Kansas, Kansas Geological Survey, Series on Spatial Analysis, No. 3, 29 p.
Olea, R. A., 1994, Fundamentals of Semivariogram Estimation, Modeling and Usage, in Stochastic Modeling and Geostatistics, (1994), J. M. Yarus and R. L.
Chambers, Eds. AAPG Computer Applications in Geology, No. 3, pp. 27-36.
Rehfeldt, K.R., Boggs, J.M, and Gelhar, L.W., 1992, Field study of dispersion in a heterogeneous aquifer 3: geostatistical analysis of hydraulic conductivity: Water Resources Research, v.28, p.3309 -3324.
Royle, A. G., 1979, Why Geostatistics?, Engineering and Mining Journal, Vol. 180, pp. 92-101.
Websites
Easton, V.J., and McColl, J.H., Statistics Glossary v1.1, (HTML Editing by Ian Jackson), http://www.cas.lancs.ac.uk/glossary_v1.1/main.html -a helpful and authoritative glossary, with definitions explained in plain language, accompanied by equations.
ADDITIONAL READING
Caers, J. (2001). "Geostatistical Reservoir Modeling Using Statistical Pattern Recognition." Petroleum Science and Engineering, V. 29, No. 3-4, p. 177-188.
Capen, E. C. (1993). "A Consistent Probabilistic Approach to Reserves Estimates."
Society of Petroleum Engineers Hydrocarbon Economics and Evaluation Symposium. SPE Paper 25830, p. 117-122.
Chavez-Cerna, M. and Bianchi-Ramirez, C. (1998). " Abstract: Geostatistics Applied to a Reservoir Study in Northwestern Peru Talara Basin ." AAPG Bulletin, v.
82, p. 1883-1984.
De Araújo Simões-Filho, I. and Queiroz De Castro, J. (2001). "Estimation of Subseismic Nonreservoir Layers within a Turbidity Oil-Bearing Sandstone, Campos Basin, Using a Geostatistical Approach." Journal of Petroleum Science and
Engineering, v. 32, No. 2-4, p. 79-86.
Emanuel, A. S., Behrens, R. A., Hewett, T.A. and Alamed,a G. K. (1988).
"Reservoir Performance Prediction Methods Based on Fractal Geostatistics:
Abstract." AAPG Bulletin, v. 72, p. 181-181.
Haldorsen, H. H. and Damsleth, L. W. (1990). " Stochastic Modeling." Journal of Petroleum Technology, April 1990, p. 404-412.
Ioannidis, M. A., Kwiecien, M. J. and Chatzis, I. (1997). "Statistical Analysis of the Porous Microstructure as a Method for Estimating Reservoir Permeability."
Petroleum Science and Engineering, v. 16, No. 4, p. 251-261.
MacDonald, A. C. and Aasen, J. O. (1993). "Parameter Estimation for Stochastic Models of Fluvial Channel Reservoirs." AAPG Bulletin, v. 77, p. 1643-1644.
McDowell,R. R., Matchen, D. L., Hohn, M. E., and Vargo, A. G. (1994). "An Innovative Geostatistical Approach to Oil Volumetric Calculations: Rock Creek Field, West Virginia: Abstract ." AAPG Bulletin, v. 78, p. 1332-1332.
Murray, C. J. (1992). "Geostatistical Simulation of Petrophysical Rock Types."
AAPG Bulletin, v. 76, p. 94-94.
Nederlof, M. H. (1994). "Comparing Probabilistic Predictions with Outcomes in Petroleum Exploration Prospect Appraisal." Nonrenovable Resources, v.3, No. 3, p.
183-189.
Norris, R. J. (1996). "Focusing Stochastic Simulation for Effective Problem-Solving in Reservoir Engineering: Abstract." AAPG Bulletin, v. 80, p. 1318.
Pawar, R. J., Edwards, E. B. and Whitney, E. M. (2001). "Geostatistical Characterization of the Carpinteria Field, California." Petroleum Science and Engineering, v. 31, No. 2-4, p. 175-192.
Smyth, M. and Buckley, M.J. (1993). " Statistical Analysis of the Microlithotype Sequences in the Bulli Seam, Australia, and Relevance to Permeability for Coal Gas." International Journal of Coal Geology, v. 22, p. 167-187.
Tetzlaff, D. (1996). "Probabilistic Estimates from Reservoir-Scale Sedimentation Models." Numer. Exp. Stratigr., Lawrence. KS. 1996, p. 145-146.
TERMINOLOGY
In compiling this list of geostatistical terminology, only the most commonly encountered terms were selected. No attempt was made to duplicate the more extensive glossary by Richardo Olea (1991). Some definitions may differ slightly from those of Olea.
