Optimal processing for seismic noise correlations

A wide spectrum of processing schemes is commonly applied during the calculation of seismic noise correlations. This is intended to suppress large-amplitude transient and monochromatic signals, to accelerate convergence of the correlation process or to modify raw correlations into more plausible approximations of interstation Green's functions. Many processing schemes, such as one-bit normalization or various other nonlinear normalizations, clearly break the linear physics of seismic wave propagation. This naturally raises the question: To what extent are the resulting noise correlations physically meaningful quantities? In this contribution, we demonstrate that commonly applied processing methods may indeed introduce an unphysical component into noise correlations. This affects not only noise correlation amplitudes but also, to a lesser extent, time-dependent phase information. The profound consequences are that most processed correlations cannot be entirely explained by any combination of Earth structure and noise sources, and that inversion results may thus be polluted. The positive component of our analysis is a new and easily applicable method that allows us to modify any existing processing such that it becomes optimal in the sense of (1) completely avoiding the unphysical component while (2) approximating the result of the original processing as closely as possible. The resulting optimal schemes can be derived purely on the basis of observed noise, without any knowledge of or assumptions on the nature of noise sources. In addition to the theoretical analysis, we present illustrative real-data examples from the Irish National Seismic Network and the Lost Hills array in Central California. We anticipate that optimal processing schemes may be most useful in applications that exploit complete correlation waveforms, amplitudes and weak arrivals, or small (time-dependent) phase shifts.