1. 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.

2. 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.

3. 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 north-south on the map. The survey was recorded in October, 1995 for PanCanadian Petroleum.

4. Base map from the Blackfoot surveyN
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.

5. Correlating the logs with the seismic data
This figure shows the correlation of well 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.

6. 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.

7. Line from inverted seismic volume
This figure shows Xline 18 from the inverted volume. The color key indicates impedance.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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 cross-validation error as our measure of success.
The next two slides show these different errors for the kriging example.

16. Kriging error variance
The error variance map. As expected the error is small at the wells and gets larger away from the wells.

17. Kriging cross-validation error
The cross-validation error map, displaying the absolute error at each well in % porosity. The standard deviation is 3.25%.

18. 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.

19. 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 :
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.

20. The multiattribute map transformX Y
Attribute map 2
Attribute map M
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.

21. 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

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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.

29. 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.

30. 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.

31. 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.

32. 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.

33. 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%.

34. 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 %

35. 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 cross-validation error was smallest when we used cokriging with the multiattribute transform.

36. 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.

37. 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.