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PROGRAM:

NAME


gravfft - Compute gravitational attraction of 3-D surfaces in the wavenumber (or
frequency) domain

SYNOPSIS


gravfft ingrid [ ingrid2 ] outfile [ n/wavelength/mean_depth/tbw ] [ density|rhogrid ] [
n_terms ] [ [f[+]|g|v|n|e] ] [ w|b|c|t |k ] [ [f|q|s|nx/ny][+a|d|h
|l][+e|n|m][+twidth][+w[suffix]][+z[p]] [ ] [ te/rl/rm/rw[+m] ] [ [level] ] [ wd] [
zm[zl] ] [ -fg ]

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

DESCRIPTION


gravfft can be used into three main modes. Mode 1: Simply compute the geopotential due to
the surface given in the topo.grd file. Requires a density contrast (-D) and possibly a
different observation level (-W). It will take the 2-D forward FFT of the grid and use
the full Parker's method up to the chosen terms. Mode 2: Compute the geopotential
response due to flexure of the topography file. It will take the 2-D forward FFT of the
grid and use the full Parker's method applied to the chosen isostatic model. The
available models are the "loading from top", or elastic plate model, and the "loading from
below" which accounts for the plate's response to a sub-surface load (appropriate for hot
spot modeling - if you believe them). In both cases, the model parameters are set with -T
and -Z options. Mode 3: compute the admittance or coherence between two grids. The output
is the average in the radial direction. Optionally, the model admittance may also be
calculated. The horizontal dimensions of the grdfiles are assumed to be in meters.
Geographical grids may be used by specifying the -fg option that scales degrees to meters.
If you have grids with dimensions in km, you could change this to meters using grdedit or
scale the output with grdmath. Given the number of choices this program offers, is
difficult to state what are options and what are required arguments. It depends on what
you are doing; see the examples for further guidance.

REQUIRED ARGUMENTS


ingrid 2-D binary grid file to be operated on. (See GRID FILE FORMATS below). For
cross-spectral operations, also give the second grid file ingrd2.

-Goutfile
Specify the name of the output grid file or the 1-D spectrum table (see -E). (See
GRID FILE FORMATS below).

OPTIONAL ARGUMENTS


-Cn/wavelength/mean_depth/tbw
Compute only the theoretical admittance curves of the selected model and exit. n
and wavelength are used to compute (n * wavelength) the total profile length in
meters. mean_depth is the mean water depth. Append dataflags (one or two) of tbw in
any order. t = use "from top" model, b = use "from below" model. Optionally specify
w to write wavelength instead of frequency.

-Ddensity|rhogrid
Sets density contrast across surface. Used, for example, to compute the gravity
attraction of the water layer that can later be combined with the free-air anomaly
to get the Bouguer anomaly. In this case do not use -T. It also implicitly sets
-N+h. Alternatively, specify a co-registered grid with density contrasts if a
variable density contrast is required.

-En_terms
Number of terms used in Parker expansion (limit is 10, otherwise terms depending on
n will blow out the program) [Default = 3]

-F[f[+]|g|v|n|e]
Specify desired geopotential field: compute geoid rather than gravity
f = Free-air anomalies (mGal) [Default]. Append + to add in the slab implied
when removing the mean value from the topography. This requires zero topography
to mean no mass anomaly.

g = Geoid anomalies (m).

v = Vertical Gravity Gradient (VGG; 1 Eotvos = 0.1 mGal/km).

e = East deflections of the vertical (micro-radian).

n = North deflections of the vertical (micro-radian).

-Iw|b|c|t |k
Use ingrd2 and ingrd1 (a grid with topography/bathymetry) to estimate
admittance|coherence and write it to stdout (-G ignored if set). This grid should
contain gravity or geoid for the same region of ingrd1. Default computes
admittance. Output contains 3 or 4 columns. Frequency (wavelength), admittance
(coherence) one sigma error bar and, optionally, a theoretical admittance. Append
dataflags (one to three) from w|b|c|t. w writes wavelength instead of wavenumber,
k selects km for wavelength unit [m], c computes coherence instead of admittance, b
writes a fourth column with "loading from below" theoretical admittance, and t
writes a fourth column with "elastic plate" theoretical admittance.

-N[f|q|s|nx/ny][+a|[+d|h|l][+e|n|m][+twidth][+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.

-Nnx/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 +twidth. 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.

-Q Writes out a grid with the flexural topography (with z positive up) whose average
was set by -Zzm and model parameters by -T (and output by -G). That is the
"gravimetric Moho". -Q implicitly sets -N+h

-S Computes predicted gravity or geoid grid due to a subplate load produced by the
current bathymetry and the theoretical model. The necessary parameters are set
within -T and -Z options. The number of powers in Parker expansion is restricted to
1. See an example further down.

