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Overview of Statistical Analysis of Spatial DataGeog 210C
Modeling Semivariograms
Chris Funk
Lecture 13
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Why do we care about semi-variograms-I
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Why do we care about semi-variograms-II
In order to interpolate we need to assign weights to
each of our data points
If we can specify the error variance expected for any
point, based on the distance to our observations,
then we can solve for the ‘optimal’ set of weights
Assuming second order stationarity
The way we do this is to
Fit a variogram model
Use that model to fill in a covariance matrix
Invert the covariance matrix, and use it to solve for our
weights
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Why do we care about semi-variograms-III
A general recipe for interpolation:
A recipe for simple kriging
Simple kriging system
Simple kriging solution
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Sample Semivariogram
Given L lag distance classes; let {hl, l = 1, . . ., L} denote the set of
average distances between data pairs in each class
Calculate the sample semivariance γ(hl) of hl-specific scatter-plot of
lagged y-attribute values, for each distance class hl:
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Semivariogram / Covariogram / Correlogram Models
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Semi-variogram model defines covariance matrix
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Example: Consider three RVs Y(s1), Y(s2) and Y (s3), and assume that their
corresponding distances h12 = ||s2 − s1||, h23 = ||s3 − s2|| and h13 = ||s3 −
s1|| are three of the distance values in the abscissa of
the covariogram plot, i.e., h12, h23, h13 are three of the hl values in the set {hl,
l = 1, . . . , L}. Based on the above interpretation, the estimated (co)variance
matrix between these three RVs is:
where σ(0) is an estimate of the overall Y-variance
(corresponding to distance h11 = h22 = h33 = 0)
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Valid Semivarogram Models-1
Pure nugget effect
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indicates complete absence of spatial correlation
could correspond to measurement error and/or microstructure, i.e., features
occurring at scales smaller than sampling interval
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Valid Semivarogram Models-2
Spherical model
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linear behavior at origin
range parameter r
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Valid Semivarogram Models-3
Exponential Model
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linear behavior at origin; rises faster than spherical; reaches sill
asymptotically
effective range parameter r; distance at which 95% of sill reached
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Valid Semivarogram Models-4
Gaussian
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quadratic behavior at origin; reaches sill asymptotically
effective range parameter r; distance at which 95% of sill
reached
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Valid Semivarogram Models-5
Stable Semivariogram
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special cases: (i) ω = 1 → exponential model, (ii) ω = 2 → Gaussian model
effective range parameter r; distance at which 95% of sill reached
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Valid Semivarogram Models-6
Linear
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unbounded (without sill) semivariogram model; implies self-similarity
link to Brownian motion in 1D
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Valid Semivarogram Models-7
Power
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unbounded (without sill) semivariogram model; implies self-similarity
• for ω = 0 : γ(h) = σ(0) → pure nugget effect;
• for ω = 1 : γ(h) = σ(0)h → linear semivariogram;
• for ω > 2 : non-stationary random field
link to self-affine random fractals: D = E + 1 − ω/2, where D = fractal dimension,
and E = topological dimension of space (E = 1, 2, 3)
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Valid Semivarogram Models-8
Hole-effect of cosine
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indicates periodic spatial variability
distance from origin to first peak = size of underlying cyclic features unbounded (without
sill) semivariogram model; implies self-similarity
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Valid Semivarogram Models-9
Cardinal Sine
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indicates periodic spatial variability
distance from origin to first peak = size of underlying cyclic features unbounded (without
sill) semivariogram model; implies self-similarity
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Combinations of covariogram models
Nugget + Exponential
C. Funk Geog 210C Spring 2011
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Data – April 2001 Precip
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Standard Errors
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Temperature Trends
C. Funk Geog 210C Spring 2011
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Precip Trends
C. Funk Geog 210C Spring 2011
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