# grdflexure¶

Compute flexural deformation of 3-D surfaces for various rheologies

## Synopsis¶

**gmt grdflexure** *input*
**-D***rm*/*rl*[/*ri*]/*rw*
**-E**[*Te*[**k**][/*Te2*[**k**]]]
**-G***outgrid*
[ **-A***Nx*/*Ny*/*Nxy* ]
[ **-C****p**|**y***value* ]
[ **-F***nu_a*[/*h_a*[**k**]/*nu_m*] ]
[ **-H***rhogrid*
[ **-L***list* ]
[ **-M***tm* ]
[ **-N***params* ]
[ **-Q** ]
[ **-S***beta* ]
[ **-T***t0*[/*t1*/*dt*[**+l**]]|*file* ]
[ **-V**[*level*] ]
[ **-W***wd*][**k**]
[ **-Z***zm*][**k**]
[ **-h**headers ]
[ **-f**flags ]
[ **--PAR**=*value* ]

**Note:** No space is allowed between the option flag and the associated arguments.

## Description¶

**grdflexure** computes the deformation due to a topographic load \(h(\mathbf{x})\)
for five different types of rheological foundations, all involving *constant thickness thin plates*:

An elastic plate overlying an inviscid half-space,

An elastic plate overlying a viscous half-space (Firmoviscous or Kelvin-Voigt),

An elastic plate overlying a viscous layer over a viscous half-space (Firmoviscous or Kelvin-Voigt),

A viscoelastic plate overlying an inviscid half-space (Maxwell solid),

A general linear viscoelastic model with an initial and final elastic plate thickness overlying an inviscid half-space.

These conditions will require the *elastic* [1; \(\Phi_e(\mathbf{k})\)],
*firmoviscous* [2,3; \(\Phi_{fv}(\mathbf{k},t)\)],
*viscoelastic* [4; \(\Phi_{ve}(\mathbf{k},t)\)],
and *general linear* (viscoelastic) response functions [5; \(\Phi_{gl}(\mathbf{k},t)\)]
If the (visco)elastic plate vanishes (zero thickness) then we obtain Airy isostasy
(1,4) or a purely *viscous* response (2,3). Temporal evolution can
also be modeled by providing incremental load grids for select times and specifying a
range of model output times. A wide range of options allow specifying the desired
rheology and related constants, including in-plate forces.

## Required Arguments¶

*input*Supplies the topographic load information in one of many forms:

A single 2-D binary grid file with the topography of the load (in meters); (See Grid File Formats).

If

**-T**is used,*input*may be a filename*template*(See Name Template for details). The load times will thus coincide with the times given via**-T**(but not all times need to have a corresponding file).A file list given as

*flist***+l**, where*flist*is an ASCII table with one load time and topography grid filename per record (e.g., as produced by grdseamount**-M**). These load times can be different from the evaluation times given via**-T**. For load time format, see**-T**.**Note**: If*flist*has three trailing words the the second will be interpreted as a load density grid name and used for that layer instead of the fixed*rl*setting in**-D**. The last trailing word is a formatted age string.A file list with extension “.lis” does not need the

**+l**modifier and will be automatically recognized as a file list.

**-D***rm*/*rl*[/*ri*]/*rw*Sets density for mantle, load, infill, and water (or air). If

*ri*differs from*rl*then an approximate solution will be found. If*ri*is not given then it defaults to*rl*. Values may be given in km/m^3 or g/cm^3.

**-E**[*Te*[**k**][/*Te2*[**k**]]Sets the elastic plate thickness (in meter); append

**k**for km. If the elastic thickness exceeds 1e10 it will be interpreted as a flexural rigidity*D*(by default,*D*is computed from*Te*, Young’s modulus, and Poisson’s ratio; see**-C**to change these values). If just**-E**is given and**-F**is used it means no plate is given and we will return a purely viscous response with or without an asthenospheric layer. Select a general linear viscoelastic response by supplying both an initial and final elastic thickness*Te2*; this response also requires**-M**.

