Ordinary Kriging as the most appropriate interpolation method for porosity in the
Sava depression Neogene sandstones
Obični kriging kao najprimjerenija interpolacijska metoda za poroznost u
neogenskim pješčenjacima Savske depresije
Davorin BALIĆ1 and Tomislav MALVIĆ1,2
1INA-Oil
industry, Oil & Gas Exploration and Production, Reservoir Engineering & Field Development, Šubićeva 29,
10000 Zagreb, e-mail: davorin.balic@ina.hr (Reservoir Geologist); tomislav.malvic@ina.hr (Adviser)
2University
of Zagreb, Faculty of Mining, Geology and Petroleum Engineering, Pierottijeva 6, 10000 Zagreb, visiting
lecturer
Abstract: Porosity can be mapped by many computer-based interpolated methods. All of them are
defined by less and more advanced mathematical equations. There is several works performed on
data from the Sava depression that tested appropriation of several such mapping methods,
available in almost all computer mapping software packages. Mathematically simpler methods are
quicker, but often less precise in cases based on smaller input dataset (approximately 10-15 hard
data or lesser). There is another option to use geostatistical deterministical methods like kriging. In
kriging methods the most often and very suitable is the Ordinary kriging technique, developed from
equations of the Simple Kriging as the basic kriging approach. It can be proven, using matrix
equations, that kriging variance (as the measure of error) is decreased in the Ordinary than in the
Simple Kriging.
The presented analysis was directed in the porosity dataset as the observed input derived from the
reservoir. Three simpler methods had been tested (Inverse Distance, Nearest Neighborhood,
Moving Average) to obtain porosity maps. Furthermore is proven that variance minimizing in better
in the Ordinary Kriging than in the Simple Kriging. Finally, the map interpolated by the Ordinary
Kriging and belonging numerical result had been point out as the most appropriate tool (and the
interpolation technique) that could be applied for porosity distribution description in reservoir
sandstones of Neogene age (from Early Pannonian to Early Pontian). Due to lithological,
depositional and tectonic properties similarities in all hydrocarbon reservoirs located in the Sava
depression, presented result and given recommendations can be applied for entire depression.
Key words: Ordinary Kriging, Simple Kriging, porosity, sandstones, Sava depression, Neogene
Sažetak: Poroznost se može kartirati upotrebom brojnih računalnih interpolacijskih metoda. Sve
one su temeljene (manje ili više) na naprednim matematičkim jednadžbama. Postoji nekoliko
primjera načinjenim na podatcima iz Savske depresije kojima je isprobana upotrebljivost takvih
metoda, a sve su bile dostupne u gotovo svim računalnim paketima za kartiranje. Matematički
jednostavnije metode su brže, no često i manje precizne u slučajevima kada ulazni skup sadrži
mali broj podataka (približno 10-15 ili manje). Postoji i druga mogućnost koja obuhvaća upotrebu
geostatističkih determinističkih metoda poput kriginga. Kod te metode vrlo je česta i primjerena
upotreba tehnike običnoga kriginga, razvijene iz jednadžbi jednostavnoga kriginga (koji je osnovna
tehnika kriginga). Moguće je dokazati, upotrebom matričnih jednadžbi, da je varijanca kriginga (kao
mjera pogrješke) manja u tehnici običnoga nego li jednostavnoga kriginga.
Prikazan analiza bila je usmjerena na skup vrijednosti poroznosti kao ulazne podatke iz ležišta.
Testirane su tri jednostavnije metode (inverzne udaljenosti, najbližega susjedstva, pokretne
sredine) kako bi se načinile karte poroznosti. Nadalje, dokazano je i kako je varijanca pogrješke
manja kod običnoga nego li jednostavnoga kriginga. Na kraju, karta interpolirana običnim
krigingom te numerički rezultat su istaknuti kao najprimjereniji alat (i interpolacijska tehnika) kojom
se može opisati raspodjela poroznosti u pješčenjačkim ležištima neogenske starosti (od donjega
panona do donjega ponta). Zahvaljujući sličnim litološkim, taložnim i tektonskim svojstvima u svim
ležištima ugljikovodika Savske depresije, prikazani rezultat te preporuke mogu se primijeniti na
području cijele depresije.
Ključne riječi: obični kriging, jednostavni kriging, poroznost, pješčenjaci, savska depresija,
neogen
1. INTRODUCTION
Porosity can be mapped by many computer-based interpolated methods. All of them are defined
by less and more advanced mathematical equations. There is several works performed on data
from the Sava depression that tested appropriation of several such mapping methods, available in
almost all computer mapping software packages. Mathematically simpler methods are quicker, but
often less precise in cases based on smaller input dataset (approximately 10-15 hard data or
lesser). There is another option to use geostatistical deterministical methods like kriging. In kriging
methods the most often and very suitable is the Ordinary Kriging technique, developed from
equations of the Simple Kriging as the basic kriging approach. It can be proven, using matrix
equations, that kriging variance (as the measure of error) is smaller in the Ordinary than in the
