Calculation of electromagnetic sensitivities in the time domain

Transcription

1 Geophys. J. Int. (1998) 133, Calculation of electromagnetic sensitivities in the time domain Andreas Hördt University of Cologne, Institut für Geophysik und Meteorologie, Aussenstelle Explorationgeophysik, Godesberger Strasse 1, 5968 Köln, Germany. Accepted 1997 December 12. Received 1997 December 4; in original form 1997 July 3 SUMMARY The speed of calculating sensitivities for 3-D conductivity structures for timedomain electromagnetic methods is significantly improved by applying the reciprocity theorem directly in the time domain. The sensitivities are obtained by convolving the electric field in the subsurface due to a transmitter at the surface with the electric field impulse response due to another transmitter, which replaces the original receiver. The acceleration compared to the classical perturbation method is approximately P/R, where P is the number of model parameters and R is the number of receiver positions. If the sensitivity has to be calculated very close to the receiver, approximate sensitivities can be obtained using an integral condition. Comparisons with the classical perturbation approach show that the method gives accurate results. Examples using transmitter receiver configurations from a long-offset transient electromagnetics survey demonstrate the usefulness of sensitivities for the evaluation of resolution properties. Key words: electromagnetic methods, Fréchet derivatives, inversion, time domain. INTRODUCTION Electromagnetic sensitivities are crucial for the inversion of electromagnetic data and for the evaluation of the resolution properties of a method. The calculation of sensitivities is usually the most time-consuming part of an inversion procedure. Recently, several methods were published where the sensitivity calculation is significantly accelerated. de Lugao & Wannamaker (1996) use reciprocity within a 2-D finite element code for magnetotelluric modelling. McGillivray et al. (1994) calculate sensitivities for inversion of frequency-domain electromagnetic data using an adjoint equation method, and Farquharson & Oldenburg (1996) derive approximate sensitivities based on the same approach. Mackie & Madden (1993) completely avoid the explicit calculation of sensitivities using a conjugate gradient approach for 3-D inversion of magnetotelluric data. This approach is adopted by Zhang, Mackie & Madden (1995) for 3-D inversion of DC data and by Newman & Alumbaugh (1997) for 3-D inversion of controlled-source frequency-domain data. All of the methods mentioned above work either in the frequency domain or for DC data. The frequency-domain methods could in principle be converted to the time domain by either Fourier transforming the time-domain data to the frequency domain, or transforming the frequency-domain sensitivities to the time domain. The wide frequency band required for an accurate representation of a time-domain response (Newman, Hohmann & Anderson 1986) would, however, limit the benefit of such an approach RAS This paper presents a method for the calculation of sensitivities directly in the time domain. Like the methods mentioned above, it uses reciprocity, and can be considered the time-domain equivalent of the adjoint equation method presented by McGillivray et al. (1994). For the forward modelling, I use the finite-difference code of Druskin & Knizhnerman (1988). This allows the calculation of electric field impulse and step responses in a 3-D earth, which are needed for the sensitivity calculation. The examples shown here use transmitter receiver configurations and a time range typical of the long-offset transient electromagnetics (LOTEM) method, which was described in detail by Strack (1992). THEORY The geometry for the calculation of sensitivities is shown in Fig. 1. The transmitter is located at the origin, and the receiver is at r. In order to derive an expression for the sensitivity of the electric field at r with respect to a change of conductivity in a small volume V at point r, it is convenient to start with the time-domain integral equation (Hohmann 1988): E(r,t)=E p (r,t)+ P V P t G(r, r,t t )E(r,t )s (r )dt dv. (1) a Here, the total electric field has been decomposed into a primary field, E, which is the field without a conductivity p change in V, and the scattered electric field generated by the anomalous currents in V. s is the anomalous conductivity, a and G is the tensor Green s function, relating the current at r 713

