| GRDMATH(1gmt) | GMT | GRDMATH(1gmt) |
grdmath - Reverse Polish Notation (RPN) calculator for grids (element by element)
grdmath [ -Amin_area[/min_level/max_level][+ag|i|s |S][+r|l][ppercent] ] [ -Dresolution[+] ] [ -Iincrement ] [ -M ] [ -N ] [ -Rregion ] [ -V[level] ] [ -bibinary ] [ -dinodata ] [ -fflags ] [ -hheaders ] [ -iflags ] [ -nflags ] [ -r ] [ -x[[-]n] ] operand [ operand ] OPERATOR [ operand ] OPERATOR ... = outgrdfile
Note: No space is allowed between the option flag and the associated arguments.
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.
Choose among the following 209 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 |
| ARC | 2 1 | Return arc(A,B) on [0 pi] |
| AREA | 0 1 | Area of each gridnode cell (in km^2 if geographic) |
| 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) |
| BCDF | 3 1 | Binomial cumulative distribution function for p = A, n = B, and x = C |
| BPDF | 3 1 | Binomial probability density function for p = A, n = B, and x = C |
| 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 back-azimuth 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) |
| CDIST2 | 2 1 | As CDIST but only to nodes that are != 0 |
| CEIL | 1 1 | ceil (A) (smallest integer >= A) |
| CHICRIT | 2 1 | Chi-squared critical value for alpha = A and nu = B |
| CHICDF | 2 1 | Chi-squared cumulative distribution function for chi2 = A and nu = B |
| CHIPDF | 2 1 | Chi-squared probability density function for chi2 = A and nu = B |
| COMB | 2 1 | Combinations n_C_r, with n = A and r = 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) |
| 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 |
| DENAN | 2 1 | Replace NaNs in A with values from B |
| DILOG | 1 1 | dilog (A) |
| DIV | 2 1 | A / B |
| DUP | 1 2 | Places duplicate of A on the stack |
| ECDF | 2 1 | Exponential cumulative distribution function for x = A and lambda = B |
| ECRIT | 2 1 | Exponential distribution critical value for alpha = A and lambda = B |
| EPDF | 2 1 | Exponential probability density function for x = A and lambda = B |
| 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 |
| FCDF | 3 1 | F cumulative distribution function for F = A, nu1 = B, and nu2 = C |
| FCRIT | 3 1 | F distribution critical value for alpha = A, nu1 = B, and nu2 = 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) |
| FPDF | 3 1 | F probability density function for F = A, nu1 = B, and nu2 = C |
| 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 |
| LCDF | 1 1 | Laplace cumulative distribution function for z = A |
| LCRIT | 1 1 | Laplace distribution critical value for alpha = 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 |
| LDISTG | 0 1 | As LDIST, but operates on the GSHHG dataset (see -A, -D for options). |
| 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 |
| LMSSCLW | 2 1 | Weighted LMS scale estimate (LMS STD) of A for weights in B |
| LOWER | 1 1 | The lowest (minimum) value of A |
| LPDF | 1 1 | Laplace probability density function for z = 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 |
| MEANW | 2 1 | Weighted mean value of A for weights in B |
| MEDIAN | 1 1 | Median value of A |
| MEDIANW | 2 1 | Weighted median value of A for weights in B |
| 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 |
| MODEW | 2 1 | Weighted mode value (Least Median of Squares) of A for weights in B |
| 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 |
| PCDF | 2 1 | Poisson cumulative distribution function for x = A and lambda = B |
| 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 |
| PERM | 2 1 | Permutations n_P_r, with n = A and r = B |
| 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) |
| POINT | 1 2 | Compute mean x and y from ASCII file A and place them on the stack |
| POP | 1 0 | Delete top element from the stack |
| POW | 2 1 | A ^ B |
| PPDF | 2 1 | Poisson distribution P(x,lambda), with x = A and lambda = B |
| PQUANT | 2 1 | The B'th Quantile (0-100%) of A |
| PQUANTW | 3 1 | The C'th weighted quantile (0-100%) of A for weights in B |
| 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 |
| RCDF | 1 1 | Rayleigh cumulative distribution function for z = A |
| RCRIT | 1 1 | Rayleigh distribution critical value for alpha = A |
| RINT | 1 1 | rint (A) (round to integral value nearest to A) |
| RMS | 1 1 | Root-mean-square of A |
| RMSW | 1 1 | Root-mean-square of A for weights in B |
| RPDF | 1 1 | Rayleigh probability density function for z = A |
| ROLL | 2 0 | Cyclicly shifts the top A stack items by an amount B |
| 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) |
