spectrum1dgmt - Online in the Cloud

PROGRAM:

NAME

spectrum1d - Compute auto- [and cross- ] spectra from one [or two] time-series

SYNOPSIS

spectrum1d [ table ] segment_size] [ [xycnpago] ] [ dt ] [ [h|m] ] [ [name_stem ] ] [ ] [
] [ -b<binary> ] [ -d<nodata> ] [ -f<flags> ] [ -g<gaps> ] [ -h<headers> ] [ -i<flags> ]

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

DESCRIPTION

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.

In addition, a single file with all of the above as individual columns will be written to
stdout (unless disabled via -T).

REQUIRED ARGUMENTS

-Ssegment_size]
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.

OPTIONAL ARGUMENTS

table One or more ASCII (or binary, see -bi) files holding X(t) [Y(t)] samples in the
first 1 [or 2] columns. If no files are specified, spectrum1d will read from
standard input.

-C[xycnpago]
Read the first two columns of input as samples of two time-series, 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.

-Ddt dt Set the spacing between samples in the time-series [Default = 1].

-L Leave trend alone. By default, a linear trend will be removed prior to the
transform. Alternatively, append m to just remove the mean value or h to remove the
mid-value.

-N[name_stem]
Supply an alternate name stem to be used for output files [Default = "spectrum"].
Giving no argument will disable the writing of individual output files.

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

-T Disable the writing of a single composite results file to stdout.

-W Write Wavelength rather than frequency in column 1 of the output file[s] [Default =
frequency, (cycles / dt)].

-bi[ncols][t] (more ...)
Select native binary input. [Default is 2 input columns].

-bo[ncols][type] (more ...)
Select native binary output. [Default is 2 output columns].

-d[i|o]nodata (more ...)
Replace input columns that equal nodata with NaN and do the reverse on output.

-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 (0 is first column).

-^ or just -
Print a short message about the syntax of the command, then exits (NOTE: on Windows
use just -).

-+ or just +
Print an extensive usage (help) message, including the explanation of any
module-specific option (but not the GMT common options), then exits.

-? or no arguments
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.

ASCII FORMAT PRECISION

The ASCII output formats of numerical data are controlled by parameters in your gmt.conf
file. Longitude and latitude are formatted according to FORMAT_GEO_OUT, whereas other
values are formatted according to FORMAT_FLOAT_OUT. 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 FORMAT_FLOAT_OUT setting.

EXAMPLES

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

gmt 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, use

paste data.t data.g | gmt spectrum1d -S256 -D1.5 -Ndata -C > results.txt

TUTORIAL

The output of spectrum1d is in units of power spectral density, and so to get units of
data-squared you must divide by delta_t, where delta_t is the sample spacing. (There may
be a factor of 2 pi somewhere, also. If you want to be sure of the normalization, you can
determine a scale factor from Parseval's theorem: the sum of the squares of your input
data should equal the sum of the squares of the outputs from spectrum1d, if you are simply
trying to get a periodogram. [See below.])

Suppose we simply take a data set, x(t), and compute the discrete Fourier transform (DFT)
of the entire data set in one go. Call this X(f). Then suppose we form X(f) times the
complex conjugate of X(f).

P_raw(f) = X(f) * X'(f), where the ' indicates complex conjugation.

P_raw is called the periodogram. The sum of the samples of the periodogram equals the sum
of the samples of the squares of x(t), by Parseval's theorem. (If you use a DFT subroutine
on a computer, usually the sum of P_raw equals the sum of x-squared, times M, where M is
the number of samples in x(t).)

Each estimate of X(f) is now formed by a weighted linear combination of all of the x(t)
values. (The weights are sometimes called "twiddle factors" in the DFT literature.) So,
no matter what the probability distribution for the x(t) values is, the probability
distribution for the X(f) values approaches [complex] Gaussian, by the Central Limit
Theorem. This means that the probability distribution for P_raw(f) approaches chi-squared
with two degrees of freedom. That reduces to an exponential distribution, and the variance
of the estimate of P_raw is proportional to the square of the mean, that is, the expected
value of P_raw.

In practice if we form P_raw, the estimates are hopelessly noisy. Thus P_raw is not
useful, and we need to do some kind of smoothing or averaging to get a useful estimate,
P_useful(f).

There are several different ways to do this in the literature. One is to form P_raw and
then smooth it. Another is to form the auto-covariance function of x(t), smooth, taper and
shape it, and then take the Fourier transform of the smoothed, tapered and shaped
auto-covariance. Another is to form a parametric model for the auto-correlation structure
in x(t), then compute the spectrum of that model. This last approach is what is done in
what is called the "maximum entropy" or "Berg" or "Box-Jenkins" or "ARMA" or "ARIMA"
methods.

Welch's method is a tried-and-true method. In his method, you choose a segment length,
-SN, so that estimates will be made from segments of length N. The frequency samples (in
cycles per delta_t unit) of your P_useful will then be at k /(N * delta_t), where k is an
integer, and you will get N samples (since the spectrum is an even function of f, only N/2
of them are really useful). If the length of your entire data set, x(t), is M samples
long, then the variance in your P_useful will decrease in proportion to N/M. Thus you need
to choose N << M to get very low noise and high confidence in P_useful. There is a
trade-off here; see below.

There is an additional reduction in variance in that Welch's method uses a Von Hann
spectral window on each sample of length N. This reduces side lobe leakage and has the
effect of smoothing the (N segment) periodogram as if the X(f) had been convolved with
[1/4, 1/2, 1/4] prior to forming P_useful. But this slightly widens the spectral bandwidth
of each estimate, because the estimate at frequency sample k is now a little correlated
with the estimate at frequency sample k+1. (Of course this would also happen if you simply
formed P_raw and then smoothed it.)

Finally, Welch's method also uses overlapped processing. Since the Von Hann window is
large in the middle and tapers to near zero at the ends, only the middle of the segment of
length N contributes much to its estimate. Therefore in taking the next segment of data,
we move ahead in the x(t) sequence only N/2 points. In this way, the next segment gets
large weight where the segments on either side of it will get little weight, and vice
versa. This doubles the smoothing effect and ensures that (if N << M) nearly every point
in x(t) contributes with nearly equal weight in the final answer.

Welch's method of spectral estimation has been widely used and widely studied. It is very
reliable and its statistical properties are well understood. It is highly recommended in
such textbooks as "Random Data: Analysis and Measurement Procedures" by Bendat and
Piersol.

In all problems of estimating parameters from data, there is a classic trade-off between
resolution and variance. If you want to try to squeeze more resolution out of your data
set, then you have to be willing to accept more noise in the estimates. The same trade-off
is evident here in Welch's method. If you want to have very low noise in the spectral
estimates, then you have to choose N << M, and this means that you get only N samples of
the spectrum, and the longest period that you can resolve is only N * delta_t. So you see
that reducing the noise lowers the number of spectral samples and lowers the longest
period. Conversely, if you choose N approaching M, then you approach the periodogram with
its very bad statistical properties, but you get lots of samples and a large fundamental
period.

The other spectral estimation methods also can do a good job. Welch's method was selected
because the way it works, how one can code it, and its effects on statistical
distributions, resolution, side-lobe leakage, bias, variance, etc. are all easily
understood. Some of the other methods (e.g. Maximum Entropy) tend to hide where some of
these trade-offs are happening inside a "black box".

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