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Spatial interpolation techniques

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Manisha Shrivastava

Spatial interpolation techniques are used to estimate values at unsampled locations based on known sample points. There are two main types of interpolation methods: global methods which apply a single mathematical function to all points, and local methods which apply functions to subsets of points. Specific interpolation methods covered include Thiessen polygons, triangulated irregular networks, spatial moving average, trend surfaces, inverse distance weighting, and Kriging. Kriging is similar to inverse distance weighting but uses minimum variance to calculate weights.

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Spatial interpolation techniques By:-Manisha
Definition: “Spatial interpolation is the procedure of estimating the values of properties at unsampled sites within an area covered by existing observations.” What is Spatial Interpolation??
classification • Global methods: – single mathematical function applied to all points – tends to produces smooth surfaces • Local methods: – single mathematical function applied repeatedly to subsets of the total observed points – link regional surfaces into composite surface
• Exact methods: – honour all data points such that the resulting surface passes exactly through all data points – appropriate for use with accurate data • Approximate methods: – do not honour all data points – more appropriate when there is high degree of uncertainty about data points
Interpolation methods –Thiessen polygons –Triangulated Irregular Networks (TINs) –Spatial moving average –Trend Surfaces –Kriging –Inverse distance weighting (IDW)
• Thiessen (Voronoi) polygons: – assume values of unsampled locations are equal to the value of the nearest sampled point • Vector-based method – regularly spaced points produces a regular mesh – irregularly spaced points produces an network of irregular polygons Thiessen Polygons
Thiessen polygon construction
Example Thiessen polygon Source surface with sample points Thiessen polygons with sample points
• Another vector-based method often used to create digital terrain models (DTMs) – adjacent data points connected by lines (vertices) to create a network of irregular triangles calculate real 3D distance between data points along vertices using trigonometry calculate interpolated value along facets between three vertices TINs
EXAMPLE- TIN
• Vector and raster method: – most common GIS method – calculates new value of each location based on range of values associated with neighbouring points – Neighbourhood determined by a filter Spatial moving average
Example SMA (circular filter) Source surface with sample points 21x21 circular filter SMA 41x41 circular filter SMA
Trend surfaces • Uses a polynomial regression to fit a least-squares surface to the data points – normally allows user control over the order of the polynomial used to fit the surface – as the order of the polynomial is increased, the surface being fitted becomes progressively more complex higher order polynomial will not always generate the most accurate surface, it dependent upon the data the lower the RMS error, the more closely the interpolated surface represents the input points
Week 17 GEOG2750 – Earth Observation and GIS of the Physical Environment 14 data point interpolated point Fitting a single polynomial trend surface
Spatial Analysis Lecture #8 (Exploring Continuous Data) Example trend surfaces Goodness of fit (R2) = 45.42 % Goodness of fit (R2) = 92.72 % Goodness of fit (R2) = 82.11 % Linear Quadratic Cubic Source surface with sample points
Inverse Distance Weighting (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
Example: IDW
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
Spatial interpolation techniques
Example: Kriging
Spatial interpolation techniques
Spatial interpolation techniques

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Spatial interpolation techniques

  • 2. Definition: “Spatial interpolation is the procedure of estimating the values of properties at unsampled sites within an area covered by existing observations.” What is Spatial Interpolation??
  • 3. classification • Global methods: – single mathematical function applied to all points – tends to produces smooth surfaces • Local methods: – single mathematical function applied repeatedly to subsets of the total observed points – link regional surfaces into composite surface
  • 4. • Exact methods: – honour all data points such that the resulting surface passes exactly through all data points – appropriate for use with accurate data • Approximate methods: – do not honour all data points – more appropriate when there is high degree of uncertainty about data points
  • 5. Interpolation methods –Thiessen polygons –Triangulated Irregular Networks (TINs) –Spatial moving average –Trend Surfaces –Kriging –Inverse distance weighting (IDW)
  • 6. • Thiessen (Voronoi) polygons: – assume values of unsampled locations are equal to the value of the nearest sampled point • Vector-based method – regularly spaced points produces a regular mesh – irregularly spaced points produces an network of irregular polygons Thiessen Polygons
  • 8. Example Thiessen polygon Source surface with sample points Thiessen polygons with sample points
  • 9. • Another vector-based method often used to create digital terrain models (DTMs) – adjacent data points connected by lines (vertices) to create a network of irregular triangles calculate real 3D distance between data points along vertices using trigonometry calculate interpolated value along facets between three vertices TINs
  • 11. • Vector and raster method: – most common GIS method – calculates new value of each location based on range of values associated with neighbouring points – Neighbourhood determined by a filter Spatial moving average
  • 12. Example SMA (circular filter) Source surface with sample points 21x21 circular filter SMA 41x41 circular filter SMA
  • 13. Trend surfaces • Uses a polynomial regression to fit a least-squares surface to the data points – normally allows user control over the order of the polynomial used to fit the surface – as the order of the polynomial is increased, the surface being fitted becomes progressively more complex higher order polynomial will not always generate the most accurate surface, it dependent upon the data the lower the RMS error, the more closely the interpolated surface represents the input points
  • 14. Week 17 GEOG2750 – Earth Observation and GIS of the Physical Environment 14 data point interpolated point Fitting a single polynomial trend surface
  • 15. Spatial Analysis Lecture #8 (Exploring Continuous Data) Example trend surfaces Goodness of fit (R2) = 45.42 % Goodness of fit (R2) = 92.72 % Goodness of fit (R2) = 82.11 % Linear Quadratic Cubic Source surface with sample points
  • 16. Inverse Distance Weighting (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
  • 18. 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