2 714 A. Hördt The approach is called the adjoint Green s function method, because the right-hand side of eq. (4) solves an equation adjoint to the original forward problem (McGillivray & Oldenburg 199). If only one component of the electric field is considered, only one column of the adjoint Green s tensor will be needed, which turns out to be the electric field solution of an auxiliary problem, which may be called the auxiliary electric field (McGillivray et al. 1994). Since many 3-D algorithms, and in particular that of Druskin & Knizhnerman (1988), calculate all field components at any point without a significant increase of computer time, the sensitivities for one transmitter and one receiver can be calcu- lated with only two forward solutions: one to calculate the electric field generated by the original source, and one to calculate the Green s functions or impulse responses, using the original receiver as a source. For one transmitter and many receivers, the number of forward solutions is proportional to the number of receivers. If instead the sensitivity is calculated by perturbing the resistivity in a small volume, the number of forward solutions is proportional to the number of model parameters. In the following examples, the transmitter will be an electric dipole in the x-direction, and the receiver as well as the point where the sensitivity is calculated will be in the y z plane. In addition, the conductivity structure is symmetric with respect to the y z plane, so that the Green s function reduces to a scalar, and eq. (3) can be written as Figure 1. Geometry for calculating sensitivities. The transmitter (Tx) is an electric dipole at the surface. The solid and dashed lines indicate the two locations used in the following figures. to the electric field at r. The main difference from the equivalent equation in the frequency domain is that instead of a multiplication of the Green s function with the electric field, eq. (1) involves a convolution. Eq. (1), or the frequency-domain version, is usually the basis for 3-D forward modelling where the background model is a layered half-space or any other structure for which analytical solutions can be obtained (Newman et al. 1986). Here, eq. (1) is used to derive an expression for the sensitivities. Hence, the background model can be any 3-D conductivity structure, and the forward modelling to calculate E or E is not necessarily p carried out with an integral equation method. For the calculation of sensitivities, which are the partial derivatives of the electric fields with respect to s, the anomalous E(r, r,t) S(r, r,t)= 1E (r,t), (5) conductivity s can be assumed to be very small. This means t p a that Born s approximation applies and the total electric field where the * operator denotes convolution. In the following, in the volume V can be replaced by the primary field. This when writing E as a scalar, it will be taken to mean the includes the assumption that the electric fields are Fréchet x-component. E(r, r, t)/ t is the Green s function, calculated differentiable, that is the remainder term when applying Born s as the time derivative of the electric field step response. Note approximation is second order in model perturbation. More that for calculating E/ t, the transmitter has unit moment. rigorous treatments of this problem are given by Chave (1984) For magnetic field sensitivities, the starting equation is the and by Boerner & West (1989). In addition, if V is sufficiently same as eq. (1), except that the magnetic field Green s function small, it can be assumed that both s and the electric field are is calculated. Assuming we are measuring the time derivative constant in V, and we obtain of the vertical magnetic field, we obtain E(r, t) E (r,t) p = P t G(r, r,t t )E (r,t )dt, (2) H(r,t) V s (r ) p = H p (r,t) + P P t GH(r, r,t t )E(r,t )s (r )dt dv. a t t a V or, in the limit s, a (6) S(r, r,t)= 1 E(r,t) The derivation of the sensitivity equation is the same as V s(r ) = P t G(r, r,t t )E (r,t )dt. (3) p for the electric field step response. Because we are measuring the time derivative of the magnetic field, GH denotes the time The sensitivity defined here may be considered as a differderivative of the magnetic field impulse response due to an ential sensitivity, because there is no volume associated with electric source. Again, reciprocity is used to calculate GH. the point where it is calculated. To obtain the sensitivity with Instead of an electric dipole source at depth for which the respect to conductivity changes in a cell, it has to be integrated magnetic response at the surface is calculated, the electric over the volume of that cell. response at depth is calculated for a magnetic source at the The tensor Green s function can be calculated by placing an surface. electric dipole at r and calculating the electric field impulse In order to avoid confusion with impulse responses and step response at r, that is to calculate G place an electric dipole xy responses, it is useful to look at reciprocity in the frequency in the y-direction at r and calculate the electric field impulse domain. There, the magnetic field derivative recorded in a loop response in the x-direction at r (Ward & Hohmann 1988). At due to an electric source is the same as the electric field due this point, reciprocity is used. It states that instead of placing to a loop source if a transmitter at r and calculating the field at r, the transmitter may operate as a receiver, and the receiver as a transmitter D= ivmm (7) (Hohmann 1988): G(r, r )=G(r, r). (4) (Ward & Hohmann 1988), where D is the moment of the electric source and M is the moment of the magnetic source.

