1. WFM 6202: Remote Sensing and GIS in Water Management
Geographic Information System (GIS)
Lecture 6
Geo-spatial Interpolation
Dr. A.K.M. Saiful Islam
Institute of Water and Flood Management (IWFM)
Bangladesh University of Engineering and Technology (BUET)
January, 2016

2. 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.

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

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

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

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

7. 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

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

9. 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

10. Interpolation examples
Our interpolated surface (represented in 1-D by the blue line) would look like this

11. Interpolation examples: Sample rate
Here our interpolated surface is much closer to reality at the local level, but we pay for this in the form of higher data gathering cost

12. Trend Surface

13. Interpolation examples
Elevation:
Source: LUBOS MITAS AND HELENA MITASOVA, University of Illinois
©2005 Austin Troy

14. 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]

15. 1st Order Trend Surface Trend Surface
In one dimension: z varies as a linear function of x x z
z = b0 + b1x + e

16. 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

17. Surface interpolation
Let say we have our ground water pollution samples
This gives us

18. Trend Surface

19. Example: Approximate Interpolation
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.
Equation
Slope
intercept

20. Inverse Distance Weighted (IDW)

21. 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

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

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

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

25. Geostatistics

26. 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.

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

28. 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.

29. Semivariance

30. 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(

31. 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)

32. 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.

33. 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)

34. Calculating the Semivariance (Irregularly Spaced Points Regularly 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.

35. Calculating the Semivariance (Irregularly Spaced Points Regularly 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

36. Variogram

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

38. 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.

39. 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.

40. 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.

41. 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

42. Kriging
Interpolation

43. 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.

44. 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.

45. 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:
Ordinary Kriging models:
Spherical, Circular, Exponential,
Gaussian and Linear.
Universal Kriging models:
Linear with Linear drift, and
Linear with Quadratic drift

46. Ordinary Kriging

47. 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.

48. 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:

49. 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,

50. 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 .

51. Ordinary Kriging
A fourth variable is introduced called the Lagrange multiplier

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

54. 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.

55. Thiessen polygons
This method builds polygons, rather than a raster surface, from control points
“grows” polygons around sample points that are supposed to represent areas of homogeneity
Source: Jens-Ulrich Nomme

56. Density Functions
We can also use sample points to map out density raster surfaces. This need to require a z value in each, it can simply be based on the abundance and distribution of points.

57. Lecture 11
Density Functions
These settings would give us a raster density surface, based just on the abundance of points within a “kernel” or data frame. In this case, a z value for each point is not necessary.

58. Spline Method Another option for interpolation method
This fits a curve through the sample data assign values to other locations based on their location on the curve
Thin plate splines create a surface that passes through sample points with the least possible change in slope at all points, that is with a minimum curvature surface
SPLINE has two types: regularized and tension
Tension results in a rougher surface that more closely adheres to abrupt changes in sample points
Regularized results in a smoother surface that smoothes out abruptly changing values somewhat

59. Surface Fitting for random Points
Triangular network called as Triangulated Irregular Network (TIN) is applied

60. 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 1985.
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.

61. Example
Here are some sample elevation points from which surfaces were derived using the three methods
©2005 Austin Troy

62. Example: Spline
Note how smooth the curves of the terrain are; this is because Spline is fitting a simply polynomial equation through the points
©2005 Austin Troy

63. Example: IDW
Done with P =2. Notice how it is not as smooth as Spline. This is because of the weighting function introduced through P
©2005 Austin Troy

64. Example: Kriging
This one is kind of in between—because it fits an equation through point, but weights it based on probabilities
©2005 Austin Troy

65. Geo-statistical Analyst of ArcGIS
This training will be on:
Histogram
Normal QQ plot
Trend Analysis
Creating a prediction map using the geo-statistical wizard
Semivariogram / covariance modeling
Searching neighbor
Creating a prediction standard error map
Display Formats
Input Data
Groundwater well data of Dinajpur district of Bangladesh

66. Steps in Geo-statistical Analyst
Representation of the Data
Explore the Data
Fit a model (create surface)
Perform Diagnostics

67. Representation of Data
Representing the data is a vital first step in assessing the validity of the data and identifying external factors that may ultimately play a role in the distribution of data.

68. Explore the Data
Distribution of the data, looking for data trends, looking for global and local outliers, examining spatial autocorrelation, understanding the co-variation among multiple data sets.

69. Explore data Histogram Q-Q plot Trend Analysis Semivariogram
Voronoi map
Cross covariance

70. Histogram
Show frequency distribution as a bar graph that displays how often observed values fall within certain intervals or classes

71. Normal distribution Skewness is zero for normal distribution
Normally distributed
Positively skewed

72. Q-Q Plot
Normal QQ Plot is created by plotting data values with the value of a standard normal where their cumulative distributions are equal

73. Trend Analysis
The Trend Analysis tool provides a three-dimensional perspective of the data.
The locations of sample points are plotted on the x,y plane. Above each sample point, the value is given by the height of a stick in the z dimension.
The unique feature of the Trend Analysis tool is that the values are then projected onto the x,z plane and the y,z plane as scatter plots.
This can be thought of as sideways views through the three-dimensional data.
Polynomials are then fit through the scatter plots on the projected planes.

74. Voronoi map
Voronoi maps are constructed from a series of polygons formed around the location of a sample point. Voronoi polygons are created so that every location within a polygon is closer to the sample point in that polygon than any other sample point. After the polygons are created, neighbors of a sample point are defined as any other sample point whose polygon shares a border with the chosen sample point.
For example, in the following figure, the bright green sample point is enclosed by a polygon, given as red. Every location within the red polygon is closer to the bright green sample point than any other sample point (given as small dark blue dots). The blue polygons all share a border with the red polygon, so the sample points within the blue polygons are neighbors of the bright green sample point.