-Tte/rl/rm/rw[+m]
Compute the isostatic compensation from the topography load (input grid file) on an
elastic plate of thickness te. Also append densities for load, mantle, and water in
SI units. Give average mantle depth via -Z. If the elastic thickness is > 1e10 it
will be interpreted as the flexural rigidity (by default it is computed from te and
Young modulus). Optionally, append +m to write a grid with the Moho's geopotential
effect (see -F) from model selected by -T. If te = 0 then the Airy response is
returned. -T+m implicitly sets -N+h

-Wwd Set water depth (or observation height) relative to topography [0]. Append k to
indicate km.

-Zzm[zl]
Moho [and swell] average compensation depths. For the "load from top" model you
only have to provide zm, but for the "loading from below" don't forget zl.

-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 +uunit 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


The FFT solution to elastic 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 a 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 the effect of the water layer above the bat.grd bathymetry using 2700 and 1035
for the densities of crust and water and writing the result on water_g.grd (computing up
to the fourth power of bathymetry in Parker expansion):

gmt gravfft bat.grd -D1665 -Gwater_g.grd -E4

Now subtract it to your free-air anomaly faa.grd and you will get the Bouguer anomaly. You
may wonder why we are subtracting and not adding. After all the Bouguer anomaly pretends
to correct the mass deficiency presented by the water layer, so we should add because
water is less dense than the rocks below. The answer relies on the way gravity effects are
computed by the Parker's method and practical aspects of using the FFT.

gmt grdmath faa.grd water_g.grd SUB = bouguer.grd

Want an MBA anomaly? Well compute the crust mantle contribution and add it to the
sea-bottom anomaly. Assuming a 6 km thick crust of density 2700 and a mantle with 3300
density we could repeat the command used to compute the water layer anomaly, using 600
(3300 - 2700) as the density contrast. But we now have a problem because we need to know
the mean Moho depth. That is when the scale/offset that can be appended to the grid's name
comes in hand. Notice that we didn't need to do that before because mean water depth was
computed directly from data (notice also the negative sign of the offset due to the fact
that z is positive up):

gmt gravfft bat.grd=nf/1/-6000 -D600 -Gmoho_g.grd

Now, subtract it to the sea-bottom anomaly to obtain the MBA anomaly. That is:

gmt grdmath water_g.grd moho_g.grd SUB = mba.grd

To compute the Moho gravity effect of an elastic plate bat.grd with Te = 7 km, density of
2700, over a mantle of density 3300, at an average depth of 9 km

gmt gravfft bat.grd -Gelastic.grd -T7000/2700/3300/1035+m -Z9000

If you add now the sea-bottom and Moho's effects, you will get the full gravity response
of your isostatic model. We will use here only the first term in Parker expansion.

gmt gravfft bat.grd -D1665 -Gwater_g.grd -E1
gmt gravfft bat.grd -Gelastic.grd -T7000/2700/3300/1035+m -Z9000 -E1
gmt grdmath water_g.grd elastic.grd ADD = model.grd

The same result can be obtained directly by the next command. However, PAY ATTENTION to
the following. I don't yet know if it's because of a bug or due to some limitation, but
the fact is that the following and the previous commands only give the same result if -E1
is used. For higher powers of bathymetry in Parker expansion, only the above example
seams to give the correct result.

gmt gravfft bat.grd -Gmodel.grd -T7000/2700/3300/1035 -Z9000 -E1

And what would be the geoid anomaly produced by a load at 50 km depth, below the a region
whose bathymetry is given by bat.grd, a Moho at 9 km depth and the same densities as
before?

gmt gravfft topo.grd -Gswell_geoid.grd -T7000/2700/3300/1035 -Fg -Z9000/50000 -S -E1

To compute the admittance between the topo.grd bathymetry and faa.grd free-air anomaly
grid using the elastic plate model of a crust of 6 km mean thickness with 10 km effective
elastic thickness in a region of 3 km mean water depth:

gmt gravfft topo.grd faa.grd -It -T10000/2700/3300/1035 -Z9000

To compute the admittance between the topo.grd bathymetry and geoid.grd geoid grid with
the "loading from below" (LFB) model with the same as above and sub-surface load at 40 km,
but assuming now the grids are in geographic and we want wavelengths instead of frequency:

gmt gravfft topo.grd geoid.grd -Ibw -T10000/2700/3300/1035 -Z9000/40000 -fg

To compute the gravity theoretical admittance of a LFB along a 2000 km long profile using
the same parameters as above

gmt gravfft -C400/5000/3000/b -T10000/2700/3300/1035 -Z9000/40000

REFERENCES


Luis, J.F. and M.C. Neves. 2006, The isostatic compensation of the Azores Plateau: a 3D
admittance and coherence analysis. J. Geothermal Volc. Res. Volume 156, Issues 1-2, Pages
10-22, http://dx.doi.org/10.1016/j.jvolgeores.2006.03.010 Parker, R. L., 1972, The rapid
calculation of potential anomalies, Geophys. J., 31, 447-455. 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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