Improved methods for quantitative analysis of three-dimensional porphyroblastic textures

Because metamorphic crystallization occurs on geological time scales, it cannot be observed directly. Instead, the conditions under which it takes place and the interplay of physical mechanisms responsible for its occurrence must be inferred from its products. One avenue of research examines the spatial distribution of porphyroblasts within a rock volume. Kretz (1969) pioneered the technique of testing a set of porphyroblasts for ordering or randomness using a variety of statistical approaches. Kretz (1974) and Carlson (1989, 1991) linked the occurrence of ordering to intergranular diffusion as the rate-limiting process in crystal growth. When diffusion is rate-limiting, a depleted zone will develop around each growing porphyroblast, in which some of the nutrients required for crystal growth are less abundant than in regions unaffected by diffusion. The occurrence of such a depleted zone may lead to two textural consequences. First, new nucleation events within the depleted zone may be inhibited by the lowered chemical affinity for reaction, producing a tendency toward ordering of porphyroblast centers. Second, porphyroblasts in close proximity to one another will compete for the limited local nutrient supply, resulting in diminished growth compared to crystals growing in isolation. Other potential rate-controlling factors, such as advective transfer of nutrients, addition of energy (heat) to the system, deformation effects, dissolution of the reactant phase, or incorporation of material into the growing porphyroblast at its interface, have no obvious bearing on ordering or competition.

Except in those rare cases in which porphyroblasts grow along a restricted two- dimensional horizon (e.g., Carlson, 1989, 1991), reliable testing for ordering requires three-dimensional data on crystal sizes and locations. Although such data can be obtained laboriously through serial sectioning (Daniel and Spear, 1999) or dissection (Kretz, 1993), high-resolution X-ray computed tomography (HRXCT) provides a powerful means to nondestructively measure sizes and locations of thousands of porphyroblasts in hand-sample– sized rocks (Carlson and Denison, 1992; Denison and Carlson, 1997). With appropriate statistical methods (Denison et al., 1997; Hirsch et al., 2000), such data can be used to assess crystallization mechanisms. Furthermore, the specimens are preserved and thus available for future study; HRXCT can also assist further relevant geochemical analysis by reliably identifying crystal central sections (e.g., Chernoff and Carlson, 1997, 1999).

Previous textural analyses of almandine-rich garnet porphyroblasts in regional metamorphic settings have produced evidence for both diffusion and interface control. Carlson (1989) found evidence for thermally accelerated diffusion control of garnet porphyroblast growth in the Picuris Mountains of New Mexico using chemical zoning profiles and normalized radius-rate analysis. These results were corroborated by HRXCT studies indicating ordering of porphyroblast centers (Carlson and Denison, 1992; Denison and Carlson, 1997) and size correlations consistent with competition (Chernoff and Carlson, 1997; Hirsch et al., 2000). Denison and Carlson (1997) and Hirsch et al. (2000) also reported evidence for ordering of garnet porphyroblasts from various other settings. Spear and Daniel (2001) also report evidence for diffusion control among garnet porphyroblasts in Harpswell Neck, Maine, based on detailed analysis of zoning profiles. Conversely, Daniel and Spear (1999) found that correlation functions and zoning profiles from garnet porphyroblasts from the Everett Formation in northwestern Connecticut gave mixed results, but the overall preponderance of their evidence led them to conclude that interface control was operative.

The HRXCT data in this study were acquired at the High-Resolution X-ray Computed Tomography (CT) Facility at the University of Texas at Austin (UTCT), which is described in detail by Ketcham and Carlson (2001). The facility features two CT subsystems, both of which were used in this study. The ultra-high-resolution subsystem employs a 200 kV microfocal X-ray source and routinely obtains data at the scale of tens of μm for small specimens (<5 cm in-plane dimension). The high-resolution subsystem utilizes a 420 kV source for imaging denser objects with diameters of up to 30 cm at resolutions on the order of hundreds of μm.

