Earthquake magnitude scaling using seismogeodetic data
Brendan W. Crowell,
Now at Department of Earth and Space Sciences, University of Washington, Seattle, Washington, USA
Cecil H. and Ida M. Green Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California, La Jolla, California, USA
Corresponding author: B. W. Crowell, Cecil H. and Ida M. Green Department of Earth and Space Sciences, University of Washington, Johnson Hall Rm-070, Box 351310, 4000 15th Avenue NE, Seattle, WA 98195-1310, USA. (firstname.lastname@example.org)
 The combination of GPS and strong-motion data to estimate seismogeodetic waveforms creates a data set that is sensitive to the entire spectrum of ground displacement and the full extent of coseismic slip. In this study we derive earthquake magnitude scaling relationships using seismogeodetic observations of either P wave amplitude or peak ground displacements from five earthquakes in Japan and California ranging in magnitude from 5.3 to 9.0. The addition of the low-frequency component allows rapid distinction of earthquake size for large magnitude events with high precision, unlike accelerometer data that saturate for earthquakes greater than M 7 to 8, and is available well before the coseismic displacements are emplaced. These results, though based on a limited seismogeodetic data set, support earlier studies that propose it may be possible to estimate the final magnitude of an earthquake well before the rupture is complete.
 Magnitude estimates obtained from earthquake early warning (EEW) systems based on P wave arrivals that predict the arrival and intensity of destructive S waves are problematic for large earthquakes [Allen et al., 2009]. There exist many different methodologies for EEW that relate the initial P wave arrivals directly to magnitude, such as the maximum predominant period [Nakamura, 1988], the predominant period (τc) [Wu and Kanamori, 2005], and the displacement amplitude (Pd) [Wu and Zhao, 2006]. In these studies that were carried out prior to the Mw 9.0 2011 Tohoku-oki earthquake, the use of Pd to predict magnitude has been particularly troublesome because the relationship appears to saturate at high magnitudes (M > 7). For the 1999 Mw 7.6 Chi-Chi, 1999 Mw 7.1 Hector Mine, and 2003 Mw 8.3 Tokachi-oki earthquakes, Pd is significantly lower than expected for these magnitudes based on empirical relationships [Brown et al., 2011; Wu and Kanamori, 2005; Wu and Zhao, 2006]. The predominant period does not saturate for large earthquakes but tends to be less precise than Pd magnitude estimates, although a small systematic bias exists for M > 7 events [Kanamori, 2005; Brown et al., 2011]. The Tohoku-oki earthquake and its aftershocks add several important events to the large earthquake database, and initial results indicate that peak amplitude parameters and predominant period have different sensitivity depending on the length of the time window over which the parameters are estimated [Hoshiba and Iwakiri, 2011]. Similar to the previous studies, Hoshiba and Iwakiri  used band-pass filtered seismic waveforms at 3–0.075 Hz in order to remove low-frequency drifts due to sensor rotations and tilts, effectively putting an upper limit on the predominant period measurements. Furthermore, the error incurred in filtering seismic waveforms is complex and can be large in both the time and frequency domains [Melgar et al., 2013a].
 Seismogeodetic waveforms (displacements and velocities) estimated from a combination of GPS and strong-motion accelerometer data are accurate enough to detect P waves for earthquakes of magnitude 4.5–5.5 or greater in the near-source region [Geng et al., 2013]. Furthermore, the waveforms contain the full spectrum of seismic motion in three dimensions from the high-frequency shaking to the permanent coseismic offset. The combination with GPS data allows the accelerometer data to be doubly integrated while retaining the low-frequency component of displacement [Bock et al., 2011]. Here we present two scaling relationships for rapid estimation of earthquake magnitude using Pd and peak ground displacement (PGD) derived from seismogeodetic data recorded during five earthquakes in Japan and southern California. Using these data we are able to differentiate between earthquakes at the upper end of the magnitude scale (>M 7) with no saturation. This is a promising new approach for improved magnitude scaling for large earthquakes compared to current seismological practice.
