On the Selection of an Interpolation Method for Creating a Terrain Model (TM) from LIDAR Data

Transcription

1 On the Selection of an Interpolation Method for Creating a Terrain Model (TM) from LIDAR Data Tarig A. Ali Department of Technology and Geomatics East Tennessee State University P. O. Box 70552, Johnson City TN Ali@etsu.edu ABSTARCT The last ten years have witnessed great advancements in mapping technologies, which have benefited the geospatial research community. This includes the development of Light Detection And Ranging devices (LIDAR) for measuring topography changes, development of more precise Global Positioning Systems (GPS s) and deployment of high-resolution satellites such as IKONOS. Since a number of useful parameters can be derived from Terrain Models (TM s) besides their representation of the terrain surface; their significance is hence growing quickly. Additionally, TM s are also being used in coastal areas to generate Instantaneous Shorelines (IS s) by intersecting them with Water Surface Models (WSM s). Moreover, the production of TM s from LIDAR data is getting popular as LIDAR data are becoming more available and convenient for the interpolation method of choice. This paper explores the suitability of some interpolation methods for creating a TM from LIDAR data. The quality of TM s created by various interpolation methods in representing the terrain surface has been tested by studying TM s created from LIDAR data for a study area with different values of uncertainty. These values of uncertainty have been artificially introduced to the LIDAR data before creating the TM s. These TM s have then been compared and analyzed along representative profiles. 1

2 1. Introduction Digital terrain modeling is very important procedure as it provides a sound representation of topography in digital format, which can easily be incorporated into both digital mapping systems and GIS. Digital terrain representation can generally be used for three purposes; analysis, visualization, or both. Terrain model (TM) is a digital representation of the terrain surface, which can be either a discrete representation of the terrain surface normally known as grid with regular spacing (GRID), or a continuous representation such as a triangular surface of irregular nature known as triangular irregular network (TIN). After the LIDAR mapping technology emerged, a new representation of the terrain surface come out that is the irregular point configurations (IPC), which is a direct product of the LIDAR system. The GRID model has a typical raster data structure, the attribute that made this type of terrain representation very popular as the analysis framework of raster data type is well developed (DeMers, 2002; Jain, 1989). For example map algebra provides a rigorous framework for analyzing terrain representations stored in GRID format both theoretically and practically (Tomlin, 1990). Therefore GRID terrain representations best suit analysis and to some extent visualization purposes. The TIN model, on the other hand better suits visualization purposes because of the continuous nature the triangular facets of the model add to the digital representation. Furthermore, not much information can be derived from TIN models since a comprehensive analysis framework for triangulated models does not yet exist. In a TIN model, the sample points are simply connected by lines to form triangles, which are represented by planes, which give continuous representation of the terrain surface. Creating a TIN, despite its simplicity requires decisions about how to pick the sample points from the original data set, and further how to triangulate it. Since this paper is concerned with creating TIN models from LIADR data, specifically the IPC product, the very important points (VIP) algorithm was found to be a good choice for picking the candidate points for the TIN model. When it comes to triangulating the sample points, the decision is not hard as most of the available triangulation methods are sufficient for producing a TIN that provides excellent means of visualization of topography. Among the existing triangulation methods that are in use, the Delaunay Triangulation (DT) is very common and popular for its rigorous structure although it produces triangles that are not hierarchical (Fowler and Little, 1979; Peucker et al., 1978). 2

