Crustal Deformation Modeling
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Elasto-static Deformation due to Internal Dislocations
The elastic deformation of the crust due to the slip of finite faults can be evaluated using the elastic Green function. Okada (1998) derived the analytic solution for rectangular faults of arbitrary position and orientation in the half space. Meade (2007) derived the analytic solution for a triangular fault, allowing the modeling of more general surfaces. The solution for a complex geometry can be obtained by linear superposition. In general, these analytic solutions correspond to the convolution of the elastic Green function (Love 1936) to a particular source geometry. Our approach here is to evaluate the elastic deformation numerically using a semi-analytic Green function in the Fourier domain (Barbot et al. 2008b, 2009a, 2009b). The method consists in identifying the equivalent body forces (and equivalent surface tractions) that represent the effect of dislocations and in a second step evaluating the convolution between the Green function and the body forces in the Fourier domain. This method takes advantage of the convolution theorem of the Fourier transform whereby the convolution is simply a product in the Fourier domain.
The purpose of the elasto-static deformation code is to validate the formulation of the Green function in the Fourier domain and validate the modeling of strike-slip and dip-slip faults. The Green function is used extensively in other problems, including deformation in a fully heterogeneous semi-infinite solid (Barbot et al. 2009a), and time-dependent 3-D deformation due to nonlinear viscoelasticity, poroelasticity and fault creep (Barbot et al. 2009b). In these applications, the elastic Green function is used to 1) evaluate the instantenous velocity and 2) compute the initial stress condition due to initial fault slip.
The purpose of the elasto-static deformation code is to validate the formulation of the Green function in the Fourier domain and validate the modeling of strike-slip and dip-slip faults. The Green function is used extensively in other problems, including deformation in a fully heterogeneous semi-infinite solid (Barbot et al. 2009a), and time-dependent 3-D deformation due to nonlinear viscoelasticity, poroelasticity and fault creep (Barbot et al. 2009b). In these applications, the elastic Green function is used to 1) evaluate the instantenous velocity and 2) compute the initial stress condition due to initial fault slip.
Elasto-Static Deformation Code: STATICU (last release 02/2010)
The source code of STATICU, used to evaluate static deformation due to fault in a uniform (homogeneous) elastic half space can be downloaded here
Some programs for post processing and visualization with gmt can be downloaded here
The code is written in Fortran 90 and compiles successfully with the intel compiler "ifort", the IBM compiler "xlf" and the GNU fortran compiler "g95". The compilation with "g95" on pcc machines, however, leads to spurious errors at execution time. This problem does not appear on intel machines.
The current version of STATICU allows output of the displacement vector in text form at specific point locations designated by their spatial coordinates (x,y,z) in the computation domain. A map-view output is available in both text and binary formats. The binary format uses the GMT (Generic Mapping Tools) defaults. The GMT output is particularly useful for rapid post-processing and plotting, but the binary format is platform-dependent.
The convolution between the equivalent body forces representing the effect of the internal dislocations is performed in the Fourier domain. The Fourier transforms can be evaluated with three different implementations of the Fast Fourier Transform (FFT), including the Cooley-Tukey FFT (written by Norman Brenner of MIT Lincoln Laboratory, June 1968), the Fast Fourier Transform of the West (FFTW) and the SGI FFT. The FFTW and the SGI FFT are both available in parallel version.
The current version of STATIC is fully parallelized and compatible with both shared-memory and distributed-memory architectures. The parallelization is implemented with MPI and OpenMP. The choice over which implementation to use is done at compilation time.
Some programs for post processing and visualization with gmt can be downloaded here
The code is written in Fortran 90 and compiles successfully with the intel compiler "ifort", the IBM compiler "xlf" and the GNU fortran compiler "g95". The compilation with "g95" on pcc machines, however, leads to spurious errors at execution time. This problem does not appear on intel machines.
The current version of STATICU allows output of the displacement vector in text form at specific point locations designated by their spatial coordinates (x,y,z) in the computation domain. A map-view output is available in both text and binary formats. The binary format uses the GMT (Generic Mapping Tools) defaults. The GMT output is particularly useful for rapid post-processing and plotting, but the binary format is platform-dependent.
The convolution between the equivalent body forces representing the effect of the internal dislocations is performed in the Fourier domain. The Fourier transforms can be evaluated with three different implementations of the Fast Fourier Transform (FFT), including the Cooley-Tukey FFT (written by Norman Brenner of MIT Lincoln Laboratory, June 1968), the Fast Fourier Transform of the West (FFTW) and the SGI FFT. The FFTW and the SGI FFT are both available in parallel version.
The current version of STATIC is fully parallelized and compatible with both shared-memory and distributed-memory architectures. The parallelization is implemented with MPI and OpenMP. The choice over which implementation to use is done at compilation time.
Example deformation model and benchmark
The following simulation illustrates the possiblity to model complex rupture geometry. I consider the rupture of the 1992 Landers, CA earthquake. Figure A shows the surface displacement model evaluated with program STATICU. The computational grid dimension is 512x512x512 with a 300m sampling. Figure B shows the numerical error.
The arrows indicate horizontal displacement and the color represent vertical displacement (positive up). Figure is generated using the post-processing software provided above. Some more results and comparison with analytical solution for the Landers earthquake can be found in bm_landers.pdf [US mirror].
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The arrows indicate horizontal displacement and the color represent vertical displacement (positive up). Figure is generated using the post-processing software provided above. Some more results and comparison with analytical solution for the Landers earthquake can be found in bm_landers.pdf [US mirror].