# kriging interpolation

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Applications in GIS (Kriging Interpolation) Dr. S.M. Malaek Assistant: M. Younesi Interpolating a Surface From Sampled Point Data Interpolating a Surface From Sampled Point Data Assumes a continuous surface that is sampled Interpolation Estimating the attribute values of locations that are within the range of available data using known data values Extrapolation Estimating the attribute values of locations outside the range of available data using known data values Interpolating a Surface From Sampled Point Data Estimating a point here: interpolation Sample data Interpolation Interpolating a Surface From Sampled Point Data Estimating a point here: extrapolation Sample data Extrapolation Sampling Strategies for Interpolation Regular Sampling Random Sampling Interpolating a Surface From Sampled Point Data Interpolating a Surface From Sampled Point Data Elevation profile Sample elevation data A B IfA = 8 feetand B = 4 feetthen C = (8 + 4) / 2 = 6 feet C Linear Interpolation Interpolating a Surface From Sampled Point Data Non-Linear Interpolation Elevation profile Sample elevation data A B C Often results in a more realistic interpolation but estimating missing data values is more complex Interpolating a Surface From Sampled Point Data Global Interpolation Uses all known sample points to estimate a value at an unsampled location Sample data Interpolating a Surface From Sampled Point Data Local Interpolation Uses a neighborhood of sample points to estimate a value at an unsampled location Sample data Uses a local neighborhood to estimate value, i.e. closest n number of points, or within a given search radius Trend Surface Trend Surface Global method Inexact Can be linear or non-linear predicting a z elevation value [dependent variable] with x and y location values [independent variables] Trend Surface 1st Order Trend Surface In one dimension: z varies as a linear function of x x z z = b0 + b1x + e Trend Surface 1st Order Trend Surface In two dimensions: z varies as a linear function of x and y z = b0 + b1x + b2y + e x y z Trend Surface Inverse Distance Weighted (IDW) Inverse Distance Weighted Local method Exact Can be linear or non-linear The weight (influence) of a sampled data value is inversely proportional to its distance from the estimated value Inverse Distance Weighted (Example) 4 3 2 100 160 IDW: Closest 3 neighbors, r = 2 200 = - =((

((

===1 ) , (1) , (11iwithi inipinipiiz y x z orddzy x zInverse Distance Weighted (Example) 4 3 2 A = 100 B = 160 C = 200 1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500 Weights A BC Inverse Distance Weighted (Example) 1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500 .0625 * 100 = 6.25.1111 * 160 = 17.76 .2500 * 200 = 50.00 WeightsWeights * Value A BC 74.01 / .4236 = 175 Total = .4236 6.25 +17.76 + 50.00 = 74.01 4 3 2 A = 100 B = 160 C = 200 Geostatistics Geostatistics Geostatistics:The original purpose of geostatistics centered on estimating changes in ore grade within a mine. The principles have been applied to a variety of areas in geology and other scientific disciplines. A unique aspect of geostatistics is the use of regionalized variables which are variables that fall between random variables and completely deterministic variables.Geostatistics Regionalized variables describe phenomena with geographical distribution (e.g. elevation of ground surface). The phenomenon exhibit spatial continuity.Geostatistics It is notalways possible to sample every location. Therefore, unknown values must be estimated from data taken at specific locations that can be sampled.

