Teacher s Aide Geologic Characteristics of Hole-Effect Variograms Calculated from Lithology-Indicator Variables 1

Mathematical Geology, Vol. 33, No. 5, 2001 Teacher s Aide Geologic Characteristics of Hole-Effect Variograms Calculated from Lithology-Indicator Variables 1 Thomas A. Jones 2 and Yuan Z. Ma 3 Variograms calculated from binary variables, such as from two lithologies, tend to show sinusoidal forms with decreasing amplitudes for increasing lag distances. This cyclicity is observed often when analyzing drill-hole data for rock sequences with alternating lithologies, and the variograms are thus labeled hole-effect variograms. Such variograms show a variety of forms: (1) Low to moderate variation in lithologic-body dimensions causes variograms to have strong cyclicity with decaying amplitude. (2) Variograms with one or more peaks and troughs usually result from a binary variable for which lithologies are about equally abundant but possibly large variations exist in the size of lithologic bodies. (3) Variograms show poor cyclicity if one lithology has highly variable body sizes and the other has moderately variable body dimensions. (4) Variograms that attain a plateau at short lag distances represent extremely high or low sandstone fraction, high variability in size of the most abundant lithology, and low variability in the other. Information about the dimensions of lithologic bodies makes it possible to approximate characteristics of the variogram of the lithology variable without numerous wells. Conversely, a hole-effect variogram of lithology may be used to estimate lithologic dimensions. KEY WORDS: geostatistics, binary variable, indicator variogram, cyclicity, sinusoids, body dimensions. INTRODUCTION Geostatistical analysis and geologic modeling are commonly used tools in the petroleum industry. The products from such work are typically geologic models of porosity and permeability, so most application effort to date has focused on such continuous variables. Geostatistical models of discrete lithofacies variables are important, however, because of their use as templates to build accurate porosity and permeability models. 1 Received 7 October 1999; accepted 31 May 2000. 2 ExxonMobil Upstream Research Company, P. O. Box 2189, Houston, Texas 77252-2189. e-mail: tajones@upstream.xomcorp.com 3 ExxonMobil Upstream Research Company, P. O. Box 2189, Houston, Texas 77252-2189. e-mail: yuan.ma@exxon.sprint.com 615 0882-8121/01/0700-0615$19.50/1 C 2001 International Association for Mathematical Geology

616 Jones and Ma This paper discusses a discrete, binary variable called a lithology-indicator variable, or more simply an indicator variable, because it indicates the presence or absence of the lithology of interest. For instance, the lithology-indicator variable may code observed sandstone as 1 and shale as 0. A variogram may then be computed from these binary data. Figure 1 shows an experimental variogram, strongly cyclical with lag distance, that results from such an indicator variable. This is a common form for rock-type information observed down drill holes if lithologies occur in repeated layers, and has been termed a hole-effect variogram (e.g., David, 1977). A substantial amount of data is desired to calculate a variogram. This is little problem for down-hole calculations, but it can introduce significant difficulties for lateral variograms. Only a few, widely scattered wells may be available, so the resulting experimental variogram does not contain much detail or resolution, and no features smaller than the well spacing result. However, more geologic information is contained in hole-effect variograms than seems apparent at first glance. We use simple arguments and explanations, along with numerical demonstrations, to show how properties of a hole-effect variogram can be inferred from distributions of the sizes of lithologic bodies. The discussion is in terms of one dimension, although the concepts extend to two or three dimensions. Cyclicity and amplitudes in hole-effect variograms are affected strongly by relative abundance of each lithology and by the sizes of lithologic (e.g., sandstone Figure 1. Experimental indicator variogram γ (h) showing hole-effect cyclicity with distance h.

