Geomorphometry II: spatial and temporal terrain analysis

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Geomorphometry IIspatial and temporal terrain
analysis

• Geospatial Analysis and Modeling
• Lecture notes
• Helena Mitasova, NCSU MEAS

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Outline

• summary parameters volumes, surface areas
• first and second order point parameters
• methods for slope, aspect and curvatures using
  polynomial and spline approximation
• computing parameters from noisy data, level of
  detail
• combining parameters to map landforms and terrain
  features
• raster time series analysis, quantification of
  coastal change

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Terrain surface parameters

• Quantification of terrain surface properties
• Derived from discrete representation of zf(x,y)
• Summary parameters quantify property for entire
  surface or its subregion volume, surface area,
  fractal dimension, roughness
• Point geometry parameters quantify geometry at
  each point of the surface gradient, curvatures
• Cumulative flow parameters

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Volume
Applied as global or zonal Volume - integral
approximated as sum of cuboids V?dx.dy.dz
dx.dy ? dz dx.dy grid cell area dz z0-zi,
cell value difference z0 constant, plane or
curved surface computed as sum of grid cell
values - elevations
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Surface area
Applied as global or zonal

• Surface area
• sum of 3D triangle areas
• depends on the direction of the diagonal
  minimum and maximum surface - see the code
• alternative 8 triangles connecting grid center
  with neighboring ones
• Accuracy is dependent on resolution and accuracy
  of the input data.

http//jtaylor1142001.net/calcjat/Solutions/VCross
Product/VCPATriangle.htm http//cnx.org/content/m1
3603/latest/ http//en.wikipedia.org/wiki/Triangle
also see the code for r.surf.area
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Point topographic parameters

• Measure geometric properties at any point of the
  topographic surface steepness, orientation,
  shape.
• Derivation based on differential geometry.
• For the surface represented by a continuous
  bivariate function z f(x,y) we can compute

gradient ?z (?z/?x, ?z/?y) (fx,fy) where
?z/?x and ?z/?y are first order partial
derivatives of f(x,y) it is a vector field that
represents the maximum rate of change in
elevation and points upslope in the direction of
steepest slope
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Gradient field

• Gradient ?z (fx,fy)
• Negative gradient is used to map water flow
  direction

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Slope and aspect

• Gradient ?z (fx,fy)
• Slope angle is gradient magnitude,
• tan ? ? (fx2 fy2) or ? 100 . ? (fx2
  fy2)
• angle between a horizontal plane and tangent to
  the surface
• measures maximum rate of change in elevation
• controls flow velocity

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Slope and aspect

• Slope angle is gradient magnitude

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Slope and aspect

• Gradient ?z (fx,fy)
• Slope angle is function of gradient magnitude
• tan ? ? (fx2 fy2)
• Aspect angle is gradient direction
• tan ? fy/fx
• steepest slope direction, controls direction of
  flow
• hillslope orientation
• note for fx0, tan ? ?, ? ?
• actual values are often rotated with 0 deg North
  increasing clockwise

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Slope and aspect

• Gradient ?z (fx, fy)
• Slope angle is gradient magnitude
• tan ? ? (fx2 fy2)
• Aspect angle is gradient direction
• tan ? fy/fx
• Change in the direction ? is given by directional
  derivative
• ?z/?s fx cos ? fy sin ?
• where (cos ?, sin ?) is a unit vector in
  direction ?

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Slope and aspect

• Example slope and aspect map

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Computing slope and aspect

• Estimating partial derivatives
• Steepest elevation difference
• Polynomial approximation on 3x3 neighborhood,
  many versions using second order polynomials
• Spline approximation on raster or directly from
  point data

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Computing slope and aspect

• Estimating partial derivatives using polynomial
  approximation on 3x3 neighborhood
• z(x,y)a0a1 x a2 y a3 xy a4 x2 a5 y2

i-1,j1 ----- i,j1 ----- i1,j1


i-1,j ------- i,j --------- i1,j


i-1,j-1 ----- i,j-1 ----- i1,j-1
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Computing slope and aspect

• Estimating partial derivatives using polynomial
  approximation on 3x3 neighborhood
• z(x,y)a0a1 x a2 y a3 xy a4 x2 a5 y2

?x
fitting this polynomial to the 9 grid
points using weighted least squares leads to
i-1,j1 ----- i,j1 ----- i1,j1


i-1,j ------- i,j --------- i1,j


i-1,j-1 ----- i,j-1 ----- i1,j-1
?y
fx(zi-1,j-1 - zi1,j-1) 2(zi-1,j - zi1,j)
(zi-1,j1 -zi1,j1) / 8?x
fy(zi-1,j-1 - zi-1,j1) 2(zi,j-1 - zi,j1)
(zi1,j-1 -zi1,j1) / 8?y
for zi,j z(0,0). It does not pass through
z(0,0) so it has smoothing effect see the code
for r.slope.aspect
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Curvatures

