Joint MT and CSEM data inversion using a ...

GEOPHYSICS, VOL. 76, NO. 3 (MAY-JUNE 2011); P. F203–F214, 13 FIGS., 1 TABLE. 10.1190/1.3560898

Joint MT and CSEM data inversion using a multiplicative cost function approach

A. Abubakar1, M. Li1, G. Pan1, J. Liu1, and T. M. Habashy1

electromagnetic receivers on the seafloor, this method has been applied in several field surveys. The high contrast in resistivity between saline-filled rocks and hydrocarbons (oil and gas) makes this method well suited for detecting resistive targets such as hydrocarbon reservoirs; see Constable et al. (1986), Srnka (1986), Chave et al. (1991), MacGregor and Sinha (2000), Eidesmo et al. (2002). To make use of CSEM data one has to use nonlinear inversion approaches, such as the Gauss-Newton method (Abubakar et al., 2008; Abubakar et al., 2009), (preconditioned) nonlinear conjugate gradient methods (Commer and Newman, 2008a) or the limited memory quasi-Newton method (Plessix and Mulder, 2008). A history and brief technical overview of this CSEM technique can be found in Constable (2010). The magnetotelluric (MT) method has a long history of use in diverse applications such as mining and geothermal explorations. It also has been applied in offshore hydrocarbon explorations, e.g., Constable et al. (1998), Hoversten et al. (1998), Constable et al. (2009), Smirnov and Pedersen (2009). MT data largely are insensitive to thin, high resistivity targets that are associated with hydrocarbon deposits trapped in thin planar sedimentary layers. This is because MT signals, which have characteristics of horizontally polarized plane waves, are more sensitive to conductive targets. However, it is known that MT has a larger depth of investigation than CSEM. Hence, MT can be used effectively to construct resistivity estimates of the subsurface (i.e., the background medium). MT recently has been used for improving seismic depth imaging workflows (Colombo and De Stefano, 2007; Virgilio et al., 2009). Similar to CSEM, interpretation of MT data has to be done using full nonlinear inversion approaches such as those described by Rodi and Mackie (2001), Kumar et al. (2007), Abubakar et al. (2009). MT and CSEM data can provide complementary information: MT mainly provides information on background resistivity structures, whereas CSEM identifies thin resistive targets. Constable and Weiss (2006) also point out the practical importance of MT data: The same receivers can collect CSEM and MT data; hence

ABSTRACT We have developed an inversion algorithm for jointly inverting controlled-source electromagnetic (CSEM) data and magnetotelluric (MT) data. It is well known that CSEM and MT data provide complementary information about the subsurface resistivity distribution; hence, it is useful to derive earth resistivity models that simultaneously and consistently fit both data sets. Because we are dealing with a large-scale computational problem, one usually uses an iterative technique in which a predefined cost function is optimized. One of the issues of this simultaneous joint inversion approach is how to assign the relative weights on the CSEM and MT data in constructing the cost function. We propose a multiplicative cost function instead of the traditional additive one. This function does not require an a priori choice of the relative weights between these two data sets. It will adaptively put CSEM and MT data on equal footing in the inversion process. The inversion is accomplished with a regularized Gauss-Newton minimization scheme where the model parameters are forced to lie within their upper and lower bounds by a nonlinear transformation procedure. We use a line search scheme to enforce a reduction of the cost function at each iteration. We tested our joint inversion approach on synthetic and field data.

INTRODUCTION The controlled-source electromagnetic (CSEM) method has the potential of providing useful information in applications such as offshore hydrocarbon exploration. With a horizontal electric dipole transmitter towed by a ship and multicomponent

Manuscript received by the Editor 27 October 2010; revised manuscript received 00 0000; published online 5 May 2011. 1 Schlumberger-Doll Research, Cambridge, Massachusetts, U.S.A. E-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]. C 2011 Society of Exploration Geophysicists. All rights reserved. V