**-G***outgrid*[=*ID*][**+d***divisor*][**+n***invalid*][**+o***offset*|**a**][**+s***scale*|**a**][:*driver*[*dataType*][**+c***options*]]

If

-Tis set thenoutgridmust be a filename template (See Name Template for details). Optionally, append =IDfor writing a specific file format (See full description). The following modifiers are supported:

+d- Divide data values by givendivisor[Default is 1].

+n- Replace data values matchinginvalidwith a NaN.

+o- Offset data values by the givenoffset, or appendafor automatic range offset to preserve precision for integer grids [Default is 0].

+s- Scale data values by the givenscale, or appendafor automatic scaling to preserve precision for integer grids [Default is 1].Note: Any offset is added before any scaling.

+saalso sets+oa(unless overridden). To write specific formats via GDAL, use =gdand supplydriver(and optionallydataType) and/or one or more concatenated GDAL-cooptions using+c. See the “Writing grids and images” cookbook section for more details.

## Optional Arguments¶

**-A***Nx*/*Ny*/*Nxy*Specify in-plane compressional or extensional forces in the

*x*- and*y*-directions, as well as any shear force [no in-plane forces]. Compression is indicated by negative values, while extensional forces are specified using positive values. Values are expected in Pa·m since**N**is the depth-integrated horizontal stresses.

**-C****p**|**y***value*Append

**p**or**y**to change the current value of Poisson’s ratio [0.25] or Young’s modulus [7.0e10 N/m^2], respectively.

**-F***nu_a*[/*h_a*[**k**]/*nu_m*]Specify a firmoviscous model in conjunction with an elastic plate thickness specified via

**-E**. Just give one viscosity (*nu_a*) for an elastic plate over a viscous half-space, or also append the thickness of the asthenosphere (*h_a*) and the lower mantle viscosity (*nu_m*), with the first viscosity now being that of the asthenosphere. Give viscosities in Pa·s. If used, give the thickness of the asthenosphere in meter; append**k**for km. Cannot be used in conjunctions with**-M**.

**-H***rhogrid*Supply optional variable load density grid. It can be a single grid or a grid name template (see Name Template for details). Requires

*rho_l*be set to - in**-D**.**Note**: If*input*is given as a list file then the optional density grids must be given as part of the list and not via**-H**.

**-L***list*Write the names and evaluation times of all grids that were created to the text file

*list*. Requires**-T**. The leading numerical column will be time in years, while the last trailing text word is formatted time. The output records thus contain*time flexuregrid timetag*.

**-N**[**a**|**f**|**m**|**r**|**s**|*nx/ny*][**+a**|**d**|**h**|**l**][**+e**|**n**|**m**][**+t***width*][**+v**][**+w**[*suffix*]][**+z**[**p**]]Choose or inquire about suitable grid dimensions for FFT and set optional parameters. Control the FFT dimension:

**-Na**lets the FFT select dimensions yielding the most accurate result.**-Nf**will force the FFT to use the actual dimensions of the data.**-Nm**lets the FFT select dimensions using the least work memory.**-Nr**lets the FFT select dimensions yielding the most rapid calculation.**-Ns**will present a list of optional dimensions, then exit.**-N***nx/ny*will do FFT on array size*nx/ny*(must be >= grid file size). Default chooses dimensions >= data which optimize speed and accuracy of FFT. If FFT dimensions > grid file dimensions, data are extended and tapered to zero.Control detrending of data: Append modifiers for removing a linear trend:

**+d**: Detrend data, i.e. remove best-fitting linear trend [Default].**+a**: Only remove mean value.**+h**: Only remove mid value, i.e. 0.5 * (max + min).**+l**: Leave data alone.Control extension and tapering of data: Use modifiers to control how the extension and tapering are to be performed:

**+e**extends the grid by imposing edge-point symmetry [Default],**+m**extends the grid by imposing edge mirror symmetry**+n**turns off data extension.Tapering is performed from the data edge to the FFT grid edge [100%]. Change this percentage via

