Geostatistics Interpolation
Why: So far, none of the interpolation methods discussed so
far provides direct estimates of the quality of the predication
made in terms of an estimation variance for the predicted
values at certain location.
 In all methods, the only way to determine the goodness of
the prediction would be to compute estimates for a set of
extra checkpoints that had not been used in the original
interpolation.
 A further objection of all methods so far is that there is no a
priori method of knowing whether the best values have
been chosen for the weighting parameters or if the size of
the search neighborhood is appropriate. Moreover, no
method studied so far provides sensible information on
(Davis, 1989):
 The number of points needed to compute the local
coverage
 The size, orientation, and shape of the neighborhood
from which those points are drawn
 Whether there are better ways to estimate the
interpolation weights than as a simple function of
distance,
 The errors associated with the interpolated values
Solution: Optimal methods of interpolation using Geostatistics
methods.
Naser El-Sheimy
Why Geostatistics
3
Consider the following problem of local
4
2
estimation: Estimating the elevation of
point (p) using the elevation of some
1
points
p
 It is reasonable, as discussed before,
to evolve an interpolation procedure
7
5
6
which weight point (1) “nearest point”
higher than any other farthest points.
 A whole range of methods can be produced to decide on the
weights according to each sample, mostly based on the
distance (and possibly direction) of the sample from the
point being estimated.
Problems with this approach:
1. Which weighting factors are the best to choose (e.g. 1/d or
1/d2)?
2. How far do we go in including points?
3. How reliable is the estimated elevation?
4. Can we expect the same method to be equally valid on all
types of terrain?
 On the other hand, the idea of weighting samples by some
measure of their similarity to what is being estimated is
intuitively appealing
 ‘Similarity’ can be measured by the covariance between
the samples or by their correlation (BUT WHAT ABOUT
STATIONARITY)
 Let us look instead at the ‘difference in elevation’ between
samples and at the same time try to describe these differences
as a function of the distance between them and use that in
describing the similarity between samples: This is the
concept of Variogram and Geostatistics
Naser El-Sheimy
Introduction: Geostatistics
 Geostatistics is a branch of applied statistics developed by
George Matheron of the Centre de Morophologie
Mathematicque in Fontainebleau, France (1960).
 A unique aspect of geostatistics is the use of regionalized
variables which are variables that fall between random
variables and completely deterministic variables.
 Typical regionalized variables are functions describing
variables that have geographic distributions (e.g. elevation
of ground surface).
 Unlike random variables, regionalized variables exhibit
spatial continuity; however, the change in the variable is so
complex that they cannot be described by any deterministic
function.
 Furthermore, although regionalized variables are spatially
continuos, it is not always possible to sample every
location. Therefore, unknown values must be estimated
from data taken at specific locations that can be sampled.
 The size, shape, orientation, and spatial arrangement of
these samples constitute the support of the regionalized
variables and influence the capability to predict the
unknown samples.
In brief, geostatistical analysis usually has the following steps:
1. Estimating the form of the regionalized variables (i.e. the
spatial similarity between the samples)
2. Estimation of the surface using Point or Universal kriging.
Naser El-Sheimy
Hypothesis of Variograms
Regionalized variable theory uses the semivariance to express the
degree of relationship (similarity) between
3
points on a surface.
2
In the opposite figure, it seems sound to
expect that the elevation of point (7) will
be ‘very different’ from that at point (p),
whilst point (1) will have a value ‘not very
different’ from that at point (p).
1
p
7
6
Let us assume that the difference in
elevation between two points depends only on the distance between
them and their relative orientation (e.g. SN, NW, etc.).
Suppose that we took a number of point-pairs that are (d) distance
apart and we calculate the difference in elevation between these
point-pairs. And once again but with distance (2d) and so on.
Therefore, we can build up a histogram of the differences in
elevation and how it relates to the distance between the point-pairs.
By this way we can build an implicit assumption of the distance
weighing technique into a statistical form.
But investigating histograms is a very tedious job, therefore let us
resort to the usual trick of summarizing the histogram by its
parameters: the mean and the variance (of the differences).
The quantity which of interest to us in these case is the variance of
the differences in elevation between samples, but in these case we
will call it semivariance or variogram (because it varies with the
distance and the direction of the distance ‘d’)
Naser El-Sheimy
4
5
Hypothesis of Variograms
The variogram is a measure of how quickly things change on the
average. The underlying principle is that, on the average, two
observations closer together are more similar than two observations
farther apart. Because the underlying processes of the data often
have preferred orientations, values may change more quickly in one
direction than another. As such, the variogram is a function of
direction.
Having defined a semivariogram or variogram, what sort of behavior
do we except it to have?
 The units of the semivariance should have the same units as the
variance of the elevation difference (i.e. m2, ft2, etc)
 The magnitude of the semivariance between points depends on the
distance and orientation between the points:
 A smaller distance yields a smaller semivariance and a larger
distance results in a larger semivariance.
 The semivariance = 0 at a distance d = 0. However, as points are
compared to increasingly distant points, the semivariance
increases. At some distance, called the range or span of the
regionalized variable.
 The range is the greatest
distance over which the value
at a point on the surface is
related to the value at another
point (neighborhood).
Semivariogram Models (Ideal
Model)
Sill
Linear
Sphere
Exponential
d
Range or Span
Lag d
Naser El-Sheimy
Calculating the Semivariance
1. Regularly Spaced Points (Omnidirection)
Consider regularly spaced points distance (d) apart, the
semivariance can be estimated for distances that are multiple of
(d):
 (d ) 
1 N (d ) ( z  z ) 2
2 N (d ) i
1 i ih
Where,
zi
is the measurement of a regionalized variable taken at location i
zi+h is another measurement taken (h) intervals away
Nd is number of separating distance = number of points – Number of
Lags
d
\
d
[1]
[2]
[3]
[4]
[2] Calculations for lag (d), [3] for lag 2d, and [4] for lag 3d
2. Irregularly Spaced Data
The difference from the omnidirectional
variograms is that h is a vector rather than a
scalar. For example, if d={d1, d2}, then each
pair of compared samples should be separated
by d1 m in E-W direction and by d2 m in S-N
direction, see the opposite figure:
Naser El-Sheimy
d2
d1
Irregularly Spaced Data (cont.)
 In practice, it is difficult to find enough sample points which are
separated by exactly the same lag vector [d]. Thus, the set of all
possible lag vectors is usually partitioned into classes.
Directional variograms
 Vectors that end in the same cell are grouped into one class and
variograms value is estimated separately for each class. The
number of directions may be different (4, 8, 16, etc.)
 Variograms is estimated using the same equation as the
omnidirectional one:
N (d )
1
 (d ) 
( zi  z ) 2

