Improved implementation of the fk and Capon methods for array analysis of seismic noise

Abstract

S U M M A R YThe frequency-wavenumber (fk) and Capon methods are widely used in seismic array studies of background or ambient noise to infer the backazimuth and slowness of microseismic sources. We present an implementation of these techniques for the analysis of microseisms (0.05-2 Hz) which draws on array signal processing literature from a range of disciplines. The presented techniques avoid frequency mixing in the cross-power spectral density and therefore yield an accurate slowness vector estimation of the incoming seismic waves. Using synthetic data, we show explicitly how the frequency averaged broad-band approach can result in a slowness-shifted spectrum. The presented implementation performs the slowness estimations individually for each frequency bin and sums the resulting slowness spectra over a specific frequency range. This may be termed an incoherently averaged signal, or IAS, approach. We further modify the method through diagonal loading to ensure a robust solution. The synthetic data show good agreement between the analytically derived and inferred error in slowness. Results for real (observed) data are compared between the approximate and IAS methods for two different seismic arrays. The IAS method results in the improved resolution of features, particularly for the Capon spectrum, and enables, for instance, Rg and Lg arrivals from similar backazimuths to be separated in the case of real data.

The frequency dependence and locations of short‐period microseisms generated in the Southern Ocean and West Pacific

“…The earthquake waveforms are extracted in 100 s lengths, where the P wave onset is positioned 50 s into the window. Each segment is tapered with a Hann window, and an implementation of the IAS Capon beamforming [ Gal et al ., ] is used to estimate slowness and back azimuth. The recovered back azimuths are compared to those expected for the ISC locations, while slowness values are compared to theoretically predicted values from the “ak135” model [ Kennett et al ., ].…”

“…The beamforming algorithm averages over multiple narrowband slowness spectra, and its stability is ensured via diagonal loading of the cross‐power spectral density. It has been shown that IAS Capon recovers slowness vectors accurately and is able to resolve multiple arrivals [ Gal et al ., ] in a given time window. The analysis is performed in eight frequency bands with widths of 0.05 Hz in the range of 0.325 to 0.725 Hz.…”

Cross-correlation beamforming

“…Futhermore, ω o is the frequency for which the beampower is computed. Either (2) is computed for only this frequency sample or the beampower is averaged over a small frequency bin around ω o (see, e.g., Gal et al 2014). Equation 3 states that a wavenumber vector is composed for each combination of slowness (here the reciprocal of horizontal apparent velocity) and backazimuth, which together define a plane wave.…”

Spiral‐Arm Seismic Arrays

“…Logistic considerations rarely allow an array design to be implemented perfectly on the ground, and the tolerance of response of the spiral-arm arrays to minor variations in site position has allowed both PSAR and SQspa to adapt to local constraints and still achieve close to the expected theoretical resolution. We have shown the linear stack response because this is directly related to the geometry of the array, but enhancement of the precision of slowness estimates can readily be made by using nonlinear stacking schemes that can exploit the vector character of the wavefield (e.g., Kennett, 2000;Gal et al, 2014).…”