2D Strain Rate Calculators

This library contains several methods to compute geodetic strain from a GPS velocity field. It is intended to be used as a means to compare various strain modeling techniques on the same input data (see https://files.scec.org/s3fs-public/0930SandwellUCERF_strain.pdf for an example showing the variability of strain modeling techniques). Several of the strain methods are borrowed from other authors, GitHub repositories, and papers. If you use the contributed strain techniques, credit should be given to the original authors accordingly.

Requirements and Installation:

• Python3, numpy, scipy, matplotlib, pygmt
• GMT5 or higher: https://www.generic-mapping-tools.org/
• Third-party Matlab or Fortran codes may be required depending on the specific strain technique you select, as detailed further below
• Optional: Jupyter notebook is needed to run the tutorial

First, clone the Strain_2D repository onto your computer and get into the top level directory.

To handle dependencies, it is easiest to create a conda environment for this software, although that step is not strictly required if you have collected all the dependencies manually. The easiest method to create an environment (after cloning the repo onto your computer and navigating to the top-level directory) looks like: conda env create -f requirements.yml conda activate Strain_2D

Then, to install this library within the newly created conda environment, run in the top level directory of the repo: pip install .

We have not yet tested the installation on Windows. Let us know if you have used the software on Windows!

### Usage: The main executable for this library is strain_rate_compute.py. The behavior of the program is controlled using a config file that specifies inputs/outputs, general strain options, and any required parameters for various strain rate techniques. You can print a sample config file from the main executable strain_rate_compute.py under the flag --print_config (also try --help for more information).

An example run-string would be: strain_rate_compute.py config.txt

Input velocities must be in a text file. They must be a space-separated table with format as follows: ```

lon(deg) lat(deg) VE(mm) VN(mm) VU(mm) SE(mm) SN(mm) SU(mm) name(optional)

In general, users are expected to clean their velocities before using Strain_2D, according to their own needs in their own projects. Within Strain_2D, the velocities will be filtered to include only those within the bounding box provided in the config parameterrange_data```.

Outputs:

Output strain components and derived quantities (invariants, eigenvectors) are written as pixel-node-registered netCDF files and/or text files, and plotted in PyGMT.

To see what's inside the large netCDF file, you can call: ```bash

!/bin/bash

infile="gpsgridder_strain.nc" # any output .nc file ncdump -h $infile ```

To convert one layer of the netCDF file (such as dilatation) into a valid pixel-node-registered grdfile, you can extact with GDAL if you have it. ```bash

!/bin/bash

outfile="dilatation.grd" gmtrange="-121.01/-113.99/30.99/37.01" # technically half a pixel outside of each bound, for pixel-node-registration gmt grdedit $infile=gd?HDF5:"$infile"://dilatation -R$gmtrange -T -G$outfile # send layer out pixel-node-registered grdfile, using GDAL. gmt grdedit $outfile -Ev # top-bottom flip for latitude increasing vs column number increasing ```

Contributing

If you're using this library and have suggestions, let me know! I'm happy to work together on the code and its applications. We are working on the contributing.md for guidelines.

See the section on API below if you'd like to contribute a method.

Supported strain methods:

For more information on the parameters required by each method, see Full Parameter Documentation. 1. delaunay_flat: Delaunay Triangulation, the simplest method. The equations can be found in many papers, including Cai and Grafarend (2007), Journal of Geodynamics, 43, 214-238. No parameters are required to use this method.

1. delaunay: a generalization of the Delaunay Triangulation for a spherical earth. This Python implementation is based on Savage et al., JGR October 2001, p.22,005, courtesy of Bill Hammond's matlab implementation. No parameters are required to use this method.

2. visr: The "VISR" method is a fortran code for the interpolation scheme of Zheng-Kang Shen et al., Strain determination using spatially discrete geodetic data, Bull. Seismol. Soc. Am., 105(4), 2117-2127, doi: 10.1785/0120140247, 2015. http://scec.ess.ucla.edu/~zshen/visr/visr.html. You can download the source code, which must be compiled and linked on your own system, for example by : gfortran -c voronoi_area_version.f90 / gfortran visr.f voronoi_area_version.o -o visr.exe. Four additional config parameters are required to use this method.

3. simple_visr: A simplified Python implementation of the VISR method (Shen et al., 2015), contributed by Tobias Kohne.

4. gpsgridder: based on a thin-sheet elastic interpolation scheme from Sandwell, D. T., and P. Wessel (2016), Interpolation of 2-D vector data using constraints from elasticity, GRL. The implementation of the code is in GMT. Three additional config parameters are required to use this method.

5. locavggrad: the weighted nearest neighbor algorithm of Mong-Han Huang and implemented in Handwerger, A. L., Huang, M. H., Fielding, E. J., Booth, A. M., & Bürgmann, R. (2019). A shift from drought to extreme rainfall drives a stable landslide to catastrophic failure. Scientific reports, 9(1), 1-12. Two additional config parameters are required to use this method.

6. wavelets: a wavelet-based matlab program from Tape, Muse, Simons, Dong, Webb, "Multiscale estimation of GPS velocity fields," Geophysical Journal International, 2009 (https://github.com/carltape/surfacevel2strain). Needs a matlab installation and some manual run steps.

