spectrum1d - compute auto- [and cross- ] spectra from one [or two] timeseries.

spectrum1d[x[y]file]-Ssegment_size] [-C] [-Ddt] [-Nname_stem] [-V] [-W] [-bi[s][n] ] [-bo[s] ]

spectrum1dreads X [and Y] values from the first [and second] columns on standard input [orx[y]file]. These values are treated as timeseries X(t) [Y(t)] sampled at equal intervals spaceddtunits apart. There may be any number of lines of input.spectrum1dwill 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 onname_stem. If the-Coption is used, eight files are created; otherwise only one (xpower) is written. The files (which are ASCII unless-bois 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]fileASCII (or binary, see-bi) file holding X(t) [Y(t)] samples in the first 1 [or 2] columns. If no file is specified,spectrum1dwill read from standard input.-Ssegment_sizeis 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 ifsegment_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.

-CRead 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.-DdtSet the spacing between samples in the timeseries [Default = 1].-Nname_stemSupply the name stem to be used for output files [Default = "spectrum"].-VSelects verbose mode, which will send progress reports to stderr [Default runs "silently"].-WWrite Wavelength rather than frequency in column 1 of the output file[s] [Default = frequency, (cycles /dt)].-biSelects binary input. Appendsfor single precision [Default is double]. Appendnfor the number of columns in the binary file(s). [Default is 2 input columns].-boSelects binary output. Appendsfor single precision [Default is double].

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, try spectrum1d data.g-S256-D1.5-Ndata 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, try paste data.t data.g | spectrum1d-S256-D1.5-Ndata-C

gmt, grdfft

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.

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