# GRDMATH

NAME
SYNOPSIS
DESCRIPTION
OPTIONS
NOTES ON OPERATORS
GRID VALUES PRECISION
GRID FILE FORMATS
GEOGRAPHICAL AND TIME COORDINATES
EXAMPLES
REFERENCES

## NAME

grdmath − Reverse Polish Notation calculator for grid files

## SYNOPSIS

grdmath [ −F ] [ −Ixinc[unit][=|+][/yinc[unit][=|+]] ] [ −M ] [ −N ] [ −Rwest/east/south/north[r] ] [ −V ] [ −bi[s|S|d|D[ncol]|c[var1/...]] ] [ −fcolinfo ] operand [ operand ] OPERATOR [ operand ] OPERATOR ... = outgrdfile

## DESCRIPTION

grdmath will perform operations like add, subtract, multiply, and divide on one or more grid files or constants using Reverse Polish Notation (RPN) syntax (e.g., Hewlett-Packard calculator-style). Arbitrarily complicated expressions may therefore be evaluated; the final result is written to an output grid file. When two grids are on the stack, each element in file A is modified by the corresponding element in file B. However, some operators only require one operand (see below). If no grid files are used in the expression then options −R, −I must be set (and optionally −F). The expression = outgrdfile can occur as many times as the depth of the stack allows.
operand

If operand can be opened as a file it will be read as a grid file. If not a file, it is interpreted as a numerical constant or a special symbol (see below).

outgrdfile

The name of a 2-D grid file that will hold the final result. (See GRID FILE FORMATS below).

OPERATORS

Choose among the following 145 operators. "args" are the number of input and output arguments.

SYMBOLS

The following symbols have special meaning:

 PI 3.1415926... E 2.7182818... EULER 0.5772156... XMIN Minimum x value XMAX Maximum x value XINC x increment NX The number of x nodes YMIN Minimum y value YMAX Maximum y value YINC y increment NY The number of y nodes X Grid with x-coordinates Y Grid with y-coordinates Xn Grid with normalized [-1 to +1] x-coordinates Yn Grid with normalized [-1 to +1] y-coordinates

## OPTIONS

 −F Force pixel node registration [Default is gridline registration]. (Node registrations are defined in GMT Cookbook Appendix B on grid file formats.) Only used with −R −I. −I x_inc [and optionally y_inc] is the grid spacing. Optionally, append a suffix modifier. Geographical (degrees) coordinates: Append m to indicate arc minutes or c to indicate arc seconds. If one of the units e, k, i, or n is appended instead, the increment is assumed to be given in meter, km, miles, or nautical miles, respectively, and will be converted to the equivalent degrees longitude at the middle latitude of the region (the conversion depends on ELLIPSOID). If /y_inc is given but set to 0 it will be reset equal to x_inc; otherwise it will be converted to degrees latitude. All coordinates: If = is appended then the corresponding max x (east) or y (north) may be slightly adjusted to fit exactly the given increment [by default the increment may be adjusted slightly to fit the given domain]. Finally, instead of giving an increment you may specify the number of nodes desired by appending + to the supplied integer argument; the increment is then recalculated from the number of nodes and the domain. The resulting increment value depends on whether you have selected a gridline-registered or pixel-registered grid; see Appendix B for details. Note: if −Rgrdfile is used then grid spacing has already been initialized; use −I to override the values. −M By default any derivatives calculated are in z_units/ x(or y)_units. However, the user may choose this option to convert dx,dy in degrees of longitude,latitude into meters using a flat Earth approximation, so that gradients are in z_units/meter. −N Turn off strict domain match checking when multiple grids are manipulated [Default will insist that each grid domain is within 1e-4 * grid_spacing of the domain of the first grid listed]. −R xmin, xmax, ymin, and ymax specify the Region of interest. For geographic regions, these limits correspond to west, east, south, and north and you may specify them in decimal degrees or in [+-]dd:mm[:ss.xxx][W|E|S|N] format. Append r if lower left and upper right map coordinates are given instead of w/e/s/n. The two shorthands −Rg and −Rd stand for global domain (0/360 and -180/+180 in longitude respectively, with -90/+90 in latitude). Alternatively, specify the name of an existing grid file and the −R settings (and grid spacing, if applicable) are copied from the grid. For calendar time coordinates you may either give (a) relative time (relative to the selected TIME_EPOCH and in the selected TIME_UNIT; append t to −JX|x), or (b) absolute time of the form [date]T[clock] (append T to −JX|x). At least one of date and clock must be present; the T is always required. The date string must be of the form [-]yyyy[-mm[-dd]] (Gregorian calendar) or yyyy[-Www[-d]] (ISO week calendar), while the clock string must be of the form hh:mm:ss[.xxx]. The use of delimiters and their type and positions must be exactly as indicated (however, input, output and plot formats are customizable; see gmtdefaults). −V Selects verbose mode, which will send progress reports to stderr [Default runs "silently"]. −bi Selects binary input. Append s for single precision [Default is d (double)]. Uppercase S or D will force byte-swapping. Optionally, append ncol, the number of columns in your binary input file if it exceeds the columns needed by the program. Or append c if the input file is netCDF. Optionally, append var1/var2/... to specify the variables to be read. The binary input option only applies to the data files needed by operators LDIST, PDIST, and INSIDE. −f Special formatting of input and/or output columns (time or geographical data). Specify i or o to make this apply only to input or output [Default applies to both]. Give one or more columns (or column ranges) separated by commas. Append T (absolute calendar time), t (relative time in chosen TIME_UNIT since TIME_EPOCH), x (longitude), y (latitude), or f (floating point) to each column or column range item. Shorthand −f[i|o]g means −f[i|o]0x,1y (geographic coordinates).

