Article

A Bayesian Method for Incorporating Self-Similarity Into Earthquake Slip Inversions

Wiley
Journal of Geophysical Research: Solid Earth
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Abstract

Distributions of coseismic slip help illuminate many properties of earthquakes, including fault geometry, stress changes, frictional properties, and potential future hazard. Slip inversions take observations and calculate slip at depth, but there are a number of commonly adopted assumptions such as minimizing the second spatial derivative of slip (the Laplacian) that have little physical basis and potentially bias the result. In light of growing evidence that fault slip shows fractal properties, we suggest that this information should be incorporated into slip inversions as a form of regularization, instead of Laplacian smoothing. We have developed a Bayesian approach to efficiently solve for slip incorporating von Karman regularization. In synthetic tests, our approach retrieves fractal slip better than Laplacian regularization, as expected, but even performs comparably, or better, when the input slip is not fractal. We apply this to the 2014 Mw 6.0 Napa Valley earthquake on a two-segment fault using Interferometric Synthetic Aperture Radar (InSAR) and Global Positioning System (GPS) data. We find the von Karman and Laplacian inversions give similar slip magnitude but in different locations, and the von Karman solution has much tighter confidence bounds on slip than the Laplacian solution. Differences in earthquake slip due to the regularization technique could have important implications for the interpretation and modeling of stress changes on the causative and neighboring faults. We therefore recommend that choice of regularization method should be routinely made explicit and justified and that von Karman regularization is a better default than Laplacian.

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... independent from the data) are applied to stabilize the inverse problem due to its ill-posed nature. These conditions are commonly imposed subjectively (Minson et al. 2013 ;Dettmer et al. 2014 ;Amey et al. 2018 ) and are typically referred to as smoothness conditions. According to Mai & Beroza ( 2002 ), a direct effect of the prior spatial correlation condition is that the larger it is, the slow er the spatial deca y of cumulative slip will be, potentially introducing bias into the results. ...
... Bayesian Inference (BI) and Bayesian Model Selection (BMS) provide robust frameworks for selecting spatial correlation models and estimating uncertainties (Mackay 2003 ;Sambridge et al. 2006 ) of complex processes such as spatio-temporal rupture on faults. Since, in FFI, the prior spatial correlation can be treated as an a priori hypothesis (Fukuda & Johnson 2008 ;Amey et al. 2018 ;Benavente et al. 2019 ), different prior spatial correlation schemes can be compared based on their consistency with data using BMS. Evidence-based methods, such as the Akaike Bayesian Information Criterion (Akaike 1980 ), naturally follow the Occam's Razor or parsimony principle. ...
... where M is the number of parameters and S corresponds to the prior covariance matrix, where prior spatial correlation between parameters can be included using ACF matrices (e.g. Amey et al. 2018 ). s p corresponds to an average vector for the logarithm of slip. ...
Article
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The spatial correlation of coseismic slip is a necessary input for generating stochastic seismic rupture models, which are commonly employed in seismic and tsunami hazard assessments. To date, the spatial correlation of individual earthquakes is characterized using finite fault models by finding the combination of parameters of a von Kármán autocorrelation function that best fits the observed autocorrelation function of the finite fault model. However, because a priori spatial correlation conditions (i.e., not in the data) are generally applied in finite fault model generation, the results obtained using this method may be biased. Additionally, robust uncertainty estimates for spatial correlations of coseismic slip are generally not performed. Considering these limitations in the classic method, here, a method is developed based on a Bayesian formulation of Finite Fault Inversion (FFI) with positivity constraints. This method allows for characterizing the spatial correlation of coseismic slip and its uncertainties for an earthquake by using samples of coseismic slip from a posterior probability density function (PDF). Furthermore, a Bayesian model selection criterion called Akaike Bayesian Information Criterion (ABIC) is applied to objectively choose between different prior spatial correlation schemes before computing the posterior, to reduce subjectivity due to this prior condition. The ABIC is calculated using an approximate analytical expression of Bayesian evidence. The method is applied to simulated P-waves, demonstrating that model selection allows for objectively estimating the most suitable prior spatial correlation scheme in FFI. Additionally, the target spatial correlation of coseismic slip is accurately recovered using samples from the posterior PDF, as well as their uncertainties. Moreover, in the simulated experiment, it is shown that a non-robust choice of the prior spatial correlation scheme can significantly bias the estimated spatial correlations of coseismic slip. We apply our method to observed P-waves from the 2015, Illapel earthquake (Mw = 8.3), finding that the spatial correlation of coseismic slip of this earthquake is better described by a von Kármán ACF, with mean correlation lengths of around 47 km and Hurst parameter of 0.58. We conclude that using our method reduces biases associated with prior spatial correlation conditions and allows for robust estimation of spatial correlations of coseismic slip and their uncertainties.
... Coseismic slip distribution inversion is a means to refine the slip state of the fault fracture surface (Wright et al. 2004;Wang et al. 2011;Amey et al. 2018). The coseismic slip distribution is the basis for analyzing the moment magnitude, Coulomb stress change, fault structure environment, rock friction characteristics, and postearthquake deformation mechanism (King et al. 1994;Elliott et al. 2016;Xu et al. 2020a;Jiang et al. 2021), and it can also be used as input data for seismic dynamics simulation and tsunami simulation (Cubas et al. 2015;Williamson et al. 2017). ...
... Coseismic slip distribution inversion can be described via a linear relationship between fault slip parameters and coseismic surface deformation (Okada 1985). Because the slip distribution inversion is usually ill-posed, the discretized coefficient matrix is ill-conditioned (Xu 1998;Amey et al. 2018). To alleviate this circumstance, it is usually necessary to add additional prior constraints or regularization terms to obtain stable parametric solutions (Tikhonov 1963;Funning et al. 2005;Wang and Gu 2020). ...
... However, this method also encountered a complex calculation process. Amey et al. (2018Amey et al. ( , 2019 proposed a von Karman Bayesian slip distribution inversion method based on the self-similar characteristics of faults. This method also refers to the research of Bagnardi and Hooper (2018) and processes the parameters with an adaptive step size, which improves the calculation efficiency and accuracy to a certain extent. ...
Article
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For slip distribution inversion with Bayesian theory, traditionally, the Markov Chain Monte Carlo (MCMC) method is well applied to generate a posterior probability density function with a sampling strategy. However, its computational cost may be expensive, and it fails to accommodate large volume data sets and estimate higher dimensional parameters of interest. In this study, we introduce variational inference theory into the study of coseismic slip distribution, and present a variational Bayesian slip distribution inversion approach. Furthermore, synthetic tests show that the variational Bayesian approach can efficiently and accurately invert the designed slip distribution; therefore, we conclude that the proposed algorithm is appropriate to invert the slip distribution parameters, which might be superior to MCMC sampling due to its excellent convergence speed and low computational burden. Taking the Dingri earthquake on March 20, 2020, as an example, we further verify the practicability of the variational Bayesian method in actual earthquakes. Additionally, the inversion results show that the main fault slip region of the Dingri earthquake occurs at depths of 2 ~ 8 km on the surface, the maximum slip amount is 0.54 m, and the coseismic release seismic moment is 5.58 × 1017 Nm, corresponding to a moment magnitude of Mw 5.79.
... We applied a Bayesian method incorporating von Karman regularization to solve for slip distribution at depth and associated standard deviation (Amey et al., 2018). Von Karman smoothing is arguably more physically meaningful than Laplacian smoothing, as it accounts for fractal (self-similar) properties of fault slip as evidenced by Aviles et al. (1987), Robertson et al. (1995), Mai and Beroza (2002), Ben-Zion (2008), Powers and Jordan (2010), and Passelègue et al. (2016). ...
... Von Karman smoothing is arguably more physically meaningful than Laplacian smoothing, as it accounts for fractal (self-similar) properties of fault slip as evidenced by Aviles et al. (1987), Robertson et al. (1995), Mai and Beroza (2002), Ben-Zion (2008), Powers and Jordan (2010), and Passelègue et al. (2016). Practically, the von Karman solution outperforms the Laplacian solution in that the former gives tighter confidence bounds on slip, as shown in Amey et al. (2018). Once the fault geometry has been determined (Section 2.4.1), ...
... Once the fault geometry has been determined (Section 2.4.1), we solve for slip and rake angle for each patch using the slipBERI code (Amey et al., 2018), as well as a hyperparameter α 2 controlling the slip variance for each separate fault. Thus, we can explore the full range of solutions for a range of variances, instead of assuming the variance of slip in advance. ...
Article
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In the continents, the importance of earthquakes that occur away from major block‐bounding faults is still debated. The 21 May 2021 MW ∼ 7.4 Maduo earthquake occurred on a secondary fault away from previously‐identified major block boundaries. Here we use 7 years of Sentinel‐1 Interferometric Synthetic Aperture Radar (InSAR) time series (between October 2014 and November 2021) to determine the distribution of coseismic slip and early postseismic afterslip following the Maduo earthquake, and the preceding interseismic strain accumulation. We devised a 13‐segment 3‐D fault geometry constrained by the SAR range offsets and the distribution of relocated aftershocks and used a Bayesian method incorporating von Karman regularization to solve for coseismic slip and afterslip models. We also used teleseismic waveforms as a standalone inversion to show the rupture evolution in space and time during the earthquake, finding that it propagates bilaterally with three notable rupture episodes. Our preferred coseismic self‐similar slip model shows a moderate shallow slip deficit, with the majority of moment release occurring in the depth interval of 1–10 km. The coseismic slip deficit is taken up in part by afterslip at shallow (<4 km) depths that grows linearly with time during the first ∼6 months, and at >10 km depths where afterslip grows logarithmically with time. We suggest that this heterogeneity is likely controlled by spatial variations in fault friction related to lithology. We discuss the implications for seismic hazard away from major tectonic block boundaries in light of our observations of the earthquake cycle on this intrablock fault.
... Coseismic slip distribution inversion is an important means of studying the delicate characteristics of seismic fault planes. The coseismic slip distribution is helpful in revealing many characteristics of a seismic event, including the fault geometry, stress variation, friction characteristics, and potential seismic hazards (Jónsson et al. 2002;Wright et al. 2003;Wang et al. 2011;Amey et al. 2018). Additionally, the slip distribution results are the driving force for the calculation of the Coulomb stress, the basis for the analysis of interseismic stress accumulation and the study of postseismic aftershock slip, and the basis for framing the seismic void area (Parsons et al. 2006;Xu et al. 2018;. ...
... methods are based on the least-squares principle (Teunissen and Amiri-Simkooei 2008;Amiri-Simkooei 2016). For the inversion of a coseismic slip distribution, Green's function matrix is seriously rank deficient, and even a small amount of data noise will lead to nonphysical slip oscillation of the least-squares solution; thus, it is an ill-posed problem (Hansen 1998;Amey et al. 2018;Wang and Gu 2020). Therefore, a regularization method is usually used to solve this problem; a Laplace smoothing constraint is most commonly used (Wright et al. 2003;Funning et al. 2005). ...
... Although the ill-posed nature of the linear solution for the slip distribution is effectively avoided, a Laplacian smoothing constraint is still used for the prior information. However, while Laplace smoothing can prevent such large stress drops, this does not necessarily mean it is the best function to describe the nature of slip (Amey et al. 2018). It is merely a mathematical constraint, rather than being based on any observed fundamental feature of slip distributions (Amey et al. 2018). ...
Article
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Given the problem that the relative weights of multitype data sets are not considered in the current studies of Bayesian von Karman regularized slip distribution inversion, we propose to add hyperparameters that control the variance-covariance information of different data sets within the Bayesian framework, present a detailed and complete theoretical method, and successfully apply it to the earthquake that occurred in Norcia, Italy, in 2016. Synthetic tests show that compared with the Bayesian von Karman regularization method with equal weights for different data, the improved Bayesian von Karman regularization method can more effectively invert the real slip distribution of this dip-slip earthquake, its inversion results are more stable, and the data fitting accuracy is higher, thus verifying the advantages of the improved method. In the slip inversion of the Norcia earthquake, the hyperparameters of the global navigation satellite system (GNSS) and interferometric synthetic aperture radar (InSAR) data sets converged to different values, which shows it is indeed necessary to determine different relative weights of different data sets. Moreover, the inversion results show that this earthquake ruptured to the surface, the fault slip region was mainly concentrated in a depth range of 0-6 km, the maximum slip was 3.87 m, and the released coseismic seismic moment was 8.95 × 10 18 Nm, corresponding to a moment magnitude of M W ¼ 6.60. These results are consistent with the existing research, thus verifying the practicability of the improved method.
... We then have an observation equation, prior of the model parameters and hyperprior, which generate the joint posterior of the model parameters and hyperparameters. The joint posterior allows us to evaluate the optimal set of the model parameters and hyperparameters with uncertainties (Minson et al. 2013;Kubo et al. 2016;Amey et al. 2018). The fully Bayesian inversion is usually regarded as a nonapproximated version of ABIC (Malinverno & Briggs 2004;Gelman et al. 2013). ...
... We may eliminate the hyperparameters by integration (marginalization) from the joint posterior for evaluating the model-parameter distribution (Fukuda & Johnson 2008). Projecting the joint posterior onto low-dimensional profiles is usual in MCMC implementations (Duputel et al. 2014;Amey et al. 2018;Bagnardi & Hooper 2018). Operations of reshaping the joint posterior into these tractable forms can all be regarded as instances of the dimensionality reduction of the joint posterior. ...
... The performance of the MAP is recognized as not necessarily high in the statistical literature, in both the Bayesian inference without hyperparameters (Lin et al. 2006) and the fully Bayesian inference (Iba 1996). Meanwhile, the MAP is also considered a generalization of the maximum likelihood estimation (termed generalized maximum likelihood estimation; Carlin & Louis 2008) with many practical applications (Carlin & Louis 2008;Amey et al. 2018;Goto et al. 2019). ...
