4.07 - Slip Inversion

Following the 2007, August 15th, Mw 8.0, Pisco earthquake in central Peru, Sladenet al. (J Geophys Res 115: B02405, 2010) have derived several slip models of this event. They inverted teleseismic data together with geodetic (InSAR) measurements to look for the co-seismic slip distribution on the fault plane, considering those data sets separately or jointly. But how close to the real slip distribution are those inverted slip models? To answer this crucial question, the authors generated some tsunami records based on their slip models and compared them to DART buoys, tsunami records, and available runup data. Such an approach requires a robust and accurate tsunami model (non-linear, dispersive, accurate bathymetry and topography, etc.) otherwise the differences between the data and the model may be attributed to the slip models themselves, though they arise from an incomplete tsunami simulation. The accuracy of a numerical tsunami simulation strongly depends, among others, on two important constraints: (i) A fine computational grid (and thus the bathymetry and topography data sets used) which is not always available, unfortunately, and (ii) a realistic tsunami propagation model including dispersion. Here, we extend Sladen’s work using newly available data, namely a tide gauge record at Callao (Lima harbor) and the Chilean DART buoy record, while considering a complete set of runup data along with a more realistic tsunami numerical that accounts for dispersion, and also considering a fine-resolution computational grid, which is essential. Through these accurate numerical simulations we infer that the InSAR-based model is in better agreement with the tsunami data, studying the case of the Pisco earthquake indicating that geodetic data seems essential to recover the final co-seismic slip distribution on the rupture plane. Slip models based on teleseismic data are unable to describe the observed tsunami, suggesting that a significant amount of co-seismic slip may have been aseismic. Finally, we compute the runup distribution along the central part of the Peruvian coast to better understand the wave amplification/attenuation processes of the tsunami generated by the Pisco earthquake.

The Mw 7.1 Anchorage earthquake on 30 November 2018 beneath the south‐central Alaska is a rare intermediate‐depth event larger than Mw 7 that occurred in a complex subduction region, where the young Yakutat oceanic terrane wedges in the continental‐oceanic plate collisional region between the Pacific oceanic plate and the North American plate. We use both ascending and descending Sentinel‐1 satellite Interferometric Synthetic Aperture Radar (InSAR) images to construct the coseismic displacement associated with this earthquake, which shows a nearly circular deformation pattern with a subsidence of ~4 cm in line of sight direction. Combining coseismic GPS data, we determine the focal mechanism of this event dominated by normal faulting with N‐S striking of 186°and westward dipping of 64°by using a uniform slip model. Then we find a preferred slip model with both geodetic data and teleseismic data, suggesting the main slips are concentrated on a depth of 55–75 km. The total released moment of our preferred slip model is 5.32 × 1019N·m, equivalent to Mw 7.1. The rupture process includes two peaks terminating at about 18 s and indicates a unilateral rupture with its front propagating northwestward direction at an average speed of 2.5 km/s. In comparison with the detailed seismic image in this region, this event just occurred in the Yakutat terrane beneath a low velocity zone, suggesting it was caused by slab tear but not slab boundary breaking and determining the lower boundary of shallow thrust‐slip in the Alaska subduction zone.

Obtaining initial parameter estimates for chaotic dynamical systems using linear associative memories

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.

SUMMARYWe estimated the slip distribution on the fault of the 2005 Nias earthquake (Mw 8.6) by inversions of local GPS and coastal uplift/subsidence data and tsunami waveform data. The 2005 Nias earthquake occurred approximately three months after the 2004 Sumatra–Andaman earthquake (Mw 9.1) at the southern extension off Sumatra Island, Indonesia. The tsunami from the 2005 earthquake caused significantly less damage than the 2004 tsunami, yet was recorded at tide gauges and ocean bottom pressure gauges around the Indian Ocean, including the coasts of Africa and Antarctica. The elastic and gravitational coupling between the solid earth and the ocean causes not only a traveltime delay but also the change of waveforms of far-field tsunamis relative to the prediction based on the long-wave theory. We corrected the computed tsunami Green's functions for the elastic and gravitational coupling effect in the tsunami waveform inversion. We found a diffused slip (∼2 m over an area of 400 km × 100 km) at deeper parts (20–54 km) of the fault with a large localized slip (7 m over 100 km × 100 km) slightly south of the epicentre. The large slips at deeper parts of the fault were responsible for the small tsunami generation. Inversion using far-field tsunami data yielded a slip distribution similar to that obtained using local geodetic data alone and that from the joint inversion of local geodetic and far-field tsunami data, which is also similar to slip distributions from previous studies based on local geodetic data. This demonstrates that far-field tsunami waveforms, once corrected for propagation effects, can be used to estimate the slip distribution of large submarine earthquakes leading to results that are similar to those obtained using sparse local geodetic data.