## NOTES ON OPERATORS

(1) The operator SDIST calculates spherical distances between the (lon, lat) point on the stack and all node positions in the grid. The grid domain and the (lon, lat) point are expected to be in degrees. Similarly, the SAZ and SBAZ operators calculate spherical azimuth and back-azimuths in degrees, respectively. A few operators (PDIST, LDIST, and INSIDE) expects their argument to be a single file with points, lines, or polygons, respectively. Be aware that LDIST in particular can be slow for large grids and numerous line segments. Note: If the current ELLIPSOID is not spherical then geodesics are used in the calculations.

(2) The operator PLM calculates the associated Legendre polynomial of degree L and order M (0 <= M <= L), and its argument is the sine of the latitude. PLM is not normalized and includes the Condon-Shortley phase (-1)^M. PLMg is normalized in the way that is most commonly used in geophysics. The C-S phase can be added by using -M as argument. PLM will overflow at higher degrees, whereas PLMg is stable until ultra high degrees (at least 3000).

(3) The operators YLM and YLMg calculate normalized spherical harmonics for degree L and order M (0 <= M <= L) for all positions in the grid, which is assumed to be in degrees. YLM and YLMg return two grids, the real (cosine) and imaginary (sine) component of the complex spherical harmonic. Use the POP operator (and EXCH) to get rid of one of them, or save both by giving two consecutive = file.grd calls.
The orthonormalized complex harmonics YLM are most commonly used in physics and seismology. The square of YLM integrates to 1 over a sphere. In geophysics, YLMg is normalized to produce unit power when averaging the cosine and sine terms (separately!) over a sphere (i.e., their squares each integrate to 4 pi). The Condon-Shortley phase (-1)^M is not included in YLM or YLMg, but it can be added by using -M as argument.

(4) All the derivatives are based on central finite differences, with natural boundary conditions.

(5) Files that have the same names as some operators, e.g., ADD, SIGN, =, etc. should be identified by prepending the current directory (i.e., ./LOG).

(6) Piping of files is not allowed.

(7) The stack depth limit is hard-wired to 100.

(8) All functions expecting a positive radius (e.g., LOG, KEI, etc.) are passed the absolute value of their argument.

## GRID VALUES PRECISION

Regardless of the precision of the input data, GMT programs that create grid files will internally hold the grids in 4-byte floating point arrays. This is done to conserve memory and furthermore most if not all real data can be stored using 4-byte floating point values. Data with higher precision (i.e., double precision values) will lose that precision once GMT operates on the grid or writes out new grids. To limit loss of precision when processing data you should always consider normalizing the data prior to processing.

## 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 2- or 4-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. When reading grids, the format is generally automatically recognized. If not, the same suffix can be added to input grid file names. See grdreformat(1) and Section 4.17 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 grdreformat(1) and Section 4.18 of the GMT Technical Reference and Cookbook for more information, particularly on how to read splices of 3-, 4-, or 5-dimensional grids.

## GEOGRAPHICAL AND TIME COORDINATES

When the output grid type is netCDF, the coordinates will be labeled "longitude", "latitude", or "time" based on the attributes of the input data or grid (if any) or on the −f or −R options. For example, both −f0x −f1t and −R90w/90e/0t/3t will result in a longitude/time grid. When the x, y, or z coordinate is time, it will be stored in the grid as relative time since epoch as specified by TIME_UNIT and TIME_EPOCH in the .gmtdefaults file or on the command line. In addition, the unit attribute of the time variable will indicate both this unit and epoch.

## EXAMPLES

To take log10 of the average of 2 files, use

grdmath file1.grd file2.grd ADD 0.5 MUL LOG10 = file3.grd

Given the file ages.grd, which holds seafloor ages in m.y., use the relation depth(in m) = 2500 + 350 * sqrt (age) to estimate normal seafloor depths:

grdmath ages.grd SQRT 350 MUL 2500 ADD = depths.grd

To find the angle a (in degrees) of the largest principal stress from the stress tensor given by the three files s_xx.grd s_yy.grd, and s_xy.grd from the relation tan (2*a) = 2 * s_xy / (s_xx - s_yy), use

grdmath 2 s_xy.grd MUL s_xx.grd s_yy.grd SUB DIV ATAN2 2 DIV = direction.grd

To calculate the fully normalized spherical harmonic of degree 8 and order 4 on a 1 by 1 degree world map, using the real amplitude 0.4 and the imaginary amplitude 1.1:

grdmath −R0/360/-90/90 −I1 8 4 YML 1.1 MUL EXCH 0.4 MUL ADD = harm.grd

To extract the locations of local maxima that exceed 100 mGal in the file faa.grd:

grdmath faa.grd DUP EXTREMA 2 EQ MUL DUP 100 GT MUL 0 NAN = z.grd
grd2xyz
z.grd −S > max.xyz

## REFERENCES

Abramowitz, M., and I. A. Stegun, 1964, Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55, Dover, New York.
Holmes, S. A., and W. E. Featherstone, 2002, A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. Journal of Geodesy, 76, 279-299.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992, Numerical Recipes, 2nd edition, Cambridge Univ., New York.
Spanier, J., and K. B. Oldman, 1987, An Atlas of Functions, Hemisphere Publishing Corp.