**+t***width*. When**+n**is in effect, the tapering is applied instead to the data margins as no extension is available [0%].Control messages being reported:

**+v**will report suitable dimensions during processing.Control writing of temporary results: For detailed investigation you can write the intermediate grid being passed to the forward FFT; this is likely to have been detrended, extended by point-symmetry along all edges, and tapered. Append

**+w**[*suffix*] from which output file name(s) will be created (i.e.,*ingrid_prefix.ext*) [tapered], where*ext*is your file extension. Finally, you may save the complex grid produced by the forward FFT by appending**+z**. By default we write the real and imaginary components to*ingrid*_real.*ext*and*ingrid*_imag.*ext*. Append**p**to save instead the polar form of magnitude and phase to files*ingrid*_mag.*ext*and*ingrid*_phase.*ext*.

**-M***tm*Specify a viscoelastic model in conjunction with a plate thickness specified via

**-E**. Append the Maxwell time*tm*for the viscoelastic model (in years); add**k**for kyr and**M**for Myr. Cannot be used in conjunctions with**-F**.

**-Q**Do not make any flexure calculations but instead take the chosen response function given the parameters you selected and evaluate it for a range of wavenumbers and times; see the note on transfer functions below.

**-S***beta*Specify a starved moat fraction in the 0-1 range, where 1 means the moat is fully filled with material of density

*ri*while 0 means it is only filled with material of density*rw*(i.e., just water) [1].

**-T***t0*[/*t1*/*dt*[**+l**]]|*file*Specify

*t0*,*t1*, and time increment (*dt*) for a sequence of calculations [Default is one calculation, with no time dependency]. For a single specific time, just give start time*t0*. Default*unit*is years; append**k**for kyr and**M**for Myr. For a logarithmic time scale, append**+l**and specify*n*steps instead of*dt*. Alternatively, give a*file*with the desired times in the first column (these times may have individual units appended, otherwise we assume year). We then write a separate model grid file for each given time step; see**-G**for output and Name Template for the file template format.

**-V**[*level*]Select verbosity level [

**w**]. (See full description) (See cookbook information).

**-W***wd*[**k**]Set reference water depth for the undeformed flexed surface in m. Must be positive. [0]. Append

**k**to indicate km. If**-W**is used and your load exceeds this depth then we scale the subaerial part of the load to account for the change in surrounding density (air vs water).

**-Z***zm*[**k**]Specify reference depth to flexed surface (e.g., Moho) in m; append

**k**for km. Must be positive. [0]. We subtract this value from the flexed surface before writing the results.**-f**flagsGeographic grids (dimensions of longitude, latitude) will be converted to meters via a “Flat Earth” approximation using the current ellipsoid parameters.

**-h**[**i**|**o**][*n*][**+c**][**+d**][**+m***segheader*][**+r***remark*][**+t***title*] (more …)Skip or produce header record(s).

**-^**or just**-**Print a short message about the syntax of the command, then exit (

**Note**: on Windows just use**-**).**-+**or just**+**Print an extensive usage (help) message, including the explanation of any module-specific option (but not the GMT common options), then exit.

**-?**or no argumentsPrint a complete usage (help) message, including the explanation of all options, then exit.

**--PAR**=*value*Temporarily override a GMT default setting; repeatable. See gmt.conf for parameters.

## Name Template¶

The format statements allowed in grid file templates require you to follow these rules:

To use the formatted time-tag as part of the file name you must use just a single %s format as part of the template (e.g., smt_%s.grd).

If you want to control the numerical formatting of the names but still have the common time unit appended then you must compose a template that has a floating point format before a later %c format for the unit. For example, smt_%05.1f%c.grd will create names like smt_001.1M.grd names. The times will be scaled to match the unit.

If you do not want any units then simply give a template with just one floating point format, e.g., smt_%05.1f_name.grd. The times will be used as is (i.e, unscaled).

For details on the format statements, see printf C language format syntax.