3 The new terrain representation method, which is the IPC, better suits visualization purposes and at the same time provides dense-points input for creating GRID or TIN model for the terrain. At this point, issues related to the selection of an interpolation method to create a TM should be addressed. Remember that, our goal is to produce a TM that best represents the terrain surface. Therefore, we need to make sure that all of the important morphological information on the terrain are captured in the TM especially if it will be used for analysis purposes. Interpolation is generally defined as the process of estimating the values of a specific attribute at unsampled locations based on the values of the attribute at the sampled locations. When that specific attribute is elevation, the process is known as spatial interpolation, the process used to create a TM from points with known elevation. Spatial interpolation methods can be classified in different ways and are generally defined based on geometric or geostatistical properties. Spatial interpolation methods can generally be classified as local or global, exact or approximate, and deterministic or stochastic methods. Local interpolation methods work on a small portion of the data points while global methods work across the whole data set. The exact interpolation methods produce a terrain surface that passes through all of the data points. Examples of exact interpolation methods include Kriging and some spline methods. Approximate methods produce terrain surfaces that follow only a global trend in the data points in which there is some degree of uncertainty. Stochastic interpolation methods incorporate the concept of geo-statistics to produce terrain surfaces with specific levels of uncertainty. Examples of these methods include Fourier analysis and Kriging. Deterministic methods on the other hand do not use the concept of geo-statistics. The quality of an interpolation method is measured by the disagreement between the true value and the interpolated value at all or selected locations (Bonham-Carter, 1994; Burrough, 1994; Lam, 1983). LIDAR is a topographic data acquisition technology that uses pulses of laser to map the terrain surface (Wehr and Lohr, 1999). The LIDAR laser scanner is commonly carried in an aircraft along with units of an Inertial Navigation System (INS) and Global Positioning System (GPS). The LIDAR system works by processing the return time for each returned laser pulse to calculate the distances or changes in terrain surface. The LIDAR system has demonstrated its ability to map different types of terrain surfaces including bare ground surface, urban areas, rural areas, or canopy. LIDAR systems can also be used to capture reflectance data in addition to their ability to 3

4 collect three-dimensional point data. Therefore LIDAR system currently provides two unique products, (1) the three-dimensional point data, which I previously called the IPC, and (2) reflectance data, on which digital imagery processing and digital photogrammetric techniques can be applied to derive useful information about the terrain (Schenk, 1999; Jain, 1989). However the two products are valuable for digital mapping and Geographic Information Systems (GIS) applications, this paper addresses the issue of creating terrain surfaces from the IPC product. Specifically, the paper discusses the issues that relate to the selection of an interpolation method for creating a TM from the IPC product of LIDAR. The test data used in the experiments presented in this paper are for approximately one-mile square study area in Lake Erie coast (see Figure 1 below). This LIDAR data set has been collected under the Airborne LIDAR Assessment of Coastal Erosion (ALACE) project (USGS, NOAA and NASA). The density of points on the ground is highly variable, but densities are typically 30 points per 100 square meters. The variability is due to the greater number of points along the edges of a pass, and where two passes overlap. The average horizontal accuracy is 75 centimeters due to uncertainties related to the altitude of the aircraft and the vertical accuracy is about 50 centimeters. In this paper, the quality of terrain surface representations that have been created for the study area using selected interpolation methods has been studied. This was done by corrupting the sample LIDAR data with different values of uncertainty that have been artificially introduced to the data before interpolating. Two common interpolation methods, the inverse-distance and Kriging have been used in this study to create terrain surface models in GRID format. On the other hand, after picking the candidate points from the LIDAR data using the VIP algorithm, the Delaunay triangulation was utilized to create TIN s from the corrupted points. Those TM s produced from the corrupted LIDAR data with different values of uncertainty including both GRID and TIN representations have been studied by comparing them at two representative profiles and four points in the data set (see Figure 2 for the location of the two profiles and the four points). 4

5 Units: Meters Figure 1: The LIADR test data set 2. Methodology 2.1 Elevations uncertainty generation In this study, the positional information of the test points of LIDAR are assumed to have negligible uncertainties compared to the elevations. This assumption is made since in this study we are only interested in examining the elevation values of the test points, which contribute directly to the resulted TM s. This assumption has reduced the complexity of the problem from three- to one- dimensional. To put the problem into perspective, consider the situation where different LIDAR systems have been used to map the same area. This is conceptually equivalent to the case when these elevations are measured with different data acquisition methods, which have different accuracy values and for each of which the measurements are taken several times. Based on that, the different values of uncertainty will produce different corresponding elevation values. The following random effect model (REM) is used to represent the data models generated by introducing specific values of uncertainty into the original LIDAR data set (Schaffrin, 2001): 5

