Interpolation Local Interpolation Methods –IDW – Inverse Distance Weighting –Natural Neighbor –Spline – Radial Basis Functions –Kriging – Geostatistical –Local Polynomials –Hybrids (interpolation and rule-based modeling)

Geospatial Analysis and Modeling MEA592 – Helena Mitasova When we need interpolation Bivariate - values measured at 2D points to surface : Scattered points, profiles, isolines to raster, Denser/smoother TIN or higher resolution raster

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Scattered points to raster Given Points Regular Grid

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Interpolation: principles Problem definition find a function which passes through (or close to) the given set of discrete points and then use this function to compute the values at a different set of points – usually a regular grid

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Interpolation: principles Problem definition find a function which passes through (or close to) the given set of discrete points and then use this function to compute the values at a different set of points – usually a regular grid Bivariate formulation Given m-points (x i, y i, z i ), i=1,m find such F(x,y) that for each i=1,m z i =F(x i,y i ) - interpolation z i =F(x i,y i ) + e(x i,y i ) - approximation and compute z k =F(x k,y k ) where (x k,y k ) are grid points Why is this a difficult problem?

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Interpolation: principles Problem does not have a unique solution:

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Interpolation: principles Problem does not have a unique solution

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Interpolation: principles Problem does not have a unique solution

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Interpolation: principles Problem does not have a unique solution: we need additional conditions r r j

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Interpolation: conditions Additional conditions define different types of interpolation: Locality condition –each point influences the surface only up to certain distance –or value at a given point will be similar to values at nearby points Geostatistical conditions –surface is one realization of a random function with spatial covariance Smoothness –Function should pass through or close to the data points and be as smooth as possible

Interpolation IDW - the simplest and most common method IDW Z i is value of known point D ij is distance to known point Z j is the unknown point n is a user selected exponent (often 1,2 or 3) Any number of points may be used up to all points in the sample; typically 3 or more

Interpolation IDW n = 1 = 39.5 n = 2 Z i = 35.5

Interpolation IDW –Factors affecting interpolated surface: Size of exponent, n affects the shape of the surface (larger n means the closer points are more influential) A larger number of sample points results in a smoother surface and the variance decreases. Cell size, increase size creates a rougher looking surface –Properties of interpolated surface: Conservative, output will never over- or under-estimate the minimum or maximum input values. As you moves away from observed points the surface will converge on the local arithmetic mean value. Surface is rough, with peaks and valleys at the observed points. The larger the exponent the greater the extent of the peaks and valleys.

Effect of IDW Powers

Interpolation: IDW n = 2 Sample = 12 n = 2 Sample = 24 n = 4 Sample = 12

Geospatial Analysis and Modeling MEA592 – Helena Mitasova Interpolation: points to 2m DEM Voronoi polygons Average weighted by inverse distance

Interpolation: Spline Family There a number of methods in the Radial Basis Function (RBF) or Spline family. A spline is a mathematical curve that often takes the form of a polynomial curve (approximately a degree 3 polynomial or sin/cos wave). RBF methods are a special case of splines. Multiquadric and spline functions The surface must have minimum curvature—the cumulative sum of the squares of the second derivative terms of the surface taken over each point on the surface must be a minimum. The second derivative is a measure of roughness.

Interpolation: Spline Family RBFs are used to produce smooth surfaces from a large number of data points. –The functions produce good results for gently varying surfaces such as elevation. RBFs are inappropriate when large changes in the surface values occur within short distances or when you suspect the sample data is prone to measurement error or uncertainty.

Basic Concept of RBFs RBFs are formed over each data location. An RBF is a function that changes with distance from a location. Z(x,y) = w 1 ϕ 1 + w 2 ϕ 2 + … Weights are found by requiring that, when the prediction is moved to a location with a measured value, the data value is predicted exactly. This forms N equations with N unknowns and can be solved uniquely. But there is a penalty for roughness, so you create a smooth surface.

Spline in Geostatistical Analyst RBF methods are a series of exact interpolation techniques; that is, the surface must go through each measured sample value. There are five different basis functions: –Thin-plate spline –Spline with tension – also in Spatial Analyst –Completely regularized spline – also in Spatial Analyst –Multiquadric function –Inverse multiquadric function Each basis function has a different shape and results in a slightly different interpolation surface.

Interpolation:Spline Family Spline –Factors affecting interpolated surface: Method, Regularized is smoother then Tension. Weight A larger number of sample points results in a smoother surface and the variance decreases. Cell size, increase size creates a rougher looking surface –Properties of interpolated surface: No Conservative: output can over- or under-estimate the minimum or maximum input values. As you moves away from observed points the surface can decrease or increase significantly from the expect. Surface is smooth, with peaks and valleys away from the observed points.

IDW vs.Spline

Natural Neighbor Natural neighbor interpolation finds the closest subset of input samples to a query point and applies weights to them based on proportionate areas in order to interpolate a value. Its basic properties are that it's local, using only a subset of samples that surround a query point, and that interpolated heights are guaranteed to be within the range of the samples used. It does not infer trends and will not produce peaks, pits, ridges or valleys that are not already represented by the input samples. It adapts locally to the structure of the input data, requiring no input from the user pertaining to search radius, sample count, or shape. It works equally well with regularly and irregularly distributed data.

Natural Neighbor The natural neighbors of any point are those associated with neighboring Voronoi (i.e. Thiessen) polygons. Initially, a Voronoi diagram is constructed of all the given points, represented by the olive colored polygons. A new Voronoi polygon, beige color, is then created around the interpolation point (red star). The proportion of overlap between this new polygon and the initial polygons are then used as the weights. By comparison, a distance based interpolator such as IDW (Inverse Distance Weighted) would assign similar weights to the northern most point and to the north-eastern point based on their similar distance from the interpolation point. Natural neighbor interpolation, however, assigns weights of 19.12% and 0.38% respectively which is based on the percentage of overlap.