# Digital Elevation Model Quality Assessment Methods: A Critical Review

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## Abstract

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## 1. Introduction

- New platforms extended the capabilities of the aircraft: satellites covering larger areas (first TIROS observation satellite in 1960, across-track stereoscopy from different orbits with SPOT-1 in 1986, along-track stereoscopy with SPOT-5 in 2002), UAVs (fixed-wing for large area surveys, multirotor for small area surveys requiring agility and precision), and mobile mapping systems on land vehicles.
- New processing methods were developed, such as aerotriangulation and automatic image matching that made it possible to process large quantities of images for the industrial production of orthophoto mosaics and altimetric databases with few ground control points [5], and the synergy between digital photogrammetry and computer vision led to very powerful 3D reconstruction methods such as SfM (structure from motion).

- DEM is multi-user information, so that multiple quality criteria must be met, and although a particular user may specify application-driven requirements, the case of multi-user databases is obviously more complex.
- The Earth’s surface is a material object that speaks to our senses, so that one can complain about unrealistic relief modelling even in an unknown region; this leads to strong requirements including aesthetic ones.

## 2. Prerequisite: Definition of the Nominal Terrain

## 3. DEM as a Cartographic Product

#### 3.1. A Very Peculiar Surface

^{24}kg) and the intensity of the associated gravitational field (g ~9.8 m·s

^{−2}), where the attraction of all material particles towards the mass center gives the Earth’s spherical shape. As a result, the Earth’s surface has almost no hidden part when seen from above. It is comparable to a bas-relief which has no hidden part when seen from the front. This property is not cancelled out by the centrifugal force produced by the Earth’s rotation, which transforms the sphere into an ellipsoid.

#### 3.2. Planimetric Grid Sructure

#### 3.2.1. Grid Structure

- The mesh size (and therefore the grid density) has an impact on the DEM absolute accuracy, but also on the ability of the DEM to describe landforms as shown below in Section 4.2. This ability is intuitive, since a smaller mesh is expected to allow a better rendering of small topographic objects, but a signal processing approach may also provide criteria to optimize resampling and to detect DEM errors [50]. The link between mesh size and resolution, and the way DEM quality assessment should consider these concepts, are addressed in Section 6.
- The mesh shape has an influence on the adequacy of the surface modelling to local landforms. Between the two most common grid structures, namely constant size squares and variable size triangles, the latter is clearly better suited to describe a surface morphology in which drainages, ridges and other slope breaks have variable orientations. However, there are several solutions to build a network of triangles from a given point cloud. For example, the Delaunay triangulation is advantageous to avoid very elongated triangles (which are less suitable for spatial analysis), but it does not necessarily give the most accurate DEM nor the most respectful one of terrain shapes: it must therefore be constrained so that the edges of the triangles are placed along slope discontinuities such as drainages or other slope break lines. This is an advantage of triangle networks, which are able to adapt to the topography.

#### 3.2.2. Interpolation Algorithm

#### 3.3. Numerical Representation of Elevation Values

## 4. Main DEM Quality Assessment Approaches

#### 4.1. External Quality Assessment

#### 4.1.1. Comparison with Ground Control Data

- It must be much more accurate than the evaluated DEM.
- It must be dense enough and well distributed over the area to allow meaningful statistical analysis, and even a spatial analysis of the error.

#### 4.1.2. Simulation-Based DEM Production Method Validation

- The topographic restitution method can be tested over a variety of landscapes, leading to more comprehensive conclusions than over a particular DEM.
- It can also be tested with a variety of sensor parameters and orbit configurations, leading to recommendations for optimum image acquisition conditions if the images are to be processed for DEM production.

- The input DEM can be reprocessed, either to exaggerate the elevation amplitude, or to introduce topographic details such as microrelief or buildings (note that all these changes in the input DEM can be parametrically controlled, allowing analytical sensitivity studies). Fractal resampling can be implemented to produce a more realistic input DEM [102], and geomorphology provides criteria to verify this realism requirement in accordance with the principles of internal validation which we will see further [103].
- Sensor parameter uncertainties can be introduced to consider the fact that the DEM production method always uses an approximation of the exact parameters.

#### 4.2. Internal Quality Assessment

#### 4.2.1. Visual Control

#### 4.2.2. Quantitative Internal Quality Assessment

^{2}= 0.9836 is observed over 12 orders (Figure 10a), which means that the relief described by the DEM is nearly fractal. However, it appears that excluding the values obtained for orders 1 and 2, which are slightly higher than the linear trend, and plotting the results for orders 3 to 12 only, increases the linearity with R

^{2}= 0.9965 (Figure 10b). This suggests that the Topodata resampling process slightly increases the number of short streams with regards to a fractal hypothesis, which could be confirmed in the evaluation of stream extraction among several DEMs (Topodata and SRTM included) by [121].