6 2-1 y = Ax + e, e ~ (0, σ P ) (1) o Where, y is the new vector of observation (after randomizing); A is the matrix of parameters coefficients; x is the vector of parameters; e is the simulated random error vector (simulated uncertainty); and σ o 2 is the variance component. Where the following model represents the prior information, which are represented by the original LIDAR data set: β 2 = x e, e ~ (0, σ Q ) (2) o + o o o o Where, β o is the original observation vector; x is the vector of parameters of the prior information β o ; e o Q o 2 σ o is the original observation error (random); is the approximated cofactor matrix; and is the variance component. For the purpose of this study, five different levels of random error were generated assuming that they reflect the levels of uncertainty in the observed elevation data. To produce random error with the levels of ± 0. 1, ± 0.3, ± 0.7, and ± 1.0 meters respectively, simulation was used. The corresponding corrupted data models have then been produced using the REM equations shown above (see equations 1 and 2). 6

7 2.2 Creating GRID models using inverse-distance interpolation Similar to other statistical interpolation methods, inverse distance interpolates the function values at the points with unknown elevations using weighted average of elevations at the sample points using the general formula shown below: n F(s) = λ i F(si ) (3) i= 1 Where F(s) is the interpolated elevation value at the location s such that s is a function of x and y, F (s i ) is the value of the elevation at the sample point s i, i is in {1, n}, and λ i are the weights computed using the distances between the known and the unknown locations as follows: λ i,0 1 d = n j= 1 i,0 1 d j (4) Where, d i, 0 d j is the distance between sampled location i and the unsampled location; and is the distance between all sampled locations j {j: 1, n} and the unsampled location. 2.3 Creating GRID models using Kriging interpolation Kriging is a weighted moving average technique in which the estimation of the elevations at the unknown locations is commonly done in two steps, weights determination and estimation of the elevation values. The determination of weights is done using a procedure called variogram modeling. The estimation of the elevation values is done using the general formula of equation 3 above. Variogram modeling is always the first step for using Kriging interpolation because they describe the spatial variability between the elevation values of the data-points. As a measure of similarity between the elevation values (Z) of a distance h apart, the variogram is represented by g (h) and is given by: 7

8 1 2 2g(h) = (ZX ZX+ h) (5) N(h) Where N (h) is the number of sample pairs at distance h apart and Z x and Z x+ h are two points at distance h apart. The variogram modeling involves two stages these are (1) the construction of the experimental variogram, and (2) fitting a model on the variogram. Since the LIDAR data are of irregular sample patterns, the distances among points have been arranged in class intervals with constant increment. The emphasis on the weights is that data points further away from the point to be estimated exert a smaller influence on the estimated value than closer points. The aim of the process is to minimize the estimation variance, which is achieved by correct choice of the weighting coefficients, which have one constraint, that their sum must be equal to unity. 2.4 Creating TIN models The TIN model is a significant alternative to the regular raster GRID and has been adopted in various GIS and digital mapping systems generally for visualization purposes. In this study TIN points have been selected from among the LIDAR points using the Very Important Points (VIP) algorithm and triangulated using the Delaunay Triangulation (DT) approach. The DT approach works be creating triangles such that for every triangle in the triangulation there is only one circle that pass through the three points that compose that triangle and no other point falls inside the circle (Fowler and Little, 1979). 3. Results and analysis 3.1 Simulation for inverse-distance interpolation (IDI) The GRID of the elevation estimates obtained with the inverse-distance interpolation method using the original data set is shown in Figure 2 below in GRID format. Figure 3 shows two profiles drawn along line A and B as shown in Figure 2 on the GRID model resulted using the IDI on the original data set. The four GRID representations created from the REM s obtained using the four uncertainty levels have been studied at four points (1, 2, 3, and 4 as illustrated in Figure 2) and the results are plotted in Figure 4 below. 8

