Editor: Clayton V. Deutsch, Department of Petroleum Engineering, School of Earth Sciences, Stanford University
Assistant Editor: Christopher J. Di Maggio, Department of Geological and Environmental Sciences, School of Earth Sciences, Stanford University
Although this issue is devoted to review articles of the Sixth International Geostatistics Congress in Wollongong, I hope to return to the earlier style of concise problem statements, discussions, and an early, informal place to relate practical application hints. The geostatistical community in North America would be well served by a newsletter that disseminates such information in a timely and convenient fashion.
The cost and tedium of photocopying, labeling and mailing this newsletter has hampered its timely distribution. Our move to an electronic format should alleviate these problems. We will not, however, become a mailing list where you are drowned with poorly posed questions, self-promotion, commercial advertisements and excess repetition; Chris and I will edit and arrange the newsletter to avoid these pitfalls. Another pitfall we hope to avoid is becoming Stanford-centric. You can help us avoid this problem by sending in contributions. It is most convenient for us to walk down the hall in search of material.
Send in your comments, suggestions and contributions to ensure a lively and interesting newsletter.
Clayton V. Deutsch
In the future, this site will provide easy access to backissues, useful software and a directory of NACOG members, related organizations and other geostatistics references.
If this new format presents a difficulty for you because of internal network restrictions at your institution, please provide me with your email address (dimaggio@pangea.stanford.edu) and I will send you the newsletter file for viewing.
Your help will be invaluable in developing this site and this newsletter. Please mail or email any questions, comments, suggestions or submissions to the editors at the addresses below.
Clayton V. Deutsch
Department of Petroleum Engineering
Stanford University
Stanford, CA 94305-2220
clayton@pangea.stanford.edu
Christopher J. Di Maggio
Department of Geological and Environmental Sciences
Stanford University
Stanford, CA 94305-2115
dimaggio@pangea.stanford.edu
Christopher J. Di Maggio
Assistant Editor
School of Mines, University of Guanajuato, Mexico
Precision of Indicator Kriging and Cokriging by Vinicio Suro-Perez
The presentation demonstrated the deviation of the conditional probability distributions derived by indicator kriging and indicator cokriging from the "true" conditional distributions calculated from exhaustive data. The deviation is explained as being due to the use of two-point statistics in estimation when the data exhibits multipoint characteristics. Another reason for the difference in the distributions could also be the order relations corrections applied in traditional indicator kriging.
Geostatistics Applied to Mine Waste Remediation and Characterization (at Leadville, Colorado) by Antonio Nieto Vega
The process of identifying lead contaminated soil areas using geostatistical tools was presented. Four datasets from Leadville County, Colorado were statistically analyzed and four different populations, related to residential domains, were identified. Variogram analysis was performed to characterize spatial variability on the data sets. Significant spatial correlation was found within the study domain and this was utilized to estimate the level of contamination at unsampled locations using ordinary kriging and indicator kriging. A cross validation study found indicator kriging to be the best estimator with the minimum overestimation bias.
Comparative Study of Volume Variance Techniques in Geostatistics by Kadri Dagdelen
Recoverable reserve estimation is strongly dependent on the variance correction applied to the local grade distributions. The talk covered various methods for correcting variance and the resultant impact on recoverable reserve estimates. The affine correction, indirect lognormal correction and simulation methods were evaluated.
Redefining the spatial support of environmental data in the Regional HydroEcological Simulation System by Jennifer Dungan
The transport processes involved in ecological modeling were enunciated as was the use of probability field simulation to obtain vegetation maps. A novel feature was the use of Bayesian analysis (of prior information) on varying support to develop the conditional cdfs. This replaced the conditional distribution obtained by kriging.
Determining locations for second-phase sampling in environmental site characterization by Sean A. McKenna
The use of geostatistical techniques to identify additional sampling locations was the focus of this talk. The presenter pointed out that the optimal locations are a function of the remediation action level, the costs of sampling and remediation, as well as the degree of risk the governing regulatory body is willing to accept in not meeting the remediation goals. Many techniques have been proposed in the literature to determine these optimal locations (e.g., kriging variance, variability between simulations, probability indicator kriging, etc.). The results of these various techniques in terms of the estimated optimal locations and in terms of their accuracy in making the correct remediation decisions and lowering overall costs were compared on an exhaustive data set. The differences between the results from different techniques were discussed and the impact on environmental remediation plan highlighted.
Ranking of Stochastic Realizations by Sanjay Srinivasan
Various schemes for ranking multiple stochastic realizations were described and demonstrated with a case study. Since flow simulation is computationally demanding, ranking of geostatistical realizations to identify the low-side, expected and high-side realizations enables the uncertainty in reservoir flow performance to be predicted using a reduced number of flow simulations. The presentation demonstrated that reservoir management is improved when expected and bounding cases are considered rather than using a limited number of ``random'' realizations.
