1TREND2D(1) GMT TREND2D(1)
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6 trend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to
7 xyz[w] data
8
10 trend2d [ table ] -Fxyzmrw -Nn_model[+r] [ xyz[w]file ] [ -Ccondi‐
11 tion_number ] [ -I[confidence_level] ] [ -V[level] ] [ -W ] [ [
12 -bbinary ] [ -dnodata ] [ -eregexp ] [ -fflags ] [ -hheaders ] [
13 -iflags ] [ -:[i|o] ]
14 Note: No space is allowed between the option flag and the associated
16 arguments.
17
19 trend2d reads x,y,z [and w] values from the first three [four] columns
20 on standard input [or xyz[w]file] and fits a regression model z =
21 f(x,y) + e by [weighted] least squares. The fit may be made robust by
22 iterative reweighting of the data. The user may also search for the
23 number of terms in f(x,y) which significantly reduce the variance in z.
24 n_model may be in [1,10] to fit a model of the following form (similar
25 to grdtrend):
26 m1 + m2*x + m3*y + m4*x*y + m5*x*x + m6*y*y + m7*x*x*x + m8*x*x*y +
27 m9*x*y*y + m10*y*y*y.
28 The user must specify -Nn_model, the number of model parameters to use;
30 thus, -N4 fits a bilinear trend, -N6 a quadratic surface, and so on.
31 Optionally, append +r to perform a robust fit. In this case, the pro‐
32 gram will iteratively reweight the data based on a robust scale esti‐
33 mate, in order to converge to a solution insensitive to outliers. This
34 may be handy when separating a "regional" field from a "residual" which
35 should have non-zero mean, such as a local mountain on a regional sur‐
36 face.
37
39 -Fxyzmrw
40 Specify up to six letters from the set {x y z m r w} in any
41 order to create columns of ASCII [or binary] output. x = x, y =
42 y, z = z, m = model f(x,y), r = residual z - m, w = weight used
43 in fitting.
44 -Nn_model[+r]
46 Specify the number of terms in the model, n_model, and append +r
47 to do a robust fit. E.g., a robust bilinear model is -N4+r.
48
50 table One or more ASCII [or binary, see -bi] files containing x,y,z
51 [w] values in the first 3 [4] columns. If no files are speci‐
52 fied, trend2d will read from standard input.
53 -Ccondition_number
55 Set the maximum allowed condition number for the matrix solu‐
56 tion. trend2d fits a damped least squares model, retaining only
57 that part of the eigenvalue spectrum such that the ratio of the
58 largest eigenvalue to the smallest eigenvalue is condition_#.
59 [Default: condition_# = 1.0e06. ].
60 -I[confidence_level]
62 Iteratively increase the number of model parameters, starting at
63 one, until n_model is reached or the reduction in variance of
64 the model is not significant at the confidence_level level. You
65 may set -I only, without an attached number; in this case the
66 fit will be iterative with a default confidence level of 0.51.
67 Or choose your own level between 0 and 1. See remarks section.
68 -V[level] (more ...)
70 Select verbosity level [c].
71 -W Weights are supplied in input column 4. Do a weighted least
73 squares fit [or start with these weights when doing the itera‐
74 tive robust fit]. [Default reads only the first 3 columns.]
75 -bi[ncols][t] (more ...)
77 Select native binary input. [Default is 3 (or 4 if -W is set)
78 input columns].
79 -bo[ncols][type] (more ...)
81 Select native binary output. [Default is 1-6 columns as set by
82 -F].
83 -d[i|o]nodata (more ...)
85 Replace input columns that equal nodata with NaN and do the
86 reverse on output.
87 -e[~]"pattern" | -e[~]/regexp/[i] (more ...)
89 Only accept data records that match the given pattern.
90 -f[i|o]colinfo (more ...)
92 Specify data types of input and/or output columns.
93 -h[i|o][n][+c][+d][+rremark][+rtitle] (more ...)
95 Skip or produce header record(s).
96 -icols[+l][+sscale][+ooffset][,...] (more ...)
98 Select input columns and transformations (0 is first column).
99 -:[i|o] (more ...)
