Part B: Surface analysis

1
Chapter 6

• Part B Surface analysis gridding and
  interpolation

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Surface analysis - Gridding Interpolation

• Gridding (a form of systematic interpolation)
• to convert a sample of data points to a complete
  coverage (set of values) for the study region
• to convert from one level of data resolution or
  orientation to another (resampling)
• to convert from one representation of a
  continuous surface to another, e.g. TIN to grid
  or contour to grid
• Quality of process
• Input data quality/accuracy
• Data density
• Data distribution
• Spatial variability of data
• Model applied typically weighted average

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Surface analysis - Gridding Interpolation

• Some other key issues
• Validity of continuity assumption/handling of
  breaklines/stratifications and missing data
• Validity of weighted average model
• How to choose weights?
• How many points to include in summation?
• Shape and size of neighbourhood region
• Selection of model and parameters
• Evaluation of quality
• Availability of related datasets

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Surface analysis - Gridding Interpolation

• Comparison of interpolation methods

Method Speed Type
Inverse distance weighting (IDW) Fast Exact, unless smoothing factor specified
Natural neighbour Fast Exact
Nearest-neighbour Fast Exact
Kriging -Geostatistical models (stochastic) Slow/Medium Exact if no nugget (assumed measurement error)
Conditional simulation Slow Exact
Radial basis Slow/Medium Exact if no smoothing value specified
Modified Shepard Fast Exact, unless smoothing factor specified
Triangulation with linear interpolation Fast Exact
Triangulation with spline interpolation Fast Exact
Profiling Fast Exact
Minimum curvature Medium Exact/Smoothing
Spline functions Fast Exact (smoothing possible)
Local polynomial Fast Smoothing
Polynomial regression Fast Smoothing
Moving average Fast Smoothing
Topogrid/Topo to Raster Slow/Medium Not specified
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Surface analysis - Gridding Interpolation

• Contour generation from grids
• Simple linear interpolation
• Smoothing splines or iterative thresholding

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Surface analysis - Gridding Interpolation

• Deterministic interpolation

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Surface analysis - Gridding Interpolation

• Deterministic interpolation IDW
• General model
• IDW version
• The values are computed for all or selected i
  (e.g. 12)
• The value for alpha is typically 1 or 2
• The division by the sum of weighted distances
  ensures that the weights add up to 1

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Surface analysis - Gridding Interpolation

• Deterministic interpolation IDW

Simple IDW calculations for point 5,5
each data point contributes
Grid point (5,5)
Calculation of weights using alpha1 and alpha2
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Surface analysis - Gridding Interpolation

• Sample data for main examples (x,y,z)

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Surface analysis - Gridding Interpolation

• Deterministic interpolation IDW

IDW grid alpha2, no smoothing
Source data
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Surface analysis - Gridding Interpolation

• IDW
• All grid points in rectangle are assigned a value
• Sample points are exactly fitted
• Surface looks rather peaky with odd
  hollows/bulls eyes - affected by point
  locations, alpha and grid resolution
• How do we know what values to choose - number of
  neighbours, alpha, grid res., break lines?
• Can we judge the goodness of fit of the
  resultant surface?
• Generally no, but could test the fit by (a)
  additional sampling (b) comparison with other
  sources of data (e.g. aerial photographs) (c)
  computational procedures

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Surface analysis - Gridding Interpolation

• Deterministic interpolation Nearest neighbour
• Simple assignment on nearest neighbour basis
• Uniform within zones, stepped between zones

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Surface analysis - Gridding Interpolation

• Deterministic interpolation Natural neighbour
• Assumes each existing point, i, has a natural
  sphere of influence its Voronoi polygon with
  area Ai
• Assumes a newly added point, p, (e.g. grid point)
  captures part of these natural areas for itself.
  These captured parts have areas Aip
• Weights are based on initial and captured Voronoi
  regions

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Surface analysis - Gridding Interpolation

• Deterministic interpolation Natural neighbour

Nat N grid extent limited to convex hull
Source data
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Surface analysis - Gridding Interpolation

• Deterministic interpolation Radial basis/splines
• A large family of interpolation functions
• Exact (or smoothing)
• Exhibit well-defined characteristics e.g.
  properties of smoothness and fit
• Thin plate splines widely used in Earth Sciences
• Core model is a form of weighted transformed
  distances

bias or offset value
z-value at point p
Radial basis function, for radius ri
Weight for ith point
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Surface analysis - Gridding Interpolation

• Deterministic interpolation Radial basis/splines
• Computation
• Compute matrix D of all interpoint distances
• Apply radial basis function to all elements of
  D??
• For grid point p, compute distance to all data
  points, as a vector r
• Apply radial basis function to all elements of
  r??
• Form augmented matrix equation shown and solve by
  inversion

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Surface analysis - Gridding Interpolation

• Deterministic interpolation Thin Plate Spline

Using 12 point neighbourhoods
Using all 62 data points
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Surface analysis - Gridding Interpolation

• Deterministic interpolation Modified Shepard
• Hybrid IDW-like system
• Typically uses local IDW or quadratic fit plus
  longer range IDW with separate parameter

Example using 8 local points for quadratic fit
and 16 for longer range IDW
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Surface analysis - Gridding Interpolation

• Deterministic interpolation
• Triangulation with linear interpolation (i.e. 2D
  linear model)

Linear grid extent limited to convex hull
Source data
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Surface analysis - Gridding Interpolation

• Deterministic interpolation Minimum curvature
• Seeks smoothed elastic membrane fit to surface
• Fit to residuals after linear regression of
  original dataset
• Linear component then added back

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Surface analysis - Gridding Interpolation