Article
Bayesian inversion generates a posterior distribution of model parameters from an observation equation and prior information both weighted by hyperparameters. The prior is also introduced for the hyperparameters in fully Bayesian inversions and enables us to evaluate both the model parameters and hyperparameters probabilistically by the joint posterior. However, even in a linear inverse problem, it is unsolved how we should extract useful information on the model parameters from the joint posterior. This study presents a theoretical exploration into the appropriate dimensionality reduction of the joint posterior in the fully Bayesian inversion. We classify the ways of probability reduction into the following three categories focused on the marginalization of the joint posterior: (1) using the joint posterior without marginalization, (2) using the marginal posterior of the model parameters and (3) using the marginal posterior of the hyperparameters. First, we derive several analytical results that characterize these categories. One is a suite of semi-analytic representations of the probability maximization estimators for respective categories in the linear inverse problem. The mode estimators of categories (1) and (2) are found asymptotically identical for a large number of data and model parameters. We also prove the asymptotic distributions of categories (2) and (3) delta-functionally concentrate on their probability peaks, which predicts two distinct optimal estimates of the model parameters. Secondly, we conduct a synthetic test and find an appropriate reduction is realized by category (3), typified by Akaike’s Bayesian information criterion. The other reduction categories are shown inappropriate for the case of many model parameters, where the probability concentration of the marginal posterior of the model parameters no longer implies the central limit theorem. The main cause of these results is that the joint posterior peaks sharply at an underfitted or overfitted solution as the number of model parameters increases. The exponential growth of the probability space in the model-parameter dimension makes almost-zero-probability events finitely contribute to the posterior mean and distributions of categories (1) and (2) be pathological. One remedy for this pathology is counting all model-parameter realizations by integrating the joint posterior over the model-parameter space of exponential multiplicity. Hence, the marginal posterior of the hyperparameters for categories (3) becomes appropriate and can conform to the law of large numbers even with numerous model parameters. The exponential rarity of the posterior mean and ABIC estimates implies the exponential time complexity of ordinary Monte Carlo methods in population mean and ABIC computations. We also present a geophysical application to estimate a continuous strain-rate field from spatially discrete global navigation satellite system data, demonstrating denser basis function expansions of the model-parameter field lead to oversmoothed estimates in naive fully Bayesian approaches, while detailed fields are resolved with convergence by the reduction of category (3). We often naively believe a good solution can be constructed from a finite number of samples with high probabilities, but the high-probability domain could be inappropriate, and exponentially many samples become necessary for generating appropriate estimates in the high-dimensional fully Bayesian posterior probability space.
... Ragon et al. (2018) further extended the work of Minson et al. (2013) and accounted for the uncertainty in fault geometry. Instead of Laplacian regularization, Amey et al. (2018) developed an inversion package slipBERI, and incorporated self-similarity, characterizing the seismic slip distribution in real earthquakes, as a prior assumption within the Bayesian inversion of earthquake slip. ...
... Here, we design a synthetic static slip to compare the performance of our method, GICMo, and a state-of-the-art method, slipBERI (Amey et al., 2018). The geodetic inversion package, slipBERI, solves for fault slip with GNSS and unwrapped InSAR phases in a Bayesian approach using von Karman regularization, and simultaneously solves for a hyperparameter that controls the degree of regularization. ...
... A series of overlapping elliptical cracks are simulated in Figure 3a and forward inversion is performed to calculate the surface displacement due to the slip increment between adjacent cracks. We aimed to compare the results based on various geodetic inversion algorithms: (a) the one-ellipse model, as described in Section 2.1, (b) a von Karman regularization algorithm (Amey et al., 2018), and (c) the two-ellipse model with different crack centers. Inversion results are shown in Figures 3b-3d, and the modeled phase and residuals are shown in Figures S2 and S3 in Supporting Information S1. ...
Article
Full-text available
Improved imaging of the spatio‐temporal growth of fault slip is crucial for understanding the driving mechanisms of earthquakes and faulting. This is especially critical to properly evaluate the evolution of seismic swarms and earthquake precursory phenomena. Fault slip inversion is an ill‐posed problem and hence regularization is required to obtain stable and interpretable solutions. An analysis of compiled finite fault slip models shows that slip distributions can be approximated with a generic elliptical shape, particularly well for M ≤ 7.5 events. Therefore, we introduce a new physically informed regularization to constrain the spatial pattern of slip distribution. Our approach adapts a crack model derived from mechanical laboratory experiments and allows for complex slipping patterns by stacking multiple cracks. The new inversion method successfully recovered different simulated time‐dependent patterns of slip propagation, that is, crack‐like and pulse‐like ruptures, directly using wrapped satellite radar interferometry (InSAR) phase observations. We find that the new method reduces model parameter space, and favors simpler interpretable spatio‐temporal fault slip distributions. We apply the proposed method to the 2011 March–September normal‐faulting seismic swarm at Hawthorne (Nevada, USA), by computing ENVISAT and RADARSAT‐2 interferograms to estimate the spatio‐temporal evolution of fault slip distribution. The results show that (a) aseismic slip might play a significant role during the initial stage and (b) this shallow seismic swarm had slip rates consistent with those of slow earthquake processes. The proposed method will be useful in retrieving time‐dependent fault slip evolution and is expected to be widely applicable to studying fault mechanics, particularly in slow earthquakes.
... We then have an observation equation, prior of the model parameters and hyperprior, which generate the joint posterior of the model parameters and hyperparameters. The joint posterior allows us to evaluate the optimal set of the model parameters and hyperparameters with uncertainties (Minson et al. 2013;Kubo et al. 2016;Amey et al. 2018). The fully Bayesian inversion is usually regarded as a non-approximated version of ABIC (Malinverno & Briggs 2004;Gelman et al. 2013). ...
... We may eliminate the hyperparameters by integration (marginalisation) from the joint posterior for evaluating the model-parameter distribution (Fukuda & Johnson 2008). Projecting the joint posterior onto low-dimensional profiles is usual in MCMC implementations (Duputel et al. 2014;Amey et al. 2018;Bagnardi & Hooper 2018). Operations of reshaping the joint posterior into these tractable forms can all be regarded as instances of the dimensionality reduction of the joint posterior. ...
... The performance of the MAP is recognised as not necessarily high in the statistical literature, in both the Bayesian inference without hyperparameters (Lin et al. 2006) and the fully Bayesian inference (Iba 1996). Meanwhile, the MAP is also considered a generalisation of the maximum likelihood estimation (termed generalised maximum likelihood estimation; Carlin & Louis 2008) with many practical applications (Carlin & Louis 2008;Amey et al. 2018;Goto et al. 2019). ...
Preprint
Bayesian inversion generates a posterior distribution of model parameters from an observation equation and prior information both weighted by hyperparameters. The prior is also introduced for the hyperparameters in fully Bayesian inversions and enables us to evaluate both the model parameters and hyperparameters probabilistically by the joint posterior. However, even in a linear inverse problem, it is unsolved how we should extract useful information on the model parameters from the joint posterior. This study presents a theoretical exploration into the appropriate dimensionality reduction of the joint posterior in the fully Bayesian inversion. We classify the ways of probability reduction into the following three categories focused on the marginalisation of the joint posterior: (1) using the joint posterior without marginalisation, (2) using the marginal posterior of the model parameters and (3) using the marginal posterior of the hyperparameters. First, we derive several analytical results that characterise these categories. One is a suite of semianalytic representations of the probability maximisation estimators for respective categories in the linear inverse problem. The mode estimators of categories (1) and (2) are found asymptotically identical for a large number of data and model parameters. We also prove the asymptotic distributions of categories (2) and (3) delta-functionally concentrate on their probability peaks, which predicts two distinct optimal estimates of the model parameters. Second, we conduct a synthetic test and find an appropriate reduction is realised by category (3), typified by Akaike's Bayesian information criterion (ABIC). The other reduction categories are shown inappropriate for the case of many model parameters, where the probability concentration of the marginal posterior of the model parameters is found no longer to mean the central limit theorem...
... The surface displacement is the summation of all contributions from the interface points experiencing either a coupling regime or an SSE. In this work, to determine the plate interface aseismic slip history in these terms from continuous GPS (or any other geodetic) measurements, we introduce and solve a constrained optimization problem based on the adjoint elastostatic equations with a Tikhonov regularization term (Calvetti et al. 2000;Asnaashari et al. 2013) and a projection operator built with the von Karman autocorrelation function (Mai & Beroza 2002;Amey et al. 2018). The new method, called ELADIN (ELastostatic ADjoint INversion), simultaneously determines the distribution of the interplate coupling and slow slip from surface displacements. ...
... One rigorous framework to overcome this problem and to determine the uncertainty of such an inverse problem solution are the Bayesian approaches. The incorporation of prior information through probability density functions (pdf) allows determining the posterior model covariance and pdfs, as well as imposing model restrictions by means of truncated prior pdfs (Tarantola & Valette 1982;Yabuki & Matsu'Ura 1992;Nocquet et al. 2014;Nishimura et al. 2004;Minson et al. 2013;Amey et al. 2018;Nocquet 2018). For instance, Minson et al. (2013) samples the posterior pdf using a Monte Carlo Markov Chain that enables to apply non-negativity constraints and any prior pdf. ...
... Although the Laplacian operator reduces unphysical and rough slip solutions (and thus unreliable large stress drops), this is not the most convenient regularization strategy to preserve the real nature of the fault slip, which has a self-similar spectral signature (Mai & Beroza 2002). Recently, Amey et al. (2018) proposed to use the von Karman autocorrelation function to build the model covariance matrix such that the penalization term should lead to self-similar slow-slip solutions. ...
Article
Full-text available
To shed light on the prevalently slow, aseismic slip interaction between tectonic plates, we developed a new static slip inversion strategy, the ELADIN (ELastostatic ADjoint INversion) method, that uses the adjoint elastostatic equations to compute the gradient of the cost function. ELADIN is a 2-step inversion algorithm to efficiently handle plausible slip constraints. First it finds the slip that best explains the data without any constraint, and then refines the solution by imposing the constraints through a Gradient Projection Method. To obtain a selfsimilar, physically-consistent slip distribution that accounts for sparsity and uncertainty in the data, ELADIN reduces the model space by using a von Karman regularization function that controls the wavenumber content of the solution, and weights the observations according to their covariance using the data precision matrix. Since crustal deformation is the result of different concomitant interactions at the plate interface, ELADIN simultaneously determines the regions of the interface subject to both stressing (i.e., coupling) and relaxing slip regimes. For estimating the resolution, we introduce a mobile checkerboard analysis that allows to determine lower-bound fault resolution zones for an expected slip-patch size and a given stations array. We systematically test ELADIN with synthetic inversions along the whole Mexican subduction zone and use it to invert the 2006 Guerrero Slow Slip Event (SSE), which is one of the most studied SSEs in Mexico. Since only 12 GPS stations recorded the event, careful regularization is thus required to achieve reliable solutions. We compared our preferred slip solution with two previously published models and found that our solution retains their most reliable features. In addition, although all three SSE models predict an upward slip penetration invading the seismogenic zone of the Guerrero seismic gap, our resolution analysis indicates that this penetration might not be a reliable feature of the 2006 SSE.
... The surface displacement is the summation of all contributions from the interface points experiencing either a coupling regime or an SSE. In the present work, to determine the plate interface aseismic slip history in these terms from continuous GPS (or any other geodetic) measurements, we introduce and solve a constrained optimization problem based on the adjoint elastostatic equations with a Tikhonov regularization term (Calvetti et al., 2000;Asnaashari et al., 2013) and a projection operator built with the von Karman autocorrelation function (Mai and Beroza, 2002;Amey et al., 2018). The new method, called ELADIN (ELastostatic ADjoint INversion), simultaneously determines the distribution of the interplate coupling and slow slip from surface displacements. ...
... One rigorous framework to overcome this problem and to determine the uncertainty of such an inverse problem solution are the Bayesian approaches. The incorporation of prior information through probability density functions (pdf) allows determining the posterior model covariance and pdfs, as well as imposing model restrictions by means of truncated prior pdfs (Tarantola and Valette, 1982;Nocquet, 2018;Minson et al., 2013;Yabuki and Matsu'Ura, 1992;Amey et al., 2018;Nocquet et al., 2014;Nishimura et al., 2004). For instance, Minson et al. (2013) samples the posterior pdf using a Monte Carlo Markov Chain that enables to apply non-negativity constraints and any prior pdf. ...
... Although the Laplacian operator reduces unphysical and rough slip solutions (and thus unreliable large stress drops), this is not the most convenient regularization strategy to preserve the real nature of the fault slip, which has a self-similar spectral signature (Mai and Beroza, 2002). Recently, Amey et al. (2018) proposed to use the von Karman autocorrelation function to build the model covariance matrix such that the penalization term should lead to self-similar slow-slip solutions. ...
Preprint
Full-text available
Understanding the interaction between tectonic plates from geodetic data is relevant to the assessment of seismic hazard. To shed light on that prevalently slow aseismic interaction, we developed a new static-slip inversion strategy, the ELADIN (ELastostatic ADjoint INversion) method, that uses the adjoint elastostatic equations to compute the gradient of the cost function. To handle plausible slip constraints, ELADIN is a 2-step inversion algorithm. First it finds the slip that best explains the data without any constraint, and then refines the solution by imposing the constraints through a Gradient Projection Method. To obtain a selfsimilar, physically-consistent slip distribution that accounts for sparsity and uncertainty in the data, ELADIN reduces the model space by using a von Karman regularization function that controls the wavenumber content of the solution, and weights the observations according to their covariance using the data precision matrix. Since crustal deformation is the result of different concomitant interactions at the plate interface, ELADIN simultaneously determines the regions of the interface subject to both stressing (i.e., coupling) and relaxing slip regimes. For estimating the resolution, we introduce a mobile checkerboard that allows to determine lower-bound fault resolution zones for an expected slip-patch size and a given stations array. We systematically test ELADIN with synthetic inversions along the whole Mexican subduction zone and use it to invert the 2006 Guerrero Slow Slip Event (SSE), which is one of the most studied SSEs in Mexico. Since only 12 GPS stations recorded the event, careful regularization is thus required to achieve reliable solutions. We compared our preferred slip solution with two previously published models and found that our solution retains their most reliable features. In addition, although all three SSE models predict an upward slip penetration invading the seismogenic zone of the Guerrero seismic gap, our resolution analysis indicates that this penetration might not be a reliable feature of the 2006 SSE.
... However, such linear inversions are always an ill-posed problem (Aster et al. 2018;Wang et al. 2019;Zhao et al. 2022). To stabilize the solutions, some regularization techniques, such as L-curve method (Hearn and Burgmann 2005;Hansen 1992;Jiang et al. 2013), cross-validation technique (Matthews and Segall 1993), Akaike's Bayesian information criterion (Yabuki and Matsu'ura 1992;Funning et al. 2005;Yi et al. 2017), VCE method (Xu et al. 2010(Xu et al. , 2016Fan et al. 2017), j R i method (Barnhart and Lohman 2010) and Bayesian approach (Fukuda and Johnson 2008;Minson et al. 2013;Amey et al. 2018;Sun et al. 2023), are usually incorporated into the LS method. ...
... We fix the fault size in our study under the condition of the relationship between fault size and magnitude moment (Wells and Coppersmith 1994). We could also resolve the fault size (Fielding et al. 2013) or vary them with circular harmonics (Amey et al. 2018), which will be our future study. As we gain more prior information about the problem, the inversion will be more efficiency and robust (Mellors et al. 1997;Käufl et al. 2016). ...
