Fractal dimension methods

The lvec

Selecting a sensible sequence of scales (i.e., the length vector lvec) is critical and should be the same when comparing surfaces and objects. The lvec should range between the resolution and extent of the surface or mesh, or across a range of scales relevant to the question being address. If you do not enter an lvec, each function will estimate one for you based on some rules of thumb for that particular method (see Table). For example, hvar, sd, cubes and boxes will generate a sequence from 2- to 10-times the resolution to the extent on a log2-scale. area will generate a sequence from 2 times the resolution to 1/8 the extent on the log2-scale.

The area method has increased bias with increased values inside lvec, because of the way surface area is calculated at each scale (using Jenness, 2004). The method calculates surface area of each cell by using the height of the cell and its 8 surrounding cells to create a surface. For border cells, the cells outside of the DEM are assumed to be of equal height as the border cells. This means that border cells may have lower surface area estimates. If the ratio of border cells to total cells inside a DEM is high, there may be an underestimation of surface area, which is the case when values inside lvec are too close to the extent. For example, if lvec contains a scale of extent/2, all the cells are border cells; if it contains extent/4, 75% of the cells are border cells. For this reason, the area method may be better suited for larger areas, relative to the lvec.

Below a fractal 128 x 128 terrain is simulated and fractal dimension is estimated with the height variation method. Because no lvec was provided, one was generated as c(8, 16, 32, 64, 128).

# simulate fractal terrain
surf <- sim_dem(L = 64, smoothness = 0.5)

# fractal dimension using height variation method
fd(surf, method="hvar", plot = TRUE, diagnose = TRUE)
#> lvec is set to c(8, 16, 32, 64).

#> $D
#> [1] 2.478227
#> 
#> $data
#>    l         h
#> 1  8 0.5603946
#> 2 16 0.8430506
#> 3 32 1.1709355
#> 4 64 1.6768663
#> 
#> $lvec
#> [1]  8 16 32 64
#> 
#> $D_vec
#> [1] 2.410824 2.526029 2.481894
#> 
#> $var
#> [1] 0.05812521
#> 
#> $method
#> [1] "hvar"

When plot=TRUE, a plot of the surface is generated with red rectangles representing lvec to help visualize the scales at which fractal dimension is estimated. When diagnostic=TRUE, the relationship between scales and mean surface heights is plotted. The red line in the best-fit line for all point and dashed lines are for each pair-wise scale with localized D estimates. For the height range method, D is 3 minus the slope. The diagnostic plot helps identify scales that are problematic (typically the extremes of lvec), which then have bias the estimate. The lvec is returned and can then be altered, entered explicitly, and fractal dimension estimated again.

hvar and sd methods are not sensitive to z-axis scaling, whereas area and cubes methods are. Here are estimates for the relative flat surface we simulated with smoothness=0.5.

# height variation
fd(surf, method = "hvar")
#> lvec is set to c(8, 16, 32, 64).
#> [1] 2.478227

# standard deviation
fd(surf, method = "sd")
#> lvec is set to c(8, 16, 32, 64).
#> [1] 2.594688

# area
fd(surf, method = "area")
#> lvec is set to c(2, 4, 8).
#> [1] 2.001035

# cube counting
fd(surf, method = "cubes")
#> lvec is set to c(2, 4, 8, 16, 32, 64).
#> [1] 2.005364

hvar and sd estimates are in the vicinity of D=2.5. However, area and cubes methods result in estimates close to 2: a flat surface. If we transform the z-scale to match the extent (L=128), we get the following.

surf_z <- surf
values(surf_z) <- values(surf_z) * ((64-1) / hr(surf_z))
hr(surf_z)
#> [1] 63

# height variation
fd(surf_z, method = "hvar")
#> lvec is set to c(8, 16, 32, 64).
#> [1] 2.478227

# standard deviation
fd(surf_z, method = "sd")
#> lvec is set to c(8, 16, 32, 64).
#> [1] 2.594688

# area
fd(surf_z, method = "area")
#> lvec is set to c(2, 4, 8).
#> [1] 2.361038

# cube counting
fd(surf_z, method = "cubes")
#> lvec is set to c(2, 4, 8, 16, 32, 64).
#> [1] 2.330863

hvar and sd estimates are identical (i.e., invariant to z-scaling) and area and cubes estimates are higher. This dichotomy raises an underlying philosophical question about fractal surfaces: does the fractal dimension of an surface change if it is transformed along a dimension?

The answer will likely depend on what you are analyzing. For example, if you are examining landscapes, then area and cubes methods will not be much use as they will tend to D=2 when surfaces a low relative to wide. Whereas, if you are examining surfaces that are as high as they are wide, then the area and cubes methods might be a better choice. If you are examining a closed 3D objects (i.e., watertight), then the cubes method is preferable as the 3D analogue to box counting on closed shapes like the coastline of the UK.

