This electronic supplement contains the following items: (1) our measurements of the log10(P/S) spectral ratios obtained from waveforms recorded at station MDJ for the two training sets (Table S1) and figures of log(P/S) values measured at 8 Hz from vertical-component waveforms at station MDJ for two training sets and probability distributions for D; (2) a section describing further details of our three-component linear discriminant function (LDF) analysis, in particular the effectiveness of different choices of the frequency components used to apply P/S spectral ratios; and (3) a tutorial section on the underlying ideas behind Mahalanobis methods for event classification, with particular reference to Figures 11, 14, and 16 in the main article.
The discrimination capability of different sets of log(P/S) measurements on MDJ seismograms, made for each set at four discrete frequencies between 1 and 15 Hz and used as inputs for LDF analysis, is listed in Table S2. When frequencies used are increased, from 1–4 Hz, 2–5 Hz, 3–6 Hz, and so on, there is an increase in the Mahalanobis distance-squared measure of separation between the means of earthquake and explosion populations, and the misclassification probability decreases. The optimum frequency set is seen to be 6–9 Hz (i.e., 6, 7, 8, and 9 Hz). At frequencies 7–10 Hz, 8–11 Hz, and 9–12 Hz, the MDJ data derived from training sets consisting of 12 earthquakes and 12 explosions still show significant discrimination capability. As the band of frequencies increases further, the Mahalanobis distance decreases, and the misclassification probability increases.
We find that frequency bands in the 6–12 Hz range (shown shaded in Table S2) provide the best discrimination power, and the 6–9 Hz band is used in the main article (i.e., the four frequencies 6, 7, 8, and 9 Hz).
Using the LDF from these four frequencies, as well as the frequency bands with strong discrimination power in the 6–12 Hz range, classifies the 12 May 2010 event as earthquake, whereas high-frequency bands 10–15 Hz classify it as an explosion. This can be expected from the P/S ratios of the 12 May 2010 event at higher frequencies, as shown in Figure 12 of the main article. However, at these higher frequencies, the LDF for the region derived from MDJ data is demonstrably not so effective. Though not as good as 6–12 Hz, the 4–8 Hz band also classifies the 12 May 2010 event as an earthquake.
Use of three discrete frequencies between 7 and 11 Hz also provides strong discrimination power. A choice of 9, 10, and 11 Hz provides discrimination power with Δ2 = 20.0 and misclassification probability = 1.27%. Adjacent choices providing slightly better results (i.e., larger Δ2) are either 7, 8, and 9 Hz (giving Δ2 = 23.8 and misclassification probability = 0.74%) or 8, 9, and 10 Hz (giving Δ2 = 23.7 and misclassification probability = 0.74%). Using three lower frequencies, namely 5, 6, and 7 Hz, leads to discrimination that is still quite good (Δ2 = 9.9 and misclassification probability = 5.77%).
Here, we first describe basic features of the log(P/S) measurement, as reported in Figure 9 of the main article for one particular frequency. The measured values for earthquakes and explosions cluster around separate mean values, which are not very far apart when measured in dimensionless terms. We then describe features of an LDF based on measurements of log(P/S) at multiple frequencies. The discriminant function values for our two training sets cluster around separate means that are much farther apart in Figures 11 and 14 of the main article, providing a more useful framework for purposes of event classification.
Thus, starting with Figure 9 of the main article, we can take the measurements made at 8 Hz for the two training sets and show them here as Figure S1. Essentially, the relevant points in Figure 9 of the main article along the 8 Hz line have been extracted, including the circle and triangle symbols. The line has been rotated clockwise 90°, and the points are now plotted together with Gaussian probability density functions (PDFs) having mean values and a standard deviation derived from the two training sets.
The utility of log(P/S) measured at 8 Hz for an event of interest for purposes of classifying the event (earthquake or explosion?) is governed largely by the degree of separation of the two distributions—that is, by the degree of overlap of the two Gaussians. In situations where the standard deviation (σ) is effectively the same for both classes (we can take σ = 0.26 as the average of the two values in Fig. S1), the overlap is quantified by the dimensionless ratio Δ given by the difference between the two means divided by the value of σ. In Figure S1, Δ = (0.52 + 0.09)/0.26 = 2.35. These two mean values are separated by not much more than two standard deviations, and there is considerable overlap.
The LDF D(r) defined in the main article reduces to a product of two scalars,
in the present 1D case, in which the scalar r is a value of log(P/S) at a single frequency. We can gain an appreciation of the effectiveness of D(r) as a discriminant by noting from the above formula that D(µEx) = (1/2) Δ2 and D(µEq) = −(1/2) Δ2, so that D(µEx)−D(µEq)=Δ2. This is the Mahalanobis distance-squared measure giving the distance between the mean of explosion values and the mean of earthquake values. The dimensionless distance Δ is measured via a unit that is determined by the standard deviation given by the data sets. In the present case, as shown in Figure S1, we have Δ2 = 5.5, which is too small to provide effective discrimination.
