Geostatistics uses spatial autocorrelation to model and predict values of regionalized variables that vary continuously over a spatial region. Spatial autocorrelation is evaluated using structure functions like the semivariogram and correlogram, which describe the spatial dependence of a variable over distance. The semivariogram plots the semivariance, or average dissimilarity between data pairs, against lag distance and can be modeled to interpolate values at unmeasured locations using kriging. Measures like the Moran's I and Geary's C statistics quantify spatial autocorrelation. Stationarity and isotropy are important assumptions that may require detrending or modeling anisotropy.

1. Geo-statistics
Notes on Introduction to Spatial Autocorrelation
Geostatistical methods were developed for interpreting data that varies
continuously over a predefined, fixed spatial region. The study of geostatistics
assumes that at least some of the spatial variation observed for natural phenomena
can be modeled by random processes with spatial autocorrelation.
D}
i
:
{z(i) 
Geostatistics is based on the theory of regionalized variables, variable
distributed in space (or time). Geostatiscal theory supports that any
measurement of regionalozed variables can be viewed as a realization of a
random function (or random process, or random field, or stochastic process)

2. Spatial Structure
Geostatisticals techniques are designed to evaluate the spatial structure of a
variable, or the relationship between a value measured at a point in one
place, versus a value from another point measured a certain distance away.
Describing spatial structure is useful for:
 Indicating intensity of pattern and the scale at which that pattern is exposed
 Interpolating to predict values at unmeasured points across the domain (e.g. kriging)
 Assessing independence of variables before applying parametric tests of significance

3. +
Z1
+
Z3
+
Zn
+
Z4 +
Zi
+
Z2
Regionalized Variable Zi
Variables are spatially correlated,
Therefore:
Z(x+h) can be estimated from
Z(x) by using a regression model.
** This assumption holds true with a
recognized increased in error, from
other lest square models.
Function Z in domain D
= a set of space dependent values
Histogram of samples zi
Z(x)
Z(x+h)
Cov(Z(x),Z(x+h))
D}
i
:
{z(i) 

4. Regionalized Variables take on values according to spatial location.
Given a variable z, measured at a location i , the variability in z can be broken down into
three components:
Where:



 )
(
)
(
)
( i
i
i s
f
z

)
(i
f A “structural” coarse scale forcing or trend

)
(i
s A random” Local spatial dependency

 error variance (considered normally distributed)
Usually removed by detrending
What we are interested in
Coarse scale forcing or trends can be removed by fitting a surface to the trend
using regression and then working with regression residuals
Regionalized Variable

5. X
1
3
2
1
:
:
µ
Y
3
2
5
3
:
:
µ
x y x·y x2 y2
b
a
Tan 

θ
a
b
As θ decreases, a/b goes to 0
θ1
a
b
Correlation: calculated using the geometric rather than the arithmetric mean
θ2
One distribution Two distributions
2
1 
 Tan
Tan 
data deviates
Deviations
Product
of
Deviations
Sum
of
squares
If you were to calculate correlation
by hand …. You would produce these
Terms.

6. Spatial Autocorrelation
Positive autocorrelation:
Negative autocorrelation:
No autocorrelation:
A variable is thought to be autocorrelated if it is possible to predict its value at
a given location, by knowing its value at other nearby locations.
 Autocorrelation is evaluated using structure functions that assess the
spatial structure or dependency of the variable.
 Two of these functions are autocorrelation and semivariance which are
graphed as a correlogram and semivariogram, respectively.
 Both functions plot the spatial dependence of the variable against the spatial
separation or lag distance.

7. Moran I (1950)
The numerator is a covariance (cross-product) term; the denominator is a variance term.
Where:
n is the number of pairs
Zi is the deviation from the mean for value at location i (i.e., Zi = yi – y for variable y)
Zj is the deviation from the mean for value at location j (i.e., Zj = yj – y for variable y)
wij is an indicator function or weight at distance d (e.g. wij = 1, if j is in distance class d
from point i, otherwise = 0)
Wij is the sum of all weights (number of pairs in distance class)
2
i
i
ij
i j
j
i
ij
(d)
Z
W
Z
Z
w
n
I



• A cross-product statistic that is used to describe autocorrelation
• Compares value of a variable at one location with values at all other locations
Values range from [-1, 1]
Value = 1 : Perfect positive correlation
Value = -1: Perfect negative correlation