^{2}coefficient in the control of the compliance with Horton’s law. This is why the internal quality assessment method, although not using any reference data, can really be considered as a quantitative approach, with objective criteria to decide if a DEM is acceptable or not.

## 5. DEM Validation at Different Levels

#### 5.1. From Point Cloud to Grid Surface Model

- In terms of absolute vertical accuracy, the input data are essential: the errors of the input point cloud are generally autocorrelated (due to orbit, relief…) so that they remain in the resampled DEM whatever the selected interpolation method.
- In terms of shape realism, the interpolation plays a major role since it can remove or create artifacts (noise, stripe, pixelation, etc.). The resampling step may filter the noise of the input point cloud and therefore improve the quality in terms of elevation accuracy. In contrast, the interpolation implemented for resampling may produce an exaggerated smoothing effect or a raster grid effect, resulting in a quality degradation in terms of shape rendering. Indeed, the choice of an interpolation algorithm affects not only the elevation absolute accuracy but also the geomorphometric indices [150], and this effect depends on the local terrain morphology [151,152,153,154]. The effect of an interpolator on the DEM quality depends on both the DEM production technique and the application-oriented user requirements [43,155,156]. Hengl and Evans [70] distinguish three aspects for classifying interpolation methods, namely, the smoothing effect (exact or approximate interpolation), the proximity effect (local or global interpolation) and the stochastic hypothesis. For example, an exact interpolation method, such as linear interpolation, is recommended if the data are very dense and accurate, while a smoothing method should be used if the data are noisy. Mitas and Mitasova [157] state that the description of the smoothing and tension effects and the consideration of ridges and streams are the most important evaluation criteria for an interpolation method.

#### 5.2. From Grid Surface Model to Derived Topographic Features

- The accuracy of the thickness, which can be estimated through the RMSE of the DEM difference, is:$${\mathrm{RMSE}}_{thickness}=\sqrt{{\mathrm{RMSE}}_{\mathrm{DEM}1}^{2}+{\mathrm{RMSE}}_{\mathrm{DEM}2}^{2}}$$$${\mathrm{RMSE}}_{thickness}=\sqrt{2}{\mathrm{RMSE}}_{DEM}$$
- whatever the accuracy of the two DEMs, any horizontal registration error Δx between them will lead to a vertical error Δz on the thickness, which is a function of the slope θ:$$\u2206\mathrm{z}=\u2206\mathrm{x}\mathrm{tan}\mathsf{\theta}$$

#### 5.3. Challenge of Global DEM Quality Assessment

- Allow a user to predict the quality of the DEM on a given site for a given application.
- Guide product improvements by post-processing or by merging data from different sources.
- Consider the variety of geographic conditions as we have just seen, to guarantee that the conclusions on the global DEM quality do not depend on particular conditions, or to limit the scope of the conclusions to a particular landscape.
- Consider different quality criteria as suggested in Section 4 (including the artifacts produced by resampling, which have been shown to also have a significant impact on the geomorphometric quality of the final DEM), so that a variety of user needs is considered. Indeed, a global DEM is typically a multi-user database, for which it is difficult to define quality standards that are suitable for all potential users.

## 6. Resolution Dependency in DEM Validation

#### 6.1. From Scale to Resolution

- The point cloud generally has a variable density, with low density areas due to physical reasons (e.g., airborne lidar point clouds have a lower density under forests than on bare ground); moreover, a cloud (for photogrammetry) or a forest (for repeat-pass SAR interferometry) can prevent the calculation of elevations and create holes in the DEM so that the resolution is locally lost.
- Whatever the input point cloud, the interpolator imposes its form to some extent, mainly in low density areas.

_{k}, a phase $\mathsf{\phi}$

_{k}, and a frequency f

_{k}:

_{max}, which requires a DEM density of at least two points per cycle in each dimension, and thus a sampling frequency of at least twice the highest frequency contained in the signal (the so-called Nyquist condition).