With this understanding of the contribution to discrimination capability obtained from measurements made at a single frequency, it is a straightforward extension to work with the definition of a multidimensional (i.e., multifrequency) LDF, D = D(r) as given in the main article. Now r is a vector of log(P/S) values for each event, obtained at different frequencies. We have vector means µEq and µEx derived from the two training sets, and instead of a single scalar standard deviation (σ), we have a covariance matrix S (sometimes called a dispersion matrix), effectively the same for the two training sets.
It is still true that the distance-squared measure of the separation of the two means µEq and µEx is given by D(µEx) − D(µEq) = Δ2, in which this scalar value is now given by
Δ2 = [S−1(µEx − µEq)]T(µEx − µEq).
Parts of the main article concern a discussion of how best to choose a set of frequencies, which increases the value of Δ2, that is, increases the separation between the two means. We have found good results (i.e., good separation between populations) when using the four frequencies 6, 7, 8, and 9 Hz. For vertical-component data (Fig. 11 of the main article) and three-component data (Fig. 14 of the main article), the underlying Mahalanobis distance-squared measures are Δ2 = 20.6 and 25.6, respectively. Figure S2 illustrates the degree of separation of the earthquake and explosion populations for vertical-component data, and Figure S3 shows the separation for three-component data.
The mean values of these Gaussian distributions are ±(1/2)Δ2, and the standard deviation is Δ for both of them. Specifically, their probability densities are
The probability of misclassifying an earthquake as an explosion is the area under the Gaussian associated with earthquakes, taken over D values associated with explosions, that is
in which erfc is the complementary error function.
Throughout our discussion and in these equations, we see various roles played by the Mahalanobis distance Δ. It is the underlying standard deviation of the LDF. When squared it gives the separation between earthquake and explosion means, and it enters directly, as a single parameter, into an expression for the misclassification rate.
Figures S2 and S3 look similar, but there is a difference in the misclassification rates, which are 1.15% and 0.57%, respectively. The misclassification rate based on vertical-component data is about twice that based on three-component data.
Our discussion has been simplified by assuming (1) that the covariance matrices for the two training sets are effectively the same, and (2) that an event of interest is, a priori, equally likely to be an earthquake or an explosion. Assumption (1) appears to be satisfied by the observed scatter in data obtained for the two training sets. Concerning (2), our assumption boils down to using D[(µEx+µEq)/2] as the deciding value, and this value is zero. If an event of interest needs to be classified on a basis of assuming unequal a priori probabilities, those probabilities can be translated into a different deciding value for D.
Table S1 [Plain Text Comma-Separated Values; ~7 KB]. Measurements of the log10(P/S) spectral ratios obtained from waveforms recorded at station MDJ for the two training sets.
Table S2. Three-component discrimination analysis using four frequencies. The fourth column of this table examines the F-distribution of the data for the null hypothesis that the mean of the two populations are significantly different. The F-distribution for each frequency band is listed and compared with the corresponding F-distribution table value for F(x) = 95% with (4, 19) degrees of freedom, which is 2.90. The F-distribution values in all 12 cases exceed the table value, indicating that the null hypothesis is true, that is, the means are not equal. The fifth column examines the χ2 approximation obtained in each frequency band for the null hypothesis that the dispersion matrices of two populations are the same. The χ2 distribution table value for F(x) = 95% with 10 degrees of freedom is 18.31, and the χ2 approximation value less than half the table values indicates that the null hypothesis is true, that is, dispersion matrices are the same. For all other bands in this table, all 12 earthquakes and all 12 explosions in the two training sets were classified correctly.
Figure S1. The log(P/S) values measured at 8 Hz from vertical-component waveforms at station MDJ for two training sets are shown as circles (earthquakes) and triangles (explosions), together with the normal (Gaussian) PDFs inferred from these two data sets. Note two length scales; the Gaussian widths, and the distance between the means (explosions and earthquakes).
Figure S2. Probability distributions for the LDF that best-separates the earthquake and explosion populations, using MDJ data. Shown here are the underlying Gaussians derived from vertical components recorded for our two training sets (earthquakes and explosions), as developed in Figure 11 of the main article.
Figure S3. Similar to Figure S2, but now the underlying Gaussians are for the three-component data as developed in Figure 14 of the main article. The standard deviation, here Δ, is slightly larger than in Figure S2, but the distance between the means, which equals Δ2, is significantly greater in this figure than in Figure S2. Hence, better classification capability is obtained with the three-component data, even though the Gaussians are wider.
[ Back ]
