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# grdmath

grdmath - Reverse Polish Notation (RPN) calculator for grids (element by element)

## Synopsis

grdmath [ -Ixinc[unit][=|+][/yinc[unit][=|+]] ] [ -M ] [ -N ] [ -R[unit]xmin/xmax/ymin/ymax[r] ] [ -V[level] ] [ -bi[ncols][type][w][+L|+B] ] [ -f[i|o]colinfo ] [ -icols[l][sscale][ooffset][,...] ] [ -h[i|o][n][+c][+d][+rremark][+rtitle] ] [ -icols[l][sscale][ooffset][,...] ] [ -n[b|c|l|n][+a][+bBC][+tthreshold] ] [ -r ] operand [ operand ] OPERATOR [ operand ] OPERATOR ... = outgrdfile

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

## 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. Grid operations are element-by-element, not matrix manipulations. 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 -r). The expression = outgrdfile can occur as many times as the depth of the stack allows in order to save intermediate results. Complicated or frequently occurring expressions may be coded as a macro for future use or stored and recalled via named memory locations.

## Required Arguments

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).

## Optional Arguments

-Ixinc[unit][=|+][/yinc[unit][=|+]]
  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 s to indicate arc seconds. If one of the units e, f, k, M, n or u is appended instead, the increment is assumed to be given in meter, foot, km, Mile, nautical mile or US survey foot, respectively, and will be converted to the equivalent degrees longitude at the middle latitude of the region (the conversion depends on PROJ_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 B. GMT File Formats for details. Note: if -Rgrdfile is used then the 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[unit]xmin/xmax/ymin/ymax[r] (more ...)
  Specify the region of interest.

-V[level] (more ...)
  Select verbosity level [c].

-bi[ncols][type] (more ...)
  Select binary input. The binary input option only applies to the data files needed by operators LDIST, PDIST, and INSIDE.

-f[i|o]colinfo (more ...)
  Specify data types of input and/or output columns.

-g[a]x|y|d|X|Y|D|[col]z[+|-]gap[u] (more ...)
  Determine data gaps and line breaks.

-h[i|o][n][+c][+d][+rremark][+rtitle] (more ...)
  Skip or produce header record(s).

-icols[l][sscale][ooffset][,...] (more ...)
  Select input columns.

-n[b|c|l|n][+a][+bBC][+c][+tthreshold] (more ...)
  Select interpolation mode for grids.

-r (more ...)
  Set pixel node registration [gridline]. Only used with -R -I.

-^
  Print a short message about the syntax of the command, then exits (NOTE: on Windows use just -).
+
  Print an extensive usage (help) message, including the explanation of any module-specific option (but not the GMT common options), then exits.
-?
  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.

## Operators

Choose among the following 161 operators. “args” are the number of input and output arguments.