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MT data come at a relatively low cost because they can be recovered from time-series data when the source is turned off. Usually, CSEM and MT data are inverted separately. In the joint interpretation workflow, MT data first is inverted and then the results are used as an initial model for CSEM data inversion. Mackie et al. (2007) propose a joint inversion method where one inverts both data sets simultaneously. By using this approach, they can obtain reasonable results with a reduced number of sources (for CSEM) and receivers. Commer and Newman (2008b) also show that one can significantly improve the inversion results by using this simultaneous joint inversion approach. However, Mackie et al. (2007) and Commer and Newman (2008b) point out that in this joint inversion approach one has to carefully choose the relative weighting between MT and CSEM data. We propose to use a multiplicative cost function for the simultaneous joint inversion of MT and CSEM data. This multiplicative cost function will effectively put equal weighting for MT and CSEM data. This approach has been used previously for multifrequency electromagnetic inversions; see Abubakar and van den Berg (2000). Unlike Mackie et al. (2007) and Commer and Newman (2008b), we use the regularized GaussNewton approach as our inversion scheme. The general description of the forward and inversion algorithms can be found in Abubakar et al. (2008). The forward algorithm is implemented based on the theory described in Druskin and Knizhnerman (1994) and Ingerman et al. (2000). The inversion algorithm uses the Gauss-Newton framework described in Habashy and Abubakar (2004) augmented with the multiplicative regularization technique introduced in van den Berg et al. (1999). The inversion algorithm is equipped with two regularization functions to produce either a smooth (using a standard L2 -norm function) or a blocky (using a weighted L2 norm function) conductivity distribution (van den Berg and Abubakar, 2001). To enhance the robustness of the algorithm, we incorporated a nonlinear transformation for constraining the minimum and maximum values of the conductivity distribution and a linesearch procedure for enforcing the error reduction in the cost function in the optimization process; see Habashy and Abubakar (2004). To illustrate the performance of our inversion method, we consider synthetic data inversion using isotropic and transverse isotropic (TI) media. We also will show the inversion results of field data collected in the Norwegian shore. The survey is carried out between the Møre basin and the Møre margin in the north Norwegian Sea. The geological environment is quite complex, covering the margin between oceanic and continental crust (the Møre margin), and the western part of the Møre basin, in which Eocene basalts are present within the section; see Blystad et al. (1995). As pointed out by Virgilio et al. (2009), one of the objectives of this survey is to determine the thickness of prebasalt sediments, and, if possible, identify the crystalline basement.

dMT ¼ sMT ðmÞ; where

(2)

h   dCSEM ¼ d CSEM rSi ; rRj ; xk ; i ¼ 1; 2; …; I; j ¼ 1; 2; …; J; k ¼ 1; 2; …; K1 T

is the vector of CSEM data and

h   iT dMT ¼ d MT rRj ; xk ; j ¼ 1; 2; …; J; k ¼ 1; 2; …; K2 (4) is the vector of MT impedance data, in which rSi , rRj , and xk are the source positions, the receiver positions, and the frequency of operations, respectively. The superscript T denotes the transpose of a vector. The vectors sCSEM ðmÞ and sMT ðmÞ represent simulated data computed by solving Maxwell equations using a finite-difference method in the frequency domain. In CSEM the source is an electric dipole oriented parallel to the tow line, and the receiver can be any component of the magnetic or electric field vector. In MT the source is a plane wave and the receivers measure the horizontal components of electric and magnetic fields. In this work we use the 2.5D approximation for CSEM data, see Abubakar et al. (2008), and 2D approximation for MT data. The forward algorithm uses the framework described in Druskin and Knizhnerman (1994); however, we use a multifrontal LU decomposition (Davis and Duff, 1997) to invert the stiffness matrix. By using this direct matrix inversion technique, we can simulate multisource experiments at nearly the cost of simulating only one single-source experiment. This feature is quite important for the inversion because it uses data from more than one source position=orientation. The direct matrix inversion technique is accomplished using the optimal grid technique (Ingerman et al. 2000) to extend the boundaries of the computational domain to infinity, and a diagonal anisotropic material averaging formula described in Keller (1964) to assign appropriate conductivity values on the finite-difference grid nodes. The unknown vector of model parameters is defined as:

      m ¼ mh x‘ ; zq ; mv x‘ ; zq ; ‘ ¼ 1; …; L; q ¼ 1; …; Q ;

(5)

