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[1] We validate that changes of ground deformation recorded by GPS contain useful information for earthquake forecasting. A moving rate of variation filter is used to extract short-term signals from GPS time series in New Zealand, California, and Japan. The precursory information of these signals for large earthquakes is evaluated using Molchan's error diagram. The results suggest that the GPS signals provide a probability gain of 2–4 for forecasting large earthquakes against a Poisson model. Further tests show that the GPS signals are not triggered by large earthquakes, and that the probability gain is not derived from forecasting aftershocks. This demonstrates that noncatalog information, such as GPS data, can be used to augment probabilistic models based on seismic catalog data to improve forecasting of large earthquakes.

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[2] Recent extensive studies on catalog-based probabilistic forecasting of earthquakes, such as the Regional Earthquake Likelihood Model (RELM) [Field, 2007] and the Collaboratory for the Study of Earthquake Predictability [Jordan, 2006], indicate that only a certain amount of precursory information (probability gain) is available in earthquake catalogs [Zechar and Jordan, 2008; Shebalin et al., 2012; Zechar and Nadeau, 2012]. Whether noncatalog measurements contain precursory information for large earthquakes has not yet been established.

[3] One candidate for these noncatalog observations is stationary GPS measurements of ground deformation. Links between these measurements and the occurrences of large earthquakes have been observed [Granat and Donnellan, 2002; Granat, 2006; Ogata, 2007]. Wang and Bebbington [2013] developed a nonlinear filter to extract anomalous changes in deformation, which may be related to regional strain changes and presage earthquake occurrences. The class of GPS measurements with the greatest variation in deformation rate showed some precursory information for large earthquakes.

[4] However, systematic studies are still needed to test whether the filtered GPS signals can be used to improve earthquake forecasts. Here we explore GPS data from three regions with different tectonic settings, adopting the nonlinear filter developed by Wang and Bebbington [2013] to extract subtle changes in GPS data, and examine its predictive power using Molchan's error diagram [Molchan, 1991; Molchan and Kagan, 1992].

2 Data

[5] Earthquake (Tables S1–S3 in the supporting information) and GPS data are from three regions:

[6] Central North Island, New Zealand. Located at the boundary of the Australian Plate and containing the Taupo Volcanic Zone (TVZ), the tectonic environment is dominated by subduction-related rifting. The selected earthquakes (10 in total, from http://geonet.org.nz/) are those with magnitudes ≥ 5.1, depths < 100 km, between 2 March 2004 and 31 December 2008 in the rectangular area 37°S–40°S and 174°E–177°E.

[7] Southern California. The seismicity is dominated by the southern part of the San Andreas Fault, which mainly accommodates transform/strike-slip motion between the North American Plate and the Pacific Plate. The 20 earthquakes are obtained from the Advanced National Seismic System, with magnitudes ≥ 4.7, depths ≤ 30 km, latitude 33°N–37°N and longitude 116°W–120°W from 1 January 1999 to 30 June 2009.

[8] Kanto region, Japan. An active and complex tectonic setting, with the Pacific Plate subducting beneath the junction region of the Eurasian, North American, and Philippine Sea plates [Kato et al., 1998]. Earthquakes (16 in total) are selected from the Japan Meteorological Agency catalog, with magnitudes ≥ 5.5, depths ≤ 50 km, latitudes 35°N–37°N, and longitudes 139°E–142°E from 1 January 1997 to 31 December 2010.

[9] The geographical and temporal limits were chosen for data quality and sufficiency, in particular to have both complete GPS records and as many earthquakes as possible. Minimum magnitudes are taken as breakpoints in the frequency-magnitude plot; see Wang and Bebbington [2013] for a sensitivity analysis. As GPS measurements are daily, only the largest magnitude earthquake on a given day is considered, the rest being removed.

[10] Figure 1 shows the continuous GPS stations used. New Zealand and Southern Californian GPS data are from http://geonet.org.nz/ and http://sopac.ucsd.edu/, respectively. Japanese data are from the Geospatial Information Authority of Japan. Sinusoidal annual and semiannual variations are removed from all GPS time series [Sagiya et al., 2000].

3 Methods

[11] The prediction algorithm is based on an alarm function. It divides the entire time period into n + 1 “cells” of equal length ℓ (days) and predicts that an earthquake with minimum magnitude M_{0} (that of the catalog used for a given region) will occur in cell i + 1 whenever the alarm function exceeds some predefined threshold value u in cell i.