7. geostats: a method based on traditional geostatistical anlysis, using kriging to do the interpolation. This method requires the user to set one model type and three parameters. The model is the type of correlation structure the user expects to exist in the underlying field. A "Nugget" model is a white-noise model, "Exponential" models a continuous but non-differentiable process, and "Gaussian" is both continuous and differentiable. All models require a nugget, which represents the level of point-wise variance in the observations, which for GNSS velocities is the same as data uncertainty. The Gaussian and Exponential models also require a "sill" and "range" to be specified. The sill characterizes the overall mean variance in the dataset, and should be specified as the total variance of the observations minus the nugget. Note that the nugget can be thought of as the variance of the data uncertainties, and the sill is the variance of the data itself. The range is the (isotropic) spatial correlation length scale. Currently, we only implement an isotropic version, although anisotropic and even spatially-varying methods can be used.

Not included methods:

If you have another strain method that you'd be willing to contribute, I would love to work with you to include it! More methods results in a more robust estimate of strain rate variability. I have not included the following techniques for the reasons given:

1. Spline: based on Python's built-in spline interpolation of the velocity field. I have found that it doesn't always produce reasonable results.
2. NDInterp: based on Numpy's linear interpolation of the velocity field. It turns out to be basically the same as Delaunay.
   

Units and Sign Conventions

This library uses the following units, definitions, and sign conventions: * station velocities: units of mm/yr in the internal format. * strain rates: units of nanostrain / yr (1e-9 / yr). * Shear strain is tensor shear strain rate (not engineering shear strain rate)

math \epsilon_{xx} = \frac{du}{dx}. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \epsilon_{xy} = \frac{1}{2} * (\frac{dv}{dx} + \frac{du}{dy}). \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \epsilon_{yy} = \frac{dv}{dy}.

• Rotation: units of (1e-9 radians)/yr, or radians / Ga. Positive means clockwise. math W = \frac{1}{2} * (\frac{dv}{dx} - \frac{du}{dy})

• Second invariant: units of (nanostrain/yr)^2 (very unusual units; usually plotted on log scale for clarity). math I2 = \frac{1}{2} * (\epsilon_{ii} * \epsilon_{jj} - {\epsilon_{ij}}^2) When you expand the Einstein summation notation, this becomes: math I2 = \epsilon_{xx} * \epsilon_{yy} - {\epsilon_{xy}}^2 Which is the same as: math I2 = \frac{1}{2} * ( tr(\epsilon_{ij})^2 - tr(\epsilon_{ij}^2) )

• Dilatation: units of nanostrain/year.

  • Positive means extension and negative means shortening. math Dilatation = \epsilon_{xx} + \epsilon_{yy}

• Max Shear: units of nanostrain / year.

  • Tensor shear strain rate (not engineering shear strain rate) math \epsilon_{max} = \frac{1}{2} * \sqrt{{(\epsilon_{xx} - \epsilon_{yy})}^2 + 4{\epsilon_{xy}}^2}

Internal Library API

If you're interested in contributing your own type of strain computation, I am happy to add new methods to the library. You would need to build a Python 'compute' function that matches the API of the other compute functions in the repository (see strain/models/strain_2d.py for template).

``` class yourstrainmethod(strain2d.Strain2d): def init(self, Params): # perform various types of parameter-setting and input-verification, # different for each technique return

def compute(self, myVelfield):
    [Ve, Vn, rot_grd, exx_grd, exy_grd, eyy_grd, velfield, residual_velfield] = your_strain_method_compute(myVelfield... etc);
    return [Ve, Vn, rot_grd, exx_grd, exy_grd, eyy_grd, velfield, residual_velfield];

```

where: * myVelfield is a 1D list of StationVel objects (very easy to use). * lons is a 1D array of longitudes, in increasing order * lats is a 1D array of latitudes, in increasing order * Ve and Vn are 2D arrays of geodetic velocities, if the method computed interpolated velocities (not every method does this) * rotgrd, exxgrd, exygrd, eyygrd are 2D arrays that correspond to lons/lats grids * exx/exy/eyy have units of nanostrain/yr * rot has units of radians/Ga with positive meaning clockwise

Example:

To reproduce the figure in the README:
1. In the example/ directory, run: strain_rate_compute.py --print_config to print an example config file (similar to the one provided here in the repo). 2. Run: strain_rate_compute.py example_strain_config.txt to compute delaunay strain rate using the parameters in the newly-created config file. 3. Change the method in the config file to try other methods, and re-run each time for different strain calculations with different parameters. Try with gpsgridder, delaunay, and loc_avg_grad. If you have the visr executable on your system, try with visr as well. 4. Run: strain_rate_comparison.py example_strain_config.txt to compute average strain rate maps from several results. 5. Navigate into Display_output/ 6. Run: ./comparison_rows_example.sh -125/-121/38/42 to view a GMT plot with several strain rate calculations in Northern California together.

Citing

If you use this library in your research, you can cite this work with:

Maurer and Materna (2023), Quantification of geodetic strain rate uncertainties and implications for seismic hazard estimates, Geophysical Journal International, 234, 3, p. 2128-2142, doi: 10.1093/gji/ggad191.

The code itself can be cited with a DOI:

Materna, K., and J. Maurer (2023). Strain2D (version 1.1.1). https://code.usgs.gov/kmaterna/Strain2D. Archived at DOI: 10.5066/P9JJW0DY

or

Materna, K., J. Maurer, and L. Sandoe (2021). Strain-2D (version 1.0.0). https://github.com/kmaterna/Strain_2D. Archived at DOI: 10.5281/zenodo.5240908