- Open Access
An adjoint data assimilation method for optimizing frictional parameters on the afterslip area
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013
Received: 13 March 2013
Accepted: 7 August 2013
Published: 6 December 2013
Afterslip sometimes triggers subsequent earthquakes within a timescale of days to several years. Thus, it may be possible to predict the occurrence of such a triggered earthquake by simulating the spatio-temporal evolution of afterslip with estimated frictional parameters. To demonstrate the feasibility of this idea, we consider a plate interface model where afterslip propagates between two asperities following a rate-and-state friction law, and we adopt an adjoint data assimilation method to optimize frictional parameters. Synthetic observation data are sampled as the slip velocities on the plate interface during 20 days. It is found that: (1) all frictional parameters are optimized if the data sets consists not only of the early phase of afterslip or acceleration, but also of the decaying phase or deceleration; and (2) the prediction of the timing of the triggered earthquake is improved by using adjusted frictional parameters.
The prediction of the occurrence time of earthquakes, based on physics-based models, is a highly challenging goal of earthquake science (e.g., Kato, 2008). Difficulties are rooted in the following facts: (1) the governing equations may not be sufficiently accurate to forecast the temporal evolution of the stress and the slip velocity; (2) it generally takes several hundred years to obtain observations over multiple cycles; and (3) differential equations are so stiff, and therefore the period when we might observe any precursory signals would be quite limited, so that, to date, unambiguous precursory signals have never been observed.
Nevertheless, this does not mean that all earthquakes are entirely unpredictable. An earthquake leads to a stress increase around the rupture region, which may either trigger another earthquake or induce afterslips. Continuous GPS measurements have revealed that afterslip lasts for several years following giant (M 8–9) earthquakes (e.g., Heki et al., 1997). Such afterslip would load another asperity (a strongly-coupled area on the fault), if it exists, and consequently another earthquake would be induced there, depending on the stress state on the asperity, the amplitude of perturbation and the relative distance between asperities. For example, Uchida et al. (2009) suggested that afterslip east of the 2003 Tokachi-oki earthquake rupture region had propagated eastward and triggered the 2004 Kushiro-oki earthquake. Thus, if we trace the spatio-temporal evolution of afterslip, it may be possible to predict the occurrence of such a triggered earthquake.
In order to predict the occurrence time of a subsequent earthquake, it is essential to know the reasonable frictional property on the fault and to develop accurate physical models. Frictional properties in afterslip areas have been estimated based on kinematic inversions of GPS data (Miyazaki et al., 2004; Hsu et al., 2006) or dynamic model-based simulations, which provide constraints on the evolution of slip and the stress state (e.g., Marone et al., 1991; Hearn et al., 2002; Montési, 2004; Perfettini and Avouac, 2004, 2007; Perfettini et al., 2005; Johnson et al., 2006; Fukuda et al., 2009; Mitsui et al., 2010). It should be noted that all of these studies have employed a rate- and state-dependent friction law (e.g. Dieterich, 1979; Ruina, 1983). Although kinematic inversion is a simple technique, it is limited to estimating only (a − b)σeff, where a and b are constitutive frictional parameters (Eq. (1)) and σeff is the effective normal stress. Fukuda et al. (2009) developed a Markov chain Monte Carlo (MCMC)-based method (e.g., Metropolis et al., 1953). Their study represented the Kurile Trench with a one-degree-of-freedom spring-slider system, and estimated (a−b)σeff, aσeff and L (L is the characteristic length (Eqs. (1)–(2)). Mitsui et al. (2010) applied a Sequential Importance Sampling (SIS) method (e.g., Liu et al., 2000) to a two-degrees-of-freedom cell model and demonstrated with synthetic afterslip data that the method worked for estimating a−b and L. These previous studies are notable because the probability density functions (PDFs) are successfully reproduced given a numerical model with 106 ensemble members (Fukuda et al., 2009). However, if we recall that earthquake simulations with a continuum medium require significant computational resources, (e.g., a single calculation by Ohtani et al. (submitted to Geophysical Research Letters, 2013) required approximately 107s of CPU time), the realization of 106 ensemble members is not practical. Thus, a computationally-efficient method is required for estimating frictional parameters which enables a realistic model.
