1. Spatial Interpolation
Inverse Distance Weighting
The Variogram
Kriging
Much thanks to Bill Harper for his insights in Practical Geostatistics and personal conversation

2. Objectives In this session we will evaluate a dataset and attempt to:
Explore the theory and implementation of inverse distance weighting
Evaluate issues with IDW interpolation
Explore the theory and implementation of the semi-variogram and it’s applicability to interpolation
Explore the theory and implementation of kriging and it’s applicability to interpolation

3. Data Set Simulated Borehole data (PG 2000) General Statistics
Iron concentration
Need to interpolate iron content for unsampled areas
General Statistics
47 samples
Mean value: 36.3
S.D.: 3.73

4. General Statistics
Histogram shows the relative distribution of the data
Generally follows a normal distribution
Other observations
Minor skew, no big deal

5. Data Set
The best unbiased estimate for the standard deviation is (see right)
Therefore, we are 90% confident that a point drawn at random would be:
30 < T < 42.6
This is based on consulting a students t distribution with 47 samples

6. Subset of Area (northwest area)
Subset of borehole data
Upper left side
General Statistics
7 samples
Mean value: 40
S.D.: 2.82
Getting somewhat better

7. Therefore, we are 90% confident that a point drawn at random would be:
The best unbiased estimate for the standard deviation is (see right)
Therefore, we are 90% confident that a point drawn at random would be:
34.2 < T < 45.7
This is based on consulting a students t distribution with 7 samples
Now, the question is, do some of the points exhibit more influence than others?
Probably, so lets evaluate the point taking nearness into account

8. Inverse Distance Weighting
IDW works by using an unbiased weight matrix based on the distances from an unknown value to known values.
Weights may be defined a number of different ways

9. IDW ArcGIS provides a nice interface to view points
This example looks at 7 neighbors
Now, lets look at it the “old fashioned way…”

10. IDW
Using 7 neighboring points allows us to interpolate a value based on distances
Interpolated value is 39.9
So, our calculation is the same as that in ArcGIS – its just math….

11. IDW – standard Error
We will compute it, without considering the autocorrelation in the data:
Standard error 2.75
Therefore, we are 90% confident that a point drawn at random would be:
34.7 < T < 45.1
This is based on consulting a students t distribution with 7 samples
Caveat: we are treating IDW like weighted mean, and the standard deviation like a weighted standard deviation. In reality, you shouldn’t develop confidence intervals for data that is autocorrelated

12. IDW Methods So which is best??? Power = 2, search = 230

13. 10 Questions to Evaluate1 What function of distance should we use?
How do we handle different continuity in different directions?
How many samples should we include in the estimation?
How do we compensate for irregularly spaced or highly clustered sampling?
How far should we go to include samples in our estimation process?
Should we honor the sample values?
How reliable is the estimate when we have it?
Why is our map too smooth?
What happens if our sample data is not Normal?
What happens if there is a strong trend in the values?
1Clark and Harper Practical Geostatistics Ecosse North America, Llc

14. Answering the 10 Questions
The Variogram

15. What is a Semi-Variogram
The semi-variogram is a function that relates semi- variance (or dissimilarity) of data points to the distance that separates them.
If we can understand the difference between an unknown quantity and a known quantity, we we can estimate the unknown point 1 d1

16. Estimating via semi-variogram
Lets assume the relationship between the unknown and known point depends on distance – 121 feet NE/SW
If these two points have the same relationship as the other points, we can look at the other points that are 121 feet NE/SW

17. Computing the standard differences
For all 31 pairs we can compute the standard deviation
We are assuming a mean of 0, and a normal distribution

18. Computing the standard differences
The single point we are looking at is 37% Fe.
If our original samples come from a normal distribution, the differences will be normal, so we be 90% confident that a point drawn at random would be:

19. Taking the semi-variogram further
Chances are, we won’t get to sample our data on a regular grid.
We have to algebraically define some function of distance with the differences in value
Therefore, we will assign h to the distance

20. Variograms å Variogram: g(h) = ½ var [ Z(x) – Z(x+h) ]
= ½ E [ {Z(x) – Z(x+h)}2 ]
In practice:
g(h) =
Where:
N(h) is the total number of
pairs of observations
separated by a distance h.
The fitted curve minimizes
the variance of the errors. å = + -
N(h) 1 i 2
h)]
Z(xi
[Z(xi)
2N(h)

21. Variogram components
Nugget variance: a non-zero value for g when h = 0. Produced by various sources of unexplained error (e.g. measurement error).
Sill: for large values of h the variogram levels out, indicating that there no longer is any correlation between data points. The sill should be equal to the variance of the data set.
Range: is the value of h where the sill occurs (or 95% of the value of the sill).
In general, 30 or more pairs per point are needed to generate a reasonable sample variogram.
The most important part of a variogram is its shape near the origin, as the closest points are given more weight in the interpolation process.

22. Variogram models
Variogram models must be “positive definite” so that the covariance matrix based on it can be inverted (which occurs in the kriging process). Because of this, only certain models can be used.