The nonlinear inverse problems are described by the following operator equations:

where x‘ and zq denote the center of the discretization cell. The unknown model parameters mh ðrÞ ¼ rh ðrÞ=r0 ðrÞ and mv ðrÞ ¼ rv ðrÞ=r0 ðrÞ are the normalized horizontal and vertical conductivities where r0 ðrÞ is the conductivity distribution of the initial model used in the inversion. In this paper we assume that the conductivity is TI-anisotropic (defined as rh ¼ rxx ¼ ryy , rv ¼ rzz with all other components of the conductivity tensor set to zero, where x, y, and z denote the Cartesian axes). Following Abubakar and van den Berg (2000) and Abubakar et al. (2008), we simultaneously invert CSEM and MT data using a multiplicative cost function defined as

dCSEM ¼ sCSEM ðmÞ;

Un ðmÞ ¼ /CSEM ðmÞ  /MT ðmÞ  /Rn ðmÞ;

INVERSION METHOD

(1)

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(6)

Joint MT and CSEM data inversion where the CSEM data cost function is given by h i2 I P J  P  CSEM CSEM  CSEM w d  s ð m Þ   i;j;k i;j;k i;j;k K1 1 X i¼1 j¼1 CSEM / ðmÞ ¼  I P J  P 2K1 k¼1  CSEM CSEM 2 wa;b;k da;b;k  a¼1 b¼1

  2 ¼ W CSEM dCSEM  sCSEM ðmÞ

(7)

and the MT data cost function is given by

/MT ðmÞ ¼

h i2 J  P  MT MT  ð m Þ wj;k dj;k  sMT  j;k

K2 1 X j¼1 2K2 k¼1

 J  P  MT MT 2 wb;k db;k  b¼1   2 ¼ W MT dMT  sMT ðmÞ ;

2

CSEM W i;j;k

6 6 ¼6 4

31 2

7 7 7   I P J 2 P  CSEM CSEM  5 2K1 wa;b;k da;b;k  wCSEM i;j;k

(9)

a¼1 b¼1

and

2 6 6 W MT j;k ¼ 6 4

for the weighted L2 -norm regularizer as introduced in van den Berg and Abubakar (2001). In the above equations, D denotes the inversion domain and mn denotes the unknown parameter at iteration nth. The L2 -norm regularizer is known to favor smooth profiles, while the weighted L2 -norm regularizer is known for its ability to preserve edges. Note that for the L2 -norm regularizer, the weight bj;n ðx; zÞ is independent of the spatial position. This weighted L2 -norm regularization factor in equations 12 and 14 belongs to the same class as the well-known total variation regularization (Vogel and Oman, 1996; Charbonnier et al., 1997; Farquharson and Oldenburg, 1998; Farquharson, 2008). The positive parameter dn is chosen to be

d2n ¼

(8)

in which wCSEM and wMT i;j;k j;k are real-valued data weighting matrices. The normalized data weighting matrices W CSEM and W MT are given by

7 wMT j;k 7 7:   J 2 P  MT MT  5 2K2 wb;k db;k 

(10)

(16)

where the gradient vector gn is given by gn ¼ /MT ðmn Þr/CSEM ðmÞm¼mn

n h  CSEM H io12 wCSEM ¼ diag JCSEM J0 ; 0

(11)

þ /CSEM ðmn Þr/MT ðmÞm¼mn þ /CSEM ðmn Þ/MT ðmn Þr/Rn ðmÞm¼mn n H  CSEM T CSEM  CSEM o ¼ /MT ðmn Þ JCSEM W W d  sCSEM ðmn Þ n n o H  MT T MT  MT W W d  sMT ðmn Þ þ /CSEM ðmn Þ JMT n þ /CSEM ðmn Þ/MT ðmn ÞLð1Þ ðmn Þmn ;

where JCSEM is the CSEM Jacobian matrix of the initial model. 0 The regularization cost function is given by j¼x;z D

(15)

Hn pn ¼ gn ;

In our implementation, wMT j;k is chosen to be an identity matrix while wCSEM is based on the so-called Jacobian weighting defined as i;j;k

Xð

/CSEM ðmn Þ/MT ðmn Þ ; DxDz

where Dx and Dz are discretization cell widths. A discussion about the choice of dn can be found in van den Berg et al. (2003). To minimize Un ðmÞ in equation 6, we use a Gauss-Newton minimization framework described in Habashy and Abubakar (2004). At the nth iteration, we obtain a set of linear equations for the search vector pn that determines the minimum of the approximated quadratic cost function, namely,