[12] The alarm function is specified in two steps: the moving rate of variation (MRV) function and the prediction strategy. First, baselines between two GPS stations are used to cancel reference frame errors, leaving three time series, namely, N(t), E(t), and U(t) for the north-south, east-west, and vertical components of the GPS measurements, respectively. For each series (taking N(t), for example) at each time point t, we obtain the slope, T_{N}(t), of the linear regression line fitted to the data {N(s) : t − 9 ≤ s ≤ t}. Then, for each t and each baseline, we calculate the range of deformation rates, R_{N}(t) = max{T_{N}(s) : t − 9 ≤ s ≤ t} − min{T_{N}(s) : t − 9 ≤ s ≤ t}, observed in the previous 10 days [Wang and Bebbington, 2013]. For simplicity, we call R_{N}(t) the “moving rate of variation (MRV).” The MRVs R_{E}(t) and R_{U}(t) for east-west and vertical components are calculated in the same way. Secondly, on each cell i, we define two GPS signal functions:

Vi1=maxRNt2+REt2+RUt/42,i−1ℓ+1≤t≤iℓ(1)

Vi2=maxRNt2+REt2,i−1ℓ+1≤t≤iℓ.(2)

The weight on the vertical component of 1/4 in equation (1) and 0 in equation (2) recognizes that the GPS measurement errors are much larger in the vertical than in the horizontal components. The corresponding alarm functions are defined by Ai+11=Vi1 and Ai+12=Vi2.

[13] To evaluate the performance of the prediction algorithm, an earthquake is counted as predicted (missed) if it falls (does not fall) in an alarmed interval. Let P(E|A) be the probability of an earthquake occurring in a cell given that the alarm function exceeded the threshold. From Bayes theorem P(E|A) = P(A|E)P(E)/P(A), where P(A|E) is the proportion of successful alarms, P(A) is the fraction of alarmed cells, and P(E) is the unconditional probability of an earthquake occurring in a random cell. Molchan's error diagram plots (Figure 2) the fraction of earthquakes missed ν(u) = 1 − P(A|E) against the fraction of alarmed cells τ(u) = P(A). In application, the alarm threshold u would be selected via retrospective analysis. The probability gain P(A|E)/P(A) [Zechar and Jordan, 2008] indicates how many times more information than a random guess (a Poisson model) is produced by the prediction algorithm [Molchan, 1991; Zechar and Jordan, 2008].

4 Data Analysis and Results

[14] For the New Zealand data, ℓ=10 days, while for Southern California, ℓ=20 days, which were optimized for each case [Wang and Bebbington, 2013]. Using these results as our “training set” to fix “parameters,” such as the signals defined in equations (1) and (2), and the cell size ℓ, we will carry out an independent test of the method for predicting earthquakes from GPS signals in the Kanto region (the “testing set”). Since the Kanto earthquakes are subduction-related, we use ℓ=10 days as for New Zealand. As the New Zealand region includes the TVZ, instead of using the baselines between two stations, we averaged the GPS measurements from three stations and calculated the movement of this average relative to the station TAUP [Wang and Bebbington, 2013]. For all three regions, the Molchan error diagram lies below the diagonal line (Figures 3a, 3b, and 3c), showing a positive correlation between the GPS signal-based alarms and earthquake occurrences.

[15] In the New Zealand and Southern California cases, single GPS baselines were chosen on the basis of GPS data quality (record length, completeness, and homoscedasticity) and the positions of stations relative to the tectonic environment. The GPS records for the Kanto region are of higher quality, and most large earthquakes in this area occurred offshore to the east, so we were able to carry out a more systematic analysis: We analyzed all the baselines between the 66 stations in the selected area with GPS records extending at least back to 1997 in order to validate the methodology and see what influence the choice of baseline has on the results. In practice, retrospective analysis could be used to select the best baselines. The resulting error diagrams suggest that while many baselines show positive correlation with earthquakes, some show stronger predictive power than others (Figures S1–S4 in the supporting information). The directions of the baselines which contain higher precursory information tend to be oriented toward the earthquake source, which may be linked to the physical mechanism of earthquake generation. Moreover, stations farther to the east, and hence closer to the earthquakes, with relatively shorter baselines, are more likely to exhibit higher probability gains. Approximately 130 (6.2%, statistically significant at α < 0.01) of the baselines show a probability gain over 2, with a false alarm rate lower than 50% (Figure S5 in the supporting information).

[16] The range of probability gains in this study achieved with ≤50% false alarm rates is 2–4. The common ranges of probability gains for forecasting earthquakes using different methods and data are 1.8–4 [Helmstetter and Sornette, 2003; Rhoades et al., 2010; Shcherbakov et al., 2010; Zechar and Nadeau, 2012; Zhang and Zhuang, 2011], 2–10 [Newman and Turcotte, 2002; Shebalin et al., 2011, 2012], and ≥ 10 [Zechar and Jordan, 2008]. Thus, the probability gains from GPS measurements lie in the common and lower range of such gains. However, these probability gains are not based on earthquake catalog data and so should ideally be additive with them.

