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Applications in GIS (Kriging Interpolation)

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

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Applications in GIS (Kriging Interpolation)

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  1. Applications in GIS(Kriging Interpolation) Dr. S.M. Malaek Assistant: M. Younesi

  2. Interpolating a Surface From Sampled Point Data

  3. 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

  4. Estimating a point here:interpolation Sample data Interpolating a Surface From Sampled Point Data Interpolation

  5. Sample data Estimating a point here:extrapolation Interpolating a Surface From Sampled Point Data Extrapolation

  6. Interpolating a Surface From Sampled Point Data • Sampling Strategies for Interpolation Regular Sampling Random Sampling

  7. If A = 8 feet and B = 4 feet then C = (8 + 4) / 2 = 6 feet Sample elevation data A C B Elevation profile Interpolating a Surface From Sampled Point Data Linear Interpolation

  8. Sample elevation data Often results in a more realistic interpolation but estimating missing data values is more complex A C B Elevation profile Interpolating a Surface From Sampled Point Data Non-Linear Interpolation

  9. Sample data Interpolating a Surface From Sampled Point Data Global Interpolation Uses all known sample points to estimate a value at an unsampled location

  10. Sample data Uses a local neighborhood to estimate value, i.e. closest n number of points, or within a given search radius Interpolating a Surface From Sampled Point Data Local Interpolation Uses a neighborhood of sample points to estimate a value at an unsampled location

  11. Trend Surface

  12. 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]

  13. z z = b0 + b1x + e x Trend Surface 1st Order Trend Surface In one dimension:z varies as a linear function ofx

  14. z y z = b0 + b1x + b2y + e x Trend Surface 1st Order Trend Surface In two dimensions:z varies as a linear function of xandy

  15. Trend Surface

  16. Inverse Distance Weighted (IDW)

  17. 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

  18. 100 IDW: Closest 3 neighbors, r = 2 4 3 160 2 200 Inverse Distance Weighted(Example)

  19. Weights A BC 1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500 A = 100 4 B = 160 3 2 C = 200 Inverse Distance Weighted(Example)

  20. Weights Weights * Value A BC 1 / (42) = .0625 1 / (32) = .1111 1 / (22) = .2500 .0625 * 100 = 6.25 .1111 * 160 = 17.76 .2500 * 200 = 50.00 Total = .4236 A = 100 4 6.25 +17.76 + 50.00 = 74.01 B = 160 3 74.01 / .4236 = 175 2 C = 200 Inverse Distance Weighted(Example)

  21. Geostatistics

  22. 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.

  23. Geostatistics • Regionalized variables describe phenomena with geographical distribution (e.g. elevation of ground surface). • The phenomenon exhibit spatial continuity.

  24. 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.

  25. Semivariance

  26. Semivariance • Regionalized variable theory uses a related property called the semivariance to express the degree of relationship between points on a surface. • Thesemivariance is simply half the varianceof the differences between all possible points spaced a constant distance apart. Semivariance is a measure of the degree of spatial dependence between samples (elevation(

  27. 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.

  28. 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):

  29. Semivariance • Ziis the measurement of a regionalized variable taken at location i , • Zi+his another measurement taken h intervals awayd • Nhis number of separating distance = number of points –Lag (if the points are located in a single profile)

  30. Calculating the Semivariance(IrregularlySpaced 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 separated in E-W direction and in S-N direction.

  31. Calculating the Semivariance(IrregularlySpaced 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

  32. Variogram

  33. Variogram • The plot of the semivariances as a function of distance from a point is referred to as a semivariogram orvariogram.

  34. 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.

  35. 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.

  36. 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.

  37. Variogram (Models( Exponential Model spherical and exponential with the same range and sill spherical and exponential with the same sill and the same initial slope

  38. Kriging Interpolation

  39. Kriging Interpolation • 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.

  40. Kriging Interpolation • 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.

  41. Ordinary Kriging

  42. 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.

  43. Punctual (Ordinary) Kriging • In our case Z, at point p, Ze (p) to be calculated using a weighted average of the known values or control points: • This estimated value will most likely differ from the actual value at point p, Za(p), and this difference is called the estimation error:

  44. Punctual (Ordinary) Kriging • If no drift exists and the weights used in the estimation sum to one, then the estimated value is said to be unbiased. The scatter of the estimates about the true value is termed the error or estimation variance,

  45. 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.

  46. Punctual (Ordinary) Kriging • A fourth variable is introduced called the Lagrange multiplier

  47. Punctual (Ordinary) Kriging • Once the individual weights are known, an estimation can be made by • And an estimation variance can be calculated by

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