Transcript Local Interpolation Techniques

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
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Linear Interpolation;
2nd Exact Fitted Surface Interpolation;
Quintic Interpolation.
Interpolation from grid/irregular data:
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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
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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/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
Contours Using IDW with w =1/D
Contours Using Inverse Distance
Squared (1/D2)
3360000
3358000
3356000
3354000
3352000
3350000
621000 622000 623000 624000 625000 626000 627000 628000 629000 630000 631000
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?