Finite Element Models of Elastic Volcano Deformation

The migration of magma within a volcano produces a deformation signature at the Earth’s surface. Inverse models of geodetic data estimate parameters that characterize the magma migration. These characterizations are tied to the specific model that relates migration to the observed deformation. A model is a simplified representation of a natural system. A modeler is tasked with the challenge of designing a model that represents the system, in the context of the available data and purpose of the model. This chapter presents a systematic approach to quantitatively simulate geodetic data with finite element models (FEMs) in the framework of a deformation modeling protocol. This chapter will (1) address the design and execution of FEMs that can account for the geophysical complexity of a volcano deformational system and (2) define techniques for including FEMs in both linear and nonlinear inverse methods to characterize a magmatic system based on observed geodetic data. With these techniques, researchers can estimate magmatic migration within active volcanoes and understand how uncertainties in the data propagate into predictions. These estimates comprise some measure of central tendency, a sense of uncertainty, and a quantification of biases

Rayleigh Wave Dispersion Curve Inversion: Occam Versus the L1-Norm

We compare inversions of Rayleigh wave dispersion curves for shear wave velocity depth profiles based on the L2-norm (Occam\u27s Inversion) and L1-norm (TV Regularization). We forward model Rayleigh waves using a finite-element method instead of the conventional technique based on a recursion formula and root-finding. The forward modeling naturally leads to an inverse problem that is overparameterized in depth. Solving the inverse problem with Occam\u27s Inversion gives the smoothest subsurface model that satisfies the data. However, the subsurface need not be smooth and we therefore also solve the inverse problem with TV Regularization, a procedure that does not penalize discontinuities. The use of such a regularization scheme for such an overparameterized inverse problem means blocky subsurface models can be obtained without fixing the layer boundaries in advance. This represents an entirely new philosophy for surface wave inversion

Finite Element Models of Elastic Earthquake Deformation

The Earth’s surface deforms in response to earthquake fault dislocations at depth. Deformation models are constructed to interpret the corresponding ground movements recorded by geodetic data such GPS and InSAR, and ultimately characterize the seismic ruptures. Conventional analytical and latest numerical solutions serve similar purpose but with different technical constraints. The former cannot simulate the heterogeneous rock properties and structural complexity, while the latter directly tackles these challenges but requires more computational resources. As demonstrated in the 2015 M7.8 Gorkha, Nepal earthquake and the 2016 M6.2 Amatrice, Italy earthquake, we develop state-of-art finite element models (FEMs) to efficiently accommodate both the material and tectonic complexity of a seismic deformational system in a seamless model environment. The FEM predictions are significantly more accurate than the analytical models embedded in a homogeneous half-space at the 95% confidence level. The primary goal of this chapter is describe a systematic approach to design, construct, execute and calibrate FEMs of elastic earthquake deformation. As constrained by coseismic displacements, FEM-based inverse analyses are employed to resolve linear and nonlinear fault-slip parameters. With such numerical techniques and modeling framework, researchers can explicitly investigate the spatial distribution of seismic fault slip and probe other in-depth rheological processes

Interpretation of Rayleigh-Wave Ellipticity Observed with Multicomponent Passive Seismic Interferometry at Hekla Volcano, Iceland

The 2010 eruption of Eyjafjallajökull has drawn increased attention to Iceland’s Eastern Volcanic Zone (EVZ) due to the threat it poses to the heavily used air-traffic corridors of the northern Atlantic Ocean. Within the EVZ, Hekla is historically one of the most active volcanoes and has exhibited a decadal eruption pattern for the past 40 years. Hekla most recently erupted in 2000 and is thus ripe for another decadal eruption. Because Hekla is generally aseismic, except for a brief time period (hours) leading up to an eruption, monitoring has previously depended on precursory deformation signals (Linde et al., 1993). As a result, seismic tomography of the internal structure of the volcano using phase arrivals of local earthquakes is not possible. Motivated by Hekla’s practically aseismic behavior in inter-eruptive periods, we installed a temporary network of four broadband seismometers around the volcanic edifice in late August 2010 with the intention of investigating the applicability of passive seismic interferometry (PSI) for imaging and monitoring the volcano

Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting

In classical tsunami-generation techniques, one neglects the dynamic sea bed displacement resulting from fracturing of a seismic fault. The present study takes into account these dynamic effects. Earth's crust is assumed to be a Kelvin-Voigt material. The seismic source is assumed to be a dislocation in a viscoelastic medium. The fluid motion is described by the classical nonlinear shallow water equations (NSWE) with time-dependent bathymetry. The viscoelastodynamic equations are solved by a finite-element method and the NSWE by a finite-volume scheme. A comparison between static and dynamic tsunami-generation approaches is performed. The results of the numerical computations show differences between the two approaches and the dynamic effects could explain the complicated shapes of tsunami wave trains.Comment: 16 pages, 10 figures, Accepted to Mathematics and Computers in Simulation. Other author's papers can be downloaded at http://www.cmla.ens-cachan.fr/~dutyk

FEM-based linear inverse modeling using a 3D source array to image magma chambers with free geometry. Application to InSAR data from Rabaul Caldera (PNG).