As described in the companion paper (Ketcham, this volume), previous HRXCT-based textural studies (Carlson and Denison, 1992; Denison and Carlson, 1997; Hirsch, 2000) utilized custom-written software that implemented a two-dimensional, slice-based interpretation method on the data volume produced by scanning. This study instead utilizes the BLOB3D software to do this analysis in three dimensions, permitting more accurate delineation of porphyroblast size, shape, and degree of impingement with neighboring crystals of the same type. Program operation is described in the companion paper.

All statistical analyses were conducted with the REDUCE3D software described by Hirsch (2000) and Hirsch et al. (2000). The statistical problem can be separated into two parts: a merit function that is sensitive to some aspect of spatial ordering or clustering, and a method for testing the significance of a particular result.

Two statistical approaches are used here to test for the presence of ordering and clustering of porphyroblasts. The first is to utilize a set of three single-valued statistics (Carlson, 1989; Denison et al., 1997), each of which is sensitive to different aspects of the porphyroblast distribution: the ordering index (OI), clustering index (CI), and impingement index (II). For each statistic, a random distribution of porphyroblast centers would yield a value of unity. Values of the OI greater than unity indicate ordering, and values less than unity suggest clustering. The CI and II have the opposite sense, with values greater than unity indicating clustering and less than unity indicating ordering.

The second approach is to use a set of correlation functions that test for randomness over a range of length scales (Daniel and Spear, 1999; Hirsch et al., 2000; Raeburn, 1996). Each proceeds by inspecting the number of crystal centers that occur within each of a set of spherical shells centered on each porphyroblast. By convention, the shell radii span a range of test distances from less than one to several times the mean crystal radius. Function values are calculated by either counting or measuring some property of the porphyroblasts within each shell, and normalizing the result based on the expected value for a random Poissonian distribution of zero- volume points in space. The Pair Correlation Function (PCF) examines the number of crystal centers in each shell and thus tests for ordering as would be caused by local suppression of nucleation, or clustering resulting from any factor that localizes nuclei. The Mark Correlation Function (MCF) tests for correlation of a “mark” variable linked to central positions, in this case crystal size (radius). The MCF is therefore sensitive to growth suppression resulting from competition for nutrients. For these functions, a value below 1.0 at a given test distance indicates ordering compared to the random case at that length scale, and a value above 1.0 indicates clustering.

Because a porphyroblast cannot nucleate within another crystal of the same material, true randomness of crystal centers is impossible (unless all nucleation is simultaneous), and some ordering is therefore inevitable. Thus, the mere presence of ordering is not enough to differentiate between growth mechanisms. It is also not possible, as a rule, to prove based on texture alone that a particular mechanism was responsible for crystal growth, as there are likely to be multiple possible means to achieve a certain size or spatial distribution. For example, Carlson et al. (1995) and Denison and Carlson (1997) found that various crystal size distribution (CSD) shapes could be generated using solely a diffusion-controlled reaction, by varying parameters such as heating path and degree and mode of clustering of nucleation sites.

Instead, it is easier to prove, statistically speaking, that a certain mechanism could not have been responsible for an observed texture. Generally, this approach consists of stating a null hypothesis and using a model based on this hypothesis to construct a 95% confidence region of statistic values that are congruent with it—a “null-hypothesis envelope.” If a measurement falls outside of this envelope, the null hypothesis is regarded as disproven (with the specified level of confidence) and the mechanism may be ruled out. The validity of this approach is naturally strongly dependent upon the correctness of the model used to construct the null-hypothesis envelope. For example, Daniel and Spear (1999) constructed confidence intervals for correlation functions based on models of both interface-controlled and diffusion-controlled growth. Hirsch et al. (2000) argued, however, that the Daniel and Spear (1999) approximation for the diffusion-controlled case is flawed because it neglects influential effects such as competition and thermal acceleration, and demonstrated that it rejects even simulations based on the Carlson et al. (1995) diffusion model as nondiffusional. Unfortunately, because of the localized complexities caused by competition, a more robust calculation of null-hypothesis envelopes for diffusion control would require running a large series of forward diffusional models, which would be computationally prohibitive.