Kano et al. (2010) utilized an efficient dynamic modelbased method, “An Adjoint Data Assimilation Method” (e.g., Lewis and Derber, 1985). Adjoint methods have been extensively applied to realistic models with 106–1010 variables in the fields of meteorology and oceanography for real-time and long-term forecasting. Kano et al. (2010) first applied an adjoint method to earthquake simulations with a three-degrees-of-freedom cell fault model and estimated a − b, a and L. They confirmed that the method is computationally much more efficient than MCMC and SIS, whilst providing nearly identical results. They also found that a−b is always constrained. However, the initial phase data of decaying afterslip is required to constrain L, and a was not constrained under any conditions.
Although Kano et al. (2010) estimated frictional parameters for three cell faults, they did not investigate the potential impact on earthquake prediction. Nevertheless, their work has motivated us to test the feasibility of predicting future afterslip propagation and its triggering of another earthquake with a realistic continuum fault model based on an adjoint method. In this work, we perform a synthetic data assimilation experiment with the fault model. A rate-and state-dependent friction law and the slowness law (Dieterich, 1979) are used as governing equations for data assimilation. Although this fault model is still simplified in many aspects, it constitutes an important step toward predicting an actual triggered earthquake by evaluating the potential impact of efficient data assimilation.
In Section 2, the numerical model, synthetic afterslip data and the data assimilation system are described. Numerical results are given in Section 3. Section 4 provides a summary and discusses potential challenges for practical application.
2.1 Forward model
2.1.1 Fault model and frictional properties
2.1.2 Governing equations
2.1.3 True state
We adopt a one model realization with frictional parameters ((A − B (kPa), A (kPa), L (mm)) = (−100,40.0,40.0) in LA, (−80.0,40.0,40.0) in SA, and (5.00, 40.0, 40.0) in the afterslip regions, as a ‘true’ state. This is obtained from forward time integration of the above governing equations.
The true state indicates that large earthquakes occur quasi-periodically in the LA region at approximately 100-year intervals, and medium-sized earthquakes occur in the SA region with an average interval of 60 years, depending on the stress interactions between the two asperities (throughout this paper, we regard a fault slip as an earthquake when the slip velocity exceeds 1 cm/s). This is a typical timescale for thrust-type earthquakes at the Kurile Trench (e.g., Sawai et al., 2009).
2.2 Synthetic data and simulation run
We investigate whether synthetic observations are sufficient to specify the frictional parameters over the afterslip region. Synthetic slip velocity observation on the fault is sampled from the true state during days 1–20 at all cells in the afterslip area. We then added a Gaussian error with zero mean and standard deviation 1.0 × 10−8 m/s (Miyazaki et al., 2004). We assimilate these synthetic slip velocity data through the adjoint data assimilation method to check whether the “first-guess” parameter values are updated to their true values.
Table of true, first-guess and estimated values of frictional parameters, cost function, and occurrence time of the triggered earthquake.
A− B (kPa)
Occurrence time of the triggered earthquake (year)
2.3 Adjoint data assimilation method
The adjoint-based data assimilation is outlined in this section. More detailed explanations are given in Lewis et al. (2006) and Ito et al. (2011). The basic idea is to define a cost function that quantifies the total misfit between the model results and the observations. The adjoint model, which is the adjoint of the dynamic model operator linearized about the time trajectory, effectively transforms the misfit into the gradient of the cost function with respect to control variables, i.e. the frictional parameters in this study. The control variables are updated iteratively in order to minimize the cost function, thereby improving estimates and prediction. Note that, in each iteration, the control variables are constant.
3.1 Optimization of frictional parameters
Table 1 shows the results of data assimilation, “estimated values” and first-guess values. As a result of data assimilation, frictional parameters are updated to (A − B (kPa), A (kPa), L (mm)) = (5.37, 40.5, 40.8), which are close to the true parameters, (5.00, 40.0, 40.0). Figure 2 shows that the updated state of the slip velocity is consistent with the observed values at points A–E, whereas those in the simulation run were not. We confirmed, by follow-up experiments, that this result is true within the range of first-guess values which are one-order larger, or smaller, than the true values. This suggests that the rough estimate of first-guess values is sufficient to recover the true frictional parameters. The reason why all frictional parameters are determined, unlike in Kano et al. (2010), is presumably that the observation data are available during the accelerating phase of the afterslip. Moreover, we found that only the observations at point B constrain all frictional parameters. Thus, we suggest that if there is at least one observation which contains both the acceleration phase and the decaying phase, all frictional parameters are constrained.