Geologic Characteristics of Hole-Effect Variograms 617 and shale) bodies. Information about dimensions of these sandstone and shale bodies makes it possible to approximate characteristics (e.g., wavelength of cyclicity, sandstone fraction, lag distance at which amplitude attenuates to a plateau, and height of first peak) of the indicator variogram without numerous wells. Conversely, a hole-effect variogram of lithology may be used to estimate such properties as sandstone fraction, mean dimensions of sandstone and shale bodies, and total variation. METHOD USED FOR ANALYSIS Assume that we are studying two lithologies distributed along a line. In map view, the line may be a traverse along depositional strike in which all points are correlative stratigraphically. Alternatively, we may observe a vertical series of lithologies down a well. For convenience, consider a strike-oriented traverse through sand-filled channels in a shale background and assume that the dimensions are widths of the channels and shale bodies. Designate two lithologies as A (channel sandstone) and B (shale). Because we assume binary occurrences of the lithologic indicator variable, each occurrence of lithology A is followed by an occurrence of lithology B, and vice versa. That is, the traverse can only contain the sequence...ababa..., as in Figure 2. It might seem reasonable for several A bodies to occur together as channels intersect each other. However, with sparse data it is difficult to identify channels as intersecting, so only a large extent of channel-facies A, not several intersecting channels, may be identifiable. A single, wide channel could not be distinguished from a complex of narrow channels. For such cases, lithology A might be redefined to represent sandstone in channel complexes. The widths of lithology A (sandstone) observed along the traverse are generally not constant, but have a statistical distribution. Designate the mean width of the A-channels as µ A, the standard deviation of the widths as σ A, and the coefficient of variation of the A-lithology widths as CV A = σ A /µ A. Similarly, designate the mean, standard deviation, and coefficient of variation of widths of the lithology-b shales as µ B, σ B, and CV B. Define the fraction of lithology A in the study area as p; the fraction of lithology B is 1 p. Observed widths of A may be treated as drawings from a probability distribution with mean µ A and standard deviation σ A, and similarly for lithology B. Figure 2. Example traverse showing succession of A-lithofacies bodies alternating with B-lithofacies bodies. Tick marks indicate observations of lithofacies along traverse.

618 Jones and Ma The width random variables may take on any convenient form; two obvious choices are normal and uniform distributions. The fixed limits of the uniform distribution make it convenient for explanatory use. Relations that are reported here for the uniform distribution were also found to apply to normally distributed widths. The model of alternating lithologies described above was used for computer simulations and for analysis. For each computer simulation, values of µ A, µ B, σ A, and σ B were specified, and then 200 A and B widths were sampled randomly from the two width distributions. A traverse was defined with a sample spacing of 1 m. These lithologies were then put into the traverse (Fig. 2). Many sample points, spaced 1 m apart, are thus observed in the traverse. A variogram was then calculated from these values. Recall that the experimental variogram (more strictly, semivariogram), γ (h), is a measure of dissimilarity of measurements observed at a specified distance h apart, and is defined as γ (h) = [Z(x + h) Z(x)] 2 /2N(h) (1) where Z(x) is an observation at location x, Z(x + h) is an observation at location x + h, the sum is taken over all pairs separated by h, and N(h) is the number of such pairs in the sum (e.g., David, 1977; Hohn, 1988). The variogram indicates the relative likelihood of transitions between different lithologies (that is, A BorB A) at two points separated by a lag distance h. However, the variogram value does not distinguish the type of transition because both lithologic transitions affect Equation (1) in the same way. For details about relations between indicator variograms and transition probabilities, see Carle and Fogg (1996). OBSERVATIONS AND ANALYSIS To analyze characteristics of hole-effect variograms, more than 200 traverses were generated, each based on sampling from the A and B distributions with specified means and variances, and variograms then were calculated from the traverses. Table 1 lists statistics for variograms shown in various figures of this paper. Three cases are identified for discussion. Both Sandstone and Shale Bodies with Constant Widths Consider the case in which each occurrence of lithology A is exactly a meters wide along the traverse, and each occurrence of B is b meters wide, that is, µ A = a, µ B = b, and σ A = σ B = 0. In this case, lateral facies changes are perfectly

Geologic Characteristics of Hole-Effect Variograms 619 Table 1. Statistics Used to Generate Variograms in Figures a Figure µ A µ B σ A σ B p V h 1 10 5.0 0 1.7 0.67 0.22 63 3 10 5 0 0 0.67 0.22 4A 10.0 4.9 0.7 0.6 0.67 0.22 >150 4B 10.0 5.0 2.4 1.8 0.67 0.22 35 5A 19.7 20.6 11.8 11.3 0.49 0.25 45? 110? 5B 20.5 10.7 11.8 5.7 0.66 0.23 38 5C 19.1 5.8 11.2 2.9 0.77 0.18 25 5D 19.6 3.3 11.7 1.4 0.85 0.12 5 a Symbols µ A,µ B,σ A, and σ B represent means and standard deviations of body dimensions, p is fraction of lithology A (sandstone), V = p(1 p) is variance, h is apparent range (distance of effective convergence to V ), and? implies ambiguity in determining h. periodic, as indicated by the variogram shown in Figure 3 with a = 10 and b = 5. This variogram is not realistic for geological data, but it illustrates concepts. Note the constant wavelength and amplitude, and the trapezoidal forms. Assume m alternating occurrences of A and B along a traverse, for a total of m(a + b) lithologic bodies. From the binary distribution, the proportion of Figure 3. Variogram showing effects of constant widths of A- and B-lithologies.

