Earthquake extraction and correlation energy at Long Beach, California seismic survey

Theory

In passive seismic interferometry, receivers record data from passive sources such as ambient seismic noise. Cross-correlating the two receiver recordings essentially turns the receivers into a source-receiver pair, thereby synthesizing new seismic responses. Specifically, cross-correlation recovers the Green's function, or the impulse response, and its time-reversed version between the two receivers, convolved with the autocorrelation of a source function such as noise (Wapenaar et al., 2010). In equation form

$$[G(x_B,x_A,t) + G(x_B,x_A,-t)] \ast S_N(t) = \langle u(x_B,t) \ast u(x_A,-t) \rangle \mbox \ \ \ ,$$ (3)

where $G$ is the Green's function between two receiver locations ($x_A,x_B$ ), $S_N(t)$ is the autocorrelation of the source function (here it is noise), and $u$ is the observed wavefield at a given receiver location. The result of cross-correlation is the traveltime difference of the recorded waves between the two given receivers.

A single cross-correlation of two recordings from two stations will likely produce noisy results. To improve signal-to-noise ratio, cross-correlations between multiple simultaneous recordings from two stations are stacked (or averaged) in the time-lag space (Curtis et al., 2006). This allows stationary signal phases to emerge, since these signals stack coherently and non-stationary phases stack incoherently. Therefore, stacking a month's worth of correlations should produce a better result than stacking a day's worth of correlations.

The method of passive seismic interferometry is effective only under certain conditions. The most important is that the principle of energy equipartition is satisfied. This means that noise must arrive at a receiver from all azimuths with the same amount of energy. If this is not satisfied, then correlation results may be far from ideal.

The station density and recording length of the array at Long Beach makes this dataset ideal for testing the limits of passive seismic interferometry in an urban environment. Higher station density provides more high-frequency information and should therefore provide better subsurface resolution. Longer recording times provide days to months of stacked correlations, as opposed to hours of stacked correlations. With longer stacked correlations, the convergence of the Green's function will improve and perhaps overcome the negative effects of cultural noise that is typically problematic in land data.

Earthquake extraction and correlation energy at Long Beach, California seismic survey