3.2 Prediction of the triggered earthquake in SA
4. Discussion and Summary
We have developed an efficient data assimilation system to estimate frictional parameters on a fault with the aim of predicting earthquakes triggered by afterslip propagation. Synthetic observation data of the slip velocity on the plate interface are assimilated to a simplified fault model with two asperities through an adjoint data assimilation method. We confirmed that all frictional parameters are well constrained from observations during 20 days, facilitating the enhanced prediction of the triggered earthquake. Moreover, if at least one observation contains both the acceleration phase and the deceleration phase, all frictional parameters are constrained.
In terms of a practical application, several points need to be further verified. First, predictions derived from a data assimilation system are strongly dependent on the base dynamic model. Currently, the dynamic model does not consider the geometry of the plate boundary, structural heterogeneity and visco-elasticity. We assumed uniform fric-tional parameters over the entire afterslip region. Moreover, it would be essential to estimate the frictional parameters in asperity (Hiyoshi et al., Possible estimates of frictional properties on earthquake and afterslip rupture surfaces with 4D-VAR method, manuscript in preparation, 2013) and other control variables, i.e. the initial slip velocity and the initial state variable (Kano et al., 2010). Our future scope includes improvement of the base model, especially the model of friction, to explain the data well and the optimization of heterogeneous parameters.
We thank T. Ochi for useful comments that helped to improve the manuscript. This work was supported by JSPS Fellows (23-1546), by MEXT KAKENHI (21340127) and by MEXT projects of “New Research Project for the Evaluation of Seismic Linkage around the Nankai Trough” and of “Observation and Research Program for Prediction of Earthquakes and Volcanic Eruptions”. We used Generic Mapping Tools (Wessel and Smith, 1998) to produce the figures.
- Ariyoshi, K., T. Matsuzawa, and A. Hasegawa, The key frictional parameters controlling spatial variations in the speed of postseismic-slip propagation on a subduction plate boundary, Earth Planet. Sci. Lett., 256, 136–146, 2007.View ArticleGoogle Scholar
- DeMets, C., R. G. Gordon, D. F. Argus, and S. Stein, Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions, Geophys. Res. Lett., 21, 2191–2194, 1994.View ArticleGoogle Scholar
- Dieterich, J. H., Modeling of rock friction 1. Experimental results and constitutive equations, J. Geophys. Res., 84, 2161–2168, 1979.View ArticleGoogle Scholar
- Fukuda, J., K. M. Johnson, K. M. Larson, and S. Miyazaki, Fault friction parameters inferred from the early stages of afterslip following the 2003 Tokachi-oki earthquake, J. Geophys. Res., 114, B04412, doi:10.1029/2008 JB006166, 2009.Google Scholar
- Hearn, E. H., R. Bürgmann, and R. E. Reilinger, Dynamics of ?Izmit earthquake postseismic deformation and loading of the Düzce earthquake hypocenter, Bull. Seismol. Soc. Am., 92, 172–193, doi:10.1785/0120000832, 2002.View ArticleGoogle Scholar
- Heki, K., S. Miyazaki, and H. Tsuji, Silent fault slip following an interplate thrust earthquake at the Japan Trench, Nature, 386, 595–597, 1997.View ArticleGoogle Scholar
- Hsu, Y.-J., M. Simons, J.-P. Avouac, J. Galetzka, K. Sieh, M. Chlieh, D. Natawidjaja, L. Prawirodirdjo, and Y. Bock, Frictional afterslip following the 2005 Nias-Simeulue earthquake, Sumatra, Science, 312, 1921–1926, doi:10.1126/science.1126960, 2006.View ArticleGoogle Scholar
- Ito, K., Y. Ishikawa, Y. Miyamoto, and T. Awaji, Short-time-scale processes in a mature hurricane as a response to sea surface fluctuations, J. Atmos. Sci., 68(10), 2250–2272, 2011.View ArticleGoogle Scholar
- Johnson, K. M., R. Bürgmann, and K. M. Larson, Frictional properties on the San Andreas fault near Parkfield, California, inferred from models of afterslip following the 2004 earthquake, Bull. Seismol. Soc. Am., 96, S321–S338, doi:10.1785/0120050808, 2006.View ArticleGoogle Scholar
