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Nonstationary Bayesian kriging: A predictive technique to generate spatial corrections for seismic detection, location, and identification

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, California 94551
Sandia National Laboratory, P.O. Box 5800, Albuquerque, New Mexico 87185

Abstract

Seismic characterization works to improve the detection, location, and identification of seismic events by correcting for inaccuracies in geophysical models. These inaccuracies are caused by inherent averaging in the model, and, as a result, exact data values cannot be directly recovered at a point in the model. Seismic characterization involves cataloging reference events so that inaccuracies in the model can be mapped at these points and true data values can be retained through a correction. Application of these corrections to a new event requires the accurate prediction of the correction value at a point that is near but not necessarily coincident with the reference events. Given that these reference events can be sparsely distributed geographically, both interpolation and extrapolation of corrections to the new point are required. In this study, we develop a closed-form representation of Bayesian kriging (linear prediction) that incorporates variable spatial damping. The result is a robust nonstationary algorithm for spatially interpolating geophysical corrections. This algorithm extends local trends when data coverage is good and allows for damping (blending) to an a priori background mean when data coverage is poor. Benchmark tests show that the technique gives reliable predictions of the correction value along with an appropriate uncertainty estimate. Tests with travel-time residual data demonstrate that combining variable damping with an azimuthal coverage criterion reduces the large errors that occur with more classical linear prediction techniques, especially when values are extrapolated in poor coverage regions. In the travel-time correction case, this technique generates both seismic corrections along with uncertainties and can properly incorporate model error in the final location estimate. Results favor the applicability of this nonstationary algorithm to other types of seismic corrections such as amplitude and attenuation measures.

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