312

b¼1

/Rn ðmÞ ¼

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n b2j;n ðx; zÞ joj ln½mh ðx; zÞj2

o þjoj ln½mv ðx; zÞj2 þ d2n dxdz;

(12)

in which ox and oz denote spatial differentiation with respect to x and z. The weights bj;n ðx; zÞ are given by 1 o b2j;n ðx; zÞ ¼ Ð n      oj ln mh;n ðx; zÞ 2 þ oj ln mv;n ðx; zÞ 2 þ d2 dxdz

(17)

where H denotes the transpose conjugate of a matrix. In the above equation we used the fact that /Rn ðmn Þ ¼ 1. The matrices JCSEM and JMT are the Jacobian matrices and their explicit n n expression can be found in Abubakar et al. (2008). The gradient of the regularization cost function is given by

h

i Lð1Þ ðmn Þv

‘;q

¼

1

X

mn;‘;q j¼x;z

oj bj;n ½oj lnðvÞ

 ‘;q

:

(18)

n

D

(13)

for the L2 -norm regularizer and

1 1 ;       V oj ln m h;n ðx; zÞ 2 þ oj ln m v;n ðx; zÞ 2 þ d2 n ð V ¼ dxdz (14)

b2j;n ðx; zÞ ¼

D

The Hessian matrix is given by

h H  CSEM T CSEM CSEM i Hn ¼ /MT ðmn Þ JCSEM W W Jn n h i    H T MT MT MT þ /CSEM ðmn Þ JMT W W J n n þ /CSEM ðmn Þ/MT ðmn ÞLð2Þ ðmn Þ:

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(19)

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In the above equation, to make the Hessian matrix nonnegative definite, we neglect the second-order derivatives of the cost function /CSEM ðmÞ and /MT ðmÞ as well as the nonsymmetric terms. The second derivative of the regularization cost function is given by

h

i L ð mn Þ  v

  X v ¼ oj bj;n oj : mn;‘;q j¼x;z mn ‘;q 1

ð 2Þ

‘;q

(20) For this multiplicative cost function the gradient is exact; however, the Hessian matrix is an approximate one because we neglected the second-order derivatives and the nonsymmetric terms. After the Gauss-Newton search vector pn is obtained by solving the linear system of equation 16 using a conjugate gradient least-squares (CGLS) technique, Golub and Van Loan (1996), the unknown model parameters are updated using a nonlinear transformation and the line-search procedure described in Habashy and Abubakar (2004). This multiplicative cost function in equation 6 will make MT and CSEM cost functions equally important. This is achieved by using the MT data cost function at the previous iteration /MT ðmn Þ as the weight for the CSEM cost function, and vice versa; see equations 17 and 19. This strategy is reasonable when we do not have a priori information on which data set is more reliable or should be dominant in the inversion process. When such a priori information is available, a constant relative weighting that puts emphasis on one of the cost functions preferably should be used. As for the weight for the regularization factor, a large weight at the beginning of the optimization process is used because the values of /CSEM ðmÞ and /MT ðmÞ still are large. In this case, the search direction is predominantly a steepest descent, which is more appropriate to use in the initial steps of the iteration process because it has a tendency to suppress large swings in the search direction. As the iteration proceeds, the optimization process gradually will reduce the error in the data misfit while the

Figure 1. (a) True and (b) initial models of the isotropic test case. The color bars are in terms of the logarithm of the resistivity (Ohm-m).

regularization factor /Rn ðmÞ remains at a nearly constant value close to unity. In this case, the search direction corresponds to a Newton search direction, which is more appropriate to use as we get closer to the minimum of the data misfit cost function of either /CSEM ðmÞ or /MT ðmÞ where the quadratic model of the cost function becomes more accurate. If noise is present in the data, the data misfit cost functions /CSEM ðmÞ and /MT ðmÞ will plateau to values determined by the signal-to-noise ratio; hence, the weight on the regularization factor is nonzero. In this way, the noise will, at all times, be suppressed in the inversion process and the need for a larger regularization when data contain noise will be automatically fulfilled. The optimization process is terminated if one of the following stopping criteria is satisfied: The value of the data misfit U gets below a prescribed error quantity, or the difference between the data misfit U at two successive iterates is within a prescribed error quantity, or the change in the model m at two successive iterates is within a prescribed error quantity, or the total of iterations exceeds a prescribed maximum.

