Bayesian inversion for facies detection: An extensible level set framework

In many cases, it has been assumed that the variability in hydrologic parameters can be adequately described through a simple geostatistical model with a given variogram. In other cases, variability may be best described as a series of jumps'' in parameter behavior, e.g., those that occur at geologic facies contacts. When using indirect measurements such as pump tests to try to map such heterogeneity (during inverse modeling), the resulting images of the subsurface are always affected by the assumptions invoked. In this paper, we discuss inversion for parameter fields where prior information has suggested that major variability can be described by boundaries between geologic units or facies. In order to identify such parameter fields, we propose a Bayesian level set inversion protocol framework, which allows for flexible zones of any shape, size, and number. We review formulas for defining facies locations using the level set method and for moving the boundaries between zones using a gradient-based technique that improves fit through iterative deformation of the boundaries. We describe the optimization algorithm employed when multiple level set functions are used to represent a field with more than two facies. We extend these formulas to the investigation of the inverse problem in a Bayesian context in which prior information is taken into account and through which measures of uncertainty can be derived. We also demonstrate that the level set method is well suited for joint inversion problems and present a strategy for integrating different data types (such as hydrologic and geophysical) without assuming strict petrophysical relations. Our framework for joint inversion also contrasts with many previous methods in that all data sources (e.g., both hydrologic and geophysical) contribute to boundary delineation at once.