Article
Full-text available
Inverting fault geometry and slip distribution simultaneously with geodetic observations based on Bayesian theory is becoming increasingly prevalent. A widely used approach, proposed by (Fukuda and Johnson, Geophys J Int 181:1441–1458, 2010) (F-J method), employs the least-squares method to solve the linear parameters of slip distribution after sampling the nonlinear parameters, including fault geometry, data weights and smoothing factor. Here, we present a modified version of the F-J method (MF-J method), which treats data weights and the smoothing factor as hyperparameters not directly linked to surface deformation. Additionally, we introduce the variance component estimation (VCE) method to resolve these hyperparameters. To validate the effectiveness of the MF-J method, we conducted inversion tests using both synthetic data and a real earthquake case. In our comparison of the MF-J and F-J methods using synthetic experiments, we found that the F-J method's inversion results for fault geometry were highly sensitive to the initial values and step sizes of hyperparameters, whereas the MF-J method exhibited greater robustness and stability. The MF-J method also exhibited a higher and more stable acceptance rate, enabling convergence to simulated values and ensuring greater accuracy of the parameter estimation. Furthermore, treating the fault length and width as unknown parameters and solving them simultaneously with other fault geometry parameters and hyperparameters using the MF-J method successfully resolved the issue of non-uniqueness in fault location solutions caused by the excessively large no-slip areas. In the 2017 Mw 7.3 Sarpol-e Zahab earthquake case study, the MF-J method produced a fault slip distribution with low uncertainty that accurately fit surface observation data, aligning with results from other research institutions. This validated the method's applicability and robustness in real-world scenarios. Additionally, we inferred that the second slip asperity was caused by early afterslip.
... To estimate the source parameters of the Mihoub earthquake and their uncertainties, we use Bayesian modelling and elastic dislocations in a homogenous and elastic half space (Okada 1985). The uncertainty is represented by the posterior probability distribution in Bayesian modelling, which solves inverse problems under a probabilistic framework (Amey et al. 2018). In the first stage of modelling, fault parameters are estimated using uniform slip on a single fault plane via the GBIS code (Bagnardi and Hooper 2018). ...
... Therefore, in the second stage of modelling, we explore the distribution of coseismic slip with varying rakes on the fault plane predicted by the uniform slip model using Bayesian inversion with von Karman smoothing implemented in the slipBERI inversion package provided by Amey et al. (2018). The fault is enlarged to 18 km along strike and 10 km downdip from the surface and discretized into patches of ˜1.5 × 1.5 km in size (with 12 × 8 patches along J Seismol Vol:. ...
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Synthetic aperture radar interferometry (InSAR) is a powerful technique for quantifying the co- and postseismic deformation of large earthquakes at the Earth’s surface. However, surface deformation caused by small- to moderate-sized earthquakes is rarely revealed by InSAR because their coseismic slip occurs mostly at significant depths (> 5 km), with limited deformation on the Earth’s surface. In this work, we investigate the surface deformation associated with the Mw 5.4 May 28, 2016, Mihoub (Algeria) earthquake and its source parameters. Interferograms calculated from Sentinel-1 TOPSAR images of both ascending and descending orbits show that, despite its small size, the earthquake produced evident deformation on the Earth’s surface, suggesting that the coseismic slip took place at a relatively shallow depth. We model the coseismic displacement fields extracted from InSAR time series using Bayesian approaches in two stages: 1) we model the coseismic slip data using uniform slip on a single fault to constrain the fault parameters. 2) We explore a variable slip model with varying rakes on the discretized fault obtained in the first stage. The modelling results indicate that the earthquake was associated with a ~ 0.5 m shallow oblique reverse slip, mostly between depths of 1.5 and 4.5 km, on a NE–SW-trending and SE-dipping thrust fault, which is in good agreement with the focal mechanism solutions of the earthquake deduced from seismology. This study demonstrates that the multitemporal InSAR (MTI) method may constrain surface displacements when the coseismic interferograms of moderate- to small-sized earthquakes are noisy and hence difficult to unwrap. The newly identified seismogenic Mihoub fault has implications for seismic hazard assessment in northern Algeria.
... We generate 10 7 samples corresponding to the posterior distribution. With such a large number of iterations (10 7 ), the PPDs are expected to follow a Gaussian distribution for each of the fault parameters (Sun et al. 2011 ;Amey et al. 2018 ;Bagnardi & Hooper 2018 ). Ho wever , oftentimes, the PPDs deviate from symmetric, unimodal Gaussian distribution, probably due to sparse geodetic data, localized structural complexity, or subjective a priori model constraints (Sun et al. 2011 ;Amey et al. 2018 ;Bagnardi & Hooper 2018 ). ...
... With such a large number of iterations (10 7 ), the PPDs are expected to follow a Gaussian distribution for each of the fault parameters (Sun et al. 2011 ;Amey et al. 2018 ;Bagnardi & Hooper 2018 ). Ho wever , oftentimes, the PPDs deviate from symmetric, unimodal Gaussian distribution, probably due to sparse geodetic data, localized structural complexity, or subjective a priori model constraints (Sun et al. 2011 ;Amey et al. 2018 ;Bagnardi & Hooper 2018 ). Our final solution to the fault parameters includes mean values and 1 σ uncertainties of posterior distributions (Table 1 ). ...
Article
Geodetic networks enable us to investigate interseismic crustal deformation along the northwest Himalaya. Using 144 GNSS surface velocities and a Bayesian inversion model, we estimate the slip rate and fault geometry of the Main Himalayan Thrust (MHT) along six arc-normal transects in the northwest Himalaya. We consider that the fault plane consists of three sections along the décollement, namely the locking zone (0−12 km), the transition zone (10−22 km) and the creeping zone (≥22 km). The MHT is found to be completely locked from the surface down to an average depth of 6 ± 2 km. The locking-to-creeping transition zone along the décollement extends from the edge of the fully locked area to a deeper depth (14 ± 3 km) to the tip of the creeping zone of the MHT (17 ± 2 km) with a slip rate of 1.6 ± 0.9 to 3.7 ± 1.1 mm yr−1. Considering the range of uncertainties between 1−2 mm yr−1 for the GNSS velocities, the inverted slip rate along the transition zone of MHT turns out to be insignificant. Thus, the locking zone along the northwest Himalaya extends from the MFT to ∼111 ± 6 km in the north with a locking depth of ∼17 ± 2 km. The deeper part of the MHT is inferred to be creeping with an average slip rate of ∼19.1 ± 1.9 mm yr−1 along the northwest Himalaya. In addition, we have also illustrated a splay-fault model to account for the fault kinematics along the splay faults and the main décollement. The splay-fault model indicates a distributed slip rate at the locking-to-creeping transition zone and about ∼15 per cent smaller slip rate of the MHT than that of the single-fault model. Further, the checkerboard test and the uniform slip model exhibit the reliability of the current GNSS network and the inversion model (single- and splay-fault models). Overall, the updated fault kinematics inevitably contribute to the improvement of seismic hazard evaluation along the northwest Himalaya.
... This study solves for the coseismic, distributed slip and rake on the fault plane (Okada 1992) detailed in Table 2 in a Bayesian manner using the software slipBERI (availability detailed in acknowledgements) and using the same downsampled dataset upon which we performed the initial inversion to determine the fault geometry. We spatially regularise the inversion using the von Karman correlation function to constrain the slip to be self-similar (Amey et al. 2018) in light of many lines of evidence suggesting that fault slip is fractal (e.g. Mai and Beroza 2002;Candela et al. 2012;Milliner et al. 2015). ...
... In the inversion, von Karman autocorrelation is used as a prior probability to solve for a slip distribution that both fits the geodetic data and shows self-similarity (full details on method in Amey et al. 2018). Additionally, we solve for slip using a trans-dimensional inversion scheme in which we solve for the location of a slipping area as well as the magnitude (full details of method given in Amey et al. 2019). ...
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This study investigates the distribution of coseismic slip of the 19th August 2018 Mw 7.2 Lombok earthquakes, Indonesia, using Interferometric Synthetic Aperture Radar (InSAR) and Global Positioning System (GPS) data. Two look directions on ascending, with a maximum displacement of 27 cm and 35 cm, and one on descending of Sentinel-1 SAR data, with a maximum displacement of 12 cm, are used. In addition, static offsets from the GPS data, which are located at the most western part of the island and the northern part of the island, detect ~ 2 cm and ~ 5 cm coseismic displacement due to the earthquake. Using combined InSAR and GPS data, this study estimates the fault location, fault geometry and the coseismic slip distribution by a joint inversion using a trans-dimensional Bayesian method. This method solves for the contiguous area of the fault that is allowed to slip in the inversion whilst also solving for the magnitude. This dampens the spurious smoothing that can occur in distributed slip solutions, in particular with far-field geodetic data or deep sources. The maximum slip of ~ 3.5 m is located at deeper portion of the fault at ~ 21 km, adjacent to the epicenter of the earthquake. This study demonstrates that the coseismic slip of the 19th August 2018 earthquakes occurred on a structure further south towards Lombok, a parallel fault structure with Flores back arc thrust.
... Firstly, we used both ascending and descending C-band Sentinel-1 satellite images to obtain the coseismic displacement of the 2020 Nima earthquake, and correct the atmospheric phase noise with the generic atmospheric correction online service (GACOS) products. Then, we inverted the fault geometry and slip distribution by employing the Bayesian inversion method (Bagnardi and Hooper, 2018;Amey et al., 2018) with the constraint of the coseismic displacements. Finally, we discuss the potential earthquake hazards and source of extensional activities in the region by using the coseismic coulomb stress change analysis and interseismic GNSS measurements. ...
... In the second step, we use the Slip BayEsian Regularised Inversion (slipBERI) package based on the Bayesian method to inverse slip distribution (Amey et al., 2018). It takes the distributed slip characteristic of fault as a priori information and provides a physics-based von Karman regularization. ...
Article
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Normal faults and conjugate strike-slip faults have been considered to play an important role in response to the extension deformation of central Tibet. On 22 July 2020, a Mw 6.3 earthquake struck the Nima county, central Tibet, in China, which provides a rare opportunity to get insights into how the normal faults in central Tibet accommodates the east-west extension caused by the Indian-Eurasian convergence. In this study, the Sentinel-1 images are collected to measure the coseismic deformation associated with this 2020 event and image its slip distribution. To mitigate the atmospheric phase effects, the generic atmospheric correction online service (GACOS) model is used to correct the coseismic inteferograms. The final coseismic deformation results show mainly negative displacement with a maximum value of ∼ 30 cm in the line of sight (LOS) direction. After that, a Bayesian inversion method is used to invert the fault model. Our results reveal the optimal seismogenic fault of this event with a strike angle of 31.3°, a dip angle of 51.6°, and show that its slip distribution is dominated by normal slip with a maximum value of 2.55 m at a depth of 4.79–9.53 km, which suggest it’s a blind normal rupture with high east-trending dip angle. The total released geodetic moment is, equivalent to Mw 6.3. In addition, we analyze the Coulomb stress change due to the 2008 Gaize and this 2020 Nima events, suggesting the 2020 event should not be triggered by the 2008 event. Finally, we estimate an interseismic slip rate of 4.7±1.2 mm/yr on the Yibug Caka-RigainPun Co (YCRC) fault with published global navigation satellite system (GNSS) measurements. Given that these high frequent normal slip events but a low crustal extension rate of 4.7±1.2 mm/yr in this region, we speculate that the asthenosphere material upwelling should be also a possible reason for E-W extensional activities in central Tibet.
... This method is used to quantitatively assess the uncertainty of the prediction using the posterior probability density function (PDF) of model parameters while explicitly integrating prior information. Amey et al. (2018) applied the Bayesian inversion method to the inference of the slip distribution in the 2014 Napa Valley earthquake with self-similarity of the fault slip (Mai and Beroza 2002) as prior information. In addition, many previous studies have used Bayesian inversion (Sambridge and Mosegaard 2002;Fukuda and Johnson 2008;Dettmer et al. 2014;Ohno et al. 2022). ...
... We continued the HMC sampling with 20,000 samples because the parameters should satisfy both criteria; nevertheless, these figures indicate the convergence with 3000 samples. The burn-in samples are often determined using a trace plot of VR or posterior probability (Amey et al. 2018); therefore, we also removed the first 5% of the chain similarly. Although the observation error of GNSS may appear to result from the horizontal error being smaller than the vertical error, we assumed the standard deviation in likelihood, σ h and σ v , representing the same error value, σ h = σ v = 2 cm to simplify the inversion. ...
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A rapid source fault estimation and quantitative assessment of the uncertainty of the estimated model can elucidate the occurrence mechanism of earthquakes and inform disaster damage mitigation. The Bayesian statistical method that addresses the posterior distribution of unknowns using the Markov chain Monte Carlo (MCMC) method is significant for uncertainty assessment. The Metropolis–Hastings method, especially the Random walk Metropolis–Hastings (RWMH), has many applications, including coseismic fault estimation. However, RWMH exhibits a trade-off between the transition distance and the acceptance ratio of parameter transition candidates and requires a long mixing time, particularly in solving high-dimensional problems. This necessitates a more efficient Bayesian method. In this study, we developed a fault estimation algorithm using the Hamiltonian Monte Carlo (HMC) method, which is considered more efficient than the other MCMC method, but its applicability has not been sufficiently validated to estimate the coseismic fault for the first time. HMC can conduct sampling more intelligently with the gradient information of the posterior distribution. We applied our algorithm to the 2016 Kumamoto earthquake (M JMA 7.3), and its sampling converged in 2 × 10 ⁴ samples, including 1 × 10 ³ burn-in samples. The estimated models satisfactorily accounted for the input data; the variance reduction was approximately 88%, and the estimated fault parameters and event magnitude were consistent with those reported in previous studies. HMC could acquire similar results using only 2% of the RWMH chains. Moreover, the power spectral density (PSD) of each model parameter's Markov chain showed this method exhibited a low correlation with the subsequent sample and a long transition distance between samples. These results indicate HMC has advantages in terms of chain length than RWMH, expecting a more efficient estimation for a high-dimensional problem that requires a long mixing time or a problem using nonlinear Green’s function, which has a large computational cost. Graphical Abstract
... When the priors of each model parameter are consistent and independent, it is achieved by perturbing each parameter in m i , where m j is a uniformly distributed random value generated in the range [ −1,1], and m j is the maximum random walk step of each parameter m j . If a model parameter of the new model test exceeds the range of uniform prior probability, the value m j TRIAL * is replaced by (Amey et al. 2018 ;Bagnardi & Hooper 2018 ;Wei et al. 2023a ): ...
