Planetary gravity, crustal thickness, displacement, stress, and strain calculations in spherical harmonics.
Displacement_strain_planet (DSP) provides several functions and example scripts for generating, among others, gravity, crustal thickness, displacement, lateral density variations, stress, and strain maps on a planet given a set of input constraints such as from observed gravity and topography data.
These functions solve the system of equations introduced in Banerdt (1986) under different assumptions. Various improvements have been made to the model including the possibility to account for finite-amplitude correction and filtering (Wieczorek & Phillips, 1998), lateral density variations at any arbitrary depth and within the surface, flexure, or moho reliefs (Wieczorek et al., 2013), and density difference between the surface topography and crust (Broquet & Wieczorek, 2019), or the addition of equations and constraints (Broquet & Andrews-Hanna, 2024).
The full documentation can be found at https://ab-ares.github.io/Displacement_strain_planet/.
Some of these functions rely heavily on the pyshtools package of Wieczorek & Meschede (2018) that is used to perform the spherical harmonic transforms, Legendre polynomial expansions, and finite-amplitude calculations.
This code is still under development. If you find any bugs or errors in the code, please report them in GitHub or to adrienbroquet at arizona.edu.
For this code, we work on the develop branch and merge it to the main branch (with a new version number) everytime significant addtions/improvements are made. If you plan on making contributions, please base everything on the develop branch.
Moho-relief calculations have been benchmarked to the ctplanet package of Mark Wieczorek.
Displacement calculations have been benchmarked to the analytical model of Broquet & Wieczorek (2019).
Strain calculations reproduce results published in the literature (e.g., Banerdt & Golombek 2000).
Thin_shell_matrix Solve the Banerdt (1986) system of 5 equations under the mass-sheet approximation and assuming that, if any, internal density variations are contained within a spherical shell. The system links 8 parameters expressed in spherical harmonics: the topography, geoid at the surface, geoid at the moho depth, net acting load on the lithosphere, tangential load potential, flexure of the lithosphere, crustal root variations, and internal density variations. Minor corrections have been made in the geoid equations, and to the elastic shell transfer functions following Beuthe (2008).
Thin_shell_matrix_nmax Solve the Banerdt (1986) system of 5 equations with finite-amplitude correction and accounting, if needed, for the presence of density variations within the surface or moho reliefs.
DownContFilter Compute the downward minimum-amplitude or -curvature filter of Wieczorek & Phillips (1998).
corr_nmax_drho Calculate the difference in gravitational potential exterior to relief referenced to a spherical interface (with or without laterally varying density) between the mass-sheet case and when using the finite-amplitude algorithm of Wieczorek & Phillips (1998).
SH_deriv Compute on the fly first and second-order spherical harmonic derivatives with respect to colatitude and longitude.
SH_deriv_store Compute and store first and second-order spherical harmonic derivatives with respect to colatitude and longitude.
Displacement_strains_shtools Calculate the Banerdt (1986) equations to determine strains and stresses from displacements with some corrections following Beuthe (2008). Spherical harmonic derivates are estimated using shtools.
Displacement_strains Calculate the Banerdt (1986) equations to determine strains and stresses from displacements with some corrections following Beuthe (2008). Spherical harmonic derivates are computed using SH_deriv_store. This deprecated method is slow and we recommend using Displacement_strains_shtools.
Principal_strainstress_angle Calculate principal strains, stresses and their principal angles.
Strainstress_from_principal Calculate strains or stresses, from their principal values.
Plt_tecto_Mars Plot the Knampeyer et al. (2006) dataset of extensional and compressional tectonic features on Mars.
Run_demo 
A jupyter notebook that contains example scripts to determine flexure, moho-relief, and strains on Mars under different assumptions, including Airy and Pratt isostasy, or due to the sole presence of a mantle plume.
Mars_crust_displacement A script that demonstrates how to calculate the moho-relief and strains on Mars, as a function of the mean planetary crustal thickness and elastic thickness. The contributions from crustal root variations and displacement are shown assuming an elastic thickness of the lithosphere. We make use of the inferred displacement to predict the principal horizontal strains and principal angle, which are compared to extensional tectonic features mapped by Knampeyer et al. (2006).
Mars_SouthPolarCap_displacement A script that demonstrates how to calculate iteratively the flexure underneath the south polar cap of Mars as a function of elastic thickness and ice density. This computation is similar to that done in e.g., Broquet et al. (2021), accepted in JGR:Planets.
Venus_crust_displacement A script that demonstrates how to calculate the moho-relief and strains on Venus, as a function of the mean planetary crustal thickness and elastic thickness.
If you would like to modify the source code, download the Displacement_strain_planet repository and install using pip (or pip3 depending on your installation).
git clone https://github.com/AB-Ares/Displacement_strain_planet.git
cd Displacement_strain_planet/
pip install .Alternatively, you can install Displacement-strain-planet via pip
pip install Displacement-strain-planet cd examples
jupyter notebook Run_demo.ipynb
python Mars_crust_displacement.pyAdrien Broquet (adrienbroquet@arizona.edu)
You can cite the latest release of the package as: Adrien Broquet. (2024). Displacement_strain_planet: 0.5.0 (Version 0.5.0). Zenodo. http://doi.org/10.5281/zenodo.4916799
@misc{Broquet2024,
author = {Broquet, A.},
title = {{Displacement{\_}strain{\_}planet}: Version 0.5.0},
url = {https://github.com/AB-Ares/Displacement_strain_planet},
doi = {10.5281/zenodo.4916799},
year = {2024}}I would like to thank the SHTools developers for putting together their great open-source code, and Mark Wieczorek, Jeff Andrews-Hanna, and Antony Trinh for stimulating discussion.