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

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

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

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

8. 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

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

10. Local Interpolation Interpolating a Surface From Sampled Point Data
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

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

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

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

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

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

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.
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(

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
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)

30. Semivariance
Sill: The distance at which the semivariance approaches a flat region. sill, is referred as the range or span of the regionalized variable.
Range: The range or span defines a neighborhood within which all data points are related to one another.

31. 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 separated in E-W direction and in S-N direction.

32. 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

33. Variogram

34. Variogram
The plot of the semivariances as a function of distance from a point is referred to as a semivariogram or variogram.

35. 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.

36. 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.

37. a is called the range of influence of a sample.
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.

38. spherical and exponential with the same range and sill
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

39. Kriging
Interpolation

40. 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.

41. 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.

42. Ordinary Kriging

43. 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.

44. 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:

45. 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,

46. 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 .

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

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