Article
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More accurate inversion of source fault geometry and slip parameters under the constraint of the Bayesian algorithm has become a research hotspot in the field of geodetic inversion in recent years. In nonlinear inversion, the determination of the weight ratio of the joint inversion of multi-source data is more complicated. In this context, this paper proposes a simple and easily generalized weighting method for inversion of source fault parameters by joint geodetic multi-source data under the Bayesian framework. This method determines the relative weight ratio of multi-source data by RMSE (Root Mean Square Error) value and can be extended to other nonlinear search algorithms. To verify the validity of the method in this paper, this paper first sets up four sets of simulated Downloaded from https seismic experiment schemes. The inversion results show that the joint inversion weighting method proposed in this paper has a significant decrease in the large residual value compared with the equal weight joint inversion and the single data source joint inversion method. The east-west deformation RMSE is 0.1458 mm, the north-south deformation RMSE is 0.2119 mm, and the vertical deformation RMSE is 0.2756 mm. The RMSE of the three directions is lower than that of other schemes, indicating that the proposed method is suitable for the joint inversion of source parameters under Bayesian algorithm. To further verify the applicability of the proposed method in complex earthquakes, the source parameters of the Maduo earthquake were inverted using the method of this paper. The focal depth of the inversion results in this paper is closer to the focal depth released by the GCMT agency. In terms of strike angle and dip angle, the joint inversion in this paper is also more inclined to the GCMT results. The joint inversion results generally conform to the characteristics of left-lateral strike-slip, which shows the adaptability of this method in complex earthquakes.
... Nevertheless, when a slip distribution features non-smooth patterns, the bias of estimating such a slip distribution with the smoothing assumption could be propagated into the estimation of the fault geometry. To better characterize a slip distribution, some other regularization approaches for slip distribution have been proposed, including, but not limited to, sparsity-promoting solutions (Evans & Meade, 2012;Hallo & Gallovič, 2020;Tomita et al., 2021) and von Karman regularization capturing the fractal properties of a fault (Amey et al., 2018). The sparsity-promoting solutions are mostly applied on planar fault geometry, and are difficult to use on non-planar fault geometry, and von Karman regularization depends on the setting of the correlation parameters. ...
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A precise finite‐fault model including the fault geometry and slip distribution is essential to understand the physics of an earthquake. However, the conventional linear inversion of geodetic data for a finite‐fault model cannot fully resolve the fault geometry. In this study, we developed a Bayesian inversion framework that can comprehensively solve a non‐planar fault geometry, the corresponding fault slip distribution with spatially variable directions, and objective weighting for multiple data types. In the proposed framework, the probability distributions of all the model parameters are sampled using the Monte Carlo method. The developed methodology removes the requirement for manual intervention for the fault geometry and data weighting and can provide an ensemble of plausible model parameters. The performance of the developed method is tested and demonstrated through inversions for synthetic oblique‐slip faulting models. The results show that the constant rake assumption can significantly bias the estimates of fault geometry and data weighting, whereas additional consideration of the variability of slip orientations can allow plausible estimates of a non‐planar fault geometry with objective data weighting. We applied the method to the 2013 Mw 6.5 Lushan earthquake in Sichuan province, China. The result reveals dominant thrust slips with left‐lateral components and a curved fault geometry, with the confidence interval of the dip angles being between 20°–25° and 56°–58°. The proposed method provides useful insights into the scope of imaging a non‐planar fault geometry, and could help to interpret more complex earthquake sources in the future.
... The profile-parallel components were used to constrain the kinematic parameters (dip-slip rate, locking depth, and fault dip) based on an edge dislocation model (Savage, 1983). A Bayesian approach that uses a Markov Chain Monte Carlo (MCMC) sampling technique was adopted to invert the fault parameters (Amey et al., 2018). For the Mishmi thrust, the estimated dip-slip rate decreases from ~21 mm/yr to ~17 mm/yr, and the optimal locking depth of the Mishmi thrust decreases from ~27 km to ~22 km after correcting the viscoelastic deformation effects (Fig. 10c~e). ...
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The Eastern Himalayan Syntaxis (EHS) played important role in accommodating the intracontinental deformation and material flow in southeastern Tibet. Quantifying the regional kinematics around the EHS is critical for understanding the deformation mechanism of the Tibetan Plateau. In this study, we compiled all available GPS measurements and constructed an elastic block model to analyze the fault kinematics around the EHS. The results suggest that the deformation characteristics around the EHS can be well explained by a linear spherical block model that considers the intra-block homogeneous strain. The fault slip rates from the block model agree well with the late Quaternary slip rates, suggesting that the GPS-derived slip rates can generally represent the long-term slip rates. However, some relatively large residuals, especially in the frontal EHS region, are still detectable, suggesting that various other factors, such as the postseismic viscoelastic relaxation, may also influence the deformation. We simulated the postseismic viscoelastic relaxation deformation caused by the 1950 Mw 8.7 Assam earthquake endured as the largest intracontinental earthquake ever recorded. The results indicate significant postseismic viscoelastic deformation due to the Assam earthquake even after 65 years. The interseismic convergence rates across the Mishmi thrust and the Arunachal Himalaya derived from pure elastic solution may be overestimated if ignoring the viscoelastic relaxation effects. The spatial variations of convergence rates across the central and eastern Himalayas could be partly related to the transient deformation following great Himalayan earthquakes.
... To recapitulate, there are many variations of Bayesian inversion methods applied to slip or slip deficit estimates. Basically, these range from hyper-constrained least square methods with Thikhonov regularization (e.g., Mazzotti et al., 2001;Métois et al., 2013Métois et al., , 2016Moreno et al., 2010;Ortega-Culaciati et al., 2021;Yáñez-Cuadra et al., 2022) to fully Bayesian sampling approaches (e.g., Amey et al., 2018;Dal Zilio et al., 2020;Jolivet et al., 2015). While the first approach is subject to a strong bias, the second approach is computationally more expensive and does not give an analytical solution, hence they do not provide adequate estimates of evidence. ...
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Inversions of geodetic data are regularly used to estimate interseismic locking in subduction zones. However, the ill‐posed nature of these problems motivates us to include prior information, physically consistent with processes of the subduction seismic cycle. To deal with model instabilities, we present an inversion method to estimate both plate‐locking and model uncertainties by inverting Global Navigation Satellite System derived velocities based on a Bayesian model selection scheme. Our method allows us to impose positivity constraints via a multivariate folded‐normal distribution, with a specified covariance matrix. Model spatial correlations are explored and ranked to find models that best explain the observed data and for a better understanding of locking models. This approach searches for hyperparameters of the prior joint multivariate probability density function (PDF) of model parameters that minimize the Akaike Bayesian Information Criterion (ABIC). To validate our approach, we invert synthetic displacements from analytic models, yielding satisfactory results. We then apply the method to estimate the plate‐locking in Central Chile (28°–39°S) and its relation to the coseismic slip distribution of earthquakes with magnitudes Mw > 8.0, on the subduction zone since 2010. We also search among different prior PDFs for a single ductile‐fragile limit depth. Our results confirms a spatial correlation between locked asperities and the 2010 Mw 8.8 Maule and 2015 Mw 8.3 Illapel earthquake rupture zones. The robustness of our locking model shows potential to improve future seismic and tsunami hazard estimations.
... Although we focused on the estimation using a weakly informative prior PDF for the slip distribution, the accurate consideration of model uncertainty that the method allows for should also be effective in estimations introducing strong prior PDFs. Moreover, by taking a fully Bayesian approach, the method can be flexibly combined with not only the widely used constraints such as the smoothing approach but also recently proposed sophisticated implicit (e.g., trans-dimensional inversion: Dettmer et al., 2014) and explicit (e.g., von Karman regularization: Amey et al., 2018Amey et al., , 2019 regularization schemes, which is expected to increase the quality of estimation. The approach of generating multiple models of the underground structure can be improved further by focusing on the genuine estimation errors of underground structure models, which remains as an important future challenge. ...
Article
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We consider a Bayesian multi‐model fault slip estimation (BMMFSE), which incorporates many candidates of the underground structure (Earth structure and plate boundary geometry) model characterized by a prior probability density function (PDF). The technique is used to study long‐term slow slip events (L‐SSEs) that occurred beneath the Bungo Channel, southwest Japan, in around 2010 and 2018. We here focus on the two advantages of BMMFSE: First, it allows for estimating slip distribution without introducing relatively strong prior information such as smoothing constraints, by combining a fully Bayesian inference and better consideration of model uncertainty to avoid overfitting. Second, the posterior PDF for the underground structure is also obtained during the fault slip estimation, which can be used as priors for the estimation of slip distribution for recurring events. The estimated slip distribution obtained using BMMFSE agreed better with the distribution of deep tectonic tremors at the down‐dip side of the main rupture area than those based on stronger prior constraints when the corresponding Coulomb failure stress changes are compared. This finding suggests a mechanical relationship between the L‐SSE and the synchronized tremors. The use of the posterior PDF of the underground structure estimated for the 2010 L‐SSE as prior PDF for the 2018 event resulted in more consistent estimation with the data, indicated by a smaller value of an information criterion.
... The inversion algorithm takes into account errors in the data and prior information on model parameters, and aims to rapidly estimate optimal model parameters ( Figure S1) and associated uncertainties through efficient sampling of the posterior PDFs (probability density functions). Such sampling is performed using an MCMC ( Markov chain Monte Carlo) method incorporating the Metropolis-Hastings algorithm and with an automatic step-size selection 39 . www.nature.com/scientificreports/ ...
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The 2013 Lushan Ms 7.0 earthquake occurred on the Longmenshan thrust tectonic zone, a typical blind reverse-fault type earthquake that caused the death of nearly 200 people. The investigation of the fault geometry and fault slip distribution of this earthquake is important for understanding the seismogenic tectonic type and seismic activity mechanism of the Longmenshan Fault Zone. In this paper, for the fault geometry of the Ms 7.0 earthquake in Lushan, the geometric parameters of the planar fault are inverted based on the rectangular dislocation model using GPS coseismic displacement data, and on this basis, a curved fault steeply-dipping on top and gently-dipping at depth is constructed by combining the aftershock distribution. The GPS and leveling data are used to invert the slip distribution of the curved fault for the Lushan Ms 7.0 earthquake. The results show that the fault is dominated by reverse slip with a small amount of sinistral rotation, and there is a peak slip zone with a maximum slip of 0.98 m, which corresponds to a depth of ~ 13.50 km, and the energy released is 1.05 × 10 ¹⁹ N/m with a moment magnitude of Mw 6.63. Compared with the planar rectangular dislocation model, the curved fault model constructed by using triangular dislocation elements can not only better approximate the fault slip, but also better explain the observed surface displacement, and the root mean square error of the GPS and leveling data fitting is reduced by 1.3 mm and 1.9 mm, respectively. Both the maximum slip and moment magnitude of the fault based on the inversion of the curved structure are slightly larger than the results based on the planar structure.
... Over the past years, many efforts have been made to improve the efficiency and accuracy of the Bayesian slip inversion, and some progress such as using another form of regularizations (Amey et al. 2018), extending the parameter model to simultaneously estimate the rupture area (Cambiotti et al. 2017), or developing more efficient sampling strategies (Minson et al. 2013;Dettmer et al. 2014) has been achieved. Nevertheless, these improvements mainly focus on the MCMC method and are still confined to the fundamental sampling technique. ...
Article
The Bayesian slip inversion offers a powerful tool for modeling the earthquake source mechanism. It can provide a fully probabilistic result and thus permits us to quantitively assess the inversion uncertainty. The Bayesian problem is usually solved with Monte Carlo methods, but they are computationally expensive and are inapplicable for high-dimensional and large-scale problems. Variational inference is an alternative solver to the Bayesian problem. It turns Bayesian inference into an optimization task and thus enjoys better computational performances. In this study, we introduce a general variational inference algorithm, automatic differentiation variational inference (ADVI), to the Bayesian slip inversion and compare it with the classic Metropolis-Hastings (MH) sampling method. The synthetic test shows that the two methods generate nearly identical mean slip distributions and standard deviation maps. In the real case study, the two methods produce highly consistent mean slip distributions, but the ADVI-derived standard deviation map differs from that produced by the MH method, possibly because of the limitation of the Gaussian approximation in the ADVI method. In both cases, ADVI can give comparable results to the MH method but with a significantly lower computational cost. Our results show that ADVI is a promising and competitive method for the Bayesian slip inversion.
... For the linear finite slip model inversion, we assumed that the slips on adjacent patches were continuous to avoid an unrealistic stress drop (Amey et al., 2018). Therefore, we regularized the linear inversion by applying the Laplacian constraint to stabilize the distributed slip (1 × 1 km patches). ...
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The extreme rainfall weather during the Indian monsoon season has increased in recent decades changing the hydrological mass distribution that may be responsible for triggering regional earthquakes. In the western Himalayas, a shallow Mw 5.7 earthquake took place right after the withdrawal of the 2019 unusual Indian summer monsoon resulting in a rapid increase of ~4.5 km³ of water storage at the nearby Mangla reservoir within three months. Through a joint inversion of space geodetic and teleseismic data, we found that the mainshock occurred on the décollement structure of the Main Himalayan Thrust in the western Himalayas, which was previously underdetermined. The significant water loading, and pore pressure diffusion led to Coulomb stress increases of ~10 kPa on the maximum coseismic slip zone promoting fault failure. Our findings demonstrate that climate change could influence certain reservoir-associated earthquakes in the Himalayas. We provide recommendations to improve the regulations for the reservoir operations in Mangla and probably other contexts with similar tectonic settings in the Himalayas during the monsoon season.
... Determining the distribution of coseismic slip is helpful to reveal numerous properties of seismic events, including the fault geometry, stress changes, friction properties, and potential seismic hazards Amey et al., 2018;Zhao et al., 2018). According to the optimal fault geometry parameters obtained from the above Bayesian estimation, we further extend the fault plane to 30 km and 20 km along the strike and downdip directions, respectively. ...