620 Jones and Ma lithology A is p = ma/m(a + b) = a/(a + b) = µ A /(µ A + µ B ) The proportion of lithology B is given by 1 p = b/(a + b) = µ B /(µ A + µ B ) Now consider the wavelength, peaks, and troughs of variograms. Equation (1) defines γ (h) to consist of a sum S divided by twice the number of terms in that sum, N(h): γ (h) = S/2N(h). For lag distance h, the number of terms in S is given by N(h) = m(a + b) h. Assume first that 1 < b a. Forh b, most pairs Z(x) and Z(x + h) have the same values, putting 0 values in the sum S. Only at transitions A Bor B A does a value of 1 go into S, and only a single entry occurs at each such lithology interface. There are h A B transitions and h B A transitions, so S = m(h + h) h and γ (h) h/(a + b). This is consistent with results given by Carle and Fogg (1996) for the slope of γ (h) near the origin. For b h a, the variogram sum remains constant at S = m(b + b) h, giving γ (h) b/(a + b) and a virtually flat top to the peak, spanning from h = min(a, b) = min(µ A,µ B )toh =max(a, b) = max(µ A,µ B ). If a = b, peaks are pointed and the cycles are triangular-shaped rather than trapezoidal. As h increases from a to a + b, γ (h) decreases linearly: γ (h) [b (h a)]/(a + b). At h = a + b, S = 0 because this step always finds A AorB B transitions at points x and x + h. The same result is obtained at h = 2(a + b), 3(a + b), etc., giving the wavelength λ = a + b = µ A + µ B Similar relations are obtained for a b. The peak heights equal min(a, b)/(a + b) = min(µ A,µ B )/(µ A +µ B )=min(p, 1 p). Sand and Shale Bodies with Small Standard Deviations of Width The variograms γ (h) described in the previous section represent a perfectly periodic function with constant cyclic amplitude and wavelength. The variogram in Figure 1, constructed from a traverse with variation in the A and B dimensions, is more typical; amplitudes of the sinusoidal cycle decay with increasing h, until γ (h) fluctuates narrowly around a constant value.

Geologic Characteristics of Hole-Effect Variograms 621 As above, the expected proportion of lithology A is p = µ A /(µ A + µ B ). In addition, the wavelength remains approximately µ A + µ B. This is because the varying body widths cause boundaries between bodies to have a spatial distribution, but the average positions of the boundaries tend to remain at the same distances. Consider the troughs of the variograms. Troughs appear at multiples of the wavelength: h = µ A + µ B,2(µ A +µ B ), etc. Recall that in the previous case, γ (µ A + µ B ) = 0. If the standard deviations are nonzero, random changes in lithology occur around the transitions at spacings h equal to multiples of the wavelength µ A + µ B and therefore γ (h) > 0. Moreover, the degree of randomness added at the lithology interfaces is greater for longer lags because the variation in random widths adds up. Hence γ [m(µ A + µ B )] >γ [n(µ A +µ B )] for integers m > n 1. Similarly, variogram peaks occur every µ A + µ B lag-distance units, with the first peak at h = 0.5λ. The heights of the peaks also change as a function of h. The first peak typically has greatest height, with maximum possible value min(p, 1 p). If variation in body dimensions is great, then shifts of the body boundaries tend to quickly raise the lows and decrease the peaks, causing rapid convergence of γ (h) to a plateau that conceptually corresponds to a sill (Fig. 1). The value of a sill equals the global variance of the variable (David, 1977); for a binary variable, this variance V equals the product of the lithology fractions: V = p(1 p). Similar to traditional variograms, γ (h) effectively converges to V at some distance that is analogous to the range of a spherical variogram. Because convergence of an experimental, hole-effect variogram to the variance often has fluctuations, defining the range of correlation sometimes can be ambiguous. For instance, the range of the variogram in Figure 4B might be estimated to be either 27 or 35 m, depending on whether the second trough is considered significant. We will use the term apparent range, denoted as h, to represent the approximate limit of predictability in the hole-effect variogram, that is, the distance at which the variogram has apparently converged to the variance. In Figure 1, h is approximately in the range 60 64. The two variograms in Figure 4 were generated from distributions with similar mean widths, but the standard deviations used in Figure 4A are less than those used for Figure 4B (Table 1). These differences cause the amplitudes to decrease at different rates, with h > 150 (out of range of figure) for Figure 4A and h = 35 for 4B. In fact, the rate of attenuation of cyclic amplitude toward V = p(1 p) depends on variability in body dimensions and proportions of the facies, being more rapid for large standard deviations and large differences between mean widths of the two lithologies. Traverses were generated with a variety of dimensional statistics; the resulting variograms were analyzed to determine an empirical relation between the dimensional statistics and h (discussed below).

