1. Part 2: Interpolation and geostatistics
Lecture 11
------Using GIS--
Lecture 10:
Part 1: GPS
Part 2: Interpolation and geostatistics
By Austin Troy & Brian Voigt
©2011 All lecture materials by Austin Troy & Brian Voigt except where noted

2. Part 1: Global Positioning System
------Using GIS--
Part 1:
Global Positioning System
Many materials for this part of the lecture adapted from Trimble Navigation Ltd’s GPS Web tutorial at
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

3. GPS Stands for Global Positioning System
Lecture 11
GPS
Stands for Global Positioning System
GPS is used to get an exact location on the surface of the earth, in three dimensions.
GPS is a very important data input source, used for surveying, military operations, engineering, vehicle tracking, flight navigation, car navigation, ship navigation, unmanned vehicle guidance, agriculture, and of course, mapping
For mapping, a GPS tells us “where” and allows us to input “what”
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

4. Lecture 11
GPS
GPS is a worldwide radio-navigation system formed from 24 satellites and their ground stations.
Uses satellites in space as reference points for locations here on earth
Ground stations help satellites determine their exact location in space. There are five monitor stations: Hawaii, Ascension Island, Diego Garcia, Kwajalein, and Colorado Springs.
Source: Wikipedia
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

5. Lecture 11
How does GPS work?
GPS derives position relative to satellite “reference points,” using triangulation
The GPS unit on the ground figures out its distance to each of several satellites using the time it takes for a radio signal to travel to the satellite
To do this, the exact position of the satellites at a given time, must be known; otherwise they can’t serve as reference points
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

6. How does GPS work? y km z km x km Lecture 11
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

7. Lecture 11
How does GPS work?
We need at least 3 satellites as reference points to “triangulate” our position.
Based on the principle that where we know our exact distance from a satellite in space, we know we are somewhere on the surface of an imaginary sphere with radius equal to the distance to the satellite.
With two satellites we know we are in the plane where the two intersect. With three or more, we can get two possible points, and one of those is usually impossible from a practical standpoint and can be discarded
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

8. How does GPS work? Here’s how the sphere concept works
Lecture 11
How does GPS work?
Here’s how the sphere concept works
A fourth satellite narrows it from 2 possible points to 1 point
Source: Trimble Navigation Ltd.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

9. Lecture 11
How does GPS work?
This method assumes we can find exact distance from our GPS receiver to a satellite. How does that work?
Simple answer: see how long it takes for a radio signal to get from the satellite to the receiver.
Since we know speed of light, we can answer this
This gets complicated when you think about the need to perfectly synchronize satellite and receiver.
A tiny error in synchronization can result in hundreds of meters of positional error
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

10. Lecture 11
How does GPS work?
The difficult part is measuring travel time, because the amount of time elapsed is tiny (about .06 seconds for an overhead satellite), and we require a way to know precisely WHEN the signal left the satellite
To do this requires comparing lag in exactly similar patterns, one from satellite and one from receiver.
Analogy: sitting in a stadium 1000 feet from the speaker and pressing “play” on your Walkman (containing an REO Speedwagon cassette, of course) at exactly the same time as the guy in the sound booth presses play for the same song.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

11. How does GPS work? delayed:“I can’t fight this feeling any more,”
Lecture 11
How does GPS work?
delayed:“I can’t fight this feeling any more,”
Local: “I can’t fight this feeling any more,”
Only, instead of using cheesy eighties rock power ballads, GPS uses something called “pseudo-random code.”
This code has to be extremely complex (hence almost random), so that patterns are not linked up at the wrong place on the code—that would generate the wrong time delay and hence the wrong distance
Source: Trimble Navigation Ltd.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

12. Lecture 11
How does GPS work?
So how do we know that the two Speedwagon fans are pressing “play” at exactly the same time? Do Speedwagon fans all think alike? Hardly.
We must assume that satellite and receiver generate signal at exactly the same time; if they’re off by 1/1000th of a second, that means 200 m of error
The satellites have expensive atomic clocks that keep perfect time—that takes care of their end.
But what about the ground receiver?
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