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The ENE striking Longmu Co fault and the North Altyn Tagh left-lateral slip fault have led to the complex regional structure in the northwestern Tibetan Plateau, resulting in a series of normal faulting and strike slip faulting earthquakes. Using both the ascending and descending Sentinel-1A/B radar images, we depict the coseismic deformation caused by the 2020 Yutian Mw 6.4 earthquake with a peak subsidence of ~ 20 cm. We determine the seismogenic fault geometry by applying the Bayesian approach with a Markov Chain Monte Carlo sampling method, which can better characterize the posterior probability density functions of the source model parameters. The estimation results reveal that the earthquake is a normal faulting event with a moderate strike slip component. Based on the optimal fault geometry model, we extend the fault plane and invert for the distributed coseismic slip model. The optimal slip model shows that the coseismic slip is mainly concentrated at shallow depths of 3–10 km with a maximum slip of ~ 1.0 m. Our preferred geodetic coseismic model exhibits no surface rupture, which may likely be due to the shallow slip deficit in the uppermost crust. We calculate the combined loading effect of the Coulomb failure stress changes induced by the coseismic dislocations and postseismic viscoelastic relaxation of the 2008 Mw 7.1, 2012 Mw 6.4 and 2014 Mw 6.9 Yutian events. Our study demonstrates that the three preceding major Yutian shocks were insufficient to trigger the 2020 Yutian earthquake, which we consider perhaps reflects the natural release of elastic strain accumulated mainly through localized tectonic movement. We attribute the 2020 Yutian event to the release of extensional stress in a stepover zone controlled by the Longmu Co and the North Altyn Tagh sinistral strike slip fault systems. The seismic risk in the southwest end of the North Altyn Tagh fault has been elevated by the Yutian earthquake sequences, which require future attention.
... One approach is that the number and size of sub-faults are optimized based on the spatial resolution of observational data prior to the MCMC sampling (Kimura et al., 2019); however, a spatially smooth fault slip distribution cannot be resolved by this method. Another idea is the introduction of a complicated regularization such as von Karman regularization, which assumes the slip distribution should show fractal properties (e.g., Amey et al., 2018). ...
Article
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Geodetic fault slip inversions have generally been performed by employing a least squares method with a spatially uniform smoothing constraint. However, this conventional method has various problems: difficulty in strictly estimating non‐negative solutions, assumption that unknowns follow the Gaussian distributions, unsuitability for expressing spatially non‐uniform slip distributions, and high calculation cost for optimizing many hyper‐parameters. Here, we have developed a trans‐dimensional geodetic slip inversion method using the reversible‐jump Markov chain Monte Carlo (rj‐MCMC) technique to overcome these problems. Because sub‐fault locations were parameterized by the Voronoi partition and were optimized in our approach, we can estimate a slip distribution without the need for spatially uniform smoothing constraints. Moreover, we introduced scaling factors for observational errors. We applied the method to the synthetic data and the actual geodetic observational data associated with the 2011 Tohoku‐oki earthquake and found that the method successfully reproduced the target slip distributions including a spatially non‐uniform slip distribution. The method provided posterior probability distributions with the unknowns, which can express a non‐Gaussian distribution such as large slip with low probability. The estimated scaling factors properly adjusted the initial observational errors and provided a reasonable slip distribution. Additionally, we found that checkerboard resolution tests were useful to consider sensitivity of the observational data for performing the rj‐MCMC method. It is concluded that the developed method is a powerful technique to solve the problems of the conventional inversion method and to flexibly express fault‐slip distributions considering the complicated uncertainties.
... For example, the introduction of a type of sparsitypromoting constraint to an estimation of slip distribution in long-term slow slip events in the Nankai Trough region in south-west Japan resulted in significantly different up-and down-dip limits for slip distribution from those obtained using smoothness constraints (Nakata et al. 2017). More sophisticated implicit (e.g., trans-dimensional inversion (Dettmer et al. 2014)) and explicit (e.g., von Karman regularization (Amey et al. 2018(Amey et al. , 2019) regularization schemes have been introduced to fault slip estimations, and it is reported that these methods significantly outperformed the traditional method based on smoothing constraint in flexibility. ...
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The model prediction errors that originate from the uncertainty of underground structure is often a major contributor of the errors between the data and the model predictions in fault slip estimation using geodetic or seismic waveform data. However, most studies on slip inversions either neglect the model prediction errors or do not distinguish them from observation errors. Several methods that explicitly incorporated the model prediction errors in slip estimation, which has been proposed in the past decade, commonly assumed a Gaussian distribution for the stochastic property of the model prediction errors to simplify the formulation. Moreover, the information on both the slip distribution and the underground structure is expected to be successfully extracted from the data by incorporating the stochastic property of the model prediction errors. In this study, we propose a novel flexible Bayesian inference framework for estimating fault slips that can accurately incorporate non-Gaussian model prediction errors. This method considers the uncertainty of the underground structure, including fault geometry, based on the ensemble modeling of the uncertainty of Green’s functions. Furthermore, the framework allows the estimation of the posterior probability density function (PDF) of the parameters of the underground structure, by calculating the likelihood of each sample in the ensemble. We performed numerical experiments for estimating the slip deficit rate (SDR) distribution on a 2D thrust fault using synthetic data of surface displacement rates, which is the simplest problem setting that can essentially demonstrates the fundamental idea and validate the advantage of the proposed method. In the experiments, the dip angle of the fault plane was the parameter used to characterize the underground structure. The proposed method succeeded in estimating a posterior PDF of SDR that is consistent with the true one, despite the uncertain and inaccurate information of the dip angle. In addition, the method could estimate a posterior PDF of the dip angle that has a strong peak near the true angle. In contrast, the estimation results obtained using a conventional approach, which introduces regularization based on smoothing constraints and does not explicitly distinguish the prediction and observation errors, included a significant amount of bias, which was not noticed in the results obtained using the proposed method. The estimation results obtained using different settings of the parameters suggested that inaccurate prior information of the underground structure with a small variance possibly results in significant bias in the estimated PDFs, particularly the posterior PDFs for SDR, those for the underground structure, and the posterior predicted PDF of the displacement rates. The distribution shapes of the model prediction errors for the representative model parameters in certain observation points are significantly asymmetric with large absolute values of the sample skewness, suggesting that they would not be well-modeled by Gaussian approximations.
... Prior information is incorporated using probability density functions (pdf) to determine the posterior distribution, which quantifies our inference's uncertainty. A prior distribution is established through its interpretation as a modeling device of the probabilistic prior knowledge available on source movements and imposing model restrictions using truncated pdfs [6,7,8,9]. ...
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We propose an efficient Bayesian approach to infer a fault displacement from geodetic data in a slow slip event. Our physical model of the slip process reduces to a multiple linear regression subject to constraints. Assuming a Gaussian model for the geodetic data and considering a truncated multivariate normal prior distribution for the unknown fault slip, the resulting posterior distribution is also truncated multivariate normal. Regarding the posterior, we propose an algorithm based on Optimal Directional Gibbs that allows us to efficiently sample from the resulting high-dimensional posterior distribution of along dip and along strike movements of our fault grid division. A synthetic fault slip example illustrates the flexibility and accuracy of the proposed approach. The methodology is also applied to a real data set, for the 2006 Guerrero, Mexico, Slow Slip Event, where the objective is to recover the fault slip on a known interface that produces displacements observed at ground geodetic stations. As a by-product of our approach, we are able to estimate moment magnitude for the 2006 Guerrero Event with uncertainty quantification.
... Inversion of geodetic data for the spatial distribution of slip on a fault is also subject to fundamental limitations, notably due to the St. Venant principle that implies a decreasing resolution with increasing distance between source and observations. However, the deployment of increasingly large and dense geodetic observatories, the development of better analytic standards in inverse theory (Aster et al., 2012;Funning et al., 2014;Fukahata & Wright, 2008;Hang et al., 2020;Nocquet, 2018;Yabuki & Matsu'ura, 1992), and the joint inversion of complementary data sets, both geodetic and seismological, has increased the accuracy of slip distributions (Atzori & Antonioli, 2011;Amey et al., 2018;Barbot et al., 2013;Duputel et al., 2014;DeVries et al., 2017;Evans & Meade, 2012;Gombert et al., 2017Gombert et al., , 2018McGuire & Segall, 2003;Minson et al., 2014;Sathiakumar et al., 2017). For example, the large uncertainties associated with shallow slip near the trench during the 2011 Mw = 9.1 Tohoku, Japan, earthquake were largely reduced by considering tsunami data (e.g., Bletery et al., 2014;Jiang & Simons, 2016;Yamazaki et al., 2011). ...
Article
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The source characteristics of slow and fast earthquakes provide a window into the mechanical properties of faults. In particular, the average stress drop controls the evolution of friction, fault slip, and event magnitude. However, this important source property is typically inferred from the analysis of seismic waves and is subject to many epistemic uncertainties. Here, we investigate the source properties of 53 earthquakes and 17 slow‐slip events on thrust and strike‐slip faults in various tectonic settings using slip distributions constrained by geodesy in combination with other data. We determine the width, potency, and potency density of slow and fast earthquake sources based on static slip distributions. The potency density, defined conceptually as the ratio of average slip to rupture radius, is a measure of anelastic deformation with limited bias from rigidity differences across depths and tectonic settings. Strike‐slip earthquakes have the highest potency density, varying from 20 to 500 microstrain. The potency density is on average lower on continental thrust faults and megathrusts, from 10 to 200 microstrain, with an algebraic decrease with centroid depth, indicative of systematic changes in dominant rupture processes with depth. Slow slip events represent an end‐member style of rupture with low potency density and large rupture width. Significant variability in potency density of slow‐slip events affects their moment‐duration scaling. The variations of source properties across tectonic settings, depth, and rupture styles can be used to better constrain numerical simulations of seismicity and to assess the source characteristics of future earthquakes and slow slip events.
... The distribution of slip on the fault can be smooth or rough depending on the rupture physics, so we only allow as much roughness as required by the data (Shirzaei & Bürgmann 2013;Amey et al. 2018;Marchandon et al. 2018). We also require a relative uniform direction of fault slip, so we penalize slip in directions orthogonal to the overall dip direction and we strictly forbid slip that deviates more than 45 • from the dip direction (Barbot et al. 2013). ...
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Inverse problems play a central role in data analysis across the fields of science. Many techniques and algorithms provide parameter estimation including the best-fitting model and the parameters statistics. Here, we concern ourselves with the robustness of parameter estimation under constraints, with the focus on assimilation of noisy data with potential outliers, a situation all too familiar in Earth science, particularly in analysis of remote-sensing data. We assume a linear, or linearized, forward model relating the model parameters to multiple data sets with a priori unknown uncertainties that are left to be characterized. This is relevant for global navigation satellite system and synthetic aperture radar data that involve intricate processing for which uncertainty estimation is not available. The model is constrained by additional equalities and inequalities resulting from the physics of the problem, but the weights of equalities are unknown. We formulate the problem from a Bayesian perspective with non-informative priors. The posterior distribution of the model parameters, weights and outliers conditioned on the observations are then inferred via Gibbs sampling. We demonstrate the practical utility of the method based on a set of challenging inverse problems with both synthetic and real space-geodetic data associated with earthquakes and nuclear explosions. We provide the associated computer codes and expect the approach to be of practical interest for a wide range of applications.
... Thus, for an arbitrary C p one can choose L as the Cholesky factorization of its inverse. This property can be used, for instance, to realize more general covariances such as Von Karman regularization, recently employed by Amey et al. (2018). The posterior PDF can then be computed from eqs (2) and (3) using Bayes' theorem (eq. 1) as ...
Article
Obtaining slip distributions for earthquakes results in an ill-posed inverse problem. While this implies that only limited and uncertain information can be recovered from the data, inferences are typically made based only on a single regularized model. Here, we develop an inversion approach that can quantify uncertainties in a Bayesian probabilistic framework for the finite fault inversion (FFI) problem. The approach is suitably efficient for rapid source characterization and includes positivity constraints for model parameters, a common practice in FFI, via coordinate transformation to logarithmic space. The resulting inverse problem is nonlinear and the most probable solution can be obtained by iterative linearization. In addition, model uncertainties are quantified by approximating the posterior probability distribution by a Gaussian distribution in logarithmic space. This procedure is straightforward since an analytic expression for the Hessian of the objective function is obtained. In addition to positivity, we apply smoothness regularization to the model in logarithmic space. Simulations based on surface wave data show that smoothing in logarithmic space penalizes abrupt slip changes less than smoothing in linear space. Even so, the main slip features of models that are smooth in linear space are recovered well with logarithmic smoothing. Our synthetic experiments also show that, for the data set we consider, uncertainty is low at the shallow portion of the fault and increases with depth. In addition, a simulation with a large station azimuthal gap of 180° significantly increases the slip uncertainties. Further, the marginal posterior probabilities obtained from our approximate method are compared with numerical Markov Chain Monte Carlo sampling. We conclude that the Gaussian approximation is reasonable and meaningful inferences can be obtained from it. Finally, we apply the new approach to observed surface wave records from the great Illapel earthquake (Chile, 2015, Mw = 8.3). The location and amplitude of our inferred peak slip is consistent with other published solutions but the spatial slip distribution is more compact, likely because of the logarithmic regularization. We also find a minor slip patch downdip, mainly in an oblique direction, which is poorly resolved compared to the main slip patch and may be an artefact. We conclude that quantifying uncertainties of finite slip models is crucial for their meaningful interpretation, and therefore rapid uncertainty quantification can be critical if such models are to be used for emergency response.
... The acceptance of the trial sample is based on the ratio of the likelihoods (Equation 3) of two samples (p trial /p current ). The trial sample is accepted as a sample of the posterior density distribution if the ratio is greater than a random value drawn between 0 and 1 (Amey et al., 2018;Minson et al., 2013). The algorithm is tuned during the initial burn-in stage, in order to achieve an optimal acceptance rate (i.e., the ratio between the accepted samples and the total probed samples) (∼15-25%) and only post burn-in samples are considered representative samples of the posterior distribution (Bagnardi & Hooper, 2018). ...
Conference Paper
We focus on the eruption of Mt. Etna which took place on 24 December, 2018. The eruption occurred after a month of unrest and was accompanied by a seismic swarm that culminated in the M4.9 earthquake on the 26th, with epicentre on the eastern flank of the volcano. We jointly analyse ground deformation and gravity data to estimate the geometrical and kinematic parameters of the source structure, together with the density of the intruding material. The data used in this study were recorded by stations in the INGV-OE monitoring network (21 GPS stations and 2 gravity stations equipped with superconducting gravimeters), during the interval of 23 to 28 December (pre to post eruption). We assume a dike-type source for the forward calculation in the defined objective function. A pattern search algorithm (PSA) is used for the iterative minimization of the misfit error. In order to estimate the posterior probability density function (PDFs) of the model parameters, we also use a Markov Chain Monte Carlo (MCMC) approach. Indeed, the calculated PDFs provide more information about the uncertainties of the model parameters, which helps to understand overall tendencies of the solutions. We first test the constrained inversion of the gravity data, to calculate the density of eruptive magmatic body, by fixing the geometrical parameters of the dike, previously retrieved through inversion of the deformation data only. Using this approach, it is possible to suitable explain the deformation data and the gravity change observed at the station in the near field (MNT), while the gravity change at the other station (SLN) remain unexplained. We then invert jointly both deformation and gravity datasets, in order to adequately fit all the observations. The final model gives a density value of ~1.8-2.0 g/cm3. This value is significantly lower than the density of bubble-free magma and indicates either the involvement of gas in the intrusive process, or the formation of dry fissures during the emplacement of the dyke.
