The goal of this set of lectures will be to describe our experiences with the use of Bayesian methods in solving some geophysical inverse problems involving one spatial dimension and to raise a series of issues with their wider application for geophysical imaging applications involving two or three spatial dimensions.
In particular, we plan to
(a) give an short overview of two practical applications of Bayesian methods in geophysical inverse problems that illustrate how these methods allow one to quantify the uncertainties of a 1D Earth model given measurement data. These examples will include the inversion of vertical offset seismic profile data for imaging structure ahead of the bit while drilling [1] and (2) the inversion of dc electrical measurements to model a saturation front moving away from an injection well [2].
(b) describe in some detail the basic Bayesian machinery used in solving 1D linear and nonlinear inverse problems ((a) the classical linear Gaussian case and (b) the use of Monte Carlo and nonlinear least squares methods for dealing with the nonlinear case.). Issues such as model parameterization, model likelihood, and computational costs will be highlighted. A good reference for this material is [3].
(c) highlight some of the issues that one encounters when applying Bayesian methods in 2D and 3D inverse problems. These issues include the problem of large model and data size as well as efficiently representing natural geological structures. I will give an overview of the use of wavelets as a potential ingredient in dealing with the model size issue [4]. I will show some early results from the use of Bayesian methods in solving a 2D linear tomography inverse problem and finally introduce the nonlinear versions of this problem that involve ray-tracing and full-waveform tomography.
[1] A. Malinverno and S. Leaney, “A Monte-Carlo Method to Quantify Uncertainty in the Inversion of Zero-Offset VSP Data”, 70th Ann. Internat. Mtg: SEG, pp. 2393-2396, 2000.
[2] A. Malinverno and C. Torres-Verdin, “Bayesian Inversion of DC Electrical Measurements with Uncertainties applied to Reservoir Monitoring”, Inverse Problems 16, pp. 1343-1356, 2000.
[3] A. Tarantola, Inverse Problem Theory, SIAM, 2004.
[4] Bennett and Malinverno, “Fast Model Updates Using Wavelets”, SIAM Multiscale Modeling and Simulation, Vol 3, pp. 106-138, 2005.