Here is the review of answers on my mail "Kriging versus Inv. Dist.
Weighting" from Nov 15:
Dear all,
This is my first try at geostat mailing list, and maybe my question will not
be very "professional".
I work with data set of porosity in one oil reservoir. Interpolations were
done with three interpolation methods: Inverse distance weighting, Kriging
(ordinary) and Cokriging (collocated). I done spatial analysis with
semivariogram modelling for (co)Kriging.
After all, I calculated true error for every included point as difference
between real value and estimated value at the same place. I was confused
when I saw that Kriging error was higher of Inverse Distance Weighting
error! The lowest errors were gained by Cokriging (with the same
semivariogram modell as used in Kriging).
What could be reason for that? Maybe 14 points is too low set for proper
modelling of directional semivariogram analysis (directions=0 and 90
degrees). I tested several lag distances and distance with the highest range
was chosen. If chosen distance is too low interpolation map contains mostly
areas of "bull-eyes". Also, input points are moderately clustered.
Thank you and best regards,
Tomislav
***********************************************************
As it is usual in Oil reservoir modelling, it seem you have a very dense
geophysical information (may be acoustic impedance) and low dense
information from wells.
In these case is natural that the best result are obtained from collocated
cokriging, similar result can be obtained from kriging with external drift
and the homologous in the simulation scope.
I think that 14 pint is to short for correct variogram modelling, and it is
recommendable to use cross variogram in collocate cokriging. A solution for
know the shape of the variogram could be using the background information,
if it is close correlate to the wells information, bat I´m not sure that it
is a good solution.
Form more help please be more specific
King regards
Adrian Martínez Vargas
Instituto Superior Minero Metalúrgico.
Moa Holguín Cuba.
CP 83329
***********************************************************
A few thoughts
1. As the "user" you have to make a number of choices when you use kriging
( I will assume that you were using Ordinary Kriging).
a. You need to begin with some simple exploratory analysis (histogram of
data, coded plot of the data (data value vs position coordinate(s), you
might want to fit to a trend surface). This latter is to help decide whether
you have to deal with a non-stationarity in the mean. In general this is to
help understand and interpret what you see in the variogram fitting and
cross-validation stages.
2. You have to estimate the variogram/covariance
i. While it is pretty common to begin with just the ordinary (second
moment) sample variogram, there are a few others that are sometimes useful.
Note that the software may have default values for the lag distance and
number of lags plotted, these may or may not be the best choices for your
data set. In general don't plot beyond about half the the maximum inter-data
location distance (the number of pairs drops off significantly). You should
at least look at directional sample variograms
(if your data set is too small these may not be very good, too few pairs).
3. You have to fit the variogram model
i. This includes determining/choosing the type(s) , note that you
may want to use a nested model, e.g., spherical, exponential, gaussian,
power, etc
ii. This includes determining the variogram model parameters, e.g.,
range(s), sill(s), nugget, angle(s) of anisotropy and the major/minor
range(s) (these latter if using an anisotropic model)
iii. The simplest form of "fitting" is of course "visual", you may
want to also consider weighted least squares if that is included in your
software.
4. You have to evaluate how well your variogram model fits the data,
cross-validation is at least one way. If you use cross-validation then there
are multiple statistics (some of these are especially sensitive to the
choices for the search neighborhood) and in some cases the results may be
contradictory. While the theory is based on the idea that there is a single
"correct" variogram model, with only a finite data set there is no unique
choice for the variogram. Use cross-validation to compare two choices rather
than as a way to absolutely optimize the choice.
b. You have to decide on the search neighborhood (this step is necessary
both for the cross-validation and for the subsequent actual kriging)
i. shape (usually this is limited in the software to circular or
elliptical)
ii. radius (for circular), lengths of axes (if elliptical)
iii. angle of orientation (if elliptical)
iv. is the search neighborhood sectioned? (usually taken 1, 2, 4 or
8 sectors)
v. minimum and maximum number of data locations to be used in the
search neighborhood
In the case of sectors, will empty sectors be allowed?
Now to your questions and concerns
It appears that you are describing a simple form of cross-validation and you
are looking only at the "mean error". As you know the "mean" sort of
balances positive and negative values so that the mean could be zero (or
close to zero) even though the individual errors are large in magnitude..
