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**PROGRAM:**

**NAME**

grdflexure - Compute flexural deformation of 3-D surfaces for various rheologies

**SYNOPSIS**

**grdflexure**

__topogrd__

__rm__/

__rl__[/

__ri__]/

__rw__

__Te__[

**u**]

__outgrid__[

__Nx__/

__Ny__/

__Nxy__] [

**p**

__poisson__] [

**y**

__Young__] [

__nu_a__[/

__h_a__/

__nu_m__] ] [

__list__] [ [

**f**|

**q**|

**s**|

__nx__/

__ny__][

**+a**|

**d**|

**h**|

**l**][

**+e**|

**n**|

**m**][

**+t**

__width__][

**+w**[

__suffix__]][

**+z**[

**p**]] [

__beta__] [

**-T**

__t0__[

**u**][/

__t1__[

**u**]/

__dt__[

**u**]|

__n__][

**+l**] ] [ [

__level__] ] [

__wd__] [

__zm__] [

**-fg**]

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

**DESCRIPTION**

**grdflexure**computes the flexural response to loads using a range of user-selectable

rheologies. User may select from elastic, viscoelastic, or firmoviscous (with one or two

viscous layers). Temporal evolution can also be modeled by providing incremental load

grids and specifying a range of model output times.

**REQUIRED** **ARGUMENTS**

__topogrd__

2-D binary grid file with the topography of the load (in meters); See GRID FILE

FORMATS below. If

**-T**is used,

__topogrd__may be a filename template with a floating

point format (C syntax) and a different load file name will be set and loaded for

each time step. The load times thus coincide with the times given via

**-T**(but not

all times need to have a corresponding file). Alternatively, give

__topogrd__as

=*flist*, where

__flist__is an ASCII table with one

__topogrd__filename and load time per

record. These load times can be different from the evaluation times given via

**-T**.

For load time format, see

**-T**.

**-D**

__rm__

**/**

__rl__

**[/**

__ri__

**]/**

__rw__

Sets density for mantle, load, infill (optional, otherwise it is assumed to equal

the load density), 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__.

**-E**

__Te__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).

**-G**

__outfile__

If

**-T**is set then

__grdfile__must be a filename template that contains a floating

point format (C syntax). If the filename template also contains either %s (for

unit name) or %c (for unit letter) then we use the corresponding time (in units

specified in

**-T**) to generate the individual file names, otherwise we use time in

years with no unit.

**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.

**-Cp**

__poisson__

Change the current value of Poisson's ratio [0.25].

**-Cy**

__Young__

Change the current value of Young's modulus [7.0e10 N/m^2].

**-F**

__nu_a__

**[/**

__h_a__

**/**

__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.

**-N[f|q|s|**

__nx/ny__

**][+a|[+d|h|l][+e|n|m][+t**

__width__

**][+w[**

__suffix__

**]][+z[p]]**

Choose or inquire about suitable grid dimensions for FFT and set optional

parameters. Control the FFT dimension:

**-Nf**will force the FFT to use the actual dimensions of the data.

**-Nq**will inQuire about more suitable dimensions, report those, then continue.

**-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 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__.

**-L**

__list__Write the names and evaluation times of all grids that were created to the text

file

__list__. Requires

**-T**.

**-M**

__tm__Specify a viscoelastic model in conjunction with an elastic plate thickness

specified via

**-E**. Append the Maxwell time

__tm__for the viscoelastic model (in ).

**-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__

**[u][/**

__t1__

**[u]/**

__dt__

**[u]|**

__n__

**][+l]**

Specify

__t0__,

__t1__, and time increment (

__dt__) for sequence of calculations [Default is

one step, with no time dependency]. For a single specific time, just give start

time

__t0__. The 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.

**-W**

__wd__Set reference depth to the undeformed flexed surface in m [0]. Append

**k**to

indicate km.

**-Z**

__zm__Specify reference depth to flexed surface (e.g., Moho) in m; append

**k**for km. Must

be positive. [0].

