small at early times.

In order to test our program we computed electromagnetic responses for several1-D and 3-D models using algorithms described by San Filipo and Hohmann (1985) and Newman et al. (1986). When random noise is added to the step response of a model, the inversion algorithm successively unde r- and over-compensates the estimated resis­tivity values with depth. For example, Fig. 5 shows central-loop solutions for different amounts of random noise added to the fields computed for a 100 Q m half-space model. The crosses represent the discret e solution; the smooth curve is a three­point running average of the discrete values. Note how the average does a better job of tracking the true resistivity than does the discrete solution.

In Fig. 6 two-layer model results are presented. TEM responses were compu ted for two models at

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103 \ ~" 8' \ 6 """," . 1l:: E I­lI)

iii '" 2'" / 0 0

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Fig. 6. Im age solutions for two-layer models; (a) and (c) ar e for 100 In X 100 In loops; (b) and (d ) ar e Cor 150 m gro unded wires. A continuously va ryi ng. es ti mated resistivity cur ve is ob ta ined by averaging discrete so lutions ( X) and is superimp osed o n th e t rue resistivi ty variation of each mod el (s traigh t lines).

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Fig. 7. Image solutions for three-layer models; (a) and (c) are for 200 m X 200 m loops; (b) and (d) are for 300 m grounded wires. A continuously varying, estimated resistivity curve is obtained by averaging discrete solutions (x) and is superimposed on the true resistivity variation of each model (straight lines).

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fig. 8. Im age solutions for three-layer models ; (a) a nd (c) a re for 200 m X200 m loops; (b) and (d) arc 300 J11 grounded wires. A con ti nu ous ly va rying, estimat ed resistivity cu rve is obtained by averaging di scr et e solu tio ns ( X) a nd is supe rim pos ed on the true res istivity vari a tio n of each model (straight lines) .

ID0l , . , 11, 11 1 1111111 LL1.l..1.l1l----'-'_LJJ1JlJ

395

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200r 1000 n·m 400

Z(m)

I600~ I

800 II 10n·m I

1000 I-------- ....

1200

1400 r 300.Q,·m 1600

1800

2000

3000,--­

2400 I­

1800 I­X(ml

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600 t o 1-2foo I

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-1200 f­

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o 0 1800 2400

! I I! I -2400 -1200 o 1200 2400 -2400 -1200 0 \200 2400

Y{ml Y{ml Fig, 9. Model geometry and source positions for a 3·D conductor (position dashed) in a two-layer Earth. Only one of the six square transmitting loops is shown.

delay times ranging from IJ.Ls to 1 s for 100 m X 100 m loops and 150 m grounded wires. In parts (a) and (b) the true resistivity abruptly in­creases by a factor of 10 at a depth of 400 m. In parts (c) and (d) it abruptly decreases by a factor of 10 at the same depth. Note that when the true resistivity decreases with depth the estimate does a better job of tracking the true variation than when it increases. The gradual transition from low to high resistivities characterizes all conductive­over-resistive two-layer model results. The accu­racy of the computed solutions is not strongly dependent on the source dimensions or the num­ber of receivers used when interpreting noise-free, layered-Earth responses.

In Figs. 7 and 8 three-layer model results are presented for the situation where resistivity de­creases with depth (type Q), resistivity increases with depth (type A), there is a buried resistive layer (type K), and where there is a buried con­ductive layer (type H). As in the case of the two-layer models, loop- and grounded-source re­sponses were computed for each three-layer model at delay times ranging from 1 p.s to 1 s. In this case the sources were made twice as large as in the two-layer case to demonstrate that the results are not strongly dependent on the source dimensions as long as the time range measured is sufficient to

extract both shallow (top layer) and deep (basal layer) information. For example, in Fig. 7c and d it would be difficult to ascertain that the basal layer has a resistivity of 1000 Q m if measure­ments had not been made out to at least 1 s.

From the three-layer model results it is ap­parent that conductive-to-resistive transitions are poorly resolved. In part, this is a consequence of the limitations of electromagnetic methods. Cur­rents are not exclusively confined to one layer at a given time and they diffuse quickly through the resistive layers. Consequently, TEM measure­ments are rather insensitive to the presence of thin resistive units or to the transition from a less resistive to a more resistive unit.

In practice, the transmitter and receivers .should be profiled along the surface to resolve lateral variations in the geoelectric section. We computed several layered-Earth models containing 3-D inho­mogeneities and interpreted the responses with our new algorithm.

The first 3-D model has 500 m of 1000 Q m material overlying a 10 Q m block embedded in a 300 n m basal half space. The block, illustrated in Fig. 9, is 2000 ill X 2000 m in plan and 600 m in depth extent. Central-loop soundings, using 1200 m X 1200 m loops at the locations noted in the figure, were initially computed in the frequency

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Fig. 10. Central-loop (step response) apparent resistivities corresponding to the model and source positions shown in Fig. 9. Discrete va lues are plotted ( X) at each time and jo ined together using a cubic spline interpolant.

domain and then transfo rmed into the step re­ shallow (early-time) resistivity is 1000 n m, while sponse using a digital filtering technique. In Fig. the resistivity deep in the section (late time) is 300 10 we show apparent resistivity curves computed n m. We also detect the presence of the 3-D for each sound ing using the techniqu e described conductor in the middle of the profile and can ear lier. From these curves we can tell that the estimate the depth to the top of this inhomogene­

b ~ . Q ( . Q ' ,..;.,..

