spectrum1d − compute auto− [and cross− ] spectra from one [or two] timeseries.
spectrum1d [ x[y]file ] −Ssegment_size] [ −C[xycnpago] ] [ −Ddt ] [ −Nname_stem ] [ −V ] [ −W ] [ −b[i|o][s|S|d|D[ncol]|c[var1/...]] ] [ −f[i|o]colinfo ]
spectrum1d reads X [and Y] values from the first [and second] columns on standard input [or x[y]file]. These values are treated as timeseries X(t) [Y(t)] sampled at equal intervals spaced dt units apart. There may be any number of lines of input. spectrum1d will create file[s] containing auto− [and cross− ] spectral density estimates by Welch’s method of ensemble averaging of multiple overlapped windows, using standard error estimates from Bendat and Piersol.
The output
files have 3 columns: f or w, p, and e. f or w is the
frequency or wavelength, p is the spectral density estimate,
and e is the one standard deviation error bar size. These
files are named based on name_stem. If the
−C option is used, up to eight files are
created; otherwise only one (xpower) is written. The files
(which are ASCII unless −bo is set) are as
follows:
name_stem.xpower
Power spectral density of X(t). Units of X * X * dt.
name_stem.ypower
Power spectral density of Y(t). Units of Y * Y * dt.
name_stem.cpower
Power spectral density of the coherent output. Units same as ypower.
name_stem.npower
Power spectral density of the noise output. Units same as ypower.
name_stem.gain
Gain spectrum, or modulus of the transfer function. Units of (Y / X).
name_stem.phase
Phase spectrum, or phase of the transfer function. Units are radians.
name_stem.admit
Admittance spectrum, or real part of the transfer function. Units of (Y / X).
name_stem.coh
(Squared) coherency spectrum, or linear correlation coefficient as a function of frequency. Dimensionless number in [0, 1]. The Signal-to-Noise-Ratio (SNR) is coh / (1 - coh). SNR = 1 when coh = 0.5.
x[y]file
ASCII (or binary, see −bi) file holding X(t) [Y(t)] samples in the first 1 [or 2] columns. If no file is specified, spectrum1d will read from standard input.
−S |
segment_size is a radix-2 number of samples per window for ensemble averaging. The smallest frequency estimated is 1.0/(segment_size * dt), while the largest is 1.0/(2 * dt). One standard error in power spectral density is approximately 1.0 / sqrt(n_data / segment_size), so if segment_size = 256, you need 25,600 data to get a one standard error bar of 10%. Cross-spectral error bars are larger and more complicated, being a function also of the coherency. |
−C |
Read the first two columns of input as samples of two timeseries, X(t) and Y(t). Consider Y(t) to be the output and X(t) the input in a linear system with noise. Estimate the optimum frequency response function by least squares, such that the noise output is minimized and the coherent output and the noise output are uncorrelated. Optionally specify up to 8 letters from the set { x y c n p a g o } in any order to create only those output files instead of the default [all]. x = xpower, y = ypower, c = cpower, n = npower, p = phase, a = admit, g = gain, o = coh. |
||
−D |
dt Set the spacing between samples in the timeseries [Default = 1]. |
||
−N |
name_stem Supply the name stem to be used for output files [Default = "spectrum"]. |
||
−V |
Selects verbose mode, which will send progress reports to stderr [Default runs "silently"]. |
||
−W |
Write Wavelength rather than frequency in column 1 of the output file[s] [Default = frequency, (cycles / dt)]. |
||
−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. [Default is 2 input columns]. |
||
−bo |
Selects binary output. Append s for single precision [Default is d (double)]. Uppercase S or D will force byte-swapping. Optionally, append ncol, the number of desired columns in your binary output file. [Default is 2 output columns]. |
||
−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). |
The ASCII output formats of numerical data are controlled by parameters in your .gmtdefaults4 file. Longitude and latitude are formatted according to OUTPUT_DEGREE_FORMAT, whereas other values are formatted according to D_FORMAT. Be aware that the format in effect can lead to loss of precision in the output, which can lead to various problems downstream. If you find the output is not written with enough precision, consider switching to binary output (−bo if available) or specify more decimals using the D_FORMAT setting.
Suppose data.g is gravity data in mGal, sampled every 1.5 km. To write its power spectrum, in mGal**2-km, to the file data.xpower, use
spectrum1d data.g −S 256 −D 1.5 −N data
Suppose in addition to data.g you have data.t, which is topography in meters sampled at the same points as data.g. To estimate various features of the transfer function, considering data.t as input and data.g as output, use
paste data.t data.g | spectrum1d −S 256 −D 1.5 −N data −C
Bendat, J. S.,
and A. G. Piersol, 1986, Random Data, 2nd revised ed., John
Wiley & Sons.
Welch, P. D., 1967, The use of Fast Fourier Transform for
the estimation of power spectra: a method based on time
averaging over short, modified periodograms, IEEE
Transactions on Audio and Electroacoustics, Vol AU-15, No
2.