# cube counting
fd(mcap, method = "cubes", plot=TRUE, diagnose=TRUE)
#> lvec is set to c(0.053, 0.106, 0.212, 0.423).

#> $D
#> [1] 2.315246
#> 
#> $data
#>            l   n
#> 4 0.05291204 134
#> 3 0.10582408  31
#> 2 0.21164817   8
#> 1 0.42329634   1
#> 
#> $lvec
#> [1] 0.42329634 0.21164817 0.10582408 0.05291204
#> 
#> $D_vec
#> [1] 2.111893 1.954196 3.000000
#> 
#> $var
#> [1] 0.5638126
#> 
#> $method
#> [1] "cubes"

Here is an example of 2D box counting for the planar projection of the mcap mesh.

# project coral as xy coordinates
mcap_2d <- mesh_to_2d(mcap)

# box counting
fd(mcap_2d, method = "boxes", plot=TRUE, diagnose=TRUE)

#> $D
#> [1] 1.39879
#> 
#> $data
#>            l   n
#> 6 0.01219243 138
#> 5 0.02438486  65
#> 4 0.04876971  28
#> 3 0.09753943  11
#> 2 0.19507886   4
#> 1 0.39015772   1
#> 
#> $lvec
#> [1] 0.39015772 0.19507886 0.09753943 0.04876971 0.02438486 0.01219243
#> 
#> $D_vec
#> [1] 1.086157 1.215013 1.347923 1.459432 2.000000
#> 
#> $var
#> [1] 0.3523516
#> 
#> $method
#> [1] "boxes"

Non-fractal surfaces

Natural surfaces often have anomalies such as peaks, troughs and edges that can result in biased fractal dimension estimates. It is up to the user to determine which method is best for their purposes, and to demonstrate that a biological or ecological result is robust to the choice of method. The function detect_drop() can help to investigate if a DEM has a lot of overhangs or sudden drops. It returns a raster where values mark the areas with sudden drops depending on a given threshold (Default threshold = 0.1).

dem1 <- dem_crop(horseshoe, x0 = -470.8104, y0 = 1270.625, L = 2, plot = TRUE)

drop1 <- detect_drop(dem1, d = 0.1)
plot(drop1)

# This DEM does not have many drops
fd(dem1, method = "hvar", lvec = c(1, 0.5, 0.25, 0.125), diagnose=TRUE)

#> $D
#> [1] 2.35037
#> 
#> $data
#>       l          h
#> 1 0.125 0.09603534
#> 2 0.250 0.16825392
#> 3 0.500 0.24666635
#> 4 1.000 0.37923605
#> 
#> $lvec
#> [1] 0.125 0.250 0.500 1.000
#> 
#> $D_vec
#> [1] 2.190997 2.448079 2.379465
#> 
#> $var
#> [1] 0.1331159
#> 
#> $method
#> [1] "hvar"

dem2 <- dem_crop(horseshoe, x0 = -466.8104, y0 = 1266.625, L = 2, plot = TRUE)

drop2 <- detect_drop(dem2, d = 0.1)
plot(drop2)

fd(dem2, method = "hvar", lvec = c(1, 0.5, 0.25, 0.125), diagnose=TRUE)

#> $D
#> [1] 2.198017
#> 
#> $data
#>       l         h
#> 1 0.125 0.1849993
#> 2 0.250 0.3998837
#> 3 0.500 0.7186458
#> 4 1.000 0.9706139
#> 
#> $lvec
#> [1] 0.125 0.250 0.500 1.000
#> 
#> $D_vec
#> [1] 1.887939 2.154299 2.566383
#> 
#> $var
#> [1] 0.3418205
#> 
#> $method
#> [1] "hvar"

In the example above, dem2 has much more and higher drops compared to dem1. We can see its effect on the height variation method by looking at the diagnostics plot. The dots in the diagnostic plot for dem2 follow a curve, and do not fall on a straight line. The variance of D in dem2 (var = 0.44) is also larger than for dem1 (0.24). Moreover, the frequency of drops correlates slightly negatively with the fractal dimension estimates when using the hvar method. Therefore, caution must be used with the interpretation of fractal dimension when the DEM has many drops or overhangs. Changes in D depending on the scale is common for non-fractal surfaces and therefore it is crucial to carefully choose the lvec and keep it the same when comparing across many surfaces.

References

• Jenness, J.S. Calculating Landscape Surface Area from Digital Elevation Models. Wildlife Society Bulletin, Vol. 32, No. 3 (Autumn, 2004), pp. 829-839.