# grdfft¶

Mathematical operations on grids in the spectral domain

## Synopsis¶

**gmt grdfft** *ingrid* [ *ingrid2* ]
**-G***outfile*|*table*
[ **-A***azimuth* ]
[ **-C***zlevel* ]
[ **-D**[*scale*|**g**] ]
[ **-E**[**r**|**x**|**y**][**+w**[**k**]][**+n**] ]
[ **-F**[**r**|**x**|**y**]*params* ]
[ **-I**[*scale*|**g**] ]
[ **-N***params* ]
[ **-Q**]
[ **-S***scale*|**d** ]
[ **-V**[*level*] ]
[ **-f**flags ]
[ **--PAR**=*value* ]

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

## Description¶

**grdfft** will take the 2-D forward Fast Fourier Transform and perform
one or more mathematical operations in the frequency domain before
transforming back to the space domain. An option is provided to scale
the data before writing the new values to an output file. The horizontal
dimensions of the grid are assumed to be in meters. Geographical grids
may be used by specifying the **-f**flags 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.

## Required Arguments¶

*ingrid*[=*ID*|?*varname*][**+b***band*][**+d***divisor*][**+n***invalid*][**+o***offset*][**+s***scale*]

2-D binary grid file to be operated on. For cross-spectral operations, also give the second grid file

ingrid2. Optionally, append =IDfor reading a specific file format [Default is =nf] or ?varnamefor a specific netCDF variable [Default is the first 2-D grid found by GMT] (See full description). The following modifiers are supported:

+b- Select aband(for images only) [Default is 0].

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

+n- Replace data values matchinginvalidwith NaN.

+o- Offset data values by the givenoffset[Default is 0].

+s- Scale data values by the givenscale[Default is 1].Note: Any offset is added after any scaling.

**-G***outfile*Specify the name of the output grid file (see Grid File Formats) or the 1-D spectrum table (see

**-E**).

## Optional Arguments¶

**-A***azimuth*Take the directional derivative in the

*azimuth*direction measured in degrees CW from north.

**-C***zlevel*Upward (for

*zlevel*> 0) or downward (for*zlevel*< 0) continue the field*zlevel*meters.

**-D**[*scale*|**g**]Differentiate the field, i.e., take \(\frac{\partial}{\partial z}\) of the grid

*z*. This is equivalent to multiplying by \(k_r\) in the frequency domain (\(k_r\) is radial wave number). Append a scale to multiply by \(k_r \cdot\)*scale*) instead. Alternatively, append**g**to indicate that your data are geoid heights in meters and output should be gravity anomalies in mGal. [Default is no scale].

**-E**[**r**|**x**|**y**][**+w**[**k**]][**+n**]Estimate power spectrum in the radial direction [

**r**]. Place**x**or**y**immediately after**-E**to compute the spectrum in the x or y direction instead. No grid file is created. If one grid is given then f (i.e., frequency or wave number), power[f], and 1 standard deviation in power[f] are written to the file set by**-G**[standard output]. If two grids are given we write f and 8 quantities: Xpower[f], Ypower[f], coherent power[f], noise power[f], phase[f], admittance[f], gain[f], coherency[f]. Each quantity is followed by its own 1-std dev error estimate, hence the output is 17 columns wide. Give**+w**to write wavelength instead of frequency, and if your grid is geographic you may further append**k**to scale wavelengths from meter [Default] to km. Finally, the spectrum is obtained by summing over several frequencies. Append**+n**to normalize so that the mean spectral values per frequency are reported instead.

**-F**[**r**|**x**|**y**]*params*Filter the data. Place

**x**or**y**immediately after**-F**to filter*x*or*y*direction only; default is isotropic [**r**]. Choose between a cosine-tapered band-pass, a Gaussian band-pass filter, or a Butterworth band-pass filter.- Cosine-taper:
Specify four wavelengths

*lc*/*lp*/*hp*/*hc*in correct units (see**-f**flags) to design a bandpass filter: wavelengths greater than*lc*or less than*hc*will be cut, wavelengths greater than*lp*and less than*hp*will be passed, and wavelengths in between will be cosine-tapered. E.g.,**-F**1000000/250000/50000/10000**-f**flags will bandpass, cutting wavelengths > 1000 km and < 10 km, passing wavelengths between 250 km and 50 km. To make a highpass or lowpass filter, give hyphens (-) for*hp*/*hc*or*lc*/*lp*. E.g.,**-Fx**-/-/50/10 will lowpass*x*, passing wavelengths > 50 and rejecting wavelengths < 10.**-Fy**1000/250/-/- will highpass*y*, passing wavelengths < 250 and rejecting wavelengths > 1000.- Gaussian band-pass:
Append

