Gaspari-Cohn Localisation Taper — gaspari

Evaluates the Gaspari & Cohn (1999) fifth-order piecewise-rational compactly-supported correlation function on a matrix of distances. This is the classical distance-based localisation taper: a smooth bump that equals 1 at zero distance, decays to 0 at twice the localisation radius, and is identically 0 beyond. Used to taper the Kalman gain when the parameters and observations carry a spatial (or otherwise metric) coordinate.

Usage

gaspari_cohn(distances, radius)

Arguments

distances: Matrix (npar x nobs). Non-negative parameter-to- observation distances.
radius: Numeric scalar (> 0). Localisation radius $c$; the taper reaches 0 at distance $2c$.

Value

Matrix (npar x nobs) of taper weights in $[0, 1]$.

Details

With $z = d / c$ (distance over localisation radius $c$): $$G(z) = \begin{cases} -\tfrac{1}{4}z^5 + \tfrac{1}{2}z^4 + \tfrac{5}{8}z^3 - \tfrac{5}{3}z^2 + 1 & 0 \le z \le 1 \\ \tfrac{1}{12}z^5 - \tfrac{1}{2}z^4 + \tfrac{5}{8}z^3 + \tfrac{5}{3}z^2 - 5z + 4 - \tfrac{2}{3}z^{-1} & 1 < z \le 2 \\ 0 & z > 2. \end{cases}$$

References

Gaspari, G. & Cohn, S.E. (1999). Construction of correlation functions in two and three dimensions. Quarterly Journal of the Royal Meteorological Society, 125(554), 723–757.

Examples

d <- matrix(c(0, 0.5, 1, 1.5, 2, 3), nrow = 2L)
gaspari_cohn(d, radius = 1.0)
#>           [,1]       [,2] [,3]
#> [1,] 1.0000000 0.20833333    0
#> [2,] 0.6848958 0.01649306    0