Generalized interferometry – I: theory for interstation correlations

Abstract

We develop a general theory for interferometry by correlation that (i) properly accounts for heterogeneously distributed sources of continuous or transient nature, (ii) fully incorporates any type of linear and nonlinear processing, such as one-bit normalization, spectral whitening and phase-weighted stacking, (iii) operates for any type of medium, including 3-D elastic, heterogeneous and attenuating media, (iv) enables the exploitation of complete correlation waveforms, including seemingly unphysical arrivals, and (v) unifies the earthquake-based two-station method and ambient noise correlations. Our central theme is not to equate interferometry with Green function retrieval, and to extract information directly from processed interstation correlations, regardless of their relation to the Green function. We demonstrate that processing transforms the actual wavefield sources and actual wave propagation physics into effective sources and effective wave propagation. This transformation is uniquely determined by the processing applied to the observed data, and can be easily computed. The effective forward model, that links effective sources and propagation to synthetic interstation correlations, may not be perfect. A forward modelling error, induced by processing, describes the extent to which processed correlations can actually be interpreted as proper correlations, that is, as resulting from some effective source and some effective wave propagation. The magnitude of the forward modelling error is controlled by the processing scheme and the temporal variability of the sources. Applying adjoint techniques to the effective forward model, we derive finite-frequency Fréchet kernels for the sources of the wavefield and Earth structure, that should be inverted jointly. The structure kernels depend on the sources of the wavefield and the processing scheme applied to the raw data. Therefore, both must be taken into account correctly in order to make accurate inferences on Earth structure. Not making any restrictive assumptions on the nature of the wavefield sources, our theory can be applied to earthquake and ambient noise data, either separately or combined. This allows us (i) to locate earthquakes using interstation correlations and without knowledge of the origin time, (ii) to unify the earthquake-based two-station method and noise correlations without the need to exclude either of the two data types, and (iii) to eliminate the requirement to remove earthquake signals from noise recordings prior to the computation of correlation functions. In addition to the basic theory for acoustic wavefields, we present numerical examples for 2-D media, an extension to the most general viscoelastic case, and a method for the design of optimal processing schemes that eliminate the forward modelling error completely. This work is intended to provide a comprehensive theoretical foundation of full-waveform interferometry by correlation, and to suggest improvements to current passive monitoring methods.

Passive seismic monitoring with nonstationary noise sources

“…As shown by Ermert et al (2016), kernel-based noise source inversion leads to results that are qualitatively similar to beamforming, in the sense that nearly identical source regions are mapped by both methods. This is to be expected because sensitivity kernels roughly act as beams pointing in the direction of possible noise sources, and with strength proportional to the noise source amplitudes.…”

“…This paper is the first real data application of theory established in Hanasoge (2014) and extends upon this via methods found in Ermert et al (2016). It is organized as follows:…”

“…We deliberately do not apply non-linear processing techniques, such as spectral whitening or one-bit normalization (Bensen et al, 2007;Groos et al, 2012), even though the latter has been shown to not add a systematic bias to the measurement in the case of Gaussian wavefield statistics and for a variety of other distributions (Hanasoge and Branicki, 2013;Fichtner et al, 2016). We deliberately do not apply non-linear processing techniques, such as spectral whitening or one-bit normalization (Bensen et al, 2007;Groos et al, 2012), even though the latter has been shown to not add a systematic bias to the measurement in the case of Gaussian wavefield statistics and for a variety of other distributions (Hanasoge and Branicki, 2013;Fichtner et al, 2016).…”

“…To extract robust information on the psd of the noise sources, we follow the approach of Ermert et al (2016) and measure causal-to-acausal energy ratios in the daily correlations, defined as…”

“…To reduce the computational costs of noise source inversion, we adopt the ray-theoretical simplification of the noise source kernels proposed by Ermert et al (2016). The raytheoretical kernel corresponding to the finite-frequency kernel in Figure 4a, is shown below This paper presented here as accepted for publication in Geophysics prior to copyediting and composition.…”