2 Data and Methods
 We focus on five earthquakes recorded at collocated high-rate (1 Hz) GPS and (100–200 Hz) strong-motion stations including thrust, strike-slip, and normal fault mechanisms: the 2003 Mw 8.3 Tokachi-oki, Japan; the 2010 Mw 7.2 El Mayor-Cucapah, Baja California, Mexico; the 2011 Mw 9.0 Tohoku-oki, Japan; and the largest two events (Mw 5.3 and 5.4) from the August 2012 Brawley seismic swarm, southern California. The Japanese data were recorded by GNSS Earth Observation Network System (GEONET), a 1200+ stations real-time GPS network operated by the Geospatial Information Authority of Japan and by accelerometers from the K-NET and KiK-net networks operated by the National Research Institute for Earth Science and Disaster Prevention. The GPS data for the California earthquakes are from the Plate Boundary Observatory (UNAVCO) and Southern California Integrated GPS Network (U.S. Geological Survey and Scripps Institution of Oceanography) projects. The California seismic data are from the California Integrated Seismic Network and the Wildlife Liquefaction Array operated by the University of California Santa Barbara. We identified pairs of GPS and accelerometer instruments that were separated by 10 m to 2.6 km. The distances are within the collocation limits of Emore et al. , who demonstrated good agreement of 1 Hz GPS and strong-motion data with instrument separations of up to 4 km. The strong-motion sensors are EpiSensor accelerometers and the broadband sensors are STS 2 instruments, both on 24 bit Quanterra data loggers, and 100 Hz K-NET95 strong-motion accelerometers for K-NET. Seismogeodetic waveforms are estimated for each earthquake using the analysis methods reviewed in the supporting information.
 For this study, P waves were manually picked on the vertical seismogeodetic velocity waveforms, using archived seismic picks as an aid, when available. The final number of stations used in the scaling analysis is 57 for the Tohoku-oki, 43 for the Tokachi-oki, 12 for the El Mayor-Cucapah, and 3 for the two Brawley events, a total of 118 observations. We test several definitions of Pd, using the maximum displacement in either a 3 or 5 s window after the P wave arrival and using either the horizontal or vertical components of motion. The time window must be small enough to minimize the early warning shadow zone [Allen and Kanamori, 2003] while being large enough to provide a sufficient amount of source information.
 For comparison, we apply a low-cut, two-pass, four-pole Butterworth filter (13 s period) to the strong-motion data to be consistent with the methods of previous studies [Wu and Zhao, 2006; Wu and Kanamori, 2005] and compute the Pd value using the same P wave picks. Figure 1 shows a comparison of displacements between the GPS solutions, doubly integrated strong-motion data, seismogeodetic data, and high-pass filtered and doubly integrated strong-motion data for stations FKS013/0211 during the Tohoku-oki earthquake (233 km from epicenter), HKD056/0511 during the Tokachi-oki earthquake (219 km), WES/P494 during the El Mayor-Cucapah earthquake (61 km), and WLA/P506 during the largest Brawley Seismic Swarm event (11 km). Notably, the long-period motions after the P wave arrivals, in particular the static components, are significantly underestimated in the high-pass filtered seismic waveforms for the larger earthquakes, highlighting the important contribution of the seismogeodetic data. As expected, for the smaller Brawley swarm earthquakes where the static displacements are at the several millimeter levels [Geng et al., 2013] as can be seen in Figure 1, the seismogeodetic and seismic motions closely match, indicating that the seismogeodetic method is applicable across this wide range of earthquake magnitudes. For more seismogeodetic waveform examples, see Bock et al. , Geng et al. , and Melgar et al. [2013a].
3.1 Seismogeodetic Pd Scaling
 We adopt a linear regression model used for seismic Pd scaling by Wu and Zhao  where seismogeodetic Pd is expressed as a function of magnitude and hypocentral distance, R:
 For this study, Mw is the moment magnitude estimate from the Global centroid moment tensor (CMT) catalog for the Japanese events and the Southern California Earthquake Data Center for the California events. The regression assumes that Pd attenuates linearly on a log-log scale for earthquakes of a similar magnitude. We solve for the three constants, A, B, and C, through least squares. Table S1 in the supporting information shows the values of A, B, and C, as well as the standard errors and the magnitude uncertainty (standard deviation of predicted minus actual magnitudes) for all eight combinations (seismogeodetic or strong motion; 3 or 5 s; horizontal or vertical components). The best combination in terms of lowest standard errors was 5 s scaling with horizontal components using seismogeodetic data. In this case,
 For all cases, 5 s scaling performed better than 3 s scaling (both data types), and horizontal scaling performed better than vertical scaling for seismogeodetic (strong-motion performed better vertically). For the seismogeodetic data, the horizontal scaling provided about a factor of 5 improvement in the magnitude uncertainty over vertical scaling and roughly a factor of 2 improvement in the standard errors. Scaling just with the strong-motion data provided only modest gains using the vertical over the horizontal.
