3-D frequency-domain CSEM inversion using unconstrained optimization Eldad Haber (1) , Uri Ascher (2) , Douglas W. Oldenburg (3)* , Roman Shekhtman (3) , & Jiuping Chen (3) , (1) EMI-Schlumberger, Richmond, CA; (2) Dept. of Computer Science, UBC, Vancouver, Canada; (3) UBC- Geophysical Inversion Facility, Dept. of Earth&Ocean Sciences, UBC, Vancouver, Canada; Summary We develop an inversion algorithm for computing fre- quency domain electromagnetic inversion for conductive bodies in the low frequency regime. The algorithm is based on an inexact Gauss-Newton method where only sensitivities times vector are calculated. Because of the anticipated heavy computational load, one has to answer many practical questions in order to wisely use the generic numerical inverse methodology. In this presentation, we address some of these questions. A synthetic model stimulating a CSAMT survey has been carried out as a demonstration of the application of this generic inverse method. Introduction In this paper we discuss the solution of electromagnetic inverse problems in the frequency domain using an un- constrained Inexact Gauss-Newton formulation. The approach is complementary to our constrained ap- proach, presented in a companion abstract, and it serves a few purposes. First, because electromagnetic data in- version is a complicated procedure, it allows us to exam- ine methods for noise estimation, data weighting, model weighting and other practical aspects of the inversion pro- cedure that can subsequently be incorporated into the constrained approach. Second, because there is no proof of convergence for the constrained approach, we may take an unconstrained step in the case that a constrained step fails. Finally, this code serves as a base for comparison with the constrained approach and it is useful in its own right. The Forward Problem The frequency-dependent Maxwell equations can be writ- ten as ∇× E − ıωµH = 0 (1a) ∇× H − (σ − ıωǫ)E = sr (ω) (1b) where ω is the frequency, E and H are the electric and magnetic fields, µ is the permeability, σ is the conductiv- ity, ǫ is the permittivity and sr is a source. The boundary conditions over the entire boundary of the spatial domain, ∂Ω, are n × H =0. (2) We let ˆ σ = σ − ıωǫ. As discussed in Haber and As- cher(2001a) and Haber et al.(2000a), this form is not fa- vorable for iterative solvers, especially when |ω ˆ σ| is small (for example in the air). We therefore reformulated the problem prior to discretizing it further such that it is more amenable to applying standard iterative solvers. A Helmholtz decomposition with Coulomb gauge is ap- plied, decoupling the curl operator: E = A + ∇φ, ∇· A =0 in Ω A · n =0 on ∂Ω. After adding a stabilization term and differentiating (Haber and Ascher, 2001a), this leads to the system ∇× µ -1 ∇× A −∇µ -1 ∇· A + iω ˆ σ(A + ∇φ)= iωs (3a) ∇· (ˆ σ(A + ∇φ)) = ∇· s (3b) in Ω, subject to n ×∇× A =0, n · A =0, (3c) n ·∇φ =0, (3d) on the boundary ∂Ω. Following Haber and Ascher(2001a) and Haber et al.(2000a), we use a finite volume approach for the dis- cretization of (3) on an orthogonal, staggered grid. We choose to discretize A on cell faces and φ at cell cen- ters. Note that the modified conductivity ˆ σ is averaged harmonically at cell faces, whereas the permeability is av- eraged arithmetically at edges. We write the fully discretized system as  L μ + iωM ˆ σ iωM ˆ σ ∇ h ∇ h · M ˆ σ ∇ h · M ˆ σ ∇ h  A φ  =  iωs ∇ h · s  (4) where ∇ h · , ∇ h × and ∇ h are matrices arising from the discretization of the corresponding continuous operators, M ˆ σ arises from the operator ˆ σ(·) and Lμ is the discretiza- tion of the operator ∇× µ -1 ∇× −∇µ -1 ∇· . This linear system can be solved using standard iterative methods (Saad, 1996) and effective preconditioners can be designed for it (Haber et al., 2000a; Aruliah and As- cher, 2002). Briefly, for small enough ω, the system is dominated by its diagonal blocks and therefore a good preconditioner can be obtained by using an approxima- tion of the matrix  Lμ 0 0 ∇ h · M ˆ σ ∇ h  . (5)