A Conjugate Direction Method for Geophysical Inversion Problems

In geophysical tomography, algebraic methods are often used to linearize the nonlinear problem of determining the characteristics of an underground region given measurements of the earth's attenuation to electromagnetic or seismic waves. In this way, a set of linear equations is developed such that the unknowns are the picture elements (pixels) of the region being scanned.Classically, these linear equations have been solved using the algebraic reconstruction technique (ART) algorithm. In this paper, a new algorithm that is a member of the set of conjugate direction (CD) methods is developed and comparisons are made between this algorithm and the ART algorithm for data arising from simulated electromagnetic probing. This new method, which we call the constrained conjugate gradient (CCG) algorithm, is shown to have a much faster convergence to a final solution than the ART algorithm. In addition, for applications involving high-contrast anomalies (for example, tunnel detection) the CCG is shown to have superior performance in locating the anomalous region for almost all test cases considered.

[1]  E. F. Laine,et al.  Cross-borehole electromagnetic probing to locate high-contrast anomalies , 1979 .

[2]  Y. Censor,et al.  Strong underrelaxation in Kaczmarz's method for inconsistent systems , 1983 .

[3]  M. Oristaglio,et al.  INVERSION OF SURFACE AND BOREHOLE ELECTROMAGNETIC DATA FOR TWO‐DIMENSIONAL ELECTRICAL CONDUCTIVITY MODELS* , 1980 .

[4]  Å. Björck,et al.  Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations , 1979 .

[5]  A. Devaney Geophysical Diffraction Tomography , 1984, IEEE Transactions on Geoscience and Remote Sensing.

[6]  J. Richmond Scattering by a dielectric cylinder of arbitrary cross section shape , 1965 .

[7]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[8]  Michael A. Saunders,et al.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares , 1982, TOMS.

[9]  R. Gordon A tutorial on art (algebraic reconstruction techniques) , 1974 .

[10]  Constantine Balanis,et al.  Algorithm and Filter Selection in Geophysical Tomography , 1986, IEEE Transactions on Geoscience and Remote Sensing.

[11]  J. Richmond,et al.  TE-wave scattering by a dielectric cylinder of arbitrary cross-section shape , 1966 .

[12]  K. Tanabe Projection method for solving a singular system of linear equations and its applications , 1971 .

[13]  S. Treitel,et al.  A REVIEW OF LEAST-SQUARES INVERSION AND ITS APPLICATION TO GEOPHYSICAL PROBLEMS* , 1984 .

[14]  A. Howard,et al.  Synthesis of EM geophysical tomographic data , 1986, Proceedings of the IEEE.

[15]  Constantine A. Balanis,et al.  A stable geotomography technique for refractive media , 1984, IEEE Transactions on Geoscience and Remote Sensing.

[16]  Gabor T. Herman,et al.  Quadratic optimization for image reconstruction, II , 1976 .