Operator | args | Returns
ABS | 1 1 | abs (A)
ACOS | 1 1 | acos (A)
ACOSH | 1 1 | acosh (A)
ACOT | 1 1 | acot (A)
ACSC | 1 1 | acsc (A)
ADD | 2 1 | A + B
AND | 2 1 | B if A == NaN, else A
ASEC | 1 1 | asec (A)
ASIN | 1 1 | asin (A)
ASINH | 1 1 | asinh (A)
ATAN | 1 1 | atan (A)
ATAN2 | 2 1 | atan2 (A, B)
ATANH | 1 1 | atanh (A)
BEI | 1 1 | bei (A)
BER | 1 1 | ber (A)
BITAND | 2 1 | A & B (bitwise AND operator)
BITLEFT | 2 1 | A << B (bitwise left-shift operator)
BITNOT | 1 1 | ~A (bitwise NOT operator, i.e., return two’s complement)
BITOR | 2 1 | A | B (bitwise OR operator)
BITRIGHT | 2 1 | A >> B (bitwise right-shift operator)
BITTEST | 2 1 | 1 if bit B of A is set, else 0 (bitwise TEST operator)
BITXOR | 2 1 | A ^ B (bitwise XOR operator)
CAZ | 2 1 | Cartesian azimuth from grid nodes to stack x,y (i.e., A, B)
CBAZ | 2 1 | Cartesian backazimuth from grid nodes to stack x,y (i.e., A, B)
CDIST | 2 1 | Cartesian distance between grid nodes and stack x,y (i.e., A, B)
CEIL | 1 1 | ceil (A) (smallest integer >= A)
CHICRIT | 2 1 | Critical value for chi-squared-distribution, with alpha = A and n = B
CHIDIST | 2 1 | chi-squared-distribution P(chi2,n), with chi2 = A and n = B
CORRCOEFF | 2 1 | Correlation coefficient r(A, B)
COS | 1 1 | cos (A) (A in radians)
COSD | 1 1 | cos (A) (A in degrees)
COSH | 1 1 | cosh (A)
COT | 1 1 | cot (A) (A in radians)
COTD | 1 1 | cot (A) (A in degrees)
CPOISS | 2 1 | Cumulative Poisson distribution F(x,lambda), with x = A and lambda = B
CSC | 1 1 | csc (A) (A in radians)
CSCD | 1 1 | csc (A) (A in degrees)
CURV | 1 1 | Curvature of A (Laplacian)
D2DX2 | 1 1 | d^2(A)/dx^2 2nd derivative
D2DY2 | 1 1 | d^2(A)/dy^2 2nd derivative
D2DXY | 1 1 | d^2(A)/dxdy 2nd derivative
D2R | 1 1 | Converts Degrees to Radians
DDX | 1 1 | d(A)/dx Central 1st derivative
DDY | 1 1 | d(A)/dy Central 1st derivative
DEG2KM | 1 1 | Converts Spherical Degrees to Kilometers
DILOG | 1 1 | dilog (A)
DIV | 2 1 | A / B
DUP | 1 2 | Places duplicate of A on the stack
ERF | 1 1 | Error function erf (A)
ERFC | 1 1 | Complementary Error function erfc (A)
EQ | 2 1 | 1 if A == B, else 0
ERFINV | 1 1 | Inverse error function of A
EXCH | 2 2 | Exchanges A and B on the stack
EXP | 1 1 | exp (A)
FACT | 1 1 | A! (A factorial)
EXTREMA | 1 1 | Local Extrema: +2/-2 is max/min, +1/-1 is saddle with max/min in x, 0 elsewhere
FCRIT | 3 1 | Critical value for F-distribution, with alpha = A, n1 = B, and n2 = C
FDIST | 3 1 | F-distribution Q(F,n1,n2), with F = A, n1 = B, and n2 = C
FLIPLR | 1 1 | Reverse order of values in each row
FLIPUD | 1 1 | Reverse order of values in each column
FLOOR | 1 1 | floor (A) (greatest integer <= A)
FMOD | 2 1 | A % B (remainder after truncated division)
GE | 2 1 | 1 if A >= B, else 0
GT | 2 1 | 1 if A > B, else 0
HYPOT | 2 1 | hypot (A, B) = sqrt (A*A + B*B)
I0 | 1 1 | Modified Bessel function of A (1st kind, order 0)
I1 | 1 1 | Modified Bessel function of A (1st kind, order 1)
IFELSE | 3 1 | B if A != 0, else C
IN | 2 1 | Modified Bessel function of A (1st kind, order B)
INRANGE | 3 1 | 1 if B <= A <= C, else 0
INSIDE | 1 1 | 1 when inside or on polygon(s) in A, else 0
INV | 1 1 | 1 / A
ISFINITE | 1 1 | 1 if A is finite, else 0
ISNAN | 1 1 | 1 if A == NaN, else 0
J0 | 1 1 | Bessel function of A (1st kind, order 0)
J1 | 1 1 | Bessel function of A (1st kind, order 1)
JN | 2 1 | Bessel function of A (1st kind, order B)
K0 | 1 1 | Modified Kelvin function of A (2nd kind, order 0)
K1 | 1 1 | Modified Bessel function of A (2nd kind, order 1)
KEI | 1 1 | kei (A)
KER | 1 1 | ker (A)
KM2DEG | 1 1 | Converts Kilometers to Spherical Degrees
KN | 2 1 | Modified Bessel function of A (2nd kind, order B)
KURT | 1 1 | Kurtosis of A
LDIST | 1 1 | Compute minimum distance (in km if -fg) from lines in multi-segment ASCII file A
