Journal of Earth Science, Vol. 22, No. 3, p. 386–395, June 2011 Printed in China DOI: 10.1007/s12583-011-0191-8
ISSN 1674-487X
Three-Dimensional Conjugate Gradient Inversion of Magnetotelluric Impedance Tensor Data Lin Changhong* (林昌洪), Tan Handong (谭捍东), Tong Tuo (佟拓) State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences, Beijing 100083, China; Key Laboratory of Geo-detection, Ministry of Education, Beijing 100083, China; School of Geophysics and Information Technology, China University of Geosciences, Beijing 100083, China ABSTRACT: We developed a three-dimensional (3D) conjugate gradient inversion algorithm for inverting magnetotelluric impedance tensor measurements. In order to show the importance of including diagonal components of magnetotelluric impedance tensor in 3D inversion, synthetic data were inverted using the 3D conjugate gradient inversion, and the inversion results were compared and analyzed. The results from the 3D inversion of synthetic data indicate that both the off-diagonal and the diagonal components are required in inversions to obtain better inversion results when there are no enough data sites to recover the target resistivity structure. These examples show that lots of information about 3D structure is also contained in the diagonal components; as a result, diagonal components should be included in 3D inversions. The inversion algorithm was also used to invert the impedance tensor data acquired in the Kayabe area in Japan. Inversions with the synthetic and real data demonstrated the validity and practicability of the inversion algorithm. KEY WORDS: magnetotelluric, impedance tensor, 3D inversion, conjugate gradients, diagonal components.
INTRODUCTION For a general 3D model, magnetotelluric (MT) impedance usually have four components, i.e., xx, xy, yx, and yy. The xy and yx components are called This study was jointly supported by the National Natural Science Foundation of China (Nos. 40774029, 41004028), the Special Fund for Basic Scientific Research of Central Colleges (No. 2010ZY53), and the Program for New Century Excellent Talents in University (NCET). *Corresponding author:
[email protected] © China University of Geosciences and Springer-Verlag Berlin Heidelberg 2011 Manuscript received December 20, 2009. Manuscript accepted March 9, 2010.
off-diagonal elements, while the xx and yy terms are called diagonal elements. As we all know, the more constraining data are used in the inversion, the better inversion results we can get. However, the diagonal elements have not been given attention due to the fact that geophysicists used to think that most of the information about geological structure was contained in the off-diagonal components, so the major MT data used in the inversion were the off-diagonal elements. Furthermore, some other issues may also lead to the rare use of the diagonal elements in MT inversion, such as it is difficult to use the diagonal elements in the inversion because these elements are much smaller than the off-diagonal ones. Because the diagonal elements are equal to zero for a two-dimensional (2D) earth model, the 2D inversion algorithms (Rodi and
Three-Dimensional Conjugate Gradient Inversion of Magnetotelluric Impedance Tensor Data
Mackie, 2001; Siripunvaraporn and Egbert, 2000; Dai and Xu, 1997; Dai, 1994; Smith and Booker, 1991; De-Groot-Hedlin and Constable, 1990; Jupp and Vozoff, 1977) only need invert the xy and yx components (TM mode and TE mode). The recent 20 years has seen many publications about 3D MT inversion algorithms, such as rapid relaxation inversion (Lin et al., 2009; Tan et al., 2003b; Smith and Booker, 1991), conjugate gradients inversion (Lin et al., 2011, 2008; Newman and Alumbaugh, 2000, 1997; Mackie and Madden, 1993; Madden and Mackie, 1989), artificial neural network inversion (Spichak and Popova, 2000), data-space method (Siripunvaraporn et al., 2005), and quasi-linear approximation inversion (Hu et al., 2006; Zhdanov et al., 2000). Most of these 3D inversion algorithms only use the off-diagonal elements as the data for inversion. Few 3D inversion algorithms (Avdeev and Avdeeva, 