Time-domain electromagnetic migration in the solution of inverse problems

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.     

Groups, algebras, and the non-linearity of geophysical inverse problems

Mathematical methods from the theory of continuous groups are used to determine whether a non-linear inverse problem, in the form of a functional, can be transformed into a linear inverse problem. If such transformations exist they can be constructed from the solutions of a linear system of differential equations. An illustration of the methodology is given by the linearization of the functional relating basement topography to observed surface gravity. The linearized inversion of gravity data for basement topography is applied to observations from Yucca Mountain, Nevada. A 2.0 km step in the basement to the west of Yucca Mountain, corresponding to the Bare Mountain fault, matches the Bouguer gravity anomaly. The resolution and uncertainty associated with the estimates of basement topography indicate that the structure directly beneath the gravity line is well constrained.     

A method of obtaining a velocity-depth envelope from wide-angle seismic data

Summary. Due to the non-uniqueness of traveltime inversion of seismic data, it is more appropriate to determine a velocity-depth ( v-z ) envelope, rather than just a v-z function. Several methods of obtaining a v-z envelope by extremal inversion have been proposed, all of which invert the data primarily from either x-p , or T-p , or both domains. These extremal inversion methods may be divided into two groups: linear extremal and non-linear extremal. There is some debate whether the linearized perturbation techniques should be applied to the inherently non-linear problem of traveltime inversion. We have obtained a v-z envelope by extremal inversion in T-p with the constraint that the inversion paths also satisfy x-p observations. Thus we use data jointly in r-p and x-p , and yet avoid the linearity assumptions.
This joint, non-linear extremal inversion method has been applied to obtain a v-z envelope down to a depth of about 30 km in the Baltimore Canyon trough using x-t data from an Expanding Spread Profile acquired during the LASE project. We have found that the area enclosed by the v-z envelope is reduced by about 15 per cent using x-p control on the T-p inversion paths, compared to the inversion without x-p control.