## Grid Distance Units¶

If a Cartesian grid does not have meter as the horizontal unit, append **+u***unit*
to the input file name to convert from the specified unit to meter. E.g., appending
**+uk** to the load file name will scale the grid x,y coordinates from km to meter. If your
grid is geographic, convert distances to meters by supplying **-f**flags instead.
netCDF COARDS geographic grids will automatically be recognized as geographic.

## Considerations¶

The calculations are done using a rectangular Cartesian FFT operation. If your geographic region is close to either pole, you should consider using a Cartesian setup instead; you can always project it back to geographic using grdproject.

## Transfer Functions¶

If **-Q** is given we perform no actual flexure calculations and no input data file is required.
Instead, we write the chosen transfer functions \(\Phi(\mathbf{k},t)\) to 7 separate files for
7 different Te values (1, 2, 5, 10, 20, 50, and 100 km). The first two columns are
always wavelength in km and wavenumber (in 1/m) for a 1:1:3000 km range. The transfer
functions are evaluated for 12 different response times: 1k, 2k, 5k, 10k, 20k, 50k,
100k, 200k, 500k, 1M, 2M, and 5M years. For a purely elastic response function
we only write the transfer function once per elastic thickness (in column 3). The 7 files are named
grdflexure_transfer_function_te_*te*_km.txt, where *te* is replaced by the 7 elastic thicknesses
in km (and 0 if **-E**[0] was used for a viscous response only).

## Examples¶

We will use a Gaussian seamount load to demonstrate **grdflexure**. First, we make
a grid of for that shape by placing a Gaussian truncated seamount at position (300,300)
with a radius of 50 km and height of 5000 m:

```
echo 300 300 0 40 40 5000 | gmt grdseamount -R0/600/0/600+uk -I1000 -Gsmt.nc t.txt -Dk -E -F0.1 -Cg
```

To compute elastic plate flexure from the load *smt.nc*,
for a 10 km thick plate with typical densities, try:

```
gmt grdflexure smt.nc -Gflex.nc -E10k -D2700/3300/1035
```

To see how in-plane stresses affect the result, we use **-A**. Remember that we need to depth-
integrated forces, not pressures, hence we try:

```
gmt grdflexure smt.nc -Gflex.nc -E10k -D2700/3300/1035 -A-4e11/2e11/-1e12
```

To compute viscoelastic plate flexure from the load *smt.nc*,
for a 20 km thick plate with typical densities and a Maxwell time of 40kyr, try:

```
gmt grdflexure smt.nc -Gflex.nc -E20k -D2700/3300/1035 -M40k
```

To compute firmoviscous plate flexure from the load *smt.nc*,
for a 15 km thick plate with typical densities overlying a viscous mantle with viscosity 2e21, try:

```
gmt grdflexure smt.nc -Gflex.nc -E15k -D2700/3300/1035 -F2e21
```

To compute the general linear viscoelastic plate flexure from the load *smt.nc*,
for an initial Te of 40 km and a final Te of 15 km with typical densities and a Maxwell time of 100 kyr, try:

```
gmt grdflexure smt.nc -Gflex.nc -E40k/15k -D2700/3300/1035 -M100k
```

To just compute the firmoviscous response functions using the specified rheological values, try:

```
gmt grdflexure -D3300/2800/2800/1000 -Q -F2e20
```

The following are not user-reproducible but shows the kind of calculations that can be done. To compute the firmoviscous response to a series of incremental loads given by file name and load time in the table l.lis at the single time 1 Ma using the specified rheological values, try:

```
gmt grdflexure -T1M =l.lis -D3300/2800/2800/1000 -E5k -Gflx/smt_fv_%03.1f_%s.nc -F2e20 -Nf+a
```

## Theory of Response Functions¶

Deformation \(w(\mathbf{x})\) caused by topography \(h(\mathbf{x})\) applied instantaneously to the
rheological foundation at time *t = 0* and evaluated at a later time *t* is given in the Fourier domain by

where \(\mathbf{k} = (k_x, k_y)\) is the wavenumber vector, \(k_r\) its magnitude, \(H(\mathbf{k})\) is the
topographic load in the wavenumber domain, *A* is the Airy density ratio, \(\gamma\) is a constant that depends
on the infill density, and \(\Phi(\mathbf{k},t)\) is the response function for the selected rheology. The **grdflexure**
module read one or more loads *h*, transforms them to *H*, evaluates and applies the response function, and
inversely transform the results back to yield on or more *w* solutions.

