Time-domain electromagnetic migration in the solution of inverse problems

Abstract

SUMMARY Time-domain electromagnetic (TDEM) migration is based on downward extrapolation of the observed field in reverse time. In fact, the migrated EM field is the solution of the boundary-value problem for the adjoint Maxwell’s equations. The important question is how this imaging technique can be related to the solution of the geoelectrical inverse problem. In this paper we introduce a new formulation of the inverse problem, based on the minimization of the residual-field energy flow through the surface or profile of observations. We demonstrate that TDEM migration can be interpreted as the first step in the solution of this specially formulated TDEM inverse problem. However, in many practical situations this first step produces a very efficient approximation to the geoelectrical model, which makes electromagnetic migration so attractive for practical applications. We demonstrate the effectiveness of this approach in inverting synthetic and practical TDEM data. Time-domain electromagnetic (EM) migration is based on downward extrapolation of the residual field in reverse time. The basic principles of EM migration have been formulated in Zhdanov (1988), Zhdanov, Matusevich & Frenkel (1988), Zhdanov & Keller (1994) and Zhdanov, Traynin & Booker (1996). EM migration has important features in common with seismic migration (Zhdanov et al. 1988; Claerbout 1985) but differs in that for geoelectric problems EM migration is carried out on the basis of Maxwell’s equations, while in the seismic case it is based on the wave equation. We have introduced time-domain EM migration as the solution of the boundaryvalue problem in the lower half-space for the adjoint Maxwell’s equations, in which the boundary values of the migration field on the earth’s surface are determined by the observed EM field. In the paper by Zhdanov, Traynin & Portniaguine (1995) a technique for transforming the EM migration fields and their different components into resistivity images of the vertical cross-section was developed. However, the question still remains open how this imaging technique can be related to the solution of the geoelectrical inverse problem. Meanwhile, Tarantola ( 1987) demonstrated that seismic-wave migration, which was the prototype for EM migration, can be treated exactly as the first iteration in some general wave-inversion scheme. In the paper by Wang et al. (1994) this analogy was extended to the case of the diffusive transient EM field. In this paper we formulate and prove an important new result: EM migration, as the solution of the boundary-value problem for the adjoint Maxwell’s equation, can be clearly associated with the inverse-problem solution. We introduce the residual EM field as the difference between the simulated EM field for some given (background) geoelectrical model and the actual EM field. The EM energy flow of the residual field through the surface of observations can be treated as a functional of the anomalous conductivity distribution in the model. The analysis shows that the gradient of the residualfield energy-flow functional with respect to the perturbation of the model conductivity is equal to the vector crosscorrelation function between the incident (background) field and the migrated residual field, calculated as the solution of the boundary-value problem for the adjoint Maxwell’s equation. This result clearly leads to a construction of the rigorous method of solving the inverse EM problem, based on iterative EM migration in the time domain, and a gradient (or conjugate gradient) search for the optimal geoelectrical model. However, the authors have found that in the framework of this method even the first iteration, based on the migration of the residual field, generates a reasonable geoelectrical image of the subsurface structure. We call the anomalous conductivity, calculated on the first iteration, the migration apparent conductivity. We obtain a simple integral relationship between the migration apparent conductivity and actual anomalous conductivity, similar to the relationship established in the time domain for the inversion method based on the back-propagated TEM field (M. Oristaglio, personal communication, 1996). It describes the space filtering of the actual conductivity with the Green’s-type function. We believe that this relationship will