# Measuring Hyperscale Topographic Anisotropy as a Continuous Landscape Property

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Directional LTP Sampling

_{0}) (Figure 1). The integral image filtering technique offers constant-time calculations for the number, mean, and variance of elevation samples within the filter, with the trade-off of using strictly rectangular window geometry [20,28,29]. The algorithm determines filter geometry by iterating over the range of filter half-lengths specified by the user, where the total length of the filter (L), in cells, is given by:

_{0}− µ)/σ,

_{0}is the central analytical cell, µ is the window mean value, and σ is the window standard deviation.

_{0}) measure the difference between the z

_{0}and the adjacent areas, rather than relating it to the surrounding area. Therefore, windows must be centered to avoid measuring what is effectively an area-weighted gradient. This gradient exists along the spatial limits of the domain with an elevation gradient due to the disproportionate inclusion of ‘no data’ values in some orientations, skewing the directional LTP value by the elevation gradient. This produces a notable edge effect that results in artificially large anisotropy values. To mitigate this distortion, the edges were buffered by the filter half-length such that no samples were taken from beyond the extent of the data. However, buffering the edges forces z

_{0}towards the center of the image as the window size increases, limiting the maximum window size along the perimeter.

_{0}, all directions are sampled from rotation axes between [0, π] radians. A continuous sampling strategy would compute a value for all rotation axes. However, DEV is calculated over an area and must sample swaths of rotation axes. More specific to the sampling strategy described above, rotation axes zero to π radians are sampled using four windows—providing a DEV value for every π/4 radian (45°) swath (Figure 2). Conversely, isotropic sampling integrates all orientations, computing a local average for the entire neighborhood. This sampling strategy achieves the efficiency required for hyperscaled analysis through the integral image filtering technique, which comes at the cost of coarse orientation resolution. Therefore, measures of the orientation of anisotropy have been excluded from the method due to the coarse resolution orientation sampling.

#### 2.2. Anisotropy Model

_{d}) was used to characterize anisotropy magnitude (A

_{m}), using the isotropically sampled DEV value (DEV

_{i}) as the point of comparison in the difference term (Equation (3)).

_{m}= sqrt(1/n × SUM_(d = 0)^n (DEV

_{d}− DEV

_{i})^2),

_{m}is anisotropy magnitude, n is the number of oriented windows, DEV

_{d}are DEV values sampled from directionally oriented windows, and DEV

_{i}is the DEV value sampled from the isotropic window.

_{m}measures variability in DEV across all sampled orientations. A

_{m}differs from ratio models used in the semivariogram methods by quantifying the average effect of orientation, as opposed to the maximum difference at an angle, orthogonal or otherwise. It should be noted that A

_{m}can be replaced with more sensitive and common metrics such as a ratio or the min-max range using the same sampling strategy. The tools developed for this research are included as plug-in tools in the open-source WhiteboxTools library as part of the Whitebox GAT project [30]. The specific details of the algorithm implementation can be viewed in the WhiteboxTools source code.

#### 2.3. Analysis

_{m}value for each cell, similar to the method used by Lindsay et al. (2015) [20] for hyperscale LTP analysis. The A

_{max}implementation outputs a second R

_{max}raster, which stored the filter half-length at which the A

_{max}value occurred. These combined rasters provide a method to feasibly characterize the topographic anisotropy magnitude for a landscape, while also retaining information from the scale dimension.

#### 2.4. Data Sets and Study Sites

## 3. Results

#### 3.1. Topographic Anisotropy-Scale Signatures

_{m}) responds to changes in scale, analogous to how spectral signatures characterize surface reflectance response to changes in the wavelength of light. Figure 5 shows scale signatures for the WSNM study sites (Sites 1 through 4) and Figure 6 shows the scale signatures for the PDR study sites (Sites 5 through 7). A

