Continuous monitoring by ambient-seismic noise tomography

Convergence rate of ambient-seismic noise correlations

To design an installation that exploits seismic interferometry as a permanent, continuous, nearly real-time monitoring system, we need to investigate how much ambient-seismic noise we must cross-correlate to retrieve a stable EGF. The correlation coefficient between two virtual sources, one created by correlating 3 hours of data and one created by correlating all data, as a function of the center-time of the shorter recording. A high correlation coefficient means that the correlation of the partial recording is very similar to a correlation of the full signal. This analysis should be made with caution, because there is no measure of the physical correctness of the EGF in these correlation coefficients. Theoretically, the inclusion of more time in longer records could deteriorate the correlation signal as an estimation of the Green's functions. However, the comparison of correlations of a partial recording with the total recording gives a measure of the convergence rate and the stability of the estimated Green's function. The resulting correlation coefficients are averaged for a virtual source in the center of the array and a source at the north-west end of the array, as shown in Figure 7. These measures for correlation-convergence show similarity with the strengths of the microseism energy as seen in Figure 2, but the especially strong periods of microseism energy on the 26$^{th}$ and 27$^{th}$ of December are less dominant in Figure 7. When higher frequencies are particularly strong, the partial stack does not resemble the full stack very well.

corrcoefVt-shots-si
Figure 7. Correlation coefficients as a function of start time comparing a 3 hour partial window with the full stack. The top panel shows the spectragram with the correlation coefficients overlayed and the bottom panel shows only the curve of correlation coefficients.

We want to analyse the cross-correlation convergence rate as a function of inter-station distance for various frequency ranges. We compute the correlation coefficient between two virtual sources after bandpassing for a certain central frequency, and bin the coefficients as a function of inter-station distance. One virtual sources is computed by cross-correlating all data while the second is computed cross-correlating a partial recording. Furthermore, to eliminate sensitivity to a particular starting time of the partial recording, we slid the partial recording window across the complete recording. We repeat the whole procedure for two virtual sources and average the mean correlation coefficients for each distance bin and partial recording length. The frequency range $1.50 - 1.75$  Hz is shown in 8a, $1.50 - 1.75$  Hz in 8b, $1.25 - 1.50$  Hz in 8c, $1.00 - 1.25$  Hz in 8d, $0.75 - 1.00$  Hz in 8e, $0.50 - 0.75$  Hz in 8f, $0.25 - 0.50$  Hz in 8f. Dotted and dashed lines are $0.50$ and $0.95$ contours. The contour lines are not very smooth. This can probably be improved by averaging the computations for more virtual sources. These figures show a trend of faster convergence rate for lower frequencies. This trend is especially apparent with inter-station distances larger than $1000$  meters. This trend is expected, because low frequencies have a larger Fresnel zone than do higher frequencies, and therefore require fewer sources surrounding the stations. Since the same sources excite the entire frequency regime, the background correlation fluctuations will be dominated by higher frequencies, and the correlation will stabilize faster for lower frequencies. Correlations between stations at shorter distances converge much faster, but a trend for different frequencies at shorter distances is not clear. This behavior explains the results in Figure 7, where a strong presence of high frequencies actually deteriorates the convergence rate.

coef-high,coef-middle,coef-low,coef-lower,coef-verylow,coef-verylower
Figure 8. Correlation coefficient of partial versus total cross-correlated signal time. Dotted and dashed lines are $0.50$ and $0.95$ contours. Analysis made for frequency ranges; $1.50 - 1.75$  Hz in (a), $1.50 - 1.75$  Hz in (b), $1.25 - 1.50$  Hz in (c), $1.00 - 1.25$  Hz in (d), $0.75 - 1.00$  Hz in (e), $0.50 - 0.75$  Hz in (f), $0.25 - 0.50$  Hz in (f).

Continuous monitoring by ambient-seismic noise tomography