Digital elevation models (DEMs) are a gridded representation of the surface of the Earth and typically contain uncertainties due to data collection and processing. Slope and aspect estimates on a DEM contain errors and uncertainties inherited from the representation of a continuous surface as a grid (referred to as truncation error; TE) and from any DEM uncertainty. We analyze in detail the impacts of TE and propagated elevation uncertainty (PEU) on slope and aspect.

Using synthetic data as a control, we define functions to quantify both TE and PEU for arbitrary grids. We then develop a quality metric which captures the combined impact of both TE and PEU on the calculation of topographic metrics. Our quality metric allows us to examine the spatial patterns of error and uncertainty in topographic metrics and to compare calculations on DEMs of different sizes and accuracies.

Using lidar data with point density of

Continuous surfaces are often projected onto an evenly sampled grid – digital elevation models (DEMs) are a common example. The accuracy of the gridded representation of the underlying surface is controlled by the spacing of the grid, the variability of the surface (i.e., the terrain itself), and the amount of uncertainty added during data collection and processing.

In recent years, an ever-growing array of DEM datasets has become available
across a range of resolutions and spatial scales. As new data acquisition and
processing strategies – such as lidar, stereo photogrammetry, and structure
from motion – mature, the number and variability of features represented in
a DEM will grow dramatically. In this paper, we refer to DEMs as high
resolution when their grid spacing is low (e.g., 1–3 m) and low resolution
when their grid spacing is high (e.g., a 10+ m DEM). Across all DEM
resolutions, it is important to quantify the uncertainties in DEMs and their
potentially large impacts on metrics calculated from the DEM

The estimation of topographic metrics, such as slope and aspect, is a common
task across scientific disciplines. Slope and aspect provide important
boundary conditions for hillslope stability analysis

A wide range of methods has been developed to accurately derive slope and
aspect from elevation data using a range of approaches, each optimized for
different use cases

In addition to TE, real-world DEMs will have some degree of measurement
uncertainty. The magnitude of that uncertainty varies across data collection
methods and data post-processing but also depends on the terrain itself and
is often difficult to estimate

The impacts of TE and DEM uncertainty on slope and aspect estimations can be quantified for any gridded data. We first use synthetic data with known properties as a control to define generalized functions applicable to any DEM and to develop a quality metric for slope and aspect calculations. Following this analysis, we turn to a high-resolution lidar dataset covering complex terrain to study the spatial structure of uncertainty in topographic metrics propagating from both TE and DEM uncertainty. This novel approach allows us to identify the optimal grid resolution that minimizes the error bounds from the combination of TE and PEU, and to analyze the implications of using suboptimal DEM resolutions for calculating slope and aspect.

In this analysis, we demonstrate the limitations of calculating slope and aspect using synthetic data surfaces. These surfaces serve as a valuable control dataset, as the precise analytical values for slope and aspect at each grid cell are known. We further add normally distributed random noise with a known mean and standard deviation to our synthetic data to investigate the impacts of DEM uncertainty on topographic metrics. Following the discussion of synthetic data, we apply the same methods to high-resolution lidar data interpolated to DEMs with varying spatial resolutions.

We generate regular

Gaussian hill elevations at

In this study, we focus on the topographic metrics (1) slope (

There exists a wide range of methods for calculating the directional
derivatives on gridded data that broadly fall into three classes:
(1) four-neighborhood methods, (2) eight-neighborhood methods, and
(3) steepest descent methods

Biases in topographic metrics can be determined both numerically, by comparing the results of calculations to known values, and analytically, by deriving the impacts of both TE and DEM uncertainty on topographic metrics.

We derive the TE from the formulas of topographic metrics by propagating the
uncertainty of the second-order finite difference approximation for the
directional derivatives

The considered topographic metrics – slope (

The sign of the TE is derived from symmetry considerations in the following way:

Equations (

Gaussian hill slope

As can be seen in Fig.

As the precise mathematical definitions of our synthetic surfaces are known, we can analytically derive their slope and aspect values at each point and compare those results to numerical calculations (see Fig. S4). In the absence of DEM uncertainty, the difference between analytically and numerically derived topographic metrics will be dominated by the TE from the second-order finite difference approximation. Additionally, modifying the absolute magnitude of the shape heights (e.g., a maximum height of 100 instead of 1) does not modify the absolute magnitudes of slope and aspect TE (see Fig. S5).

Slope and aspect calculations have very different spatial error patterns (see
Fig.

Elevation models are never perfect – there are always errors and
uncertainties due to data collection or processing. The uncertainty in
calculated topographic metrics can be constrained by propagating DEM
uncertainties into their calculations

As with TE, PEU introduces distinctive spatial patterns into the slope and
aspect estimates (Fig.

