Investigation of Sampling and Interpolation Techniques for DEMs Derived from Different Data Sources

Investigation of Sampling and Interpolation Techniques for DEMs Derived from Different Data Sources FARRAG ALI FARRAG 1 and RAGAB KHALIL 2 1: Assistant professor at Civil Engineering Department, Faculty of Engineering, Assiut University, Egypt. 2:Lecturer at Civil Engineering Department, Faculty of Engineering, Assiut University, Egypt. ملخص:.(GIS). (Interpolation methods) : 1) Inverse distance to a power, 2) Kriging, 3) Radial basis function and 4) Triangulation with linear interpolation. (GPS).(Total Station) (Control points). Interpolation ABSTRACT The Digital Elevation Model (DEM) is an important part of mapping technology. It is used for several purposes including contours derivation, geometric correction of photogrammetric and remote sensing images and Geographic Information System (GIS) applications. There are different procedures and techniques for collecting the data to generate DEMs. These techniques include digitizing contour maps, direct field observations using ground surveying methods, photogrammetric and remote sensing procedures and recently by Global Positioning System (GPS) and laser profiling and laser scanning. Interpolation is often required to create DEM from sparse number of points. In this paper the interpolation accuracy of four methods namely: 1) Inverse distance to a power, 2) Kriging, 3) Radial basis function and 4) Triangulation with linear interpolation are investigated. The investigation was practically performed using GPS and Total Station observations of the same test area for comparative purposes. Eight configurations of control points, which are different in number and distribution are analysed. KEY WORDS: Interpolation methods; DEM; Sampling; Contour map; Accuracy; GPS. 1

INTRODUCTION In a DEM, earth s surface is represented as spatially referenced regular grid points where each grid point represents a ground elevation value [2]. DEMs have a wide range of applications in surveying and mapping stated by [9]. A DEM is one of the most widely used data sets for analyzing the terrain and constructing a geographical information system [6]. The DEM also aids automatic recognition of terrain features in town planning and automatic building extraction and offers automated assessment of land resources and attributes [1]. DEMs have a major role to play in hydrological modeling, analysis of visibility and hazard mapping [2]. DEM is very important to the validity of soil erosion model [14]. The DEM technique is becoming a powerful tool for representation of both existing and proposed ground surface in the fields of civil surveying, geology, and mining engineering [4]. DEMs have been used to delineate drainage networks and watershed boundaries, to calculate slope characteristics, to enhance distributed hydrologic models and to produce flow paths of surface runoff [12]. There are different techniques for collecting the data to generate DEMs. DEMs can be created by digitizing contour maps, by direct field observations using ground surveying methods, by photogrammetric and remote sensing procedures and recently by GPS and laser profiling and laser scanning. In this paper the GPS and Total Station observation techniques were used to obtain data for DEM generation. The sampling pattern is an important factor for generating DEM. It may include regular, quasi-regular and irregular modes as stated by [1]. In regular (grid) mode the spot heights are measured in a regular geometric pattern. Quasi-regular mode, in which the data are observed along parallel lines spaced at regular intervals but with data randomly spaced along each line. Irregular mode is the one in which the data are collected at random without regard to their geometric distribution. The grid DEM is the easiest of all to manipulate using machines, even though it is time consuming specially if ground surveying techniques are used for collect the data and it can misrepresent the real surface in areas of highly variable terrain [5]. Configurations for minimizing the field observations are proposed and tested in this paper. Interpolation is one of the most important parts of DEM building [14]. It can be defined as procedure of determining the height of any intermediate point of known planimetric coordinates. Interpolation produces a regularly spaced array of Z values from irregularly spaced XYZ data for contour or surface plots. Estimates obtained from interpolation should reflect the real world physical features by incorporating spatial trends which are present in the point data [13]. In this paper the interpolation accuracy of four methods namely: 1) Inverse distance to a power, 2) Kriging, 3) Radial basis function and 4) Triangulation with linear interpolation (all of which are applicable to grid as well as scattered data), are investigated and practically tested using data obtained by GPS and Total Station observations. These methods were found to be the optimum interpolation methods as resulted by [5]. The mathematical background of the used interpolation techniques can be found in [3, 5, 7, 8, 9 and 11]. 2

