SPATIAL INTERPOLATION, SURFACE ANALYSIS & NETWORK ANALYSIS

SPATIAL INTERPOLATION Interpolation is a process of estimating the value of attributes at unsampled locations within area covered by measurements made at point locations. When the estimation is done for the sites lying outside the area covered by existing measurements, it is called extrapolation. The role of interpolation in GIS is to fill in the gaps between the observed data points, which may be having regular, clustered, or randomly spaced observed points. Interpolation is used to convert data from point values to continuous fields so that the spatial patterns sampled can be compared with the spatial patterns of other spatial entities.

Pollution Level shown using Interpolation Technique

Interpolation is required to be performed: (i) in complication where conversion of scanned images from one grid system to another grid system is performed, (ii) in transformation of a continuous surface from one kind of tessellation to another, and (iii) in conversion of data from sets of sample points to a discretised, continuous surface.

Interpolation is most commonly used process for generating contours using DEM or DTM which contain spot levels obtained by levelling or photogrammetry. Most GIS packages often provide a number of interpolation methods to generate continuous surfaces for use as map overlays or display. Thiessan Polygon Method. Triangulated Irregular Network (TIN) etc.

Triangulated Irregular Network (TIN) Triangulated irregular network is a method of constructing a surface from a set of irregularly spaced data points. In the TIN data model, the terrain is recorded as a continuous surface made up of a mosaic of non- overlapping triangular surfaces formed by connecting selectively sampled points of elevation data using a consistent method of triangle construction.

To make the interpolation simpler mathematically, most TIN models assume plane triangular surfaces. The sides of a triangular face are called the edge, and the ends of an edge are called the vertices or nodes. In a TIN model, edges represent terrain features, for example, ridges, river channels, and breaks, while the vertices describe nodal topographic features, such as peaks, pits, and passes.

The TIN data model is distinct from the DEM data model in the following respect: (i) In a TIN, every data point has (x, y, h) where h is the elevation, while in DEM (x, y) are hidden. (ii) A TIN may include plain topographical relationships between points and their proximal triangles.

In TIN data models, topological relationships play a significant role. By using the method of triangulation, the totally unstructured collected elevation points are turned into a properly organized geographic database suited for terrain modeling applications. Since a given set of data points can be triangulated in many ways as shown in Fig. different contour maps will be generated by different triangulation network of data points. Thus, a necessary procedure must be adopted to ensure the creation of an unique TIN for a given set of data points.

Different triangulation network using same data points

The TIN model is an exact interpolation method based on local data points. It uses linear equation and trigonometry to calculate the interpolated values at point other than the data points within the boundary of the TIN.

SURFACE ANALYSIS

To generate a surface as close to a real-world surface, sufficient numbers of points at suitable locations are required. If the data points are not sufficient in number or at suitable locations, interpolation is used to fill the gaps and create the surface for analysis. In practice, however, a surface-fitting algorithm is commonly used to improve the results of terrain modeling.

A local approach of interpolation that takes into account only two points in the neighbourhood of interpolated point, deficient the consideration of the characteristics of terrain continuity and smoothness. The global approach of interpolation removes such shortcomings in the local approach by utilizing all or most of the data points to characterize the surface at a point, thus allowing estimation to be made on the trend of the surface.

A widely used global surface-fitting method is trend surface analysis where the terrain surface is approximated by a polynomial expansion of the coordinates of the sampled points. Some of the commonly used polynomials are: Linear z = a + bx + cy ………………………..(1) Quadratic z = a + bx + cy + dx 2 + exy + fy 2 … (2) Cubic z = a + bx + cy + dx 2 + exy + fy 2 + gx 3 + hx 2 y + ixy 2 + jy 3 ……………………………………..(3)

where z is the estimated height at a point having coordinates (x, y), and a, b, c,…, j are polynomial coefficients. The coefficient values are determined by solving a set of simultaneous equations formed using the known values at data points, by the method of least squares. Finally, an optimum local interpolation method known as kriging, named after the pioneering work of Danie Krige, is now widely used in DTM software packages as it is flexible and can handle any type of data.