75. Cross variance
The Crosscovariance cloud shows the empirical crosscovariance for all pairs of locations between two datasets and plots them as a function of the distance between the two locations.

76. Fit a Model
A wide variety of interpolation methods available to create surface.
Two main groups of interpolation techniques:
1. Deterministic
2. Geo-statistical

77. Interpolation techniques
1. Deterministic: is used for creating surfaces from measures points based either on extent of similarity (Inverse Distance Weighted (IDW) or the degree of smoothing (radial basis functions and polynomials)
2. Geo-statistical: is based on statistics and is used for more advanced prediction of surface modeling that also includes errors or uncertainty of prediction.

78. Deterministic Methods
Four types:
Inverse Distance Weighted (IDW)
Global Polynomial
Local Polynomial
Radial Basis Functions
Can classified into two groups:
Global uses entire data set
Global polynomial
Local calculates prediction from measured point with specified neighbors:
IDW, local polynomials, radial basis functions

79. Inverse Distance Weighted (IDW)
A window of circular shape with the radius of dmax is drawn at a point to be interpolated, so as to involve six to eight surrounding observed points.

80. Global polynomial interpolation
Global Polynomial interpolation fits a smooth surface that is defined by a mathematical function (a polynomial) to the input sample points. The Global Polynomial surface changes gradually and captures coarse-scale pattern in the data.
Conceptually, Global Polynomial interpolation is like taking a piece of paper and fitting it between the raised points (raised to the height of value). This is demonstrated in the diagram below for a set of sample points of elevation taken on a gently sloping hill (the piece of paper is magenta).

81. Local Polynomial interpolation
While Global Polynomial interpolation fits a polynomial to the entire surface, Local Polynomial interpolation fits many polynomials, each within specified overlapping neighborhoods. The search neighborhood can be defined using the search neighborhood dialog

82. Radial Basis Functions (RBF)
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, completely regularized spline, multi-quadric function, and inverse multi-quadric function.
RBF methods are a form of artificial neural networks.

83. Geo-statistical Methods
Kriging and Co-kriging
Algorithm
Ordinary -A variety of kriging which assumes that local means are not necessarily closely related to the population mean, and which therefore uses only the samples in the local neighbourhood for the estimate. Ordinary kriging is the most comrnonly used method for environmental situations.
Simple - A variety of kriging which assumes that local means are relatively constant and equal to the population mean, which is well-known. The population mean is used as a factor in each local estimate, along with the samples in the local neighborhood. This is not usually the most appropriate method for environmental situations.
Universal -
Indicator
Probability
Disjunctive
Output Surfaces
Prediction and prediction standard error
Quantile
Probability and standard errors of indicators

84. 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.

85. Interpolation using Kriging
Kriging weights

86. 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)

87. Types of semivariogram models
Geostatistical Analyst provides the following functions to choose from to model the empirical semivariogram:
Circular
Spherical
Tetraspherical
Pentaspherical
Exponential
Gaussian
Rational Quadratic
Hole Effect
K-Bessel
J-Bessel
Stable

88. Semi-variogram Models

89. Trend
An example of a global trend can be seen in the effects of the prevailing winds on a smoke stack at a factory (below).
In the image, the higher concentrations of pollution are depicted in the warm colors (reds and yellows) and the lower concentrations in the cool colors (greens and blues).
Notice that the values of the pollutant change more slowly in the east–west direction than in the north–south direction.
This is because east–west is aligned with the wind while north–south is perpendicular to the wind.

90. Detrending tool

91. Anisotropy
Anisotropy is a characteristic of a random process that shows higher autocorrelation in one direction than another.
The following image shows conceptually how the process might look.
Once again, the higher concentrations of pollution are depicted in the warm colors (reds and yellows) and the lower concentrations in the cool colors (greens and blues).
The random process shows undulations that are shorter in one direction than another.

92. Example: Anisotropy and trend
Point X A B C D
Grade
4.1 10 2 1 8
Real distance from X
178
222
397
548
Anisotropic distance from X
249
191
502
591

93. Accounting for Anisotrophy

94. Searching Neighbor
The points highlighted in the data view give an indicator of the weights (absolute value in percent) associated with each point located in the moving window. The weights are used to estimate the value at the unknown location which is at the center of the cross hair.

95. Data transformation

96. Declustering method
There are two ways to decluster your data: by the cell method and by Voronoi polygons. Samples should be taken so they are representative of the entire surface. However, many times the samples are taken where the concentration is most severe, thus skewing the view of the surface. Declustering accounts for skewed representation of the samples by weighting them appropriately so that a more accurate surface can be created.

97. Bi-variate normal distribution

98. Output Surface

99. Cross Validation
Cross-validation uses all of the data to estimate the trend and autocorrelation models. It removes each data location, one at a time, and predicts the associated data value.

100. Various Surface produced using ordinary kriging

101. Model comparison
Comparison helps you determine how good the model that created a geostatistical layer is relative to another model.

102. Display Format Filled contour Contours Grids Combination of contours
Filled contour and hill shade
Hill shade

103. Exercise on Geo-statistical Analyst
Data from 21 Groundwater observation Wells as shape file “gwowell_bwdb.shp”
Weekly data from December to May for 1994 to 2003
Upazilla shape file “upazila.shp”
Tasks:
Represent data
Explore data
Fit Model
Diagnostic output
Create output maps