- Kano, M., S. Miyazaki, K. Ito, and K. Hirahara, Estimation of frictional parameters and initial values of simulation variables using an adjoint data assimilation method with synthetic afterslip data, Zishin 2, 63, 57–69, 2010 (in Japanese).Google Scholar
- Kato, N., Numerical simulation of recurrence of asperity rupture in the Sanriku region, northeastern Japan, J. Geophys. Res., 113, B06302, doi:10.1029/2007 JB005515, 2008.Google Scholar
- Kitagawa, G., Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, J. Comput. Graph. Stat., 5, 1–25, 1996.Google Scholar
- Lewis, J. and J. Derber, The use of adjoint equations to solve a variational adjustment program with advective constraint, Tellus, 37A, 309–322, 1985.View ArticleGoogle Scholar
- Lewis, J., S. Lakshmivarahan, and S. Dhall, Dynamic Data Assimilation: A Least Squares Approach, 654 pp, Cambridge University Press, 2006.Google Scholar
- Liu, J., R. Chen, and T. Logvinenko, A theoretical framework for sequential importance sampling and resampling, in Sequential Monte Carlo in Practice, 225–242, Springer-Verlag, 2000.Google Scholar
- Marone, C. J., C. H. Scholz, and R. Bilham, On the mechanics of earthquake afterslip, J. Geophys. Res., 96, 8441–8452, 1991.View ArticleGoogle Scholar
- Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys., 21, 1087–1092, 1953.View ArticleGoogle Scholar
- Mitsui, N., T. Hori, S. Miyazaki, and K. Nakamura, Constraining interpolate frictional parameters by using limited terms of synthetic observation data for afterslip: a preliminary test of data assimilation, Theor. Appl. Mech. Jpn., 58, 113–120, 2010.Google Scholar
- Miyazaki, S., P. Segall, J. Fukuda, and T. Kato, Space time distribution of afterslip following the 2003 Tokachi-oki earthquake: Implications for variations in fault zone frictional properties, Geophys. Res. Lett., 31, L06623, doi:10.1029/2003 GL019410, 2004.Google Scholar
- Montési, L. G. J., Controls of shear zone rheology and tectonic loading on postseismic creep, J. Geophys. Res., 109, B10404, doi:10.1029/2003 JB002925, 2004.View ArticleGoogle Scholar
- Okada, Y., Internal deformation due to shear and tensile faults in a halfspace, Bull. Seismol. Soc. Am., 82, 1018–1040, 1992.Google Scholar
- Perfettini, H. and J.-P. Avouac, Postseismic relaxation driven by brittle creep: A possible mechanism to reconcile geodetic measurements and the decay rate of aftershocks, application to the Chi-Chi earthquake, Taiwan, J. Geophys. Res., 109, B02304, doi:10.1029/2003 JB002488, 2004.Google Scholar
- Perfettini, H. and J.-P. Avouac, Modeling afterslip and aftershocks following the 1992 Landers earthquake, J. Geophys. Res., 112, B07409, doi:10.1029/2006 JB004399, 2007.Google Scholar
- Perfettini, H., J.-P. Avouac, and J.-C. Ruegg, Geodetic displacements and aftershocks following the 2001 Mw=8.4 Peru earthquake: Implication for the mechanics of the earthquake cycle along subduction zones, J. Geophys. Res., 110, B09404, doi:10.1029/2004 JB003522, 2005.Google Scholar
- Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd edition, 963 pp, Cambridge University Press, New York, 1996.Google Scholar
- Rice, J. R., Spatio-temporal complexity of slip on a fault, J. Geophys. Res., 98, 9885–9907, 1993.View ArticleGoogle Scholar
- Ruina, A., Slip instability and state variable friction laws, J. Geophys. Res., 88, 10359–10370, 1983.View ArticleGoogle Scholar
- Sawai, Y., T. Kamataki, M. Shishikura, H. Nasu, Y. Okamura, K. Satake, K. H. Thomson, D. Matsumoto, Y. Fujii, J. Komatsubara, and T. T. Aung, Aperiodic recurrence of geologically recorded tsunamis during the past 5500 years in eastern Hokkaido, Japan, J. Geophys. Res., 114, B01319, doi:10.1029/2007JB005503, 2009.Google Scholar
- Uchida, N., S. Yui, S. Miura, T. Matsuzawa, A. Hasegawa, Y. Motoya, and M. Kasahara, Quasi-static slip on the plate boundary associated the 2003 M8.0 Tokachi-oki and 2004 M7.1 off-Kushiro earthquakes, Japan, Gondowana Res., 16, 527–533, 2009.View ArticleGoogle Scholar
- Wessel, P. and W. H. F. Smith, New, improved version of Generic Mapping Tools released, Eos Trans. AGU, 79(47), 579, doi:10.1029/98EO00426, 1998.View ArticleGoogle Scholar