Ž .
Journal of Applied Geophysics 46 2001 45–54
www.elsevier.nlrlocaterjappgeo
Full 3-D inversion of electromagnetic data on PC
Yutaka Sasaki
Department of Earth Resources Engineering, Graduate School of Engineering, Kyushu UniÕersity, Hakozaki, Higashi, Fukuoka 812-8581, Japan
Received 5 August 1999; accepted 31 October 2000
Abstract
Ž .
Ž .
Three-dimensional 3-D electromagnetic EM inversion might be believed to require high-performance computers. Ž
. However, with the rapid progress of recent computer technology, running 3-D inversions on personal computers PC is
becoming a rational choice. This paper describes an attempt to carry out full 3-D inversions of synthetic frequency-domain EM data on a PC. In the inversion, a staggered-grid finite difference scheme is used to solve for the secondary electric field.
Ž .
The system of equations is solved using the incomplete Cholesky biconjugate gradient ICBCG method. By applying the w
Ž .
x static divergence correction proposed by Smith
Geophysics 61 1996
1319–1324 , the rates of convergence are dramatically improved, and the forward calculation per source location on a medium-size grid takes only a few minutes on a
PC. The sensitivities of the EM responses to subsurface resistivity changes are calculated from forward solutions using the reciprocity relation. An inverse problem is formulated so that a model is found that has a smooth structure and at the same
time is close to an initial model, and is solved with an iterative least-squares method. A synthetic example for an airborne EM survey shows that the lateral extents of 3-D bodies are well resolved, and the vertical coil coaxial system gives a better
lateral resolution than the horizontal coil coplanar system. An example for a ground EM survey shows that expanding the survey coverage or aperture is needed to improve the resolution at depth. q 2001 Elsevier Science B.V. All rights reserved.
Keywords: 3-D inversion; Finite difference method; Biconjugate gradient method; Electromagnetics
1. Introduction
Ž .
Inverting electromagnetic EM data for three-di- Ž
. mensional 3-D electrical conductivity structures has
been a major challenge in applied geophysics for many years. Full 3-D inversions involve rigorous
forward modelings for general 3-D structures, which are computationally very time-consuming compared
to 1-D and 2-D cases. Despite this difficulty, several
Fax: q81-92-642-3614. Ž
. E-mail address: sasakimine.kyushu-u.ac.jp Y. Sasaki .
attempts have been made to carry out 3-D inversions using different forward numerical solutions. Eaton
Ž .
Ž .
1989 and Pellerin et al.
1993 both used the
integral equation approach to formulate 3-D inverse problems for controlled-source EM surveys. Mackie
Ž .
and Madden 1993 developed an inversion proce- dure for magnetotelluric data that is based on the
finite difference and conjugate gradient methods. Xie Ž
. et al. 1995 derived a new integral equation in terms
of the magnetic field and applied it to invert for Ž
. conductivities and permittivities. Ellis 1995 per-
formed a 3-D inversion on airborne EM data using Ž
. the finite element method. Newman 1995 presented
0926-9851r01r - see front matter q 2001 Elsevier Science B.V. All rights reserved. Ž
. PII: S 0 9 2 6 - 9 8 5 1 0 0 0 0 0 3 8 - 0
an inversion scheme in which the finite difference method is used for the forward modeling and the
integral equation method for the inverse formulation. Ž
. Later, Newman and Alumbaugh
1997 modified
their inversion scheme for use on a massively paral- lel computer.
These inversion
methods required
powerful higher-end workstations or supercomputers. Thus, it
is commonly accepted that 3-D EM inversions are too expensive and cannot be a practical interpretation
tool for most field geophysicists. However, due to the ever increasing power of personal computers
Ž
. PC , running 3-D inversion schemes on PC is be-
coming a rational choice. The purpose of this paper is to demonstrate using synthetic examples that full
3-D EM inversions are feasible on PC. The inversion method consists of three key components: inversion
algorithm, forward modeling, and sensitivity calcula- tion. In this paper, these key points are first dis-
cussed and then the accuracy of the finite-difference solution used in the inversion routine is verified.
Finally, two synthetic examples of 3-D inversions involving airborne and ground EM methods are pre-
sented.
2. Linearized inversion
