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Stream networks are considered important units in many environmental decision making processes. The extraction of streams using digital elevation models (DEMs) presents many advantages. However it is very sensitive to the uncertainty of the elevation datasets used. The main aim of this paper is to implement geostatistical simulations and assess the propagated uncertainty and map the error of location streams. First, point sampled elevations are used to fit a variogram model. Next two hundred DEM realizations are generated using conditional sequential Gaussian simulation; the stream network map is extracted for each of these realizations, and the collection of stream networks is analyzed to quantify the error propagation. At each grid cell, the probability of the occurrence of a stream and the propagated error are estimated. The more probable stream network are delineated and compared with the digital stream network derived from topographic map. The method is illustrated using a small dataset (8742 sampled elevations) for Anaguid Saharan platform. All computations are run in two free softwares: R and SAGA. R is used to fit variogram and to run sequential Gaussian simulation. SAGA is used to extract streams via RSAGA library.

A digital elevation model (DEM) is a representation of terrain elevation as a function of geographic location [

Methods of errors investigation in DEMs have been widely explored [7,13,14]. The simplest methods are based on criteria such as: differences in elevations between adjacent points [15,16]; elevation histogram analyses [17,18]; systematic detection by inspection of anomalous values within a given moving window [9,19,20]. More complex methods introduce remote sensing and/or geostatistical process. [21,22] have investigated the value of Brownian processes in a fractal terrain simulation model for improving DEM accuracy. [

Techniques of error location and visualization have also been considered. [

This article proposes a methodology to assess errors of stream networks extracted of digital elevation models. It uses a small case study to demonstrate how to implement geostatistical simulations and assess the propagated uncertainty and map the error of location streams. Our secondary objective is to promote the geostatistical tools implemented in the open source environnement for computing (R), and geographical analysis tools implemented in the open source GIS (SAGA). To do so, we adopted an R based robust code written by Hengel, [

Monte Carlo simulation methods have been used by many researchers to evaluate error in GIS data, including [33-35] and have been applied to specifically address DEM uncertainty. For example, [

The basis for using the Monte Carlo method in error propagation analysis is that the original data is perturbed repeatedly by the realisation of the modeled error, and the GIS analysis is calculated from the perturbed data set. Finally, statistical summaries are drawn from the stack of analysis results based on the perturbed data sets [35-37, 41]. In practice, the Monte Carlo method in error propagation of stream networks developed in this study can be summarized in four steps.

• Calculate an experimental variogram from the data and fit a variogram model to represent the variability of the input DEM. This step is achieved by using weighted least squares (WLS) algorithm as implemented in the geoR package.

• Generate multiple realizations of the DEM using conditional simulation. The most common technique in geostatistics used to generate equiprobable realizations of a spatial feature is the Sequential Gaussian Simulation [

• Derive stream network for each realization using the “Channel Network” function, which is available also via the command line “ta_channels” SAGA library, and save the temporary result.

• Estimate probability of the occurrence of stream network. To derive a probability of mapping stream, we need to import all gridded maps of stream, then count how many times the model estimated a stream over a certain grid node. The probability of occurrence of detecting stream is simply the average value of stream from m simulations.

The Monte Carlo approach requires a significantly large number of realizations to produce a reliable estimate of the distribution function. The number of realizations m must be sufficiently large to obtain stable results. Theoretically, the accuracy of the Monte Carlo method is proportional to the square root of the number of runs m [

In this case study the analyses were done with 200 simulation runs, which was a compromise to get reasonably reliable simulation results with moderate computation load. Realizations of the DEM error were done by using sequential Gaussian simulation [

In this article we use a combination of statistical and geographical computing software to assess propagated error of detecting streams: SAGA^{1} (System for Automated Geoscientific Analyses) for geographical computing and R^{2} for statistical computing. Many spatial packages^{3} have been developed in recent years, which allow R to be also used for spatial analysis. Three important packages that are used in this paper are gstat [

The case study was carried out on a DEM of Anaguid located in southern of Tunisia. This DEM has a 30 m grid cell resolution with 341 × 369 dimension and it was interpolated with the ordinary Kriging method (^{2}. Its elevation ranges from 300 to 440 m with an average and a standard deviation of 36.86. This area is specifically suitable as it presents two contrasting landscapes: plains with terraces in the west and dissected plateau with small valleys in the east direction. Geologically, the area under study is occupied essentially by hard carbonate rocks and conglomerates deposited from the upper Cretaceous to the Neogene.

The first results of our analysis are the variograms models fitted using geoR package (

The delineation of stream networks is important in many environmental and hydrological applications. DEM error propagation can provide valuable additional information for the reliability of stream extraction. The methodology and its application presented here demonstrate an easily employed method to assess DEM error and its impact on stream networks. Previous research used techniques that require higher accuracy data sources such as higher resolution DEMs (LIDAR for example). The reality is that most DEM users do not have such data. The methodology p resented here was designed to remedy this

issue by providing a methodology based on Monte Carlo approach that was successfully implemented using R and SAGA open source software. The purpose of this methodology is to provide DEM users with a suite of tools by which they can evaluate the effect of uncertainty in DEMs and derived topographic parameters. However, some limitations have been identified in this study:

• Firstly, we have limited the number of simulations to 200 runs. It should be feasible to evaluate the increase in accuracy with an increasing number of runs, e.g. by evaluating the change in derived probability or attribute property such as estimated stream length or catchments width. If such a parameter or function does not change anymore below a certain threshold, no more simulations seem to be required.

• Secondly, we have set the grid cell size at 30m without any real justification. It is relevant to evaluate the increase in accuracy with an increasing grid cell size by plotting the error of mapping streams versus the grid spacing index, one can select the grid cell size that shows the maximum information content in the final map. The optimal grid cell size is the one where further refinement does not change the accuracy of derived streams. Future research should aim to perform a more comprehensive evaluation with adaptive cell sizes.

• Thirdly, the computational burden of this method is also an issue. The most costly operations are geostatistical simulations and extraction of stream networks. Geostatistical simulations with standard computer, takes more than 10 minutes to generate 50 simulations for this small study area (341 × 369 pixels). This means that this method is at the moment limited to small data with few hundreds of points; it would be probably of limited use for large data.

• Fourthly, the demonstrated methodology did not assess uncertainty associated with specific DEM applications such as hydrologic modeling or hazard mapping. Researchers can, however, use the uncertainty estimates provided by the proposed simulation techniques to better assess uncertainty for projects that utilize DEMs and DEM-derived data. For example, input parameters for hydrologic models (such USLE) might require elevation and slope values that are frequently obtained directly from a DEM.

• Finally our conclusions are limited to this empirical study of Anaguid with a relatively gentle topography. An obvious future direction is to conduct similar evaluations in other sites with different topographic characteristics to verify the robustness of our results.

The authors would like to thank the reviewers for their excellent editorial suggestions during the review of this manuscript.