During the past decades, we observed a strong interest in 3D DC resistivity inversion and imaging with complex topography. In this paper, we implemented 3D DC resistivity inversion based on regularized conjugate gradient method with FEM. The Fréchet derivative is assembled with the electric potential in order to speed up the inversion process based on the reciprocity theorem. In this study, we also analyzed the sensitivity of the electric potential on the earth’s surface to the conductivity in each cell underground and introduced an optimized weighting function to produce new sensitivity matrix. The synthetic model study shows that this optimized weighting function is helpful to improve the resolution of deep anomaly. By incorporating topography into inversion, the artificial anomaly which is actually caused by topography can be eliminated. As a result, this algorithm potentially can be applied to process the DC resistivity data collected in mountain area. Our synthetic model study also shows that the convergence and computation speed are very stable and fast. 1. Introduction In DC resistivity exploration method, complex topography can generate artificial anomalies which will cause difficulty for the data interpretation. Based on Qiang and Luo’s work [1] on 3D DC finite element resistivity modeling with complex topography, we conducted 3D regularized inversion and imaging for this complex model. The efficiency for 3D inversion problem depends primarily on 3 factors: efficient inversion algorithm, method for computing sensitivity matrix and the solver for a large liner system. Tripp et al. [2] introduced a method for calculating the sensitivity matrix, based on the relationship between electric potential and model parameter. This developed method for computing sensitivity matrix was successfully applied to a 2D DC resistivity inversion problem. Park and Van [3] introduced 3D inversion based on finite difference method. Sasaki [4] also described similar 3D inversion algorithm but based on finite element method. These introduced 3D inversion methods work well to recover shallow resistivity anomalies but fail to produce high-resolution image for the deep anomalous bodies. The synthetic model studies show that the recovered resistivity imaging is quite different from the true model if the anomaly is located deep underground. Zhang et al. [5] conducted the research on 3D DC resistivity inversion using conjugate gradient method. Loke and Barker [6] introduced the E-SCAN (pole-pole array) 3D inversion technique where the Fréchet derivative matrix can be
References
[1]
J.-K. Qiang and Y.-Z. Luo, “The resistivity FEM numerical modeling on 3-D undulating topography,” Chinese Journal of Geophysics, vol. 50, no. 5, pp. 1606–1613, 2007 (Chinese).
[2]
A. C. Tripp, G. W. Hohmann, and C. M. Swift Jr., “Two-dimensional resistivity inversion,” Geophysics, vol. 49, no. 10, pp. 1708–1717, 1984.
[3]
S. K. Park and G. P. Van, “Inversion of pole-pole data for 3-D resistivity structure beneath arrays of electrodes,” Geophysics, vol. 56, no. 7, pp. 951–960, 1991.
[4]
Y. Sasaki, “3-D resistivity inversion using the finite-element method,” Geophysics, vol. 59, no. 12, pp. 1839–1848, 1994.
[5]
J. Zhang, R. L. Mackie, and T. R. Madden, “3-D resistivity forward modeling and inversion using conjugate gradients,” Geophysics, vol. 60, no. 5, pp. 1313–1325, 1995.
[6]
M. H. Loke and R. D. Barker, “Practical techniques for 3D resistivity surveys and data inversion,” Geophysical Prospecting, vol. 44, no. 3, pp. 499–523, 1996.
[7]
X. P. Wu and G. M. Xu, “Derivation and analysis of partial derivative matrix in resistivity 3-D inversion,” Oil Geophysical Prospecting, vol. 34, pp. 363–372, 1999.
[8]
X.-P. Wu and G.-M. Xu, “Study on 3-D resistivity inversion using conjugate gradient method,” Chinese Journal of Geophysics, vol. 43, no. 3, pp. 420–427, 2000.
[9]
X.-P. Wu, “3-D resistivity inversion under the condition of uneven terrain,” Chinese Journal of Geophysics, vol. 48, no. 4, pp. 932–936, 2005.
[10]
H.-F. Liu, J.-X. Liu, R.-W. Guo, X.-K. Deng, and B.-Y. Ruan, “Efficient inversion of 3D IP data for continuous model with complex geometry,” Journal of Jilin University, vol. 41, no. 4, pp. 1212–1218, 2011.
[11]
N. G. Papadopoulos, M.-J. Yi, J.-H. Kim, P. Tsourlos, and G. N. Tsokas, “Geophysical investigation of tumuli by means of surface 3D electrical resistivity tomography,” Journal of Applied Geophysics, vol. 70, no. 3, pp. 192–205, 2010.
[12]
P. I. Tsourlos and R. D. Ogilvy, “An algorithm for the 3-D inversion of tomographic resistivity and induced polarisation data: preliminary results,” Journal of the Balkan Geophysical Society, vol. 2, pp. 30–45, 1999.
[13]
J. G. Huang, B. Y. Ruan, and G. S. Bao, “Resistivity inversion on 3-D section based on FEM,” Journal of Central South University of Technology, vol. 35, pp. 295–299, 2004.
[14]
T. Günther, C. Rücker, and K. Spitzer, “Three-dimensional modeling and inversion of dc resistivity data incorporating topography—II inversion,” Geophysical Journal International, vol. 166, no. 2, pp. 506–5517, 2006.
[15]
G. A. Oldenborger and P. S. Routh, “The point-spread function measure of resolution for the 3-D electrical resistivity experiment,” Geophysical Journal International, vol. 176, no. 2, pp. 405–414, 2009.
[16]
A. Dey and H. F. Morrison, “Resistivity modeling for arbitrarily shaped three-dimensional structures,” Geophysics, vol. 44, no. 4, pp. 753–760, 1979.
[17]
R. G. Ellis and D. W. Oldenburg, “The pole-pole 3-D DC-resistivity inverse problem: a conjugate- gradient approach,” Geophysical Journal International, vol. 119, no. 1, pp. 187–194, 1994.
[18]
D. J. LaBrecque, M. Miletto, W. Daily, A. Ramirez, and E. Owen, “The effects of noise on Occam's inversion of resistivity tomography data,” Geophysics, vol. 61, no. 2, pp. 538–548, 1996.
[19]
M.-J. Yi, J.-H. Kim, Y. Song, S.-J. Cho, S.-H. Chung, and J.-H. Suh, “Three-dimensional imaging of subsurface structures using resistivity data,” Geophysical Prospecting, vol. 49, no. 4, pp. 483–497, 2001.
[20]
C. C. Pain, J. V. Herwanger, M. H. Worthington, and C. R. E. de Oliveira, “Effective multidimensional resistivity inversion using finite-element techniques,” Geophysical Journal International, vol. 151, no. 3, pp. 710–728, 2002.
[21]
A. Pidlisecky, E. Haber, and R. Knight, “RESINVM3D: a 3D resistivity inversion package,” Geophysics, vol. 72, no. 2, pp. H1–H10, 2007.
[22]
L. Marescot, S. P. Lopes, S. Rigobert, and A. G. Green, “Nonlinear inversion of geoelectric data acquired across 3D objects using a finite-element approach,” Geophysics, vol. 73, no. 3, pp. F121–F133, 2008.
[23]
M. S. Zhdanov, Geophysical Inverse Theory and Regularization Problems, Elsevier, New York, NY, USA, 2002.