This article is subject to a CC Attribution 3.0 License.In this study, we present a method to fully integrate a family of finite element models (FEMs) into the regularized linear inversion of InSAR data collected at Rabaul caldera (PNG) between February 2007 and December 2010. During this period the caldera experienced a long-term steady subsidence that characterized surface movement both inside the caldera and outside, on its western side. The inversion is based on an array of FEM sources in the sense that the Green’s function matrix is a library of forward numerical displacement solutions generated by the sources of an array common to all FEMs. Each entry of the library is the LOS surface displacement generated by injecting a unity mass of fluid, of known density and bulk modulus, into a different source cavity of the array for each FEM. By using FEMs, we are taking advantage of their capability of including topography and heterogeneous distribution of elastic material properties. All FEMs of the family share the same mesh in which only one source is activated at the time by removing the corresponding elements and applying the unity fluid flux. The domain therefore only needs to be discretized once. This precludes remeshing for each activated source, thus reducing computational requirements, often a downside of FEM-based inversions. Without imposing an a-priori source, the method allows us to identify, from a least-squares standpoint, a complex distribution of fluid flux (or change in pressure) with a 3D free geometry within the source array, as dictated by the data. The results of applying the proposed inversion to Rabaul InSAR data show a shallow magmatic system under the caldera made of two interconnected lobes located at the two opposite sides of the caldera. These lobes could be consistent with feeding reservoirs of the ongoing Tavuvur volcano eruption of andesitic products, on the eastern side, and of the past Vulcan volcano eruptions of more evolved materials, on the western side. The interconnection and spatial distribution of sources find correspondence in the petrography of the volcanic products described in literature and in the dynamics of the single and twin eruptions that characterize the caldera. As many other volcanoes, Rabaul caldera is an active and dangerous volcanic system whose dynamics still need to be understood to effectively predict the behavior of future eruptions. The good results obtained from the application of the method to Rabaul caldera show that the proposed linear inversion based on the FEM array of sources is suitable to generate models of magmatic systems. The method can image in space and time the complex free geometry of the source that generates the deformation, widening our understanding of deformational sources and their dynamics. This takes source modeling a step towards more realistic source models.Peer Reviewe

Homogeneous vs heterogeneous subduction zone models: Coseismic and postseismic deformation

A finite-element model (FEM) incorporating geologic properties characteristic of a subduction zone is compared with FEMs approximating homogeneous elastic half-spaces (HEHS)s to investigate the effect of heterogeneity on coseismic and postseismic deformation predictions for the 1995 Colima-Jalisco M_w =8.0 earthquake. The FEMs are used to compute a coefficient matrix relating displacements at observation points due to unit dislocations of contact-node pairs on the fault surface. The Green's function responses are used to solve the inverse problem of estimating dislocation distributions from coseismic GPS displacements. Predictions from the FEM with heterogeneous material properties, loaded with either of the HEHS dislocation distributions, significantly overestimate coseismic displacements. Postseismic deformation predictions are also sensitive to the coseismic dislocation distribution, which drives poroelastic and viscoelastic relaxation. FEM-generated Green's functions, which allow for spatial variations in material properties, are thus preferable to those that assume a simple HEHS because the latter leads to dislocation distributions unsuitable for predicting the postseismic response

Did a submarine landslide contribute to the 2011 Tohoku tsunami?

Many studies have modeled the Tohoku tsunami of March 11, 2011 as being due entirely to slip on an earthquake fault, but the following discrepancies suggest that further research is warranted. (1) Published models of tsunami propagation and coastal impact underpredict the observed runup heights of up to 40 m measured along the coast of the Sanriku district in the northeast part of Honshu Island. (2) Published models cannot reproduce the timing and high-frequency content of tsunami waves recorded at three nearshore buoys off Sanriku, nor the timing and dispersion properties of the waveforms at offshore DART buoy #21418. (3) The rupture centroids obtained by tsunami inversions are biased about 60 km NNE of that obtained by the Global CMT Project. Based on an analysis of seismic and geodetic data, together with recorded tsunami waveforms, we propose that, while the primary source of the tsunami was the vertical displacement of the seafloor due to the earthquake, an additional tsunami source is also required. We infer the location of the proposed additional source based on an analysis of the travel times of higher-frequency tsunami waves observed at nearshore buoys. We further propose that the most likely additional tsunami source was a submarine mass failure (SMF—i.e., a submarine landslide). A comparison of pre- and post-tsunami bathymetric surveys reveals tens of meters of vertical seafloor movement at the proposed SMF location, and a slope stability analysis confirms that the horizontal acceleration from the earthquake was sufficient to trigger an SMF. Forward modeling of the tsunami generated by a combination of the earthquake and the SMF reproduces the recorded on-, near- and offshore tsunami observations well, particularly the high-frequency component of the tsunami waves off Sanriku, which were not well simulated by previous models. The conclusion that a significant part of the 2011 Tohoku tsunami was generated by an SMF source has important implications for estimates of tsunami hazard in the Tohoku region as well as in other tectonically similar regions

Water waves generated by a moving bottom

Tsunamis are often generated by a moving sea bottom. This paper deals with the case where the tsunami source is an earthquake. The linearized water-wave equations are solved analytically for various sea bottom motions. Numerical results based on the analytical solutions are shown for the free-surface profiles, the horizontal and vertical velocities as well as the bottom pressure.Comment: 41 pages, 13 figures. Accepted for publication in a book: "Tsunami and Nonlinear Waves", Kundu, Anjan (Editor), Springer 2007, Approx. 325 p., 170 illus., Hardcover, ISBN: 978-3-540-71255-8, available: May 200