Soil erosion mitigation through spatial simulation & prediction of forest duff consumption during prescribed fires by Stan Miller
An intesting application of spatial simulation is to predict the fire consumption of forest floor organics (duff), material which helps to retain soils is presented. The study focused on the investigation of soil erosion following forest fires. Sequential Gaussian simulation has been used to simulate pre-burn duff thickness, moisture content, and post-burn duff thickness in a selected study area. The resultant spatial realizations of pre-burn duff thickness and moisture content were used to predict post-burn duff thickness.
Non-Ergodic Variogram and Local Kriging by Marco Alfaro
The talk focussed on the dangers of using the non-ergodic variogram when data are clustered. The worst case example is that of a string of data with mostly zeros along the string, but a few values above zero in the middle of the string. The proposed solution to this problem is to consider a family of variograms from which a single variogram is chosen for each local estimation problem. The choice of each local variogram would be concerned with minimizing error at locations with large kriging weights . A program that used a power-law variogram was demonstrated. The exact power value was determined by doing cross-validation within every search neighborhood.
Geostatistical modeling for the semi-arid land surface by Fernando Avila
The talk briefed the audience on an extensive environmental monitoring program currently underway in a basin on the Arizona-Sonora border. The goal is to monitor ecological and hydrological changes within the basin. The talk dealt with using Hilbert space to look at both point and non- point support variables as continuous functions. A preliminary model treating monitoring instruments as filters utilizing the concepts of "data space" and the "model space" is being developed. Within the model space, data is only seen through the filter of the measuring instruments.
Data Aggregation in Geostatistics and the Prediction of Non-linear Transformation of Random Functions by Miguel Ancona
Current research on a generalization of "constrained kriging" as proposed by Noel Cressie in 1993 was the topic of this talk. The presenter was concerned that for variables with measurement error the unbiasedness constraint is too restrictive and the mean squared error (MSE) should be retained.
The Fifth International Geostatistics Congress took place September 22-27 in Wollongong, 80 kilometers from Sydney, Australia. The venue, on the expensive side, provided superb conference accommodations and magnificient sea views but few side activities which held the delegates together fostering ample after-sessions discussions. Two hundred and ten delegates from 25 different countries attended with over 100 papers presented orally and ten others through posters. The range of topics presented was extremely large from abstract methodology to analysis of ryegrass distribution and modeling of fluvial reservoirs. The quality of papers was equally variable but befits the concept of an open congress establishing the state of a discipline, including samples at the poor end.
Morning plenary sessions were dedicated to papers labeled "theory" with afternoon sessions focusing on mining, petroleum and environmental applications. Many morning session papers were repeats of established theory and did not deserve plenary audience, while some outstanding developments using actual data were relegated to afternoon sessions. I would only mention the paper by Pierre Goovaerts "Accounting for Local Uncertainty in Environmental Decision-Making Processes" and that by Ian Glacken "Change of Support and Use of Economic Parameters for Block Selection," both breaking new ground in basing the selection problem in economical terms accounting for spatial uncertainty. The snobbish bias towards "theory" started at GEOSTAT Troia 1992 should be stopped; excellent case studies and innovative developments tuned to a particular problem or even a specific data set deserve wider attention, for they can seed novel general methodology. Plenary sessions should be reserved for the best papers, not only for theory-labeled abstracts and heavy-weight authors. The problem, for which I have no solution, is how to uncover the gems in some 100 papers reviewed by as many different reviewers with widely variable dedication to their task; all this within the few months preceding a congress. Remember, an abstract - even an extended abstract - is often no indication of the actual content and quality of a paper.
Notable observations I drew from GEOSTAT Wollongong 96 were, in no particular order as I write this review in a plane back to the U.S.:
The Fifth Congress voted for the centurymark GEOSTAT 2000 to be hosted by South Africa in Capetown, a heartfelt decision that underlines South Africa's (and Danie Krige's) original contribution to geostatistics. I have a dream, that of an opening session with Professors Danie Krige and Georges Matheron jointly dedicating this new century to all the young bright minds who will make geostatistics into what we only dreamt it could be.
A.G. Journel
United Airlines flight 862
Sydney-SFO
September 27, 1996
Professor Andre G. Journel (Andre) gave the opening address at the congress. His presentation on "the abuse of principles in model building and the quest for objectivity" set the tone for the conference: questioning and challenging. R. Mohan Srivastava's (Mo) closing presentation "Matheronian geostatistics: where is it going?" placed an emphasis on more science and professionalism; a worthy goal for the future.