#### 6.2. Stability of Topographic Indices with Regards to Resolution

#### 6.3. Relevant Resolution for Landform Modelling

## 7. Discussion

_{z}and therefore to predict DEM external quality. However, this approach remains limited for several reasons:

- σ
_{z}is a very optimistic indicator and it may give the illusion that the accuracy of a global DEM is the same all over the world. σ_{z}is a very limited indicator as we have shown, excellent for predicting the vertical accuracy of the DEM but unsuitable for guaranteeing a realistic shape rendering. - σ
_{z}also depends on slope and landcover, and this double influence cannot be predicted quantitatively because the slope is not necessarily known before the DEM is calculated and the landcover can be described qualitatively at best. - The interpolation, which is applied to resample the DEM in the required grid structure, also has an impact on the quality by creating artifacts.

- Mathematical modelling of the error expected for different DEM production techniques as a function of landscape characteristics and acquisition/processing parameters.
- Definition of invariant properties of the Earth’s surface to support internal quality assessment methods, based on the advances of geoscience and even comparative planetology.
- Analysis of scale effects, which has become a challenging research issue with the development of very high-resolution products, leading to increased needs for gigantic data volumes management and for the characterization of extremely complex phenomena.

## 8. Conclusions

- The quality of a DEM makes sense for a given application, i.e., depends on user needs.
- The nominal terrain, i.e., the physical surface, which is supposed to be modelled, must be explicitly defined for the quality to make sense.
- The quality of a DEM can be assessed using different methods, with or without ground control data, according to quantitative criteria.
- Artifact detection can be carried out with no ground control data, by revealing the non-compliance of the DEM with physical or statistical characteristics of the Earth’s surface.
- Visual analysis can complement quality control provided that the DEM is visualized appropriately.
- Testing mapping methods on simulated images allows a more in-depth validation of these methods.
- The quality must be considered at a given scale. However, the concept of scale is an abuse of language that comes from the paper map and must be replaced by equivalent concepts adapted to the digital world.
- The quality of a global DEM cannot be easily deduced from an analysis of local data nor from a theoretical approach only: it also requires a great deal of experience based on the analysis of many DEMs on a variety of landscapes, or on simulation-based studies.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Cartographic representation of terrestrial relief in maps in Southern France: Cassini map, 18th century (

**a**); IGN map, 1957 (

**b**).

**Figure 7.**Unrealistic background landscapes in Renaissance paintings (example of Mona Lisa from Leonardo da Vinci).

**Figure 8.**Comparison of hillshade (

**top**) and hypsometry (

**bottom**) for DEM artifact detection (from left to right: original DEM, DEM with Gaussian noise, 8-bit coded DEM, DEM subsampled 4 times). Source: DEM produced by contour-line interpolation in a 1:20,000 topographic map of Lebanon.

**Figure 9.**Detail of the hydrographic network extracted from SRTM (red) and Topodata (green). Source: [108].

**Figure 10.**Fractal behaviour of the hydrographic network as illustrated by the variation of the logarithm of the number of streams for each Strahler order as a function of this order, for orders 1 to 12 (

**a**) and 3 to 12 (

**b**). Source: [108].

**Figure 11.**Control of the compliance of a DEM with Benford’s law for two metrics: elevation and slope (source: [125]).

**Figure 12.**Aspect directional histograms in Topodata and SRTM and identification of main directions (source: [108]).

**Figure 13.**Main types of errors in the drainage network extracted from a DEM (blue line: real network; red line: network extracted from the DEM).

**Figure 14.**Visual comparison between hillshaded SRTM 3″ (

**left**) and SRTM 1″ (

**right**) DEMs displayed on NASA website to promote the new 1″ product.

**Figure 15.**Effect of DEM subsampling on elevation and slope histograms. The distributions of elevation (

**a**) and slope (

**b**) are computed for an input DEM (obtained from a SPOT-4 stereo pair over Lebanon with 10 m ground sampling distance) and for subsampled DEMs with the increasing ratios of 2, 4, 8. A higher subsampling ratio leads to larger and less numerous cells (source: [90]).

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**MDPI and ACS Style**

Polidori, L.; El Hage, M.
Digital Elevation Model Quality Assessment Methods: A Critical Review. *Remote Sens.* **2020**, *12*, 3522.
https://doi.org/10.3390/rs12213522

**AMA Style**

Polidori L, El Hage M.
Digital Elevation Model Quality Assessment Methods: A Critical Review. *Remote Sensing*. 2020; 12(21):3522.
https://doi.org/10.3390/rs12213522

**Chicago/Turabian Style**

Polidori, Laurent, and Mhamad El Hage.
2020. "Digital Elevation Model Quality Assessment Methods: A Critical Review" *Remote Sensing* 12, no. 21: 3522.
https://doi.org/10.3390/rs12213522