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Lecture 14

2. – Spatial Interpolation –Part 1 Chapter 12
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3. INTERPOLATION
Procedure to predict values of attributes at unsampled points
Why?
Can’t measure all locations:
Time
Money
Impossible (physical- legal)
Changing cell size
Missing/unsuitable data
Past date (eg. temperature)
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4. Sampling Sampling Techniques:
A shortcut method for investigating a whole population
Data is gathered on a small part of the whole parent population or sampling frame, and used to inform what the whole picture is like
Techniques:
Systematic
Random
Cluster
Adaptive
Stratified
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5. Difficult to stay on lines
Systematic sampling pattern
Easy
Samples spaced uniformly at fixed X, Y intervals
Parallel lines
Advantages
Easy to understand
Disadvantages
All receive same attention
Difficult to stay on lines
May be biases
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6. Advantages Disadvantages Random Sampling
Select point based on random number process
Plot on map
Visit sample
Advantages
Less biased (unlikely to match pattern in landscape)
Disadvantages
Does nothing to distribute samples in areas of high
Difficult to explain, location of points may be a problem
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7. Advantages Cluster Sampling
Cluster centers are established (random or systematic)
Samples arranged around each center
Plot on map
Visit sample
(e.g. US Forest Service, Forest Inventory Analysis (FIA)
Clusters located at random then systematic pattern of samples at that location)
Advantages
Reduced travel time
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8. Advantages Disadvantages Adaptive sampling
More sampling where there is more variability.
Need prior knowledge of variability, e.g. two stage sampling
Advantages
More efficient, homogeneous areas have few samples, better representation of variable areas.
Disadvantages
Need prior information on variability through space
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9. Stratified
A wide range of data and fieldwork situations can lend themselves to this approach - wherever there are two study areas being compared, for example two woodlands, river catchments, rock types or a population with sub-sets of known size, for example woodland with distinctly different habitats.
Random point, line or area techniques can be used as long as the number of measurements taken is in proportion to the size of the whole.
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10. Example
if an area of woodland was the study site, there would likely be different types of habitat (sub-sets) within it. Random sampling may altogether ‘miss' one or more of these.
Stratified sampling would take into account the proportional area of each habitat type within the woodland and then each could be sampled accordingly; if 20 samples were to be taken in the woodland as a whole, and it was found that a shrubby clearing accounted for 10% of the total area, two samples would need to be taken within the clearing. The sample points could still be identified randomly (A) or systematically (B) within each separate area of woodland.
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11. INTERPOLATION
Many methods - All combine information about the sample coordinates with the magnitude of the measurement variable to estimate the variable of interest at the unmeasured location
Methods differ in weighting and number of observations used
Different methods produce different results
No single method has been shown to be more accurate in every application
Accuracy is judged by withheld sample points
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12. INTERPOLATION Raster surface
Outputs typically:
Raster surface
Values are measured at a set of sample points
Raster layer boundaries and cell dimensions established
Interpolation method estimate the value for the center of each unmeasured grid cell
Contour Lines
Iterative process
From the sample points estimate points of a value Connect these points to form a line
Estimate the next value, creating another line with the restriction that lines of different values do not cross.
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13. Sampled locations and values
Example Base
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Elevation contours
Sampled locations and values