101 Swap 1st and 2nd column on input and/or output.
102 -^ or just -
104 Print a short message about the syntax of the command, then
105 exits (NOTE: on Windows just use -).
106 -+ or just +
108 Print an extensive usage (help) message, including the explana‐
109 tion of any module-specific option (but not the GMT common
110 options), then exits.
111 -? or no arguments
113 Print a complete usage (help) message, including the explanation
114 of all options, then exits.
115
117 The domain of x and y will be shifted and scaled to [-1, 1] and the
118 basis functions are built from Chebyshev polynomials. These have a
119 numerical advantage in the form of the matrix which must be inverted
120 and allow more accurate solutions. In many applications of trend2d the
121 user has data located approximately along a line in the x,y plane which
122 makes an angle with the x axis (such as data collected along a road or
123 ship track). In this case the accuracy could be improved by a rotation
124 of the x,y axes. trend2d does not search for such a rotation; instead,
125 it may find that the matrix problem has deficient rank. However, the
126 solution is computed using the generalized inverse and should still
127 work out OK. The user should check the results graphically if trend2d
128 shows deficient rank. NOTE: The model parameters listed with -V are
129 Chebyshev coefficients; they are not numerically equivalent to the m#s
130 in the equation described above. The description above is to allow the
131 user to match -N with the order of the polynomial surface. For evaluat‐
132 ing Chebyshev polynomials, see grdmath.
133 The -Nn_modelr (robust) and -I (iterative) options evaluate the signif‐
135 icance of the improvement in model misfit Chi-Squared by an F test. The
136 default confidence limit is set at 0.51; it can be changed with the -I
137 option. The user may be surprised to find that in most cases the reduc‐
138 tion in variance achieved by increasing the number of terms in a model
139 is not significant at a very high degree of confidence. For example,
140 with 120 degrees of freedom, Chi-Squared must decrease by 26% or more
141 to be significant at the 95% confidence level. If you want to keep
142 iterating as long as Chi-Squared is decreasing, set confidence_level to
143 zero.
144 A low confidence limit (such as the default value of 0.51) is needed to
146 make the robust method work. This method iteratively reweights the data
147 to reduce the influence of outliers. The weight is based on the Median
148 Absolute Deviation and a formula from Huber [1964], and is 95% effi‐
149 cient when the model residuals have an outlier-free normal distribu‐
150 tion. This means that the influence of outliers is reduced only
151 slightly at each iteration; consequently the reduction in Chi-Squared
152 is not very significant. If the procedure needs a few iterations to
153 successfully attenuate their effect, the significance level of the F
154 test must be kept low.
155
157 The ASCII output formats of numerical data are controlled by parameters
158 in your gmt.conf file. Longitude and latitude are formatted according
159 to FORMAT_GEO_OUT, absolute time is under the control of FOR‐
160 MAT_DATE_OUT and FORMAT_CLOCK_OUT, whereas general floating point val‐
161 ues are formatted according to FORMAT_FLOAT_OUT. Be aware that the for‐
162 mat in effect can lead to loss of precision in ASCII output, which can
163 lead to various problems downstream. If you find the output is not
164 written with enough precision, consider switching to binary output (-bo
165 if available) or specify more decimals using the FORMAT_FLOAT_OUT set‐
166 ting.
167
169 To remove a planar trend from data.xyz by ordinary least squares, use:
170 gmt trend2d data.xyz -Fxyr -N2 > detrended_data.xyz
172 To make the above planar trend robust with respect to outliers, use:
174 gmt trend2d data.xzy -Fxyr -N2+r > detrended_data.xyz
176 To find out how many terms (up to 10 in a robust interpolant are sig‐
178 nificant in fitting data.xyz, use:
179 gmt trend2d data.xyz -N10+r -I -V
181
183 gmt, grdmath, grdtrend, trend1d
184
186 Huber, P. J., 1964, Robust estimation of a location parameter, Ann.
187 Math. Stat., 35, 73-101.
188 Menke, W., 1989, Geophysical Data Analysis: Discrete Inverse Theory,
190 Revised Edition, Academic Press, San Diego.
191
193 2019, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe
1945.4.5 Feb 24, 2019 TREND2D(1)