23. Semi-variogram models
We can enter some numbers in Mathcad and see how the variogram changes.

24. Effect of lag size on variograms
Variogram with a lag size of 5m and a lag tolerance of 2.5m.
Variogram with a lag size of 10m and a lag tolerance of 5m.

25. Anisotropy
There may be higher spatial autocorrelation in one direction than in others, which is called anisotropy:
The figure shows a case of geometric anisotropy, which is incorporated in the variogram model by means of a linear transformation.

26. Semi-variogram tips We are assuming a normal distribution
Gives us a picture of the relationship of data values with distance.
If you don’t have a good spatial structure in the semi-variogram, don’t revert to IDW – this is stupid!!!

27. Comparing Software for Computing the Semi-Variogram
Practical Geostatistics 2000
ArcGIS Geostatistical Analyst

28. Assessing Fit of the Variogram
Cressie Goodness of Fit
For each point used to create the variogram, match how well the model actually fits it

29. Kriging 3 components: structural (constant
Kriging is based on the idea that you can make inferences regarding a random function Z(x), given data points Z(x1), Z(x2), …Z(xn).
3 components: structural (constant
mean), random spatially correlated
component and residual error.
Z(x) = m(x) + g(h) + e”

30. Kriging This is our variogram from the borehole data
To discuss the mathematics of kriging, we will look at a simple example of 3 points, and get back to our data in a moment

31. Numerical Example of Iron Ore Data From Practical Geostatistics 2000
Kriging
Numerical Example of Iron Ore Data
From Practical Geostatistics 2000

32. Data Set Iron Ore Data, based on sample set from PG 2000
Three point example for simplicity

33. Calculating Distances
The first thing we do is determine the distances between each point
Also calculate difference in Z values between all points

34. Semi Variogram
We apply the GLM, based on other test performed on the data
The values chosen give the best Cressie statistics for fit on all data points
Note: Mathcad is not great at creating semivariograms!!!

35. Computing Weights
Using basic matrix algebra, we can solve for the weights.
The weights will add to one, due to our eventual “slight of hand” with the last row.

36. Solving the Unknown
Basic matrix algebra will solve for the unknown value
We also compute the standard error and variance

37. Solving Our Borehole Data
Start with our original example
Since we have 7 points rather than 3, the screens will be “busier”

38. Borehole Data
The ability to create semi-variograms in MathCad is pretty bad, but this allows us to visualize the mathematics
Here we are using the spherical model

39. Borehole Data
Again, we can see with this dataset the weights also add up to one

40. Solution
Here we’ve computed the value of the unknown point, and the standard error
This was based on the limited set of 7 points, now we’ll do it with the rest.

41. Predicting the Point
ArcGIS has a good interface for evaluating the weights of the points, in addition to predicting a test location

42. Kriging Results ESRI Geostatistical Analyst PG 2000 Interpolated value
41.26
Standard error
2.16
PG 2000
41.14
2.11

43. Standard Errors
Based on Kriging results, we can assume the “true” value of the unknown point, with 90% confidence as:
37.6 < < %Fe
So, we are getting better results, better looking maps, and smaller confidence intervals

44. IDW vs. Kriging Kriging IDW
Kriging appears to give a more “natural” look to the data
Kriging avoids the “bulls eye” effect
Kriging also give us a standard error
IDW

45. Results

46. Review of 10 Questions to ask1
What function of distance should we use?
The variogram shows us the spatial structure, and association of the data, and will give us a hint as to what function to possibly use.
How do we handle different continuity in different directions?
Here again, the variogram will tell us whether there is any spatial association, and we can determine which direction by evaluating whether anisotropy exists.
How many samples should we include in the estimation?
Again, we can look at the variogram
How do we compensate for irregularly spaced or highly clustered sampling?
The variogram defines the relationship between points and their distances from other points. Calculating weights in Kriging takes the distances among all points into account.
1Clark and Harper Practical Geostatistics Ecosse North America, Llc

47. 10 Questions to ask1
How far should we go to include samples in our estimation process?
By looking at the variogram we can identify the sill (that area where the spatial correlation has little value). The range tells us the distance where the points are no longer correlated.
Should we honor the sample values?
Still lots of debate on this one. IDW says yes, that’s why we get the bullseye. The nugget effect in Kriging allows us to say no. But, we can set the nugget to zero with Kriging.
How reliable is the estimate when we have it?
Kriging allows us to compute the standard error
Why is our IDW map too smooth?
In IDW when you include points far away they become part of the weights. Since the weights have to add up to one, you are basically taking power away from the closer ones.
1Clark and Harper Practical Geostatistics Ecosse North America, Llc

48. 10 Questions to Ask What happens if our sample data is not Normal?
Basically, make the data normal…
What happens if there is a strong trend in the values?
First, remove the trend, then re-interpolate the points (see ESRI Calif. Ozone example, or Clark and Harper Wolfcamp Data)

49. Conclusions
It is possible to interpolate an unknown point based on other points in a data set
While it can be done with descriptive statistics, other methods are clearly better
The variogram helps answer many questions related to our data, and provides a wealth of information related to the spatial structure of the data
More robust (geostatistical) methods for interpolation appear to provide better results