3 In the time domain, this means that D= mm t needs to be fulfilled. In other words, the magnetic field impulse response for an electric source is equivalent to the electric field step response for a magnetic source. However, the Green s P 2 (8) T ime-domain EM sensitivities 715 that case, the electric field step response has a non-zero value at t= (Kaufmann & Keller 1983), which means the time derivative has a singularity at t=. To handle the singularity and derive the condition for the value to be implemented at t=, we can write E dt= lim t eap e E t dt+ P 2 E t dt B. function in eq. (6) is the time derivative of the magnetic field e (11) impulse response, which is equivalent to the time derivative of The first term in brackets includes the singularity at t= the electric field step response. Therefore, GH can be determined and is E(t=). The second term describes the behaviour after by placing a loop source at the surface and calculating the the singularity, where the Green s function and the electric time derivative of the electric field step response at depth. field are continuous, and in the limit e it is equal to In the following examples, the time derivative of the vertical E(t=2) E(t=). Combining the terms, we obtain magnetic field broadside to the electric dipole will be considered, and the conductivity structure will have proper symmetry, so P 2 E dt=e(2)=edc. that the sensitivity equation again reduces to a scalar: t (12) EM(r, r,t) This equation now includes the singularity of the Green s SM(r, r,t)= 1E (r,t). (9) t p function at t=. It is also valid if the receiver is in the subsurface, because then E(t=)=. For the convolution Now, EM/ t is the time derivative of the electric field step eq. (12) means that integrating all coefficients used for E/ t response due to a magnetic loop source and E is the same p should give the DC electric field. Therefore, electric field as was used in eq. (5). E The sensitivities derived here are point sensitivities in the = tk EDC Dt n E i sense that the assumption of a very small volume was made t, (13) t= i=1 to assume a constant electric field inside. For inversions, larger where Dt is the sampling interval used for the convolution. cells will have to be used, for which this assumption is no This can be used for the implementation of both eqs (5) longer valid. In that case, the sensitivity can be obtained by (electric field sensitivities) and (9) (magnetic field sensitivities). integrating over the volume of the cell. For magnetic field sensitivities, EDC is the static electric field of a loop source, which is exactly zero. Therefore, eq. (13) IMPLEMENTATION reduces to There are different ways to implement the time derivative and E = tk n E i the convolution in eqs (5) and (9) or in the tensor versions t. (14) t= i=1 of those equations. The optimum way may depend on the For the electric field sensitivities, EDC can be calculated capabilities of the modelling routine and the requirements on analytically, assuming a homogeneous half-space in the area accuracy. For example, some modelling routines may not be close to the receiver. able to calculate impulse responses directly. In those cases, the If eq. (1) is used for the magnetic field sensitivities, and the time derivative can be performed numerically, either before time derivative is carried out after the convolution, eq. (13) convolution, or after convolution of the two-step responses. does not help. For that case, however, reciprocity allows the Instead of eq. (9) we may implement following statement: EM(r, r,t) SM= 1E (r,t)= t p t [EM(r, r,t)1e p (r,t)]. (1) P 2 EM(t)dt=c P 2 HE dt=che,dc, (15) t If the time derivative is performed before the convolution, a critical situation for the implementation occurs for points close where EM is the electric field in the subsurface for a magnetic to the receiver (i.e. if r r is small). In this case, the decay source at the surface, and HE is the magnetic field at the of E/ t for t is very fast, and has to be calculated at very surface for an electric source in the subsurface. The constant c early times. For a finite difference program this can be a accounts for the source and receiver moments (see eq. 8). problem, because the grid is usually designed for a particular Eq. (15) states that the area under the step-on electric field time and depth range, and there is a minimum time t after transient for a magnetic source equals the static magnetic field min which the response can be calculated with reasonable accuracy. for the electric source. Again, for points close to the receiver, However, we need a good approximation for the sensitivities a homogeneous half-space can be assumed, and HDC can be close to the receiver, because those points do influence the calculated analytically. In this case, HDC does not even depend inversion