Admissibility (of semivariogram models): for a given covariance model, the kriging variance must be 0, this condition is also known as positive definite.
Anisotropy: refers to changes in a property when measured along different axes.
In geostatistics, anisotropy refers to covariance models that have major and minor ranges of different distances (correlation scale or lengths). This condition is easiest seen when a variogram shows a longer range in one direction than in another. In this module, we discuss two types of anisotropy:
Geometric anisotropic covariance models have the same sill, but different ranges;
Zonal anisotropic covariance models have the same range, but different sills.
Auto-correlation: a method of computing a spatial covariance model for a regionalized variable. It measures a change in variance (variogram) or correlation (correlogram) with distance and/or azimuth.
Biased estimates: seen when there is a correlation between standardized errors and estimated values (see Cross-Validation). A histogram of the standardized errors is skewed, suggesting a bias in the estimates, so that there is a chance that one area of a map with always show estimates higher (or lower) than expected.
Block kriging: Kriging with nearby sample values to make an estimated value for an area; making a kriging estimate over an area, for example estimating the average value at the size of the grid cell. The grid cell is divided into a specified number of sub-cells, a value is kriged to each sub-cell, and then the average value is placed at the grid node.
Cokriging: the process of estimating a regionalized variable from two or more variables, using a linear combination of weights obtained from models of spatial auto-correlation and cross-correlation. The multivariate version of kriging.
Conditional bias: a problem arising from insufficient smoothing which causes high values of an attribute to be overstated, while low values are understated.
Conditional simulation: a geostatistical method to create multiple (and equally probable) realizations of a regionalized variable based on a spatial model. It is conditional only when the actual control data are honored. Conditional simulation is a variation of conventional kriging or cokriging, and can be considered as an extrapolation of data, as opposed to the interpolations produced by kriging. By relaxing some of the kriging constraints (e.g. minimized square error), conditional simulation is able to reproduce the variance of the control data. Simulations are not estimations; their goal is to characterize variability or risk. The final “map”
captures the heterogeneity and connectivity mostly likely present in the reservoir.
Post processing conditional simulation produces a measure of error (standard deviation) and other measures of uncertainty, such as iso-probability and uncertainty maps.
Correlogram: a measure of spatial dependence (correlation) of a regionalized variable over some distance. The correlogram can also be calculated with an azimuthal preference.
Covariance: a measure of correlation between two variables. The kriging system uses covariance, rather than variogram or correlogram values, to determine the kriging weights, . The covariance can be considered as the inverse of the variogram, and equal to the value of the sill minus the variogram model (or zero minus the correlogram).
Coregionalization: the mutual spatial behavior between two or more regionalized variables.
Cross-correlation: a technique used to compute a spatial cross-covariance model between two regionalized variables. This provides a measure of spatial correlation between the two variables. It produces a bivariate analogue of the variogram.
Cross-validation: a procedure to check the compatibility between a data set, its spatial model and neighborhood design. First, each sampled location is kriged
with all other samples in the search neighborhood. The estimates are then compared against the true sample values. Significant differences between estimated values and true values may be influenced by outliers or other anomalies. This technique is also used to check for biased estimates produced by poor model and/or neighborhood design.
Drift: often used to describe data containing a trend. Drift usually refers to short scale trends at the size of the neighborhood.
Estimation variance: the kriging variance at each grid node. This is a measure of global reliability, not a local estimation of error.
Experimental variogram: a measure of spatial dependence (dissimilarity or increasing variability) of a regionalized variable over some distance and/or direction. This is the variogram that is based upon the sample data; upon which the model variogram will be fitted.
External drift: a geostatistical linear regression technique that uses a spatial model of covariance when a secondary regionalized variable (e.g. seismic attribute) is used to control the shape of the final map created by kriging or simulation.
Geostatistics: the statistical method used to analyze spatially (or temporally) correlated data and to predict the values of such variables distributed over distance or time.
h-Scatterplot: a plot obtained by selecting a value for separation distance, h, then plotting the pairs Z(x) and Z(x+h) as the two axes of a bivariate plot. The shape and correlation of the cloud is related to the value of the variogram for distance, h.
Histogram: a plot, which shows the frequency or number of occurrences (Y-axis) of data, falling into size classes of equal width (X-axis).
Indicator variable: a binary transformation of data to either 1 of 0, depending on whether the value of the data point surpasses or falls short of a specified cut-off value.
Interpolation: estimation technique in which samples located within a certain search neighborhood are weighted to form an estimate, such as the kriging technique.