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ity from the expression 500'7T(pl t ' )l/2, where t' is the time at which the apparent resistivity values directly over the body diverge from the actual resistivity (pI) of the top layer. We derived this relationship from two-layer model results. At sta­tion 0, t ' - 0.1 ms and p' is 1000 Q m, yielding a

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depth of burial of 497 ill, which compares well with the true depth of 500 m.

It requires essentially the same computational effort to calculate estimated resistivities as it does apparent resistivities. and in general the former provide more information directly, The image

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Fig. 11. Image solutions corresponding to the model and source positions shown in Fig. 9. The discrete (X) and averaged estimates are shown together with the true resistivity variation beneath each station. Station -1000 is directly over the edge of the 3-D body.

398

solutions are superimposed in Fig. 11 on the true model at each central-loop station. The true model is simply the variation in resistivity beneath each station and is therefore ambiguous at station -1000. There are two problems with the interpre-

STATION -2200

tation. First, it appears that the conductive zone extends under stations which are not over the 3-D body, e.g., station 1800. Hence the conductor is spatially smeared out in the interpretation. The second problem involves an overshoot in the resis-

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Fig. 12. Layered-Earth inversion results (X) corresponding 10 the image interpretation in Fig. 11. The top and bottom layer resistivities were held fixed while the other resistivities and all thicknesses were allowed to vary in the inversion. The true resistivity variation is superimposed on the inverse solution except at station -1000.

• ­tivity beneath the conductor. This feature is par­ticularly evident at stations 0 and 600, which are over the 3-D body. Even with these problems the solution does a good job of resolving the variation of resistivity with depth and across the model.

These same responses also were interpreted using a non-linear least-squares, layered-Earth, in­version algorithm (Newman et al., 1987). Even though the background resistivities of 1000 and 300 n m were fixed in the inversion, the results are quite misleading, as shown in Fig. 12. The error in the interpretation is a consequence of attempting to fit I-D responses to 3-D data. There are obvious similarities between the image and layered -Earth interpretations. However, the layered-Earth inversion results are more strongly biased by 3-D effects. Furthermore, the computa­tional time used in the layered-Earth inversion was roughly 30 min, whereas the approximate (image) inversion took only - 3 min on a VAX 11/785 computer.

In Figs. 13 and 14 we show results for a similar 3-D model consisting of a 800 m wide X 200 m

399

thick X 800 m long conductor in a 300-n-m half space. The step response was estimated from volt­ages computed for a current waveform terminated with a linear ramp (pu lse response). The essential point here is to compare the result of using a single central-loop receiver with that for several receivers located both inside and outside the 400 ill X 400 m transmitting loops. In these figures we have contoured estimated resistivities in cross-sec­tion for a series of source positions along a centered pr ofile taken across the model. The shapes of the contour patterns are similar, but while the central-loop solution (Fig. 13) is closer to the true resistivity of 10 n rn in the block, the overshoot beneath the body is slightly worse than in the case of using multip le receivers (Fig. 14). Overshoots are characteristic in our results when 3-D conduc­tors are present.

Using multiple receivers allows us to estimate the direction of the inhomogeneity relative to each source position. Based on the misfit between the field of the image and that measured on either side of the source at each image depth , we obtain

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\jJi\ Fig. 13. Estimated resistivity section for 3-D body in an otherwise homogeneous half sp ace. The cross-section is taken along a profile cutting directly across the center of the 3-D body shown. The image solution was computed using the central-loop respons e for source positions 400 m apart.

400

•

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an indication of any significant change in resistiv­ity in either direction along the survey profile. The arrows in Fig. 14 identify the shallowest depth at which the program discerns that the misfit be­

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tween measured and image fields is indicative of an inhomogeneity. Because the actual currents in the model expand laterally as they diffuse down­ward, the true depth of the conductor is less than

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I

401

the depth suggested by these arrows. The larger the separation of the sounding and the body, the larger the amount of time needed before the ex­panding current system is influenced by the body, and the greater its apparent depth. Because the responses computed about station 0 are symmetric and the effect of the 3-D body on the responses computed about stations ±1600 is very weak, no arrows appear beneath these stations.

It has long been recognized that spatial asym­metry in EM responses can be indicative of 3-D effects. The image solution provides a means of quantifying this information. Random noise in the data can make it appear that there are multiple inhomogeneities in both directions along the pro­file. This situation is easily identified in practice because the directional indicators are not coherent with increasing depth or from one source position to the next.