result. The aim of the application of the following on the resistivity of the half-space (Strack 1992). The resulting conditions is to obtain reasonable approximations for the equation to determine EM(t=) is sensitivities for points close to the receiver, without having to EM = c design an extra grid for the finite difference code. HE,DC n EM. (16) t= Dt i For the implementation, this comes down to determining i=1 what value to use for E/ t. This will represent the time t= RESULTS range before t, where no accurate results can be obtained min from the modelling routine. Special care must be taken if both transmitter and receiver are at the surface of a half-space. In The sensitivities in the following examples are calculated for a half-space of 5 V m resistivity. The transmitter is an x-directed

4 716 A. Hördt electric dipole located at the origin. The parameters were taken 25 m (dashed line in Fig. 1) and at 125 m (solid line in Fig. 1) from two real surveys in France in 1994 and 1996 (Hördt & horizontal distance from the transmitter. The main feature is Petrat 1996). The goal of the surveys was to monitor variations that at 25 m distance from the transmitter ( left column) the in the gas level of an underground gas storage site. For that step response has a sharp behaviour in time, but is broader at purpose, data were recorded with the TEAMEX multichannel 125 m distance. The impulse response behaves in the opposite data acquisition system (patent No. PCT/DE 91/238). The way: it is sharp at 125 m and broader at 25 m. This is of data have the high redundancy required for a 2-D or 3-D course what was expected, because the behaviour is determined inversion. by the distance r r between transmitter and receiver, and Fig. 2 shows the electric fields and impulse responses required for calculation of the impulse responses, the transmitter was to calculate the sensitivities for a receiver 15 m away from put at 15 m. the transmitter for two points in the ground. The top row Fig. 3 shows the sensitivities. They were obtained by conshows the electric field step responses at 3 m depth and at volving the curves in Fig. 2 in the left column (for 25 m) and right column (for 125 m) with each other. The curves are very similar to each other, and this is a nice confirmation of the theory because in the ideal case they should be equal. The reason is reciprocity: since everything is on the z y plane, and there is symmetry with respect to the x z plane between transmitter and receiver, the sensitivities at 25 m horizontal distance from the transmitter should be equal to those at 25 m horizontal distance from the receiver. The small differences remaining in the actual calculation in Fig. 3 are due to differences in the finite difference forward modelling. The unit of sensitivity is V2 A 1m 3, which is equal to Vm 1(S m 1) 1 m 3. It means the sensitivity gives the increase oftheelectricfieldinvm 1 due to a change of conductivity in Sm 1 in a small volume. Since the sensitivities are negative in Fig. 3, an increase of conductivity would result in a decrease of the electric field. For this configuration and the given symmetries, this is what would be expected from the physics. Fig. 4 shows the magnetic field impulse responses required Figure 2. Top row: electric field step responses. E (r,t) in eq. (5) p to calculate magnetic field sensitivities. They were determined for r =(, 25 m, 3 m) (left column) and for r =(, 125 m, 3 m) using a loop source at the surface at m from the (right column). Bottom row: electric field impulse responses. transmitter and calculating the time derivative of the electric E(r, r,t)/ t in eq. (5) for r=(, 15, ) m; r is the same as in the top row. field at 3 m depth. In this case, the distance between trans- mitter and receiver is m instead of 15 m, because the finite difference code uses a staggered grid with a horizontal grid distance of 125 m. As for the electric field impulse responses, the curve close to the receiver (at 125 m) has a sharper decay than close to the transmitter (at 25 m). A feature common to both curves is that they change sign and that the area underneath is zero, which can be theoretically understood by taking the time derivative of eq. (15). The sensitivities are obtained by convolving the magnetic field impulse responses with the electric field step responses. The latter are identical to those shown in the top row in Fig. 2. The result is shown in Fig. 5. In this case, there is no reciprocity, Figure 3. Electric field sensitivities at 3 m depth for two different and the curves look quite different. The maximum amplitude distances from the transmitter. occurs at different times and is larger close to the transmitter. Figure 4. Magnetic field impulse responses. EM(r, r, t)/ t in eq. (9) for r =(, 25 m, 3 m) ( left) and r =(, 125 m, 3 m) (right).

