What’s the Point? Interpolation & Extrapolation with a Regular Grid DEM David Kidner, Mark Dorey & Derek Smith University of Glamorgan School of Computing Pontypridd WALES, U.K. CF37 1DL

Geocomputation’99 July 25 th - 28 th What’s the Point? Digital Terrain Modelling and “Grids” –What’s the Point ? Interpolation Algorithms –Tests and Results Extrapolation Algorithms Data Compression –Tests and Results Conclusions

Geocomputation’99 July 25 th - 28 th A Digital Elevation Model (DEM) Regular grid of elevations represents the heights at discrete samples of a continuous surface –vertices are sampled or interpolated independently –represented in a 2D matrix No direct topological relationship between points 2D Grid imposes an implicit representation of surface form –usually as a linear relationship between vertices Simple and convenient

Which One’s the DEM? (a) Discrete Elevation Samples (b) Implicit (Linear) Continuous Surface

Geocomputation’99 July 25 th - 28 th Interpolation DEM resolution should be dependent upon the variability of each terrain surface –… but rarely is The requirement of the DEM is to represent the terrain surface such that elevations can be retrieved or inferred for any given location –i.e. usually by interpolation The method of interpolation is often ignored Required for most, if not all applications

Geocomputation’99 July 25 th - 28 th What’s the Point? Does Interpolation matter? –What’s the height at D ? –Where’s the 60m Contour(s) ? D

Geocomputation’99 July 25 th - 28 th Interpolation for Visibility Analysis A B C D What’s the profile through the “cell” ?

Geocomputation’99 July 25 th - 28 th Interpolation for Visibility Analysis (a) Linear with Diagonal (b) Linear without Diagonal (c) Bilinear 20m Object (a) Completely Obscured (b) Completely Visible (c) Partially Visible

Geocomputation’99 July 25 th - 28 th Interpolation Algorithms Very small interpolation errors may lead to very large application errors –visibility analysis, hydrological modelling, contouring, etc. Interpolation is flawed if we only consider the grid cell of the point to be estimated Most GIS only consider the 4 vertices of the grid cell ! –bilinear interpolation

Geocomputation’99 July 25 th - 28 th Interpolation Alternatives For the most part, we can use polynomial interpolation of the form: h i =a 00 + a 10 x + a 01 y + a 20 x 2 + a 11 xy + a 02 y 2 + a 30 x 3 + a 21 x 2 y + a 12 xy 2 + a 03 y 3 + a 31 x 3 y + a 22 x 2 y 2 + a 13 xy 3 + a 32 x 3 y 2 + a 23 x 2 y 3 + a 33 x 3 y 3 + … + a mn x m y n solved from the set of simultaneous equations which are set up, one for each point.

Geocomputation’99 July 25 th - 28 th Interpolation Alternatives Level Plane (1 coefficient) Linear Plane (3) Double Linear and Bilinear (4) Biquadratic (8 or 9) Bicubic (12 or 16) Biquintic (36) Jancaitis Biquadratic, Piecewise Cubics, etc.

Linear 1 Linear 2 Double Linear Bilinear Biquadratic Bicubic Jancaitis Biquintic (9 term) (16 term) (36 term)

Geocomputation’99 July 25 th - 28 th Results (1) Test Surface Functions (Franke, 1979; Akima, 1996)

Geocomputation’99 July 25 th - 28 th Results (1) Test Surface Functions

Geocomputation’99 July 25 th - 28 th Results (1) Test Surface Functions

Geocomputation’99 July 25 th - 28 th Results (1) Test Surface Functions Higher-order interpolation algorithms will always out-perform linear techniques

Geocomputation’99 July 25 th - 28 th Based on Ordnance Survey data for S. Wales: 1:50,000 Scale (50 m) DEMs and 1:10,000 Scale (10 m) DEMs –Higher-order interpolation algorithms will always out- perform linear techniques –By 3% to 10% (of the RMSE) –Less correlation as to which algorithm performs best Results (2) Actual Terrain

Geocomputation’99 July 25 th - 28 th Extrapolation Interpolation outside the spatial extent Extrapolation can be considered to be at the heart of the best techniques for spatial data compression –i.e. what is the next symbol in the series –or “standing on the surface and given my field of view, what is the elevation if I take one step backwards?”

Geocomputation’99 July 25 th - 28 th Why do we need DEM compression? Seen as yesterday’s problem ? –expensive hardware; small capacity disks, etc. File/Internet Transfer Higher Resolutions O.S. 50m DEM O.S. 10m DEM 2m LiDAR DEM

DEM Extrapolation & Prediction

DEM Transformation for Compression Elevation Range Frequency 5% 10% ORIGINAL DEM Elevation Correction Range (Prediction Errors) Frequency 20% 40% DEM OF ERROR CORRECTIONS

Terrain Extrapolators

a fg Z Linear Predictor: Z = a + g - f e.g. 462 = Z = = 465 (Correction = -3)

12-Point Predictor: e.g. Z = w1*458+w2* w12*469 = 464 (Correction = -2) hi b e lmnop a fg Z

Geocomputation’99 July 25 th - 28 th Frequency Distribution of 15x15 Corrections

O.S. South Wales 1201x801 DEM

Prediction (extrapolation) Corrections

Geocomputation’99 July 25 th - 28 th Data Compression Results GZIPped DEM requires a storage capacity of 261% of the best extrapolator and Arithmetic Coding method

Geocomputation’99 July 25 th - 28 th Summary Mathematical modelling has now largely been forgotten by today’s GIS developers Many GIS techniques are of limited value and may propagate through to application error (e.g. visibility analysis) For DEM Interpolation –don’t use linear algorithms Mathematical modelling offers significant savings for spatial data compression