i h
2 N (d ) i1
The only difference here is the points used in the equation are
defined as the points located at the tail and head of vector [d].
Naser El-Sheimy
Understanding semivariances
 The semivariance is not only equal to the average of the squared
differences between pairs of points spaced at distance h, it is also
equal to the variance of these differences. That is, the
semivariance can also be defined as:
2

 (Z i  Z ih )  
1  N (d ) 
 (d ) 
 (Z  Z ih ) 
 
2 N (d )  i1  i
N (d )
 




 Note that the mean of the regionalized variable Zi is also the
mean of the regionalized variable at Zi+h because these are the
same observations, merely taken in a different order. That is
 Zi  Zi h

N
N
d
d
Therefore, their differences must be zero. That is:
 Zi  Zi  h  ( Zi  Zi  h )


0
N
N
N
d
d
d
Note that this strictly true if the regionalized variable is
stationary. If the data are not stationary, the mean of the sequence
changes with d and the d) equation must be modified.
Semivariance
h
Autocorrelation
1
Autocorrelation
Distance d
Naser El-Sheimy
Estimation of the surface using Point Kriging
 Kriging is named after a South African engineer, D. G. Krige.
 Kriging estimates are best linear unbiased estimates of the surface
at the specified locations, provided the surface is stationary and
the correct form of the variogram has been determined.
Principle (the use of variograms in Kriging):
Given: (3) points of known elevation
1
Required: Estimate the elevation of point
(p) (call it Ze(p)) using a weighted average
of the known samples:
p
Ze ( p)  Wi  Z ( pi )
2
this is called
linear estimator, and if the weights sum to 1 we call it unbiased
linear estimator (remember if all weight are the same we will end up
with the mean of all samples).
For simplicity let us consider the following situation:
 Ze(p) = Z1
that is all weights, except W1, equal zeros.
 This estimated value will most likely differ from the actual
value at point p, Za(p), and this difference is called the
estimation error:
  p  Z e ( p)  Z a ( p)  Z1  Z a ( p) and
the variance of the estimation error =  2p  (Z1  Z a ( p)) 2 which is
the is the same form as the semivariance at a distance d1p (in fact it
is twice the semivariance  (d1 p ) .
Naser El-Sheimy
3
Estimation of the surface using Point Kriging (Cont.)
Now if we use the three known elevations, Z1, Z2, and Z3, to
estimate the unknown elevation at point P, Ze(P). Three weights
must be determined W1, W2, and W3, to make an estimate. Once
the individual weights are known, estimation can be made by
Ze ( p)  W Z W Z W Z such that W1+ W2+ W3 = 1
1 1
2 2
3 3
The estimate and estimation error-depend on the weights chosen.
Ideally, kriging tries to choose the optimal weights that produce the
minimum estimation error.
That is kriging minimize the estimation variance with respect to
weights, i.e:
 2
0
Wi
i=1 to n (n is 3 in our example)
 This will provide n-equation in n-unknowns (W1, W2, and W3).
These weights will provide an estimator that has minimum
value of the estimation variance.
 However, they will not necessarily add up to one (there is
nothing in the above equations that constraints the sum of the
weight to one).
 Effectively we need to satisfy the equation Wi  1 . Thus, we
actually will have (n+1) equation in (n) unknowns.
 To rectify this we must introduce another unknown, in the form
of Lagrange multiplier, , to balance up the system (i.e. get n+1
equations in n+1 unknowns.
Naser El-Sheimy
Estimation of the surface using Point Kriging (Cont.)
Therefore instead of minimizing the estimation variance, we actually
minimize:  2   (Wi 1) with respect to W1, W2, ……..Wn and 
After differentiating  2   (Wi 1) with respect to W1, W2, W3 and
, the following set of equation is obtained:
W  (d )  W  (d )  W  (d )     (d )
1 11
2 12
3 13
1p
W  (d )  W  (d )  W  (d )     (d )
1 21
2 22
3 23
2p
W  (d )  W  (d )  W  (d )     (d )
1 31
2 32
3 33
3p
W W W 1
1
3
3
 Where (dij) is the semivariance between control points i and j
corresponding to the distance between them, dij.
 Since dij = dji the left-hand matrix is symmetrical, with zeroes
along the diagonal since the distance from a point to itself is
zero. The values of the semivariances are taken from the known
or estimated semivariogram.
 Separating these equations into matrix form yields
 ( d )
11


