Deterministic frequency‐wavenumber methods and direct measurements of rupture propagation during earthquakes using a dense array: Theory and methods

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Abstract
We present an approach for measuring the spatial extent, duration, directions and speeds of rupture propagation during an earthquake using array signal processing techniques. Unlike most previous approaches, no assumptions are made regarding the directions and speeds of propagation. The solution we obtain is unique, and its accuracy, precision and resolution can be estimated. We use preprocessing and coprocessing techniques that improve the accuracy/precision of existing frequency‐wavenumber methods. Subarray spatial averaging modifies frequency‐wavenumber techniques so that they are sensitive to multiple deterministic (transient) as well as stochastic (stationary) signals. This technique is also advantageous because it ensemble averages the cross spectral matrix and reduces the variance of the frequency‐wavenumber spectrum. Seismogram alignment is a preprocessing procedure that accounts for nonplanar wavefronts due to lateral velocity variations beneath an array. This procedure also allows us to study signals of interest using much shorter time windows than has previously been possible. The use of short time windows is important for reducing uncertainty in estimates of arrival times of seismic phases. We tested subarray spatial averaging and seismogram alignment with a variety of frequency‐wavenumber techniques and found that the multiple signal classification (MUSIC) method gave the best resolution of multiple signals. We also note that standard theoretical estimates of uncertainties in peak locations of frequency‐wavenumber spectra are much smaller than those typically observed using seismic arrays and we present a different formula that more accurately describes observed uncertainties. Using synthetic P and S body‐wave seismograms from extended earthquake sources we show that the above array signal processing techniques can be combined with ray theory to obtain accurate estimates of the locations and rupture times of an earthquake's high‐frequency seismic sources. We show how to estimate uncertainties in source locations and rupture times due to limitations of the data, uncertainties in source parameters, and uncertainties in velocity structure. We find that the size of the uncertainties can be very sensitive to a fault's geometry relative to an array and we suggest criteria for optimizing an array's location. We show that high‐frequency source locations and their associated rupture times can be used to estimate an earthquake's spatial extent, duration, directions and speeds of rupture propagation.

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