Simple Kriging. The presented analysis was directed in the porosity dataset as the observed input
derived from the reservoir. Three simpler methods had been tested (Inverse Distance, Nearest
Neighbourhood and Moving Average) to obtain porosity maps. Furthermore is proven that variance
minimizing in better in the Ordinary Kriging than in the Simple Kriging. Finally, the map interpolated
by the Ordinary Kriging and belonging numerical result had been point out as the most appropriate
tool (and the interpolation technique) that could be applied for porosity distribution description in
reservoir sandstones of Neogene age (from Early Pannonian to Early Pontian). Due to lithological,
depositional and tectonic properties similarities in all hydrocarbon reservoirs located in the Sava
depression, presented result and given recommendations can be applied for entire depression.
2. SIMPLE INTERPOLATION
The Inverse Distance Weighting method (Figure 1) estimates values from a relatively simple
mathematical expression. The influence of each point is inversely proportional to the distance from
an estimated location. The number of points included in the estimation, is defined by circle radii
drawn around a selected location.
Figure 1: Porosity map interpolated by Inverse Distance Weighting (from: Balić et al., 2008)
The Nearest Neighbourhood method (Figure 2) assigns the value of the closest point to each
grid node. It is not so much an interpolation method as zonal assignment technique. This method is
useful in case of relative large zones without data (blind areas), which need to be schematically
mapped.
Figure 2: Porosity map interpolated by Nearest Neighbourhood (from: Balić et al., 2008)
The Moving Average method (Figure 3) calculates the values of grid nodes from averaged data
measured in a particular ellipsoid or circle that surrounds each grid node. There is also a need to
define the lowest number of data that can be averaged. This is not exact interpolation but rather a
kind of simple method characterised by its trivial smoothing character.
Figure 3: Porosity map interpolated by Moving Average (from: Balić et al., 2008)
Results are presented on different maps (Balić et al., 2008). Almost all can be recognised by their
very different morphologies. The result of Inverse Distance Weighting method (Figure 1) indicates
the real porosity distribution in the reservoir and is also similar to the kriging map (Figure 4) which
is considered as the most authentic algorithm in the sense of geology and geomathematics.
Figure 4: Porosity map interpolated by kriging with isotropic variogram model and range 1100 m
(from: Balić et al., 2008)
However, the maps obtained by other two methods, which are not exact interpolators, are
inappropriate. These are Nearest Neighbourhood (Figure 2) and Moving Average (Figure 3)
maps, where the presented isoporosity cannot be meaningfully interpreted. Even if the radii of the
searching ellipsoid is changed (moving average), the gradual transition among lines has not been
achieved. The Nearest Neighbourhood result strongly emphasises the polygonal presentation.
Therefore, these two methods cannot be applied to porosity interpolation in Neogene sandstone
reservoirs of the Sava depression. Numerical estimation of maps is performed a cross-validation
equation. The following values were obtained for the different methods (starting with the lowest
error, from Balić et al., 2008): Ordinary Kriging (366.93), Moving Average (369.26), Inverse
Distance Weighting (371.97) and eventually Nearest Neighbourhood (389.00).
3. ADVANCED MAPPING
3.1. Mathematical fundamentals of kriging
The principle of kriging is shown most simply with sets of equations that define the method. Kriging
is applied to the estimation of the values of a regionalised variable at a selected location (Zk),
based on surrounding existing values (Zi). Each such location is assigned a relevant weighting
coefficient (i), and the calculation of this is the most demanding part of the kriging algorithm. The
value of a regionalised variable can be defined as:
Z i  Z  xi 
(Equation 1)
Where is:
xi
- is the value at the known location.
Moreover, the value of a regionalised variable estimated by kriging based on n points is:
n
Z k   i  Z i
(Equation 2)
i 1
Where are:
i
- is the weighting coefficient for a particular location “i”;
Zi
- are known values, the so-called “control points” (hard data);
Zk
- is the value estimated by kriging.
These equations represent the system of linear kriging equations.
Previous equation also can be written as the matrix:
A    B
(Equation 3)
In matrices A and B the values are expressed as variogram values, i.e., these values depend only
on the distances and orientations between the control points and not on their values. The third
matrix includes weighting coefficients, which are simply estimated from a system with “n” equations
with “n” unknown variables.
3.2. Simple Kriging theory
Simple Kriging, as its name implies, is the simplest kriging technique. The full matrix equation is:
  ( Z1  Z1 )  ( Z1  Z 2 ) ...  ( Z1  Z n )   1   ( Z1  Z ) 
 ( Z  Z )  ( Z  Z ) ...  ( Z  Z )    ( Z  Z )
2
1
2
2
2
n 
2