• Deterministic interpolation Local polynomial
• Fits a local polynomial (e.g. quadratic) to each
  grid point, using window (e.g. circle) of given
  size or number of points
• Smoothing

Example local quadratic distance-weighted OLS,
with 12 points per fit
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Surface analysis - Gridding Interpolation

• Deterministic interpolation some other methods
  
• Triangulation with non-linear interpolation -
  Clough-Tocher bi-cubic patches fitted to every
  triangle, such that edge and vertex joins are
  smooth
• Bi-linear suitable for dense uniform datasets,
  e.g. grids with some missing cell values simple
  weighted averaging of directly or indirectly
  neighbouring cells/points
• Profiling input dataset is contour vectors.
  Typically interpolation is linear between closest
  pair of contours (8-point)
• Moving average uses a moving circular or
  elliptical window (similar to time series
  analysis) to obtain simple local average
• Topogrid conversion of contour vectors to
  hydrologically consistent DEM. Includes
  enforcement of local drainage, lake boundaries,
  ridges and stream lines

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Core concepts
• Geostatistics definition
• models and methods for data observed at a
  discrete set of locations, such that the observed
  value, zi, is either a direct measurement of, or
  is statistically related to, the value of an
  underlying continuous spatial phenomenon, F(x,y),
  at the corresponding sampled location (xi,yi)
  within some spatial region A. (Diggle et al)

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Core concepts
• Addresses questions such as
• how many points are needed to compute the local
  average?
• what size, orientation and shape of neighbourhood
  should be chosen?
• what model and weights should be used to compute
  the local average?
• what errors (uncertainties) are associated with
  interpolated values?

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Variograms
• Analyse the observed variation in data values by
  distance bands using a spatial autocorrelation-lik
  e measure, ?
• Typically, analyse all pairs of observations that
  lie within specific distance bands
• Compute the average values of ? for each band
• Plot these averages against distance
• Fit an experimental curve to the observed pattern

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Variograms
• Analyse the observed variation in data values by
  distance bands using a spatial autocorrelation-lik
  e measure, ?
• Semivariance measure is most often used
• Bands have width ?. N(h) is the number of pairs
  in the band with mid-point distance h

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Zinc data (x,y,z)
• Burrough McDonnell (1998) App.3 98 soil samples
  (zzinc ppm)

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Variograms

Smallest observed separation
Fitted curve
sill
Average semivariance for band 4
C1C0C (structural variance)
Range, A0
C0 Nugget
Lag (distance band)
model
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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Variogram models

Model Formula Notes
Nugget effect Simple constant. May be added to all models. Models with a nugget will not be exact
Linear No sill. Often used in combination with other functions. May be used as a ramp, with a constant sill value set at a range, a
Exponential Exp() k is a constant, often k1 or k3. Useful when there is a larger nugget and slow rise to the sill
Spherical Sph() Useful when the nugget effect is important but small. Given as the default model in some packages.
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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Core concepts
• Geostatistical methods assume a general form of
  model which is a mix of
• (a) a deterministic model m(x,y)
• (b) a regionalised statistical variation from
  m(x,y)
• (c) a random noise (Normal error) component
  (actually two components which we cannot
  generally separate)
• (b) is chosen by analysing the semivariance, ?

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Further key
  concepts
• Sample size
• Support
• Declustering
• Thresholding
• Stationarity
• Transformation
• Anisotropy
• Madograms and Rodograms
• Periodograms and Fourier analysis

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Kriging
• Examine the data (z values) for spatial trends
  and Normality and transform variables if
  necessary, e.g. loge(z) or loge(z1)
• Compute the experimental variogram and fit a
  suitable model IFF the pattern warrants this
  (i.e. the data is not just noise)
• Check the model by cross validation and examine
  size of mean squared deviation
• Decide on method to use Kriging or Conditional
  simulation
• Generate a grid from the selected model and plot
• Compare the results to (a) the input data points
  and (b) other sources of data/information

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Kriging
• Ordinary Kriging basic procedure is exactly the
  same as for radial basis functions, but with
  function applied being the modelled variogram
• Computation
• Compute matrix D of all interpoint distances
• Apply variogram function to all elements of D??
• For grid point p, compute distance to all data
  points, as a vector r
• Apply variogram function to all elements of r??
• Form augmented matrix equation shown and solve by
  inversion

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Ordinary Kriging
  (OK)
• Estimated value at p is then
• Estimated variance at p is
• where m is the Lagrangian multiplier and the ci
  are the modelled semivariance values (the ?
  values)

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Ordinary Kriging

Zinc data Predicted values Zinc data Estimated standard deviation

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Kriging
• Goodness of fit methods
• Residual plots, standard deviation plots
• Examination for artefacts
• Simple cross-validation
• Jack-knifing
• Re-sampling
• Examination of independent datasets
• Stratification

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Kriging other
  models
• Universal Kriging Kriging with a trend
• Indicator Kriging thresholding
• Stratified Kriging regionalised approach
• Co-Kriging use of associated data

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Conditional
  simulation (Gaussian sequential method)
• Similar to OK approach. 3 core steps
• Analyse the data/transform as necessary, fit
  model variogram, define grid to use (possibly
  multi-level)
• Randomly select a grid node to visit (e.g. apply
  a random walk process) as execute step 3
• Apply OK estimation at node based on local
  observed data points. Take estimated value and
  apply Normal random process with mean as estimate
  and variance as estimated variance. Return to
  step 2
• This process is then repeated until all nodes
  have been visited. Then iterated M times

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Surface analysis - Gridding Interpolation

• Geostatistical interpolation Conditional
  simulation, M100 iterations

Zinc data Predicted values Zinc data Estimated standard deviation