Abstract
An important component of F3DWI is a numerical method for solving the 3D elastodynamic equation. In this chapter, I will cover the details of a 3D finite-difference code for solving the (visco)elastic wave equation, the Anelastic Wave Propagation (AWP) code developed and maintained by SCEC scientists. The AWP code is a fourth-order staggered-grid finite-difference code that has been thoroughly tested, validated and optimized for computational efficiency. I will cover the mathematical formulation behind this code in Sect. 2.1 and give a tutorial about how to use this code in the context of F3DWI in Sect. 2.2. In Sect. 2.3, En-Jui and I will show you how to use this code in a realistic application, which involves the validation of the three SCEC Community Velocity Models for Southern California.
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Notes
- 1.
The TeraGrid was a grid-computing infrastructure combining HPC resources at 11 different partner sites. It was supported by the US National Science Foundation and operated from 2004 through 2011. The TeraGrid project was succeeded by the Extreme Science and Engineering Discovery Environment (XSEDE), a partnership involving HPC resources from 17 institutions, which has been supported by NSF since 2011.
- 2.
The overall problem size is fixed and the total number of processors is increased. Therefore the workload on each processor decreases with the number of processors.
- 3.
The workload for each processor is constant and the overall problem size increases in proportion with the number of processors.
- 4.
The “M8” earthquake simulation was the largest earthquake simulation coordinated by SCEC. It used a uniform mesh with 40-m grid spacing and the simulation volume was 810-km long, 405-km wide and 85-km deep. The total number of finite-difference grid points used in this simulation was about 436 billion. The full simulation took about 24 h and produced about 6 min of the 3D wavefields in the modeling volume.
- 5.
The Gordon Bell prize is awarded by the Association of Computing Machinery (ACM) each year to recognize outstanding achievements in high-performance computing, especially in applying HPC technology to science and engineering applications.
- 6.
The solution we are seeking must be sufficiently smooth and have regular second-order spatial derivatives. In the weak formulation, the solution we are seeking may have only first-order derivatives (Evans 2010).
- 7.
Causality implies that the complex function is analytic. The Kramers-Kronig relation indicates that the real and imaginary parts of any analytic function are related through the Hilbert transform, therefore the full function can be reconstructed by knowing just one of its parts. A corollary is that an analytic function can be reconstructed by knowing only its phase.
- 8.
In the literature these variables are also known as relaxation functions, anelastic functions, internal variables, additional functions, etc.
- 9.
- 10.
The Padé approximant provides an approximation of a function using rational functions and often gives better accuracy than a truncated Taylor series (Baker and Graves-Morris 1996).
- 11.
The solutions of some differential equations may contain both rapid-varying, transient modes and longer-period slow modes. The numerical algorithms for solving such equations need to use very small time steps to ensure numerical stability even after the rapid-varying, transient modes are no longer visible in the solutions. Such differential equations are said to be stiff (Heath 2005).
- 12.
The volumetric stress is also called the mean stress or the hydrostatic stress in the literature.
- 13.
The stress imaging formulation is sometimes called the zero-stress formulation in the literature.
- 14.
\(\mathbf {{A}}:\mathbf {{B}}=\sum _{i}\sum _{j}A_{ij}\bar {B}_{ij}\), where complex matrices \(\mathbf {{A}}\) and \(\mathbf {{B}}\) have the same dimension, \(\bar {{B}}_{ij}\) is the complex conjugate of \(B_{ij}\). For the fourth-order elastic tensor \(\mathbf {{c}}\), its double dot product with a real second-order tensor \(\boldsymbol {{\phi }}\) can be calculated as \(\mathbf {{c}}\,:\,\boldsymbol {{\phi }}=\sum _{k}\sum _{l}c_{ijkl}\phi _{kl}\) and the result is a second-order tensor.
- 15.
If the medium is isotropic, for the P wave we have \(\mathbf {{A}}\times \mathbf {{k}=\mathbf {{0}}}\) and \(|\mathbf {{k}|=}\omega /\alpha \), and for the S wave we have \(\mathbf {{A}}\cdot \mathbf {{k}}=\mathbf {{0}}\) and \(|\mathbf {{k}}|=\omega /\beta \), where \(\alpha \) and \(\beta \) are the P- and S-wave velocities.
- 16.
The characteristic length scale of the variations in the material properties is much larger than the grid spacing.
- 17.
- 18.
- 19.
- 20.
Python is a high-level programming language and is often used as a scripting language. Python interpreters are available on almost every platform. Details about the Python language can be found at https://www.python.org/ and http://en.wikipedia.org/wiki/Python_(programming_language).
- 21.