 2  

   


   

 ( Z n  Z1 )  ( Z n  Z 2 ) ...  ( Z n  Z n ) n   ( Z n  Z )
(Equation 4)
Where are:

- are the variogram values;
Z1…Zn - are known measured values at points;
Z
- is the point at which new values are estimated from known (hard) data (Z1…Zn).
Kriging uses dimensionless point data that represent the values of the regionalized variable. In
Simple Kriging, it is assumed that the regionalized variable has second order stationary, the
excepted value is everywhere constant and known, and the covariance function is known
[c(x,y)=Cov(Z(x),Z(y))]. Furthermore, when this estimation is performed at the control point, the
error can also be calculated at the point as:
  Z real  Z estimated 
(Equation 5)
If there is no external drift in the variable and the sum of all weighting coefficients is 1,
unbiasedness is achieved. The difference between all the measured and estimated values is called
the estimation error or kriging variance and it is expressed as:
n
2 
 (Z
i 1
real
 Z estimated ) i2
n
(Equation 6)
In an ideal case, kriging tries to calculate the optimal weighting coefficients that will lead to the
minimal estimation error. Such coefficients, which lead to an estimation of unbiasedness with
minimal variance, are calculated by solving of the matrix equations system. If the matrices in
previous equation are represented by linear equations, it can be written as:
(Z1-Z1)x1 + g(Z1-Z2)x2 + … + (Z1-Zn)xn = (Z1-Z)
(Z2-Z1)x1 + g(Z2-Z2)x2 + … + (Z2-Zn)xn = (Z2-Z)
……………
(Equation 7)
(Zn-Z1)x1 + g(Zn-Z2)x2 + … + (Zn-Zn)xn = (Zn-Z)
Moreover, for previous linear equation to be considered unbiased, an additional condition must be
fulfilled, i.e., the sum of all weighting coefficients is 1. This condition is achieved by adding new
conditions to the kriging matrices. Technique of Ordinary Kriging is the most-often used
unbiasedness estimation. The estimation is the selected point that can be represented by the
following equation:
 1    ( Z 1  Z 0 ) 
  

...
 

     (Z  Z ) 
n
0 
 n 
Z

(
Z

Z
)
...

(
Z

1
1
1  Zn ) 


...
...
...


  ( Z  Z ) ...  ( Z  Z ) 
n
1
n
n 

(Equation 8)
Moreover, for previous linear equation to be considered unbiased, an additional condition must be
fulfilled, i.e. the sum of all weighting coefficients is 1. This condition is achieved by adding new
conditions to the kriging matrices. Technique of Ordinary Kriging is the most-often used
unbiasedness kriging technique.
3.3. Ordinary Kriging theory
All kriging techniques (except Simple Kriging) have added some constraints to the matrices, to
minimize the error   k2 ( x ) , and these techniques are unbiasedness estimations. Generally, these
factors would describe some external limit (restriction) on the input data, which cannot simply be
observed in the measured values. The most-used kriging technique is probably Ordinary Kriging,
and we therefore analyse the constraint factor in Ordinary Kriging equations, called the Lagrange
multiplicator.
As discussed above, if the sum of all weighting coefficient is 1, kriging expression can be written
as:
(Z1-Z1)x1 + g(Z1-Z2)x2 + … + (Z1-Zn)xn + m = (Z1-Z)
(Z2-Z1)x1 + g(Z2-Z2)x2 + … + (Z2-Zn)xn + m = (Z2-Z)
……………
(Equation 9)
(Zn-Z1)x1 + g(Zn-Z2)x2 + … + (Zn-Zn)xn + m = (Zn-Z)
1 + 2 + … + n + 0 = 1
If such a system of linear equations is shown as kriging matrices it can be written as:
 (Z1  Z1 )  (Z1  Z 2 )
 ( Z  Z )  ( Z  Z )
2
1
2
2



 ( Z n  Z 1 )  ( Z n  Z 2 )

