Digital Terrain Model (DTM)

Idea of DTM

Aim: height interpolation at any point based on measured/known points

Interpolation method

•Continuous interpolation preferred (0 order, 1st order, 2nd order continuity)

•Good approximation of the surface of the earth

Digital Elevation Modeling Journal

A digital terrain model (DTM) is a topographic model of the bare earth that can be manipulated by computer program (Wikipedia)

Layout of base points

Regular layout base points (tesselation/GRID)

Irregular base points

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Break linesextremal points

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Boundary restrictions(e.g. lakes)

Collection of elevation dataTopographic survey (irregular points and breaklines)

Photogrammetry (grid, contours)

Contour line digitizing (contours + extremal points + breaklines)

Radar measurements (SRTM Shuttle Radar Topography Mission)1” resolution (30 m) US only3” resolution (100 m)

Leveling of grid points

GTOPO30 30” resolution

Sample data

Creation of a DTM

Regular layout (Rectangular Grid, DEM)

•Inverse Distance Weight (IDW)

•Kriging

Triangulated Irregular Network (TIN)

•Optimal, non overlapping triangle network, minimal sum of perimeters

•Delaunay triangulation

•Interpolated points from irregular base points

•Original base points are used

•Surface interpolation (trends)

IDW (Shepard 1968)

n

iii fwyxF

1

),(w – weightf – function value at the base point

n

j

pj

pi

i

t

tw

1

t – distance between base point and interpolated point

p – usual value is 2

Distance limit

Direction restriction (quarters)

Kriging (Krige 1951)

n

ii

n

iii wvwv

11

1ˆ Linear combination the elevation ofbase points

Conditions for the weight used:

Unbiased estimationEstimate minimal standard deviation

Variograms (geostatistics)

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1

2

)(21

)(hn

iPP hiiZZ

hnh

h – distance from base point

Effective distance, (h) doesn’t change as h increased

Least squares method

Surface interpolation

Polynom interpolation

One continuous surface (global solution)

Dynamic surfaces (local, patchwork)

Spline interpolation

...),( 243210 xaxyayaxaayxf

2nd order continuity between cubic polynoms

Sample

33

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Sample 2

Delaunay triangulation (1934)

Minimize the sum of the perimeter of the non overlapping triangles

Algorithm (incremental):

Start from an optimal triangle contains all the base pointsthen add a new point and divide the triangle

Condition: no points in the inscribedcircle of the triangle

Sample

Voronoi cells

DTM manipulation:

•Add point

•Add breakline

•Add triangle or polygon

•Erase part

Dual problem of DelaunayTriangulation. Areas nearest to the base points.

Areas of DTM applications

Contour line interpolation

Cross sections

Viewshed analysis

Slope category map

Aspect (slope direction)

Watershed analysis

Flow directions

Modeling (e.g. erosion)

Planning of roads, railways, pipelines

Visualization of the terrainVolume calculation

Reduction (terrain correction) of gravity measurements

Rectification of airbone or satellite photos

Hydrology example

3D view of DTM