622 Jones and Ma Figure 4. Two variograms computed from distributions with similar mean widths, but standard deviations for curve in part (B) are greater than for part (A) (Table 1).

Geologic Characteristics of Hole-Effect Variograms 623 Sandstone and Shale Bodies with Greater Standard Deviations of Width Greater variability in lithologic-body dimensions causes some of the general cyclic forms in hole-effect variograms to break down. Characteristics of the lithology-indicator variogram depend on the means and standard deviations of body size for the two lithologies. Such hole-effect variograms (Fig. 4B) commonly are characterized by three significant extrema (peaks and troughs) at the near-lag distances. However, larger differences between the two lithologic proportions and greater variability in body dimensions can degrade the cyclic forms quickly. Figure 5 shows four variograms that are based on conditions more extreme than in Figure 4. Variograms are affected significantly by sandstone fraction, the sizes of the lithologic bodies, and the standard deviations of their dimensions. Similar values were used for µ A and σ A for all variograms, but µ B and σ B decrease from curves A to D. This implies an increase of sandstone fraction p from A to D, causing lower sills from A to D. The same effect occurs for constant µ B and decreasing µ A. Moderate sandstone fractions and moderate to high variations tend to yield variograms showing some sinusoidal form (Fig. 5, curves A, B). However, the Figure 5. Variograms computed with similar means and standard deviations for A-lithology widths, but with varying means and standard deviations for B-lithology widths (Table 1).

624 Jones and Ma combination of large sandstone bodies in a background of small shale bodies tends to yield a variogram with a more or less flat plateau and without significant cyclicity (Fig. 5, curve C). Variograms for which the A-lithology bodies are large and variable in size, but B-lithology body dimensions are smaller and less variable (Fig. 5, curve D; Table 1), climb rapidly to V, reaching it at h approximately equal to µ B. With an extremely high sand fraction and large standard deviation of body size, cyclicity disappears from the variograms. The reverse situation (low sandstone fraction and large shale bodies) gives similar variograms through symmetry. The statistics (Table 1) associated with Figure 5 show that the first peak in curve A is misplaced. Rather than being at h = 0.5λ = 20, it is located at approximately h = 30. This is caused by large coefficients of variation in the body dimensions. Similar large variation also causes troughs to be shifted and distorted, and the variogram to rapidly converge to V for shorter lag distances. The rapid move toward randomness distorts the sinusoidal pattern and overrides the simple peak and trough relations. To investigate where the relation breaks down, we calculated many variograms with various combinations of means and standard deviations. Figure 6 consists of two crossplots, with max(cv A,CV B ) on both ordinates. The abscissas plot (A) percent deviation of the first peak from 0.5λ, and (B) percent deviation of the first trough from λ. The points in the figure that have negative deviations may be disregarded. They consist of peaks or troughs with shorter distances than expected, caused by ambiguity of determining the peak (flat-topped first peak in Fig. 1) and a nonsinusoidal start of a flat variogram (Fig. 5, curve D). Note also that deviations in the range (0, 10) percent may not be significant, given the resolution of typical available information, especially if the variograms have short wavelengths. Figure 6A shows little deviation from expected peak locations if max(cv A, CV B ) is approximately less than 0.5, but large deviations occur for greater coefficients of variation in body size of one lithology. The three solid dots at deviations of approximately 26% all have strong curvature on the front of the peaks, artificially making them appear to represent greater h. Similarly, Figure 6B shows little deviation from expected trough locations for variograms in which max(cv A, CV B ) < 0.5, but large deviations occur for max(cv A, CV B ) > 0.55. For both peaks and troughs, if max(cv A, CV B ) > 0.5, the variograms tend to approach the variance rapidly and distort the sinusoidal form. However, peak locations can be ambiguous (e.g., flat-topped) if CVs are very small. Lithology fraction can also affect peak and trough locations, but to a lesser degree than variability of body size. When the variability is not strong (less than the breakdown value of 0.5 in Figure 6), equal sandstone and shale fractions tend to preserve the sinusoidal wavelength. Large differences in two lithology fractions tend to deform the sinusoidal waves. The open circles in Figure 6 represent variograms with 0.4 p 0.6. The solid dots represent 0.3 p < 0.4 and 0.6 < p 0.7, and the cross symbols represent p < 0.3 and p > 0.7.