2006; Siripunvaraporn et al., 2005; Newman and Alumbaugh, 2000) can invert the tensor data including both the diagonal and offdiagonal elements. In fact, the diagonal elements can be easily calculated once the off-diagonal elements (both xy and yx terms) are acquired in the field. If better inversion results can be obtained by adding the diagonal elements in inversion, more attention should be paid to these components. The study on the diagonal elements can give us some insights when we choose the field MT data in 3D inversion in the future. Based on the analysis of the conjugate gradient algorithm, a 3D conjugate gradient inversion algorithm that can invert the off-diagonal impedance data, Zxy and Zyx, has been implemented in Lin et al. (2008). After further study on the relationship between the diagonal components, Zxx and Zyy, and the conjugate gradient algorithm, we developed a 3D conjugate gradient inversion algorithm that can invert all the impedance tensor elements: Zxx, Zxy, Zyx, and Zyy. In this article, synthetic data from several testing models are inverted using the 3D conjugate gradient inversion, and the inversion results are compared. These examples show the importance of including the diagonal elements in 3D inversion. We will first give an overview of the 3D conjugate gradient inversion of impedance tensor data. Second, different results from inverting different components will be analyzed. Finally, an inversion example using real data will be
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given. OVERVIEW OF THE 3D CONJUGATE GRADIENT INVERSION OF THE IMPEDANCE TENSOR The objective function of the 3D conjugate gradient inversion of impedance tensor measurements is defined as ψ (m) = ( Z obs − F ( m)) T V −1 ( Z obs − F ( m)) + (1) λ( m0 − m) T LT L(m0 − m)
where Zobs is a vector for the observed impedance tensor, F(m) is the forward operator to calculate the impedance tensor, V is the variance of data error, λ is the regularization parameter, L is a simple second-difference Laplacian, and m0 is the priori model. The corresponding gradient of the objective function can be expressed as g = −2 ATV −1e + 2λLT L(m0 − m) (2) where A is the Jacobian matrix, and e is the data error vector. Forward modeling uses the staggered-grid finite difference method (Tan et al., 2003a). With a staggered grid, a discretization of the integral form of the Maxwell equations (3) v∫ H ⋅ dl = ∫∫ J ⋅ dS = ∫∫ σE ⋅ dS
v∫ E ⋅ dl = ∫∫ iμ ωH ⋅ dS 0
(4)
leads to the following linear system KH=s (5) where K is a sparse complex-symmetric matrix, H is a vector composed of the three magnetic field components, and s is the source vector related to the boundary conditions. The magnetic field will be obtained by solving equation (5). Then, the electric field will be obtained by equation ▽×H=σE. Finally, the impedance tensor response of the 3D model can be calculated from the magnetic and electric fields (Lin et al., 2008; Tan et al., 2004). A flowchart for the 3D conjugate gradient inversion of impedance tensor is listed as follows (Fig. 1): (1) fields and responses are calculated through forward modeling from an given initial model, (2) the objective function and its gradient are obtained by multiplying the transpose of the Jacobian matrix and one vector, (3) the iterative step is obtained by multiplying the Jacobian matrix and another vector, (4) the model is updated using the step, and (5) the fields and
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2008). ATV-1e and Ap can be computed by solving the pseudoforward modeling problem two times separately. Therefore, the inversion algorithm only requires one forward and four pseudoforward modeling per frequency to produce the model update at each iteration. SYNTHETIC EXAMPLES To obtain synthetic MT data, we designed some 3D synthetic geological models. The corresponding impedance tensor responses were calculated using the staggered-grid finite difference modeling program. One percent Gaussian random noise was added to the data. All results were computed on a PC with a 2.2 GHz Intel® CoreTM 2 Duo CPU and 2 GB of physical memory.
