Patchwise Interpolation Techniques

1. Patchwise Interpolation Techniques

2. Local Interpolation Techniques

3. Local Versus Global Interpolation Techniques • Global methods: • Local variations have been considered as random, • unstructured noise that had to be minimized. • Local methods: • Only use information from the nearest data points:

4. General Procedure • Define a search area or neighborhood around the point to be interpolated; • Find the data points within this neighborhood; • Choose a mathematical model to represent the variation over this limited number of points; • Evaluate the height at the interpolation point under consideration. • Z = f(Zi) where Zi is the point in the search area

5. Local Interpolation: Special Considerations • The size, shape, and orientation of the neighbourhood; • The number of data points to be used; • The distribution of the data points: • Regular grid, irregularly distributed/TIN; • The kind of interpolation function to use; • The possible incorporation of external information on trends or different domains; • All these methods smooth the data to some degree: • They compute some kind of average value within a window.

6. Local Interpolation Techniques • Interpolation from TIN data • Linear Interpolation; • 2nd Exact Fitted Surface Interpolation; • Quintic Interpolation. • Interpolation from grid/irregular data: • Nearest neighbour assignment; • Linear Interpolation; • Bilinear interpolation; • Cubic convolution; • Inverse distance weighting (IDW); • Optimal functions using geostatistics (Kriging).

7. Interpolation within a TIN • TIN local interpolation methods honor the Z values at the triangle nodes • Exact interpolation techniques • Alternatives: • Linear • Second exact fit surface • BivariateQuintic

8. TIN Linear Interpolation: Assumptions • Considers the surface as a continuous faceted surface formed by triangles • The normal to the surface is constant • Height calculated based solely on the Z values for the nodes of the triangle within which the point lies • Produces continuous but notsmooth surface

9. Linear Interpolation on TIN Continuous but notsmooth surface

10. Linear Interpolation: Concept / Procedure • Fit a plane through the triangle facet including the interpolation point. • Use the fitted plane to estimate the elevation at the interpolation point.

12. 2nd Degree Exact Fit Surface • Assumes the triangles represent tilted flat plates • Rationale: a better approximation can be achieved using curved or bent triangle plates, particularly if these can be made to join smoothly across the edges of the triangles. • Exact and smooth technique • Results in a very crude approximation

13. 2nd Degree Exact Fit Surface: Procedure • Find the three neighbour triangles closest to the faces of the triangle containing the point of interest • Fit a second-degree polynomial trend to the points of the triangles • The fitted surface is exactly passing through all six points

16. 2nd Exact Fit Surface: Notes • Contour curved rather than straight lines • abrupt changes in direction crossing from one triangular plate to another

17. Grid Interpolation Techniques • Use points sampled in a grid pattern • Alternatives • Nearest Neighbor Assignment. • Linear interpolation. • Inverse Distance Weighting. • Cubic convolution. • Bilinear interpolation. • Krigging

18. Nearest Neighbour (NN) Interpolation • Assigns the value of the nearest mesh point in the input lattice or grid to the output mesh point or grid cell. • No actual interpolation is performed based on values of neighbouring mesh points.

19. NN Procedure • Define the radius distance • Search the area • Quadrant search • Octant search

20. NN Procedure • Find the nearest point • Assign the height of the point to the interpolated point • Notes: • No control over distribution and number of points used • NN does not yield a continuous surface.

21. Inverse Weighted Distance (IWD) • Points closer to interpolation point should have more influence • The technique estimates the Z value at a point by weighting the influence of nearby data point according to their distance from the interpolation point. • An exact method for topographic surfaces • Fast • Simple to understand and control

22. Inverse Weighted Distance: Computation

23. Weighted Distance: Possible Weights

24. IDW: Example • Interpolating a height point using W = 1/D Point distance z value w wz 1 300 105 1/300 0.3499 2 200 70 1/200 0.35 3 100 55 1/100 0.55 Swi = S(1/di) = 0.0183 Swizi = 105/300+70/200+55/100= 1.2499 Substituting in formula: 1.2499 ¸ 0.0183 Z = 68.1764 using 1/D Z = 62.85 using 1/D2 Z = 57.96 using 1/D3

25. Contours Using IDW with w =1/D

26. Contours Using Inverse Distance Squared (1/D2)

27. Inverse Distance Squared Surface

28. Conclusions • Interpolation of environmental point data is important skill • Many methods classified by • Local/global, approximate/exact, gradual/abrupt and deterministic/stochastic • Choice of method is crucial to success • Error and uncertainty • Poor input data • Poor choice/implementation of interpolation method • Is it possible to use explanatory variables to improve interpolation, and if so, how?