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

Abstract

S U M M A R YMathematical 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 soiutions of a linear system of differential equations. A n 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 k m 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 Rain-Rate Retrieval Algorithm for Attenuated Radar Measurements

“…One of the difficulties for any regularization method in a nonlinear inverse problem is the optimal weighting of the penalty term as a trade-off between the error and the spatial structure (resolution) of the solution (Vasco 1997;Ceccherini 2005). The discrete L-curve method is popular for determining the value of the regularization parameter in iterative regularization methods; here, a compromise is made between the minimization of the constraints and the minimization of the differences between the measured and simulated spectra weighted by the noise (Hansen 1998;Schimpf and Schreier 1997).…”

Intersections, ideals, and inversion

“…Both of these methodologies suffer from limita-tions, and neither makes full use of the analytic structure of the equations defining the inverse problem. In specific cases it is possible to transform a non-linear problem to a linear one (Vasco 1997) and then apply conventional techniques, but this in not possible in general. For a large class of inverse problems, differential equations with undetermined coefficients and many integral equations, the discrete form of the equations are polynomial (Everett 1994).…”

Invariance, groups, and non-uniqueness: The discrete case

“…Techniques of Backus & Gilbert have been extended to estimate bounds on linear functionals in order to explore non‐uniqueness (Oldenburg 1983). Alternatively, it may be possible to transform a non‐linear inverse problem into a linear problem using either statistical or algebraic means (Vasco 1995, 1997). Such transformations are not unique, there can be more than one, but they can be devised such that the mappings to the linear problem and back are one‐to‐one.…”

“…Another advantage of Lie groups is that they may be used to treat continuous problems in the form of differential and integral equations (Olver 1986). For example, in Vasco (1997) Lie groups are used to determine if an inverse problem involving non‐linear functionals may be transformed into a linear inverse problem.…”