Connecting beamforming and kernel-based noise source inversion

Beam forming and backprojection methods offer a data-driven approach to image noise sources, but provide no opportunity to account for prior information or iterate through an inversion framework. In contrast, recent methods have been developed to locate ambient noise sources based on cross-correlations between stations and the construction of finite-frequency kernels, allowing for inversions over multiple iterations. These kernel-based approaches show great promise, both in mathematical rigour and in results, but are less physically intuitive and interpretable. Here we show that these apparently two different classes of methods, beamforming and kernel-based inversion, arc achieving exactly the same result in certain circumstances. This paper begins with a description of a relatively simple beamforming or backprojection algorithm, and walks through a series of modifications or enhancements. By including a rigorously defined physical model for the distribution of noise sources and therefore synthetic correlation functions, we come to a framework resembling the kernel-based iterative approaches. Given the equivalence of these approaches, both communities can benefit from bridging the gap. For example, inversion frameworks can benefit from the numerous image enhancement tools developed by the beam forming community. Additionally, full-waveform inversion schemes that require a window selection for the comparisons of misfits can more effectively target particular sources through a windowing in a beamform slowness domain, or might directly use beamform heatmaps for the calculation of misfits. We discuss a number of such possibilities for the enhancement of both classes of methods, testing with synthetic models where possible.