| SDIST2 | 2 1 | As SDIST but only to nodes that are != 0 |
| SAZ | 2 1 | Spherical azimuth from grid nodes to stack lon, lat (i.e., A, B) |
| SBAZ | 2 1 | Spherical back-azimuth 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 |
| STDW | 2 1 | Weighted standard deviation of A for weights in B |
| 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 |
| TCDF | 2 1 | Student's t cumulative distribution function for t = A, and nu = B |
| TCRIT | 2 1 | Student's t distribution critical value for alpha = A and nu = B |
| TN | 2 1 | Chebyshev polynomial Tn(-1<t<+1,n), with t = A, and n = B |
| TPDF | 2 1 | Student's t probability density function for t = A, and nu = B |
| TRIM | 3 1 | Alpha-trim C to NaN if values fall in tails A and B (in percentage) |
| UPPER | 1 1 | The highest (maximum) value of A |
| VAR | 1 1 | Variance of A |
| VARW | 2 1 | Weighted variance of A for weights in B |
| WCDF | 3 1 | Weibull cumulative distribution function for x = A, scale = B, and shape = C |
| WCRIT | 3 1 | Weibull distribution critical value for alpha = A, scale = B, and shape = C |
| WPDF | 3 1 | Weibull density distribution P(x,scale,shape), with x = A, scale = B, and shape = C |
| WRAP | 1 1 | wrap A in radians onto [-pi,pi] |
| 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) |
| ZCDF | 1 1 | Normal cumulative distribution function for z = A |
| ZPDF | 1 1 | Normal probability density function for z = A |
| ZCRIT | 1 1 | Normal distribution critical value for alpha = A |
The following symbols have special meaning:
| PI | 3.1415926... |
| E | 2.7182818... |
| EULER | 0.5772156... |
| EPS_F | 1.192092896e-07 (single precision epsilon |
| XMIN | Minimum x value |
| XMAX | Maximum x value |
| XRANGE | Range of x values |
| XINC | x increment |
| NX | The number of x nodes |
| YMIN | Minimum y value |
| YMAX | Maximum y value |
| YRANGE | Range of y values |
| YINC | y increment |
| NY | The number of y nodes |
| X | Grid with x-coordinates |
| Y | Grid with y-coordinates |
| XNORM | Grid with normalized [-1 to +1] x-coordinates |
| YNORM | Grid with normalized [-1 to +1] y-coordinates |
| XCOL | Grid with column numbers 0, 1, ..., NX-1 |
| YROW | Grid with row numbers 0, 1, ..., NY-1 |
| NODE | Grid with node numbers 0, 1, ..., (NX*NY)-1 |
The operator LDISTG is a version of LDIST that operates on the GSHHG data. Instead of reading an ASCII file, it directly accesses one of the GSHHG data sets as determined by the -D and -A options.
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.
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.
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. (more ...)
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.
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.
The coastline database is GSHHG (formerly GSHHS) which is compiled from three sources: World Vector Shorelines (WVS), CIA World Data Bank II (WDBII), and Atlas of the Cryosphere (AC, for Antarctica only). Apart from Antarctica, all level-1 polygons (ocean-land boundary) are derived from the more accurate WVS while all higher level polygons (level 2-4, representing land/lake, lake/island-in-lake, and island-in-lake/lake-in-island-in-lake boundaries) are taken from WDBII. The Antarctica coastlines come in two flavors: ice-front or grounding line, selectable via the -A option. Much processing has taken place to convert WVS, WDBII, and AC data into usable form for GMT: assembling closed polygons from line segments, checking for duplicates, and correcting for crossings between polygons. The area of each polygon has been determined so that the user may choose not to draw features smaller than a minimum area (see -A); one may also limit the highest hierarchical level of polygons to be included (4 is the maximum). The 4 lower-resolution databases were derived from the full resolution database using the Douglas-Peucker line-simplification algorithm. The classification of rivers and borders follow that of the WDBII. See the GMT Cookbook and Technical Reference Appendix K for further details.
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.
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 YLM 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
To creat a dumb file saved as a 32 bits float GeoTiff using GDAL, run
gmt grdmath -Rd -I10 X Y MUL = lixo.tiff=gd:GTiff
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.
gmt, gmtmath, grd2xyz, grdedit, grdinfo, xyz2grd
2019, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe
| May 21, 2019 | 5.4.5 |