... EO derived measurements of coseismic ground deformation (coupled with other a priori constraints from seismological, optical, topographic and field mapping) can be used as inputs for geophysical inverse models (e.g., Elliott et al. 2016). These models aim to approximate the Earth, typically as an elastic medium in a half-space, from which the parameters of the earthquake source (Okada 1985) such as fault location and orientation (e.g., Bagnardi and Hooper 2018) and the distribution of slip are determined (e.g., Simons et al. 2002;Feng et al. 2010;Amey et al. 2018). More complicated models can include other rheologies such as viscoelastic processes (e.g., Pollitz et al. 2000), in particular when looking at postseismic deformation (e.g., Deng et al. 1998). ...
Article
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In this paper, we illustrate some of the current methods for the exploitation of data from Earth Observing satellites to measure and understand earthquakes and shallow crustal tectonics. The aim of applying such methods to Earth Observation data is to improve our knowledge of the active fault sources that generate earthquake shaking hazards. We provide examples of the use of Earth Observation, including the measurement and modelling of earthquake deformation processes and the earthquake cycle using both radar and optical imagery. We also highlight the importance of combining these orbiting satellite datasets with airborne, in situ and ground-based geophysical measurements to fully characterise the spatial and timescale of temporal scales of the triggering of earthquakes from an example of surface water loading. Finally, we conclude with an outlook on the anticipated shift from the more established method of observing earthquakes to the systematic measurement of the longer-term accumulation of crustal strain.
... To tackle the multiplicity and improve the stability of the inverse problem, the Bayes inversion framework is widely used by many researchers [12], [13], which combines the prior information with the conditional probability. Thus, the posterior probability distribution suitable for different situations can be obtained by considering the prior information as a Gaussian, Cauchy, or Laplacian distribution [14], [15]. Other methods such as L 1 norm and L 2 norm, which can solve the ill-posed problem by adding regularization terms, are also widely researched [16]. ...
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Prestack amplitude variation with offset (AVO) inversion can provide abundant reservoir information underground, which is always implemented trace-by-trace. However, it cannot guarantee the lateral accuracy of the inversion results. To utilize the lateral difference of the angle gathers and improve the lateral resolution, the difference angle gathers are introduced. Based on the Bayes inversion framework, the objective function considering the difference angle gathers is first constructed. Then, the effect of difference angle gathers on inversion results is analyzed, which is essential to improve the accuracy of the inversion results. To further figure out the applicable conditions of difference angle gathers, different forward operators are analyzed including a nonlinear operator and a linear operator. The used nonlinear operator is the exact Zoeppritz's equation. The linear operator is a linear perturbation equation based on the elastic inverse-scattering theory. Due to the difference of angle gathers in adjacent traces, the linear forward operator may cause a deviation of the updated parameters. By comparison, the exact Zoeppritz's equation as the nonlinear forward operator has better applicability and precision. Based on the proposed method, the elastic parameters are obtained from seismic data. Numerical examples show that the inverted elastic parameters of the proposed method have a higher horizontal resolution, and the details in the inversion profile can be better highlighted.
... The acceptance of the trial sample is based on the ratio of the likelihoods (Equation 3) of two samples (p trial /p current ). The trial sample is accepted as a sample of the posterior density distribution if the ratio is greater than a random value drawn between 0 and 1 (Amey et al., 2018;Minson et al., 2013). The algorithm is tuned during the initial burn-in stage, in order to achieve an optimal acceptance rate (i.e., the ratio between the accepted samples and the total probed samples) (∼15-25%) and only post burn-in samples are considered representative samples of the posterior distribution (Bagnardi & Hooper, 2018). ...
Article
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Plain Language Summary Volcanic eruptive activity is often preceded and accompanied by ground deformation and changes in the local gravity field, in turn driven by the dynamics of magma in the plumbing system of the volcano. The analysis of deformation and gravity data can provide unique information on intrusive events leading to paroxysmal activity. Here, we use ground deformation andgravity data to shed new light on the December 2018 eruption of Etna, which ensued from fissures in the upper SE flank of the volcano. Data inversion indicates that the eruption was fed by the intrusion of light magma along a sheet‐like structure, cutting the SE sector of the volcanic edifice. However, some features of the available observations suggest that the mechanism, which led to the eruptive event involved, besides the elastic deformation of the volcanic edifice in response to the magma intrusion, inelastic accommodation of the strain, and formation of dry fractures.
Article
When conducting coseismic slip distribution inversion with interferometric synthetic aperture radar (InSAR) data, there is no universal method to objectively determine the appropriate size of InSAR data. Currently, little is also known about the computing efficiency of variance component estimation implemented in the inversion. Therefore, we develop a variance component adaptive estimation algorithm to determine the optimal sampling number of InSAR data for the slip distribution inversion. We derived more concise variation formulae than conventional simplified formulae for the variance component estimation. Based on multiple sampling data sets with different sampling numbers, the proposed algorithm determines the optimal sampling number by the changing behaviors of variance component estimates themselves. In three simulation cases, four evaluation indicators at low levels corresponding to the obtained optimal sampling number validate the feasibility and effectiveness of the proposed algorithm. Compared with the conventional slip distribution inversion strategy with the standard downsampling algorithm, the simulation cases and practical applications of five earthquakes suggest that the developed algorithm is more flexible and robust to yield appropriate size of InSAR data, thus provide a reasonable estimate of slip distribution. Computation time analyses indicate that the computational advantage of variation formulae is dependent of the ratio of the number of data to the number of fault patches and can be effectively suitable for cases with the ratio smaller than five, facilitating the rapid estimation of coseismic slip distribution inversion.
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Quantification of non‐uniqueness and uncertainty is important for transient electromagnetism (TEM). To address this issue, we develop a trans‐dimensional Bayesian inversion schema for TEM data interpretation. The trans‐dimensional posterior probability density (PPD) offers a solution to model selection and quantifies parameter uncertainty resulting from the model selection from all possible models rather than determining a single model. We use the reversible‐jump Markov chain Monte Carlo sampler to draw ensembles of models to approximate PPD. In addition to providing reasonable model selection, we address the reliability of the inversion results for uncertainty analysis. This strategy offers reasonable guidance when interpreting the inversion results. We make the following improvements in this paper. First, in terms of algorithmic acceleration, we use the nonlinear optimization inversion results as the initial model and implement the multi‐chain parallel method. Second, we develop double factors to control the sampling step size of the proposed distribution, so that the sampling models cover the high‐probability region of the parameter space as much as possible. Finally, we provide the potential scale reduction factor‐ η convergence criteria to assess the convergence of the samples and ensure the rationality of the output models. The proposed methodology is first tested on synthetic data and subsequently applied to a field dataset. The TEM inversion results show that probability inversion can provide reliable references for data interpretation through uncertainty analysis.
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Earthquake source parameters can be estimated using seismological observations, but the identification of the fault responsible is often complicated by location uncertainties and the inherent ambiguity between nodal planes. Satellite Interferometric Synthetic Aperture Radar (InSAR) can be used to observe ground deformation and model fault geometry but is limited by climate conditions (water vapour) and ground coverage (dense vegetation). In the tropics, the atmosphere is dynamic and most regions are densely vegetated, making detecting coseismic deformation challenging. Here, we perform a systematic inspection of coseismic interferograms from Sentinel-1 SAR images, to assess their suitability for detecting coseismic deformation in Costa Rica. Using data from the seismological network, we target seven earthquakes between 2016 and 2020 with depths 20\le \, 20 km and magnitudes Mw 5.3–6.2. For each event, we use the seismic parameters to compute line-of-sight displacements for ascending and descending geometries and for both nodal planes and generate 12- and 24-d coseismic interferograms where available. We obtain interferograms with coseismic displacement signals for three of the seven earthquakes. We invert the geodetic data to retrieve the earthquake source parameters but the lack of offshore geodetic coverage causes trade-offs between parameters and large uncertainties. The Jacó and Golfito earthquakes likely occurred on the subduction interface and the geodetic locations were 6–9 km closer to the coast than previous seismic estimates. The Burica earthquake occurred on a shallow steeply dipping thrust fault in the outer forearc. For the other earthquakes, no coseismic deformation was detected due to atmospheric noise or poor coherence. These results demonstrate the suitability of 12-d Sentinel-1 interferograms for monitoring shallow earthquakes of magnitude > Mw 5.7 in Central America. This approach can be used to begin a surface deformation catalogue for the region, which will ultimately help improve the understanding of active deformation processes and improve hazard maps.
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With the rapid advancements in synthetic aperture radar (SAR) satellites and associated processing algorithms over recent decades, interferometric SAR (InSAR) has emerged as a routine method for monitoring large-scale ground deformation and interpreting geophysical processes. Statistical inference serves as a major component in InSAR technique developments and applications. This article provides an overview of InSAR deformation measurement and InSAR-constrained geophysical inversion, using a statistical inference point of view. Its objectives are to facilitate understanding of the method by addressing its underlying mathematical challenges. We begin by introducing the concept of statistical inference and the structure of our content organization framework. Next, we investigate the distinct concerns associated with statistical inference in InSAR deformation measurement and InSAR-constrained geophysical inversion. Finally, we propose several significant directions for future research. Table 1 includes abbreviations used throughout this article. Additionally, we highlight relevant resources, such as mathematical background, open source codes, and data repositories, in an appendix, which is available as supplementary material at https://doi.org/10.1109/MGRS.2023.3344159 .
Preprint
Inferring from the occurrence pattern of slow slip events (SSEs) the probability of triggering a damaging earthquake within the nearby velocity weakening portion of the plate interface is critical for hazard mitigation. Although robust methods exist to detect long-term SSEs consistently and efficiently, detecting short-term SSEs remains a challenge. In this study, we propose a novel statistical approach, called singular spectrum analysis isolate-detect (SSAID), for automatically estimating the start and end times of short-term SSEs in GPS data. The method recasts the problem of detecting SSEs as that of detecting change-points in a piecewise signal. This is achieved by obscuring the deviation from piecewise-linearity in the underlying SSE signals using added noise. We verify its effectiveness on a range of model-generated synthetic SSE data with different noise levels, and demonstrate its superior performance compared to two existing methods. We illustrate its capability in detecting short-term SSEs in observed GPS data using 36 GPS stations in southwest Japan via the co-occurrence of non-volcanic tremors, hypothesis tests and fault estimation.
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Neural networks are commonly used for post-stack and pre-stack seismic inversion. With sufficient labelled data, the neural network-based seismic inversion results are more accurate than that use traditional seismic inversion methods. However, in the case of insufficient labeled data, the accuracy of neural networks-based seismic inversion results decreases and is even lower than those based on traditional inversion methods. In addition, the seismic inversion results based on neural networks generally suffer from lateral discontinuity. It further reduces the accuracy of the inversion results. To tackle these problems, we propose a pre-stack seismic amplitude variation with offset (AVO) inversion method based on Closed-Loop Multi-task conditional Wasserstein Generative Adversarial Network (CMcWGAN), which is a GAN-based AVO inversion method. CMcWGAN enables simultaneous and accurate inversion of P-wave velocity ( Vp ), S-wave velocity ( Vs ), and density ( ρ ). Moreover, it uses the low-frequency information of elastic parameters as a conditional input to alleviate the problem of lateral discontinuity in inversion results. Experimental results of simulated data show that the inversion results based on CMcWGAN have higher accuracy than those based on traditional AVO inversion methods. In addition, when the seismic angle gather is noisy, CMcWGAN has better robustness than the traditional methods. CMcWGAN can also obtain reasonable AVO inversion results in field seismic angle gather data.inversion results. Experimental results of simulated data show that the inversion results based on CMcWGAN have higher accuracy than those based on traditional AVO inversion method. In addition, when the seismic angle gather is noisy, CMcWGAN has better robustness than traditional method. CMcWGAN can also get reasonable AVO inversion results in field seismic angle gather data.
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A finite-fault earthquake slip model can characterize the kinematics of rupture, which is essential for earthquake mechanism studies and seismic hazard assessments. The finite-fault model of an earthquake can be inverted from a wide range of geodetic measurements and seismic recordings. The linear least squares method minimizes misfits between observations and forward modeling and is a common approach in finite-fault inversion problems. This method may not yield the most plausible finite-fault model and has four main limitations. First, total parameter spaces are challenging to explore, and non-Gaussian parameter uncertainties cannot be evaluated. Second, to improve the stability of the fault slip inversion, fault slip smoothing operators (regularization techniques) are usually applied; however, determining the strength of smoothing a fault slip distribution is subjective. Third, the fault geometry needs to be predetermined, and different fault geometry settings can result in varying in- version results. Fourth, it is difficult to account for uncertainties in forward modeling because of imprecise earth ve- locity models. In contrast, Bayesian inversion determines the probability density distribution of the model parameters, providing a globally optimized solution to all model parameters and characterizing trade-offs between pairs of model parameters. Therefore, the Bayesian approach effectively overcomes the problems encountered in linear inversion. With the rapid improvement in computer technology, Bayesian inversion has become highly developed, especially in the past decade. This review reports the results of recent Bayesian finite-fault inversion studies and explains the theory and methodology of Bayesian finite-fault inversion.
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A devastating tsunami struck Palu Bay in the wake of the 28 September 2018 Mw = 7.5 Palu earthquake (Sulawesi, Indonesia). With a predominantly strike‐slip mechanism, the question remains whether this unexpected tsunami was generated by the earthquake itself, or rather by earthquake‐induced landslides. In this study we examine the tsunami potential of the co‐seismic deformation. To this end, we present a novel geodetic data set of Global Positioning System and multiple Synthetic Aperture Radar‐derived displacement fields to estimate a 3D co‐seismic surface deformation field. The data reveal a number of fault bends, conforming to our interpretation of the tectonic setting as a transtensional basin. Using a Bayesian framework, we provide robust finite fault solutions of the co‐seismic slip distribution, incorporating several scenarios of tectonically feasible fault orientations below the bay. These finite fault scenarios involve large co‐seismic uplift (>2 m) below the bay due to thrusting on a restraining fault bend that connects the offshore continuation of two parallel onshore fault segments. With the co‐seismic displacement estimates as input we simulate a number of tsunami cases. For most locations for which video‐derived tsunami waveforms are available our models provide a qualitative fit to leading wave arrival times and polarity. The modeled tsunamis explain most of the observed runup. We conclude that co‐seismic deformation was the main driver behind the tsunami that followed the Palu earthquake. Our unique geodetic data set constrains vertical motions of the sea floor, and sheds new light on the tsunamigenesis of strike‐slip faults in transtensional basins.