Details about UTM can be found at http://www.dmap.co.uk/utmworld.htm and also on Wikipedia.
- 22.
Parallelization based on the domain-decomposition approach requires us to exchange stress and particle velocity fields on the boundaries of the sub-meshes between neighboring MPI processes. The communication overhead is therefore proportional to the number of grid points on the boundaries of each sub-mesh. Suppose the number of grid points in each dimension of a sub-mesh is \(NXT\), \(NYT\) and \(NZT\), the total number of grid points on the boundaries of each sub-mesh is therefore \(S=2\left (NXT\times NYT+NXT\times NZT+NYT\times NZT\right )\). The total number of grid points of the whole sub-mesh is \(V=NXT\times NYT\times NZT\). For a given \(V\), we need to minimize \(S\) in order to minimize the communication overhead. To minimize \(S\), we can replace the \(NZT\) in the expression for \(S\) with \(V/\left (NXT\times NYT\right )\) and set the first-order partial derivatives of \(S\) with respect to \(NXT\) and \(NYT\) to zeros to obtain two equations for \(NXT\) and \(NYT\). The solution of these two equations gives \(NXT=NYT\). The same analysis can also be applied to \(NXT\) and \(NZT\) and the solution gives \(NXT=NZT\). Therefore the minimum number of boundary grid points of the sub-mesh for a given \(V\) is obtained when \(NXT=NYT=NZT\) (i.e., a cube-shaped sub-mesh).
- 23.
Under the bash shell, all directories listed in the PATH environment variable are automatically searched for executables. The bin directory can be added to PATH by adding the line export PATH=/home/pochen/wrk/F3DWI/bin:$PATH to the .bashrc or .bash_profile file under the home directory.
- 24.
The GMT is a set of command-line binary executables for generating and processing figures related to geographic data sets. More detailed information can be found at http://gmt.soest.hawaii.edu/.
- 25.
The sizeof function used on line 25–27 returns the number of bytes for a given data type. On most systems, the float data type is four-byte long. At the current stage the awp code only reads binary material property files written in four-byte-long floating-point numbers.
- 26.
The MPI functions MPI_Reduce and MPI_Allreduce can combine the values provided in the input buffer of each MPI process, using a specified operation, such as taking the minimum or maximum, and return the combined value to the output buffer of either the root process (in the case of MPI_Reduce) or all processes (in the case of MPI_Allreduce).
- 27.
On large computing systems, the computational tasks (i.e., batch jobs) are allocated among the available computing resources through a job scheduling software. One example is the Portable Batch System (PBS) software. To interact with the job scheduling system (e.g., submit jobs, monitor the progress of jobs, cancel previously submitted jobs), we often need to write short scripts.
- 28.
The CFL condition is a necessary condition for an explicit time-integration scheme to be stable (Courant et al. 1967). Roughly speaking, for a given mesh with grid spacing \(h\), the time-step length \(\Delta t\) must be sufficiently small in order for the numerical scheme to be causal.
- 29.
In F3DWI, a geodetic location is specified using real-valued longitude and latitude in degrees, the western hemisphere has negative longitude and the southern hemisphere has negative latitude.
- 30.
The depth is specified in km and should always be larger than or equal to 0. The free-surface is at 0 km depth.
- 31.
The A&R moment tensor format is the format specified on page 112 in (Aki and Richards 2002). Note that the coordinate system for this format (\(x\): North, \(y\): East, \(z\): Down) is different from the one used in F3DWI. Necessary coordinate transforms are done automatically in F3DWI.
- 32.
The A&R focal mechanism format is the format specified on page 101 in (Aki and Richards 2002). The strike, dip and rake angles should be in degrees in the source-receiver file.
- 33.
We can execute any of the scripts in py and any sequential binary executables in bin from inside the project JOBS directory, where the parameter file and other input/output files are stored. In such a case, the current directory is the project JOBS directory.
- 34.
A good introduction to PBS can be found at http://hpc.sissa.it/pbs/pbs.html.
- 35.
On your system, you may have different parallel job launchers, such as mpirun, mpirun_rsh, mpirun.lsf, aprun, runjob. You may need to look into the user guide of your cluster or consult with your system administrator.
- 36.
9 components \(\times \) 45 grid points \(\times \) 500 time steps \(\times \) 4 bytes = 810000 bytes. You can verify the exact byte size of the file by typing the command ls -l SX96PVRDM001001000 in the terminal.
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Chen, P., Lee, EJ. (2015). Anelastic Wave Propagation (AWP). In: Full-3D Seismic Waveform Inversion. Springer Geophysics. Springer, Cham. https://doi.org/10.1007/978-3-319-16604-9_2
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