Combining geostatistics and
multiattribute transforms –
A channel sand case study
Brian Russell #, Larry Lines #,
Dan Hampson*, and Todor Todorov*.
#CREWES
Consortium, University of Calgary
*Hampson-Russell Software Ltd.
Introduction
• In this talk, we will look at a new approach
to integrating well log and seismic data,
which involves post-stack inversion,
geostatistics, and multiattribute transforms.
• This method will be applied to data slices
extracted from multiple 3D volumes.
• We will illustrate this approach using the
Blackfoot dataset.
The Blackfoot survey
Channel
Alberta
N
Calgary
This map shows the location of the Blackfoot survey area, with the portion
used in this study outlined in red. The objective, a Glauconitic channel
within the Lower Cretaceous Mannville formation, is shown running northsouth on the map. The survey was recorded in October, 1995 for
PanCanadian Petroleum.
Base map from the Blackfoot survey
N
Xline 18
This map shows the 12 wells in the seismic survey area, and Xline 18.
Note that we have rotated the map from the previous display.
Correlating the logs with the seismic data
This figure shows the correlation of well 14-09 with the seismic data, where
the synthetic trace is in blue the and the seismic trace is in red. The sonic
and porosity logs are on the right. The top and base of sand are also shown.
Seismic line from volume
This figure shows Xline 18 from the seismic volume, showing correlated
sonic logs from two intersecting wells, and the picked channel top.
Line from inverted seismic volume
This figure shows Xline 18 from the inverted volume. The color key
indicates impedance.
Acoustic impedance slice
Channel
This map shows the arithmetic average of the acoustic impedance
over a 10 ms window below the channel top event. Notice the
channel in green (low impedance) on the left of the map.
Wells showing average porosity
High
Porosity
This map shows the average porosity over the zone of interest at
each well. Notice the high porosity (purple) values in the center left.
Initial crossplot
This is the crossplot between the well porosities and acoustic impedance
values. The red line is the regression fit, and the correlation is –0.65.
Regression applied to inversion slice
Channel
This map shows the application of a the regression line from the
previous slide to the inversion slice. Notice that the wells do not tie.
Map-based geostatistics
• Map-based geostatistics involves producing three types
of maps:
– Optimal maps (Best Linear Unbiased Estimates)
from sparse well data (kriging).
– Maps that incorporate both sparse well data and a
secondary seismic attribute (cokriging and kriging
with external drift, or KED).
– Conditional simulations of a range of equally
probable maps.
• In this talk, we will focus on the kriged and cokriged
maps.
• Both of these maps are based on the variogram.
Variograms
(a) The figure above shows the
variogram from the 12 wells on
the map, using a spherical
variogram.
(b) This figure shows the seismic
variogram, again with a spherical
fit. By using the Markov-Bayes
linear assumption, we can scale
this variogram for both the kriged
and cokriged maps.
Kriged result
Channel
This map shows the result of applying kriging to the 12 wells within the
3D seismic survey, using the scaled seismic-to-seismic variogram.
Estimating the error
• To see the error associated with kriging, we usually
display the error variance. But this is simply the
“theoretical” error, and will go to zero as the variance of
the input values goes to zero.
• A better measure of the error is the cross-validation
error, which is found by successively leaving out well
values and comparing their correct values to the
predicted value.
• We will use the standard deviation of the crossvalidation error as our measure of success.
• The next two slides show these different errors for the
kriging example.
Kriging error variance
The error variance map. As expected the error is small at the wells
and gets larger away from the wells.
Kriging cross-validation error
The cross-validation error map, displaying the absolute error at each
well in % porosity. The standard deviation is 3.25%.
Cokriged result
Channel
This map shows the collocated cokriging result, using impedance as the
secondary variable. Note the “imprint” of the kriged map. The standard
deviation of the validation error is 2.91%, better than for kriging.
The multiattribute transform
• In the multiattribute transform using multilinear
regression, we compute M+1 weights such that the log
value L(x,y) at a particular map value is a weighted sum
of M attributes Ai :
L( x , y )  w 0  w1 A1 ( x , y )    w M AM ( x , y )
• The solution to this problem can be found by using a
standard least-squares technique.
• A key problem is deciding which attributes to use.
• Another important consideration is which of the attributes
are statistically significant.
• The next slide shows this approach pictorially.
The multiattribute map transform
X
L  log value
Attribute map 1
Y
w1 A1
Attribute map 2
w 2 A2
Attribute map M
w m Am
This figure shows the multiattribute map transform approach in
schematic form. We need to compute the weights wi which, when
multiplied by the attribute values, will produce the log value.
