Numerical Analysis. TOPIC Interpolation TOPIC Interpolation.
Patchwise Interpolation Techniques
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Transcript of Patchwise Interpolation Techniques
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Patchwise Interpolation Techniques
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Local Interpolation Techniques
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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:
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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
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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.
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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).
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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
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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
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Linear Interpolation on TIN
Continuous but not smooth surface
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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.
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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
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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
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2nd Exact Fit Surface: Notes
• Contour curved rather than straight lines • abrupt changes in direction crossing from one
triangular plate to another
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Grid Interpolation Techniques
• Use points sampled in a grid pattern• Alternatives
– Nearest Neighbor Assignment.– Linear interpolation.– Inverse Distance Weighting.– Cubic convolution.– Bilinear interpolation.– Krigging
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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.
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NN Procedure• Define the radius
distance
• Search the area– Quadrant search
– Octant search
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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.
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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
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Inverse Weighted Distance: Computation
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Weighted Distance: Possible Weights
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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
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Contours Using IDW with w =1/D
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Contours Using Inverse Distance Squared (1/D2)
621000 622000 623000 624000 625000 626000 627000 628000 629000 630000 631000
3350000
3352000
3354000
3356000
3358000
3360000
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Inverse Distance Squared Surface
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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?