_{m}was sampled with filter half-lengths ranging from 7 to 500 cells, incremented by one, for both data sets. This corresponds to filter half-lengths of 7 to 500 m and 70 to 5000 m for the WSNM and PDR data sets, respectively. While a unique scale signature exists for every cell, the seven sites are representative of the varied topographic conditions. Recall that the study areas (as well as study Sites 2, 3, 4, 5, and 6) were chosen for containing similarly sized and oriented, repeating anisotropic landforms. The scale signature for Site 1, a flat area, shows little anisotropy. The signatures for both Sites 3 and 4 also showed minimal anisotropy, despite featuring several linear shaped dunes. The small-scale activity (<90 m) for Site 4 (Figure 6) corresponds with the road network surrounding the location. Conversely, the scale signature for Site 2 captures the proportionally increasing size of the dunes, and the larger dune complex with a multimodal distribution and Sites 5 and 6 show multimodal distributions corresponding to the drumlins and the outcrop respectively, as well as the larger scale topography containing them (i.e., the valley and floodplain). Site 7 featured a topographically rough area, and produced a multimodal signature with relatively small magnitude anisotropy values.

#### 3.2. Spatial Distribution of Topographic Anisotropy

_{max}) for each cell, as well as secondary rasters of the filter half-length at which the maximum anisotropy magnitude value occurred (R

_{max}). Applying the information from the scale signatures allows the user to avoid the computation of redundant scales that are unlikely to update the A

_{max}value. Combining information from both the A

_{max}and R

_{max}rasters provides a self-selecting, hyperscale characterization of the spatial distribution of topographic anisotropy in the landscape, with a means to confirm the observed pattern through the R

_{max}raster.

_{max}was calculated for each grid cell of the two DEMs using filter half-lengths ranging from 50 to 300 cells, incremented by 5 cells. This corresponds to filter lengths (in ground units) ranging from 101 to 601 m and 1001 to 6001 m for the WSNM and PDR datasets respectively. For reference, the execution times (excluding data input and output) using an Intel i7-4770 CPU was 6.17 and 1.94 minutes for the WSNM (~116 million cells excluding nodata) and PDR (~40 million cells excluding nodata) data sets respectively. Figure 7 provides examples of the A

_{max}and R

_{max}raster outputs for both study DEMs, and Figure 8 provides detailed A

_{max}raster examples for the areas surrounding study Sites 2 through 6 to demonstrate the distribution of A

_{max}with respect to local anisotropic textures. The WSNM dune field R

_{max}raster demonstrates that maximum anisotropy occurred consistently at small scales (<100 m) for the more linear dunes, and occurs at larger scales in the region populated with more crescentic dunes (Figure 7b). The algorithm was also generally able to identify the linear texture created by the dunes, which appear with intermediate and high magnitude anisotropy values (Figure 7a and Figure 8b). This variation in anisotropy magnitude occurred when the dune geometry deviated from a linear pattern to saddle points where dunes merged and smaller scale features with different orientations (e.g., where a dune is transected by a road).

_{max}raster (Figure 7d). Although the PDR A

_{max}and R

_{max}outputs were more spatially varied compared to the WSNM outputs, the highly anisotropic cells were consistently co-located with anisotropic features such as drumlins, scours, outcrops, valleys and river channels (Figure 8d,e). Relatively larger scale topographic features such as valleys were moderately anisotropic while smaller scale features such as drumlins and river channels were more strongly anisotropic (Figure 8d,e). Flat areas were typically isotropic (Figure 7c). Buffering around the edges produced slightly elevated anisotropy values due to the exclusion of larger filter sizes from the analysis.

## 4. Discussion

_{max}analysis used the maximum anisotropy values across a range of scales for each cell to create spatially referenced raster images, effectively mapping the highest anisotropy value from every cell’s scale signature. Although the multimodal distributions found in the scale signatures suggest that the maximum anisotropy is not the only significant characteristic of signatures (e.g., number of peaks indicates signature complexity), high anisotropy values were still consistently co-located with anisotropic landforms. Furthermore, the low spatial variation in the WSNM R

_{max}raster (Figure 7b) indicates that the A

_{max}values occurred at consistent scales, suggesting that the dune landforms were driving the anisotropy values by influencing the landscape texture. Conversely, the highly varied mosaic of scales in the PDR R

_{max}image (Figure 7d) revealed the absence of an optimal single window size with which to measure anisotropy, especially for complex terrain. This also applies to extensive areas where larger scale landscape shaping processes occur simultaneously with smaller scale processes. This emphasizes the value of hyperscale methodological approaches by removing the responsibility of scale selection from the user, and minimizing potential errors as a result. The A

_{max}method self-selects the optimal window size for each individual grid cell, resulting in a hyperscale characterization of topographic anisotropy.

