Data Analysis

We point out that there are several regions in our four year window where robust estimates of the SDM could not be calculated, as this procedure [Egbert 1997] relies upon having at least two observatories operating; on several occasions either one or both of the observatories were not operational, or at least one group of channels (electric or magnetic) at a single site were down. Since there are many disjoint time windows to analyze, and our primary interest is identifying any variation in EM fields which may possibly be associated with the M6 Parkfield 2004 earthquake, we focused first on the days [137-299], 2004. This 163-day window features no swapping of instruments, auxiliary power supplies, or data loggers. Signal-to-noise ratios for all primary array channels are very good, with the exception of Hy at SAO, which was slightly degraded. This slight degradation did not significantly impact our robust SDM estimates. The SDM is estimated in each frequency band from a multivariate time-series of band averaged Fourier coefficients, having dimension NxM, when N is the number of array channels, and M is the number of time windows which were Fourier transformed out of a day's data (typically 898 for Periods 4-30s). The actual robust SDM estimate is calculated as per the iterative method of Egbert 1997, resulting in a collection of 25 covariance matrices, (one matrix for each frequency band) for each of our 163 days. In order to compress and display this data, we reduce each SDM to a list of its four dominant eigenvalues. These are calculated in units of signal to noise ratio. For all data presented in this summary we consider the array to be comprised of the eight primary channels, 100m electrodes and horizontal magnetometers. Figure 2.36 is a plot of these daily eigenvalues.

Figure 2.36: Eigenvalues of the SDM, the y-axis is log(Period (s)), and the x-axis is day-of-year. The color bar represents units (10*log10(SNR)). Note the variation in amplitude in the highest frequency of the third eigenvalue.
\begin{figure}\begin{center}
\epsfig{file=Kappler07_1.eps, width=14cm,height=7cm}\end{center}\end{figure}

Since the dominant eigenvector on one day is not necessarily collinear, or even approximately collinear with the dominant eigenvector calculated on some other day, Figure 2.36 is subject to variation resulting from the fact that it was made by simply ordering the eigenvalues in descending fashion. The lack of guaranteed day-to-day continuity in eigenvalues suggests that a time series plot obtained by plotting daily eigenvalues of the SDM is difficult to interpret physically. This motivates us to look at the prevailing modes of the eigenvector decomposition. A two-week window comprising days 137-151 was sub-selected, and each day's dominant eigenvectors were averaged according to the ranks of their corresponding eigenvalues. In this we obtain four dominant average modes. These modes are simply linear combinations of array channels, which tend to exhibit coherence, on average. By calculating daily the power observed in each mode, we obtain the following time series:

Figure 2.37: Power observed in dominant modes of the SDM. Note the large increase in power around days 205-210, which correspond to a major solar storm. Sundays are marked with thin black vertical lines, and the earthquake with the thick black dashed line
\begin{figure}\begin{center}
\epsfig{file=Kappler07_2.eps, width=7cm,height=14cm}\end{center}\end{figure}

It is apparent from Figure 2.37 that the third eigenvalue shows a weekly period, implying that most of the power in this mode can be attributed to cultural noise. At least some of the unusual noise in the highest frequency band of the third eigenvalue seems to have dispersed into the fourth, and possibly other average modes. This may suggest that signal of this period is not coherent with typical cultural noise. The two dominant modes of the SDM are normally associated with the plane-wave natural micropulsation sources [Egbert 1989]. We suggest that any signals associated with seismic activity along the fault - should they exist - are unlikely to be coherent with the MT source-field. As a method of looking more closely at the recorded time-series, we calculate residual fields by subtracting off the contributions of these natural fields. We select a method of residual calculation that uses sensors at a remote site to predict fields at PKD, where the intersite transfer function itself is calculated as an average over the two week interval (days 137-151). Our residual calculation intentionally uses no Parkfield data, except in the transfer function calculation, which is performed over a time-window distant from the earthquake, ensuring that the residuals will not unintentionally identify the signature of an EM process at work only at PKD by lumping such a signature in with the predicted signal. We use a method based on the average eigenmodes calculated from the SDM, which is essentially a method of similar triangles. Recall that an eigenvector of the SDM is simply a linear combination of array channels.


\begin{displaymath}
EV_{k} = \sum_{i=1}^{N_{PKD}} p_{i,k}P_{i} +\sum_{i=1}^{N_{SAO}} s_{i,k}S_{i}
\end{displaymath} (20.1)

where s$_{i,k}$, p$_{i,k}$ are scalar coefficients for the k$^{th}$ eigenmode, and P$_{i}$, S$_{i}$ are array data for SAO and PKD respectively. The index of summation N$_{site}$ reflects the number of channels operating at the site. Numeric values for the p$_{i,k}$ and s$_{i,k}$ are fixed during the calculation of the k$^{th}$ average eigenmode, and yield the fixed PKD to SAO component ratio:


\begin{displaymath}
PS_{k} = \frac{\sum_{i=1}^{N_{PKD}} {p_{i,k}}^{2}}{\sum_{i=1}^{N_{SAO}} {s_{i,k}}^{2}}
\end{displaymath} (20.2)

Now choosing an eigenmode, i.e. fixing k (we thus omit further reference to k), we can project actual SAO data onto the chosen mode, observing a set of coefficients:


\begin{displaymath}
\sum_{i=1}^{N_{SAO}} {[s_{i}^{obs}]}^{2}
\end{displaymath} (20.3)

We then predict the PKD data assuming the existence of a complementary vector, collinear with $\Sigma$ p$_i$ P$_i$ obeying the ratio:


\begin{displaymath}
\frac{\sum_{i=1}^{N_{PKD}} {p_{i}}^{2}}{\sum_{i=1}^{N_{SAO}}...
...[p_{i}^{pred}]}^{2}}{\sum_{i=1}^{N_{SAO}} {[s_{i}^{obs}]}^{2}}
\end{displaymath} (20.4)

By solving for the predicted PKD signal for the first two eigenmodes, and subtracting from the recorded PKD data, we obtain residual fields.

Figure 2.38: Magnetic Fields Calculated from SAO (a) and Residual Magnetic fields from subtracting predicted fields from observed data (b)
\begin{figure}\begin{center}
\epsfig{file=Kappler07_3.eps}\end{center}\end{figure}

Figure 2.38 illustrates the signal (predicted) and residual fields for the sensor Hy (oriented N-S) at PKD for three months around the earthquake. We choose to show the band at a period of 85s because this period is in the midst of the band where magnetic anomalies were reported by Fraser-Smith et al. We see that no significant variation is evident in either the natural fields or in the local residual fields.

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