Attribute slices
• We have already seen one of the slices that will be used
in our multiattribute transform: the impedance slice.
• The next three slides will show the other slices used.
Each attribute (except trace length) was derived from
the seismic volume by taking an RMS average over a
10 msec window below the picked top of sand.
• The following attribute slices were extracted:
–
–
–
–
–
–
Seismic amplitude
Amplitude envelope
Instantaneous phase
Instantaneous frequency
Integrated seismic trace
Trace length – the total length of the trace over the window
Seismic amplitude slices
(a) Seismic amplitude slice.
(b) Amplitude envelope slice.
The map in (a) shows the RMS average of the seismic amplitude over
a 10 ms window below the channel top event, whereas the map in (b)
shows the RMS average of the amplitude envelope over the same
window. Notice that the two slices are very close in appearance.
Instantaneous phase and frequency slices
(a) Instantaneous phase slice.
(b) Instantaneous frequency slice.
The map in (a) shows the RMS average of the instantaneous phase over
a 10 ms window below the channel top event, whereas the map in (b)
shows the RMS average of the instantaneous frequency over the same
window.
Integrated trace and trace length slices
(a) Integrated trace slice.
(b) Total trace length slice.
The map in (a) shows the RMS average of the integrated trace over a 10
ms window below the channel top event, whereas the map in (b) shows
the total trace length over the same window.
Computational issues
• We are now ready to compute the multilinear
regression map, which will be a linear combination of
the previous maps.
• We will also include a non-linear option by computing
transforms (inverse, square root, etc) of the data.
• First, we will look at the correlation coefficients between
the wells and each of the attribute slices.
• We then compute the best combination of attributes
using a technique called step-wise regression.
• Finally, we decide which attributes are significant using
a validation technique in which the target well is left out
in jackknife fashion.
Correlation coefficients for all the slices
This table shows the correlation coefficients between the well
porosity values and all of the attribute slices, sorted by decreasing
correlation coefficient.
Validation error plot
This is the validation error for the 5 attributes used in the multiattribute
process. The red line leaves out the target well and shows that only the
first 3 attributes should be used.
Validation error and weights
Only use
the first
three
attributes
based on
validation
error.
Weights
The table at the top shows the numerical values from the previous slide
and also shows that the best non-linear fit is between the square root of
porosity and the inverse attributes. The bottom table shows the weights.
Multilinear regression result
Channel
This map is the result of applying the multilinear regression
weights, shown in the previous slide, to the attributes. Note that
the result is in pseudo-porosity.
New crossplot
This is the new crossplot between the well porosity and the
pseudo-porosity from the multiattribute transform. Note that the
correlation coefficient has gone up to 0.81.
Combining multilinear regression
with geostatistics
• We will now combine the multilinear regression
result with the well values using geostatistics.
• That is, the multiattribute transform will replace the
inversion slice as the secondary variable.
• The first step is to re-compute the seismic to
seismic variogram.
• We will then compute the cokriging result.
• Finally, we will perform a statistical analysis of the
results.
New seismic variogram
This is the seismic variogram used in the final cokriging process.
Note that this is an exponential fit, rather than the spherical fit used in
the earlier variograms.
Cokriging with the multiattribute transform
Channel
This map shows the result of applying the collocated cokriging process
using the multiattribute transform as the secondary variable. The
standard deviation of the validation error is now 2.3%.
Map review
(a) Kriging, Std. Dev.= 3.25 %
(b) Cokriging with impedance, Std. Dev.=2.91 %
Note the increase in geological
information as we move from
(a) the kriged map with wells
alone to (b) cokriging with
inversion, and finally to (c)
cokriging with the multiattribute
transform. However, all three
maps match the wells.
(c) Cokriging with multiple attributes,
Std. Dev.= 2.33 %
Conclusions
• In this paper, we have combined geostatistics with
multiattribute map analysis.
• Traditional cokriging uses a single secondary
attribute. When we used impedance as this
attribute, we saw a strong “imprint” from the wells.
• Using a multiattribute transform, we were able to get
a better fit between the wells and the porosity map.
• After a second pass of cokriging, the final map was
more realistic from an exploration point of view.
• Statistically, the standard deviation of the crossvalidation error was smallest when we used
cokriging with the multiattribute transform.
Future Work
• In this paper, we concentrated on the
multi-linear transform. In future work, use
of the Probabilistic Neural Network will be
explored.
• We will also incorporate the converted
wave volume from the Blackfoot dataset.
Acknowledgements
• We wish to thank our colleagues at both
CREWES and at Hampson-Russell Software
for their input to this study.
• We also wish to thank the sponsors of the
CREWES consortium.