Sarah Minson, Ph.D. 
US Geological Survey

This seminar will provide an introduction to Bayesian analysis and its advantages and disadvantages relative to traditional optimization approaches for solving geophysical inverse problems. Bayesian methods have particular value for solving ill-posed inverse problems for two reasons. First, under-determined inverse problems by definition have more than one solution. The solution to the Bayesian inverse problem is the probability density function which describes the ensemble of all possible models which are consistent with the observed data and any constraints imposed on the model. This is a much more informative solution than computing one solution to the inverse problem via traditional optimization. Second, Bayesian inverse solutions may be obtained without evaluating the inverse of the design matrix, so that no matter how ill-posed the inverse problem is, regularization is never required (although it can be used if desired).

Demonstrations of how to apply Bayesian methods to real geophysical problems will be given. Although these examples will be drawn primarily from earthquake source modeling, the methods and tools presented are completely generic and applicable to a wide array of geophysical inverse problems, and the discussion will be broadly directed to the modeling community. Two end-member Bayesian approaches will be discussed. One is the completely generalized approach in which any desired cost function and a priori constraints can be used to fit the data. In practice, this often requires that the solution be computed via Monte Carlo simulation, which can have extreme computational expense for large numbers of free parameters. (Monte Carlo sampling strategies will be discussed as well.) The second approach is to choose the cost function and prior constraints carefully so as to obtain an analytical solution to the inverse problem. The cost of computing the solution becomes trivial but, depending on the physical system being modeled, some compromise in the inversion design may be required to keep the solution analytical.
Please note that the first 15 sec has no audio due to recording errors.

[All 2012-13 webinars] [YouTube]

1. Minson
2. webinar
3. 2013