Gaussian hill (

When we compare our analytically derived topographic metric uncertainty
patterns (Fig.

Both TE and PEU yield distinct spatial patterns, implying that the error and
uncertainty in topographic metrics will vary throughout a DEM. Using both the
TE and PEU magnitudes, we can examine whether TE or PEU is dominant at an
arbitrary grid spacing, given a known DEM uncertainty
(Fig.

TE and PEU magnitudes for two noise levels,

As can be seen in Fig.

The trade-off between TE and PEU can also be thought of as a quality ratio.
Simply put, the impact of DEM uncertainty on topographic metrics is modulated
by the grid spacing – i.e., 1 cm of vertical noise on a 10 m grid will
have a very different impact than 1 cm of vertical noise on a 1 cm grid. As
both TE and PEU have distinct spatial patterns, a combined quality ratio (QR)
can capture a holistic view of the overlapping influence of TE and PEU. The
QR can be defined from the combination of the TE and PEU as

The QR can also used as a normalized metric to compare DEMs across grid
spacings and noise levels (Figs.

Gaussian hill slope

As can be seen in Fig.

There are distinct differences between the optimal slope and aspect grid
spacings for the same dataset. For any given noise level, the optimal grid
spacing to calculate aspect is higher than that for slope. The difference in
optimal grid spacing is driven by the relative magnitudes of the PEU between
slope and aspect calculations – the PEU for aspect calculations is always
higher than that for slope due to differences in the formulas for slope and
aspect (see Fig.

In practice, datasets will not have homogeneous noise across grid spacings –
coarser parameterizations of a surface will include more uncertainty as
fine-scale features are aggregated into single pixels. Additionally, noise in
real-world datasets is influenced by non-uniform landscape features such as
slope, aspect, terrain relief, and vegetation cover
(e.g.,

When we compare the heterogeneous noise result to the analysis shown in
Fig.

Optimal grid spacing for slope

As can be seen from Fig.

As can be seen in Figs.

When homogeneous noise is added to synthetic surfaces, the large spikes in
aspect distributions related to the cardinal directions (e.g., at
45/90/135

Slope distributions are systematically shifted towards higher values as more noise is added to the synthetic surfaces. This effect is due to the presence of more large “steps” in the synthetic data, where smooth transitions across elevation gradients are replaced with more stepped hills and dips. Across noise levels and grid sizes, distributions with similar QRs maintain the same general shape. This indicates that the QR captures how “wrong” the aggregate distribution is when compared to the original synthetic surface.

If different window sizes are used to calculate slope and aspect (e.g.,

The Santa Cruz Island (SCI; see Fig.

Slope

Topographic setting of Santa Cruz Island (SCI), showing
elevation

Using LAStools

In addition to producing DEMs, we generate pixel-wise standard deviations estimated from the point cloud. That is, we determined the standard deviation of all ground-classified lidar points falling into each grid cell for each grid resolution. While vegetation cover may impact ground-classification results, our calculations are based on ground-classified points only. Because of the high point-cloud density, this standard deviation metric should reflect terrain variability and not vegetation cover. This measure serves as a DEM uncertainty and is highly spatially heterogeneous. Elevation and standard deviation maps for additional grid resolutions can be found in the Supplement (Figs. S22–S24). An island-wide canopy-height model can be found in Fig. S12.

Using our standard deviation grids, we can then calculate both TE and PEU on
SCI (see Fig. S25). Using these two grids as inputs, we can also derive the
QR across the entire SCI (Fig.

Selected metrics for the Pozo catchment in the southwestern part of
SCI, calculated on the optimal grid resolution for this catchment (3 m
grid):

Using our pixel-wise standard deviations – and derived QRs – at each grid resolution, we can determine the island-wide optimal grid resolution for the calculation of topographic metrics which minimizes the combined influence of TE and PEU on our lidar dataset.

Median QRs for all slopes, low slopes (

Unlike the synthetic data examined in Fig.

It should be noted that a 4 m optimal grid resolution is likely too
large for many applications, particularly those interested in
microtopography or other small-scale features. In those cases, slope and
aspect values calculated from a higher-resolution DEM should be used with
caution – PEU will continue to grow as grid resolution increases, and the
rate of that increase will accelerate (see Fig.

