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)

## 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 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.

## 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

Other meteorology functions: Derivate(), EOF(), GeostrophicWind(), WaveFlux(), thermodynamics

## 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)
#> $amplitude #> [1] 0.5346706 0.2782641 0.1818072 1.1082726 1.5542779 4.0782414 0.0297764 #> [8] 1.9349096 1.5767522 0.6752076 #> #>$phase
#>  [1] 4.53676694 1.93019927 1.84107225 1.22020775 0.77217833 0.05856863
#>  [7] 0.40566550 0.20271090 0.36914895 0.54555395
#>
#> $k #> [1] 1 2 3 4 5 6 7 8 9 10 #> #>$r2
#>  [1] 1.044884e-02 2.830151e-03 1.208140e-03 4.489403e-02 8.829839e-02
#>  [6] 6.079128e-01 3.240707e-05 1.368412e-01 9.087038e-02 1.666365e-02
#>

# 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))
#> [1] TRUE

# The Wave envelopes shows where the signal is the most "wavy".
plot(x, type = "l", col = "grey")
#> Warning: "add" is not a graphical parameter

# 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)]
#>         lon   lat       gh     gh.1
#>       <num> <num>    <num>    <num>
#>    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 = 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]
#>         lat     k          x        y   lon
#>       <num> <num>      <num>    <num> <num>
#>    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 = 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)]
#>         lon   lat       gh     gh.1  gh.no12
#>       <num> <num>    <num>    <num>    <num>
#>    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 = after_stat(level))) +
coord_polar()

# Much faster
jan[, gh.no12 := FilterWave(gh, -2:0), by = .(lat)]
#>         lon   lat       gh     gh.1  gh.no12
#>       <num> <num>    <num>    <num>    <num>
#>    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 = after_stat(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
#>       <num> <num>    <num>    <num>    <num>     <num>
#>    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 = after_stat(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
#>       <num> <num>    <num>    <num>    <num>     <num>     <num>
#>    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")

`