Finite Element Method for the Space and Time Fractional Fokker–Planck Equation

The detailed spatial and temporal information of surface deformation detected during volcanic unrest by InSAR images suggests a degree of complexity of volcanic systems (e.g., source geometries and distribution of material properties) that cannot be correctly represented by simple models of a pressure source embedded in an elastic, homogeneous, isotropic half-space.The inversion of deformation data, performed for the characterization of the source of deformation, is based on the model we choose to represent the volcanic system. Therefore the quality of the chosen model influences the source size and its temporal changes estimated through the inversion, and thus their interpretation. In fact, our assumptions about geometries and/or magma and rock properties affect the estimations of changes in magma volumes and reservoir pressure. To obtain a more reliable interpretation of surface signals, it is thus paramount to have more realistic models, where the distribution of material properties is constrained by multiple data sets, with greater flexibility in the definition of sources in space and time.Assuming we could invert InSAR data with models that can deal with a complex and arbitrarily shaped deformation source, how unique could this solution be? How much could we say about the evolution of the deformation source over time? Furthermore, how much information about the spatial complexity of the source and its evolution in space and time would be missed?To answer these questions, we characterize the deformation source from the inversion of InSAR data based on a finite element method (FEM) forward model without an a-priori source geometry. The deformation source is bound by estimating the strength of an amorphous cluster of deformation sources distributed over a grid. This uses the principle of superposition already applied to point or cuboid volume elements, embedded in a homogeneous half-space. Also, the numerical model integrates the cluster-source with a heterogeneous distribution of material properties and the topography.In our study, we quantify the ambiguity in the estimation of arbitrary geometries of sources of deformation composed by clusters of Finite Element Method unit sources distributed over a grid. The regularized least-squares solutions of the steady-state PDEs inverse model are obtained using a COMSOL Multiphysics-based routine. Through the inversion of the InSAR time series of the unrest at Uturuncu volcano (Bolivia), we quantify the ability of the employed cluster-source approach to identify the changes of deformation sources in time.&#160;This research is financed by an individual fellowship of the European Union&#8217;s Horizon 2020 research and innovation programme under the Marie Sk&#322;odowska-Curie grant agreement No. 793811.

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A finite element/Fourier treatment of the Fokker–Planck equation

Journal of Computational Physics ◽

10.1016/j.jcp.2012.06.004 ◽

2012 ◽

Vol 231 (18) ◽

pp. 6192-6206 ◽

Cited By ~ 4

Author(s):

J. Andrew Spencer ◽

Jeong-Young Ji ◽

Eric D. Held

Keyword(s):

Finite Element ◽

Planck Equation ◽

Fokker Planck Equation ◽

Fokker Planck

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