We investigate the spectral characteristics and source properties of earthquake rupture models with self‐similar, self‐healing slip pulses. We compare pulse‐like and crack‐like rupture models across various idealized and heterogeneous source scenarios to identify conditions under which far‐field displacement spectra characterized by two corner frequencies (double‐corner frequency spectra) emerge. Five primary mechanisms are identified as causes of second high‐frequency corners: (a) slip‐pulse width, (b) rupture directivity, (c) gradual rupture arrest, (d) characteristic length of slip heterogeneity, and (e) Mach waves from supershear rupture. Spectra associated with slip‐pulse width are most evident at small take‐off angles from the fault normal, while those linked to rupture directivity and characteristic slip heterogeneity appear at large take‐off angles. Estimated stress drops depend strongly on rupture mode and speed. Pulse‐like models exhibit greater variability in estimated stress drops due to unknown rupture speeds than crack models, with pulse‐like ruptures systematically underestimating moment‐based stress drops by up to 40%. Comparisons to a recent study of Mw 3.0 to 5.5 earthquakes in Japan indicate a relationship between normalized corner frequency and scaled energy that fits better with pulse‐like rupture models than with crack models, indicating small‐ to moderate‐size earthquakes predominantly exhibit pulse‐like rupture behavior. The gradual rupture arrest mechanism can produce double‐corner spectra; however, it is unlikely to represent realistic earthquake sources due to its spectral decay. Our results highlight the importance of incorporating pulse‐like rupture dynamics into source models and point out challenges in interpreting source spectra and stress drop estimates when rupture characteristics are uncertain.

[1] Seismic slip inversion is studied by means of singular value decomposition (SVD), with emphasis on the role of singular vectors and regularization of the solution. Because the stable part of the slip inversion result is given in terms of a linear combination of the leading singular vectors (representing directions in the model space most sensitive to data), the performance of the inversion depends simply on how well the real slip model can be expanded into these vectors. The analysis is demonstrated using a synthetic model of a symmetric bilateral rupture with two asperities. Inverting the data yields a persisting false asperity in the middle of the fault, regardless of the inversion scheme used, because such a model cannot be well represented by only the leading singular vectors. The parallel study of the singular vectors from the individual stations and the whole network is suggested in order to understand how to achieve balance between overregularized solutions (with possible slip inversion artifacts) and underregularized solutions (vulnerable to data errors). The ideas are applied to the Mw 6.3 Movri Mountain earthquake, 2008, Greece, showing a weak indication of an asymmetric bilateral rupture. We suggest that inversion methods, not working explicitly with singular vectors, may yield artifacts, the origin of which can be also often explained by the SVD technique. Therefore, practical earthquake studies should be broadly complemented by the analysis of the leading singular vectors, a simple tool to evaluate the efficiency and potential drawbacks of the slip inversion.

SUMMARYThe model prediction errors that originate from the uncertainty of underground structure are 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 modelling 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 2-D thrust fault using synthetic data of surface displacement rates, which is the simplest problem setting that can essentially demonstrate 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 noted 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-modelled by Gaussian approximations.

Parameter estimation of unstable, limit cycling systems using adaptive feedback linearization: example of delta wing roll dynamics

A notable feature of the 1995 Kobe (Hyogo-ken Nanbu) earthquake is that violent ground motions occurred in a narrow zone. Previous studies have shown that the origin of such motions can be explained by the 3D velocity structure in this zone. This indicates not only that the 3D velocity structure significantly affects strong ground motions, but also that we should consider its effects in order to determine accurately the rupture process of the earthquake. Therefore, we have performed a joint source inversion of strong-motion, geodetic, and teleseismic data, where 3D Green's functions were calculated for strong-motion and geodetic data in the Osaka basin. Our source model estimates the total seismic moment to be about 2.1 × 1019 N m and the maximum slip reaches 2.9 m near the hypocenter. Although the locations of large slips are similar to those reported by Yoshida et al. (1996), there are quantitative differences between our results and their results due to the differences between the 3D and 1D Green's functions. We have also confirmed that our source model realized a better fit to the strong motion observations, and a similar fit as Yoshida et al. (1996) to the observed static displacements.

A vacuumed artificial mouth has been assembled and tested to measure reed velocity for a Bb clarinet along the width of the reed. Reed velocity measurements may be useful for better estimation of parameters in physical models, such as the relevant surface area of the vibrating reed. Use of a vacuum system instead of a pressurized mouth chamber allows straightforward observation and manipulation of the mouthpiece apparatus. Point measurements of reed velocity were obtained via a laser-Doppler vibrometer directed at the reed surface when artificially blown. Simultaneous high-speed exposures were recorded to visualize reed motion. Preliminary results indicate that the velocity amplitude of any torsional motion in the reed is negligible compared to an asymmetric reed velocity, likely caused by natural limitations of the clarinet ligature. Velocity measurements also indicate that the reed may sometimes rebound against the mouthpiece in its oscillatory period. High-speed exposures support this conclusion by vis...

Regularized reduced rank growth curve models

Our objective was to identify the best fit mathematical models for in vitro gas production kinetics using rumen fluid and forage plants commonly used in ruminant feed to obtain better estimates of parameters that describe the rumen fermentation. Four mathematical models were tested, two unicompartmental (M1 = first order, M2 = Gompertz) and two bicompartmental (M3 = M1 + M2; M4 = M2 + M2). Two temperate grasses were evaluated, as well as four tropical grasses and three temperate [...]

The statistical behavior of some common parameter estimation methods is investigated by simulation. The methods based on the maximum likelihood principle give the best estimates of parameters in a given model. An analysis of the possibilities for detecting systematic errors by means of the maximum likelihood method showed that this method gives at best the same information as other well known tests.

Chapter 7 - Single contact point transient tyre models