1. Spatial Interpolation, geostatistics and sampling
Lecture 11
------Using GIS--
Lecture 10:
Spatial Interpolation, geostatistics and sampling
By Austin Troy
©2008 All lecture materials by Austin Troy except where noted

2. ©2008 All lecture materials by Austin Troy except where noted
What is interpolation?
Three types:
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.
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How does it Look
Let say we have our ground water pollution samples
This gives us
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How does it work
This can be displayed as a 3D trend surface in 3D analyst
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5. 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
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6. Interpolation examples
Elevation:
Elevation values tend to be highly spatially autocorrelated because elevation at location (x,y) is generally a function of the surrounding locations
Except is 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 still may be autocorrelated at regional (small scale)
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7. Interpolation examples
Elevation:
Source: LUBOS MITAS AND HELENA MITASOVA, University of Illinois
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Sampling example
Imagine this elevation cross section: If each dashed line represented a sample point (in 1-D), this spacing would miss major local sources of variation, like the gorge
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Sampling example
Our interpolated surface (represented in 1-D by the blue line) would look like this
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Sampling example
If we increased the sampling rate, we would pick up that local variation
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Sampling example
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
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12. Interpolation examples
Weather: Great Plains vs SF Bay Area
Groundwater contamination: depends on geology
Crime rates: depends on population density
New York City income: may require fine scale
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Sampling Approaches
Often a regular gridded sampling strategy is appropriate and can eliminate sampling biases
Sometimes, though, it can introduce biases if the grid pattern correlates in frequency with something in the landscape, such as trees in a plantation or irrigation lines
Random sampling can avoid this but introduces other problems including difficulty in finding sample points and uneven distribution of points, leading to geographic gaps.
This depends partially on the size of the support, or sampling unit
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Sampling Approaches
An intermediate approach is the stratified random sample
Create geographic or non-geographic subpopulations, from each of which random sample is taken
Proportional or equal probability SRS: enforce a certain sampling rate, πhj= nh/Nh for each stratum h and obs j.
Simple SRS: enforce a certain sample size nh
Disproportionate SRS: where πhj varies such that certain strata are oversampled and certain undersampled.
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Sampling Approaches
DSRS is advantageous when subpopulation variances are unequal, which is frequently the case when stratum sizes are considerably different. In DSRS we sample those strata with higher variance at higher rate. We may also use this when we have an underrepresented subpopulation that will have too few observations to model if sampled with SSRS.
Proportional samples are self-weighting because the rates are the same for each stratum
The other two have unequal sampling probabilities (unless a simple SRS has equal Nh) and may require weighting
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Sampling Approaches
When the stratifying unit is geographical (e.g. county, soil polygon, forest stand), this is called a cluster sample.
In a one stage cluster sample (OSCS) a series of geographic units are sampled and all observations within are sampled: obviously this does not work for interpolation
More relevant is a two stage cluster sample (TSCS) in which we take a sample of cluster units and then a subsample of the population of each cluster unit.
In this type of sample, variance has two components, that between clusters and that between observations
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Sampling
The number of samples we want within each zone depends on the statistical certainty with which we want to generate our surface
Do we want to be 95% certain that a given pixel is classified right, or 90% or 80%?
Our desired confidence level will determine the number of samples we need per strata
This is a tradeoff between cost and statistical certainty
Think of other examples where you could stratify….
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Sampling
A common problem with sampling points for interpolation is what is not being sampled?
Very frequently people leave out sample points that are hard to get to or hard to collect data at
This creates sampling biases and regions whose interpolated values are essentially meaningless
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Sampling example
Say we want to make an average precipitation layer and we find that in our study zone precipitation is highly spatially variable within 10 miles of the ocean
We’d a coastline layer to help us sample.
We’d have high density of sampling points within 10 miles of the ocean a much lower density in the inland zones
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Sampling example
Say we were looking at an inland area, far from any ocean, and we decided that precipitation varied with elevation. How would we set up our sampling design?
In this case, flat areas would need fewer sample points, while areas of rough topography would need more
In our sampling design we would set up zones, or strata, corresponding to different elevation zones and we would make sure that we get a certain minimum number of samples within each of those zones
This ensures we get a representative sample across, in this case, elevation;
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Sampling
The number of zones we use will determine how representative our sample is; if zones are big and broad, we do not ensure that all elevation ranges are represented
©2008 All lecture materials by Austin Troy except where noted

22. Sampling and Scale dependency
Sampling strategy for interpolation depends on the scale at which you are working and the scale dependency of the phenomenon you are studying
In many cases interpolation will work to pick up regional trends but lose the local variation in the process
The density of sample points must be chosen to reflect the scale of the phenomenon you are measuring.
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Scale dependency
If you have a high density of sample points, you will capture local variation, which is appropriate for large-scale (small-area) studies
If you have low density of sample points, you will lose sensitivity of local variation and capture only the regional variation; this is more appropriate for small-scale (large-area) studies
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24. How does interpolation work
In ArcGIS, to interpolate:
Create or add a point shapefile with some attribute that will be used as a Z value
Click Spatial Analyst>>Interpolate to Raster and then choose the method
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25. Three methods in Arc GIS
IDW
SPLINE
Kriging
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26. 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
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27. 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:
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IDW-How it works
Z value at location ij is f 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
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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 going 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
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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
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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.
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IDW-How it works
This can be done in ArcGIS 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
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IDW-How it works
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Optimizing P value
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Plot of model fits
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
Mean pred. Error near zero means unbiased
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Plot of model errors
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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.
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 recompute whole
Overall function is continuous with continuous first and second derivatives.
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Spline Method
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
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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
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40. Pros and Cons of Spline Method
Splines retain smaller features, in contrast to IDW
Produce clear overview of data
Continuous, so easy to calculate derivates for topology
Results are sensitive to locations of break points
No estimate of errors, like with IDW
Can often result in over-smooth surfaces
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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
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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
Semivariograms 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
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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
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Kriging Method
Hence, 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 semivariogram, which plots the difference in pair values (variance) against their distances.
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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
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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
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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
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48. 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?

49. Functional Forms
From Fortin and Dale Spatial Analysis:A Guide for Ecologists

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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
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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.
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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
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53. Kriging output: prediction
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54. Kriging: Ordinary vs. Universal
Known as Kriging in the presence of universal trends.
Universal kriging is used where 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
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Example
Here are some sample elevation points from which surfaces were derived using the three methods
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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
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Example: Spline
Note how smooth the curves of the terrain are; this is because Spline is fitting a simply polynomial equation through the points
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Example: Kriging
This one is kind of in between—because it fits an equation through point, but weights it based on probabilities
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59. Other methods of interpolation
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
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