Kriging is a generic name for a family of least-square linear regression algorithms that are used to estimate the value of a continuous attribute (such as terrain height) at any unsampled location using only attribute data available over the study area. Kriging treats the continuous attribute to be interpolated as a regionalized variable. It is possible to map one attribute using other attributes, if available and spatially correlated, using co-kriging approach.

Kriging Technology This applies when the data for one particular attribute are more difficult or too expensive to obtain. Therefore, the other less expensive set of attribute data becomes surrogate for the more expensive set.  Grid cell values are calculated from data points within a specified search radius.  These points are weighted based on distance and geographic orientation to the cell using a semivariogram to determine directional trends.

NETWORK ANALYSIS

In the context of GIS, a network is defined as a set of interconnected linear features through which resources can flow. Common examples of networks include highways, railways, city streets, canals, rivers, transportation routes etc. through which different vehicles and water can flow, respectively.

There are many spatial problems that require the use of network analysis for their solution. These include : (i) to find the shortest path (in terms of physical distance or least cost) that can be followed to visit a series of features in a network, known as pathfinding, (ii) to assign one or more portions of a network to be served by a facility or business location, called as allocation,

(iii) to find all portions of the network that are connected with the movement of a particular feature (e.g., city transport), called as tracing, (iv) to depict the accessibility of a location and the interactions that occur between different locations (based on a technique known as gravity modeling). This is widely used in economics, geography, engineering, and urban planning, known as spatial interaction, (v) to generate a distance matrix between different pairs of locations in the network, known as distance matrix calculation, and (vi) to determine simultaneously the locations of existing and planned facilities, as well as the allocation of demand to these facilities, known as location-allocation modeling.

Network Analysis Applications: Some of the Network analysis applications are: 1. Shortest Path Problem 2. Route Tracing 3. Salesman Travel Problem 4. Location-allocation Modeling

1. Shortest Path Problem The shortest path which is the shortest distance (or least- time path) between two points on a network, is determined by proximity analysis in a raster GIS. Obstruction to travel are added to a raster grid by increasing the value of cells that are barriers to travel. Then the result is obtained as least-cost route. Vector GIS network analysis of finding shortest path is more flexible and provides a thorough obstruction analysis such as restrictions and congestion in traffic routes.

2. Route Tracing Route tracing through the network analysis is required to identify the routes for unidirectional flow of resources with special reference to stream networks and services, such as sewerage systems and cable TV networks. The key concept in route tracing is connectivity of network links at network nodes. Knowledge of direction of flow is an important factor for route tracing. Therefore, each link in the network must be associated with direction of flow that can be defined at the time of digitizing process by keeping the directions of digitization and flow same. Now tracing the links downstream or upstream of a point on the network, is performed by moving in the direction of flow or against it.

3. Salesman Travel Problem A salesman may be required to visit a specific set of clients in a day, for which he would like to know the best (usually the quickest) route that he can follow to finish his job. Similar problems are collection of garbage and distribution of mails. Such problems are solved by analyzing the order of stops and paths between them. Getting solutions of such problems, is a complex task, and to simplify such tasks, the ordering of the stops can be determined by calculating the minimum path between each stop and every other stop in the list based on impedance met in the network. A trial and error method (also referred to as heuristic) is then applied to order the visits minimizing the total impedance from first stop to the last.

4. Location-allocation Modeling An important application of network analysis is allocation of resources. This is done by modeling of supply and demand in which help of movement of goods, people, and information or services through the network are required to match the demand with the supply. Allocation of resources is usually done by allocating links in the network to the nearest center of supply taking into account impedance values. The maximum catchments area of a particular supply centre can also be determined on the basis of the demand located along adjacent links in the network.

A situation may arise by imposing limitation on supply and demand, in which some parts of the network may not be serviced despite a demand being present in that part. Such problems can be solved by reducing the supply to some parts of the network or by identifying the optimum location for a new centre to meet the shortfall in supply relative to demand.

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