Patchwise Interpolation Techniques

Local Interpolation Techniques

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:

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

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.

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).

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– Bivariate Quintic

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 not smooth surface

Linear Interpolation on TIN

Continuous but not smooth surface

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.

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

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

2nd Exact Fit Surface: Notes

• Contour curved rather than straight lines • abrupt changes in direction crossing from one

triangular plate to another

Grid Interpolation Techniques

• Use points sampled in a grid pattern• Alternatives

– Nearest Neighbor Assignment.– Linear interpolation.– Inverse Distance Weighting.– Cubic convolution.– Bilinear interpolation.– Krigging

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.

NN Procedure• Define the radius

distance

• Search the area– Quadrant search

– Octant search

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.

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

Inverse Weighted Distance: Computation

Weighted Distance: Possible Weights

IDW: Example• Interpolating a height point using W = 1/DPoint 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/DZ = 62.85 using 1/D2

Z = 57.96 using 1/D3

Contours Using IDW with w =1/D

Contours Using Inverse Distance Squared (1/D2)

621000 622000 623000 624000 625000 626000 627000 628000 629000 630000 631000

3350000

3352000

3354000

3356000

3358000

3360000

Inverse Distance Squared Surface

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?