The Spectral Amplitude Grouping method (SAG) for analyzing crustal stress conditions. A potential for intermediate-term warnings

A new method called Spectral Amplitude Grouping (SAG) has been developed (Lund and Böðvarsson 2000). This method addresses the problem of similarity of focal mechanisms used in stress tensor inversion. SAG has shown to be useful for analysis of temporal evolution of the earthquake grouping patterns. The SIL system calculates spectral amplitudes on three component data rotated into vertical, radial and transverse components. Windows are placed on the direct P- and S-wave arrivals and transforming to the frequency domain the low frequency asymptotes, or DC-level spectral amplitudes, are estimated for the different components (Rögnvaldsson and Slunga 1993). We obtain five amplitude values at each recording station; vertical and radial P (PZ and PR) and vertical, radial and transverse S (SZ, SR and ST), which we refer to as amplitude components. These amplitude components, together with first motion directions, form the basis for the focal mechanism calculation in the SIL system and will be utilized in the spectral amplitude correlation and grouping scheme. In order to assess the similarity of the focal mechanisms of two different events all amplitude components in common for the two events are correlated using linear cross-correlation

$$r = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i (x_i - \bar{x})^2} \sqrt{\sum_i (y_i - \bar{y})^2}} \notag$$

where r is the correlation coefficient, $\bar{x}$, is the mean of the logarithms of one event's amplitude components, x_i, and $\bar{y}$ the mean ofthe logarithms of the other event's amplitude components, y_i. The logarithms are utilized to decrease the importance of the nearest stations, thereby stabilizing the correlation. We use the correlation coefficient as the measure of how similar two events are. All events are correlated with all other events and the events are then grouped according to the correlation coefficients. The grouping is controlled by three parameters; a lower limit on the correlation coefficients, r_min, a lower limit on the fraction, f_min, of fellow events in the group that a single event is allowed to have below r_min and the minimum number of events needed to have a group. After some testing we adopted r_min=0.9, f_min=0.8 and at least four events in the group, as our parameters for studying larger amounts of seismicity. If a more detailed study on fewer events is desired, the r_min and/or f_min values should be increased. We define two modes of running the correlation and grouping, the first correlates all events in a catalogue with all other events in one large run, and then performs an iterative grouping that allow us to find the optimal homogeneity within the groups. The second mode starts with a small group of events that are correlated and grouped, and then the events are correlated and grouped one by one with the previous events. This mode will not obtain the optimal group homogeneity but instead it allows us to study the time variations in correlation and grouping. During the development of the correlation and grouping scheme we discovered that it is useful also for applications other than as a preprocessor to stress tensor inversion. If we run the correlation in the second mode the temporal evolution of the earthquake grouping patterns can be studied and the groups of similar events produced by the grouping can be utilized either for composite focal mechanism calculations or as a starting group for relative relocation. We tested the correlation and grouping algorithm on a set of 636 microearthquakes, 0.0$\leq$M_L$\leq$2.7, occurring between July 1, 1998, and November 13, 1998, in Ölfus, southwestern Iceland. On November 13 there was a magnitude 5.0 earthquake in the Ölfus area. Cumulative number of events and cumulative seismic moment is plotted in Figure 6.

  

Figure 6: Correlation results in Ölfus, July 1 to November 13, 1998. A) Plotted versus time in days is the cumulative number of events (solid line, scale to the left) and the cumulative scalar seismic moment (dashed line, scale to the right). B) Number of ungrouped events (solid line) and the number of groups (dashed line) versus time in days. C) Number of ungrouped events (solid line) and number of groups (dashed line) versus the event number.

Studying the Ölfus seismicity using the second mode of correlation and grouping we obtain the plots in Figures 6B and 6C. In Figure 6B we see that the seismicity in July correlates very well, the rapid increase in number of events is not mirrored by an increase in the number of ungrouped events. Conversely, the increase in seismicity in November has an associated increase in the number of ungrouped events. The grouping pattern becomes clearer if we plot the number of ungrouped events and number of groups as a function of the event number (Figure 6C). We now clearly see a change in the grouping pattern around event 430, which corresponds to late September, where the slope of the curve significantly changes. We interpret the lack of correlation after September as an indication that the microseismicity changed characteristics. Before late September many events occur on the same fault (or a very close, similarly oriented fault) with very similar slip directions. We refer to these events as repeated events. After September, spectral amplitude correlation indicates that either the focal mechanisms are different, both compared to earlier and to current seismicity, or the events occur at different locations compared to earlier and current seismicity.

A version of this program program has been installed for real-time monitoring of earthquake grouping patterns.