Abstract

We develop a fast method for optimally designing experiments in the
context of statistical seismic source inversion. In particular, we
efficiently compute the optimal number and locations of the
receivers or seismographs. The seismic source is modeled by a point
moment tensor multiplied by a time-dependent function. The parameters
include the source location, moment tensor components, and start time
and frequency in the time function. The forward problem is modeled
by elastodynamic wave equations. We show that the Hessian of
the cost functional, which is usually defined as the square of
the weighted L _{2} norm of the di fference between the
experimental data and the simulated data, is proportional to
the measurement time and the number of receivers. Consequently, the
posterior distribution of the parameters, in a Bayesian setting,
concentrates around the "true" parameters, and we can employ Laplace
approximation and speed up the estimation of the expected Kullback-
Leibler divergence (expected information gain), the optimality criterion
in the experimental design procedure. Since the source parameters span
several magnitudes, we use a scaling matrix for efficient control of
the conditional number of the original Hessian matrix. We use a
second-order accurate finite diff erence method to compute the
Hessian matrix and either sparse quadrature or Monte Carlo sampling to
carry out numerical integration. We demonstrate the efficiency,
accuracy, and applicability of our method on a two-dimensional seismic
source inversion problem.