Paper III

Correlation of microearthquake body-wave spectral amplitudes

Introduction
During the work on Paper II, and especially during the period when I performed synthetic tests on the stress inversion algorithm, we frequently discussed the issue of the similarity of the focal mechanisms we input to the inversion. As I discussed in Section 3.4.1, the stress tensor inversion needs focal mechanisms sufficiently well distributed in terms of orientation in order to produce well constrained estimates. It is thus important to assess how similar, or redundant, the focal mechanisms are for a group of events before they are input to the inversion. When calculating the confidence limits it is necessary to disregard the redundant events, or the confidence limits will be unrealistically small. In the SIL group, we also discussed the possibility to disregard redundant events altogether in the inversion, only including one event as a representative for a group of similar events, in order to shorten computer run-times. Quantifying the similarity of focal mechanisms in terms of the strike, dip and rake angles is not trivial. In addition, the SIL focal mechanism algorithm [Rögnvaldsson and Slunga, 1993] provides us with, very valuable, information on the uncertainty in the mechanisms in terms of a range of acceptable focal mechanisms. These acceptable mechanisms further complicates the similarity test. A solution to the similarity test problem is spectral amplitude correlation and grouping. We return one step back in the chain of earthquake data processing. The focal mechanisms are calculated from spectral amplitudes and polarities [Rögnvaldsson and Slunga, 1993], so by comparing the spectral amplitude distributions of closely located events, we can assess their similarity in terms of focal mechanisms.

Summary
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

$$r = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i (x_i - \bar{x})^2} \sqrt{\sum_i (y_i - \bar{y})^2}}$$ (77)

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 of the 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 18.

Figure: Correlation results at 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 Figure 18B and C. In Figure 18B we see that the seismicity in July correlate 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 18C. 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. These events will be referred to as solitary events. High accuracy relative relocation [Slunga et al., 1995] supports the notion of a more diffuse seismicity towards the end of the period.

Using one of the groups produced by spectral amplitude correlation of the entire data set, now with $r_{min} = 0.9$ and $f_{min} = 0.9$, we compute a composite focal mechanism. There were 27 events in the group and the nodal planes of the optimal focal mechanisms have a spread of approximately ${50}^{\circ}$, both in strike and dip. Utilizing the spectral amplitudes from the 27 events we compute a median distribution of spectral amplitudes for the entire group. All events' first motion directions are stacked and then the combined amplitudes and polarities are used to compute a new focal mechanism using the SIL algorithm. The nodal planes of the composite focal mechanism are compared to the result of relative relocation, and one of the nodal planes is only ${15}^{\circ}$ away from the common fault plane defined by the relocations. The fit is not perfect, but considering the rough amplitude stacking technique it is quite good. Among the acceptable focal mechanisms we find one nodal plane that is only ${7}^{\circ}$ away from the common fault plane.

Figure: Stress tensor inversion of the 199 events from Ölfus, Iceland, July, 1998. Symbols and patterns are as in Figure 17, with the exception that I do not plot the 10% confidence limits for the stresses here. A) Inversion result using 199 events and 199 as the number of events when calculating the confidence levels. B) Result using 199 events in the inversion and 59 reduced events, according to Magee [1997], for the confidence level calculation. C) Result using 79 reduced (from spectral amplitude correlation) events, of which 10 are composite events, for both the inversion and the confidence level estimates.

Finally, we test the correlation and grouping scheme in terms of input to the stress tensor inversion. Magee [1997] discussed the issue of redundant focal mechanisms and approached the problem by assigning the nodal plane normals to ${10}^{\circ}$ bins over the lower hemisphere and using half the number of filled bins as the number of non-redundant events when calculating the confidence limits. The correlation and grouping scheme can be used to assess the similarity of closely located events, i.e. if two events have identical focal mechanisms but are far from each other (depending on station geometry and focal mechanism), they will not be identified as similar. The correlation will, thus, not be able to identify all redundant focal mechanisms, but as we shall see below, it can significantly improve the calculation of confidence limits and reduce the size of the data set. We will use the term reduced events instead of non-redundant henceforth.

We perform stress tensor inversions using both the full number of events and the binned number of events for calculating confidence limits and compare these to an inversion of a data set reduced by the correlation. Microearthquake data from 1998, 199 events in July and 182 in November, is used for the inversions. I show here the result of the July data. The binning technique of Magee [1997] estimates that there are 59 reduced events in July and the spectral amplitude correlation results in 69 ungrouped events and 10 groups, i.e. 79 reduced events. In Figure 19 we show the inversion results using the three different approaches and the inversion algorithm of Paper II. Figures 19A and 19B show the confidence regions obtained using 199 and 59 reduced events, respectively, to calculate the confidence limits. The inversion itself uses all 199 events which is shown by the identical misfits and optimal stress states. We see that there is a large difference in the size of the confidence areas. In Figure 19C is the result of using spectral amplitude correlation to estimate the number of reduced events and using only the reduced events in the inversion. Running the July inversion with the 79 reduced events instead of 199 significantly shortens the necessary computer time. We see in Figure 19C that the optimal stress state is almost identical to the state estimated using the full data set but that the misfit is slightly larger when using only 79 events in the inversion. This is not surprising since among the 199 events there are most likely some rather poorly constrained mechanisms, resulting in a large number of acceptable focal mechanisms which gives the inversion algorithm larger freedom in the choice of fault plane, see Paper II, compared to using 79 well determined events.

Concluding remarks
The spectral amplitude correlation and grouping technique has proven a successful method in terms of assessing the similarity of focal mechanisms of closely located events and grouping of similar events. An event representing the group is selected either through a composite focal mechanism or by choosing the highest quality event, i.e. the event with most stations, amplitude components and polarities. The grouping technique also allows us to form batches of a certain number of reduced events, which will be useful in studies of temporal or spatial variations in the stress field, as in Paper IV.

The issue of how different the focal mechanisms have to be in order to be considered non-redundant in the inversion has not been meticulously studied. We observed the grouping patterns and empirically found values of the grouping parameters that produced homogeneous groups, minimizing the number of events that ``almost'' made it into the groups.

In order to study the influence of spatial variation on the grouping result we will implement means of taking the events locations into account.

The spectral amplitude correlation and grouping technique seems to be a very promising tool for the study of temporal or spatial variations in earthquake repeating patterns. By repeating pattern we refer to changes in the number of repeated events (or groups of repeated events) versus the number of solitary events. A larger study of microearthquake repeating patterns will be undertaken in Paper IV.