TESTING PROCEDURE AND DATA PROCESSING An area of 40000 m 2 (200 x 200 m) near the new Assiut city (Egypt) formed the study area. Its average elevation is 146 m above mean sea level. 441 three-dimensional coordinates were collected once using Total Station (TS) and once again using GPS. The fieldwork was carried out using Topcon (GTS 712) total station and the GPS technique was performed using two Trimble GPS receivers operating in differential mode. The base receiver was set on a permanent point of known coordinates at Assiut University campus (about 10 km away from the test field), while the other receiver was moved in a marked grid of 10 m intervals which covers the test area. The orthometric height was obtained from that of ellipsoidal height at the observation stations. A grid DEM was then obtained by using the tested interpolation techniques with different configurations of control points (distribution forms). The planimetric coordinates of the points to be interpolated coincide with the points where total station and GPS observations were made. The tested area represents a moderate terrain roughness with standard deviation of 2.76m (TS) and 3.28m (GPS) in heights at 10 m grids. To test the accuracy of each interpolation technique, the number and distribution of control points were selected in two distribution forms that can be described as follows: i- The first form based on using the minimum possible number of control points with a simple common distribution. It contains five different cases similar to the cases proposed by [1] as illustrated in Figure (1). The first case consists of five control points, four of them at the four corners of the test area and one point at the center of the area. In case 2 six control points were used; these were the four corner control points plus two more control points along the center line of the model at equal distances from the point at the center. In case 3 eight control points were used; these were the four corner control points plus four control points at the midsides of the area. A central control point was added to the control points of case 3 to form case 4. In the fifth case, ten control points were used; these were the eight control points of case 3 plus two more control points along the center line of the model at equal distances from the point at the center. In each case, the rest of the known points within the test area were used as check points. ii-the second form based on using the regular spaced control points with different spacing. It contains three different cases of 20, 40 and 50 meters between control points in both x and y directions. THE RESULTS AND ANALYSIS The standard deviation, σ, of the interpolated heights of the test points for each case in both distribution forms of control points was computed as: where σ = [ v ] 2 n 3

v = residual of the interpolated heights compared with the measured heights, n = number of the interpolated points in each case for each technique. Case 1 Case 2 Case 3 Case 4 Case 5 Control points Figure (1): Number and distribution of control points in the first form For all cases, 98% of the tested points satisfied the accuracy criteria of 3σ. The results of GPS observations are summarized for form 1 and form 2 in table 1 and table 2, and are graphically represented in figure 2 and figure 3, respectively. Table 1: The standard deviation (m) of the interpolated heights of form 1(GPS data) Case # Control points Inverse distance to power Kriging Radial basis function Triangulation with linear interpolation. 1 5 2.21 1.84 1.77 1.83 2 6 1.47 1.39 1.44 1.74 3 8 1.64 1.51 1.50 1.57 4 9 1.81 1.51 1.50 1.52 5 10 1.51 1.39 1.41 1.40 Table 2: The standard deviation (m) of the interpolated heights of form 2 (GPS data) Case# Point spacing Inverse distance to power Kriging Radial basis function Triangulation with linear interpolation. 1 20 0.77 0.62 0.62 0.63 2 40 0.86 0.91 0.93 0.92 3 50 1.11 1.03 1.05 1.06 4