LDIST2 | 2 1 | As LDIST, from lines in ASCII file B but only to nodes where A != 0
LE | 2 1 | 1 if A <= B, else 0
LOG | 1 1 | log (A) (natural log)
LOG10 | 1 1 | log10 (A) (base 10)
LOG1P | 1 1 | log (1+A) (accurate for small A)
LOG2 | 1 1 | log2 (A) (base 2)
LMSSCL | 1 1 | LMS scale estimate (LMS STD) of A
LOWER | 1 1 | The lowest (minimum) value of A
LRAND | 2 1 | Laplace random noise with mean A and std. deviation B
LT | 2 1 | 1 if A < B, else 0
MAD | 1 1 | Median Absolute Deviation (L1 STD) of A
MAX | 2 1 | Maximum of A and B
MEAN | 1 1 | Mean value of A
MED | 1 1 | Median value of A
MIN | 2 1 | Minimum of A and B
MOD | 2 1 | A mod B (remainder after floored division)
MODE | 1 1 | Mode value (Least Median of Squares) of A
MUL | 2 1 | A * B
NAN | 2 1 | NaN if A == B, else A
NEG | 1 1 | -A
NEQ | 2 1 | 1 if A != B, else 0
NORM | 1 1 | Normalize (A) so max(A)-min(A) = 1
NOT | 1 1 | NaN if A == NaN, 1 if A == 0, else 0
NRAND | 2 1 | Normal, random values with mean A and std. deviation B
OR | 2 1 | NaN if B == NaN, else A
PDIST | 1 1 | Compute minimum distance (in km if -fg) from points in ASCII file A
PDIST2 | 2 1 | As PDIST, from points in ASCII file B but only to nodes where A != 0
POP | 1 0 | Delete top element from the stack
PLM | 3 1 | Associated Legendre polynomial P(A) degree B order C
PLMg | 3 1 | Normalized associated Legendre polynomial P(A) degree B order C (geophysical convention)
POW | 2 1 | A ^ B
PQUANT | 2 1 | The B’th Quantile (0-100%) of A
PSI | 1 1 | Psi (or Digamma) of A
PV | 3 1 | Legendre function Pv(A) of degree v = real(B) + imag(C)
QV | 3 1 | Legendre function Qv(A) of degree v = real(B) + imag(C)
R2 | 2 1 | R2 = A^2 + B^2
R2D | 1 1 | Convert Radians to Degrees
RAND | 2 1 | Uniform random values between A and B
RINT | 1 1 | rint (A) (round to integral value nearest to A)
ROTX | 2 1 | Rotate A by the (constant) shift B in x-direction
ROTY | 2 1 | Rotate A by the (constant) shift B in y-direction
SDIST | 2 1 | Spherical (Great circle|geodesic) distance (in km) between nodes and stack (A, B) (...)
SAZ | 2 1 | Spherical azimuth from grid nodes to stack lon, lat (i.e., A, B)
SBAZ | 2 1 | Spherical backazimuth from grid nodes to stack lon, lat (i.e., A, B)
SEC | 1 1 | sec (A) (A in radians)
SECD | 1 1 | sec (A) (A in degrees)
SIGN | 1 1 | sign (+1 or -1) of A
SIN | 1 1 | sin (A) (A in radians)
SINC | 1 1 | sinc (A) (sin (pi*A)/(pi*A))
SIND | 1 1 | sin (A) (A in degrees)
SINH | 1 1 | sinh (A)
SKEW | 1 1 | Skewness of A
SQR | 1 1 | A^2
SQRT | 1 1 | sqrt (A)
STD | 1 1 | Standard deviation of A
STEP | 1 1 | Heaviside step function: H(A)
STEPX | 1 1 | Heaviside step function in x: H(x-A)
STEPY | 1 1 | Heaviside step function in y: H(y-A)
SUB | 2 1 | A - B
SUM | 1 1 | Sum of all values in A
TAN | 1 1 | tan (A) (A in radians)
TAND | 1 1 | tan (A) (A in degrees)
TANH | 1 1 | tanh (A)
TAPER | 2 1 | Unit weights cosine-tapered to zero within A and B of x and y grid margins
TN | 2 1 | Chebyshev polynomial Tn(-1<t<+1,n), with t = A, and n = B
TCRIT | 2 1 | Critical value for Student’s t-distribution, with alpha = A and n = B
TDIST | 2 1 | Student’s t-distribution A(t,n), with t = A, and n = B
UPPER | 1 1 | The highest (maximum) value of A
XOR | 2 1 | 0 if A == NaN and B == NaN, NaN if B == NaN, else A
Y0 | 1 1 | Bessel function of A (2nd kind, order 0)
Y1 | 1 1 | Bessel function of A (2nd kind, order 1)
YLM | 2 2 | Re and Im orthonormalized spherical harmonics degree A order B
YLMg | 2 2 | Cos and Sin normalized spherical harmonics degree A order B (geophysical convention)
YN | 2 1 | Bessel function of A (2nd kind, order B)
ZCRIT | 1 1 | Critical value for the normal-distribution, with alpha = A
ZDIST | 1 1 | Cumulative normal-distribution C(x), with x = A

