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#' @title Spherical Variance or Standard Deviation of Surface
#' @description Derives the spherical standard deviation of a raster surface
#'
#' @param r A terra SpatRaster class object
#' @param d Size of focal window or a matrix to use in focal function
#' @param variance (FALSE|TRUE) Output spherical variance rather than standard deviation
#' @param ... Additional arguments passed to terra:app (can write raster to disk here)
#'
#' @details
#' Surface variability using spherical variance/standard deviation.
#' The variation can be assessed using the spherical standard deviation of the normal
#' direction within a local neighborhood. This is found by expressing the normal
#' directions on the surfaces cells in terms of their displacements in a Cartesian (x,y,z)
#' coordinate system. Averaging the x-coordinates, y-coordinates, and z-coordinates
#' separately gives a vector (xb, yb, zb) pointing in the direction of the average
#' normal. This vector will be shorter when there is more variation of the normals and
#' it will be longest--equal to unity--when there is no variation. Its squared length
#' is (by the Pythagorean theorem) given by: R^2 = xb^2 + yb^2 + zb^2
#' where; x = cos(aspect) * sin(slope) and xb = nXn focal mean of x
#' y = sin(aspect) * sin(slope) and yb = nXn focal mean of y
#' z = cos(slope) and zb = nXn focal mean of z
#'
#' The slope and aspect values are expected to be in radians.
#' The value of (1 - R^2), which will lie between 0 and 1, is the spherical variance.
#' and it's square root can be considered the spherical standard deviation.
#'
#' @return A terra SpatRaster class object of the spherical standard deviation
#'
#' @author Jeffrey S. Evans <jeffrey_evans<at>tnc.org>
#'
#' @examples
#' \donttest{
#' library(terra)
#' elev <- rast(system.file("extdata/elev.tif", package="spatialEco"))
#'
#' ssd <- spherical.sd(elev, d=5)
#'
#' slope <- terrain(elev, v='slope')
#' aspect <- terrain(elev, v='aspect')
#' hill <- shade(slope, aspect, 40, 270)
#' plot(hill, col=grey(0:100/100), legend=FALSE,
#' main='terrain spherical standard deviation')
#' plot(ssd, col=rainbow(25, alpha=0.35), add=TRUE)
#' }
#'
#' @seealso \code{\link[terra]{app}} for details on ... arguments
#'
#' @export
spherical.sd <- function(r, d, variance = FALSE, ...) {
if(!inherits(r, "SpatRaster"))
stop("r must be a terra or raster object")
if(class(d)[1] != "matrix") { d = matrix(1,d,d) }
s <- terra::terrain(r, v='slope', unit='radians')
a <- terra::terrain(r, v='aspect', unit='radians')
x <- terra::lapp(c(a,s), fun=function(x,y) {cos(x) * sin(y) } )
y <- terra::lapp(c(a,s), fun=function(x,y) {sin(x) * sin(y) } )
z <- terra::app(s, fun=cos)
xb = terra::focal(x, d, fun=mean)
yb = terra::focal(y, d, fun=mean)
zb = terra::focal(z, d, fun=mean)
if(variance == TRUE) {
return(
terra::lapp( c(xb, yb, zb), fun=function(x,y,z) { 1 - (x^2 + y^2 + z^2) }, ... )
)
} else {
return(
terra::lapp( c(xb, yb, zb), fun=function(x,y,z) { sqrt( 1 - (x^2 + y^2 + z^2) ) }, ... )
)
}
}
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