Bayesian inversion using nested trans-dimensional Gaussian processes

Abstract

\n To understand earth processes, geoscientists infer subsurface earth properties such as electromagnetic resistivity or seismic velocity from surface observations of electromagnetic or seismic data. These properties are used to populate an earth model vector, and the spatial variation of properties across this vector sheds light on the underlying earth structure or physical phenomenon of interest, from groundwater aquifers to plate tectonics. However, to infer these properties the spatial characteristics of these properties need to be known in advance. Typically, assumptions are made about the length scales of earth properties, which are encoded a priori in a Bayesian probabilistic setting. In an optimization setting, appeals are made to promote model simplicity together with constraints which keep models close to a preferred model. All of these approaches are valid, though they can lead to unintended features in the resulting inferred geophysical models owing to inappropriate prior assumptions, constraints or even the nature of the solution basis functions. In this work it will be shown that in order to make accurate inferences about earth properties, inferences can first be made about the underlying length scales of these properties in a very general solution basis. From a mathematical point of view, these spatial characteristics of earth properties can be conveniently thought of as ‘properties’ of the earth properties. Thus, the same machinery used to infer earth properties can be used to infer their length scales. This can be thought of as an ‘infer to infer’ paradigm analogous to the ‘learning to learn’ paradigm which is now commonplace in the machine learning literature. However, it must be noted that (geophysical) inference is not the same as (machine) learning, though there are many common elements which allow for cross-pollination of useful ideas from one field to the other, as is shown here. A non-stationary trans-dimensional Gaussian Process (TDGP) is used to parametrize earth properties, and a multichannel stationary TDGP is used to parametrize the length scales associated with the earth property in question. Using non-stationary kernels, that is kernels with spatially variable length scales, models with sharp discontinuities can be represented within this framework. As GPs are multidimensional interpolators, the same theory and computer code can be used to solve geophysical problems in 1-D, 2-D and 3-D. This is demonstrated through a combination of 1-D and 2-D non-linear regression examples and a controlled source electromagnetic field example. The key difference between this and previous work using TDGP is generalized nested inference and the marginalization of prior length scales for better posterior subsurface property characterization.