Theoretical Foundations of Noise Interferometry

Abstract

The retrieval of a deterministic signal from recordings of a quasi-random ambient seismic field is the central goal of noise interferometry. It is the foundation of numerous applications ranging from noise source imaging to seismic tomography and time-lapse monitoring. In this chapter, we offer a presentation of theoretical approaches to noise interferometry, complemented by a critical discussion of their respective advantages and drawbacks. The focus of this chapter is on interstation noise correlations that approximate the Green's function between two receivers. We explain in detail the most common mathematical models for Green's function retrieval by correlation, including normal-mode summation, plane-wave decomposition, and representation theorems. While the simplicity of this concept is largely responsible for its remarkable success, each of these approaches rests on different but related assumptions such as wavefield equipartitioning or a homogeneous distribution of noise sources. Failure to meet these conditions on Earth may lead to biases in traveltimes, amplitudes, or waveforms in general, thereby limiting the accuracy of the method. In contrast to this well-established method, interferometry without Green's functional retrieval does not suffer from restrictive conditions on wavefield equipartitioning. The basic concept is to model the interstation correlation directly for a given power-spectral density distribution of noise sources and for a suitable model of the Earth that may be attenuating, heterogeneous, and anisotropic. This approach leads to a coupled problem where both structure and sources affect data, much like in earthquake tomography. Observable variations of the correlation function are linked to variations in Earth structure and noise sources via finite-frequency sensitivity kernels that can be used to solve inverse problems. While being mathematically and computationally more complex, interferometry without Green's function retrieval has produced promising initial results that make successful future applications likely. We conclude this chapter with a summary of alternative approaches to noise interferometry, including interferometry by deconvolution, multi-dimensional deconvolution, and iterated correlation of coda waves.

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