13. How does GPS work? Here is where the fourth satellite signal comes in.
Lecture 11
How does GPS work?
Here is where the fourth satellite signal comes in.
While 3 perfect satellite signals can give a perfect location, 3 imperfect signals can’t, but 4 can
Imagine time to receiver as distance, with each distance from each satellite defining a circle around each satellite of that radius
If receiver clock is correct, 4 circles should meet at one point. If they don’t meet, the computer knows there is an error in the clock: “ They don’t add up”
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

14. Lecture 11
How does GPS work?
Dotted lines represent real distance, and solid lines represent erroneous distance, based on clock error—they don’t meet. Notice here we used three circles, because we’re looking in 2D, but in reality (3D) this represents four satellites, or four circles
Assuming the clock error affects all measurements equally, computer can apply a correction factor that makes circles meet in one place
Source: Trimble Navigation Ltd.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

15. Lecture 11
How does GPS work?
The receiver then knows the difference between its clock’s time and universal time and can apply that to future measurements.
Of course, the receiver clock will have to be resynchronized often, because it will lose or gain time
This is one reason why a GPS receiver needs at least four channels to get four signals
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

16. Lecture 11
How does GPS work?
So now we know how far we are from the satellites, but how do we know where the satellites are?? We can’t use them as a reference otherwise.
Because the satellites are ~ 20,200 km up they operate according to the well understood laws of physics, and are subject to few random, unknown forces.
This allows us to know where a satellite should be at any given moment.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

17. Lecture 11
How does GPS work?
There is a digital “almanac” on each GPS receiver that tells it where a given satellite is supposed to be at any given moment.
While the positions can be predicted very accurately based on simple mathematics, the DOD does monitor them using precise radar, just to make sure.
These errors are called “ephemeris” and are caused by gravitational pull of other celestial bodies
That info is relayed to the satellite, which transmits the info when it sends its pseudo random code.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

18. Lecture 11
GPS sources of error
Even after all this, there are still many factors that can generate errors and reduce positional accuracy
One of the biggest error sources is the fact that the radio signal does not travel at the exact speed of light in different parts of the atmosphere as it does in the vacuum of space.
This can be partly dealt with using predictive models of known atmospheric behavior
Source: Trimble Navigation Ltd.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

19. Lecture 11
GPS sources of error
Signals also can bounce off features, like tall buildings, cliffs and mountains, resulting in “multipath error,” where a direct signal hits, followed by a bunch of “bounced” signals which can confuse the receiver.
Good receivers have algorithms that can deal with this by determining what counts as a multi-path signal and choosing the first one as the signal to use
There are other errors as well, resulting from things like ionospheric distortions and satellite inaccuracies
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

20. GPS: Selective Availability
Lecture 11
GPS: Selective Availability
Until May of 2000, the DoD intentionally introduced a small amount of error into the signal for all civilian users, calling it “selective availability,” so non- US military users would not have the same positional accuracy as the US military.
SA resulted in about 100 m error most of the time
Turning off SA reduced error to about 30 m radius
Statement by President Clinton regarding US decision to stop degrading GPS accuracy:
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

21. Lecture 11
Differential GPS
This is a way to dramatically increase the accuracy of GPS positioning to a matter of a few meters, using basic concepts of geometry
This was used in the past to overcome SA, but with that gone, is now used for reducing the 30m error
DGPS uses one stationary and one moving receiver to help overcome the various errors in the signal
By using two receivers that are nearby each other they are getting essentially the same signals; since position of one is known, clock error can be calculated
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

22. Lecture 11
How does DGPS work?
The stationary receiver must be located on a control point whose position has been accurately surveyed: eg. USGS benchmarks
The stationary unit works backwards—instead of using timing to calculate position, it uses its position to calculate timing
It determines what the GPS signal travel time should be and compares it with what it actually is
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

23. ©2008. Austin Troy. All rights reserved
Lecture 11
How does DGPS work?
Can do this because, precise location of stationary receiver is known, and hence, so is location of satellite
Once it knows error, it determines a correction factor and sends it to the other receiver.
©2008. Austin Troy. All rights reserved
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

