Interpolation Techniques Patchwise Methods Local polynomial surface fit Local trend surfaces patchwise method Equal size patches Separate functions calculated for each patch typically polynomial ID: 378339
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Slide1
Patchwise Interpolation TechniquesSlide2
Patchwise Methods
Local polynomial surface fit
Local trend surfaces (patchwise method)
Equal size patches
Separate functions calculated for each patch (typically polynomial)Slide3
Advantages
Low order terms can be used
Derived points easily calculatedSlide4
Disadvantages
Needs more organisation of data and processing
Subdivision needs to be carried out with care
Poor distribution of data points near patch corners - affects computed parameters + derived node values Slide5
Local Interpolation TechniquesSlide6
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:Slide7
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(Z
i
) where Z
i
is the point in the search areaSlide8
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.Slide9
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).Slide10
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
QuinticSlide11
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 Slide12
Linear Interpolation on TIN
Continuous
but
not
smooth
surfaceSlide13
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.Slide14Slide15
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 approximationSlide16
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
Slide17Slide18Slide19
2nd Exact Fit Surface: Notes
Contour
curved
rather than straight lines
abrupt
changes
in direction crossing from one triangular plate to another
Slide20
Grid Interpolation Techniques
Use points sampled in a grid pattern
Alternatives
Nearest Neighbor Assignment.
Linear interpolation.
Inverse Distance Weighting.
Cubic convolution.
Bilinear
interpolation.
KriggingSlide21
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.Slide22
NN Procedure
Define the radius distance
Search the area
Quadrant search
Octant searchSlide23
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.Slide24
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 controlSlide25
Inverse Weighted Distance: ComputationSlide26
Weighted Distance: Possible WeightsSlide27
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
S
w
i
=
S(1/
di) = 0.0183
S
w
izi = 105/300+70/200+55/100= 1.2499
Substituting in formula: 1.2499 ΒΈ 0.0183
Z = 68.1764 using 1/DZ = 62.85 using 1/D2
Z = 57.96 using 1/D3Slide28
Contours Using IDW with w =1/DSlide29
Contours Using Inverse Distance Squared (1/D2)Slide30
Inverse Distance Squared SurfaceSlide31
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?