Inverse distance weighting: Non-geostatistical interpolation technique that assumes that attributes vary according to the inverse of their separation (raised to some power).
Iso-probability map: maps created by post processing conditional simulations to show the value of the regionalized variable at a constant probability threshold.
For example, at the 10th, 50th (median), or the 90th percentiles. These maps provide a level of confidence in the mapped results.
Kriging: a method of calculating estimates of a regionalized variable using a linear combination of weights obtained from a model of spatial correlation. It assigns weights to samples to minimize estimation variance. The univariate version of cokriging.
Kriging variance: see estimation variance.
Lag: a distance parameter (h) used during computation of the experimental covariance model. The lag distance typically has a tolerance of one-half the initial lag distance.
Linear estimation method: a technique for making estimates based on a linear weighted average of values, such as seen in kriging.
Model variogram: a function fitted to the experimental variogram as the basis for kriging.
Moving neighborhood: a search neighborhood designed to use only a portion of the control data point during kriging or conditional simulation.
Nested variogram model: a linear combination of two or more variogram (correlogram) models. It has more than one range showing different scales of spatial variability; for example, a short-range exponential model combined with a longer-range spherical model. Often, it involves adding a nugget component to one of the other models.
Nonconditional simulation: a method that does not use the control data during the simulation process; quite often used to observe the behavior of a spatial model and neighborhood design.
Nugget effect: a feature of the covariance model where the experimental points defining the model does not appear to intersect the y-axis at the origin. The nugget represents a chaotic or random component of attribute variability. The nugget model shows constant variance at all ranges, but is often modeled as zero variance at the control point (well location). Abbreviated as Co by convention.
Ordinary (co-)kriging: a technique in which the local mean varies and is re-estimated based on the control points in the search neighborhood ellipse (moving neighborhood).
Outliers: data points falling outside about 2.5 standard deviation of the mean value of the sample population possibly the result of bad data values or local anomalies.
Point kriging: making a kriging estimate at a specific point, for example at a grid node, or a well location.
Positive definite: see admissibility.
Random function: the random function has two components: (1) a regional structure component manifesting some degree of spatial auto-correlation (regionalized variable) and lack of independence in the proximal values of Z(x), and (2) a local, random component (random variable).
Random variable: a variable created by some random process, whose values follow a probability distribution, such as a normal distribution.
Range: the distance where the variogram reaches the sill, or when the correlogram reaches zero correlation. Also known as the correlation range or correlation scale, it represents the distance at which correlation ceases. It is abbreviated as a by convention.
Regionalized variable: a variable that has some degree of spatial auto-correlation and lack of independence in the proximal values of Z(x).
Risk map: see Uncertainty Map
Simple kriging: the global mean is constant over the entire area of interpolation and is based on all the control points used in a unique neighborhood (or is supplied by the user).
Semivariogram: a measure of spatial dependence (dissimilarity or increasing variability) of a regionalized variable over some distance; a plot of similarity between points as a function of distance between the points. The variogram can also be calculated with an azimuthal preference. The semivariogram is commonly called a variogram. See also correlogram.
Sill: the upper level of variance, where the variogram reaches its correlation range. The variance of the sample population is the theoretical sill of the variogram.
Smearing: a condition produced by the interpolation process where high-grade attributes are allowed to influence the estimation of nearby lower grades.
Stationarity: the simplest definition is that the data do not exhibit a trend; spatial statistical homogeneity. This implies that a moving window average shows homogeneity in the mean and variance over the study area.
Stochastic modeling: used interchangeably with conditional simulation, although not all stochastic modeling applications necessarily use control data.
Support: the size, shape, and geometry of volumes upon which we estimate a variable. The effect of which is that attributes of small support are more variable than those having a larger support.
Transformation: a mathematical process used to convert the frequency distribution of a data set from Lognormal to Normal.
Unique neighborhood: a neighborhood search ellipse that uses all available data control points. The practical limit is 100 control points. A unique
neighborhood is used with simple kriging.
Uncertainty map: these are maps created by post processing conditional simulations. A threshold value is selected, for example, 8 % porosity, an uncertainty map shows at each grid node, the probability that porosity is either above or below the chosen threshold.
Variogram: geostatistical measure used to characterize the spatial variability of an attribute.
Weights: values determined during an interpolation or simulation, that are multiplied by the control data points in the determination of the final estimated or simulated value at a grid node. To create a condition of unbiasness, the weights,
, sum to unity for geostatistical applications.
SUGGESTED REFERENCE
Olea, R. A., 1991, Geostatistical Glossary and Multilingual Dictionary, New York, Oxford University Press, 177 pages.