24. Lecture 11
How does DGPS work?
Since the reference receiver does not know which satellites the mobile receiver is using, it sends a message to it telling the correction factor for all
It used to be that only big companies and governments could use DGPS because they had to set up their own reference receiver station
Now there are many public agencies that maintain them, especially the Coast guard; these stations broadcast on a radio frequency, which GPS receivers with a radio receiver can pick up
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

25. Lecture 11
Differential GPS
DGPS improves accuracy much more than disabling of SA does
This table shows typical error—these may vary
Source:
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

26. Lecture 11
Surveyor DGPS
There are even more accurate types of DGPS that surveyors use
These are accurate to a matter of millimeters
This uses a very involved method that won’t be discussed here
One of the techniques they use though, “carrier-phase GPS” is beginning to make its way into consumer GPS
Use carrier-phase signal, which is much smaller cycle widths than the standard code phase signal
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

27. Lecture 11
Aviation DGPS
FAA is implementing DGPS for the continent, so all planes can get extremely accurate GPS navigation, called Wide Area Augmentation System (WAAS)
They have installed 25 ground reference stations as well as a master ground station that almost instantaneously processes and sends out satellite errors
Improves error to 7 m and, when finished, will allow GPS to be used as primary navigational tool for Category I landings, where there is some visibility.
Soon, it will allow zero-visibility landing navigation
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

28. Lecture 11
GPS Uses
Trimble Navigation Ltd., breaks GPS uses into five categories:
Location – positioning things in space
Navigation – getting from point a to point b
Tracking - monitoring movements
Mapping – creating maps based on those positions
Timing – precision global timing
You can learn about all these applications at these web links, but we mainly care about mapping
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

29. Lecture 11
GPS Uses
The uses for GPS mapping are enormous. Here are just a few examples:
Centerlines of roads
Hydrologic features (over time)
Bird nest/colony locations (over time)
Fire perimeters
Trail maps
Geologic/mining maps
Vegetation and habitat
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

30. GPS Uses: Beach Monitoring
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

31. Introduction to interpolation, geostatistics and spatial sampling
------Using GIS--
Part 2:
Introduction to interpolation, geostatistics and spatial sampling
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

32. What is interpolation? Resampling of raster cell size
Lecture 11
What is interpolation?
Resampling of raster cell size
Transforming a continuous surface from one data model to another (e.g. TIN to raster or raster to vector).
Creating a surface based on a sample of values within the domain.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

33. Lecture 11
What is interpolation?
Let say we have our ground water pollution samples
This gives us
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

34. Lecture 11
What is interpolation?
This can be displayed as a 3D trend surface in 3D analyst
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

35. Requirements of interpolation
Lecture 11
Requirements of interpolation
Interpolation only works where values are spatially dependent—that values for nearby points tend to be more similar
Where values across a landscape are geographically independent, interpolation does not work
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

36. Interpolation examples
Lecture 11
Interpolation examples
Elevation values tend to be highly spatially autocorrelated because elevation at location (x,y) is generally a function of the surrounding locations
Except in areas where terrain is very abrupt and precipitous, such as Patagonia, or Yosemite
In this case, elevation would not be autocorrelated at local (large) scale, but may be autocorrelated at regional (small scale)
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

37. Interpolation examples
Lecture 11
Interpolation examples
Elevation:
Source: LUBOS MITAS AND HELENA MITASOVA, University of Illinois
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

38. How does interpolation work
Lecture 11
How does interpolation work
Create or add point data which includes an attribute that will be used as a Z value
Spatial Analyst Tools >>> Interpolation
Inverse Distance Weighting (IDW)
Kriging
Natural Neighbor
Spline
Geostatistical Analyst
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

39. Inverse Distance Weighting
Lecture 11
Inverse Distance Weighting
IDW weights the value of each point by its distance to the cell being analyzed and averages the values.
IDW assumes that unknown value is influenced more by nearby than far away points, but we can control how rapid that decay is. Influence diminishes with distance.
IDW has no method of testing for the quality of predictions, so validity testing requires taking additional observations.
IDW is sensitive to sampling, with circular patterns often around solitary data points
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

40. Inverse Distance Weighting
Lecture 11
Inverse Distance Weighting
IDW: assumes value of an attribute z at any unsampled point is a distance-weighted average of sampled points lying within a defined neighborhood around that unsampled point. Essentially it is a weighted moving avg
Where λi are given by some weighting fn and
Common form of weighting function is d-p yielding:
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

