The stationary phase approximation in seismic interferometry: Error quantification and the effects of source correlations.

Seismic interferometry is a powerful and well-established technique that relies on cross-correlating seismic observations at different receiver locations to yield new seismic responses that, under certain conditions, provide a useful estimate of the Green's function between the given receiver locations, as if there was a source at one of these locations. The inter-receiver signals thus estimated allow us to monitor and remotely illuminate near-surface crustal structures.

Underpinning seismic interferometry is the principle of stationary phase, which states that non-trivial contributions to highly oscillatory integrals, such as those found in interferometry, arise from stationary points of the phase of these cross-correlations. This principle is widely invoked to make approximations in interferometry, both in theory, to derive and simplify interferometric formulations, as well as in practical applications, to justify the use of non-ideal source or receiver distributions. Further, it has been established that spatial variations in the source intensity must be smooth in order to apply this approximation.

While there have been some empirical explorations of the uncertainty introduced by this approximation, the errors have not yet been quantified analytically, and neither the effects of non-smooth variations in the sources, nor of statistical correlations between sources, have been formally considered. In this work, we apply a mathematical framework to seismic interferometry in two dimensions. This analysis yields an exact expression for the error in the interferometric estimate of the inter-receiver Green’s function. Moreover, we extend this approach to a scenario of inhomogeneous, statistically correlated sources, and illustrate the effects of source correlation and roughness on the phase and amplitude of the stationary-phase interferometric estimate. We provide statistical conditions to ensure that the stationary phase estimate is unbiased, and give an explicit bound for the error in the estimated spectrum. These error quantities are given in terms of parameters that are either known (such as the inter-receiver distance), or can be estimated from empirical data. Therefore, we expect these results to be applicable in practical interferometric studies that make use of the stationary phase approximation, as a tool to quantify error and uncertainty in empirical results.