9 IDI Units: Meters Figure 2: The GRID model of the original data set obtained using inverse-distance interpolation Figure 3: The two profiles drawn on the IDI GRID along lines A and B of Figure 2. 9

10 Elevations P1 P2 P3 P ORIG REM1 REM2 REM3 REM4 Random Effect Models with the Inverse-Distance Interpolation Figure 4: The elevations of the four points on the GRID s of the IDI created from the REM s 3.2 Simulation for Kriging interpolation Here only two variogram models have been used to represent the variation among the elevation data in the study area, which are (1) linear, and (2) Gaussian. The GRID of the elevation estimates obtained with the Kriging interpolation, specifically ordinary Kriging using the original data set and a linear model of the variogram is shown in Figure 5 below in GRID format. For the case of linear model variogram, two profiles along lines A and B have also been drawn as illustrated in Figure 6 below. For the same case, the four GRID representations created from the REM s obtained using the four uncertainty levels have been studied at the four locations and are plotted in Figures 7 below. Similarly, The GRID obtained using the original data set and a Gaussian model of the variogram is shown in Figure 8 below in GRID format. For this case, two profiles along lines A and B have also been drawn as illustrated in Figure 9 below. Again, the four GRID representations created from the REM s obtained using the four uncertainty levels have been studied at the four locations and are plotted in Figures 10 below. 10

11 Units: Meters Figure 5: The GRID model of the original data set obtained using ordinary Kriging (Linear Var Model) Figure 6: The two profiles drawn on the Kriging (Linear Var Model) GRID along lines A and B of Figure 2. 11

12 Elevations ORIG REM1 REM2 REM3 REM4 Random Effects Models with Kriging (Linear Var Model) P1 P2 P3 P4 Figure 7: The elevations of the four points on the GRID s of the Kriging (Linear Var Model) created from the REM s Units: Meters Figure 8: The GRID model of the original data set obtained using ordinary Kriging (Gaussian Var Model) 12

13 Figure 9: The two profiles drawn on the Kriging (Gaussian Var Model) GRID along lines A and B of Figure Elevations P1 P2 P3 P ORIG REM1 REM2 REM3 REM4 Random Effect Models with Kriging (Gaussian Var Model) Figure 10: The elevations of the four points on the GRID s of the Kriging (Gaussian Var Model) created from the REM s 13

14 3.3 Simulation for TIN The TIN model created from the elevations of the original LIDAR data set is shown in Figure 11. Two profiles along lines A and B have also been drawn on the TIN model as illustrated in Figure 12 below. Also, the four TIN models created from the REM s obtained using the four uncertainty levels have been studied at the four locations and are plotted in Figures 13 below. Units: Meters Figure 11: The TIN model of the original data set 14

15 Figure 12: The two profiles drawn along lines A and B drawn on the TIN model of the original LIDAR data Elevations ORIG REM1 REM2 REM3 REM4 Random Effect Models with TIN P1 P2 P3 P4 Figure 13: The elevations of the four points on the TIN models created from the REM s 15