5 T ime-domain EM sensitivities 717 Figure 5. Magnetic field sensitivities at 3 m depth for two different distances from the transmitter. Also, the sensitivity close to the receiver has a sign reversal, 4 V m compared to 5 V m background resistivity, which is a while the sensitivity close to the transmitter does not change 2 per cent resistivity perturbation. The sensitivity was calcu- sign with time. lated by taking the differences of the two curves calculated Figs 3 and 5 show that the time of maximum sensitivity at with and without the body, and normalizing by the body a certain depth depends on the location of the point and on volume and conductivity difference. The comparison with the the configuration. However, a comparison with simple concepts curve calculated with the convolution approach is shown in to estimate the penetration depth of the method provides some Fig. 6. The solid curve is the same as the circles in Fig. 3. Now insight into the physics. For example, consider the maximum a log log scale is used, because this is more appropriate for sensitivity in Fig. 3 which occurs at approximately 4 ms. The visualizing a large time range. The curves are in very good diffusion depth (Spies 1989), which is the depth of the maximum agreement, even though there are still visible differences. These plane wave electric field at a certain time, is approxi- are probably due to the perturbation approach, because it has mately 56 m at 4 ms, which is nearly twice as deep as the true the inherent problem that two large numbers are subtracted depth of the point considered here. Directly below the trans- from each other to obtain a small number. To increase the mitter (not shown in the figures) the sensitivity at 3 m depth differences, the body size could be increased, but this could has a maximum at approximately 6 ms. This may be compared result in leaving the region of linearity. Considering this with the depth used for 1-D imaging techniques (Eaton & trade-off, the agreement in Fig. 6 can be considered satisfactory. Hohmann 1989; Karlik 1995), which is equal to the depth of Fig. 7 shows the same comparison for the magnetic field the maximum of the electric field under the transmitter. For sensitivities. The solid curve is the same as the dashed line 6 ms we obtain 98 m, which is again significantly greater than in Fig. 5. The agreement between the two approaches is very the depth considered here. We may conclude that the maximum good, including the critical range around the sign reversal sensitivity at a given depth occurs later than the maximum at about 7 ms. Note that the grid was not really set up to electric field. The reason is that the sensitivity is obtained by give accurate results at very early times, which explains the convolving the electric field with the appropriate Green s inaccuracies before 1 ms. function, and this procedure always delays the maximum. As mentioned before, the implementation of the convolution Next, a comparison of the convolution approach with the may become critical if the location for which the sensitivity is perturbation approach is presented. For this purpose, a calculated is very close to the receiver. The problem arises body with extensions 125 m in the x- and y-directions and from an undersampling of the impulse response, because 5 m in the z-direction was introduced with its centre at close to the receiver the decay of the Green s function is very 125 m horizontal distance, 3 m depth. The resistivity is sharp. Of course, one could decrease the minimum time in the Figure 6. Comparison of electric field sensitivities calculated with the perturbation and convolution approaches for 3 m depth and 125 m horizontal distance from the transmitter.

