We are developing a new 3D code for application to electromagnetic induction tomography and to<br>environmental imaging problems. We have used the finite-difference frequency-domain formulation<br>of Beilenhoff et al. (1992) and the anisotropic PML (perfectly matched layer) approach (Berenger,<br>1994) to specify boundary conditions following Wu et al. (1997). PML deals with the fact that<br>the computations must be done in a finite domain even though the real problem is effectively of<br>infinite extent. The resulting formulas for the forward solver reduce to a problem of the form Ax<br>= y, where A is a non-Hermitian matrix with real values off the diagonal and complex values along<br>its diagonal. The matrix A may be either symmetric or nonsymmetric depending on details of the<br>boundary conditions chosen (i.e., the particular PML used in the application). The basic equation<br>must be solved for the vector x (which represents field quantities such as electric and magnetic fields)<br>with the vector y determined by the boundary conditions and transmitter location. Of the many<br>forward solvers that could be used for this system, relatively few have been thoroughly tested for the<br>type of matrix encountered in our problem. Our studies of the stability characteristics of the Bi-CG<br>algorithm raised questions about its reliability and uniform accuracy for this application. We have<br>found the stability characteristics of Bi-CGSTAB [ an alternative developed by van der Vorst (1992)<br>for such problems] to be entirely adequate for our application, whereas the standard Bi-CG was<br>quite inadequate. We have also done extensive validation of our code using semianalytical results as<br>well as other codes. The new code is written in Fortran 90 and is designed to be easily parallelized,<br>but we have not yet tested this feature of the code. An adjoint method is being developed for<br>solving the inverse problem for conductivity imaging (for mapping underground plumes), and this<br>approach, when ready, will make repeated use of the current forward modeling code.


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