BAYESIAN SEISMIC WAVE-FORM INVERSION - PARAMETER-ESTIMATION AND UNCERTAINTY ANALYSIS

The goal of geophysical inversion is to make quantitative inferences a bout the Earth from remote observations. Because the observations are finite in number and subject to uncertainty, these inferences are inhe rently probabilistic. A key step is to define what it means for an Ear th model to fit the data. This requires estimation of the uncertaintie s in the data, both those due to random noise and those due to theoret ical errors. But the set of models that fit the data usually contains unrealistic models; i.e., models that violate our a priori prejudices, other data, or theoretical considerations. One strategy for eliminati ng such unreasonable models is to define an a priori probability densi ty on the space of models, then use Bayes theorem to combine this prob ability with the data misfit function into a final a posteriori probab ility density reflecting both data fit and model reasonableness. We sh ow here a case study of the application of the Bayesian strategy to in version of surface seismic field data. Assuming that all uncertainties can be described by multidimensional Gaussian probability densities, we incorporate into the calculation information about ambient noise, d iscretization errors, theoretical errors, and a priori information abo ut the set of layered Earth models derived from in situ petrophysical measurements. The result is a probability density on the space of mode ls that takes into account all of this information. Inferences on mode l parameters can be derived by integration of this function. We begin by estimating the parameters of the Gaussian probability densities ass umed to describe the data and model uncertainties. These are combined via Bayes theorem. The a posteriori probability is then optimized via a nonlinear conjugate gradient procedure to find the maximum a posteri ori model. Uncertainty analysis is performed by making a Gaussian appr oximation of the a posteriori distribution about this peak model. We p resent the results of this analysis in three different forms: the maxi mum a posteriori model bracketed by one standard deviation error bars, pseudo-random simulations of the a posteriori probability (showing th e range of typical subsurface models), and marginals of this probabili ty at selected depths in the subsurface. The models we compute are con sistent both with the surface seismic data and the borehole measuremen ts, even though the latter are well below the resolution of the former . We also contrast the Bayesian maximum a posteriori model with the Oc cam model, which is the smoothest model that fits the surface seismic data alone.