Lastly, we present results for a structural model consisting of a 50-m layer of 100-Q-m overburden overlying a thick 1000-Q-m layer overlying a 50-Q­m basement. The basement contains an anticlinal feature having a strike extent of 3000 m and a relief of 600 m. This exploration model is intended to simulate the case of volcanic flows overlying a potential hydrocarbon trap in the underlying sedi­ments. The objective of an EM survey in this situation would be to map the thickness of the volcanics to determine the position of the anti­cline. A loop-source interpretation for 1000 m X

-3600 -3000 -2400 -1800 -1200 -600 0 600 I I I

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1000 m loops is shown in Fig. 15; a grounded­source interpretation for 900-m-long wires is shown in Fig. 16. Estimated resistivities are con­toured for a series of source positions along a centered profile taken across the structure. Note that because most of the current diffuses down­ward from the grounded sources, rather than out­ward as in the case of the loop sources, the resolu­tion in Fig. 16 is somewhat better than that in Fig. 15. Multiple receiver positions were used at each sounding. A layered-Earth inversion of the central-loop data for this model is illustrated in Newman et al. (1987).

The short-offset, moving grounded-source method is more time consuming in the field than either the loop-source or the long-offset, fixed grounded-source technique. However, in compari­son it probably offers two important advantages. The first is a more realistic l-D interpretation because most of the current diffuses straight downward from the grounded wire, rather than outward as in the case of the loop source, and because the response measured in the vicinity of the source would not be as strongly affected by lateral variations compared with long-offset mea­surements. Furthermore, as the source and re­ceivers are profiled together from station to sta­tion, we expect better lateral resolution for the same reasons. It is interesting to note that depths associated with the image solution for loop- and grounded-sources which are of comparable size

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500

1000

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Fig. 16. Estimated resistivity section for the same structural model as that of Fig. 15, but using grounded sources.

J i

402

are approximately the same for layered-Earth 6. Interpretation of field data models. In 3-D environments it is not obvious whether the short-offset, grounded-source metho d The United States Geological Survey (U SGS) possesses a greater depth of investigation than the collected several transient soundings from the other two techniques. Medicine Lake geothermal site in northern Cali-

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fornia in 1982. Sirotem responses, using the central-loop configuration, were converted to ap­parent resistivities and then inverted using a non­linear least-squares, layered-Earth, model fitting technique. Their inversion method is the same as that mentioned in the previous section. The survey and results are described by Anderson et al. (1983).

In order for us to interpret the results, we had to convert the apparent resistivities for each sounding into the step response. This process pre­sents the biggest obstacle to interpreting field measurements unless the step response has been measured directly. Also, we currently assume that the measured response is due to an ideal step-function current waveform and is measured at discrete delay times. In practice the current waveform is bipolar and repetitive, and the tran­sient is integrated over contiguous windows in time to help eliminate noise. The consequence of these violations of our assumption in the interpre­tation has not been evaluated .

Estimated resistivities are plotted together with the published I-D inversion results at several of the stations in Fig. 17. At depths corresponding to times earlier than the first delay time and later than the last delay time, we do not obtain an estimate of the resistivity. Currents are not con­centrated at these relatively shallow and deep depths during the time range measured . The accu­racy of the I-D inversion at these depths is there­fore limited.

The comparison of results is quite remarkable, attesting to the quality of the data and the validity of both interpretation methods. The best agree­ment probably occurs where the sounding reflects I-D structure. Furthermore, it is evident that cer­tain features are essential in the interpreted model, such as the 10 Q m basement under station 1 and the resistive-conductive-resistive-conductive se­quence beneath station 3. Other features are not essential, such as the very resistive basement be­neath stations 4 and 10.

7. Conclusions

Our technique for estimating the resistivity of the Earth from TEM soundings has both ad­

~

403

vantages and disadvantages. On the positive side, the solution is very fast, taking approximately the same amount of time as is needed to compute apparent resistivities. The program could be im­plemented on a smaller computer and interpreta­tions could be done on-site. Our solution is not biased by an artificial parameterization of the Earth into a finite number of layers or by the choice of an initial guess model typically required by model-fitting routines. Furthermore, our solu­tion appears to accommodate 3-D effects in a manner which is preferable to that using layered­Earth inversion. The position and resistivity of 3-D inhomogeneities are estimated and conductive units are typically well resolved. The technique appears to be insensitive to small amounts of random noise .

On the negative side, resistivity overshoots due to 3-D effects can occur, and resistive units are poorly resolved. The technique does not provide a means of incorporating a priori geoelectrical con­straints and only the transient step response can be interpreted directly. At interpreted depths cor­responding to times which are earlier or later than the actual delay times used in the field, we obtain no information regarding the resistivity of the Earth.

In addition to providing the geophysicist with a quick on-site interpretation of TEM soundings, the image solution may have its most important role in providing a reasonable resistivity distribu­tion for the first guess in a more sophisticated multi-dimensional inversion scheme.

Acknowledgments

We thank Gregory Newman for providing the layered-Earth modeling program and the layered­Earth inversion result. Bob Wheeler of the How­ard Hughes Medical Institute developed most of the graphics software used to present the results. Financial support for this research was provided by a consortium of private corporations including Amoco Production Co., ARCO Oil and Gas Co., Chevron Resources Co., CRA Exploration Pty, Ltd., Standard Oil Production Co., and Unocal Corp.

.. 404

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.. .1 \