(
d
)

21

 (d 31 )
 1


With
 (d12 )  (d13 )
 (d 21)  (d 23 )
 (d32 )  (d33 )
1
1
an
1 W    (d1 p ) 

  1 
1 W   (d 2 p )
2  


1 W   (d ) 
3  3p 


0     1 
estimation
 z2  W1 (d1 p )  W2 (d 2 p )  W3 (d3 p )  
Naser El-Sheimy
variance
of:
Universal Kriging
 In Punctual Kriging we assumed that the regionalized variable
being mapped is stationary.
 Unfortunately, in DTM it may happens that regionalized variables
are not stationary, but rather exhibit changes in their average
values from place to place.
If we attempt to compute a semivariogram for such a variable, we
will discover that it may not have the properties we have discussed.
However, if we reexamine the definition as in the following

2 

 N (d ) 
)
 (Z  Z
1

i


i

h
)

equation:  (d ) 
(Z  Z
 


2 N (d )  i1  i ih
N (d )
 




We will note that it contains two parts, the first is the difference
between pairs of points and the second being the average of these
differences.
 If the regionalized variable is stationary, the second part
vanishes, but it is not stationary, this average will have some
value.
 Therefore, the regionalized variable can be regarded as
composed of two parts, the residual and the drift. The drift is
the weighted average of points within the neighborhood around
the unknown value. The residual is the difference between the
regionalized variable and the drift.
 If the drift is removed from a regionalized variable, the residual
then becomes a regionalized variable in itself which is
stationary, and thereby, allowing a semivariogram to be made.
Naser El-Sheimy
Universal Kriging
Universal Kriging can thus be regarded as consisting of 3 steps:
1. Achieving stationarity by estimating and removing the drift
2. Kriging the stationary residuals to obtain the needed estimates,
and
3. Combining the estimated residuals with the drift to obtain
estimates of the actual surface
How it works
Method 1: Removing the drift
1. Removing the drift using the global interpolation methods
2. Following that, the residuals can be treated as a regionalized
variable and therefore punctual kriging methods can be used.
3. Note that the output of the punctual kriging is residuals. That is,
in order to obtain actual surface, the drift has to be added back.
Method 2: Estimating the drift simultaneously while kriging
1. Incorporate the drift into the system of simultaneous equations
used to find the kriging weights as additional constraints.
2. Solving this expanded set of equations will produce a set of
weights for the kriged estimate, which will include the effect of
specified drift within the local neighborhood.
Naser El-Sheimy
Universal Kriging (Cont.)
Estimating the drift require that the drift to be modeled by an
arbitrary function of the points used in the computation, e.g. low
order polynomial. That is the drift D at point P might be defined as
either:
Firs order polynomial: D p  a1X i  a2Yi
Or second order: D p  a1X i  a2Yi  a3 X iYi  a4 X i2  a5Yi2
Where, Xi, Yi are the coordinates of the ith point within the
neighborhood, and the a's are the unknown drift coefficients.
The simplest example consists of kriging a point, assuming the drift
is linear (i.e. the number of primary unknowns is three “one
elevation and two drift parameters”). The linear drift model, given
in, has two coefficients, so a minimum of three known points must
be used in the drift estimation process, or we will run out of degrees
of freedom.
The following is an example of how to estimate both the drift and
the regionalized variable by universal kriging:

















h11
h21
h31
h41
h51
 h12
 h22
 h32
 h42
 h52
 h13
 h23
 h33
 h43
 h53
 h14
 h24
 h34
 h44
 h54
 h15
 h25
 h35
 h45
 h55
1
1
1
1
1
X
1
Y
1
X
2
Y
2
X
3
Y
3
X
4
Y
4
X
5
Y
5
Naser El-Sheimy
1
1
1
1
1
0
0
0
Y  W    h1 p 
1
1  1 


X
Y  W   h2 p 

2
2  2 




 h3 p 
X
Y  W
3
3  3 
X
Y  W   h4 p 
4
4  4 
X
Y  W   h5 p 
5
5
5
 
0
0      1 


0
0      X p 
1 

 
0
0     Y p 
  2 

X