NUMERICAL RESULTS AND DISCUSSIONS Isotropic synthetic data inversion We present a synthetic data inversion example in which two thin reservoirs are embedded in a nonflat layered formation. The horizontal distance between these reservoirs is about 1 km. The widths of the reservoirs are about 5 km and the resistivities of the reservoirs are 50 Ohm-m. The resistivity distribution of this test model is shown in Figure 1a, and the color bar is given in terms of the logarithm of the resistivity. The seabed consists of three nonflat layers. Their resistivity values from top to bottom are 1.67, 3.33, and 6.67 Ohm-m. The seawater depth is 840 m and the water resistivity is 0.33 Ohm-m. For the survey, we used 21 receivers located at the seafloor. These receivers are deployed uniformly from 10 km to 10 km. They record CSEM and MT data and they are indicated by circle signs in Figure 1a. For CSEM data, we invert only the inline fields (horizontal electric fields in the tow direction and the sources are horizontal electric dipoles also oriented in the tow direction), while for MT data we use transverse electric (TE) and transverse magnetic (TM) polarization impedances. The CSEM source is towed from 10 to 10 km at 50 m above the receivers. In the inversion we only use CSEM sources located every 500 m; hence, we have 41 CSEM sources. We use three CSEM frequencies (0.25, 0.75, and 1.25 Hz) and eight MT frequencies (0.005, 0.01, 0.025, 0.05, 0.1, 0.25, 0.5, and 1 Hz). The synthetic data sets are generated using a 3D frequency-domain forward modeling code described in Zaslavsky et al. (2006). For each frequency, we also added random white noise of 2% maximum amplitude of all data points to CSEM and MT data. Hence, the noises are frequency-dependent; however, they are not offset-dependent. In the inversion we use a domain of 36 by 11.16 km with grid size 200 by 60 m. The number of unknowns is 33,480. For this case we assumed that the resistivity distribution is isotropic; hence we have m ¼ mh ¼ mv .

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Joint MT and CSEM data inversion For the initial model we only use a model with a homogeneous seabed as shown in Figure 1b. The homogeneous seabed resistivity is 6.67 Ohm-m. Figure 2a shows the reconstructed image using only MT data. We observe that the three seabed layers are properly reconstructed in shape and conductivity values; however MT inversion failed to detect the presence of the reservoir. This reflects the limitation of MT data where the low-frequency electromagnetic plane waves can have a large depth of investigation and good horizontal resolution; however they lack sensitivity to resistive layers. Note that the regions x < 10 km and x > 10 km are not well reconstructed because we do not have receivers in these regions. In addition, the resistivity of the seabed for depths larger than 8 to 9 km is not well reconstructed. The latter is

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because we already reached the limit of the MT data depth of investigation at the frequency of operations. The inversion result of CSEM data is given in Figure 2b. In this case we reconstruct a rough estimate model of the reservoir; however, the layered formation is not reconstructed properly. This mainly is because of the limited depth of investigation of CSEM data. Because the layered formation is not well reconstructed, the CSEM inversion tries to compensate by putting relatively shallow conductive and resistive objects close to the interface between the water and seabed. In addition, the depth and shape of the reservoirs are not very well reconstructed. Figure 3 shows the joint inversion results using CSEM and MT data. In Figure 3a, we first invert MT data and then use the result as an initial model for the CSEM data inversion. By comparing the inversion result in Figure 3a with the result in Figure 2b, we observe quite significant improvements. In this case, the thicknesses of the reservoirs are better reconstructed; however, the depths of both reservoirs are more or less the same. In Figure 3b, we present the result of inverting MT and CSEM data simultaneously. We observe that the reservoir and the layers both are well reconstructed. This is because MT data are sensitive to the layering structure and CSEM data mainly are sensitive to the resistive layers. At first glance it seems that both joint inversion approaches (sequential and simultaneous) produce similar results. However, close examination of the results reveals that the depths of the reservoirs are better reconstructed by using the simultaneous joint inversion approach (i.e., they are not located at the same depth). In Figure 4 we plot the true model and inverted models obtained by using joint inversion approaches only on the region 10 km < x < 10 km and 1 km < z < 4 km. From Figure 4, we also observe that the second layer and the third layer of the seabed are reconstructed better using the simultaneous joint inversion algorithm. This also is apparent when we compare the normalized data misfits (the cost functions without including the

Figure 2. Inverted models of the isotropic test case using (a) MT data and (b) CSEM data.