Table 1 indicates that the Kriging and the Radial basis function interpolation methods are the most accurate methods. The second most accurate is Triangulation with linear interpolation method. The accuracy obtained by Inverse distance to power interpolation method is the lowest in all cases of control points. Referring to Figure 2 and comparing case 1 with case 2 and 4 we can notice that the improvement in the accuracy due to one extra control point inside the area is more than that due to four extra control points at the borders. When comparing case 2 with case 5 we can notice that the same accuracy was obtained although case 5 has four extra control points at the borders. We can conclude that the control points inside the study area are more effective than the control points at the borders. Table 2 indicates that the Kriging, Radial basis function and Triangulation with linear interpolation gave the same results. This is also clear from Figure 3. The Inverse distance to power interpolation gave approximately the same results except at spacing 20 m it gave the lowest accuracy. Inverse distance Kriging Radial Basis Triangulation Standard Deviation (m) 2.20 2.10 2.00 1.90 1.80 1.70 1.60 1.50 1.40 1.30 1 2 3 4 5 Case Figure 2: Accuracy of interpolated heights of form 1 (GPS data) 5

Standard Deviation (m) 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 Inverse distance Kriging Radial Basis Triangulation 20 30 40 50 Control Points spacing (m) Figure 3: Accuracy of interpolated heights of form 2 (GPS data) The results of all techniques showed improvement in accuracy with the decrease of the spacing between control points. The error range of the heights interpolated is approximately the same for all interpolation techniques and it decreased with increasing the number of control points. This is shown in figure 4 for the form (1) and figure 5 for the form (2) of control points. Inverse distance Kriging Radial basis Triangulation 14.00 Error Range (m) 12.00 10.00 8.00 6.00 4.00 1 2 3 4 5 Case Figure 4: Error range of form 1 (GPS data) 6

Inverse distance Kriging Radial basis Triangulation 14.00 Error Range (m) 12.00 10.00 8.00 6.00 4.00 20 30 40 50 Control Points spacing (m) Figure 5: Error range of form 2 (GPS data) The same investigations were performed on the data collected using total station. It noticed that similar results were obtained for different cases in both forms of control points. The results of form 1 are summarized in table 3, and are graphically represented in figure 6. Figure 6 shown that increasing the number of control points lead to increase the interpolation accuracy in spite of the location of these control points. The results of TS data are more accurate than the GPS data but the same ordinal and trend of the interpolation methods were obtained as shown in figure 7. Inverse distance to power showed to be the lowest accurate interpolation among the tested techniques. Table 3: The standard deviation (m) of the interpolated heights of form 1(minimum control points) (TS data) Case # Control points Inverse distance to power Kriging Radial basis function Triangulation with linear interpolation. 1 5 1.88 1.54 1.50 1.52 2 6 1.54 1.37 1.35 1.41 3 8 1.22 1.24 1.26 1.31 4 9 1.29 1.24 1.26 1.16 5 10 1.25 1.25 1.26 1.22 7

Inverse distance Kriging Radial Basis Triangulation Standard Deviation (m) 1.90 1.80 1.70 1.60 1.50 1.40 1.30 1.20 1.10 1 2 3 4 5 Case Figure 6: Accuracy of interpolated heights of form 1 (TS data) Table 4: The standard deviation (m) of the interpolated heights of form 2 (TS data) Case# Point spacing Inverse distance to power Kriging Radial basis function Triangulation with linear interpolation. 1 20 0.64 0.41 0.40 0.44 2 40 0.73 0.68 0.69 0.71 3 50 0.99 0.84 0.81 0.91 Standard Deviation (m) 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 Inverse distance Kriging Radial basis Triangulation 20 30 40 50 Control Points spacing (m) Figure 7: Accuracy of interpolated heights of form 2 (TS data) 8

The volume of cut and fill from the mean elevation for both GPS and TS data were computed using both control points forms. The results of form 1 are shown in figure 8 for GPS data and in figure 9 for TS data. Comparing case 1 with case 4 and case 2 with case 5, show clearly the effect of the location of control points. Inverse distance Kriging Radial basis Triangulation Error in Volume (m3) 35000 30000 25000 20000 15000 10000 5000 0 1 2 3 4 5 Case Figure 8: Error in volume using form 1 (GPS data) Inverse distance Kriging Radial basis Triangulation 25000 Error in Volume (m3) 20000 15000 10000 5000 0 1 2 3 4 5 Case Figure 9: Error in volume using form 1 (TS data) 9