_{max}values approximately half as large as other parts of the same dune. This limitation is also expressed as narrow bands of lower A

_{max}values at rotation angles that lie in between the swath during the rapid change in orientation of the crest for the crescentic dunes. Secondly, the limited ability to characterize topographic anisotropy for complex geometric patterns. However, the landscape geometries that can be characterized are comparable to the second-order partial derivatives used to define elementary forms [38], morphometric primitives [39], and the Hessian matrices used to detect ridges in the field of computer vision [40,41]. The information generated is well-suited as a morphometric characteristic that can augment the translation of continuous landscapes to discrete landforms [38,42,43]. Alternatively, this method is also ideal to exploratory analyses, to guide the parameterization (such as scale selection) of targeted measures on known entities.

## 5. Conclusions

_{max}and R

_{max}rasters demonstrated spatial co-location of large anisotropy magnitude values with anisotropic landforms at scales consistent with the expected scale range of the landforms. Therefore, despite inherently coarse orientation sampling, this method has demonstrated the ability to characterize hyperscale topographic anisotropy in natural landscapes. Future work should compare the RMSD model of anisotropy to the more common ratio models and develop better ways to analyze and visualize voluminous multidimensional spatial data.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Graphical representation of the orientated window aggregation scheme. The grey boxes represent the windows from the 3 × 3 matrix that were aggregated to form the (

**A**) north–south oriented window, (

**B**) east–west oriented window, (

**C**) northwest–southeast oriented window, (

**D**) northeast–southwest oriented window, and (

**E**) the isotropic window. Note that each window from the 3 × 3 matrix is composed of a variable number of grid cells from the input data.

**Figure 2.**Schematic demonstrating the theoretical differences between the deviation from mean elevation (DEV) measurements from the π/4 radian swath sampling strategy, an isotropic sampling strategy, and hypothetical continuous sampling strategy using infinitesimally small swaths (effectively infinite line measurements).

**Figure 5.**Anisotropy scale signatures for the WSNM data set. The hillshade image for each study site is provided and labelled accordingly.

**Figure 6.**Anisotropy scale signatures for the PDR data set. The hillshade image for each study site is provided and labelled accordingly.

**Figure 7.**Spatial distribution of A

_{max}and R

_{max}rasters for the WNSM (

**A**and

**B**respectively), and PDR (

**C**and

**D**respectively), using filter half-lengths ranging from 50 to 300 grid cells. Both color ramps use blue to represent small magnitude values and red to represent large scale values, relative to the data set.

**Figure 8.**High-magnification A

_{max}rasters for the areas surrounding study Sites 2 through 6 using filter size half-lengths ranging from 50 to 300 grid cells. Blue represents isotropy, green and yellow represents intermediate anisotropy, while red represents strong anisotropy, scaled relative to the data set.

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**MDPI and ACS Style**

Newman, D.R.; Lindsay, J.B.; Cockburn, J.M.H.
Measuring Hyperscale Topographic Anisotropy as a Continuous Landscape Property. *Geosciences* **2018**, *8*, 278.
https://doi.org/10.3390/geosciences8080278

**AMA Style**

Newman DR, Lindsay JB, Cockburn JMH.
Measuring Hyperscale Topographic Anisotropy as a Continuous Landscape Property. *Geosciences*. 2018; 8(8):278.
https://doi.org/10.3390/geosciences8080278

**Chicago/Turabian Style**

Newman, Daniel R., John B. Lindsay, and Jaclyn M. H. Cockburn.
2018. "Measuring Hyperscale Topographic Anisotropy as a Continuous Landscape Property" *Geosciences* 8, no. 8: 278.
https://doi.org/10.3390/geosciences8080278