1
1
...  ( Z 1  Z n ) 1  1    ( Z 1  Z ) 
...  ( Z 2  Z n ) 1 2   ( Z 2  Z )

1     
   

...  ( Z n  Z n ) 1 n   ( Z n  Z )

...
1
0  m  
1
(Equation 10)
The number of weighting coefficients and control points can be very large, but contemporary
computers can successfully solve numerically demanding tasks. The estimation can be performed
simply by calculating the influence of all control points weighted by their associated coefficients
according to:
Z  1  Z1  2  Z 2  ...  n  Z n
(Equation 11)
The calculation of the estimation variance includes adding the Lagrange coefficient:
 2  1   (Z1  Z )  2   (Z 2  Z )  ...  n   (Z n  Z )  m
(Equation 12)
Here are shown the two probably most-used kriging techniques. In general, in all linear kriging
techniques (since they are constrained optimization problems), the associated equations can be
divided into two parts:
a) In one part of the equations, the spatial dependence (spatial correlation) of the measured
data is calculated, usually using a variogram;
b) The other part of the equations includes different constraints, resulting in the sum of all
weighting coefficient being equal to 1.
4. CONCLUSION
The Ordinary Kriging technique is more appropriate for the interpolation of point data (compared
with Simple Kriging). In lot of papers it was proven that geostatistical interpolation (as well as
stochastic estimation) is the best approach to mapping geological variables. It is concluded that
kriging techniques can be successfully applied with minimal of 10 to 15 data points. Any
representative statistical dataset of geological variables must include at least 30 data points. This
means that we cannot conclude the value of the area population mean (expected value) based
only on the usually available datasets. For this reason, the so-called “local mean” is most often
applied, which is calculated only from the hard data encompassed by the searching radii. This
favours the Ordinary Kriging technique. Therefore, we must be very careful to analyse the
variables with Ordinary Kriging equations. Most of the variables are the standard elements of all
kriging techniques, but there is a unique variable, the Lagrange (linear) multiplicator. It has been
shown (Figure 5) that this value can be negative, but the kriging variance cannot be negative
(mathematically impossible). If the value is randomly sampled many times from some interval (e.g.
-1,1), one value will produce the least possible kriging variance. The user only needs to select the
width of sampling interval and number of samples. This procedure can be described through four
steps (similar like in paper Malvić & Balić, 2009):
test stopping
direction of changing
“m” value
2
minimum of variance
the case when the Lagrange multiplicator
is decreased in negative values
m -0.3 -0.2 -0.1 0.01 0.1
0.2
negative
kriging variance
negative
kriging variance
m -0.3 -0.2 -0.1 0.01 0.1
0.2
direction of changing “m” value
test stopping
2
minimum of variance
the case when the Lagrange multiplicator
is increased in positive values
Figure 5: Graphical representation of how to select the most appropriate value for Lagrange multiplicator
using random sampling (from Malvić & Balić, 2009)
1. Based on experience, it can be assumed that value of the Lagrange multiplicator should occur
in the interval (-1,1). Some values can be very close to 0. For these reasons, we consider that
the starting point in the random sampling should be set at 0.01.
2. In the next step, the starting value must be decreased (in steps of e.g. 0.05), i.e. it becomes
negative. Again, the appropriate kriging variance can be calculated and registered. Such a
procedure should be repeated until the kriging variance is positive. Using this procedure, the
value of m with the least kriging variance can be selected (Figure 5 left).
3. The positive side of the Lagrange value should then be checked, again increasing the value in
steps of +0.05. Reduction in the variance will be observed, and the procedure should be
continued until this reduction vanishes. The first time when the variance is increased, the
calculation is stopped (Figure 5 right).
4. It is important to know that if, for the first positive ‘m’ (m>0), the variance immediately started
increasing, then it can be considered that the minimum kriging variance is found in the negative
interval (less than 0) and random sampling can be ceased (Figure 5 left).
We believe that these four rules completely describe the correct procedure for selecting the
Lagrange multiplicator in Ordinary Kriging equations. Consequently, this kriging technique is one of
the best interpolation algorithms for mapping geological variables.
5. REFERENCES
1. Balić, D., Velić, J. & Malvić, T.: Selection of the most appropriate interpolation method for
sandstone reservoirs in the Kloštar oil and gas field. Geologia Croatica, 61, 1, 27–35
(2008).
2. Malvić, T. & Balić, D. (2009): Linearity and Lagrange Linear Multiplicator in the Equations
of Ordinary Kriging. Nafta, 59, 1, 31-37.