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Surface approximation of Point Data using different Interpolation Techniques – A GIS approach
Surface is an abstract class used to represent continuous spatial phenomena. Although the underlying surface data is continuous, a surface can represent nested holes and islands using masks. Only one z-coordinate value is allowed for each planimetric location on a Surface. In a context of spatial mathematics, each surface object can be represented by a single-valued function z = f (x, y), where z can be an elevation value or any other kind of measurement with respect to the application domain. Factor to be used to correct mismatch between X/Y and Z units. In the context of data structuring, a surface object is a structure that maintains neighborhood relation useful for computational-based functions similar to the finite-element approach. Using the neighborhood relation, a computation unit can be formed to facilitate and simplify various calculation, e.g. spatial gradient (slope), aspect, sun reflectance, flow, isolines, line of sight, surface area, volume, etc.
Different interpolator produces good estimates, however, the selection of appropriate interpolator entirely depends upon the physiography and distribution of points in the area. Resulting surface defined over a two dimensional region, is one of the most important object in GIS. Continuous variables such as terrain elevation and atmospheric temperature are approximately modeled by TIN or grid in GIS (Sadahiro, 2001). Scalar fields modeled as surfaces in GIS are analyzed not only visually but also mathematically and even statistically. There have been proposed numerous methods in the literature which are classified as- a) Method so called geostatistics including variogram, correlogram, and krigging (Isaaks and Srivastava 1989, Cressie 1993) used for spatial analysis as well as for spatial interpolation. b) The second type of method focuses on modeling surfaces by mathematical functions. It includes trend surface analysis (Tobler 1964, Bailey and Gatrell 1995) and its extensions (Hagget 1968, Griffith 1981) used for modeling simple surfaces. 3.0 Methodology: Visiting every location in a study area to measure the height magnitude or concentration of phenomena is usually difficult or expensive. Instead, strategically dispersed sample point locations are selected and using the surface creation function, an estimated value is assigned to all other locations. The input points can be either randomly or regularly spaced points containing heights, concentration or magnitude measurements. Ground water depth was measured from operational open as well as tube wells and samples were collected for quality analysis in pre and post monsoon seasons for a period of six years i.e. 1995-2000 to monitor the changes in the given period of time. The analysis for various water qualities like EC (Electrical Condutivity), pH and SAR (Sodium Absorption Ratio) was carried out in the laboratory. Ground Water sampling locations were first marked over topographic sheets of 1: 50,000 scale and latitude and longitude were calculated for each point. These locations were entered into GIS as point coverage through digitization and transformed into geographical coordinate systems. These points were used as input in interpolation process. Each point location was assigned a unique code in their feature attribute table (PAT). The ground water depth and corresponding chemical data (EC, pH and SAR) for each point were entered as separate database file (.dbf). These associated information were linked to the corresponding point data through a common field (sampling code) for the surface approximation. Arcview (ver.3.2a) GIS software with spatial extension module was used for surface interpolation. Two types of models were used in the study to represent the surfaces: Grids and TINs so that the results of which best suits our application can be carried further for more advanced studies. Fig: 2 shows the various steps involved in surface approximation using different interpolation techniques.  Figure 2 Flow Chart Showing different steps involved in surface approximation using Grid and TIN methods 3.1 Grid Analysis: Grids represent surface using a mesh of regularly spaced points. One can estimate value anywhere within the mesh by anything nearby mesh point’s value, giving more weight to those that are close. The resulting grid theme is the best estimate of what the quantity is on the actual surface for each location. Surface interpolator makes certain assumptions about how to estimate the best values, that inturn depends how the sample points are distributed in the area. In this study two-grid surface interpolator (IDW and Spline) were used and the results from which were compared to best suit our application. 3.1.1 Inverse Distance Weighted (IDW): This interpolation technique weights the contribution of each input (control) points by a normalized inverse of the distance from the control point to the interpolated point. IDW assumes that each input point has a local influence that dimiminishes with distance. It weights the points closer to the processing points, greater than those farther away. A specified number of points, or all points within a specified radius is used to determine the output value for each location. The power parameter in the IDW interpolator controls the significance of the surrounding points upon the interpolated value. A higher power results in less influence from distant points. Each line in a barrier input line theme is used as break that limits the search for input sample points. A line can represent a cliff, ridge or some other interruption in a landscape. A choice of number of barriers will use all points specified in the No. of Neighbors or within the identified radius. (Fig-3e) shows the surface created from sampled GW height data for the year 2000 post monsoon using IDW technique by keeping output cell size of 0.000796 mm, No. of Neighbors = 12, Power = 2 and No Barrier options in order to get the surface containing 250 rows and 313 columns. This result was kept uniform for all the data processed using various techniques.  Figure 3 Comparative view of results obtained from TIN, Spline and IDW 3.1.2 Spline: This technique fits a minimum curvature surface through the input points. Conceptually, it is like bending a sheet of rubber to pass through the points, while minimizing the total curvature of the surface. It fits a mathematical function to a specified number of nearest input points, while passing through the sample points. This method is best for gently varying surfaces where change in physiography or other phenomenon is not abrupt. It is not appropriate if there are large changes in the surface within a short horizontal distance because it can overshoot estimated values. The regularized method yield a smooth surface as the weight parameters defines the weight of the third derivative of the surface in the curvature minimization expression. The estimated surface for GW depth data for the year 2000 post monsoon (Fig-3d) contains output grid cell size of 0.000796, 250 rows, 313 columns, weight 0.1 and No. of points = 12 to get surface containing 250 rows and 313 columns. Both the method produces good estimates, but neither estimates every unknown value with perfect accuracy. The interpolation solely depends on the no. of sampling locations and how evenly and effectively it is distributed over the study area. In the present, study an average of 20 sample points were taken in the entire study area. 3.1.3 Validation of Result: In order to validate the results obtained through spline, of the total available samples, few samples were randomly selected and kept as test samples. Remaining samples were used for surface approximation using spline method. It has been observed that values at test samples were nearly similar to the values at corresponding points over surface generated without considering those test samples. Again this result was found to be similar to those considering all the sampling points. This confirms that the sampling locations available were able to predict or interpolate values at points beyond or at locations where sampling points were not available. 3.2 Triangular Irregular Network (TIN) analyses: A TIN is an object used to represent a surface. Since representation of a surface can be done in many different ways, TIN (Triangulated Irregular Network) also implies a specific storage structure of surface data. TIN partitions a surface into a set of contiguous, non-overlapping, triangles. A height value is recorded for each triangle node. Heights between nodes can be interpolated thus allowing for the definition of a continuous surface. TINs can accommodate irregularly distributed as well as selective data sets. This makes it possible to represent a complex and irregular surface with a small data set. The TIN Edge (An index of a triangle edge, value 0 to 2) class provides access to information about edges of a TIN for surface analysis purposes. Each TIN Edge object is an edge of a triangle that contains the edge elements of a TIN. Z-Factor is used to correct mismatch between X/Y and Z units. TIN was created for ground water height for the year 2000 post monsoon period (fig 3b). The second step involved the conversion of TIN into grid in order to visualize the surface keeping the cell size (0.000796) and no. of rows (250) and columns (313) same to that of surface generated from grid analysis (fig 3c). |