Figure 1. Flowchart of the 3D conjugate gradient inversion. responses for the new model are calculated. This procedure is repeated until the model’s responses fit the real data to a satisfactory precision. Two main operations are needed to obtain the update step after fields, and responses are calculated through forward modeling: the multiplication of the transpose of the Jacobian matrix and one vector, ATV-1e, and the multiplication of the Jacobian matrix and another vector, Ap. The main computational overburden in the complete inversion process is in these two operations. The 3D conjugate gradient inversion algorithm for the impedance tensor does not need to compute every element of the Jacobian matrix; instead, it directly obtains the value of ATV-1e and Ap by solving the pseudoforward modeling problems (Lin et al.,
Model 1. One Conductive Prism and One Resistive Prism Model 1 (Fig. 2) consists of a 1 Ω·m conductive rectangular prism and a 1 000 Ω·m resistive prism embedded in a 100 Ω·m half-space. Both the two prisms have a dimension of 6 km×6 km×3 km with the top located 3 km below the earth’s surface. The spacing between the conductor and the resistor is 3 km. The size of the 3D grid is 69×43×46 (including 10 air layers). The impedance tensor data (Zxx, Zxy, Zyx, and Zyy) at four frequencies (3.3–0.1 Hz), which were computed at the grid center on the earth’s surface (2 747 sites), are used in the inversion. The 3D inversion was carried out in four scenarios: (1) inverting only Zxy; (2) inverting only Zyx; (3) inverting only Zxy and Zyx; (4) inverting all four impedance elements, Zxx, Zxy, Zyx, and Zyy. Table 1 shows
Figure 2. Plan view of the synthetic model 1.
Three-Dimensional Conjugate Gradient Inversion of Magnetotelluric Impedance Tensor Data Table 1
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Statistical data for test model 1
Data used in the inversion
Iteration number
Runtime (min)
Chi-square error
Zxy
51
4 742
1.025
Zyx
42
3 565
1.185
Zxy+Zyx
26
2 454
1.079
Zxx+Zxy+Zyx+Zyy
11
1 061
0.974
Figure 3. Results from the 3D inversion of synthetic data generated from model 1. The top row shows the test model; the second row shows the result of inverting only Zxy; the third row shows the result of inverting only Zyx; the fourth row shows the result of inverting both Zxy and Zyx; and the bottom row shows the result of inverting all the impedance tensor elements. The black dashed lines show the prism margins. The first column show the horizontal slices at 4.5 km depth, the second column show the vertical slices at y=0 km along the X axis, the third column show the vertical slices at x=-4.5 km along the Y axis, and the fourth column show the vertical slices at x=4.5 km along the Y axis.
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the total number of iteration, the CPU runtime, and the chi-square errors for the four scenarios. The inversion results are shown in Fig. 3. Inverting only Zxy (the second row) recovers the conductor and the resistor similarly: the prism boundary in X direction is well recovered; both the conductor and the resistor are out of the boundaries for some extent in Y direction; the top for both the prisms is shallower than the real boundary. Inverting only Zyx (the third row) recovers the conductor well but not the resistor: the boundaries of conductor in Y direction are well-recovered, while the resistor is much smaller than the real boundaries in Y direction; the conductor is out of the prism boundary for some extent along negative X direction, while the resistor is out of the boundaries for some extent along positive X direction; the tops for both the two prisms are shallower than the real boundaries. The images resulting from inverting both Zxy and Zyx (the fourth row) are much better than those from inverting either Zxy or Zyx: the boundaries of conductor in X, Y, and Z directions are recovered relatively well; the resistor is little out of the prism
boundaries only in positive Y direction. The results from inverting all the impedance elements (the bottom row) are similar to those from inverting both Zxy and Zyx. Model 2. Two Conductive Rectangular Prisms Model 2 is similar to the model 1, except that the resistivities of both the prisms are 10 Ω·m. Different from model 1 that generates data at all sites in the inversion, data at only part of the sites (Fig. 4) are generated from model 2. The impedance tensor data (Zxx, Zxy, Zyx, and Zyy) at four frequencies (3.3–0.1 Hz) were computed at 396 sites on the earth’s surface and used in the inversion. There are 11 parallel survey lines. They are labeled along positive X direction (corresponding to grid) as lines 5, 6, 8, 11, 20, 23, 27, 35, 59, 62, and 64. The sites on each line are labeled from 2 to 37 along positive Y direction. There are three lines on the top of one conductor, while there is no profile on the top of the other conductor. The size of the 3D grid is 69×43×46 (including 10 air layers).