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Understanding the nature of foreshock evolution is important for earthquake nucleation and hazard evaluation. Aseismic slip and cascade triggering processes are considered to be two end-member precursors in earthquake nucleation processes. However, to perceive the physical mechanisms of these precursors leading to the occurrence of large events is challenging. In this study, the relocated 2021 Yangbi earthquake sequences are observed to be aligned along the northwest–southeast direction and exhibit spatial migration fronts toward the hypocenters of large events including the mainshock. An apparent static Coulomb stress increase on the mainshock hypocenter was detected, owing to the precursors. This suggests that the foreshocks are manifestations of aseismic transients that promote the cascade triggering of both the foreshocks and the eventual mainshock. By jointly inverting both Interferometric Synthetic Aperture Radar and Global Navigation Satellite Systems data, we observe that the mainshock ruptured a blind vertical fault with a peak slip of 0.8 m. Our results demonstrate that the lateral crustal extrusion and lower crustal flow are probably the major driving mechanisms of mainshock. In addition, the potential seismic hazards on the Weixi–Weishan and Red River faults deserve further attention.
Preprint
Understanding the nature of foreshock evolution is important for earthquake nucleation and hazard evaluation. Aseismic slip and cascade triggering processes are considered to be two end-member precursors in earthquake nucleation processes. However, to perceive the physical mechanisms of these precursors leading to the occurrence of large events is challenging. In this study, the relocated 2021 Yangbi earthquake sequences are observed to be aligned along the NW-SE direction and exhibit spatial migration fronts towards the hypocenters of large events including the mainshock. An apparent static Coulomb stress increase on the mainshock hypocenter was detected, owing to the precursors. This suggests that the foreshocks are manifestations of aseismic transients that promote the cascade triggering of both the foreshocks and the eventual mainshock. By jointly inverting both InSAR and GNSS data, we observe that the mainshock ruptured a blind vertical fault with a peak slip of 0.8 m. Our results demonstrate that the lateral crustal extrusion and lower crustal flow are probably the major driving mechanisms of mainshock. Additionally, the potential seismic hazards on the Weixi-Weishan and Red River faults deserve further attention.
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Earthquake source inversion, which estimates the heterogeneous slip distribution on fault from geophysical data, is a fundamental technique for estimating earthquake rupture process and obtaining information about the physics of fault rupture. Source inversion requires the spatial discretization of fault, which can be performed uniformly and non-uniformly. Uniform fault discretization is a conventional approach that requires smoothing and/or non-negative constraints of slips as prior information to obtain a stable and reliable solution; however, the combination of uniform discretization and these prior constraints may distort a source-inversion solution. As a non-uniform discretization approach, source inversion using a trans-dimensional inversion approach has recently attracted attention. To study the effect of fault discretization on geodetic source inversion, through the analysis of geodetic data on the 2015 Gorkha, Nepal, earthquake and synthetic tests, we investigated what kind of solution the conventional source inversion with uniform discretization and the trans-dimensional source inversion provide and what kind of uncertainty their solutions have. We found that the combination of uniform discretization and non-negative constraint led to excessively smooth solutions with poor data fit. Even without using the non-negative constraint, the conventional inversion with uniform discretization provided distorted and sometimes over-fitted solutions, which could not be identified based on uncertainty information. In contrast, the trans-dimensional source inversion provided reasonable solutions composed only of meaningful slips, which were required to explain the data. We also found that uncertainty information depends on the source-inversion method; consequently, the evaluation of method-induced uncertainty is difficult. This suggests that we look at earthquake ruptures through the lens of source inversion with inherent method-dependent bias.
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Active crustal deformation of the Tibetan plateau results in destructive continental earthquakes and is therefore the focus of intense research interest. Increases in the numbers of Global Navigation Satellite System (GNSS) networks and stations deployed in Tibet are allowing for the characterization of crustal deformation during different phases of the earthquake cycle. Here, we present the status of a “seismic + high-rate GNSS” network deployed in eastern Tibet, including its data streams and data processing system, with the goal of supporting quasi-real-time earthquake source determination. Furthermore, we use this network to test a prototype earthquake early warning (EEW) system using data from the 2008 Mw 7.9 Wenchuan earthquake, the 2011 Mw 9.0 Tohoku earthquake, and 2200 synthetic earthquakes with moment magnitudes ranging from 6.5 to 7.5 on the southern Longmen Shan fault and Anninghe fault. The results show that our current methodology could respond to moderate-to-large earthquakes (magnitude 7+) within tens of seconds after the origin time, with implications for EEW applications in China.
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The ENE striking Longmu Co fault and the North Altyn Tagh left-lateral slip fault have led to the complex regional structure in the northwestern Tibetan Plateau, resulting in a series of normal faulting and strike slip faulting earthquakes. Using both the ascending and descending Sentinel-1A/B radar images, we depict the coseismic deformation caused by the 2020 Yutian Mw 6.4 earthquake with a peak subsidence of ~20 cm. We determine the seismogenic fault geometry by applying the Bayesian approach with a Markov Chain Monte Carlo sampling method, which enables us to find the posterior probability density functions of the source model parameters. The estimation results reveal that the earthquake have dominantly by normal slip with moderate strike slip component. Based on the optimal fault geometry model, we extend the fault plane and invert for the finite fault model dislocation, which indicate that the slip is mainly concentrated at a shallow focal depth of 3–10 km with a maximum slip of ~1.0 m. Our preferred geodetic coseismic model exhibits no surface rupture, which may likely due to the shallow slip deficit in the uppermost crust. We calculate the combined loading effect of the Coulomb failure stress changes induced by the coseismic dislocations and postseismic viscoelastic relaxation of the 2008 Mw 7.1 and 2014 Mw 6.9 Yutian events. Our study demonstrates that the two preceding major Yutian shocks were insufficient to trigger the 2020 Yutian earthquake, which we consider perhaps reflects the natural release of elastic strain accumulated mainly through localized tectonic movement. We attribute the 2020 Yutian event to the release of extensional stress in a stepover zone controlled by the Longmu Co and the North Altyn Tagh sinistral strike slip fault systems. The seismic risk in the southwest end of the North Altyn Tagh fault has been elevated by the Yutian earthquake sequences, which require future attention.
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Geological faults comprise large-scale segmentation and small-scale roughness. These multi-scale geometrical complexities determine the dynamics of the earthquake rupture process, and therefore affect the radiated seismic wavefield. In this study, we examine how different parameterizations of fault roughness lead to variability in the rupture evolution and the resulting near-fault ground motions. Rupture incoherence naturally induced by fault roughness generates high-frequency radiation that follows an ω−2 decay in displacement amplitude spectra. Because dynamic rupture simulations are computationally expensive, we test several kinematic source approximations designed to emulate the observed dynamic behavior. When simplifying the rough-fault geometry, we find that perturbations in local moment tensor orientation are important, while perturbations in local source location are not. Thus, a planar fault can be assumed if the local strike, dip, and rake are maintained. We observe that dynamic rake angle variations are anti-correlated with the local dip angles. Testing two parameterizations of dynamically consistent Yoffe-type source-time function, we show that the seismic wavefield of the approximated kinematic ruptures well reproduces the radiated seismic waves of the complete dynamic source process. This finding opens a new avenue for an improved pseudo-dynamic source characterization that captures the effects of fault roughness on earthquake rupture evolution. By including also the correlations between kinematic source parameters, we outline a new pseudo-dynamic rupture modeling approach for broadband ground-motion simulation.
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Fault slip distributions provide important insight into the earthquake process. We analyze high-resolution along-strike co-seismic slip profiles of the 1992 Mw = 7.3 Landers and 1999 Mw = 7.1 Hector Mine earthquakes, finding a spatial correlation between fluctuations of the slip distribution and geometrical fault structure. Using a spectral analysis, we demonstrate that the observed variation of co-seismic slip is neither random nor artificial, but self-affine fractal and rougher for Landers. We show that the wavelength and amplitude of slip variability correlates to the spatial distribution of fault geometrical complexity, explaining why Hector Mine has a smoother slip distribution as it occurred on a geometrically simpler fault system. We propose as a physical explanation that fault complexity induces a heterogeneous stress state that in turn controls co-seismic slip. Our observations detail the fundamental relationship between fault structure and earthquake rupture behavior, allowing for modeling of realistic slip profiles for use in seismic hazard assessment and paleoseismology studies.
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We report macroscopic stick-slip events in saw-cut Westerly granite samples deformed under controlled upper crustal stress conditions in the laboratory. Experiments were conducted under triaxial loading (σ1>σ2=σ3) at confining pressures (σ3) ranging from 10 to 100 MPa. A high-frequency acoustic monitoring array recorded particle acceleration during macroscopic stick-slip events allowing us to estimate rupture speed. In addition, we record the stress drop dynamically and we show that the dynamic stress drop measured locally close to the fault plane is almost total in the breakdown zone (for normal stress >75 MPa), while the friction f recovers to values of f > 0.4 within only a few hundred microseconds. Enhanced dynamic weakening is observed to be linked to the melting of asperities which can be well explained by flash heating theory in agreement with our postmortem microstructural analysis. Relationships between initial state of stress, rupture velocities, stress drop, and energy budget suggest that at high normal stress (leading to supershear rupture velocities), the rupture processes are more dissipative. Our observations question the current dichotomy between the fracture energy and the frictional energy in terms of rupture processes. A power law scaling of the fracture energy with final slip is observed over 8 orders of magnitude in slip, from a few microns to tens of meters.
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Earthquake slip distributions are asymmetric along strike, but the reasons for the asymmetry are unknown. We address this question by establishing empirical relations between earthquake slip profiles and fault properties. We analyze the slip distributions of 27 large continental earthquakes in the context of available information on their causative faults, in particular on the directions of their long-term lengthening. We find that the largest slips during each earthquake systematically occurred on that half of the ruptured fault sections most distant from the long-term fault propagating tips, i.e., on the most mature half of the broken fault sections. Meanwhile, slip decreased linearly over most of the rupture length in the direction of long-term fault propagation, i.e., of decreasing structural maturity along-strike. We suggest that this earthquake slip asymmetry is governed by along-strike changes in fault properties, including fault zone compliance and fault strength, induced by the evolution of off-fault damage, fault segmentation and fault planarity with increasing structural maturity. We also find higher rupture speeds in more mature rupture sections, consistent with predicted effects of low velocity damage zones on rupture dynamics. Since the direction(s) of long-term fault propagation can be determined from geological evidence, it might be possible to anticipate in which direction earthquake slip, once nucleated, may increase, accelerate and possibly lead to a large earthquake. Our results could thus contribute to earthquake hazard assessment and Earthquake Early Warning.
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Finite-fault earthquake source inversions infer the (time-dependent) displacement on the rupture surface from geophysical data. The resulting earthquake source models document the complexity of the rupture process. However, multiple source models for the same earthquake, obtained by different research teams, often exhibit remarkable dissimilarities. To address the uncertainties in earthquake-source inversion methods and to understand strengths and weaknesses of the various approaches used, the Source Inversion Validation (SIV) project conducts a set of forward-modeling exercises and inversion benchmarks. In this article, we describe the SIV strategy, the initial benchmarks, and current SIV results. Furthermore, we apply statistical tools for quantitative waveform comparison and for investigating source-model (dis)similarities that enable us to rank the solutions, and to identify particularly promising source inversion approaches. All SIV exercises (with related data and descriptions) and statistical comparison tools are available via an online collaboration platform, and we encourage source modelers to use the SIV benchmarks for developing and testing new methods. We envision that the SIV efforts will lead to new developments for tackling the earthquake-source imaging problem.
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Geodetic slip inversions for three major (Mw > 7) strike-slip earthquakes (1992 Landers, 1999 Hector Mine and 2010 El Mayor–Cucapah) show a 15–60 per cent reduction in slip near the surface (depth < 2 km) relative to the slip at deeper depths (4–6 km). This significant difference between surface coseismic slip and slip at depth has been termed the shallow slip deficit (SSD). The large magnitude of this deficit has been an enigma since it cannot be explained by shallow creep during the interseismic period or by triggered slip from nearby earthquakes. One potential explanation for the SSD is that the previous geodetic inversions lack data coverage close to surface rupture such that the shallow portions of the slip models are poorly resolved and generally underestimated. In this study, we improve the static coseismic slip inversion for these three earthquakes, especially at shallow depths, by: (1) including data capturing the near-fault deformation from optical imagery and SAR azimuth offsets; (2) refining the interferometric synthetic aperture radar processing with non-boxcar phase filtering, model-dependent range corrections, more complete phase unwrapping by SNAPHU (Statistical Non-linear Approach for Phase Unwrapping) assuming a maximum discontinuity and an on-fault correlation mask; (3) using more detailed, geologically constrained fault geometries and (4) incorporating additional campaign global positioning system (GPS) data. The refined slip models result in much smaller SSDs of 3–19 per cent. We suspect that the remaining minor SSD for these earthquakes likely reflects a combination of our elastic model's inability to fully account for near-surface deformation, which will render our estimates of shallow slip minima, and potentially small amounts of interseismic fault creep or triggered slip, which could ‘make up’ a small percentages of the coseismic SSD during the interseismic period. Our results indicate that it is imperative that slip inversions include accurate measurements of near-fault surface deformation to reliably constrain spatial patterns of slip during major strike-slip earthquakes.
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The long-term slip on faults has to follow, on average, the plate motion, while slip deficit is accumulated over shorter timescales (e.g., between the large earthquakes). Accumulated slip deficits eventually have to be released by earthquakes and aseismic processes. In this study, we propose a new inversion approach for coseismic slip, taking interseismic slip deficit as prior information. We assume a linear correlation between coseismic slip and interseismic slip deficit and invert for the coefficients that link the coseismic displacements to the required strain accumulation time and seismic release level of the earthquake. We apply our approach to the 2011 M9 Tohoku-Oki earthquake and the 2004 M6 Parkfield earthquake. Under the assumption that the largest slip almost fully releases the local strain (as indicated by borehole measurements), our results suggest that the strain accumulated along the Tohoku-Oki earthquake segment has been almost fully released during the 2011 M9 rupture. The remaining slip deficit can be attributed to the postseismic processes. Similar conclusions can be drawn for the 2004 M6 Parkfield earthquake. We also estimate the required time of strain accumulation for the 2004 M6 Parkfield earthquake to be ~25years (confidence interval of [17, 43] years), consistent with the observed average recurrence time of ~22years for M6 earthquakes in Parkfield. For the Tohoku-Oki earthquake, we estimate the recurrence time of ~500-700years. This new inversion approach for evaluating slip balance can be generally applied to any earthquake for which dense geodetic measurements are available.