Figure 3. Inverted models of the isotropic test case using joint MT and CSEM data inversion with (a) the sequential approach and (b) the simultaneous approach.

Figure 4. Zoom-in of the isotropic test case (a) true model, (b) the sequential approach, and (c) the simultaneous approach.

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data weighting matrices) for every frequency of the joint CSEM and MT data inversion run; see Table 1. In the table we show also the number of iterations ðnÞ used by each inversion run. From this table we observe that the simultaneous joint CSEM and MT data produce the lowest data misfits. It seems that this joint inversion helps the CSEM data inversion more than the MT data inversion, because the MT data misfits of the separate MT inversion do not reduce as much as the CSEM data misfits. This exercise also shows that without the use of a proper background medium (either derived from an MT inversion or from other independent measurements) the CSEM data interpretation will not produce reliable inversion results.

FORCE field data inversion We present the inversion results of the so-called FORCE data set. The survey is carried out between the Møre basin and the Møre margin in the north Norwegian Sea as shown in Figure 5. The geological environment is quite complex, covering the margin between oceanic and continental crust (the Møre margin), and the western part of the Møre basin, in which Eocene basalts are present within the section; see also Virgilio et al. (2009). One of the objectives of this survey is to determine the thickness of prebasalt sediments, and, if possible, identify the crystalline basement. More information on the geology of the area can be found in Blystad et al. (1995). The survey line extends from northwest to southeast. Fourteen MT stations were used. The frequency of the acquired MT data ranges from 0.0003 to 0.1 Hz, among which eight frequencies are used in our inversion from 0.02 to 0.09 Hz. For MT data we use TE and TM polarization impedances. In the CSEM survey, there are 14 receivers and 324 sources. The receiver locations are indicated in Figure 6a by small white rectangles, and they are located between x ¼ 15 km and

x ¼ 25 km. We only use data at three frequencies (0.125 Hz, 0.25 Hz, and 0.375 Hz) for the CSEM data inversion. The number of measurement data points is 2,928. For the inversion domain, we choose a rectangular domain with size 70 by 8.8 km. The grid size is 250  50 m. Hence, the total of unknown resistivity cells is 42,450. The initial model used in the inversion is a homogeneous seabed model with resistivity 2.5 Ohm-m as shown in Figure 6a. The unknown medium is assumed to be an isotropic one. Note that the interface between the water and seabed is not flat. Figure 6b shows the inversion results using only MT data. The inversion process stops after 24 Gauss-Newton iterations. In this figure, we observe a large resistive body on the left (its size extends beyond 8 km in depth). These results are consistent with the sensitivity of the MT data (we have deep penetration depth; however, we obtain lower resolution because of the very low operating frequency). Next, we invert the CSEM data only and the results are shown in Figure 6c. In this case the top of the salt is reconstructed clearly, but the image of the deep region beyond 4 km cannot be trusted because of the lack of sensitivity of the CSEM data. We also observe that the CSEM data has a better resolution because it operates at higher frequencies. For the CSEM inversion we use 31 iterations. Next, we invert CSEM and MT data either sequentially or simultaneously. The reconstructed images are shown in Figure 7a and 7b. The CSEM data inversion using the MT data inversion results as its initial model takes 31 iterations, whereas the simultaneous joint inversion takes 22 iterations. In both joint inversion results the salt body is well reconstructed. However, in the simultaneous joint inversion result the presence of a resistive body (which probably is an artifact) located under the salt layer is significantly weaker than in the sequential joint inversion result. When we compare these inversion results with the results from the seismic interpretation as shown in Figure 8a and 8b, or with the known geology of the area as shown in Figure 8c, we

Table 1. Normalized data misfit values (in %) for the isotropic test model inversion. CSEM CSEM, 0.25 Hz CSEM, 0.75 Hz CSEM, 1.25 Hz MT, 0.005 Hz MT, 0.01 Hz MT, 0.02 Hz MT, 0.05 Hz MT, 0.1 Hz MT, 0.25 Hz MT, 0.5 Hz MT, 1 Hz n

MT

35.169 68.436 97.403

4

5.006 3.064 2.416 1.853 1.752 1.865 1.680 1.860 11

Sequential

Simultaneous

3.296 4.473 5.273

1.822 2.270 2.163 4.476 2.513 2.343 1.871 1.592 1.759 1.642 1.792 21

31

Figure 5. Survey outline of the FORCE data set. This plot is obtained from Virgilio et al. (2009).

