Sequential Bayesian techniques enable tracking of evolving geophysical parameters via sequential observations. They provide a formulation in which the geophysical parameters that characterize dynamic, nonstationary processes are continuously estimated as new data become available. This is done by using prediction from previous estimates of geophysical parameters, updates stemming from physical and statistical models that relate seismic measurements to the unknown geophysical parameters. In addition, these techniques provide the evolving uncertainty in the estimates in the form of posterior probability density functions. In addition to the particle filters (PFs), extended, unscented, and ensemble Kalman filters (EnKFs) were evaluated. The filters were compared via reflector and nonvolcanic tremor tracking examples. Because there are numerous geophysical problems in which the environmental model itself is not known or evolves with time, the concept of model selection and its filtering implementation were introduced. A multiple model PF was then used to track an unknown number of reflectors from seismic interferometry data. We found that when the equations that define the geophysical problem are strongly nonlinear, a PF was needed. The PF outperformed all Kalman filter variants, especially in low signal-to-noise ratio tremor cases. However, PFs are computationally expensive. The EnKF is most appropriate when the number of parameters is large. Because each technique is ideal under different conditions, they complement each other and provide a useful set of techniques for solving sequential geophysical inversion problems.
Estimation of Spatial Deformation for Nonstationary Processes via Variogram Alignment
