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

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)

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

Valid Semivarogram Models-2

Spherical model

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linear behavior at origin

range parameter r

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

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

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

Valid Semivarogram Models-6

Linear

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unbounded (without sill) semivariogram model; implies self-similarity

link to Brownian motion in 1D

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)

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

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

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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