### Variable infill approximation¶

If \(\rho_i = \rho_l\) then \(\gamma = 1\), otherwise the infill density varies spatially and the Fourier solution is not valid. We avoid these complications by letting \(\rho_l = \rho_i\) and increasing the deformation amplitude by

The approximation is good except for very large loads on thin plates (*Wessel*, 2001).

### Elastic response function¶

The time-independent *elastic response function* is

where the *flexural wavenumber k* and constants \(\epsilon_s\) via in-plane stresses \(N_x, N_y, N_{xy}\) are

for subscripts \(s = \left (x, y, xy \right )\).
In the most common scenario, \(N_s\) are all zero and the elastic response function becomes *isotropic*:

### Firmoviscous response function¶

The *firmoviscous response function* \(\Phi(\mathbf{k},t)\) scales the magnitude of the deformation at a given wavenumber and time
and depends on rheological parameters and in-plane stresses:

If the foundation is an inviscid half-space, then the *relaxation parameter* \(\tau(k_r) = \infty\), there is no time-dependence,
and \(\Phi_{fv}(\mathbf{k},t) = \Phi_e(\mathbf{k})\). Otherwise, it is given by

where \(\beta(k_r)\) depends on whether we have a finite-thickness layer of thickness \(T_a\) and viscosity
\(\eta_a\) above the half-space of viscosity \(\eta_m\) (*Cathles*, 1975; *Nakada*, 1986).
If no finite layer exists then \(\beta(k_r) = 1\), otherwise

where

### Airy and viscous response function¶

In the limit \(t \rightarrow \infty, \tau \rightarrow 0\) and we approach the purely elastic solution

Otherwise, if the plate has no strength (**-E**0), then \(\Phi_e(\mathbf{k}) = 1\) and the response function is purely *viscous* and isotropic:

For \(t \rightarrow \infty\) (or for an inviscid half-space) we approach Airy isostasy: \(w(\mathbf{x}) = A h(\mathbf{x})\).

### Maxwell viscoelastic response¶

For case (4), the viscoelastic response function (only available for an inviscid substratum) is

where \(t_m\) is the *Maxwell relaxation time* (*Watts*, 2001).

### General linear viscoelastic response¶

For case (5), the general linear viscoelastic response function (with an inviscid substratum) is (*Karner*, 1982)

where subscripts *i* and *f* refers to the initial (*t = 0*) and final (\(t = \infty\)) values for rigidities (\(D_i, D_f\))
and elastic response functions (\(\Phi_i, \Phi_f\)).

## References¶

Cathles, L. M., 1975, *The viscosity of the earth’s mantle*, Princeton University Press.

Karner, G. D., 1982, Spectral representation of isostatic models, *BMR J. Australian Geology & Geophysics, 7*, 55-62.

Nakada, M., 1986, Holocene sea levels in oceanic islands: Implications for the rheological
structure of the Earth’s mantle, *Tectonophysics, 121*, 263–276,
https://doi.org/10.1016/0040-1951(86)90047-8.

Watts, A. B., 2001, *Isostasy and Flexure of the Lithosphere*, 458 pp., Cambridge University Press.

Wessel. P., 2001, Global distribution of seamounts inferred from gridded Geosat/ERS-1 altimetry, J. Geophys. Res., 106(B9), 19,431-19,441, https://doi.org/10.1029/2000JB000083.

Wessel, P., 2016, Regional–residual separation of bathymetry and revised estimates of Hawaii plume flux,
*Geophys. J. Int., 204(2)*, 932-947, https://doi.org/10.1093/gji/ggv472.

## See Also¶

gmt, gmtflexure, grdfft, gravfft, grdmath, grdproject, grdseamount