6 718 A. Hördt Figure 7. Comparison of absolute values of magnetic field sensitivities calculated with the perturbation and convolution approaches for 3 m depth and 125 m horizontal distance from the transmitter. The notch at 7 ms comes from a sign reversal of the sensitivities. modelling code, but in most cases this would mean having for sensitivities close to zero ( S <1 2 V2 A 1m 4), and to design a different grid, an additional effort which can be logarithmic for S &1 2 V2 A 1m 4. A +2 on the colour avoided. Fig. 8 gives an example of how the correction works scale corresponds to +( )=+1 18, a 2 for the magnetic field. Sensitivities were calculated for a point corresponds to ( )= at the surface, at 125 m distance from the receiver using three The most striking feature is the line below the receiver different methods. First, the sensitivity was calculated in the which separates negative from positive sensitivities. Directly same way as before, using no correction (dashed lines). The underneath the receiver, the sensitivity is zero. This can be minimum time used in the finite difference code was.1 ms. understood, since the vertical magnetic field directly above an The time-domain wavenumber (Ward & Hohmann 1988) for anomalous current is zero. At intermediate times, the situation.1 ms and 5 V m is 1/126 m, which means that a 125 m changes dramatically. Near the receiver, the pattern is replaced distance is in the critical range where the Green s function is by one with opposite sign. The original pattern, with positive undersampled. Second, the sensitivity was calculated using an and negative sensitivities before and behind the receiver has analytic expression for both the electric field step response and penetrated to greater depths. At later times the original pattern the magnetic field impulse response, using 1 ms sampling rate has been completely replaced by the new one with opposite (solid line). Third, the sensitivity was calculated using the sign. Around the transmitter, the sensitivity never changes sign, responses from the finite difference code with the improper and only decreases in amplitude. sampling, but using eq. (16) to calculate the value of the Next, a conductive body is introduced into the otherwise impulse response at t= (circles). The figure shows that this homogeneous half-space. The dimensions of the body are properly corrects for the undersampling, and the corrected 1 m 1 m 2 m and it has a conductivity of 5 V m. sensitivity is in reasonable agreement with the one that was It extends from 1 to 2 m in the y-direction, 3 to 5 m calculated analytically. in the z-direction, and is centred in the x-direction. Magnetic The sensitivities can be used for inversion, or simply to field sensitivities with and without the body are shown in give some insight into the resolution properties of the method. Fig. 1. The sensitivities within the body are higher than in the In particular, the important question of which parts of the absence of the body, which is no surprise because the magnetic subsurface contribute to the response can be answered using field should be more sensitive for changes in a conductive a visualization of sensitivities. Fig. 9 shows magnetic field body. The body seems to delay the replacement of the original sensitivities for a 5 V m half-space for three different delay pattern by a new one of opposite sign. The currents in the times. In order to display both sign changes and the large body do not decay away so quickly and maintain the original dynamic range, a special colour scale was used. It is linear sensitivity feature. Another important observation is that the Figure 8. Comparison of different methods to calculate the magnetic field sensitivity at r =(, 1375, ) m.

7 T ime-domain EM sensitivities 719 body, though having quite a large contrast of 1 : 1, does not have a huge effect on the overall sensitivity pattern. Electric field sensitivities with and without the body are shown in Fig. 11. The top panel, without the body, shows that reciprocity is fulfilled. The pattern is symmetric with respect to a line between transmitter and receiver. Sensitivities are maximum around the transmitter and receiver and quickly decrease elsewhere. In the bottom panel, the pattern is modified by the conductive body. Within and below the conductor, the sensitivity is significantly smaller, by more than two orders of magnitude, than in the absence of the body. This is a little counter-intuitive, because we might expect stronger currents to flow in the body which increase the sensitivity, just as for magnetic fields. However, the dominating effect here seems to be that the electric field of those increased currents is very small at the surface because it is screened by the high-contrast boundary at the top of the body. We also have to remember that in the middle of the symmetric body there is no vertical electric field component, and no charges accumulate at the top boundary of the body, which might increase the sensitivity. structure, and show with example inversions that this approach is feasible. For the formalism presented here, this would mean replacing E/ t in eqs (5) and (9) with the field calculated for a layered background structure, and could be applied without restrictions. ACKNOWLEDGMENTS This work was carried out during a sabbatical at Western Atlas Logging Services in Houston, Texas, following the invitation of K.-M. Strack. I enjoyed discussions with the members of the Advanced Scientific Research group, in particular with Lev Tabarovsky. I would like to thank Charles Stoyer, Colin Farquharson and Enrique Gomez-Trevino for their useful comments on the original manuscript. The project was partly sponsored by Western Atlas Logging Services and by the Deutsche Forschungsgemeinschaft (Project No.: Ho 156/4-1). I am indebted to Vladimir Druskin and Leonid Knizhnerman for their continuous support and permission to use their program. CONCLUSIONS REFERENCES It has been shown using computational examples that the use Boerner, D.E. & West, G.F., Fréchet derivatives and single of the reciprocity theorem is quite feasible and advantageous scattering theory, Geophys. J. Int., 98, for calculating 3-D sensitivities directly in the time domain. Chave, A.D., The Fréchet derivatives of electromagnetic induction. J. geophys. 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