Modeling Spatial Correlation(The Semivariogram)

©2007 Dr. B. C. Paul

Fitting Spatially Correlated Data This is the case where grabbing random

samples from around does not produce a random result Samples closer together in location parameter are

likely to be similar Real Engineering world situations

Location is location Your taking samples of a site – those close together are

likely to be similar Common with Environmental Clean-ups or Ore

Reserves We encountered this situation when we were

looking at decreasing variance of the mean I showed you a plot of half squared differences that

a computer referenced to see how much variance would average out when one went to larger samples.

The Semivariogram

Half SquaredDifferencesPlotted here(CalledSemi variance)

Distance Plotted Here

Data usually follows a line like this

Semivariograms are anotherType of model that it oftenPays to use your judgment andFit yourself rather than justTell the computer to do leastSquares.

How Do You Do A Semivariogram (With a Computer Unless you have a death wish)

Suppose I have this grid ofSamples spaced on 50 footCenters.

I tell my computer to look at allPossible pairs of points that are50 feet apart. (There are a wholeBunch of them).

For each pair the computerSubtracts one from the other toGet a difference. I squares theDifference and then adds up theResults for every pair.

Continuing My Computational Love Fest The Computer eventually totals up all the

squared differences for every sample pair in the whole grid that is 50 feet apart.

The computer then looks at its tally of how many pairs it found and divides the total by the number of pairs to get an average squared difference for pairs 50 feet apart.

The computer then divides that value by two It’s a calculus and derivation thing where an extra

two shows up and its convenient to just include it in the definition of semivariance

I have my computer do this for every other distance and compute a semivariance for that distance.

The Result

Semivariance

Distance

We now have a sort of histogram of spatialCorrelation of samples.

Model Fitting Time

Semivariance

Distance

We will try to fit a mathematical model to our pattern of spatial correlation.(Not just any line will do – has to meet specific mathematical conditionsYou don’t want to understand).

We Will Look at Fitting a Spherical Model(Because it works about 95% of the time)

Semivariance

Distance

Plot a line for theOver-all variance ofSamples.(As samples becomeVary far apart theyHave no spatialRelationship and tendTo have the sameVariance as just theBackground of theSample set.

A Range of Influence

Semivariance

Distance

Look for a linear trend in the data at the first(you will probably see your semivarianceRising to meet your variance of samples)

Working on the Range

Semivariance

Distance

The Linear trend will intersect the variance of samplesAt about 2/3rds the Range in influence.(The actual range curves up to meet the sampleVariance).

What Does Range of Influence Mean?

Samples located within the range of influence of each other are spatially correlated and when you draw one sample the value of the other sample a distance away is not a matter of random chance When you have a spatially correlated sample

set you can use that information to make more than luck of the draw guesses on the values at points that were never sampled (Can see how I could mine an ore deposit or clean

up an environmental mess better if I knew that kind of stuff)

The Cill and Nugget

Semivariance

Distance

This value is the Cill – it representsThe amount of the variation in theDeposit that shows spatial correlationOver a range of influence.

Your linear trend normally does not intersect 0 at zero distance –Most real deposits have a random element – called the nugget (it first got its name from whether your gold sampling happened to hit a gold nugget or not).

The Model

Nd )(

)*2*2

*3(*)( 3

3

Rd

R

dCNd

Semivariance is represented by gammad represents distanceN is the Nugget value fit to the graphThis applies if d = 0

Semivariance for d>0 and <R where R is the range ofInfluence. C and N represent Cill and Nugget(Cill got misspelled because the mathematicians were French)

NCd )(

Semivariance for d>R

The Model

The spherical model is called a three part model (any guesses about why?)

Our model represents the average similarity of sample values located a distance d apart We’ll look at how we can use that later on

One assumption we make is “Stationarity” Means that our model of spatial correlation continues

to fit over the entire study area May need different models for different types of

mineralized rock or contaminated soils

Some Exceptions

Sometimes the model varies depending on which direction you are moving In that case you have to have your

computer look by direction as well as distance for pairs of samples in the set.

You will fit model differently in different directions

This is called anisotropy Need more detailed study on geostatistics

to get good explanation of anisotropic models and how to fit them.

The Not Really a Grid Factor

Sometimes samples are not onA regular square grid.In that case you have theComputer start with each sampleIndividually and look for possiblePairs in a certain direction withA cone of tolerance.

The cone is in turn broken up intoSteps of distance. Any samplePairs located in the interval areTreated as if they were at a gridPoint(similar to the arbitrary limits weUse for cells in histograms).

Illustrations of Fitting

Semivariogram in X Direction

0

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Ga

mm

a

X

Semivariogram Y Direction

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Gam

ma

Y

Our Sample Variance is 100Checking for anisotropy indications

They both appear to be leveling out at 100Both appear to have a nugget of about 20Common Cills and Nuggets mean no Zonal Anisoptropy

Checking for indications of different rangeAppear about the same – no geometric anisotropy

Fitting the Model

Semivariogram Y Direction

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Gam

ma

Y

Nugget looks like a readOf 20.