 The results of the 5 s horizontal scaling are shown in Figures 2a and 2b. The data for the three largest earthquakes overlap using the strong-motion data (Figure 2b), a condition known as saturation. Furthermore, the standard deviation of the regressed moment magnitude is 0.383 for the seismogeodetic data and 0.807 for the strong-motion data (Table S1). The Pd values determined from seismogeodetic data (Figure 2a) better delineate the larger events than the strong-motion data (Figure 2b). Comparing Figures 2a and 2b, the seismogeodetic and strong-motion Pd values give comparable magnitude estimates for the El Mayor-Cucapah earthquake and the Brawley swarm events but not for the larger Japanese earthquakes. The results demonstrate that as opposed to strong-motion data alone, with seismogeodetic data there is no saturation; the distribution of Pd values for the larger Mw 9.0 event is significantly delineated with the lesser magnitude events. Some overlap of individual stations still exists between the two Japanese earthquakes closer to the hypocenter; however, the scatter of the measurements about the regression lines is tight as evidenced by the standard errors and uncertainty estimates.
 Figure 3a shows the seismogeodetic 5 s Pd magnitude estimates and the corresponding accelerometer-only estimates as a function of time after the earthquake for the two Japanese earthquakes. The magnitude estimate on Figure 3a is the mean of the 5 s Pd magnitude estimates from all stations reporting up to that time. The number of sites included in the magnitude estimate as a function of time is shown in Figure 3b. The first P wave arrivals are recorded at about 20 s after the origin time for both earthquakes. The magnitude for the Tohoku-oki earthquake starts out as an underestimate due to the anomalously low amplitude at a single station close to the hypocenter (Figure 2a—second closest Tohoku-oki station) although quickly recovers to M 8.75 50 seconds after the earthquake onset. The evolution of magnitude estimates for the Tokachi-oki earthquake starts out above M 9.0 and then quickly converges to M 8.5 by the time 10 stations are included in the magnitude estimate at 41 s. By 60 s, the magnitude estimates of both earthquakes are well within the uncertainties of the scaling relationship. Contrast this with the warnings issued by Japan Meteorological Agency (JMA) during the Tohoku-oki earthquake: M 7.2 at 26 s after the earthquake origin, M 7.7 at 47 s, and M 8.0 at 102 s [Hoshiba et al., 2011]. Magnitude estimates from the accelerometer-only Pd observations and regression laws (Figure 3a) do not converge to the correct solution, although the magnitude estimates are still within the uncertainties of the derived scaling relationship.
3.2 Peak Ground Displacement
 The PGD available from the seismogeodetic data can provide a revised magnitude estimate as more source information is available. We make two modifications for PGD scaling compared to the approach for deriving the Pd scaling law. First, we use all three components of displacement (north, east, and up) and define PGD as the maximum displacement over the time series. We use all components because the PGD measurement is not affected by vertical scatter as much as Pd and important source information lies in all three components of motion. Second, we modify the attenuation term to include dependence on magnitude. We find the following scaling law:
 The standard errors are 0.211, 0.046, and 0.010 for A, B, and C, respectively, with an earthquake magnitude standard deviation of 0.224 (Table S1), a marked improvement from Pd scaling. We also performed the regression using the GPS-only solution, and we find a 10% improvement in the standard errors and magnitude uncertainty using the seismogeodetic solution (Table S1). The majority of the improvement is for the lower magnitude earthquakes; the two Japanese earthquakes show only marginal differences between seismogeodetic PGD and GPS-only PGD since the signal-to-noise ratio is very large for most stations. Figure 2c shows the seismogeodetic PGD as a function of hypocentral distance. The scatter in measurements is much less than in Pd scaling, providing a more robust, albeit delayed, magnitude estimate. The magnitude estimation delay is dependent upon the moment release characteristics of the earthquake, the material properties, and the geometry of stations around the earthquake.