Figure 4. Plan view of the test model 2. The black dashed lines show the profiles used in the inversion. Table 2
Statistical data for test model 2
Data used in the inversion
Iteration number
Runtime (min)
Chi-square error
Zxy
31
2 686
0.996
Zyx
37
3 208
1.052
Zxy+Zyx
25
1 866
0.993
Zxx+Zxy+Zyx+Zyy
19
1 377
0.965
The 3D inversion were also carried out in four scenarios: (1) inverting only Zxy; (2) inverting only Zyx; (3) inverting only Zxy and Zyx; (4) inverting all four impedance elements, Zxx, Zxy, Zyx, and Zyy. Table 2 shows the iteration numbers, the runtimes, and the
chi-square errors for the four scenarios. The inversion results are shown in Fig. 5. Results of inverting only Zxy (the second row) recover only one conductor with data sites on the top: the prism boundaries are well recovered in both X and Z direc-
Three-Dimensional Conjugate Gradient Inversion of Magnetotelluric Impedance Tensor Data
tions; the conductor is out of the prism boundaries for some extent in Y direction. Results of inverting only Zyx (the third row) recover the conductor with data sites on the top well: the conductor is out of the
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boundaries for some extent only in X direction. Results of inverting only Zyx recover the conductor without data sites on the top badly: the lowest resistivity of the conductor is just down to 30 Ω·m, and the
Figure 5. Results from the 3D inversion of synthetic data generated from model 2. The top row shows the test model, the second row shows the result of inverting only Zxy, the third row shows the result of inverting only Zyx, the fourth row shows the result of inverting both Zxy and Zyx, and the bottom row shows the result of inverting all the impedance tensor elements. The black dashed lines show the prism margins. The first column shows the horizontal slices at 4.5 km depth, the second column show the vertical slices at y=0 km along the X axis, the third column shows the vertical slices at x=-4.5 km along the Y axis, and the fourth column shows the vertical slices at x=4.5 km along the Y axis.
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top of the conductor is much shallower than the real boundary. Results of inverting both Zxy and Zyx (the fourth row) recover the conductor with data sites on the top quite well, which is similar to that of inverting all the impedance elements (the bottom row). However, for the conductor without data sites on the top, results from inverting all the impedance elements are the best: the prism boundaries are recovered better, and the resistivity of the conductor is much closer to the real value. ANALYSIS OF THE INVERSION RESULTS When data at all sites on the earth’s surface are used in the 3D inversion (model 1), the result from all the impedance elements is quite similar to that from both Zxy and Zyx. However, both of them are better than that from either Zxy or Zyx. Under this condition, the diagonal elements, Zxx and Zyx, are not important in the 3D inversion, and the target resistivity structure can be recovered well using the off-diagonal components. In order to study the diagonal terms in the inversion, we choose data at part of the sites to be inverted. For the conductor with data sites on the top, the result from all the impedance elements is quite similar to that from both Zxy and Zyx but better than that from either Zxy or Zyx. For the conductor without any data site on the top, the result from all impedance elements is much better than any other three ones. This shows that the diagonal elements have significant effects on the conductor without data sites on the top. Therefore, the following conclusions can be drawn: (1) When there are enough data sites to recover the target geology, the diagonal elements are not important in the 3D inversion, and the target geology can be recovered well with the off-diagonal components. (2) When there are no enough data sites to recover the target geology, the inversion result is greatly affected by the diagonal elements. Under this condition, both the off-diagonal and the diagonal components are required in inversion to obtain better inversion results. FIELD EXAMPLE There are 209 MT sites in the Kayabe area in Japan. The Kayabe area was particularly interesting be-
Lin Changhong, Tan Handong and Tong Tuo
cause of its geothermal potential. Seven wells were drilled in this area, and two of them successfully produced geothermal fluid. An intensive investigation, such as a gravity survey, a Schlumberger measurement and a geochemical survey, has been conducted in this area. These surveys did not get good results. Hence, array MT measurement was conducted with most of the sounding points around the two wells producing geothermal fluid (Takasugi et al., 1992). In this article, we only choose 147 sites in the regular network area for 3D inversion. The site distribution is shown in Fig. 6. There are 13 parallel survey lines. They are labeled from south to north as lines a, b, c, d, e, f, g, h, i, j, k, l, and m. The numbers of sites on each survey line are 10, 11, 13, 13, 13, 12, 12, 10, 12, 13, 11, 8, and 9. The line orientation is SW-NE with a spacing of 100 m between each line. The site interval on the survey lines is also 100 m.