In Andre's customary manner he insisted that (1) models are required to go anywhere beyond data, (2) there is no unique model hence no objective model, (3) uncertainty is model-based, therefore (4) there can be no objective assessment of uncertainty. As an apparent counter example, Mo presented a 20 grid node example. Subject to the constraints of 10 black, 10 white, and a specific lag 1 correlation, there are exactly 276 outcomes -- a space of uncertainty defined without choosing a model. There is no inconsistency between Andre's logic and Mo's neat example. Mo and Andre agree that, in practice, the combinatorial explodes leading to an almost infinite space of uncertainty. A reservoir of modest size would have 10^8 data-support cells, 3 attributes, and 10 possible values for each attribute. There are then 10^30,000,000 possible combinations! Of course, this would be reduced by soft constraints such as spatial statistics -- we find ourselves back to a model of some kind!
Andre exhorted the congress delegates not to hide behind buzzword principles in choosing a model:
Mo presented his assessment of the health of the geostatistics community. He rightfully encouraged us (the geostatistics community) to reflect on our shortcomings and to be our own worst critics. As an example, Mo focused attention on six issues:
Simulation is a two-pass process. In the first pass, the categorical variables are simulated by a truncated Gaussian field generated by a standard sequential algorithm; in the second pass, the grade variable is co-simulated using for conditioning the categorical data previously simulated and the grade data, and utilizing the cross-variogram between categorical and grade values. Transformations are proposed via cumbersome Hermite expansions but may be achieved more efficiently with a standard normal scores transform.
Dowd describes two case studies, one with only two categories (the presence or absence of a quartz vein), and the second with four lithological categories in addition to the presence or absence of a controlling reef structure. In this latter example, a double truncated Gaussian technique is used to overcome the sequence-of-categories problem.
While the methodology conceptually shows promise, the paper raises a number of questions:
The truncated plurigaussian RF model calls for:
The second step is the determination of the four thresholds (two for each RF) defining the locus of each lithotype in the Z1-Z2 plane. If the partition is a rectangle, the relationship between the lithotype proportion, the correlation coefficient and the threshold values can be derived. But the problem is the back calculation of the thresholds given the proportions, an ill-defined problem if each lithotype is considered separately. The authors consider the simplest case, in which grouping of the lithotypes leads to infinite threshold values for one of the Gaussian functions. The other threshold values are determined first for the lithotype groups and then for each individual lithotype. This step also calls for expert knowledge and, in the general case, for a global optimization procedure.
Finally, given the variogram models of the Gaussian RFs and their correlation coefficient at lag zero, the variogram and cross-variogram models of the lithotype indicators are computed. Again, the inverse problem, that is, reproduction of the original anisotropic facies indicator variograms, requires an iterative trial-and-error process. The task of conditioning the simulations to lithotype data is done through yet another trial-and-error iterative algorithm of Geman & Geman type.
In conclusion, the algorithm faces very difficult, ill-conditioned inverse problems. As it stands now, it relies on many ad hoc decisions and is still a long way from a practically sound solution.
The case study consists of a 2D map of seismically identified faults. Faults of length less than 600 m., which seem to appear in clusters, were isolated. The density maps of these small faults was estimated distinctly from larger faults by factorial kriging. The resulting point density map is claimed to match the "true" one; details and results are not shown, while variograms are shown but do not make sense. Accounting for this density map, the small fault population was simulated using MPP. It was concluded that the resulting simulations resemble the true image. The same method was then used to infer sub-seismic faults (of lengths less than 10 m.).
The paper shows that MPP can honor spatially varying density, thus reproducing the clustering characteristics of small fault patterns. In addition to the parameter inference required by MPP, the challenge here is to infer the point density maps. This was done through factorial kriging, a process unfortunately not described in the paper. In any case, the spatial variability of small scale features could only be inferred through knowledge of the spatial variability of the density of the whole fault population, so it seems odd that the spatial variability in density of small fault centers could be extracted from only the knowledge of the density variations of the centers of the larger, seismically visible faults.
The method is straightforward, at least in 2D. It allows more flexibility in defining fracture shapes than a conventional object technique with each fracture being a single object, but its implementation in 3D would probably be tedious. Inference of the conditioning parameters required by the algorithm appears ad hoc and could be a drawback of the technique. Orientations, lengths, and especially the number of fracture intersections are not parameters which can be clearly exported from analog data. The fractal dimension of a fracture network is also a very controversial concept. Few fracture patterns are actually fractal. Crucial spatial information is lost when reducing density and clustering to such a single number as the fractal dimension.
Clayton V. Deutsch
Department of Petroleum Engineering
Stanford University
Stanford, CA 94305-2220
clayton@pangea.stanford.edu
Christopher J. Di Maggio
Department of Geological and Environmental Sciences
Stanford University
Stanford, CA 94305-2115
dimaggio@pangea.stanford.edu