Geologic Characteristics of Hole-Effect Variograms 625 Figure 6. Crossplots with maximum coefficient of variation of two lithologies in variogram, max(cv A, CV B ), on ordinate. A, Percent deviation of first peak from 0.5λ on abscissa. B, Percent deviation of first trough relative to λ on abscissa. For both plots, open circles represent 0.4 p 0.6, solid dots represent 0.3 p < 0.4 and 0.6 < p 0.7, and symbols represent p < 0.3 and p > 0.7.

626 Jones and Ma Convergence at h The rate of attenuation of cyclic patterns in hole-effect variograms depends mainly on variability of the widths of the lithologic bodies and the proportion of each facies. The exact relationship between attenuation and the geologic properties is complex and unknown. To study the relations, many variograms were constructed with a variety of dimensional parameters. In each, the apparent range of correlation was normalized by wavelength, as h /λ, and the total variability was normalized as (σ A + σ B )/(µ A + µ B ). Figure 7 shows a plot with h /λ (apparent point of convergence, normalized to wavelength) on the vertical axis and (σ A + σ B )/(µ A + µ B ) on the horizontal axis. The cloud of points defines the region that relates h to the standard deviations σ A and σ B. Open circles represent moderate values of sand fraction (0.3 p 0.7), and the black dots represent low or high sand fraction (p < 0.3 or p > 0.7). The cloud steepens and exits the plot for smaller standard deviations, but greater values of h /λ are not important in practice; limits on search distance commonly preclude using very large h. This empirical approximation should be used with caution because determination of h may be ambiguous; in Figure 5, curve A, h may be 45, but arguments could be made that h = 110. Figure 7. Crossplot of h /λ (point of apparent convergence to plateau, normalized to wavelength) on ordinate vs. (σ A + σ B )/(µ A + µ B ) on abscissa. Open circles represent moderate sand fractions (0.3 p 0.7) and solid circles represent low or high sand fractions.

Geologic Characteristics of Hole-Effect Variograms 627 RELATING VARIOGRAMS AND DIMENSIONAL DATA The variety of forms found in lithologic-indicator variograms may be summarized as follows: At an extreme of no variability, perfect cyclicity results (Fig. 3). Variograms based on low to moderate variation in body widths show a strong cyclicity with decaying amplitude (Figs. 1 and 4). A hole-effect variogram with one or more peaks and troughs (Fig. 5, curves A and B) usually results from a binary variable where p is near 0.5 and possibly moderate to large variations exist in the size of lithologic bodies for both facies. The variogram in Figure 5, curve C, represents the case where one lithology has highly variable facies bodies and the other has moderately variable body dimensions. A variogram that attains a flat sill at short lag distances represents extremely high or low sandstone fraction, high variability in size of the most abundant lithology, and low variability in the other. Spatial correlation exists only for very short lag distances (Fig. 5, curve D). Consider a situation in which it is necessary to generate a horizontal variogram, but only a few or widely separated wells have been drilled in the area. Analysis of core and seismic surveys leads to a geologic interpretation of the depositional environment, and from there to a choice of depositional analogue. Turning to a data base, company files, or experience, we can extract or infer dimensions of the observed lithofacies. The dimensional statistics µ A,µ B,σ A, and σ B may be used to sketch an approximation of a variogram along the traverse direction. Although accuracy drops off with greater lag distance h, in many situations the wavelength, peaks and troughs, and critical values may be estimated, as follows: 1. The mean dimensions may be used to estimate the lithofacies fractions, as p = µ A /(µ A + µ B ) and hence 1 p. 2. The wavelength of the variogram cycles is λ = µ A + µ B. The first peak occurs at an average distance h = 0.5λ, unless max(cv A, CV B ) > 0.5. Troughs are also equally spaced, with the first trough at h = λ, unless max(cv A, CV B ) > 0.5. 3. The maximum height of the first peak is min(p, 1 p). When standard deviations are large relative to means, the actual height may be less than the maximum, but no general relation has been found. 4. The point h at which the variogram essentially has converged to V = p(1 p) may be estimated by using the plot in Figure 7. For given (σ A +

