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 Squared Differences Plotted here (Called Semi variance) Distance Plotted Here Data usually follows a line like this Semivariograms are another Type of model that it often Pays to use your judgment and Fit yourself rather than just Tell the computer to do least Squares.

How Do You Do A Semivariogram (With a Computer Unless you have a death wish) Suppose I have this grid of Samples spaced on 50 foot Centers. I tell my computer to look at all Possible pairs of points that are 50 feet apart. (There are a whole Bunch of them). For each pair the computer Subtracts one from the other to Get a difference. I squares the Difference and then adds up the Results 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 spatial Correlation 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 conditions You 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 the Over-all variance of Samples. (As samples become Vary far apart they Have no spatial Relationship and tend To have the same Variance as just the Background of the Sample set.

A Range of Influence Semivariance Distance Look for a linear trend in the data at the first (you will probably see your semivariance Rising to meet your variance of samples)

Working on the Range Semivariance Distance The Linear trend will intersect the variance of samples At about 2/3rds the Range in influence. (The actual range curves up to meet the sample Variance).

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 represents The amount of the variation in the Deposit that shows spatial correlation Over 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 Semivariance is represented by gamma d represents distance N is the Nugget value fit to the graph This applies if d = 0 Semivariance for d>0 and <R where R is the range of Influence. C and N represent Cill and Nugget (Cill got misspelled because the mathematicians were French) 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 on A regular square grid. In that case you have the Computer start with each sample Individually and look for possible Pairs in a certain direction with A cone of tolerance. The cone is in turn broken up into Steps of distance. Any sample Pairs located in the interval are Treated as if they were at a grid Point (similar to the arbitrary limits we Use for cells in histograms).

Illustrations of Fitting Our Sample Variance is 100 Checking for anisotropy indications They both appear to be leveling out at 100 Both appear to have a nugget of about 20 Common Cills and Nuggets mean no Zonal Anisoptropy Checking for indications of different range Appear about the same – no geometric anisotropy

Fitting the Model Nugget looks like a read Of 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 Variance of samples is 100 again Checking for anisotropy Appears to have about same nugget at 20 Both appear to level off around 100 Therefore probably no zonal anisotropy Checking for Common Range of Influence I don’t think so – This must be a geometric anisotropy

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

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

Lets Try Another One 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 2/3 rd R is about 340 so R is about 510 on the X axis

Checking Out the Y Axis 2/3rds R is about 350 which implies R = 525 R= 510 and R= 525 are similar enough that an anisotropy in range is not Really worth modeling.

Trying One More Case Sample variance is again 100 Both 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) Sometimes Mineralization may be Controlled by Processes that have Different ranges of Influence. In this case we have Something with a Short range and Something with a long Range. I’m guessing I have a short range structure at around 100 Range

My Long Range Structure 2/3 rd 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 >0 but less than 500 so use the second part of the spherical model 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 13.4 Add components = 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 1/5 th of XRange so ΔY gets multiplied by 5  ΔX = ΔY = (ie 53.05*5)  Pythagorean Distance is  is >0 and < 500 so use 2 nd part of formula 58.57