14. INTERPOLATION 1st Method - Thiessen Polygon
Assigns interpolated value equal to the value found at the nearest sample location
Conceptually simplest method
Only one point used (nearest)
Often called nearest sample or nearest neighbor
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15. INTERPOLATION Thiessen Polygon Advantage: Ease of application
Accuracy depends largely on sampling density
Boundaries often odd shaped as transitions between polygons
are often abrupt
Continuous variables often not well represented
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16. Thiessen Polygon
Draw lines connecting the points to their nearest neighbors.
Find the bisectors of each line.
Connect the bisectors of the lines and assign the resulting polygon the value of the center point
Source:
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17. Thiessen Polygon
Start: 1) 2) 3)
Draw lines connecting the points to their nearest neighbors.
Find the bisectors of each line.
Connect the bisectors of the lines and assign the resulting polygon the value of the center point 1 2 3 5 4
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23. Sampled locations and values Thiessen polygons
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24. INTERPOLATION Fixed-Radius – Local Averaging
More complex than nearest sample
Cell values estimated based on the average of nearby samples
Samples used depend on search radius
(any sample found inside the circle is used in average, outside ignored)
Specify output raster grid
Fixed-radius circle is centered over a raster cell
Circle radius typically equals several raster cell widths
(causes neighboring cell values to be similar)
Several sample points used
Some circles many contain no points
Search radius important; too large may smooth the data too much
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25. Fixed-Radius – Local Averaging
INTERPOLATION
Fixed-Radius – Local Averaging
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26. Fixed-Radius – Local Averaging
INTERPOLATION
Fixed-Radius – Local Averaging
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27. Fixed-Radius – Local Averaging
INTERPOLATION
Fixed-Radius – Local Averaging
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28. INTERPOLATION Inverse Distance Weighted (IDW)
Estimates the values at unknown points using the distance and values to nearby know points (IDW reduces the contribution of a known point to the interpolated value)
Weight of each sample point is an inverse proportion to the distance.
The further away the point, the less the weight in helping define the unsampled location
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29. INTERPOLATION Inverse Distance Weighted (IDW)
Zi is value of known point
Dij is distance to known point
Zj is the unknown point
n is a user selected exponent
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30. INTERPOLATION
Inverse Distance Weighted (IDW)
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31. INTERPOLATION Inverse Distance Weighted (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
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32. INTERPOLATION
Inverse Distance Weighted (IDW)
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33. INTERPOLATION
Inverse Distance Weighted (IDW)
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34. INTERPOLATION Splines
Name derived from the drafting tool, a flexible ruler, that helps create smooth curves through several points
Spline functions are use to interpolate along a smooth curve.
Force a smooth line to pass through a desired set of points
Constructed from a set of joined polynomial functions
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35. Spline Surface created with Spline interpolation
Passes through each sample point
May exceed the value range of the sample point set
                                                                                                                                                                                                                             
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36. INTERPOLATION : Splines
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37. Interpolation vs. Prediction
Spatial prediction is more general than spatial interpolation.
Both are used to estimate values of a variable at unknown locations.
Interpolation use only the measured target variable and sample coordinates to estimate the variable at unknown locations.
Prediction methods address the presence of spatial autocorrelation.
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38. Tobler’s Law Tobler's First Law of Geography.
A formulation of the concept of spatial autocorrelation by the geographer Waldo Tobler (1930-), which states:
"Everything is related to everything else, but near things are more related than distant things."
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39. Trend Surface Interpolation
Fitting a statistical model, a trend surface, through the measured points. (typically polynomial)
Where Z is the value at any point x
Where ais are coefficients estimated in a regression model
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40. Trend Surface Interpolation
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41. Global Mathematical Functions Polynomial Trend Surface
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42. Global Mathematical Functions Polynomial Trend Surface
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43. Similar to Inverse Distance Weighting (IDW)
INTERPOLATION
Kriging
Similar to Inverse Distance Weighting (IDW)
Kriging uses the minimum variance method to calculate the weights rather than applying an arbitrary or less precise weighting scheme
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44. Cross-correlation – two variables change in concert (positive or negative)
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45. Prediction Kriging Method relies on spatial autocorrelation
Higher autocorrelations, points near each other are alike.
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46. Similar to Inverse Distance Weighting (IDW)
Prediction
Kriging
Similar to Inverse Distance Weighting (IDW)
A statistically based estimator of spatial variables
Kriging uses the minimum variance method to calculate the weights rather than applying an arbitrary or less precise weighting scheme
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47. INTERPOLATION Kriging
A statistical based estimator of spatial variables
Components:
Spatial trend
Autocorrelation
Random variation
Creates a mathematical model which is used to estimate values across the surface
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48. Semivariogram
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49. Lag Distance Kriging uses the concept of lag distance
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50. Lag Distance and Autocorelataion
The semivariance at a given lag distance is a measure of the spatial autocorrelation at that distance.
The variogram is used to determine the weights used in the kriging process.
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51. SemiVariogram
Range:
The distance where the model first flattens out is known as the range.
Sample locations separated by distances closer than the range are spatially autocorrelated, whereas locations farther apart than the range are not.
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52. SemiVariogram Nugget:
Theoretically, at zero separation distance (lag = 0), the semivariogram value is 0. However, at an infinitesimally small separation distance, the semivariogram often exhibits a nugget effect, which is some value greater than 0.
The nugget effect can be attributed to measurement errors or spatial sources of variation at distances smaller than the sampling interval or both.
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53. Semivariogram Nugget Effect:
Measurement error occurs because of the error inherent in measuring devices.
Natural phenomena can vary spatially over a range of scales.
Variation at microscales smaller than the sampling distances will appear as part of the nugget effect.
Before collecting data, it is important to gain some understanding of the scales of spatial variation.
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54. Types of Kriging
Depending on the stochastic properties of the random field and the various degrees of stationarity assumed, different methods for calculating the weights can be deducted, i.e. different types of kriging apply.
Classical methods are:
Ordinary Kriging assumes stationarity (flat/no trend)of the first moment of all random variables
Simple Kriging assumes a known stationary mean.
Universal Kriging assumes a general polynomial trend model, such as linear trend model .
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55. Kriging
A surface created with Kriging can exceed the value range of the sample points but will not pass through the points.
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56. Lecture 14