• Curvature at a point inverse of the radius of a
  circle tangent to the surface at this point in a
  given direction m-1

concave, lower value convex, higher value
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Curvatures

• Curvature at a point inverse of the radius of a
  circle tangent to the surface at this point in
  the normal plane
• In general, surface has different curvatures in
  different directions
• Principal directions have max and min curvatures,
  gaussian and mean are function of principal
  curvatures
• Geomorphometric curvatures
• profile, tangential / plan

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Curvatures profile

• Profile curvature
• in the direction of gradient, along the flow line
• measures rate of change in slope
• convex increasing slope -gt flow acceleration,
• concave decreasing slope -gt flow deceleration

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Curvatures tangential

• Tangential curvature
• perpendicular to gradient, direction of tangent
  to contourline
• measures rate of change in aspect / curvature of
  contours
• convex flow divergence,
• concave flow convergence

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Curvatures and flow

• Kp / Kt concave/concave
  convex/concave
• Flow decelerated-convergent
  accelerated-convergent

Kp / Kt concave/convex
convex/convex Flow decelerated-divergent
accelerated-divergent
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Curvatures profile, tangential

• Equations for curvatures can be derived using the
  general equation for curvature of a normal plane
  section through a point on a surface
• Denote pfx2fy2 and qp1,
• then equation for the profile curvature
• kp (fxx fx2 2fxy fx fy fyy fy2 ) / p vq3
• and the equation for tangential curvature
• kt (fxx fy2 - 2fxy fx fy fyy fx2 ) / p vq

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Curvatures

• Geomorphometric curvatures
• profile
  tangential

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Computing curvatures

• Estimate first and second order partial
  derivatives
• Polynomial approximation on 3x3 neighborhood,
  various versions
• Spline approximation on raster or from point data

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Computing curvatures

• Estimating partial derivatives using polynomial
  approximation on 3x3 neighborhood
• z(x,y)a0a1 x a2 y a3 xy a4 x2 a5 y2

?x
i-1,j1 ----- i,j1 ----- i1,j1


i-1,j ------- i,j --------- i1,j


i-1,j-1 ----- i,j-1 ----- i1,j-1
fx, fy same as for slope and aspect fxx,fyy,fxy
second order differences
?y
see the code for r.slope.aspect
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Topo parameters from splines

• simultaneous computation with interpolation -
  based on the original points
• tens or hundreds of points can be used - larger
  area than 3x3 neighborhood
• explicit equations for partial derivatives
• thin plate spline with tension - first order only
• regularized spline - first and second order
• regularized spline with tension - all orders
• tuning the level of detail by tension and
  smoothing parameters

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Topo parameters from splines

• Profile curvature from contours

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Topo parameters from splines

• Tangential curvature at 1m resolution from
  multiple return lidar points
• high tension low
  tension

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Smoothing and geometry analysis
Tension and smoothing is used to create a surface
at a desired level of detail and smooth-out the
noise.
profile curvature slope
road
tension 700
tension 100
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Tuning the level of detail
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Feature extraction and change analysis
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Terrain time series

• Setting the resolution and interpolation
  parameters
• Measures of change
• Isoline change or linear feature change (e.g.
  shoreline, crest lines) horizontal or 3D
  migration
• Elevation and Volume change
• Feature area change
• Per cell analysis for change trends
• average, stddev, count, median, mode,
• minimum,min_raster,maximum,max_raster,
• range,sum,variance,diversity,
• slope, offset, detcoeff,
• quart1,quart3,perc90, skewness, kurtosis

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Spatial coastal change indicators
New, spatially explicit indicators representing
coastal terrain evolution based on per grid cell
statistics a) core surface below which
elevation never decreased and terrain dynamics
outer envelope above which elevation never
increased (core is 67 the envelope volume)? b)
standard deviation map shows areas with most
elevation change in red Mitasova, Overton,
Recalde,Bernstein, and Freeman, to appear in
JCR Wegmann and Clements, 2004, GRASS Newsletter
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Spatial and temporal indicators
a) time at minimum and b) time at maximum
maps reflect recent re-nourishment and dune
repair efforts c) regression slope maps show
spatial pattern of elevation trends, inset
transparency added as function of correlation
coefficient, white areas have r2lt0.3
elevation increase decrease
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Elevation surface evolution as volume
Standard representation hard to interpret set of
shorelines
New approach Evolution of terrain surface is
represented as a volume with time used for 3rd
dimension. Evolution of a contour is then
represented as an isosurface. The approach
reveals often neglected high dynamics of
foredunes and stability of backshore beach
(z1.5m)?
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Summary and references

• Neteler 2008 Ch 5.4.4., 6.8.6
• Hengl, T. and Reuter, H. I., 2008,
  Geomorphometry Concepts, Software, Applications,
  Elsevier