41. Lecture 11
IDW-How it works
Z value at location ij is fn of Z value at known point xy times the inverse distance raised to a power P.
Z value field: numeric attribute to be interpolated
Power: determines relationship of weighting and distance; where p= 0, no decrease in influence with distance; as p increases distant points becoming less influential in interpolating Z value at a given pixel
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

42. Lecture 11
IDW-How it works
There are two IDW method options Variable and Fixed radius:
1. Variable (or nearest neighbor): User defines how many neighbor points are to be used to define value for each cell
2. Fixed Radius: User defines a radius within which every point will be used to define the value for each cell
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

43. Lecture 11
IDW-How it works
Can also define “Barriers”: User chooses whether to limit certain points from being used in the calculation of a new value for a cell, even if the point is near. E.g. wouldn't use an elevation point on one side of a ridge to create an elevation value on the other side of the ridge. User chooses a line theme to represent the barrier
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

44. IDW-How it works What is the best P to use?
Lecture 11
IDW-How it works
What is the best P to use?
It is the P where the Root Mean Squared Prediction Error (RMSPE) is lowest, as in the graph on right
To determine this, we would need a test, or validation data set, showing Z values in x,y locations that are not included in prediction data and then look for discrepancies between actual and predicted values. We keep changing the P value until we get the minimum level of error. Without this, we just guess.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

45. IDW-How it works This can also be done using the Geostatistical Wizard
Lecture 11
IDW-How it works
This can also be done using the Geostatistical Wizard
You can look for an optimal P by testing your sample point data against a validation data set
This validation set can be another point layer or a raster layer
Example: we have elevation data points and we generate a DTM. We then validate our newly created DTM against an existing DTM, or against another existing elevation points data set. The computer determine what the optimum P is to minimize our error
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

46. Mean pred. Error near zero means unbiased
Lecture 11
Optimize P value
Mean pred. Error near zero means unbiased
The blue line indicates degree of spatial autocorrelation (required for interpolation). The closer to the dashed (1:1) line, the more perfectly autocorrelated.
Where horizontal, indicates data independence
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

47. Lecture 11
Spline Interpolation
This fits a curve through the sample data and assigns 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.
Uses piece-wise functions fitted to a small number of data points, but joins are continuous, hence can modify one part of curve without having to re-compute whole
Overall function is continuous with continuous first and second derivatives.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

48. Lecture 11
Spline Method
Tension Method: results in a rougher surface that more closely adheres to abrupt changes in sample points
Regularized Method: results in a smoother surface that smoothes areas of abruptly changing values
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

49. Lecture 11
Spline Method
Weight: this controls the tautness of the curves. High weight value with the Regularized Type, will result in an increasingly smooth output surface. Under the Tension Type, increases in the Weight will cause the surface to become stiffer, eventually conforming closely to the input points.
Number of points around a cell that will be used to fit a polynomial function to a curve
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

50. Pros and Cons of Spline Method
Lecture 11
Pros and Cons of Spline Method
Splines retain smaller features, in contrast to IDW
Produce clear overview of data
Continuous, so easy to calculate derivatives for topology
Results are sensitive to locations of break points
No estimate of errors, like with IDW
Can often result in over-smooth surfaces
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

51. Lecture 11
Kriging Method
Like IDW interpolation, Kriging forms weights from surrounding measured values to predict values at unmeasured locations. As with IDW interpolation, the closest measured values usually have the most influence. However, the kriging weights for the surrounding measured points are more sophisticated than those of IDW. IDW uses a simple algorithm based on distance, but kriging weights come from a semivariogram that was developed by looking at the spatial structure of the data. To create a continuous surface or map of the phenomenon, predictions are made for locations in the study area based on the semivariogram and the spatial arrangement of measured values that are nearby.
--from ESRI Help
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