8. 
  


i
2
i
ij
i j
2
j
i
ij
(d)
Z
2W
]
)
y
(y
w
1)[
[(N
C
• A squared difference statistic for assessing spatial autocorrelation
• Considers differences in values between pairs of observations, rather than the
covariation between the pairs (Moran I)
GEARY C (1954)
The numerator in this equation is a distance term that gets squared.
The Geary C statistic is more sensitive to extreme values & clustering than the Moran I.
Values range from [0,3]
Value = 1 : No autocorrelation
Value = 0 : Positive autocorrelation
Value > 1 : Negative autocorrelation

9. Semivariance
2
)
(
1
)
(
2 j
i
i j
ij
d
y
y
w
n
d 
 

Where :
j is a point at distance d from i
nd is the number of points in that distance class (i.e., the sum of the weights wij for
that distance class)
wij is an indicator function set to 1 if the pair of points is within the distance class.
2
)
(
)
(
2
1
)
( i
d
i
d
n
i
y
y
d
n
d 

 



The geostatistical measure that describes the rate of change of the regionalized
variable is known as the semivariance.
Semivariance is used for descriptive analysis where the spatial structure of the data
is investigated using the semivariogram and for predictive applications where the
semivariogram is fitted to a theoretical model, parameterized, and used to predict the
regionalized variable at other non-measured points (kriging).

10. The sill is the value at which the semivariogram levels off (its asymptotic value)
The range is the distance at which the semivariogram levels off (the spatial
extent of structure in the data)
The nugget is the semivariance at a distance 0.0, (the y –intercept)
A semivariogram is a plot of the structure function that, like autocorrelation,
describes the relationship between measurements taken some distance apart.
Semivariograms define the range or distance over which spatial dependence exists.

11. Autocorrelation assumes stationarity, meaning that the spatial
structure of the variable is consistent over the entire domain of the dataset.
The stationarity of interest is second-order (weak) stationarity, requiring that:
(a) the mean is constant over the region
(b) variance is constant and finite; and
(c) covariance depends only on between-sample spacing
 In many cases this is not true because of larger trends in the data
 In these cases, the data are often detrended before analysis.
 One way to detrend data is to fit a regression to the trend, and use only
the residuals for autocorrelation analysis
Stationarity

12. Autocorrelation also assumes isotropy, meaning that the spatial structure of the
variable is consistent in all directions.
Often this is not the case, and the variable exhibits anisotropy, meaning that there
is a direction-dependent trend in the data.
Anistotropy
If a variable exhibits different ranges in different directions, then there is a geometric
anisotropy. For example, in a dune deposit, permeability might have a larger range
in the wind direction compared to the range perpendicular to the wind direction.
If the variable exhibits different sills in different directions, then there is a zonal
anisotropy. For example, a variogram in a vertical wellbore typically shows a bigger
sill than a variogram in the horizontal direction.
Some variograms are a combination of both geometric and zonal anisotropies.

13. bd
co 

γ(d)
)]
/
exp(
1
[
γ(d) 2
2
a
d
c
co 



)]
/
exp(
1
[
γ(d) a
d
c
co 



Gaussian:
Linear:
Spherical:
Exponential:
For predictions, the empirical semivariogram is converted to a theoretic one by
fitting a statistical model (curve) to describe its range, sill, & nugget.









a
d
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c
a
d
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d
a
d
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c
o
o
,
)],
2
/
(
)
2
/
3
[
γ(d)
3
3
There are four common models used to fit semivariograms:
Where:
c0 = nugget
b = regression slope
a = range
c0+ c = sill
Assumes no
sill or range

14. • Check for enough number of pairs at each lag distance (from 30 to 50).
• Removal of outliers
• Truncate at half the maximum lag distance to ensure enough pairs
• Use a larger lag tolerance to get more pairs and a smoother variogram
• Start with an omnidirectional variogram before trying directional variograms
• Use other variogram measures to take into account lag means and variances
(e.g., inverted covariance, correlogram, or relative variograms)
• Use transforms of the data for skewed distributions (e.g. logarithmic transforms).
• Use the mean absolute difference or median absolute difference to derive the range
Variogram Modeling Suggestions