**-V[**

__level__

**]**

**(more**

**...)**

Select verbosity level [c].

**-fg**Geographic grids (dimensions of longitude, latitude) will be converted to meters

via a "Flat Earth" approximation using the current ellipsoid parameters.

**-^**

**or**

**just**

**-**

Print a short message about the syntax of the command, then exits (NOTE: on Windows

use just

**-**).

**-+**

**or**

**just**

**+**

Print an extensive usage (help) message, including the explanation of any

module-specific option (but not the GMT common options), then exits.

**-?**

**or**

**no**

**arguments**

Print a complete usage (help) message, including the explanation of options, then

exits.

**--version**

Print GMT version and exit.

**--show-datadir**

Print full path to GMT share directory and exit.

**GRID** **FILE** **FORMATS**

By default GMT writes out grid as single precision floats in a COARDS-complaint netCDF

file format. However, GMT is able to produce grid files in many other commonly used grid

file formats and also facilitates so called "packing" of grids, writing out floating point

data as 1- or 2-byte integers. To specify the precision, scale and offset, the user should

add the suffix

**=**

__id__[

**/**

__scale__

**/**

__offset__[

**/**

__nan__]], where

__id__is a two-letter identifier of the grid

type and precision, and

__scale__and

__offset__are optional scale factor and offset to be

applied to all grid values, and

__nan__is the value used to indicate missing data. In case

the two characters

__id__is not provided, as in

**=/**

__scale__than a

__id__

**=**

__nf__is assumed. When

reading grids, the format is generally automatically recognized. If not, the same suffix

can be added to input grid file names. See

**grdconvert**and Section grid-file-format of the

GMT Technical Reference and Cookbook for more information.

When reading a netCDF file that contains multiple grids, GMT will read, by default, the

first 2-dimensional grid that can find in that file. To coax GMT into reading another

multi-dimensional variable in the grid file, append

**?**

__varname__to the file name, where

__varname__is the name of the variable. Note that you may need to escape the special meaning

of

**?**in your shell program by putting a backslash in front of it, or by placing the

filename and suffix between quotes or double quotes. The

**?**

__varname__suffix can also be used

for output grids to specify a variable name different from the default: "z". See

**grdconvert**and Sections modifiers-for-CF and grid-file-format of the GMT Technical

Reference and Cookbook for more information, particularly on how to read splices of 3-,

4-, or 5-dimensional grids.

**GRID** **DISTANCE** **UNITS**

If the 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. If your grid is geographic, convert

distances to meters by supplying

**-fg**instead.

**CONSIDERATIONS**

netCDF COARDS grids will automatically be recognized as geographic. For other grids

geographical grids were you want to convert degrees into meters, select

**-fg**. If the data

are close to either pole, you should consider projecting the grid file onto a rectangular

coordinate system using

**grdproject**.

**PLATE** **FLEXURE** **NOTES**

The FFT solution to plate flexure requires the infill density to equal the load density.

This is typically only true directly beneath the load; beyond the load the infill tends to

be lower-density sediments or even water (or air). Wessel [2001] proposed an

approximation that allows for the specification of an infill density different from the

load density while still allowing for an FFT solution. Basically, the plate flexure is

solved for using the infill density as the effective load density but the amplitudes are

adjusted by the factor

__A__= sqrt ((rm - ri)/(rm - rl)), which is the theoretical difference

in amplitude due to a point load using the two different load densities. The

approximation is very good but breaks down for large loads on weak plates, a fairy

uncommon situation.

**EXAMPLES**

To compute elastic plate flexure from the load

__topo.nc__, for a 10 km thick plate with

typical densities, try

gmt grdflexure topo.nc -Gflex.nc -E10k -D2700/3300/1035

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

**REFERENCES**

Cathles, L. M., 1975,

__The__

__viscosity__

__of__

__the__

__earth's__

__mantle__, Princeton 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,

__http://dx.doi.org/10.1029/2000JB000083__

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