2-D joint structural inversion of cross-hole electrical resistance and ground penetrating radar data

We present a joint structural inversion algorithm for cross-hole electrical resistance tomography (ERT) and cross-hole radar travel time tomography (RTT) that encourages coincident sharp changes on a smoothly varying background in the two models. The proposed approach is based on the combination of two iterative soft-thresholding inversion algorithms in parallel manner where the structural information is exchanged at each iteration. Iterative thresholding algorithm allows to obtain a sparse wavelet representation of the model (blocky model) by applying a thresholding operator to the wavelet coefficients of model obtained through a Gauss–Newton iteration. The structural information is introduced in the inversion system using the smoothness weighting matrices that control boundary cells and the thresholds that are estimated by maximizing a structural similarity criterion, which is a function of the two (ERT and RTT) models. A Canny edge detector is implemented to extract the structural information. The detected edges serve to build a weighting matrix that is used to alter the smoothness matrix constraint. To validate our methodology and its implementation, tests were performed on three synthetic models. The results show that the parameters estimated by our joint inversion approach are more consistent than those from individual inversions and another joint inversion algorithm. In addition, our approach appears to be robust in high noise level conditions. Finally, the proposed algorithm was applied for vadose zone characterisation in a sandstone aquifer. It achieves results that are consistent with hydrogeological information and geophysical logs available at the site. The results were also compared in terms of structural similarities to models obtained by a joint structural inversion algorithm with a cross-gradient constraint. Based on this comparison and hydrogeologic information, we conclude that the proposed algorithm allows to the RTT and ERT models to be dissimilar in the areas where the data are incompatible.

[1]  Andrew Binley,et al.  Modeling unsaturated flow in a layered formation under quasi-steady state conditions using geophysical data constraints , 2005 .

[2]  Andrew McKenzie,et al.  The physical properties of major aquifers in England and Wales , 1997 .

[3]  A. Binley,et al.  Vadose zone flow model parameterisation using cross-borehole radar and resistivity imaging , 2001 .

[4]  Ronny Ramlau,et al.  A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints , 2006, Numerische Mathematik.

[5]  Erwan Gloaguen,et al.  bh_tomo - a Matlab borehole georadar 2D tomography package , 2007, Comput. Geosci..

[6]  Yutaka Sasaki,et al.  Two‐dimensional joint inversion of magnetotelluric and dipole‐dipole resistivity data , 1989 .

[7]  Zhou Bing,et al.  Explicit expressions and numerical calculations for the Fréchet and second derivatives in 2.5D Helmholtz equation inversion , 1999 .

[8]  J. Claerbout,et al.  Robust Modeling With Erratic Data , 1973 .

[9]  Yi Huang,et al.  Radar frequency dielectric dispersion in sandstone: Implications for determination of moisture and clay content , 2003 .

[10]  M. Bosch.  Lithologic tomography: From plural geophysical data to lithology estimation , 1999 .

[11]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[12]  Sven Treitel,et al.  Cooperative inversion of geophysical data , 1988 .

[13]  Andrew Binley,et al.  Characterization of Heterogeneity in Unsaturated Sandstone Using Borehole Logs and Cross-Borehole Tomography , 2004 .

[14]  Andrew Binley,et al.  Estimating Petrophysical Data From Borehole Geophysics , 2001 .

[15]  Jens Tronicke,et al.  Cooperative inversion of 2D geophysical data sets: A zonal approach based on fuzzy c-means cluster analysis , 2007 .

[16]  A. Binley,et al.  Seasonal variation of moisture content in unsaturated sandstone inferred from borehole radar and resistivity profiles. , 2002 .

[17]  A. Roberts,et al.  A framework for 3-D joint inversion of MT, gravity and seismic refraction data , 2011 .

[18]  John F. Canny,et al.  A Computational Approach to Edge Detection , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  David L.B. Jupp,et al.  Joint Inversion of Geophysical Data , 2007 .