 Figure 3c shows the hypocentral distance as a function of the time to achieve PGD. We used PGD values up 120 s for the Tokachi event and 240 s for the Tokoku event, which is about twice the rupture time for each event and corresponds to the arrival of surface waves at the furthest stations. The most striking feature of Figure 3c is that the slopes for the Tohoku-oki, Tokachi-oki, and El Mayor-Cucapah earthquakes are all the same. We obtain rates of 3.65 ± 0.06, 3.62 ± 0.17, and 3.62 ± 0.34 km/s for the Tohoku-oki, Tokachi-oki, and El Mayor-Cucapah earthquakes, respectively. Interestingly, the x intercept time roughly corresponds to the timing of peaks in the moment release rates from previous studies. We find x intercepts of 62 ± 3 s, 17 ± 4 s, and 32 ± 13 s for the Tohoku-oki, Tokachi-oki, and El Mayor-Cucapah earthquakes, respectively, in line with peak moment release estimates of 70 s [Suzuki et al., 2011], 20–30 s [Yagi, 2004], and 27 s [Wei et al., 2011] from dynamic rupture models. For the Brawley seismic swarm events, the slope of the PGD time curve is not well defined because of the small PGD values.
 Seismogeodetic data support a warning and modeling system that can build up complexity as more time passes. Pd scaling from seismogeodesy can be combined with existing seismic early warning methodologies to create a more robust estimate of initial magnitude for large earthquakes and can be updated over time using PGD scaling for a particular event. Immediately after the full extent of permanent coseismic deformation has occurred, higher-order modeling such as finite-fault CMT determination and slip inversions [Melgar et al., 2013b] and rapid tsunami models [Melgar and Bock, 2013] can be performed using local and regional data in advance of teleseismic data availability.
 The timing of early warnings from Pd is solely dependent on the network configuration near the earthquake epicenter and the velocity structure in the region. Warning of impending S wave ground motion is possible with knowledge of the distance to the hypocenter and Pd. The distance determination would require at least four stations to determine a hypocenter or a single station distance measurement based on seismic data [Nakamura, 1988]. In the three largest events, a four-station Pd warning would be available in 32 s, 34 s, and 33 s for the Tohoku-oki, Tokachi-oki, and El Mayor-Cucapah earthquakes, respectively, and an update within 55 s would bring the magnitude in line with the final estimate. With closer stations to the epicenter, a warning can be issued earlier. The bulk of the warning time for those earthquakes is the seismic P wave travel time since the four closest stations are ~100 km from the sources.
 For a revised early estimate based on PGD, the maximum moment release time is relevant. PGD could provide a valuable early update of event size and provide earlier access to point source ShakeMap products [e.g., Wald et al., 1999] for emergency response that is more closely related to the total moment release of the event. As in the previous example, the same four stations in each earthquake reach PGD in 104 s, 62 s, and 68 s, respectively. From Figure 3, we see that a good PGD estimate is available slightly after Pd magnitude estimates are fully stable for the Tokachi-oki earthquake and about 50 s later for the Tohoku-oki earthquake. Even with the poor spatial coverage around these earthquakes (offshore for the Japanese earthquakes, GPS coverage only on the north side of the fault for El Mayor-Cucapah), PGD can still be an effective warning parameter for further away cities, such as Tokyo and Los Angeles. Coseismic displacements were found to be stable after 157 s for the Tohoku-oki, 116 s for the Tokachi-oki, and 113 s for the El Mayor-Cucapah earthquakes, so PGD offers a factor of 2 improvement in warning time and Pd a factor of 4 for a four-station warning [Melgar et al., 2012, 2013b; Crowell et al., 2012]. For quick magnitude determination from PGD there is however one drawback over Pd. Since Pd uses a set time window, determining the stations to use for magnitude estimation at a given time is as simple as including only the stations that have a P wave arrival. Since PGD has a dependence on the peak moment release time, all stations available at a given time would have to be included in magnitude estimation, resulting in a slower convergence. PGD convergence could be sped up by using an S wave mask to exclude stations that have not yet undergone strong shaking. Potentially of more importance in the correlation of PGD and moment release is the suggestion that PGD is a reliable early indicator of termination of rupture and thus provides early warning that the earthquake shaking estimate is unlikely to be exceeded.
