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Register a new userInverting elastic dislocations using the Weakly-enforced Slip Method
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Added by
Gertjan van Zwieten
Publication date
2021
fields
Physics
and research's language is
English
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Earthquakes cause lasting changes in static equilibrium, resulting in global deformation fields that can be observed. Consequently, deformation measurements such as those provided by satellite based InSAR monitoring can be used to infer an earthquakes faulting mechanism. This inverse problem requires a numerical forward model that is both accurate and fast, as typical inverse procedures require many evaluations. The Weakly-enforced Slip Method (WSM) was developed to meet these needs, but it was not before applied in an inverse problem setting. Consequently, it was unknown what effect particular properties of the WSM, notably its inherent continuity, have on the inversion process. Here we show that the WSM is able to accurately recover slip distributions in a Bayesian-inference setting, provided that data points in the vicinity of the fault are removed. In a representative scenario, an element size of 2 km was found to be sufficiently fine to generate a posterior probability distribution that is close to the theoretical optimum. For rupturing faults a masking zone of 20 km sufficed to avoid numerical disturbances that would otherwise be induced by the discretization error. These results demonstrate that the WSM is a viable forward method for earthquake inversion problems. While our synthesized scenario is basic for reasons of validation, our results are expected to generalize to the wider gamut of scenarios that finite element methods are able to capture. This has the potential to bring modeling flexibility to a field that if often forced to impose model restrictions in a concession to computability.
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Tectonic faults are commonly modelled as Volterra or Somigliana dislocations in an elastic medium. Various solution methods exist for this problem. However, the methods used in practice are often limiting, motivated by reasons of computational efficiency rather than geophysical accuracy. A typical geophysical application involves inverse problems for which many different fault configurations need to be examined, each adding to the computational load. In practice, this precludes conventional finite-element methods, which suffer a large computational overhead on account of geometric changes. This paper presents a new non-conforming finite-element method based on weak imposition of the displacement discontinuity. The weak imposition of the discontinuity enables the application of approximation spaces that are independent of the dislocation geometry, thus enabling optimal reuse of computational components. Such reuse of computational components renders finite-element modeling a viable option for inverse problems in geophysical applications. A detailed analysis of the approximation properties of the new formulation is provided. The analysis is supported by numerical experiments in 2D and 3D.
In this paper we derive the continuum limit of a multiple-species, interacting particle system by proving a $Gamma$-convergence result on the interaction energy as the number of particles tends to infinity. As the leading application, we consider $n$ edge dislocations in multiple slip systems. Since the interaction potential of dislocations has a logarithmic singularity at zero with a sign that depends on the orientation of the slip systems, the interaction energy is unbounded from below. To make the minimization problem of this energy meaningful, we follow the common approach to regularise the interaction potential over a length-scale $delta > 0$. The novelty of our result is that we leave the emph{type} of regularisation general, and that we consider the joint limit $n to infty$ and $delta to 0$. Our result shows that the limit behaviour of the interaction energy is not affected by the type of the regularisation used, but that it may depend on how fast the emph{size} (i.e., $delta$) decays as $n to infty$.
Data-driven machine-learning for predicting instantaneous and future fault-slip in laboratory experiments has recently progressed markedly due to large training data sets. In Earth however, earthquake interevent times range from 10s-100s of years and geophysical data typically exist for only a portion of an earthquake cycle. Sparse data presents a serious challenge to training machine learning models. Here we describe a transfer learning approach using numerical simulations to train a convolutional encoder-decoder that predicts fault-slip behavior in laboratory experiments. The model learns a mapping between acoustic emission histories and fault-slip from numerical simulations, and generalizes to produce accurate results using laboratory data. Notably slip-predictions markedly improve using the simulation-data trained-model and training the latent space using a portion of a single laboratory earthquake-cycle. The transfer learning results elucidate the potential of using models trained on numerical simulations and fine-tuned with small geophysical data sets for potential applications to faults in Earth.
We investigate the nature of friction in granular layers by means of numerical simulation focusing on the critical slip distance, over which the system relaxes to a new stationary state. Analyzing a transient process in which the sliding velocity is instantaneously changed, we find that the critical slip distance is proportional to the sliding velocity. We thus define the relaxation time, which is independent of the sliding velocity. It is found that the relaxation time is proportional to the layer thickness and inversely proportional to the square root of the pressure. An evolution law for the relaxation process is proposed, which does not contain any length constants describing the surface geometry but the relaxation time of the bulk granular matter. As a result, the critical slip distance is scaled with a typical length scale of a system. It is proportional to the layer thickness in an instantaneous velocity change experiment, whereas it is scaled with the total slip distance in a spring-block system on granular layers.
It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results.
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