Figure 6. MT sites used in the inversion (“.” denotes an MT site).
Figure 7. Chi-square error versus iteration number.
Three-Dimensional Conjugate Gradient Inversion of Magnetotelluric Impedance Tensor Data
The model is discretized on a 32×32×46-layer (including 10 air layers) grid. Impedance tensor data (Zxx, Zxy, Zyx, and Zyy) at 13 frequencies (64–1 Hz) at 147 sites were inverted using the 3D conjugate gradient inversion program. After 59 iterations, the chi-square error was down to 9.62, and the inversion was terminated (Fig. 7). Figure 8 shows a comparison
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between the model response and observed data for selected sites (kbb001 and kbj001), which is chosen for a later contrast with the 3D model response. Figure 9 shows the horizontal slices from the 3D inversion at different depths (10 to 653 m). These horizontal slices show that conductive bodies develop in the middle and southern areas at depths between 325–553 m. The
Figure 8. Comparison of 3D model response and observed data for selected sites. Observed data: black circles (Zxx or Zxy), white circles (Zyx or Zyy). Synthetic data: solid line (Zxx or Zxy), dashed line (Zyx or Zyy).
Figure 9. Horizontal slices of the recovered resistivity model obtained from the 3D inversion (“.” denotes an MT site).
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conductors extend upward gradually and develop widely in the survey area at depths between 10–218 m. The wells drilled in this area indicated that a possible geothermal reservoir exists, and the reservoir probably originates in the fractures developing in the intrusive rock formation. Takasugi et al. got the conclusion that the relatively resistive intrusive rock formation could be detected from the scattered MT sites, and the dense network survey clarified the existence of the relatively conductive fracture zone in the intrusive rock formation (Takasugi et al., 1992). In our 3D inversion, the sites in the dense network were used. Therefore, we infer that those conductive bodies in Fig. 9 should be the indication of the geothermal reservoir or the fracture zone.
combination with the off-diagonal components. REFERENCES CITED Avdeev, D. B., Avdeeva, A. D., 2006. A Rigorous Three-Dimensional Magnetotelluric Inversion. Progress in Electromagnetics Research, 62: 41–48 Dai, S. K., 1994. Geophysical Inversion Using Apparent Model Space Contrast Method. Acta Geophys. Sinica, 37(Suppl. II): 524–533 (in Chinese with English Abstract) Dai, S. K., Xu, S. Z., 1997. Rapid Inversion of Magnetotelluric Data for 2-D and 3-D Continuous Media. Oil Geophysical Prospecting, 32(3): 305–317 (in Chinese with English Abstract) De-Groot-Hedlin, C. D., Constable, S. C., 1990. Occam’s Inversion to Generate Smooth, Two-Dimensional Models from
CONCLUSIONS We have developed a 3D conjugate gradient inversion algorithm for inverting MT impedance tensor measurements. This algorithm can invert either all the impedance tensor elements or one or two elements of the tensor as needed. The inversion results obtained from inverting both the synthetic and real data have demonstrated the validity and practicability of this inversion algorithm. Inversion examples for the synthetic data show the importance of including the diagonal elements in 3D inversion. The results indicate that the target geology can be recovered well with the off-diagonal components when there are enough data sites to recover the target geology; both the off-diagonal terms and the diagonal terms are required in the inversion to obtain better inversion results when there are no enough data sites to recover the target geology. These examples show that lots of information about 3D structure is also contained in the diagonal components; as a result, diagonal components should be considered in 3D inversion. On the other hand, we also find that the target geology cannot be recovered when only diagonal components are inverted from some other inversion of the synthetic data. Therefore, most of the information about geology structure is still contained in the off-diagonal components although the diagonal elements contain some information about the 3D structure, and the diagonal components should be used in
Magnetotelluric
Data.
Geophysics,
55(12):
1613–1624 Hu, Z. Z., Hu, X. Y., He, Z. X., 2006. Pseudo-ThreeDimensional Magnetotelluric Inversion Using Nonlinear Conjugate
Gradients. Chinese
J.
