Topic 15: Spatial Interpolation Flashcards
Surface models and interpolation
Surface models
- A common type of spatial model in GIS is spatial interpolation and related models
- these are predictive models that are used to estimate continuous field phenomena
Spatial interpolation
- refers to operations that use known values at sampled locations in a region to predict the unknown values at all remaining locations
Interpolation and Tobler’s Law
- principle of spatial autocorrelation
- “all things are related, but nearby things are more related than distant things”
- for continuous field phenomena, this means that surface are smooth and predictable - perfect for interpolation
Deterministic vs stochastic models
Deterministic models assume that measurements at sampled points are essentially error-free
- the true surface is considered constant and consistently predictable, with no variability to account for
- prediction is entirely analytical (ie., mathematical) in nature
Stochastic (geostatistical) models make predictions from the statistical properties of the sample data, which means they can incorporate error and variability into the prediction process
- the true surface is modelled as a trend + some variable component, and the expression of a phenomena at any location is predictable only within a range of values, not an absolute value
- prediction is probalistic and requires the use of statistical techniques
- same inputs can produce slightly different patterns
Global vs. local methods
Global interpolators derive a single mathematical function that is applied across the entire prediction surface
- all sample data are used to build the prediction function (thus, modifying one sample value can change the whole map)
- work best on very smooth surfaces
- generalizes
Local interpolators use neighbourhoods of points for prediction
- prediction of a value at a location typically uses some form of a distance-weighted algorithm on measured spatial autocorrelation
- better at accommodating abrupt transitions on surface
Inverse Distance (IDW) - Deterministic Advantages
- Efficient to calculate large datasets
- Easy to understand
- Good for high density of spatial points
Inverse Distance (IDW) - Deterministic Disadvantages
- Prediction is poor on small datasets or low density of sample points
- Assumes continuous variation; doesn’t quantify spatial autocorrelation
- Does not perform well with abrupt changes
Ordinary Kriging - Stochastic Advantages
- Uses spatial autocorrelation to derive spatial weights
- Minimizes sampling bias
- weighs cluster samples less heavily than single points
- Able to quantify interpolation errors
Ordinary Kriging - Stochastic Disadvantages
- Complex (Requires geostatistics background)
- Requires care when modelling spatial correlation structures
- Assumes stationarity across the map
- Computationally intensive
What is Inverse Distance Weighting (IDW)?
- One of the most straightforward methods of interpolation
- unknown points are interpolated from nearby points within a fixed or variable distance
- the unknown point is calculated to be the weighted mean of neighbours where weight is assigned by the inverse of the distance - the greater the distance, the smoother the surface
What is ordinary kriging?
- most popular form of interpolation from the kriging family
- Similar to IDW in that it estimates the attribute at an unsampled location as a weighted average of the attributes at nearby locations
- kriging uses a more sophisticated weight, which includes distance and the spatial structure and arrangement of the sample data
- begins by examining how different the attributes of all pairs of points are, based on their distance apart
- then it uses this information to weight nearby points to estimate attributes at unsampled locations
Steps to Ordinary Kriging
Step 1: construct a model for the semivariogram
- Semivariogram is a model for spatial dependence, or how proximity influences the similarity of nearby points
- derived by fitting a curve through a plot of semivariance by distance, which reflects how much variation exists between pairs of points based on distance
Step 2: predict attribute values at unsampled locations
- Kriging uses the semivariogram to estimate the weights, which is an expression of spatial dependence (not simple distance)
- kriging quantitatively predicts how similar should the unknown value be based on location