## 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

## Notes On Operators

1. The operator SDIST calculates spherical distances in km 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. The operators LDIST and PDIST compute spherical distances in km if -fg is set or implied, else they return Cartesian distances. Note: If the current PROJ_ELLIPSOID is ellipsoidal then geodesics are used in calculations of distances, which can be slow. You can trade speed with accuracy by changing the algorighm used to compute the geodesic (see PROJ_GEODESIC).

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.nc 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. (9) The bitwise operators (BITAND, BITLEFT, BITNOT, BITOR, BITRIGHT, BITTEST, and BITXOR) convert a grid’s single precision values to unsigned 32-bit ints to perform the bitwise operations. Consequently, the largest whole integer value that can be stored in a float grid is 2^24 or 16,777,216. Any higher result will be masked to fit in the lower 24 bits. Thus, bit operations are effectively limited to 24 bit. All bitwise operators return NaN if given NaN arguments or bit-settings <= 0.

## 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 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. When reading grids, the format is generally automatically recognized. If not, the same suffix can be added to input grid file names. See grdreformat and Section Grid file format specifications 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 and Section Grid file format specifications 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 gmt.conf file or on the command line. In addition, the unit attribute of the time variable will indicate both this unit and epoch.

## STORE, RECALL and CLEAR

You may store intermediate calculations to a named variable that you may recall and place on the stack at a later time. This is useful if you need access to a computed quantity many times in your expression as it will shorten the overall expression and improve readability. To save a result you use the special operator STO@label, where label is the name you choose to give the quantity. To recall the stored result to the stack at a later time, use [RCL]@label, i.e., RCL is optional. To clear memory you may use CLR@label. Note that STO and CLR leave the stack unchanged.

## Macros

Users may save their favorite operator combinations as macros via the file grdmath.macros in their current or user directory. The file may contain any number of macros (one per record); comment lines starting with # are skipped. The format for the macros is name = arg1 arg2 ... arg2 : comment where name is how the macro will be used. When this operator appears on the command line we simply replace it with the listed argument list. No macro may call another macro. As an example, the following macro expects three arguments (radius x0 y0) and sets the modes that are inside the given circle to 1 and those outside to 0:

INCIRCLE = CDIST EXCH DIV 1 LE : usage: r x y INCIRCLE to return 1 inside circle

Note: Because geographic or time constants may be present in a macro, it is required that the optional comment flag (:) must be followed by a space.

## Examples

To compute all distances to north pole:

>     gmt grdmath -Rg -I1 0 90 SDIST = dist_to_NP.nc

To take log10 of the average of 2 files, use

>     gmt grdmath file1.nc file2.nc ADD 0.5 MUL LOG10 = file3.nc

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

>     gmt grdmath ages.nc SQRT 350 MUL 2500 ADD = depths.nc

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

>     gmt grdmath 2 s_xy.nc MUL s_xx.nc s_yy.nc SUB DIV ATAN 2 DIV = direction.nc

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:

>     gmt grdmath -R0/360/-90/90 -I1 8 4 YML 1.1 MUL EXCH 0.4 MUL ADD = harm.nc

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

>     gmt grdmath faa.nc DUP EXTREMA 2 EQ MUL DUP 100 GT MUL 0 NAN = z.nc
>     gmt grd2xyz z.nc -s > max.xyz

To demonstrate the use of named variables, consider this radial wave where we store and recall the normalized radial arguments in radians:

>     gmt grdmath -R0/10/0/10 -I0.25 5 5 CDIST 2 MUL PI MUL 5 DIV STO@r COS @r SIN MUL = wave.nc

## 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.

## See Also

gmt, gmtmath, grd2xyz, grdedit, grdinfo, xyz2grd

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