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Joint MT and CSEM data inversion observe good correlations between them, especially for the simultaneous joint inversion result. This shows that the joint MT and CSEM inversion scheme using the multiplicative cost function might greatly improve the data interpretation. In Figure 9, we show the comparison between MT measured data and simulated data for the initial and inverted models for TM and TE polarization at frequency 0.05 Hz. The top plots in Figure 9 show the real part of the impedances whereas the bottom plots show the imaginary part. In Figure 10, we also show the comparison between CSEM measured data and simulated data for the initial and inverted models for data at 0.125 Hz, 0.25 Hz, and 0.375 Hz and for the receiver located close to the center of the inversion domain. Figures 9 and 10 show excellent agreement between the measured and simulated data of the inverted model.

TI-anisotropic synthetic data inversion By using this final example, we discuss pros and cons of carrying out the joint MT and CSEM data inversion for TI-anisotropic media. For the synthetic test example, we consider a thin reservoir embedded in a seabed that consists of three nonflat layers as shown in 11a and 11b. The width of the reservoir is about 6 km and its depth is about 1.5 km below the water-seabed interface.

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The horizontal and vertical resistivities of the reservoir are 20 Ohm-m and 50 Ohm-m, respectively. The horizontal resistivities of the layers (from top to bottom) are 1, 2, and 4 Ohm-m, while their vertical resistivities are 1.33, 3.33, and 6.67 Ohm-m. The thickness of the first and second (from the top) layers are about 1.5 and 1 km. The seawater depth is 1 km and the water resistivity is 0.33 Ohm-m. For the survey, we used 21 receivers located at the seafloor. These receivers are deployed uniformly from 10 to 10 km. They record CSEM and MT data and are indicated by circles in Figure 11. For CSEM data we invert only the inline fields (horizontal fields in the tow direction) while for MT data we use TE and TM polarization impedances. The CSEM source is towed from 10 to 10 km at 50 m above the receivers. In the inversion we only use CSEM sources located every 500 m; hence, we have 41 CSEM sources. We use two CSEM frequencies (0.25 and 0.75 Hz) and eight MT frequencies (0.005, 0.01, 0.025, 0.05, 0.1, 0.25, 0.5, and 1 Hz). The synthetic data sets are generated using the 3D frequency-domain forward modeling code described in Zaslavsky et al. (2006). For each frequency, we added random white noise of 2% maximum amplitude of all data points to CSEM and MT data. In the inversion we use a domain of 36 by 6 km, with grid size 200 by 75 m. The number of unknowns is 14,400. For the

Figure 6. Initial model (a) and the FORCE data inversion results from (b) MT data and (c) CSEM data. The color bars are in terms of the logarithm of the resistivity (Ohm-m), and the axes of the figures are in meters.

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Figure 7. The FORCE data inversion results from (a) joint sequential MT and CSEM data and (b) joint simultaneous MT and CSEM data. The color bars are in terms of the logarithm of the resistivity (Ohm-m), and the axes of the figures are in meters.

Figure 8. Correlation of the joint MT and CSEM inversion results with seismic and geologic data (a) sequential inversion results, (b) simultaneous inversion results, and (c) plot of a geologic section from Virgilio et al. (2009).

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Joint MT and CSEM data inversion

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Figure 9. MT data fit of the joint MT and CSEM inversion result for (a) TM impedance (Ohm) and (b) TE impedance (Ohm) at 0.05 Hz. The top plots show the real-part of the impedances while the bottom plots show the imaginary-part.

Figure 10. CSEM data fit of the joint MT and CSEM inversion results for CSEM frequencies at (a) 0.125 Hz, (b) 0.255 Hz, and (c) 0.375 Hz. The receiver used is located around the center of the inversion domain. The top plots show the amplitude of the electric fields (in logarithmic scale) while the bottom plots show the phase of the electric fields. The electric fields are in V=m.

