How does the equation for the SpatRaster roughness index (terrain, v = "roughness") work?

The terra package offers and describes the following terrain indices:

x <- terrain(x, v="roughness")
x <- terrain(x, v="TPI")
x <- terrain(x, v="TRI")

I am confused on how this is calculated based on the package description of roughness as "the difference between the maximum and the minimum value of a cell and its 8 surrounding cells" (Hijmans et al. 2023). How does this work for edge and corner cells? I am assuming that the calculation reduces to a cell and its 5 or 3 surrounding cells in these cases?

The ruggedness (TRI) index is described as "the mean of the absolute differences between the value of a cell and the value of its 8 surrounding cells". The following is a graphic illustration of how I envision the calculation of these indices from the description provided.

Does this provide a correct interpretation of these indices?

If this interpretation is incorrect, then I am hoping someone point me in the correct direction (a reference) or explain here. I am interested in coding to calculate a slope of 16° from a DSM and an elevational difference of 1.3 m, but think that a terrain index would give a better indicator of the 1.3 m criterion for this habitat model.

## > 16° slope
habitat_slope_mat <- matrix(nrow = 2, ncol = 3)
habitat_slope_mat[1, ] <- c(0,16,0) # from,to = 0 absent
habitat_slope_mat[2, ] <- c(16,minmax(x)[2],1) # from,to = 1 present

habitat_slope <- classify(x, habitat_slope_mat, include.lowest=TRUE)

I looked at the cited references and was expecting to find the formula for this to help me think of the best way to treat the 1.3 m criterion. I have been unable to locate a written / published description that further explains the method. This paper is listed in the citations for the terrain function description:

Jones, K.H., 1998. A comparison of algorithms used to compute hill (sic) *terrain *as a property of the DEM. Computers & Geosciences 24: 315-323

The correct title for the article (DOI: 10.1016/S0098-3004(98)00032-6) is: "A comparison of algorithms used to compute hill *slope *as a property of the DEM". I cannot locate the formula for roughness in that paper and was interested in reading more on this topic.

I am not sure if this question is appropriate here, as you do not seem to be asking a coding question.

The manual points to Wilson et al (2007) for terrain indices. It also shows how you can use focal instead of terrain to compute them.

You can see for yourself what happens with small examples like this:

library(terra)
x <- rast(nrow=3, ncol=3, vals=c(1,2,3,1,2,1,1,2,8), ext=ext(0,1,0,1), crs="local")

as.matrix(x, wide=T)
#     [,1] [,2] [,3]
#[1,]    1    2    3
#[2,]    1    2    1
#[3,]    1    2    8

terrain(x, "roughness") |> as.matrix(wide=TRUE)
#     [,1] [,2] [,3]
#[1,]  NaN  NaN  NaN
#[2,]  NaN    7  NaN
#[3,]  NaN  NaN  NaN

focal(x, w=3, fun=\(x) {max(x) - min(x)}) |> as.matrix(wide=T)
#     [,1] [,2] [,3]
#[1,]   NA   NA   NA
#[2,]   NA    7   NA
#[3,]   NA   NA   NA
 
terrain(x, "TRI") |> as.matrix(wide=TRUE)
#     [,1]  [,2] [,3]
#[1,]  NaN   NaN  NaN
#[2,]  NaN 1.375  NaN
#[3,]  NaN   NaN  NaN

focal(x, w=3, fun=\(x) sum(abs(x[-5]-x[5]))/8) |> as.matrix(wide=T)
#     [,1]  [,2] [,3]
#[1,]   NA    NA   NA
#[2,]   NA 1.375   NA
#[3,]   NA    NA   NA

So the edges become missing (you could do other things via focal)

Or look at the source code.