Geophys.,
49(4):
1226–1234 (in Chinese with English Abstract) Jupp, D. L. B., Vozoff, K., 1977. Two-Dimensional Magnetotelluric Inversion. Geophys. J. Roy. Astr. Soc., 50(2): 333–352 Lin, C. H., Tan, H. D., Tong, T., 2008. Three-Dimensional Conjugate Gradient Inversion of Magnetotelluric Sounding Data. Applied Geophysics, 5(4): 314–321 Lin, C. H., Tan, H. D., Tong, T., 2009. Parallel Rapid Relaxation Inversion of 3D Magnetotelluric Data. Applied Geophysics, 6(1): 77–83 Lin, C. H., Tan, H. D., Tong, T., 2011. The Possibility of Obtaining nearby 3D Resistivity Structure from Magnetotelluric 2D Profile Data Using 3D Inversion. Chinese J. Geophys., 54(1): 245–256 (in Chinese with English Abstract) Mackie, R. L., Madden, T. R., 1993. Three-Dimensional Magnetotelluric Inversion Using Conjugate Gradients. Geophys. J. Int., 115(1): 215–229 Madden, T. R., Mackie, R. L., 1989. Three-Dimensional Magnetotelluric Modeling and Inversion. Proc. IEEE, 77(2): 318–333 Newman, G. A., Alumbaugh, D. L., 1997. Three-Dimensional Massively Parallel Electromagnetic Inversion—I. Theory. Geophys. J. Int., 128(2): 345–354 Newman, G. A., Alumbaugh, D. L., 2000. Three-Dimensional Magnetotelluric Inversion Using Non-linear Conjugate
Three-Dimensional Conjugate Gradient Inversion of Magnetotelluric Impedance Tensor Data Gradients. Geophys. J. Int., 140(2): 410–424
395
Spatial Resolution of the Resistivity Structure Revealed by
Rodi, W., Mackie, R. L., 2001. Nonlinear Conjugate Gradients
a Dense Network MT Measurement—A Case Study in the
Algorithm for 2-D Magnetotelluric Inversion. Geophysics,
Minamikayabe Area, Hokkaido, Japan. Journal of Geo-
66(1): 174–187
magnetism and Geoelectricity, 44(4): 289–308
Siripunvaraporn,
W.,
Egbert,
G.,
2000.
An
Efficient
Tan, H. D., Wei, W. B., Deng, M., et al., 2004. General Use
Data-Subspace Inversion Method for 2-D Magnetotelluric
Formula in MT Tensor Impedance. Oil Geophysical Pros-
Data. Geophysics, 65(3): 791–803
pecting, 39(1): 114–116 (in Chinese with English Ab-
Siripunvaraporn, W., Egbert, G., Lenbury, Y., et al., 2005.
stract)
Three-Dimensional Magnetotelluric Inversion: Data-Space
Tan, H. D., Yu, Q. F., Booker, J., et al., 2003a. Magnetotelluric
Method. Physics of the Earth and Planetary Interiors,
Three-Dimensional Modeling Using the Staggered-Grid
150(1–3): 3–14
Finite Difference Method. Chinese J. Geophys., 46(5):
Smith, J. T., Booker, J. R., 1991. Rapid Inversion of Two- and Three-Dimensional Magnetotelluric Data. J. Geophys. Res., 96(B3): 3905–3922 of
Magnetotelluric
Tan, H. D., Yu, Q. F., Booker, J., et al., 2003b. Three-Dimensional Rapid Relaxation Inversion for the
Spichak, V., Popova, I., 2000. Artificial Neural Network Inversion
705–711 (in Chinese with English Abstract)
Data
in
Terms
of
Three-Dimensional Earth Macroparameters. Geophys. J. Int., 142(1): 15–26 Takasugi, S., Tanaka, K., Kawakami, N., et al., 1992. High
Magnetotelluric Method. Chinese J. Geophys., 46(6): 850–854 (in Chinese with English Abstract) Zhdanov, M. S., Fang, S., Hursan, G., 2000. Electromagnetic Inversion Using Quasi-linear Approximation. Geophysics, 65(5): 1501–1513
