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r.flexure - Computes lithospheric flexural isostasy


raster, geophysics


r.flexure --help
r.flexure [-l] method=string input=name te=name te_units=string output=name [solver=string] [tolerance=float] [northbc=string] [southbc=string] [westbc=string] [eastbc=string] [g=float] [ym=float] [nu=float] [rho_fill=float] [rho_m=float] [--overwrite] [--help] [--verbose] [--quiet] [--ui]


Allows running in lat/lon: dx is f(lat) at grid N-S midpoint
Allow output files to overwrite existing files
Print usage summary
Verbose module output
Quiet module output
Force launching GUI dialog


method=string [required]
Solution method: Finite Diff. or Superpos. of analytical sol'ns
Options: FD, SAS
input=name [required]
Raster map of loads (thickness * density * g) [Pa]
te=name [required]
Elastic thicnkess: scalar or raster; unis chosen in "te_units"
te_units=string [required]
Units for elastic thickness
Options: m, km
output=name [required]
Output raster map of vertical deflections [m]
Solver type
Options: direct, iterative
Default: direct
Convergence tolerance (between iterations) for iterative solver
Default: 1E-3
Northern boundary condition
Options: 0Displacement0Slope, 0Moment0Shear, 0Slope0Shear, Mirror, Periodic, NoOutsideLoads
Default: NoOutsideLoads
Southern boundary condition
Options: 0Displacement0Slope, 0Moment0Shear, 0Slope0Shear, Mirror, Periodic, NoOutsideLoads
Default: NoOutsideLoads
Western boundary condition
Options: 0Displacement0Slope, 0Moment0Shear, 0Slope0Shear, Mirror, Periodic, NoOutsideLoads
Default: NoOutsideLoads
Eastern boundary condition
Options: 0Displacement0Slope, 0Moment0Shear, 0Slope0Shear, Mirror, Periodic, NoOutsideLoads
Default: NoOutsideLoads
gravitational acceleration at surface [m/s^2]
Default: 9.8
Young's Modulus [Pa]
Default: 65E9
Poisson's ratio
Default: 0.25
Density of material that fills flexural depressions [kg/m^3]
Default: 0
Mantle density [kg/m^3]
Default: 3300

Table of contents


r.flexure computes how the rigid outer shell of a planet deforms elastically in response to surface-normal loads by solving equations for plate bending. This phenomenon is known as "flexural isostasy" and can be useful in cases of glacier/ice-cap/ice-sheet loading, sedimentary basin filling, mountain belt growth, volcano emplacement, sea-level change, and other geologic processes. r.flexure and v.flexure are the GRASS GIS interfaces to the the model gFlex. As both r.flexure and v.flexure are interfaces to gFlex, this must be downloaded and installed. The most recent versions of gFlex are available from, and installation instructions are avaliable on that page via the file.


The parameter method sets whether the solution is Finite Difference ("FD") or Superposition of Analytical Solutions ("SAS"). The Finite difference method is typically faster for large arrays, and allows lithospheric elastic thickness to be varied laterally, following the solution of van Wees and Cloetingh (1994). However, it is quite memory-intensive, so unless the user has a computer with a very large amount of memory and quite a lot of time to wait, they should ensure that they use a grid spacing that is appropriate to solve the problem at hand. Flexural isostatic solutions act to smooth inputs over a given flexural wavelength (see , so if an appropriate solution resolution is chosen, the calculated flexural response can be interpolated to a higher resolution without fear of aliasing.

The flexural solution is generated for the current computational region, so be sure to check g.region before running the model!

input is a 2-D array of loads in a GRASS raster. These are in units of stress, and equal the density of the material times the acceleration due to gravity times the thickness of the column. This is not affected by what you choose for g, later: it is pre-calculated by the user.

te, written in standard text as Te, is the lithospheric elastic thickness.

Several boundary conditions are available, and these depend on if the solution method is finite differece (FD) or superposition of analytical solutions (SAS). In the latter, it is assumed that there are no loads outside of those that are explicitly listed, so the boundary conditions are "NoOutsideLoads". As this is the implicit case, the boundary conditions all default to this.

The finite difference boundary conditions are a bit more complicated, but are largely self-explanitory:

0-displacement-0-slope boundary condition
"Broken plate" boundary condition: second and third derivatives of vertical displacement are 0. This is like the end of a diving board.
First and third derivatives of vertical displacement are zero. While this does not lend itsellf so easily to physical meaning, it is helpful to aid in efforts to make boundary condition effects disappear (i.e. to emulate the NoOutsideLoads cases)
Load and elastic thickness structures reflected at boundary.
"Wrap-around" boundary condition: must be applied to both North and South and/or both East and West. This causes, for example, the edge of the eastern and western limits of the domain to act like they are next to each other in an infinite loop.

All of these boundary conditions may be combined in an way, with the exception of the note for periodic boundary conditions. If one does not want the boundary conditions to affect the solutions, it is recommended that one places the boundaries at lesat one flexural wavelength away from the load.

r.flexure may be run in latitude/longitude coordinates (with the "-l" flag), but its grid constraint is that it can have only one dx and one dy for the entire domain. Thus, it chooses the average dx at the midpoint between the northernmost and southernmost latitudes for which the calculations are made. This assumption can break down at the poles, where the East–West dimension rapidly diminishes.

The Community Surface Dynamics Modeling System, into which gFlex is integrated, is a community-driven effort to build an open-source modeling infrastructure for Earth-surface processes.




Wickert, A. D. (2015), Open-source modular solutions for flexural isostasy: gFlex v1.0, Geoscientific Model Development Discussions, 8(6), 4245–4292, doi:10.5194/gmdd-8-4245-2015.

Wickert, A. D., G. E. Tucker, E. W. H. Hutton, B. Yan, and S. D. Peckham (2011), Feedbacks between surface processes and flexural isostasy: a motivation for coupling models, in CSDMS 2011 Meeting: Impact of time and process scales, Student Keynote, Boulder, CO.

van Wees, J. D., and S. Cloetingh (1994), A Finite-Difference Technique to Incorporate Spatial Variations In Rigidity and Planar Faults Into 3-D Models For Lithospheric Flexure, Geophysical Journal International, 117(1), 179–195, doi:10.1111/j.1365-246X.1994.tb03311.x.


Andrew D. Wickert

Last changed: $Date 2015-11-15 (Sun, 15 Nov 2015)$


Available at: r.flexure source code (history)

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