Calculate Lagrangian correlation of the Askey form
Arguments
- v1
Prevailing wind, u-component.
- v2
Prevailing wind, v-component.
- k
Scale parameter of \(\|\boldsymbol v\|\), \(k>0\). Default is 2.
- h1
Horizontal distance matrix or array.
- h2
Vertical distance matrix or array, same dimension as
h1.- u
Time lag, same dimension as
h1.
Details
The Lagrangian correlation function of the Askey form with parameters \(\boldsymbol v = (v_1, v_2)^\top\in\mathbb{R}^2\) has the form $$C(\mathbf{h}, u)=\left(1-\dfrac{1}{k\|\boldsymbol v\|} \left\|\mathbf{h}-u\boldsymbol v\right\|\right)^{3/2}_+,$$ where \(\|\cdot\|\) is the Euclidean distance, \(x_+=\max(x,0), \mathbf{h} = (\mathrm{h}_1, \mathrm{h}_2)^\top\in\mathbb{R}^2\), and \(k > 0\) is the scale parameter controlling the magnitude of asymmetry in correlation.
References
Askey, R. (1973). Radial characteristic functions, Tech. Report No. 1262, Math. Research Center, University of Wisconsin-Madison.
See also
Other correlation functions:
cor_cauchy(),
cor_exp(),
cor_fs(),
cor_lagr_exp(),
cor_lagr_tri(),
cor_sep(),
cor_stat(),
cor_stat_rs()
Examples
h1 <- matrix(c(0, -5, 5, 0), nrow = 2)
h2 <- matrix(c(0, -8, 8, 0), nrow = 2)
u <- matrix(0.1, nrow = 2, ncol = 2)
cor_lagr_askey(v1 = 5, v2 = 10, h1 = h1, h2 = h2, u = u)
#> [,1] [,2]
#> [1,] 0.9259455 0.4974824
#> [2,] 0.3839919 0.9259455
h1 <- array(c(0, -10, 10, 0), dim = c(2, 2, 3))
h2 <- array(c(0, -10, 10, 0), dim = c(2, 2, 3))
u <- array(rep(-c(1, 2, 3), each = 4), dim = c(2, 2, 3))
cor_lagr_askey(v1 = 10, v2 = 10, h1 = h1, h2 = h2, u = u)
#> , , 1
#>
#> [,1] [,2]
#> [1,] 0.3535534 0.0000000
#> [2,] 1.0000000 0.3535534
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 0.0000000 0
#> [2,] 0.3535534 0
#>
#> , , 3
#>
#> [,1] [,2]
#> [1,] 0 0
#> [2,] 0 0
#>