628 Jones and Ma σ B )/(µ A + µ B ), estimate h /λ and multiply by λ = µ A + µ B to convert to h in distance units. 5. The attenuating amplitude of the peaks and troughs is difficult to estimate, although an exponential attenuation may be considered. Variograms generated from lithofacies size distributions certainly are approximations. However, for most applications the general forms are adequate, and exact heights of peaks and troughs are less important. Use of size distributions to approximate variograms may be especially valuable for simulated-annealing applications where numerical variograms, rather than mathematical models, may be used. Now consider the opposite situation in which we may need lithofacies dimensions to generate an object-based facies model. If a variogram should be available from multiple wells, analogues, or outcrop measurements, then it may be used to estimate the critical values, as follows: 1. Use the plateau value V observed on the variogram in the quadratic equation p(1 p) = V to obtain min(p, 1 p) = [1 (1 4V )]/2. 2. Use geologic interpretation, measured sandstone percentage, etc., to determine whether the calculated fraction in step 1 applies to lithology A (p) orb(1 p). 3. Estimate the wavelength λ = µ A + µ B from constant peak or trough spacings. Be careful if max(cv A, CV B ) > 0.5. 4. Obtain µ A and µ B by µ A = pλ and µ B = (1 p)λ. Alternatively, the maximum possible height of the first peak is min(p, 1 p) = min(µ A, µ B )/λ. Multiply the observed height by λ to get min(µ A,µ B ), and use the results in steps 1 and 2. 5. Estimate the apparent range h from the variogram to obtain h /λ using the estimated wavelength λ from step 3. Use Figure 7 to obtain (σ A + σ B )/(µ A + µ B ) and hence the non-normalized total variation (σ A + σ B ). We have found no way to estimate σ A and σ B separately. CONCLUDING REMARKS Geologic characteristics of lithofacies size distributions can be related to hole-effect variograms. This information can be useful when a variogram on a binary variable must be generated. However, recall that this paper is based on approximations, empirical calculations, and the restriction that intersecting bodies of the same lithofacies type in the traverse are not distinguishable. All figures shown here were made from artificially generated traverses, with the body dimensions sampled from uniform distributions to simplify the presentation. However, we also generated traverses using normal distributions. The wavelengths remained the same, and attenuation to V was similar. The main difference

Geologic Characteristics of Hole-Effect Variograms 629 was that the normal distributions caused the peaks and troughs to be slightly more rounded. Experimental variograms calculated with actual lithologic data may not look like variograms presented here: These are based on perfect conditions, with abundant, closely spaced data points along a traverse. Measured data for actual applications are not abundant, evenly distributed, or spaced closely enough to provide good control for short lag distances. For instance, if the sample spacing is substantially greater than the mean widths of the lithologic bodies or the wavelength, then no hole effect can be seen on the calculated variogram. In addition, more complex distributions may be observed in nature, including superimposed cycles and wavelengths dependent on thickness of complexes and lithologic units. Geostatistical methods use theoretical variogram models to describe spatial discontinuity of the property under study. The variogram model must match the form of the experimental variogram in order to honor underlying continuity conveyed in the data. Therefore, modeling hole-effect variograms is important for kriging or simulating lithology or lithofacies. Journel and Froidevaux (1982) discuss fitting hole-effect variograms for continuous variables. Ma and Jones (2001) discuss fitting lithologic hole-effect variograms with several models. ACKNOWLEDGMENTS Craig Calvert, Ken Dahlberg, Chris Donofrio, Anil Deshpande, Martha Gerdes, and Peter Glenton read various drafts of the manuscript. Anonymous reviewers made helpful suggestions. REFERENCES Carle, S. F., and Fogg, G. E., 1996, Transition probability-based indicator geostatistics: Math. Geology, v. 28, no. 4, p. 453 476. David, M., 1977, Geostatistical ore reserve estimation: Elsevier Scientific Publishing Company, New York, 364 p. Hohn, M. E., 1988, Geostatistics and petroleum geology: Van Nostrand Reinhold, New York, 264 p. Journel, A. G., and Froidevaux, R., 1982, Anisotropic hole-effect modeling: Math. Geology, v. 14, no. 3, p. 217 239. Ma, Y. Z., and Jones, T. A., 2001, Modeling hole-effect variograms of lithology-indicator variables: Math. Geology, v. 33, no. 5, p. 631 648.