A transformational approach to geophysical inverse problems

Abstract

S U M M A R YA set of coordinate transformations is used to linearize a general geophysical inverse problem. Statistical and analytic techniques are employed to estimate the parameters of such linearization transformations. In the transformed space, techniques from linear inverse theory may be utilized. Consequently, important concepts, such as model parameter covariance, model parameter resolution and averaging kernels, may be carried over to non-linear inverse problems. I apply the approach to a set of seismic cross-borehole traveltimes gathered at the Conoco Borehole Test Facility. The seismic survey was conducted within the Fort Riley formation, a limestone with thin interbedded shales. Between the boreholes, the velocity structure of the Fort Riley formation consists of a high-velocity region overlying a section of lower velocity. It is found that model parameter resolution is poorest and spatial averaging lengths are greatest in the underlying low-velocity region.

Ground‐penetrating radar velocity tomography in heterogeneous and anisotropic media

“…This approach has worked well in treating the isotropic nonlinear seismic traveltime inverse problem (Vasco, 1995). We have incorporated an isotropic finite-difference code into a nonlinear, conjugategradient-based radar traveltime inversion program (Vasco, 1995). That is, we numerically solve the eikonal equation (A-8) for isotropic dielectric and magnetic permeability matrices, as has been done in seismology for seismic velocity (Vidale, 1988;Podvin and Lecomte, 1991).…”

“…Then, an iterative nonlinear optimization program is used to minimize the penalized misfit. This approach has worked well in treating the isotropic nonlinear seismic traveltime inverse problem (Vasco, 1995). We have incorporated an isotropic finite-difference code into a nonlinear, conjugategradient-based radar traveltime inversion program (Vasco, 1995).…”

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.…”