5 Discussion

[17] To compare with a “null” case, we applied the MRV algorithm to a relatively quiescent region, Alice Springs in central Australia. Figure 4 shows the New Zealand GPS measurements with the Alice Springs data overlaid. The MRV of each component is shown alongside. The much smoother Alice Springs GPS measurements mean the MRVs are a magnitude smaller than those for New Zealand.

[18] To show that the GPS signals are not a postseismic effect generated by large earthquakes (or main shocks) and the predicted events are not thus large aftershocks of these major events [Kagan, 1996], we will perform two additional tests. Swapping the roles of earthquakes and GPS signals in the error diagram allows us to test if the GPS signals are triggered by earthquakes. We use the signal functions defined in equations (1) and (2) in section 3 and, for simplicity, consider only Vi1. For a threshold v_{c}, if Vi1>vc in cell i (1 ≤ i ≤ I), then cell i is considered to have a GPS signal. We consider, respectively, 50%, 25%, 15%, and 5% of the cells which have the largest V^{(1)}'s as signals, i.e., we choose the 50, 75, 85, and 95 percentiles of V^{(1)} as thresholds to determine the occurrences of GPS signals. The alarm function for each cell i is

Fi=∑j:tj<si−ℓ/2100.75Mj/si−tj,(3)

where s_{i} is the center of the temporal cell i, and ℓ is the length of each cell, taking a value of 10 days. Here 100.75Mj represents the stress drop during an earthquake corresponding to Benioff strain [Benioff, 1951]. The temporal decay term is that of the Omori formula [Omori, 1894]. We alarm any cell which has a value of F_{i} greater than a threshold AL_{c}. Varying AL_{c} generates the required error diagram, as shown for different quantiles v_{c} in Figures 3d–3f. Similar results are obtained using Vi2 in place of Vi1. The curves close to the diagonal line imply that the GPS signals extracted in section 4 are not predictable from earthquakes.

[19] Our aim in constructing the MRV was to extract precursory signals; hence, it is not optimized for postseismic effects. An algorithm could be designed specifically to detect postseismic effects, using a window of data after each day. The fact that no postseismic information was picked up by the MRV could also be because the stations used did not see a substantial afterslip signal given their locations relative to the earthquake sources.

[20] To see how well earthquakes forecast earthquakes in the data used, we used the alarm function in equation (3) against subsequent earthquakes. Increasing the alarm threshold from 0 to max{F_{i}}, we obtain the error diagrams in Figures 3g–3i, with curves close to the diagonal, indicating no predictive information. Recall that the pruning of earthquakes in a given day to that with the largest magnitude effectively declusters the catalogs.

[21] We have demonstrated that the apparent predictive ability of the GPS data is not the result of a main shock causing anomalous postseismic deformation that precedes large aftershocks. We have assumed that the mechanism causing aftershocks, while having the same or similar driver, operates on a more localized basis. Hence, we do not expect it to be visible in the great majority of GPS baselines. The exceptions would be those with a station very close to the earthquake epicenter. In general, using GPS data to forecast aftershocks will have less power than using main shocks to forecast aftershocks. The latter is well covered by the RELM-type experiments. This research aims to verify that the noncatalog data, in particular GPS measurements, can be used to improve forecasting of large earthquakes, rather than substituting for catalog-based probabilistic forecasting.

[22] The results above indicate that the azimuth and location of the baseline(s) affect the probability gain and may yield a means to localize the possible location, and perhaps even the mechanism, of a predicted earthquake. The spatial coherence of the baseline signals will be the primary diagnostic, and it might be possible to incorporate this in robust statistical methods to improve the hit rate and reduce false alarms.

[23] While we have shown that signals extracted from GPS measurements can improve the forecasting of large earthquakes, using such signals in practical probabilistic earthquake forecasts require a probability model incorporating these signals and the observations of the earthquake process itself. Such a model can be specified by a conditional intensity similar to a self-excited and mutually excited Hawkes process [Zhuang et al., 2005]. Multiple GPS baselines and even other geophysical observations could be included in such models, hopefully all serving to improve the probability gain of earthquake forecasts.

6 Conclusions

[24] We have validated, in three contrasting situations, that GPS deformation contains precursory information for large earthquakes. Using Molchan's error diagram, GPS signals extracted using the MRV technique show a probability gain of 2–4 against a Poisson model. The GPS signals are not a postseismic effect, and their performance is not derived from temporal clustering in the earthquakes. Hence, it is worthwhile considering incorporation of GPS data into the usual catalog-based earthquake forecasting models.

Acknowledgments

[25] T. Wang was supported by a Visiting Research Fellowship in the ERI, University of Tokyo. J. Zhuang is supported by grant-in-aid 22700299 for Young Scientists (B) from the Japan Society for the Promotion of Science. Helpful discussions with Y. Ogata, M. Matsu'ura, and M. Brenna are gratefully acknowledged.

[26] The Editor thanks Brad Lipovsky and an anonymous reviewer for their assistance in evaluating this paper.