57. INTERPOLATION (cont.) Exact/Non Exact methods
Exact – predicted values equal observed
Theissen
IDW
Spline
Non Exact-predicted values might not equal observed
Fixed-Radius
Trend surface
Kriging
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58. INTERPOLATION Kriging
A statistical based estimator of spatial variables
Components:
Spatial trend
Autocorrelation
Random variation
Creates a mathematical model which is used to estimate values across the surface
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59. Lecture 14

60. INTERPOLATION (cont.) Exact/Non Exact methods
Exact – predicted values equal observed
Theissen
IDW
Spline
Non Exact-predicted values might not equal observed
Fixed-Radius
Trend surface
Kriging
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61. Which method works best for this example?
Thiessen
Polygons
Fixed-radius –
Local Averaging
IDW: squared,
12 nearest points
Original Surface:
Trend Surface
Spline
Kriging
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62. Interpolation vs. Prediction
Spatial prediction is more general than spatial interpolation.
Both are used to estimate values of a variable at unknown locations.
Interpolation use only the measured target variable and sample coordinates to estimate the variable at unknown locations.
Prediction methods address the presence of spatial autocorrelation.
Lecture 14

63. Tobler’s Law Tobler's First Law of Geography.
A formulation of the concept of spatial autocorrelation by the geographer Waldo Tobler (1930-), which states:
"Everything is related to everything else, but near things are more related than distant things."
Lecture 14

64. Prediction Kriging Method relies on spatial autocorrelation
Higher autocorrelations, points near each other are alike.
Lecture 14

65. Cross-correlation – two variables change in concert (positive or negative)
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66. Similar to Inverse Distance Weighting (IDW)
Prediction
Kriging
Similar to Inverse Distance Weighting (IDW)
A statistically based estimator of spatial variables
Kriging uses the minimum variance method to calculate the weights rather than applying an arbitrary or less precise weighting scheme
Lecture 14

67. INTERPOLATION Kriging
A statistical based estimator of spatial variables
Components:
Spatial trend
Autocorrelation
Random variation
Creates a mathematical model which is used to estimate values across the surface
Lecture 14

68. Lag Distance Kriging uses the concept of lag distance
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69. Semivariogram
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70. SemiVariogram
Range:
The distance where the model first flattens out is known as the range.
Sample locations separated by distances closer than the range are spatially autocorrelated, whereas locations farther apart than the range are not.
Lecture 14

71. SemiVariogram Nugget:
Theoretically, at zero separation distance (lag = 0), the semivariogram value is 0. However, at an infinitesimally small separation distance, the semivariogram often exhibits a nugget effect, which is some value greater than 0.
The nugget effect can be attributed to measurement errors or spatial sources of variation at distances smaller than the sampling interval or both.
Lecture 14

72. Semivariogram Nugget Effect:
Measurement error occurs because of the error inherent in measuring devices.
Natural phenomena can vary spatially over a range of scales.
Variation at microscales smaller than the sampling distances will appear as part of the nugget effect.
Before collecting data, it is important to gain some understanding of the scales of spatial variation.
Lecture 14

73. Lag Distance and Autocorelataion
The semivariance at a given lag distance is a measure of the spatial autocorrelation at that distance.
The variogram is used to determine the weights used in the kriging process.
Lecture 14

74. Types of Kriging
Depending on the stochastic properties of the random field and the various degrees of stationarity assumed, different methods for calculating the weights can be deducted, i.e. different types of kriging apply.
Classical methods are:
Ordinary Kriging assumes stationarity (flat/no trend)of the first moment of all random variables
Simple Kriging assumes a known stationary mean.
Universal Kriging assumes a general polynomial trend model, such as linear trend model .
Lecture 14

75. Kriging
A surface created with Kriging can exceed the value range of the sample points but will not pass through the points.
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76. Lecture 14

77. INTERPOLATION (cont.) Exact/Non Exact methods
Exact – predicted values equal observed
Theissen
IDW
Spline
Non Exact-predicted values might not equal observed
Fixed-Radius
Trend surface
Kriging
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