The visual investigation of the effect of number and distribution of control points was carried out by generating perspective views of the original and interpolated DEM. The visual investigation was carried on both GPS and total station data where similar results were obtained. In order to avoid repetition the graphical representation of GPS data is only given as shown in figure 10, figure 11 and figure 12. Figure 10 represents the best case of form 1 (case 5). Figure 11 represents the worst case of form 2 (50 m spacing). Figure 12 represents the best case of form 2 (20 m spacing). In all figures the three-dimensional view of the original DEM obtained by GPS is indicated by (a), while the three-dimensional views of the resulting DEM from Inverse distance, Kriging, Radial basis function and triangulation with linear interpolation methods are indicated by (b), (c), (d) and (e) respectively. By comparing Figures 5b, 5c, 5d and 5e with the original DEM of Figure 5a it becomes clear that no one of them accurately represents the detailed undulation of the original terrain. This indicates that the proposed number and distribution of control points in form 1 is not sufficient to represent the original surface even with the most accurate interpolation technique. Although Figure 6, the worst case of form 2 (regular distribution and spacing of control points), doesn t represent the details of the original terrain, one can notice that, these results are better than those of Figure5. Figure 10: Perspective view of the original GPS-derived DEM (10a) and four derived DEMs of form 1 case 5: Inverse distance (10b), Kriging (10c), Radial basis (10d), Triangulation (10e) 10

Figure 11: Perspective view of the original GPS-derived DEM (11a) and four derived DEMs of form 2; 50 m spacing: Inverse distance (11b), Kriging (11c), Radial basis (11d), Triangulation (11e) Figure 12: Perspective view of the original GPS-derived DEM (12a) and four derived DEMs of form 2; 20 m spacing: Inverse distance (12b), Kriging (12c), Radial basis (12d), Triangulation (12e) 11

Figure 12 shows clearly that the short spacing between control points leads to good results. The Kriging interpolation technique (Figure 12c) and Radial Basis function (Figure 12d) produced surfaces that are most similar to the original surface (Figure 12a). The surface produced by the Triangulation with linear interpolation technique (Figure 12e) is almost similar to Figures 12c and 12d, and it is the next most accurate followed by the Inverse distance to power technique. CONCLUSIONS From the results and analysis the following remarks could be concluded: 1- The closer the control data to the interpolated points, the better the accuracy of interpolation. These expected results are clear from the tabulated results. 2- Regular spacing sampling provides better results compared with using few control points on the boundaries and a few points inside the tested area (form1), where such configuration is not sufficient to represent details of the surface undulation. 3- The control points that lay inside the tested area are more effective than that at the borders. 4- Kriging interpolation gave the best accuracy for surface representation followed by Radial basis function and triangulation with linear interpolation. 5- For accurate digital representation of ground surfaces of similar terrain roughness, it is recommended to use spacing between control points of 20 m. 6- The used GPS observation technique gave results close to that of the total station observations; but it is recommended to perform further studies on the accuracy of the different GPS observation techniques for DEM generated. REFERENCES 1. Algarni D. and Elhassan, I., 2001 Comparison of thin plate spline, polynomial, C`function and shepard's interpolation techniques with GPS-drived DEM, International journal of Applied Earth observation and Geoinformation, Volume 3, Issue 2, Pages 155-161. 2. Ardiansyah P. and Yokoyama R., 2002 DEM generation method from contour lines based on the steepest slope segment chain and a monotone interpolation function, ISPRS Journal of Photogrammetry and Remote Sensing, Volume 57, Issues 1-2, November, Pages 86-101. 3. Driscoll T. and Fornberg B., 2002 "Interpolation in the Limit of Increasingly Flat Radial Basis Functions", Computers and Mathematics with Applications, Volume 43, Pages 413-422. 4. Du, CH., 1996 An Interpolation Method for Grid-Based Terrain Modelling, the computer journal, Vol. 39, No. 10, pages 837-843. 12

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