WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam WFM 6202: Remote Sensing and GIS in Water Management Akm Saiful Islam Institute of Water and Flood Management (IWFM) Bangladesh University of Engineering and Technology (BUET) [Part-B: Geographic Information System (GIS)] Lecture-6: Interpolation January, 2008

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam Principle of Interpolation  Interpolation is the procedure of estimating the value of properties at unsampled points or areas using a limited number of sampled observations.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam Interpolation Techniques 1. Pointwise interpolation 1(a) Thiessen polygon 1(b) Weighted Average 2. Interpolation by curve fitting –2.1 Exact interpolation 2. 1(a). Nearest neighbor 2. 1.(b) Linear interpolation 2. 1(c) Cubic interpolation –2.2 Approximate interpolation 2.2(a) Moving Average 2.2(b) B-spline 2.2(c) Curve Fitting by Least Square Method  3. Interpolation by surface fitting –3.1 Regular grid 3.1(a) Bilinear Interpolation 3.1(b) Bicubic Interpolation –3.2 Random points 3.2(a) TIN

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 1. Pointwise Interpolation  Pointwise interpolation is used in case the sampled points are not densely located with a limited influence or continuity in surrounding observations, for example climate observations such as rainfall and temperature, or ground water level measurements at wells.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 1(a) Thiessen Polygons  Thiessen polygons can be generated using distance operator which creates the polygon boundaries as the intersections of radial expansions from the observation points.  This method is also known as Voronoi tessellation.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 1(b) Weighted Average  A window of circular shape with the radius of d max is drawn at a point to be interpolated, so as to involve six to eight surrounding observed points.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2. Interpolation by Curve Fitting  the principle of curve fitting respectively to interpolate the value at an unsampled point using surrounding sampled points.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2. Curve Fitting Curve fitting is an important type of interpolation in many applications of. Curve fitting is divided into two categories.  2.1 exact interpolation : a fitted curve passes through all given points.  2.2 approximate interpolation : a fitted curve does not always pass through all given points.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2.1 Exact interpolation There are three methods: 2.1(a) nearest neighbor : the same value as that of the observation is given within the proximal distance

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2.1 Exact interpolation 2.1(b) linear interpolation: a piecewise linear function is applied between two adjacent points.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2.1 Exact interpolation 2.1(c) cubic interpolation : a third order polynomial is applied between two adjacent points under the condition that the first and second order differentials should be continuous.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2.2. Approximate Interpolation There are three methods; 2.2(a) Moving Average: a window with a range of -d to +d is set to average the observation within the region

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2.2 Approximate Interpolation 2.2(b) B-Spline: a cubic curve is determined by using four adjacent observations

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 2.2 Approximate Interpolation 2.2(c) Curve Fitting by Least Square Method. Least square method (sometimes called regression model) is a statistical approach to estimate an expected value or function with the highest probability from the observations with random errors. The highest probability is replaced by minimizing the sum of square of residuals in the least square method. Slope Equation intercept

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 3. Interpolation by Surface Fitting  the principle of surface fitting respectively to interpolate the value at an unsampled point using surrounding sampled points.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 3. Surface Fitting Surface fitting is widely used for interpolation of points on continuous surfaces such as digital elevation model (DEM), geoid, climate model (rainfall, temperature, pressure etc.) and so on. Surface fitting is classified into two categories: –3.1 surface fitting for regular grid and –3.2 surface fitting for random points. 3.1 Surface Fitting for Regular Grid Following two methods are commonly used. 3.1(a) Bilinear Interpolation 3.1(b) Bicubic Interpolation

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 3.1 Surface Fitting for Regular Grid 3.1(a) Bilinear Interpolation Bilinear function is used to interpolate z using the following formula with respect to normalized coordinates (u, v) of the original coordinates (x, y)

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 3.1Surface Fitting for Regular Grid 3.1(b) Bicubic Interpolation Third order polynomial is used to fit a continuous surface using 4 x 4 = 16 adjacent points.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam 3.2 Surface Fitting for random Points 3.2. (a) Triangular network called as Triangulated Irregular Network (TIN) is applied

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam Compare Interpolation methods Thiessen polygons are Used for service area analysis of public facilities such as hospitals. Originally proposed to estimate aerial averages precipitation in Inverse Distance Weighted can be a good way to take a first look at an interpolated surface. However, there is no assessment of prediction errors. Accuracy depends on the selection of a power value and the neighborhood search strategy. A smaller (6) actually produce better estimations than a larger number (12). Thin-plate Splines (applies to surface) are recommended for smooth, continuous surfaces such as elevation and water table. Also used for interpolating mean rainfall surface and land demand surface.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam Geo-statistical method- Kriging Kriging is a geostatistical method for spatial interpolation. It can assess the quality of prediction with estimated prediction errors. It uses statistical models that allow a variety of map outputs including predictions, prediction standard errors, probability, etc. Semivariogram can be fitted as: 1.Ordinary Kriging models: Spherical, Circular, Exponential, Gaussian and Linear. 1.Universal Kriging models: Linear with Linear drift, and Linear with Quadratic drift

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful IslamDr. Akm Saiful Islam Semivariogram The semivariogram functions quantifies the assumption that things nearby tend to be more similar than things that are farther apart. Semivariogram measures the strength of statistical correlation as a function of distance. Semivariance: Y(h) = ½ [(Z(xi) - Z(xj)] 2 Covarience = Sill – Y(h)