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initial model we use a model with a homogeneous isotropic seabed as shown in Figure 11c and 11d. The homogeneous seabed resistivity is 6.67 Ohm-m. Figure 12a and 12b show the reconstructed horizontal and vertical resistivity images using only MT data. The inversion uses 11 iterations. We observe that the horizontal resistivity distribution of the layered formation is reconstructed pretty well. However, the MT data inversion fails to reconstruct the vertical resistivity distribution of the layered formation. In addition to that, similar to the isotropic test case, the MT data inversion could not detect the presence of the reservoir in either the horizontal or vertical resistivity images. These observations are consistent with the sensitivity of the MT measurements. Next, we show the CSEM data inversion results in Figure 12c and 12d. These results were obtained after 41 iterations. We observe that the reservoir is present in the vertical resistivity image. However, in the horizontal resistivity image, the CSEM data inversion can produce only a vague image of the first and second layers of the seabed. This is consistent with the 1D studies done in Everett and Constable (1999), Constable and Weiss (2006), Ramananjaona et al. (2008), and the 2D study in Abubakar et al. (2010), which show that the CSEM inline electric field data are mainly sensitive to the vertical resistivity. Furthermore, we notice that the CSEM data inversion only properly reconstructs the top of the reservoir. The thickness and bottom shape of the reservoir clearly are overestimated.

Next, we invert MT and CSEM data simultaneously. The inversion results after 14 iterations are shown in Figure 12e and 12f. We observe some improvements in the reconstructed shape of the reservoir and the layered formation of the seabed. Moreover the estimate of the vertical resistivity value of the reservoir

Figure 12. The TI-anisotropic inversion results: (a) MT, horizontal resistivity; (b) MT, vertical resistivity; (c) CSEM, horizontal resistivity; (d) CSEM, vertical resistivity; (e) joint, horizontal resistivity; (f) joint, vertical resistivity.

Figure 11. The (a, b) true and (c, d) initial TI-anisotropic models: (a, c) horizontal resistivity, (b, d) vertical resistivity. The color bars are in terms of the logarithm of the resistivity (Ohm-m).

Figure 13. The isotropic inversion result of the simultaneous joint inversion algorithm.

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Joint MT and CSEM data inversion is more accurate. However, we still could not detect the reservoir in the horizontal resistivity image. These results again are consistent with the physics of the measurements: MT data mainly are sensitive to the horizontal resistivity, while the CSEM inline electric field data mainly are sensitive to the vertical resistivity. MT data also are not sensitive to the presence of thin resistive targets. Hence, we observe only the presence of the resistive reservoir in the vertical resistivity image. Note also that the artifacts in the left and right regions of the images are caused by the lack of sources or receivers on top of these regions. In Figure 13, we also show the joint inversion result when we use the isotropic assumption. Although we obtain an object that resembles the reservoir, its location and shape are not accurate. Furthermore, we also observe a few artifacts appearing as layers. This shows the importance of using a correct resistivity anisotropy model for CSEM and MT data inversion.

CONCLUSIONS We presented an inversion algorithm to jointly invert the CSEM and MT data. An important issue for this simultaneous joint inversion approach is how to assign the relative weights between CSEM and MT data. In this work we proposed to minimize the product of CSEM and MT cost functions instead of the traditional additive one. In this multiplicative cost function there is no need to select the relative weighting between these two data sets. This cost function adaptively will put CSEM and MT data on equal footing in the inversion process. The effectiveness of this approach was demonstrated using synthetic and field data sets. The approach can be extended straightforwardly for 3D geometries.

ACKNOWLEDGMENTS We thank V. Druskin of Schlumberger-Doll Research (SDR) and L. Knizhnerman of the Center for Geophysical Expedition for providing the 2D finite-difference forward programs. We thank WesternGeco-Electromagnetic (WG-EM) for providing the FORCE data set, and L. Masnaghetti and R. Luetkemeier from WG-EM for their help in interpreting the inversion results and providing the seismic plot. We also acknowledge valuable comments from C.G. Farquharson, M. Jegen-Kulcsar, and two anonymous reviewers who significantly improved the quality and readability of this manuscript.

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