[20]  Andrew Binley,et al.  Applying petrophysical models to radar travel time and electrical resistivity tomograms: Resolution‐dependent limitations , 2005 .

[21]  I. Daubechies,et al.  Tomographic inversion using L1-norm regularization of wavelet coefficients , 2006, physics/0608094.

[22]  C. N Bouza,et al.  Spall, J.C. Introduction to stochastic search and optimization. Estimation, simulation and control. Wiley Interscience Series in Discrete Mathematics and Optimization, 2003 , 2004 .

[23]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[24]  L. Landweber.  An iteration formula for Fredholm integral equations of the first kind , 1951 .

[25]  S. Hubbard,et al.  Joint inversion of crosshole radar and seismic traveltimes , 2008 .

[26]  S. Friedman,et al.  Relationships between the Electrical and Hydrogeological Properties of Rocks and Soils , 2005 .

[27]  L. Jared West,et al.  Borehole time domain reflectometry in layered sandstone: Impact of measurement technique on vadose zone process identification , 2004 .

[28]  E. Haber,et al.  Joint inversion: a structural approach , 1997 .

[29]  Eero P. Simoncelli,et al.  Image quality assessment: from error visibility to structural similarity , 2004, IEEE Transactions on Image Processing.

[30]  A. Binley,et al.  Improved hydrogeophysical characterization using joint inversion of cross‐hole electrical resistance and ground‐penetrating radar traveltime data , 2006 .

[31]  Andrew Binley,et al.  High‐resolution characterization of vadose zone dynamics using cross‐borehole radar , 2001 .

[32]  D. Oldenburg,et al.  NON-LINEAR INVERSION USING GENERAL MEASURES OF DATA MISFIT AND MODEL STRUCTURE , 1998 .

[33]  Robert A. Eso,et al.  Iterative Reconstruction Algorithm For Non-linear Operators , 2008 .

[34]  Zhou Wang,et al.  Translation insensitive image similarity in complex wavelet domain , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[35]  Max A. Meju,et al.  A quadratic programming approach for joint image reconstruction: mathematical and geophysical examples , 2005 .

[36]  James C. Spall,et al.  Introduction to Stochastic Search and Optimization. Estimation, Simulation, and Control (Spall, J.C. , 2007 .

[37]  E. Kozlovskaya,et al.  An algorithm of geophysical data inversion based on non-probabilistic presentation of a priori information and definition of Pareto-optimality , 2000 .

[38]  Max A. Meju,et al.  Joint two‐dimensional cross‐gradient imaging of magnetotelluric and seismic traveltime data for structural and lithological classification , 2007 .

[39]  Douglas W. Oldenburg,et al.  Integrating geological and geophysical data through advanced constrained inversions , 2009 .

[40]  S. Greenhalgh,et al.  Zonation for 3D aquifer characterization based on joint inversions of multimethod crosshole geophysical data , 2010 .

[41]  M. Meju,et al.  Characterization of heterogeneous near‐surface materials by joint 2D inversion of dc resistivity and seismic data , 2003 .

[42]  Richard Baraniuk,et al.  The Dual-tree Complex Wavelet Transform , 2007 .

[43]  Carsten Rücker,et al.  A NEW JOINT INVERSION APPROACH APPLIED TO THE COMBINED TOMOGRAPHY OF DC RESISTIVITY AND SEISMIC REFRACTION DATA , 2006 .

[44]  J. Harris,et al.  Coupled seismic and tracer test inversion for aquifer property characterization , 1993 .

[45]  N. Linde,et al.  Local earthquake (LE) tomography with joint inversion for P‐ and S‐wave velocities using structural constraints , 2006 .

[46]  A. Dey,et al.  Resistivity modelling for arbitrarily shaped two-dimensional structures , 1979 .

[47]  Eric L. Miller,et al.  Nonlinear inverse scattering methods for thermal-wave slice tomography: a wavelet domain approach , 1998 .