Patchwise - PowerPoint Presentation

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.Slide14
Slide15

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

Slide17
Slide18
Slide19

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 ¸ 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?