*lo*/*hi*, the two wavelengths in correct units (see**-f**flags) to design a bandpass filter. At the given wavelengths the Gaussian filter weights will be 0.5. To make a highpass or lowpass filter, give a hyphen (-) for the*hi*or*lo*wavelength, respectively. E.g.,**-F**-/30 will lowpass the data using a Gaussian filter with half-weight at 30, while**-F**400/- will highpass the data.- Butterworth band-pass:
Append

*lo*/*hi*/*order*, the two wavelengths in correct units (see**-f**flags) and the filter order (an integer) to design a bandpass filter. At the given cut-off wavelengths the Butterworth filter weights will be 0.707 (i.e., the power spectrum will therefore be reduced by 0.5). To make a highpass or lowpass filter, give a hyphen (-) for the*hi*or*lo*wavelength, respectively. E.g.,**-F**-/30/2 will lowpass the data using a 2nd-order Butterworth filter, with half-weight at 30, while**-F**400/-/2 will highpass the data.

**Note**: For filtering in the time (or space) domain instead, see grdfilter.

**-G***outfile*|*table*Filename for output netCDF grid file OR 1-D data table (see

**-E**). This is optional for -E (spectrum written to standard output) but mandatory for all other options that require a grid output.

**-I**[*scale*|**g**]Integrate the field, i.e., compute \(\int z(x,y) dz\). This is equivalent to divide by \(k_r\) in the frequency domain (\(k_r\) is radial wave number). Append a scale to divide by \(k_r \cdot\)

*scale*instead. Alternatively, append**g**to indicate that your data set is gravity anomalies in mGal and output should be geoid heights in meters. [Default is no scale].

**-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. Consult module documentation for the default action:

**+d**: Detrend data, i.e. remove best-fitting linear trend.**+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*.

**-Q**Selects no wavenumber operations. Useful in conjunction with

**-N**modifiers when you wish to write out the 2-D spectrum (or other intermediate grid products) only.

**-S***scale*|**d**Multiply each element by

*scale*in the space domain (after the frequency domain operations). [Default is 1.0]. Alternatively, append**d**to convert deflection of vertical to micro-radians.

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

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

**-f**flagsGeographic 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 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.

## 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 **-f**flags 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 **-f**flags. If the data are close to either pole, you should
consider projecting the grid file onto a rectangular coordinate system
using grdproject

## Data Detrending¶

The default detrending mode is to remove a best-fitting linear plane (**+d**).
Consult and use **-N** to select other modes.

## Normalization of Spectrum¶

By default, the power spectrum returned by **-E** simply sums the contributions
from frequencies that are part of the output frequency. For *x*- or *y*-spectra
this means summing the power across the other frequency dimension, while for the
radial spectrum it means summing up power within each annulus of width *delta_q*,
the radial frequency (*q*) spacing. A consequence of this summing is that the radial
spectrum of a white noise process will give a linear radial power spectrum that
is proportional to *q*. Appending **n** will instead compute the mean power
per output frequency and in this case the white noise process will have a
white radial spectrum as well.

## Examples¶

**Note**: Below are some examples of valid syntax for this module.
The examples that use remote files (file names starting with `@`

)
can be cut and pasted into your terminal for testing.
Other commands requiring input files are just dummy examples of the types
of uses that are common but cannot be run verbatim as written.

To obtain the normalized radial spectrum from the remote data grid @white_noise.nc, after removing the mean, let us try:

gmt grdfft @white_noise.nc -Er+n -N+a > spectrum.txt

To upward continue the sea-level magnetic anomalies in the file mag_0.nc to a level 800 m above sealevel:

gmt grdfft mag_0.nc -C800 -V -Gmag_800.nc

To transform geoid heights in m (geoid.nc) on a geographical grid to free-air gravity anomalies in mGal:

gmt grdfft geoid.nc -Dg -V -Ggrav.nc

To transform gravity anomalies in mGal (faa.nc) to deflections of the vertical (in micro-radians) in the 038 direction, we must first integrate gravity to get geoid, then take the directional derivative, and finally scale radians to micro-radians:

gmt grdfft faa.nc -Ig -A38 -S1e6 -V -Gdefl_38.nc

Second vertical derivatives of gravity anomalies are related to the
curvature of the field. We can compute these as mGal/m^{2} by:

gmt grdfft gravity.nc -D -D -V -Ggrav_2nd_derivative.nc

To compute cross-spectral estimates for co-registered bathymetry and gravity grids, and report result as functions of wavelengths in km, try:

gmt grdfft bathymetry.nc gravity.grd -E+wk -fg -V > cross_spectra.txt

To examine the pre-FFT grid after detrending, point-symmetry reflection, and tapering has been applied, as well as saving the real and imaginary components of the raw spectrum of the data in topo.nc, try:

gmt grdfft topo.nc -N+w+z -fg -V -Q

You can now make plots of the data in topo_taper.nc, topo_real.nc, and topo_imag.nc.