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The GEDTF has been used to estimate the accuracy of SRTM [1]. To do that the following error model of a DEM was used: .......................................................(1) where the indices M, T, I, O denote: the total variance of the DEM error (M), variance of the terraininduced error (T), variance of the instrumentinduced error (I) and variance of the other sources of errors (O), respectively. The variance of the terraininduced error is now derived.
Figure 1. Formation of a digital terrain model. A digital representation of the Earth's surface or digital terrain model (DTM) is obtained through a process of rounding off elevations of points found within a predefined area (pixel). With reference to Figure 1, let's consider this process on a onedimensional example. A fragment of the Earth's surface, small enough to be considered as a strech of stright line from point A to B, will be represented by a horizontal line from A'B'  representing a pixel. The elevation for this pixel hAB can be set using various methods. As it can be demonstrated, the most accurate startegy is to calculate the elevation as a mean of the elevations of both ends of the pixel, e.g. hAB = avg(hA,hB), where avg is an averaging operator and hA,hB  is the elevation of surface at point A and B, respectively. It is clear that if the elevation of the pixel was set at hAB, then the elevation of almost all surface points within the pixel will be misrepresented. The grey area in Figure 1 represents the disparities   between the elevation of the pixel and the true elevation surface points within the pixel. For the first pixel on the left in Figure 1, the following is true: .....................................................(2) It is obvious from Figure 1 that the probability of all discrepancies ?(l) within a pixel is the same, because each value of appears only once. Hence, can be modeled as a uniformly distributed random variable. Therefore, the probability density function of is: ..............................(3) where ?h is the maximal elevation difference within a pixel. In our case, for the left pixel the following is valid: . The variance of a random variable is known as the second degree central moment. This is expressed in terms of our situation as follows [11]: .........................................................(4) Equation (3) yields the final expression for the variance of the disparities between the terrain and pixel elevation: ................................................................(5) Note that the pixel size d and local slope of terrain ? can be related to ?h. This can be seen in Figure 1. The following is valid: ............................................................(6) Therefore, the variance of error caused by simplifying the Earth's surface by DTM can be estimated from the following:
...........................................................(7)
Equation (7) is the sought expression for the variance of the terrain induced error  . As it can be learned from Equation (7) the variance of the terrain induced error for flat surfaces (? = 0) is zero. The variance of the remaining errors may be assessed experimentally by comparing known elevation of flat features with elevation obtained from the assessed instrument or method. The GEDTF can provide required data on such features. As it was already mentioned, this method was used for the accuracy assessment of SRTM [1]. For this purpose some 300 runways were used. It was found that the standard deviation of the instrumentinduced and other error sources of SRTM is ±1.55m. 

Copyright © 2010, K. Becek. All Right Reserved. 