The size, shape, orientation, and spatial arrangement of the sample locations are termed the support and influence the capability to predict the unknown samples. Semivariance Semivariance Regionalized variable theory uses a related property called the semivariance to express the degree of relationship between points on a surface. The semivariance is simply half the variance of the differences between all possible points spaced a constant distance apart. Semivariance is a measure of the degree of spatial dependence between samples (elevation( Semivariance semivariance :The magnitude of the semivariance between points depends on the distance between the points. A smaller distance yields a smaller semivariance and a larger distance results in a larger semivariance.Calculating the Semivariance (Regularly Spaced PointsRegularly Spaced Points( Consider regularly spaced points distance (d) apart, the semivariance can be estimated for distances that are multiple of (d) (Simple form): = =+hNih i ihz zNh12) (21) (Semivariance Zi is the measurement of a regionalized variable taken at location i , Zi+h is another measurement taken h intervals away d Nh is number of separating distance = number of points Lag (if the points are located in a single profile) = =+hNih i ihz zNh12) (21) (Calculating the Semivariance (Irregularly Spaced PointsRegularly Spaced Points( Here we are going to explore directional variograms. Directional variograms is defines the spatial variation among points separated by space lag h. The difference from the omnidirectional variograms is that h is a vector rather than a scalar. For example, if d={d1,d2}, then each pair of compared samples should be separatedin E-W direction andin S-N direction.Calculating the Semivariance (Irregularly Spaced PointsRegularly Spaced Points( In practice, it is difficult to find enough sample points which are separated by exactly the same lag vector [d]. The set of all possible lag vectors is usually partitioned into classes Variogram Variogram The plot of the semivariances as a function of distance from a point is referred to as a semivariogram or variogram.Variogram The semivariance at a distance d = 0 should be zero, because there are no differences between points that are compared to themselves. However, as points are compared to increasingly distant points, the semivariance increases.Variogram The range is the greatest distance over which the value at a point on the surface is related to the value at another point. The range defines the maximum neighborhood over which control points should be selected to estimate a grid node. Variogram (Models( It is a model semi-variogram and is usually called the spherical model. a is called the range of influence of a sample. C is called the sill of the semi-variogram.>s|.|

\|= a h Ca hahahChwherewhere332123) (Variogram (Models( Exponential Model ( )a he C h = 1 ) ( spherical and exponential with the same range and sillspherical and exponential with the same sill and the same initial slope Kriging Interpolation KrigingInterpolation Kriging is named after the South African engineer, D. G. Krige, who first developed the method. Kriging uses the semivariogram, in calculating estimates of the surface at the grid nodes.KrigingInterpolation The procedures involved in kriging incorporate measures of error and uncertainty when determining estimations. In the kriging method, every known data value and every missing data value has an associated variance. If C is constant (i.e. known value exactly), its variance is zero. Based on the semivariogram used, optimal weights are assigned to known values in order to calculate unknown ones. Since the variogram changes with distance, the weights depend on the known sample distribution. Ordinary Kriging Ordinary Kriging Ordinary kriging is the simplest form of kriging. It uses dimensionless points to estimate other dimensionless points, e.g. elevation contour plots. In Ordinary kriging, the regionalized variable is assumed to be stationary. Punctual (Ordinary) Kriging In our case Z, at point p, Ze (p) to be calculated usingaweightedaverageoftheknownvalues or control points: ) ( ) (i i ep z w p z = This estimated value will most likely differ from the actual valueatpointp,Za(p),andthisdifferenceiscalledthe estimation error: ) ( ) ( p z p za e p = cPunctual (Ordinary) Kriging Ifnodriftexistsandtheweightsusedinthe estimation sum to one, then the estimated value issaidtobeunbiased.Thescatterofthe estimatesaboutthetruevalueistermedthe error or estimation variance, np z p znii i a iez==122)] ( ) ( [Punctual (Ordinary) Kriging kriging tries to choose the optimal weights that produce the minimum estimation error . Optimal weights, those that produce unbiased estimates and have a minimum estimation variance, are obtained by solving a set of simultaneous equations . ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 3 33 3 32 2 31 12 23 3 22 2 21 11 13 3 12 2 11 1ppph h w h w h wh h w h w h wh h w h w h w= + += + += + +13 2 1= + + w w wPunctual (Ordinary) Kriging A fourth variable is introduced called the Lagrange multiplier1) ( ) ( ) ( ) () ( ) ( ) ( ) () ( ) ( ) ( ) (3 2 13 33 3 32 2 31 12 23 3 22 2 21 11 13 3 12 2 11 1= + + = + + + = + + + = + + + w w wh h w h w h wh h w h w h wh h w h w h wppp(((((

=(((((

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1) () () (0 1 1 11 ) ( ) ( ) (1 ) ( ) ( ) (1 ) ( ) ( ) (32132133 32 3123 21 2113 12 11ppphhhwwwh h hh h hh h hPunctual (Ordinary) Kriging Once the individual weights are known, an estimation can be made by3 3 2 2 1 1) ( z w z w z w p ze+ + = And an estimation variance can be calculated by ) ( ) ( ) ( 3 13 21 2 1 12+ + + =p p p zh w h w h w