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The Lepsy fault of the northern Tien Shan, SE Kazakhstan, extends E-W 120 km from the high mountains of the Dzhungarian Ala-tau, a sub-range of the northern Tien Shan, into the low-lying Kazakh platform. It is an example of an active structure that connects a more rapidly deforming mountain region with an apparently stable continental region (SCR), and follows a known Palaeozoic structure. Field-based and satellite observations reveal a ~10 m vertical offset exceptionally preserved along the entire length of the fault. Geomorphic analysis and age control from radiocarbon and optically stimulated luminescence (OSL) dating methods indicate that the scarp formed in the Holocene and was generated by at least two substantial earthquakes. The most recent event, dated to some time after ~400 years BP, is likely to have ruptured the entire 120 km fault length in a Mw 7.5{8.2 earthquake. The Lepsy fault kinematics were characterised using digital elevation models and high-resolution satellite imagery, which indicate that the predominant sense of motion is reverse right-lateral with a fault strike, dip and slip vector azimuth of ~110°, 50°S and 317{343 respectively, which is consistent with predominant N-S shortening related to the India-Eurasia collision. In light of these observations, and because the activity of the Lepsy fault would have been hard to ascertain if it had not ruptured in the recent past, we note that the absence of known active faults within low-relief and low strain-rate continental interiors does not always imply an absence of seismic hazard.
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Real-time high-rate geodetic data have been shown to be useful for rapid earthquake response systems during medium to large events. The 2014 Mw 6.1 Napa, California earthquake is important because it provides an opportunity to study an event at the lower threshold of what can be detected with GPS. We show the results of GPS-only earthquake source products such as peak ground displacement (PGD) magnitude scaling, centroid moment tensor (CMT) solution and static slip inversion. We also highlight the retrospective real-time combination of GPS and strong motion data to produce seismogeodetic waveforms that have higher precision and longer period information than GPS-only or seismic-only measurements of ground motion. We show their utility for rapid kinematic slip inversion and conclude that it would have been possible, with current real-time infrastructure, to determine the basic features of the earthquake source. We supplement the analysis with strong motion data collected close to the source to obtain an improved post-event image of the source process. The model reveals unilateral fast propagation of slip to the north of the hypocenter with a delayed onset of shallow slip. The source model suggests that the multiple strands of observed surface rupture are controlled by the shallow soft sediments of Napa Valley and do not necessarily represent the intersection of the main faulting surface and the free surface. We conclude that the main dislocation plane is westward dipping and should intersect the surface to the east, either where the easternmost strand of surface rupture is observed or at the location where the West Napa fault has been mapped in the past.
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Europe's Sentinel-1A spacecraft and its extraordinary images of slip from the South Napa earthquake herald a new era of space-based surveillance of faults.
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Online Material: Movie of wave propagation, GPS coseismic displacements, rupture velocity, waveform comparisons, geologic and 3D seismic structure, and moment rate functions. On 24 August 2014 at 10:20:44.06 UTC, a large earthquake struck the north San Francisco Bay region, approximately 10 km south‐southwest of Napa, California, causing local damage in older wood frame and masonry buildings, road surfaces, sidewalks, and masonry wall structures (Bray et al. , 2014). Using long‐period (50–20 s) three‐component, complete displacement records, the Berkeley Seismological Laboratory (BSL) estimated the scalar seismic moment at 1.32×1018 N·m for a depth of 11 km, corresponding to a moment magnitude of M w 6.0. The strike/dip/rake from the seismic moment tensor solution was 155°/82°/−172°, which is in overall agreement with the trends of structures comprising the West Napa fault system (Fig. 1). Geologic mapping revealed an approximately 14 km long surface rupture with 40–45 cm maximum observed slip on a complex multibranched fault system (Bray et al. , 2014; Earthquake Engineering Research Institute [EERI], 2014; Mike Oskin and Alex Morelan, written comm., 2014). The largest surface offsets were found on a northwest‐striking trend located approximately 1.8 km west of the mapped West Napa fault. Aftershocks are generally located west of the western branch of the surface fault, which had the largest offsets, and indicate a westward dip of the primary fault plane (Fig. 2). Figure 1. Locations of Berkeley Digital Seismic Network (BDSN) stations are shown as labeled squares. Plate Boundary Observatory (PBO) Global Positioning System (GPS) sites are shown as circles, and the positions of Interferometric Synthetic Aperture Radar (InSAR) returns are small gray squares. San Francisco and Napa Valley are indicated by SF and NV. The Berkeley Seismological Laboratory focal mechanism is shown, and the thick line shows the mapped surface trace (EERI, 2014). Figure 2. (a) Coseismic fault‐slip model based on the joint inversion of …
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Co-seismic surface deformation in large earthquakes is typically measured using field mapping and with a range of geodetic methods (e.g., InSAR, lidar differencing, and GPS). Current methods, however, either fail to capture patterns of near-field co-seismic surface deformation or lack pre-event data. Consequently, the characteristics of off-fault deformation and the parameters that control it remain poorly understood. We develop a standardized method to fully measure the surface, near-field, co-seismic deformation patterns at high-resolution using the COSI-Corr program by correlating pairs of aerial photographs taken before and after the 1992 Mw 7.3 Landers earthquake. COSI-Corr offers the advantage of measuring displacement across the entire zone of surface deformation and over a wider aperture than that available to field geologists. For the Landers earthquake, our measured displacements are systematically larger than the field measurements, indicating the presence of off-fault deformation. We show that 46 % of the total surface displacement occurred as off-fault deformation, over a mean deformation width of 154 m. The magnitude and width of off-fault deformation along the rupture is primarily controlled by the macroscopic structural complexity of the fault system, with a weak correlation with the type of near-surface materials through which the rupture propagated. Both the magnitude and width of distributed deformation are largest in stepovers, bends, and at the southern termination of the surface rupture. We find that slip along the surface rupture exhibits a consistent degree of variability at all observable length scales and that the slip distribution is self-affine fractal with dimension of 1.56. This article is protected by copyright. All rights reserved.
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On 24 August 2014, the M 6.0 South Napa earthquake shook much of the San Francisco Bay area, leading to significant damage in the Napa Valley. The earthquake occurred in the vicinity of the West Napa fault (122.313° W, 38.22° N, 11.3 km), a mapped structure located between the Rodger’s Creek and Green Valley faults, with nearly pure right‐lateral strike‐slip motion (strike 157°, dip 77°, rake –169°; http://comcat.cr.usgs.gov/earthquakes/eventpage/nc72282711#summary, last accessed December 2014) (Fig. 1). The West Napa fault previously experienced an M 5 strike‐slip event in 2000 but otherwise exhibited no previous definitive evidence of historic earthquake rupture (Rodgers et al., 2008; Wesling and Hanson, 2008). Evans et al. (2012) found slip rates of ∼9.5 mm/yr along the West Napa fault, with most slip rate models for the Bay area placing higher slip rates and greater earthquake potential on the Rodger’s Creek and Green Valley faults, respectively (e.g., Savage et al., 1999; d’Alessio et al., 2005; Funning et al., 2007).
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This paper develops a probabilistic Bayesian approach to the problem of inferring the spatiotemporal evolution of earthquake rupture on a fault surface from seismic data with rigorous uncertainty estimation. To date, uncertainties of rupture parameters are poorly understood, and the effect of choices such as fault discretization on uncertainties has not been studied. We show that model choice is fundamentally linked to uncertainty estimation and can have profound effects on results. The approach developed here is based on a trans-dimensional self-parametrization of the fault, avoids regularization constraints and provides rigorous uncertainty estimation that accounts for model-selection ambiguity associated with the fault discretization. In particular, the fault is parametrized using self-adapting irregular grids which intrinsically match the local resolving power of the data and provide parsimonious solutions requiring few parameters to capture complex rupture characteristics. Rupture causality is ensured by parametrizing rupture-onset time by a rupture-velocity field and obtaining first rupture times from the eikonal equation. The Bayesian sampling of the parameter space is implemented on a computer cluster with a highly efficient parallel tempering algorithm. The inversion is applied to simulated and observed W-phase waveforms from the 2010 Maule (Chile) earthquake. Simulation results show that our approach avoids both over- and underparametrization to ensure unbiased inversion results with uncertainty estimates that are consistent with data information. The simulation results also show the ability of W-phase data to resolve the spatial variability of slip magnitude and rake angles. In addition, sensitivity to spatially dependent rupture velocities exists for kinematic slip models. Application to the observed data indicates that residual errors are highly correlated and likely dominated by theory error, necessitating the iterative estimation of a non-stationary data covariance matrix. The moment magnitude for the Maule earthquake is estimated to be ∼8.9, with slip concentrated in two zones updip of and north and south of the hypocentre, respectively. While this aspect of the slip distribution is similar to previous studies, we show that the slip maximum in the southern zone is poorly resolved compared to the northern zone. Both slip maxima are higher than reported in previous studies, which we speculate may be due to the lack of bias caused by the regularization used in other studies.
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Great earthquakes rarely occur within active accretionary prisms, despite the intense long-term deformation associated with the formation of these geologic structures. This paucity of earthquakes is often attributed to partitioning of deformation across multiple structures as well as aseismic deformation within and at the base of the prism (Davis et al., 1983). We use teleseismic data and satellite optical and radar imaging of the 2013 M-w 7.7 earthquake that occurred on the southeastern edge of the Makran plate boundary zone to study this unexpected earthquake. We first compute a multiple point-source solution from W-phase waveforms to estimate fault geometry and rupture duration and timing. We then derive the distribution of subsurface fault slip from geodetic coseismic offsets. We sample for the slip posterior probability density function using a Bayesian approach, including a full description of the data covariance and accounting for errors in the elastic structure of the crust. The rupture nucleated on a subvertical segment, branching out of the Chaman fault system, and grew into a major earthquake along a 50 degrees north-dipping thrust fault with significant along-strike curvature. Fault slip propagated at an average speed of 3.0 km/s for about 180 km and is concentrated in the top 10 km with no displacement on the underlying decollement. This earthquake does not exhibit significant slip deficit near the surface, nor is there significant segmentation of the rupture. We propose that complex interaction between the subduction accommodating the Arabia-Eurasia convergence to the south and the Ornach Nal fault plate boundary between India and Eurasia resulted in the significant strain gradient observed prior to this earthquake. Convergence in this region is accommodated both along the subduction megathrust and as internal deformation of the accretionary wedge.
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Geodetic observations of surface displacements during and following earthquakes such as the March 11, 2011 great Tohoku earthquake can be used to constrain the spatial extent of coseismic slip and postseismic afterslip, and characterize the spectrum of earthquake cycle behaviors. Slip models are often regularized by assuming that slip on the fault varies smoothly in space, which may result in the artificial smearing of fault slip beyond physical boundaries. Alternatively, it may be desirable to estimate a slip distribution that is spatially compact and varies sharply. Here we show that sparsity promoting state vector regularization methods can be used to recover slip distributions with sharp boundaries, representing an alternative end-member result to very smooth slip distributions. Using onshore GPS observations at 298 stations during and in the ˜2 weeks following the Tohoku earthquake, we estimate a band of coseismic slip between 30 and 50 km depth extending 500 km along strike with a maximum slip of 64 m, corresponding to a minimum magnitude estimate of MW = 8.8. Our estimate of afterslip is located almost exclusively down-dip of the coseismic rupture, with a transition between 40 and 50 km depth and an equivalent moment magnitude MW = 8.2. This depth may be interpreted as coincident with the transition from velocity strengthening to velocity weakening frictional behavior, consistent with the upper limit of cold subduction estimates of the thermal structure of the Japan trench.
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The decay of seismicity with distance from strike-slip faults in California is well described by a power-law of the form R~(x2+d2)-gamma/2, where x is distance from a fault, gamma is the decay rate of seismicity, and d is a near-fault inner scale. We establish this relationship by analyzing high-resolution regional and relocated catalogs in fault-referenced coordinate systems for four classes of faults (large faults in northern CA, large faults in southern CA, small faults in southern CA, and aftershocks of large earthquakes). Results from a multi-catalog analysis of hypocentral variance are used to estimate parameter bias due to event mislocation. Scaling parameters, d and gamma, vary regionally: seismicity is more localized on faults in northern California (d=0.04± 0.01 km; gamma=1.54±0.15) than in southern California (d=0.21±0.04 km; gamma=0.95±0.05). An investigation of individual fault segments shows that the localization of small earthquakes correlates with cumulative offset, on-fault earthquake density, and aseismic slip rate and we interpret this evolution of seismicity in the context of rate-and-state and damage-zone models of faults. Scaling parameters for aftershocks of large earthquakes (gamma=1.45±0.10) and declustered catalogs of southern California large faults (gamma=1.22±0.05) also show increased localization of seismicity, suggesting how epidemic-type earthquake-triggereing models might be modified to improve earthquake forecasts.
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While many earthquakes have now been studied using interferometric synthetic aperture radar (InSAR) data, a full assessment of the quality and additional value of InSAR source parameters compared to seismological techniques is still lacking. We compile a catalog of source models obtained using InSAR and estimate the corresponding centroid moment tensor (CMT) parameters; we refer to this compilation as the ICMT archive. We compare source parameters from over 70 InSAR studies of 57 global earthquakes with those in the Global CMT (GCMT), International Seismological Centre (ISC) and Engdahl-Hilst-Buland (EHB) seismic catalogs. We find an overall good agreement between fault strike, dip and rake values in the GCMT and ICMT archives. Likewise, the differences in seismic moment between these two archives are relatively small, and we do not find support for previously suggested trends of InSAR leading to larger moments than seismic data. However, epicentral locations show substantial discrepancies, which are larger for the GCMT (median differences of similar to 21 km) than for the EHB and ISC catalogs (median differences of similar to 10 km). Since InSAR data have a high spatial resolution, and thus should map epicentral locations accurately, this allows us to obtain a first independent estimate of epicentral location errors in the seismic catalogs. Earthquake depths from InSAR are systematically shallower than those in the EHB catalog, with a median of differences of similar to 5 km. While this trend may be partly due to unmodeled crustal complexity, it is also compatible with the observation that the rupture of crustal earthquakes tends to propagate upward in the seismogenic layer.
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Principal slip surfaces in fault zones accommodate most of the displacement during earthquakes. The topography of these surfaces is integral to earthquake and fault mechanics, but is practically unknown at the scale of earthquake slip. We use new laser-based methods to map exposed fault surfaces over scales of 10 mum to 120 m. These data provide the first quantitative evidence that fault-surface roughness evolves with increasing slip. Thousands of profiles ranging from 10 mum to >100 m in length show that small-slip faults (slip