ANALYSIS OF SURFACE INTERPOLATION ALGORITHMS FOR DIFFERENT SURFACE TYPES

ABSTRACT
Digital Terrain Models (DTMs) has become a very significant tool in extracting geomorphological information from various land areas. Their use is in the fields of mapping, landscape planning, urban design and many more. Automatic stereocorrelation has been used to generate a DTM from ASTER stereo image pair (3N and 3B) using ENVI software. Elevation values were extracted and used with four different interpolation algorithms. The resulting surfaces when compared with those from the topographic map showed that the Inverse Distance Weighting (IDW) can achieve a Root Mean Square Error (RMSE) of ±10.773m and Mean Absolute Error (MAE) of ±8.714m for flat terrain as compared to RMSE of ±11.035m, MAE of ±8.999m for spline; RMSE of ±11.121m, MAE of ±9.102m for Natural Neighbour (NN) and RMSE of ±12.108m, MAE of ±9.979m for kriging interpolation method using a point density of 61.49 points per km2. Hence, IDW is best for this surface type. For undulating terrain, IDW again gave the least RMSE of ±13.549m and MAE of ±10.789m in comparison to RMSE of ±13.711m, MAE of ±10.963m for NN; RMSE of ±13.717m, MAE of ±11.028m for spline and RMSE of ±14.835m, MAE of ±11.658m for kriging interpolation method for point density 62.30 points per km2 and hence, IDW is again best for this surface type. For mountainous terrain, NN interpolation method with RMSE of ±19.044m and MAE of ±13.909m gave best results than the other interpolation types. RMSE of ±21.167m, MAE of ±15.241m was obtained for kriging; RMSE of ±21.632, MAE of ±14.687m for IDW and RMSE of ±21.721m, MAE of ±14.544m for spline for point density 141.64 points per km2 and so NN works best for mountainous terrains. It is therefore recommended that IDW interpolation algorithm should be used for both flat and undulating terrains whereas NN should also be used for mountainous terrains.


CHAPTER ONE
INTRODUCTION
1.1 Background of Study
Land surfaces are generally represented in a computer environment as DTMs (Weibel and Heller, 1999). These digital representations are modeled from terrain reliefs through collections of data samples and algorithms which can interpolate elevations of intermediate unknown points. DTMs have several possible applications such as in the field of military where usage ranges from surveillance and intelligence gathering to strategic planning in battle field as a guide in missile launching. DTMs also play an integrate part in creating relief maps. Accurate elevation data helps geologists to determine and extract various geomorphological information from various terrain characteristics.

Advancement in technology has increased extensively the capability of DTMs generation from satellite images to more accurately represent terrains, making it useful in the field of civil engineering, landscape planning, urban design and road traffic engineering. Integration of DTM data with Geographic Information System (GIS) provides opportunity to model terrain relief, analyze and visualize phenomenon related to topography.

Over the year’s digital representation of terrains have been denoted severally as DTM, Digital Elevation Model (DEM) or Digital Surface Model (DSM). Although these terms are mostly used synonymously, the difference or meaning basically lies in its mode of application (Oksanen, 2006). DSM data includes low rise and high rise buildings, roads, bridges, forest trees and structures that can be found on the surface of the earth (Maune et al. 2007). DEM data does not necessarily include objects or manmade features on the surface of the earth, but mostly represents the bare ground with natural phenomenon like rivers (Oksanen, 2006; Maune et al. 2007). A DTM on the other hand is a continuous or smooth surface which aside from the values of elevations grids, also consists of other elements that describe the topographic surface such as slope, aspect, curvature, gradient, skeleton (pits, saddles, ridges, peaks) and others (Podobnikar, 2005). The DEM is often used generically for DTM (Maune et al. 2007; Li et al, 2005).

DTMs can be represented and stored in several ways. The commonly used data formats for DTMs are, (i) the regular grid (raster) and (ii) the Triangulated Irregular Network (TIN) (Weibel and Heller, 1991; Peng et al, 2004). The TIN transforms an irregularly spaced points data thus (x, y, z) values to form contiguous, non-overlapping, triangles that represents the surface. The TIN model allows extra data in complex areas and less data in non-complex areas thereby reducing redundancy. This therefore enables it to represent information about altitude, slope and aspects. However, they can be quite demanding towards memory space and computing time and also the algorithms involved could be sophisticated (De Wulf et al, 2012).

DTMs grid according to Weibel and Heller (1991) gives a matrix structure that records topological relation between data points stored as a two-dimensional array of elevations. Although the raster format has a number of setbacks which involves a rectangular data array irrespective of the morphology of the terrain, it remains the most popular format in the foreseeable future (Pike et al, 2009). This is because, it represents a terrain in a more technically controlled manner of grid cells where each cell could have its own property (Hengl, 2006). Grid DTMs ensures simplicity of the models and low memory space requirements whilst allowing for fast and straightforward data computations (De Wulf et al, 2012).

In DTMs elevations are presented as surface values on the land surface in areas of interest. Shi et al (2005) recognized that the overall accuracy of a generated DTM depends on both the propagation error and the model error. Leberl (1973) also asserted that a DTM performance depend on the terrain and the method used in interpolating the new points from the existing measurements. This therefore suggests that, apart from a good sampling of points required to improve the quality of any DTM, a good modelling of the surface would also depend on the appropriate DTM interpolation method chosen and used.

Many research works have been conducted on the various interpolation algorithms, however, an understanding of the terrain conditions upon which the interpolation is performed have largely been ignored. Hengl et al (2009) therefore claimed that, an inexperienced user would mostly be confused as to which technique to select in order to produce a DTM that would best suit a particular purpose. There are various data sources for DTMs. These data sources are severally aerial photography, satellite imagery, cartographic maps and measured terrestrial points.

1.2 Problem Statement
Many techniques exist for interpolating to approximate a surface from elevation data exists. The accuracy of the resulting surfaces depends on the nature of the landform and the interpolating algorithm used for interpolating the surface. There is no technique defined for different landforms but the user has to experiment with different techniques to select the best that will fit each landform type or just use any randomly irrespective of whether it is the best for the circumstance or not. The implementation and determination of which interpolation type is best for each landform type poses a problem which is the objective of this research

1.3 Aim and Objectives
1.3.1 Aim
The aim of this thesis is to implement various interpolation algorithms on different terrain types and determine which algorithm is most suitable for which type of terrain.


1.3.2 Specific Objectives
The objectives of the research are;

1.      To generate DTMs from ASTER stereo imagery.

2.      To investigate how the various interpolation algorithms perform with different terrain characteristics.

3.      To determine the quality of DTM generated.

1.4 Research Questions
1.      How many interpolation points are needed for DTM generation for each interpolation type?

2.      What interpolating algorithm is best suited for the different terrains?

3.      What is the quality of the generated surface?

1.5 Organization of Thesis
The work represented here are structured into five chapters.
Chapter 1 is an introductory chapter that includes a background to the study and problem statement. The main aim and objectives are also laid out here. A number of research questions are posed to answer the objectives.

In chapter 2, DTM generation and interpolation methods are discussed. This chapter contains literature about DTM sources and the generation of DTM from ASTER. Interpolation types and methods, as well as the errors associated with interpolation and previous work that have been done are also reviewed.

The materials and the methodology applied in the current study are discussed in chapter

This include the processes involved in DTM generation, issue of contour derived DTM, DTM modelling, study area and dataset preparation for interpolation.


The results obtained are stated in chapter 4. This chapter also contains a discussion of the results.

The main findings are stated in chapter 5 as conclusions. Some recommendations are also made towards further research in this chapter

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