52. Lecture 11
Kriging Method
In other words, kriging substitutes the arbitrarily chosen p from IDW with a probabilistically-based weighting function that models the spatial dependence of the data.
The structure of the spatial dependence is quantified in the semi-variogram
Semi-variograms measure the strength of statistical correlation as a function of distance; they quantify spatial autocorrelation
Kriging associates some probability with each prediction, hence it provides not just a surface, but some measure of the accuracy of that surface
Kriging equations are estimated through least squares
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

53. Lecture 11
Kriging Method
Kriging has both a deterministic, stochastic and random error component Z(s) = μ(s) + ε’(s)+ ε’’(s), where
μ(s) = deterministic component
ε’(s)= stochastic but spatially dependent component
ε’’(s)= spatially independent residual error
Assumes spatial variation in variable is too irregular to be modeled by simple smooth function, better with stochastic surface
Interpolation parameters (e.g. weights) are chosen to optimize fn
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

54. Lecture 11
Kriging Method
The foundation of Kriging is notion of spatial autocorrelation, or tendency of values of entities closer in space to be related.
This is a violation of classical statistical models, which assumes that observations are independent.
Autocorrelation can be assessed using a semi-variogram, which plots the difference in pair values (variance) against their distances.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

55. Lecture 11
Semivariance
Semivariance(distance h) = 0.5 * average [ (value at location i– value at location j)2] OR
Based on the scatter of points, the computer (Geostatistical Analyst) fits a curve through those points
The inverse is the covariance matrix which shows correlation over space
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

56. Lecture 11
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

57. Variogram Plots semi-variance against distance between points
Lecture 11
Variogram
Plots semi-variance against distance between points
Is binned to simplify
Can be binned based on just distance (top) or distance and direction (bottom)
Where autocorrelation exists, the semivariance should have slope
Look at variogram to find where slope levels
Binning based on distance only
Binning based on distance and direction
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

58. Variogram SV value where it flattens out is called a “sill.”
Lecture 11
Variogram
SV value where it flattens out is called a “sill.”
The distance range for which there is a slope is called the “neighborhood”; this is where there is positive spatial structure
The intercept is called the “nugget” and represents random noise that is spatially independent
sill
nugget
range
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

59. Lecture 11
Steps
Variogram cloud: can use bins to make cloud plot of all points or box plot of points
Empirical variogram: choose bins and lags
Model variogram: fit function through empirical variogram
Functional forms?
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

60. Functional Forms Lecture 11
From Fortin and Dale Spatial Analysis:A Guide for Ecologists

61. Lecture 11
Kriging Method
We can then use a scatter plot of predicted versus actual values to see the extent to which our model actually predicts the values
If the blue line and the points lie along the 1:1 line this indicates that the kriging model predicts the data well
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

62. Lecture 11
Kriging Method
The fitted variogram results in a series of matrices and vectors that are used in weighting and locally solving the kriging equation.
Basically, at this point, it is similar to other interpolation methods in that we are taking a weighting moving average, but the weights (λ) are based on statistically derived autocorrelation measures.
λs are chosen so that the estimate is unbiased and the estimated variance is less than for any other possible linear combo of the variables.
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

63. Kriging Method Produces four types of prediction maps:
Lecture 11
Kriging Method
Produces four types of prediction maps:
Prediction Map: Predicted values
Probability Map: Probability that value over x
Prediction Standard Error Map: fit of model
Quantile maps: Probability that value over certain quantile
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

64. Kriging output: prediction
Lecture 11
Kriging output: prediction
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

65. Kriging: Ordinary vs. Universal
Lecture 11
Kriging: Ordinary vs. Universal
Ordinary Kriging – trend is a constant or linear function
Universal kriging – used when there is an underlying trend beyond the simple spatial autocorrelation
Generally this trend occurs at a different scale
Trend may be fn of some geographic feature that occurs on one part of the map
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

66. Lecture 11
Example
Here are some sample elevation points from which surfaces were derived using the three methods
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

67. Lecture 11
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
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

68. Lecture 11
Example: Spline
Note how smooth the curves of the terrain are; this is because Spline is fitting a simply polynomial equation through the points
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011

69. Lecture 11
Example: Kriging
This one is kind of in between—because it fits an equation through point, but weights it based on probabilities
Lecture Materials by Austin Troy, Brian Voigt, Weiqi Zhou and Jarlath O’Neil Dunne© 2011