16 4. Conclusion From the plots of the GRID s and TIN s created for the random effect models at the four test points (1, 2, 3, and 4) for the three interpolation methods, TIN model results are as good as the original data set. This is based on the elevation intervals in meters for the three interpolation methods that were (1) in the range ( ) for the inverse distance interpolation, (2) in the range ( ) for Kriging with linear variogram model, (3) in the range ( ) for Kriging with Gaussian variogram model, and (4) in the range ( ) for the TIN model. From these experiments, the inverse distance interpolation results were found to be comparable to those obtained using the triangulation model. It is also clear from the results that, linear Kriging displayed variation with the Kriging interpolation with a Gaussian variogram model for higher random errors values. By looking at the profiles drawn for the original data using the three interpolation methods, we noticed that (1) the profiles of the GRID of the inverse distance method are similar to those of the GRID of the Kriging with Gaussian variogram model, and (2) the TIN profiles do not represent the topography efficiently because the triangulation facets are not more than mosaiced planes. Therefore, the triangulation method was the most accurate and the more stable in the presence of random error in the data when single locations are considered, however it does not provide faithful profiles of the terrain. Also, the inverse distance interpolation method was found to be more reliable than Kriging and comparable to the triangulation when measuring single points. Furthermore, the inverse distance interpolation method has provided more accurate profiles compared to the TIN, which are as well comparable to those of the Kriging with Gaussian variogram model. Kriging, on the other hand was found not stable in the presence of error when it comes to single locations on the resulted GRID although it has provided reasonable profiles. This may be due to the complexity of the Kriging model and the models used to fit the variograms (linear and Gaussian). Nevertheless, from the overall evaluation of the three interpolation methods it seems that, the inverse distance is the best for this specific situation, there is major disadvantage for this interpolation method that is related to the identification of the subsets from the original data set to be used in procedure. Comparing the results of the inverse distance and Kriging interpolations, it seems that the inverse distance method worked better than the Kriging because of the fact that LIDAR data is dense, which is not the best input for Kriging, but it suits the inverse distance 16

17 method. To conclude, triangulation is the best interpolation choice for LIDAR data since it estimates single locations more robustly, however it may not produce the best profiles. Also, inverse distance is a good choice only if a reasonable data sub-setting strategy is used, however it may not provides accurate elevation values at single locations. Kriging, on the other hand is not recommended to be used on LIDAR data based on the results obtained in this experiment. References Bonham-Carter, G. 1994, Geographic Information Systems for Geoscientists, Modelling with GIS, Pergamon, New York, pp Burrough, P. 1994, Principles of Geographical Information Systems for Land Resources Assessment, Oxford Science Publications, New York, pp DeMers, M. 2002, GIS Modeling in Raster, J. Wiley, New York, 203p. Fowler, R. and J. Little 1979, Automatic extraction of irregular network digital terrain models, Computer Graphics, Vol. 13, pp Isaacs, E. and R. Srivastava 1989, An introduction to applied geostatistics, Oxford University Press, New York. Jain A. 1989, Fundamentals of Digital Image Processing. Prentice Hall, Inc., Englewood Cliffs, New Jersey. Lam, N. 1983, Spatial Interpolation Methods: A Review, in The American Cartographer, Vol. 10, No. 2, pp Peucker, T., R. Fowler, J. Little, and D. Mark 1978, The Triangulated Irregular Network, Proceedings of the American Society of Photogrammetry: Digital Terrain Models (DTM) Symposium, St. Louis, Missouri, May 9-11, pp

18 Schaffrin B. 2001, Adjustment Computations for Random Processes, Class notes for GS 862, The Ohio State University. Schenk, T. 1999, Digital Photogrammerty, Volume 1, TerraScience. Tomlin, D. 1990, Geographic Information Systems and Cartographic Modeling, Prentice Hall, Englewood Cliffs, New Jersey. Wehr, A. and U. Lohr 1999, Airborne Laser Scanning - An Introduction and Overview, Journal of Photogrammetry and Remote Sensing, Vol. 54, pp Biography Tarig A. Ali, Ph.D. Assistant Professor, Department of Technology and Geomatics, East Tennessee State University B.S., University of Khartoum, Sudan (1993) M.S., The Ohio State University (1999) Ph.D., The Ohio State University (2003) ali@etsu.edu Phone: (423) Web site: Address: P. O. Box Department of Technology and Geomatics Johnson City, TN