Www.spatialanalysisonline.com Chapter 6 Part A: Surface analysis – geometrical methods.

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Chapter 6

Part A: Surface analysis – geometrical methods

3rd edition www.spatialanalysisonline.com 2

Surface analysis – geometrical methods

Modelling surfaces - surfaces and fields Surfaces – typically scalar fields:

Continuous - z-values (magnitude) assumed to exist for every (x,y) coordinate pair

Real valued (may be integer coded, e.g. remote sensing data) and generally positive (may be negative)

Single valued (open or 2D manifold) – multiple values treated as separate surfaces or layers

Surfaces - vector fields: Magnitude and direction assumed to exist for every (x,y)

coordinate pair



Modelling surfaces - surfaces and fields

Mt St Helens – rendered grid Mt St Helens – wireframe



Modelling surfaces - surfaces and fields Surfaces - Data sources:

• Physical surfaces – national mapping agencies, field surveys. DEM, contour, TIN or raster (image) models plus associated attribute data

• Sample surveys – point/block samples converted to grids using interpolation procedures

• Remote sensing – satellite, aerial• Vector data – e.g. wind strength/direction, magnetic

survey data• Programmatically derived surfaces (theoretical models

and best fits)



Modelling surfaces – raster models {x,y,z} representation, n x m Row order – geographic vs mathematical Treatment of missing and masked data Coding of cell neighbourhoods



Modelling surfaces – raster models Advantages:

Computationally very convenient Easy to display visually (2D image and 3D models) Aligns with some data capture (remote sensing) techniques Readily available for physical surfaces (DEM)

Disadvantages Very large storage requirement Computation can be processor intensive Fixed grid size, shape, orientation Representation of certain objects (e.g. lines) may be poor



Modelling surfaces – raster models Cell neighbourhoods and derivatives

First order partial derivatives – finite difference model

Second order partial derivatives (simple version)

y

zz

yz

xzz

xz SNWE

2,

2 y

zz

yz

x

zz

xz

2,

21,01,00,10,1

yxzzzz

yxz

y

zzz

y

z

x

zzz

x

z SWSENWNESNWE

4,

*2,

*2 2

22

2

22

2




Second order partial derivatives (8-cell finite difference version)

x

zzzzzz

xz

8

22 1,10,11,11,10,11,1

y

zzzzzz

yz

8

22 1,11,01,11,11,01,1




Local surface models• Fit quadratic polynomial to local neighbourhood (OLS) z=ax2+by2+cxy+dx+ey+f (6 parameters)

• Analytically differentiate

• Aspect: A=tan‑1(e/d)

• Slope: St=tan‑1(e2+d2)

• Curvatures: see later slides

OR

• Fit partial quartic polynomial to local neighbourhood (exactly) z=ax2y2+bx2y+cxy2+dx2+ey2+fxy+gx+hy+i (9 parameters)

• Curvatures: see later slides



Modelling surfaces – vector models Principal models:

TIN• Compact, fast to process• Representational detail, complexity of processing

Contour – raster DEM datasets often derived from contour source material

Conversion to-from TIN/DEM



Modelling surfaces – vector models

A. Source raster B. Contour - derived C. TIN - derived



Modelling surfaces – mathematical models



Modelling surfaces – statistical and fractal models

A. Random uniform B. Random Normal C. Ridged multi-fractal



Modelling surfaces – hybrid (pseudo-random) models



Surface geometry – gradient, slope, aspect Gradient: vector measure – 2 components:

Slope – often computed as rise over run (tan) – varies by direction. Usually defined as maximum value at a given point (magnitude component)

Aspect – direction of maximum slope (direction component)



Surface geometry – slope models Rise over run (tan) Rise over surface distance (sin) Surface z=F(x,y) analytical differential

Surface – grid differential

Surface – averaging algorithms (D-infinity, 8-point etc.) TIN model – direct computation or conversion to grid Slope – resolution, orientation effects

22

yF

xF

S

22

22

yzz

xzz

S SNWE



Surface geometry – aspect Direction in degrees from North

Directional bias from grid orientation Classified aspect – gradation, 8-way, 4-way Aspect and lighting/thermal modelling

yz

xz

A ,tan2360

270 1



Surface geometry – profiles Single profiles

Linear transects Polygonal transects



Surface geometry – profiles Multiple profiles

Baselines are average across entire grid



Surface geometry – morphology



Surface geometry – curvature Coordinate systems

1. Original grid coordinates (x,y,z)

2. Rotated grid coordinates (x-rot,y-rot,z) in direction of aspect

3. Tangential coordinates (surface normal, surface tangential plane)

Curvature computation and naming wrt alternative coordinate systems



Surface geometry – profile curvature Math model:

Quadratic model:

Quartic model:

pqyz

xz

p

pq

yz

y

zyz

xz

yxz

xz

x

z

pr

1 ,

,

2

22

2/3

2

2

222

2

2

2 2

3/ 22 2 2 2

200

1pr

ad be cde

e d d e

2 2

2 2

200pr

dg eh fgh

g h



Surface geometry – plan curvature Math model:

Quadratic model:

Quartic model:

222 2 2

2 2

3/ 2

22

2

pl

z z z z z z zx x y x y yx y

p

z zp

x y

2 2

3/ 22 2

200pl

bd ae cde

e d

2 2

2 2

200pl

dh eg fgh

g h



Surface geometry – tangential curvature

tg

pl

z z z z z z zx x y x y yx y

pq

pq

z zp q p

x y

222 2 2

2 2

1/ 2

1/ 2

22

2

, where

, and 1



Surface geometry – additional quadratic curvatures Longitudinal:

Cross-sectional:

Min, Max and mean:

2 2

2 2

200lon

ad be cde

e d

2 2

2 2

200cro

bd ae cde

e d

22min )( cbaba

22max )( cbaba

max min( )/ 2mean



Surface geometry – directional derivatives Computed for direction :

First derivative:

Second derivative:

cos( ) sin( )dz z zds x y

d z z zx yds x

z

y

2 2 22

2 2

22

2

cos ( ) 2 cos( )sin( )

sin ( )



Surface geometry – paths Paths as plane curves Paths as space curves Parametric specification Path curvature:

Radius of curvature: 1/path curvature=1/ Smoothing

2 2

2 2

3 22 2 /

x y y xt tt t(t)x yt t



Surface smoothing Resolution increase/Grid

re-calculation Using a smoothing

interpolator (e.g. spline) Filtering or kernel

smoothing (e.g. 3x3 ‘Gaussian’ kernel)

1 2 1

2 4 2

1 2 1



Surface geometry – pit filling Hydrographic modelling

Prior to flow modelling 8-cell model and other rules Masked fill Depression-depth based filling

Error correction Arising from data collection Arising from data processing (e.g. interpolation)



Surface geometry – volumetric analysis Profiles – simple cut and fill computations Surfaces:

Single grid vs reference (base) surface (e.g. z=0) Grid pairs – grid 1 (upper), grid 2 (lower) Result – estimate positive or negative volume

(relative, and/or wrt base) Computational procedures

Numerical integration (trapezoidal rule) Exact computation from TIN Indirect computation from point or profile data



Visibility – Overview Application areas Line of sight modelling Viewshed (visible areas) modelling

Single and multi-point problems Static vs dynamic problems

Optical vs radio path visibility Euclidean model Earth curvature model Propagation modelling



Visibility – line of sight analysis Simplified form of viewshed Point source plus direction(s)

Coloured line transect(s) Tabulated data Profile plots

Point source, offset from surface

Line of sight direction lines

Lines of sight – yellow= visible from source, red=not visible

Viewshed: dark blue=visible area



Visibility – viewsheds and RF propagation Viewshed (visible areas) modelling

Input surface raster Point set raster – single, multi-point, zones etc Offsets for observation and target points Range (distance and angular) constraints Output – binary or multi-coded raster RF – selection of propagation model, parameters

(e.g. frequency, gain) and clutter modelling (typically surface offsets and obstacles)



Visibility – viewsheds and RF propagation

A. Source topography B. Simple optical viewshed (pink=not visible)

Mobile phone mast



Visibility – Isovist analysis Analysis of visibility in the plane One or more source points Complex optimisation problem

Sample point – green areas show visible street areas

Near optimal locations for cameras providing full coverage of streets



Visibility – Space syntax Analysis of visibility in the built environment



Watersheds and drainage – assumptions Uniform precipitation Flows take place entirely across surfaces

which they do not alter; unaffected by absorption or groundwater

Flows grow as a linear function with distance; not altered by slope values, just by direction

No barriers to flow Study region is complete and meaningful in

the context of the analysis



Watersheds and drainage – modelling steps Input (complete/mosaic-ed) DEM Remove pits Identify flow directions – D-8, D-infinity or MFM Output ldd grid Identify flats and extrema Accumulate hypothetical flows to generate and

merge streams – include pour points Identify watersheds and stream basins



Watersheds and drainage – D-infinity Max gradient of 8 facets identified Flows assigned to cells (pixels) in proportions:

21

22

21

11 ,

pp



Watersheds and drainage – case studyPit filled DEM Flow accumulations and watersheds



Watersheds and drainage – case studyFlats and extrema Stream basins

Www.spatialanalysisonline.com Chapter 6 Part A: Surface analysis – geometrical methods.

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