Digital Terrain Models

DTMs with adequate detail can show the variability in topography across a surface. DTMs should characterize a complete continuous land surface for a given area, and permit calculations of heights from any given point (Hengl and Evans, 2009). The most common method used for creating DTMs is to use high-density elevation points extracted from aerial photographs (Nelson et al., 2009).

A DTM may have break-lines which are features that illustrate the change in surface slope (Longley et al., 2005). These features can either contain irregularly spaced

100 points with elevation information. In this study, the breakline is the polyline at the base of the beach that defines the sharp discontinuity present between the foot of the beach and the reef flat. The height of the base of the beach is assumed to be 1.166 m above SEAFRAME datum.

There are many advantages to constructing DTMs for reef islands. Within an atoll, it may be difficult to access all the reef islands and carry out fieldwork, thus limiting observations of the topography of all the reef islands. By constructing a terrain model of the reef islands more information can be made available and generally at a larger scale. For example, a DTM of Fongafale, Funafuti, Tuvalu, was used to portray topography in great detail and show areas that may be affected by sea-level rise (Yamano et al., 2006). This was made possible by extracting cross-island transects from the DTM to show the variable morphology. The extent to which topographical features of an area can be determined depends on the resolution of a DTM; the higher the resolution, the greater the detail, whereas the lower the resolution, the less detail it provides.

In choosing an appropriate DTM format, several factors need to be considered such as the task objective and the spread of data points. Triangular based DTM (TINs) can be applied in any given situation (Yanalak and Baykal, 2003), however raster grid based DTMs are only suitable for regularly distributed elevation points (Hengl and Reuter, 2009) Spatial analysis works well with raster formats, as each elevation point is stored as one node/cell, making it easier to perform investigations (Longley

et al., 2005). The grid or cell size of raster formats depends on the average distance

between data points (Chow and Hodgson, 2008). In TINs, spatial analysis is more complex, as this format manages the information for a single node as well as for its neighbours. Another disadvantage of TINs is that these models require more computer memory to store large datasets, and therefore it takes more time to carry out analysis. One of the advantages it has over other raster formats, however, is its greater capability to present detailed information when surface relief is irregular (Yanalak and Baykal, 2003).

101 In all DTM’s spatial interpolation fills in the missing gaps required to create a continuous surface (Longley et al., 2005). Interpolation methods work by approximating the elevation of an interpolated point using the known values of neighbouring points that are distributed around the point within a given area (Yanalak and Baykal, 2003). In GIS, spatial interpolations are based on Tobler’s (1970, p. 236) First Law of Geography that states “everything is related to everything else, but near things are more related than distant things”. Currently there is a range of spatial interpolation methods available that can be applied to predict an unknown elevation value. This study will only discuss the two interpolation methods Inverse Distance Weighted (IDW) and Delaunay triangulation methods. TINs are derived from the latter interpolation method.

The IDW method uses mathematical algorithms and neighbouring points to estimate the elevations (z) at unknown locations (x0) (Burrough and McDonnell, 1998; Longley et al., 2005). This is a simple and commonly applied interpolation method that predicts the elevation of an unknown location based on the elevations of, and the distance to, surrounding measured values (Longley et al., 2005; Hengl and Reuter, 2009). Applying Tobler’s First Law, IDW estimates the interpolated value by obtaining the mean value of its close neighbours. The neighbours closer to the position of the unknown location have more weight in the estimation compared to those located at a further distance (Longley et al., 2005). This interpolation method was considered appropriate for this study for two reasons. The dataset was dense and the interpolated height values fell within the range of the measured values. However, this is not the case with some interpolation methods, such as spline. The requirement of the spline method is that the interpolation passes exactly through all input data points predicting heights above and below the points creating a smooth surface (Longley et al., 2005).

The Delaunay triangulation method interpolates elevation data following the Delaunay criterion which does not allow any data to be placed within the circumcircle of any triangle (Hengl and Reuter, 2009). This method has been applied in this study to develop a TIN from many single elevation points referred to

102 as mass points converted from contours to provide a detailed topography across reef islands.

The quality of a model cannot be determined by its visual appearance alone, but must be tested using either quantitative or qualitative methods (Burrough and McDonnell, 1998). One quantitative approach to assess the quality of a model would be to verify the uncertainties and errors within the dataset. However, sometimes surfaces that are statistically accurate do not represent the topography of a given area, therefore the best method to apply is that which best represents the true surface (Yang and Holder, 2000). The best way to determine the quality of a model is to assess if it meets both the study’s objectives and portrays a true surface of the study site, which can be defined simply by the term “fitness for use” (Fisher and Tate, 2006).

TIN models can be converted to grid formats by using GIS. By using GIS, it is possible to perform analysis on a selected format that suits the dataset, and also this selected format can then be converted to another format to meet the objective of the study (Hengl and Reuter, 2009). In this chapter, the two DTM formats are utilised. Firstly, the raster grid format is used to create a DEM to enable calculations of reef island volumes. Secondly, the DEM is converted to a TIN format to show differences in elevations between the reef islands. This is visually compared with the other TIN model developed from denser mass elevation points derived from contours.