waves() R function from [metR]

Fourier transform functions

Use fft() to fit, filter and reconstruct signals in the frequency domain, as well as to compute the wave envelope.


FitWave(y, k = 1)

BuildWave(
  x,
  amplitude,
  phase,
  k,
  wave = list(amplitude = amplitude, phase = phase, k = k),
  sum = TRUE
)

FilterWave(y, k, action = sign(k[k != 0][1]))

WaveEnvelope(y)

Arguments

y: numeric vector to transform
k: numeric vector of wave numbers
x: numeric vector of locations (in radians)
amplitude: numeric vector of amplitudes
phase: numeric vector of phases
wave: optional list output from FitWave
sum: whether to perform the sum or not (see Details)
action: integer to disambiguate action for k = 0 (see Details)

Returns

FitWaves returns a a named list with components

k: wavenumbers
amplitude: amplitude of each wavenumber
phase: phase of each wavenumber in radians
r2: explained variance of each wavenumber

BuildWave returns a vector of the same length of x with the reconstructed vector if sum is TRUE or, instead, a list with components

k: wavenumbers
x: the vector of locations
y: the reconstructed signal of each wavenumber

FilterWave and WaveEnvelope return a vector of the same length as y

Details

FitWave performs a fourier transform of the input vector and returns a list of parameters for each wave number kept. The amplitude (A), phase ( $\phi$ ) and wave number (k) satisfy:

y = \sum A cos((x - \phi)k)

The phase is calculated so that it lies between 0 and $2\pi/k$ so it represents the location (in radians) of the first maximum of each wave number. For the case of k = 0 (the mean), phase is arbitrarily set to 0.

BuildWave is FitWave's inverse. It reconstructs the original data for selected wavenumbers. If sum is TRUE (the default) it performs the above mentioned sum and returns a single vector. If is FALSE, then it returns a list of k vectors consisting of the reconstructed signal of each wavenumber.

FilterWave filters or removes wavenumbers specified in k. If k is positive, then the result is the reconstructed signal of y only for wavenumbers specified in k, if it's negative, is the signal of y minus the wavenumbers specified in k. The argument action must be be manually set to -1 or +1

if k=0.

WaveEnvelope computes the wave envelope of y following Zimin (2003). To compute the envelope of only a restricted band, first filter it with FilterWave.

Examples


# Build a wave with specific wavenumber profile
waves <- list(k = 1:10,
              amplitude = rnorm(10)^2,
              phase = runif(10, 0, 2*pi/(1:10)))
x <- BuildWave(seq(0, 2*pi, length.out = 60)[-1], wave = waves)

# Just fancy FFT
FitWave(x, k = 1:10)

# Extract only specific wave components
plot(FilterWave(x,  1), type = "l")
plot(FilterWave(x,  2), type = "l")
plot(FilterWave(x,  1:4), type = "l")

# Remove components from the signal
plot(FilterWave(x,  -4:-1), type = "l")

# The sum of the two above is the original signal (minus floating point errors)
all.equal(x, FilterWave(x,  1:4) + FilterWave(x,  -4:-1))

# The Wave envelopes shows where the signal is the most "wavy".
plot(x, type = "l", col = "grey")
lines(WaveEnvelope(x), add = TRUE)

# Examples with real data
data(geopotential)
library(data.table)
# January mean of geopotential height
jan <- geopotential[month(date) == 1, .(gh = mean(gh)), by = .(lon, lat)]

# Stationary waves for each latitude
jan.waves <- jan[, FitWave(gh, 1:4), by = .(lat)]
library(ggplot2)
ggplot(jan.waves, aes(lat, amplitude, color = factor(k))) +
    geom_line()

# Build field of wavenumber 1
jan[, gh.1 := BuildWave(lon*pi/180, wave = FitWave(gh, 1)), by = .(lat)]
ggplot(jan, aes(lon, lat)) +
    geom_contour(aes(z = gh.1, color = after_stat(level))) +
    coord_polar()

# Build fields of wavenumber 1 and 2
waves <- jan[, BuildWave(lon*pi/180, wave = FitWave(gh, 1:2), sum = FALSE), by = .(lat)]
waves[, lon := x*180/pi]
ggplot(waves, aes(lon, lat)) +
    geom_contour(aes(z = y, color = after_stat(level))) +
    facet_wrap(~k) +
    coord_polar()

# Field with waves 0 to 2 filtered
jan[, gh.no12 := gh - BuildWave(lon*pi/180, wave = FitWave(gh, 0:2)), by = .(lat)]
ggplot(jan, aes(lon, lat)) +
    geom_contour(aes(z = gh.no12, color = after_stat(level))) +
    coord_polar()

# Much faster
jan[, gh.no12 := FilterWave(gh, -2:0), by = .(lat)]
ggplot(jan, aes(lon, lat)) +
    geom_contour(aes(z = gh.no12, color = after_stat(level))) +
    coord_polar()

# Using positive numbers returns the field
jan[, gh.only12 := FilterWave(gh, 2:1), by = .(lat)]
ggplot(jan, aes(lon, lat)) +
    geom_contour(aes(z = gh.only12, color = after_stat(level))) +
    coord_polar()

# Compute the envelope of the geopotential
jan[, envelope := WaveEnvelope(gh.no12), by = .(lat)]
ggplot(jan[lat == -60], aes(lon, gh.no12)) +
    geom_line() +
    geom_line(aes(y = envelope), color = "red")

References

Zimin, A.V., I. Szunyogh, D.J. Patil, B.R. Hunt, and E. Ott, 2003: Extracting Envelopes of Rossby Wave Packets. Mon. Wea. Rev., 131, 1011–1017, tools:::Rd_expr_doi("10.1175/1520-0493(2003)131<1011:EEORWP>2.0.CO;2")

waves function