Among other things I would look at plot of the errors (i.e., plot data
locations with each point coded by the error from IWD or from kriging). I
would look at a histogram of the errors and also of the "normalized" errors
(in the case of kriging, divide each error by the kriging standard
deviation).
While it is not a theoretical contradication for the IWD results to be
better than the kriging results I suggest that this may indicate one of
several things, the variogram model is not well fitted and/or the search
neighborhood parameters need to be changed.
Note also that IWD can be sensitive to the choice of the exponent but this
will depend on the particular data set.
Finally if one compares in a theoretical sense, IWD vs kriging, the form of
the estimator is the same for both. The weights reflect to some degree the
interdependence between the value at the data location and the value at the
place where an estimate is desired. However kriging does one more thing, it
incorporates the independence between the values at the data locations in
the search neighborhood. While it would be possible to use a modified notion
of "distance", most software implementing IWD treat it as "isotropic" and
hence it does not "decluster" the data locations. Think of the estimation
location as the center of a circle (i.e., the search neighborhood),
contrast the case of two data locations close together but both the same
distance from the center vs the case where they are spread apart at an
angle of perhaps 90 degrees. In IWD the two data locations will have the
same weights in both cases but in kriging (even with an isotropic variogram)
the weights are different. When they are close together the weight is
essentially "split" between them, i.e., because they are very close they are
"highly correlated" and hence instead of two separate pieces of information
there is really only one.
A couple of possibly helpful references
1991, Myers,D.E., On Variogram Estimation. in Proceedings of the First
Inter. Conf. Stat. Comp., Cesme, Turkey, 30 Mar.-2 April 1987, Vol II,
American Sciences Press, 261-281
1991, Myers,D.E., Interpolation and Estimation with Spatially Located Data,
Chemometrics and Intelligent Laboratory Systems 11, 209-228
(The previous) one is was written as a "tutorial" and it uses the free
"GEOEAS" package)
1982, V. Kane, C. Begovich, T. Butz and D.E. Myers,Interpretation of
Regional Geochemistry. Computers and Geosciences, 8, no. 2, 117-136
(The previous one discusses optimization of the exponent in IWD)
Warrick, A.W., Zhang, R., El-Haris, M.K. and Myers, D.E., Direct comparisons
between kriging and other interpolators. in Proceedings of the Validation of
Flow and Transport Models in the Unsaturated Zone, Ruidoso, NM, 23-26 May,
1988, 505-510
Donald E. Myers
http://www.u.arizona.edu/~donaldm
***********************************************************
Hi Tomislav,
Don't be surprised. It is my experience that cross-validation
might sometimes indicate that best interpolation results are obtained
using the simplest techniques. If your observations are not
too clustered and display no anisotropy, inverse square
distance could yield good results.
Now, you didn't explain which secondary information was used
for cokriging and how many neighboring values were used
in the different interpolators.
Regards,
Pierre Goovaerts
Dr. Pierre Goovaerts
Consultant in (Geo)statistics
and Senior Chief Scientist with Biomedware Inc.
710 Ridgemont Lane
Ann Arbor, Michigan, 48103-1535, U.S.A.
E-mail: [EMAIL PROTECTED]
Phone: (734) 668-9900
Fax: (734) 668-7788
http://alumni.engin.umich.edu/~goovaert/
***********************************************************
Tomislav,
Try getting a semivariogram range from the (presumably)
dense secondary data. This could be seismic derived porosities.
Construct a semivariogram for porosity using that range and
a sill more in line with the variance that you see in
the sampled points. Then re-perform the kriging.
14 points is a bit on the low side, but unfortunately more
the norm than the exception for most oil and gas datasets,
especially for offshore locations. Would there be analogous
data that you can use? At every lag distance try to get at
least 50 pairs for calculation of the variogram values.
Regards,
Syed
***********************************************************
This paper may be of interest to you.
http://www.uky.edu/~mueller/Internal/map%20quality%20for%20SSFM.pdf
If you send me your address, i will send you another paper that will be
published in January.
tom
***********************************************************
I wish to thank everybody for very useful answer and help.
With my best regards,
Tomislav Malvic, M.Sc.
Reservoir Geologist
INA-Oil Industry plc. (INA-Naftaplin)
Subiceva 29
HR-10000 Zagreb
CROATIA
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