 One concern of the scaling relationships derived for Pd and PGD is the robustness of the parameters A, B, and C. Since the majority of data used for the regressions are from the larger Japanese earthquakes, the actual coefficients used in EEW may be different. To test the parameter robustness, we performed a bootstrap analysis with a random sample of the Japanese earthquake values and all the California earthquake values to give equal weight to all magnitude earthquakes in this study (details in the supporting information—Figures S1 and S2). We find that our best fit for both Pd and PGD falls within the bootstrap distributions for coefficients A, B, and C. We conclude that the errors in these coefficients and magnitude estimates are realistic. As the number of events for which seismogeodetic data become available increases, the accuracy of the scaling relationships should improve.
 While the time period between Pd and PGD is not investigated in this paper, Hoshiba and Iwakiri  showed the utility of larger-time windows in estimating τc for the Tohoku-oki earthquake. In our case, PGD can be viewed as a very large time window after the P wave arrival and PGD, in effect, sets the lower limit on uncertainty for linear displacement scaling.
 The high sensitivity of seismogeodetic Pd to magnitude determined over a 5 s interval following the P wave arrival time indicates that the eventual size of the earthquake rupture may be deterministic [Olson and Allen, 2005], and there may be an earthquake nucleation phase [Ellsworth and Beroza, 1995]; however, the investigation of more earthquakes is required. This model is in opposition to the cascade model of earthquake rupture where slip starts on a small patch for both small and large earthquakes and then grows if local conditions on the fault are favorable. Olson and Allen  showed using , a scaling law for earthquakes between M 3 and 8 could be achieved with seismic data with uncertainties of ~ 1 magnitude unit. Rydelek and Horiuchi  claimed that no such scaling could be obtained for earthquakes greater than M 6 in Japan. Zollo et al.  proposed that the probability of future rupture is proportional to the initial stress drop (i.e., large earthquakes have a larger initial energy release) as a mechanism for earthquake determinism. The lack of saturation of Pd scaling indicates that seismogeodetic data provide important source information that cannot be obtained with seismic data alone, nor just with geodetic displacements that are too noisy to detect Pd. On the other hand Crowell et al.  showed that geodetic measurements alone can provide a robust PGD result, and we also show statistically good agreement between GPS-only PGD and seismogeodetic PGD. Also of concern with regards to Pd scaling of large earthquakes is the linearity of scaling relationships as magnitude increases. The complicated rupture mechanisms illuminated through back projection methods for the Tohoku-oki [Simons et al., 2011] and the 2012 Mw 8.6 Sumatra earthquakes [Meng et al., 2012] hint at possible nonlinearity for scaling relations. However, too few large earthquakes have been recorded recently to form a generalized relationship.
 Seismogeodesy provides a new and unique view of the rupture process over the full spectrum of displacements of interest to seismologists and engineers and valuable information that can augment existing EEW systems for earthquakes large enough to be of societal relevance. Using the horizontal components of seismogeodetic displacement to determine Pd, we find scaling relationships between Pd and magnitude that are similar to existing methodologies but do not saturate at high magnitudes and have smaller uncertainties (~ 0.4 magnitude units). Seismogeodetic PGD has an even smaller uncertainty for magnitude determination and, although available at a later time, can provide rapid updates for robust EEW systems. The significance of this study lies in demonstrating that the lower frequency information provided by seismogeodetic data better separates the data distributions for large events to overcome the saturation present with seismic data. As additional data sets become available the validity and thus the usefulness and accuracy of scaling relationships will be tested to determine whether or not earthquake source magnitude is deterministic and under what conditions.
 We would like to thank the National Research Institute for Earth Science and Disaster Prevention for access to K-NET and KiK-net data and the Geospatial Information Authority of Japan for GEONET GPS data. California GPS data is provided by the Plate Boundary Observatory operated by UNAVCO for EarthScope (www.earthscope.org) and supported by the National Science Foundation (No. EAR-0350028 and EAR-0732947; seismic data are from the California Integrated Seismic Network and the Wildlife Liquefaction Array operated by the University of California Santa Barbara. The comments of two anonymous reviewers have been extremely helpful in improving the manuscript. This paper was funded by NASA AIST-11 grant NNX09AI67G, NASA ROSES grant NNX12AK24G, NASA Earth and Space Science Fellowship NNX12AN55H, and Southern California Earthquake Center award 12083.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.