Levels out at 100 Cill = 100 – 20 =80

2/3rds R is about 350 so Range of influence is about 525

Lets Try Another One

Semivariogram in X Direction

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Ga

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Semivariogram Y Direction

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Gam

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Y

Variance of samples is 100 againChecking for anisotropy

Appears to have about same nugget at 20Both appear to level off around 100Therefore probably no zonal anisotropy

Checking for Common Range of InfluenceI don’t think so – This must be a geometric anisotropy

Checking Out the Range

Semivariogram in X Direction

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Gam

ma

X

2/3rds R is about 350 again – so range is about 525Nugget = 20 Cill = 80

Check Out the Y Axis

Semivariogram Y Direction

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Ga

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a

Y

2/3rd R is about 150 so range Y is 225 Range is a little more than twice as far inThe X direction (or principle axis)Cill is 80 Nugget is 20

Lets Try Another OneSemivariogram in X Direction

01020304050607080

0 500 1000

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Ga

mm

a

X

Semivariogram Y Direction

020406080

100120140160

0 500 1000

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Gam

ma

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Sample Variance is still 100 but X is only getting to about 70 and Y about 130

Also X appears to hit Y axis at 10 while in Y it appears to be 30

This is appears to be a Zonal Anisotropy

I’m A Little Unsure About the Range being the Same or Different

Semivariogram in X Direction

01020304050607080

0 100 200 300 400 500 600 700 800 900 1000

Distance

Gam

ma

X

2/3rd R is about 340 so R is about 510 on the X axis

Checking Out the Y Axis

Semivariogram Y Direction

020406080

100120140160

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Distance

Gam

ma

Y

2/3rds R is about 350 which implies R = 525R= 510 and R= 525 are similar enough that an anisotropy in range is notReally worth modeling.

Trying One More CaseSemivariogram in X Direction

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Distance

Ga

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X

Semivariogram Y Direction

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Ga

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Sample variance is again 100Both directions appear to intersect Y axis around 10 and to level out at 100

Probably no zonal anisotropy

Range is not obviously different but is a little “funky”

When Range is Kinky(The Nested Structure)

Semivariogram in X Direction

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Ga

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SometimesMineralization may beControlled byProcesses that haveDifferent ranges ofInfluence.

In this case we haveSomething with a Short range andSomething with a longRange.

I’m guessing I have a short range structure at around 100 Range

My Long Range Structure

Semivariogram in X Direction

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Gam

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2/3rd R is about 525 so R long is about 787

Making Semivariogram Math Work Out

Isotropic with no nested structures Just use the 3 part model

Geometric Anisotropy (means range varies by direction) Use coordinate transform

If Rx is twice Ry use an isotropic three part model, but double the y component of distance before doing the 3 part model calculation

Handling Zonal Anisotropies and Nested Structures

Handled by adding components For zonal anisotropy add separate models together to

get the total model Nugget*cos(θ) will give variable nugget by direction A normal 3 part model that only counts distance

component in one direction will make cill change. Can have an isotropic or geometric anisotropy to

handle the other component Nested Structures

Just separately compute the long and short range models and then add them up.

As a practical matter – feed the task to a computer and let it calculate the predicted gamma values for the model you figured out.

Mathematical Examples – Computing a raw value of gamma

The following samples are a distance 50 feet apart on an isotropic semivariogram model P1 25 P2 31

Diff is 6 Squared diff. is 36 Half squared diff is 18 Total so far is 18 Number of pairs so far is 1

Continuing the computation

P1 = 25, P3 = 20 Difference is 5 Squared diff is 25 Half squared diff is 12.5 Total to this point is 30.5 Number of pairs so far is 2

P2 = 31, P4 = 38 Difference is 7 Squared diff is 49 Half squared diff is 24.5 Total to this point is 45 Number of pairs so far is 3

Continuing the computation

P3 = 20, P5 = 27 Difference is 7 Squared difference is 49 Half Squared diff is 24.5 Total to this point is 69.5 Number of pairs so far is 4

P4 = 38, P6 = 33 Difference is 5 Squared difference is 25 Half Squared diff is 12.5 Total to this point is 82 Number of pairs so far is 5

Finishing the raw gamma value for pairs 50 feet apart

82 / 5 = 16.4 16.4 would be the value plotted on the

semivariogram Most values plotted in reality will be

based on more than 5 pairs, however the calculation procedure is the same.

Eventually so called sample points on the semivariogram will be replaced with a mathematical model that will be used to compute gamma values where ever they are needed.

Examples of Semivariogram Mathematical Models

Case 1 – An isotropic spherical model with a cill of 80, a nugget of 20 and a range of 500 Let point 1 be X= 0, Y =0 Let point 2 be X= 75 Y =0 ΔX = 75 ΔY = 0 Pythagorean distance is 75 75 >0 but less than 500 so use the second part of

the spherical model

)*2*2

*3(* 3

3

RD

R

DCN )

*2

75

500

75*5.1(*8020

5003

3

Ie – 37.87

More Mathematical Models

An isotropic nested structure Nugget =20 Range 1 =50 Feet Cill 1 =20, Range 2 = 500 Feet Cill 2 = 60

Nested structures are accomplished by simply adding model components

P1 X=0, Y=0 , P2 X=53.03, Y=53.03 ΔX=53.03 ΔY=53.03 Pythagorean Distance = 75 Nugget = 20 Model 1 – 75>50 so model is C+N or 20 Model 2 – 75>0 and 75<500 so use 2 model component

)*2*2

*3(* 3

3

RD

R

DCN

)*2

75

500

75*5.1(*6020

5003

3

13.4

Add components 20+20+13.4 = 53.4

More Models

A geometric anisotropy with Nugget = 20, Cill=80 RangeX=500, Range Y=100

P1 X=0, Y=0 , P2 X=53.03, Y=53.03 Geometric Anisotropy is handled by stretching

the distance in the short axis direction YRange is 1/5th of XRange so ΔY gets multiplied by

5 ΔX = 53.03 ΔY = 265.15 (ie 53.05*5) Pythagorean Distance is 270.4 270.4 is >0 and < 500 so use 2nd part of formula

)*2*2

*3(* 3

3

RD

R

DCN 58.57