In this analysis of the airborne lidar dataset from SCI, we find that a
4 m DEM resolution minimizes the combined influence of TE and PEU.
Slope and aspect analyses performed at this grid resolution have the smallest
error bounds and thus provide the most reliable data for analysis of
topographic metrics over the entire island. However, there exist other
methods of choosing an appropriate DEM resolution for a given analysis, for
example, that of

While a single, whole-island grid resolution is useful in some applications,
it is clear from Fig.

Optimal grid resolution for minimizing error bounds on slope and
aspect calculations across SCI compared to catchment median slope. Point
sizes are scaled to catchment size, from 0.1 to 34 km

There exist large spatial variations in the optimal grid resolution across
SCI, driven by differences in terrain slope and DEM quality. When these
differences are compared to median catchment slope and median catchment
standard deviation, a clear pattern emerges
(Fig.

It is worth noting that the vast majority of the catchments with optimal slope and aspect resolutions above 10 m are small catchments concentrated on the northwestern edge of SCI (see Fig. S29). These catchments have two unique topographic features which contribute to the high optimal grid sizes. (1) They have two distinct slope regimes: one steep hill of nearly constant slope, followed by one flatter residual marine terrace with a different constant slope. (2) The catchments drain into steep, deeply incised cliffs. Interestingly, neither removing the steep cliffs (which have high elevation uncertainty) nor aggregating several small catchments into one larger analysis unit (to improve the statistical reliability of the median QR) results in higher optimal grid resolutions. We posit that the low optimal grid resolutions are a product of the unique two-stage topography in this region. However, an in-depth analysis of the specific factors which drive the optimal terrain resolution in each individual zone is outside of the scope of this paper.

As a final application, we clipped out only the river networks on SCI to test whether they had higher or lower optimal grid resolutions than the island as a whole or their individual catchments. We found that river networks, particularly those parts of the network which are in steep terrain, have higher elevation uncertainties, which leads to predicted higher optimal grid resolutions than the surrounding terrain. In applying our method of deriving optimal grid resolution, it is important to take into account the possibility of large differences in the optimal grid resolution over relatively small spatial scales and to account for the differences in elevation uncertainty in different terrain types.

As has been shown in Figs.

There are clear differences in the slope and aspect distributions with
increasingly poor QRs. This effect can also be seen in quantile–quantile
plots of the entire SCI slope distribution when compared to the optimal grid
resolution (see Fig.

As a final test of grid-resolution dependence, we resampled several grids to
nominally match the optimal grid resolution – in this case, we chose 3 m
for the Pozo catchment (see Figs.

In general, slope distributions from resampled data should be used with caution; simply resampling elevation data to the optimal grid resolution before calculating slope does not guarantee better results – the DEM has to be created from the ungridded point data to fully take advantage of optimal gridding for slope calculations.

This study presents a detailed account of uncertainties and errors in the calculation of terrain slope and aspect derived from both truncation errors and uncertainty in the underlying source DEM. We first develop our analysis on synthetic data, which act as a control dataset. We then extend our methodology onto a high point-density lidar dataset, which allows us to compare the relative impacts of gridding truncation errors and propagated elevation uncertainty on the calculation of topographic metrics.

From the analysis of both synthetic and real-world data, we identify the
following key points: (1) the relative impact of truncation error and
propagated elevation uncertainty can be captured in a single metric, which we
call the quality ratio. This metric can be used to compare the accuracy of
topographic metrics across DEM spatial resolutions and uncertainty
distributions. (2) There exists an optimal grid resolution at which to
calculate terrain slope and aspect for a given dataset that
minimizes the total impact of both truncation error and propagated elevation
uncertainty; the distribution of DEM uncertainties leads to spatial variation
in the optimal grid resolutions at which to calculate slope and aspect. For
Santa Cruz Island in southern California, we find an optimal grid resolution
of 4 m, with island-average slope (aspect) errors of 0.25

Given that grid-resolution-driven effects on regional slope and aspect distributions could have significant impacts on the interpretation of landscape morphology, we recommend that region-specific optimal DEM resolutions be determined before the calculation of topographic metrics.

All codes and data for reproducing the results can be
found in

The supplement related to this article is available online at:

TS led the development and writing of the MS, as well as the primary data analysis. AR contributed to the development of the methods and developed most of the Python codes. BB contributed to the development of the methods and processed the lidar data.

The authors declare that they have no conflict of interest.

The authors thank Fiona Clubb and Ben Purinton for comments on an earlier version of the manuscript.

This research has been supported by the State of Brandenburg (Germany) through the Ministry of Science and Education, the NEXUS project, the Deutsche Forschungsgemeinschaft (German Research Foundation) and Open Access Publication Fund of Potsdam University.

This paper was edited by Giulia Sofia and reviewed by Marco Cavalli and two anonymous referees.