Fourier transform

Perform a fourier transform of the data and return the

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)

Value

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 returns a vector of the same length as y

Details

FitWave uses fft to make a fourier transform of the data and then 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.

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, doi:10.1175/1520-0493(2003)131<1011:EEORWP>2.0.CO;2

Examples

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)]
#>         lon   lat       gh     gh.1
#>    1:   0.0 -22.5 3167.817 11.73417
#>    2:   2.5 -22.5 3166.253 11.84884
#>    3:   5.0 -22.5 3165.204 11.94097
#>    4:   7.5 -22.5 3164.823 12.01036
#>    5:  10.0 -22.5 3165.247 12.05689
#>   ---                              
#> 4028: 347.5 -90.0 2701.129  0.00000
#> 4029: 350.0 -90.0 2701.129  0.00000
#> 4030: 352.5 -90.0 2701.129  0.00000
#> 4031: 355.0 -90.0 2701.129  0.00000
#> 4032: 357.5 -90.0 2701.129  0.00000
ggplot(jan, aes(lon, lat)) +
    geom_contour(aes(z = gh.1, color = ..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]
#>         lat k          x        y   lon
#>    1: -22.5 1 0.00000000 11.73417   0.0
#>    2: -22.5 1 0.04363323 11.84884   2.5
#>    3: -22.5 1 0.08726646 11.94097   5.0
#>    4: -22.5 1 0.13089969 12.01036   7.5
#>    5: -22.5 1 0.17453293 12.05689  10.0
#>   ---                                  
#> 8060: -90.0 2 6.06501915  0.00000 347.5
#> 8061: -90.0 2 6.10865238  0.00000 350.0
#> 8062: -90.0 2 6.15228561  0.00000 352.5
#> 8063: -90.0 2 6.19591884  0.00000 355.0
#> 8064: -90.0 2 6.23955208  0.00000 357.5
ggplot(waves, aes(lon, lat)) +
    geom_contour(aes(z = y, color = ..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)]
#>         lon   lat       gh     gh.1  gh.no12
#>    1:   0.0 -22.5 3167.817 11.73417 5.858149
#>    2:   2.5 -22.5 3166.253 11.84884 4.006104
#>    3:   5.0 -22.5 3165.204 11.94097 2.689758
#>    4:   7.5 -22.5 3164.823 12.01036 2.061137
#>    5:  10.0 -22.5 3165.247 12.05689 2.261535
#>   ---                                       
#> 4028: 347.5 -90.0 2701.129  0.00000 0.000000
#> 4029: 350.0 -90.0 2701.129  0.00000 0.000000
#> 4030: 352.5 -90.0 2701.129  0.00000 0.000000
#> 4031: 355.0 -90.0 2701.129  0.00000 0.000000
#> 4032: 357.5 -90.0 2701.129  0.00000 0.000000
ggplot(jan, aes(lon, lat)) +
    geom_contour(aes(z = gh.no12, color = ..level..)) +
    coord_polar()


# Much faster
jan[, gh.no12 := FilterWave(gh, -2:0), by = .(lat)]
#>         lon   lat       gh     gh.1  gh.no12
#>    1:   0.0 -22.5 3167.817 11.73417 5.858149
#>    2:   2.5 -22.5 3166.253 11.84884 4.006104
#>    3:   5.0 -22.5 3165.204 11.94097 2.689758
#>    4:   7.5 -22.5 3164.823 12.01036 2.061137
#>    5:  10.0 -22.5 3165.247 12.05689 2.261535
#>   ---                                       
#> 4028: 347.5 -90.0 2701.129  0.00000 0.000000
#> 4029: 350.0 -90.0 2701.129  0.00000 0.000000
#> 4030: 352.5 -90.0 2701.129  0.00000 0.000000
#> 4031: 355.0 -90.0 2701.129  0.00000 0.000000
#> 4032: 357.5 -90.0 2701.129  0.00000 0.000000
ggplot(jan, aes(lon, lat)) +
    geom_contour(aes(z = gh.no12, color = ..level..)) +
    coord_polar()


# Using positive numbers returns the field
jan[, gh.only12 := FilterWave(gh, 2:1), by = .(lat)]
#>         lon   lat       gh     gh.1  gh.no12 gh.only12
#>    1:   0.0 -22.5 3167.817 11.73417 5.858149  11.17110
#>    2:   2.5 -22.5 3166.253 11.84884 4.006104  11.45865
#>    3:   5.0 -22.5 3165.204 11.94097 2.689758  11.72661
#>    4:   7.5 -22.5 3164.823 12.01036 2.061137  11.97348
#>    5:  10.0 -22.5 3165.247 12.05689 2.261535  12.19776
#>   ---                                                 
#> 4028: 347.5 -90.0 2701.129  0.00000 0.000000   0.00000
#> 4029: 350.0 -90.0 2701.129  0.00000 0.000000   0.00000
#> 4030: 352.5 -90.0 2701.129  0.00000 0.000000   0.00000
#> 4031: 355.0 -90.0 2701.129  0.00000 0.000000   0.00000
#> 4032: 357.5 -90.0 2701.129  0.00000 0.000000   0.00000
ggplot(jan, aes(lon, lat)) +
    geom_contour(aes(z = gh.only12, color = ..level..)) +
    coord_polar()


# Compute the envelope of the geopotential
jan[, envelope := WaveEnvelope(gh.no12), by = .(lat)]
#>         lon   lat       gh     gh.1  gh.no12 gh.only12  envelope
#>    1:   0.0 -22.5 3167.817 11.73417 5.858149  11.17110 10.834194
#>    2:   2.5 -22.5 3166.253 11.84884 4.006104  11.45865  9.544124
#>    3:   5.0 -22.5 3165.204 11.94097 2.689758  11.72661  8.173016
#>    4:   7.5 -22.5 3164.823 12.01036 2.061137  11.97348  6.906214
#>    5:  10.0 -22.5 3165.247 12.05689 2.261535  12.19776  5.958197
#>   ---                                                           
#> 4028: 347.5 -90.0 2701.129  0.00000 0.000000   0.00000  0.000000
#> 4029: 350.0 -90.0 2701.129  0.00000 0.000000   0.00000  0.000000
#> 4030: 352.5 -90.0 2701.129  0.00000 0.000000   0.00000  0.000000
#> 4031: 355.0 -90.0 2701.129  0.00000 0.000000   0.00000  0.000000
#> 4032: 357.5 -90.0 2701.129  0.00000 0.000000   0.00000  0.000000
ggplot(jan[lat == -60], aes(lon, gh.no12)) +
    geom_line() +
    geom_line(aes(y = envelope), color = "red")

Arguments

Value

Details

References

See also

Examples