The simpler course is to test only the null hypothesis of pseudorandom nucleation sites (i.e., sites limited only by the requirement that they are not within an already-growing porphyroblast at the time of nucleation) combined with an interface-controlled growth law, defined by the equation dR/dt = k (Kretz, 1974), where R is the radius of the growing porphyroblast, t is time, and k is a constant. Envelopes are calculated by creating a set of pseudorandom dispositions of crystals with the same size distribution as the measured data, within the same volume, and then performing the statistical calculations on them. At each test distance the 95% confidence interval is defined as the range of values obtained after deleting the top and bottom 2.5% of results. Statistic results for a sample that fall within an envelope are considered not to disprove the null hypothesis, whereas ordering or clustering may be inferred when results fall respectively below or above an envelope. Although the presence of clustering is not diagnostic of any particular growth-rate law, ordering is considered indicative of diffusion-controlled growth, insofar as it is the principal mechanism thought to produce such a texture. A possible alternative may be ordering of favorable nucleation sites in the precursor, but we consider such a mechanism unlikely in most cases for two reasons. First, it would require a perhaps coincidental similarity in the length scale of ordering between precursor and product. Second, any pre-existing ordering signal may be expected to degrade if there is a one-to-many relationship between the ordered property and the nucleation sites. For example, if a precursor phase hosting favorable nucleation sites features ordered grain centers, the dispersal of these sites around the peripheries of the grains would likely weaken or obscure the original signal.

In the creation of null-hypothesis envelopes, pseudorandom placement also needs to be constrained based on the quality of the data being analyzed. Because CT data have limited resolution and discrimination ability, in some cases it is to be expected that some strongly intergrown porphyroblasts will occasionally be misidentified as a single crystal. This in turn can have a significant effect on the statistical tests used here, as the principal signal for ordering consists of suppression of nucleation and growth around a growing crystal, for which interpenetrating crystals are the most contrary case. It is likely that similar issues exist with other data-acquisition approaches, such as serial sectioning, in which detection may vary with the sectioning-plane thickness as well as with the orientation of the porphyroblasts with respect to the sectioning plane. Specimen characteristics can also impact the ability to identify interpenetrating crystals. When porphyroblasts are euhedral, impingements are easily recognizable as departures from an ideal crystal form. Conversely, when they are subhedral or anhedral, or have irregular shapes due to alteration or resorption, it becomes more difficult to distinguish natural shape variation from cases of impingement.

Application of observability criteria consists of selecting values for a₁ and a₂ and ensuring that no crystal pairs violate ¹ and ² during pseudorandom placement of crystals when creating null hypothesis envelopes. Hirsch et al. (2000) determined values for these parameters by first computing d/d_I and l/r_s for all impinging porphyroblasts in data and corresponding simulations from Denison and Carlson (1997). Values were chosen such that the number of porphyroblasts that failed to pass both tests was minimized for the data and maximized for the simulations. The pair of values listed above was applied to all of the specimens studied so as to have a common baseline. However, because detection of impinged crystals is highly dependent on specimen and data characteristics, it is preferable to customize these parameters on a case-by-case basis, as will be done here.

In general, the effect of applying observability criteria is to move the null-hypothesis envelope further into the ordered field with increasing a₁ and a₂, making detection of ordering more difficult. In other words, application of observability criteria quantifies the apparent ordering caused by poor detection of impingement and prevents this circumstance from resulting in a spurious indication of ordering. Because some genuinely ordered textures are also excluded in the process, the degree of ordering required to get a statistically strong signal is increased.

Because one intention of this work is to illustrate how the new and improved analytical methods impact previous results, the specimens studied here are a subset of those examined by Carlson and Denison (1992) and Denison and Carlson (1997). They are all amphibolite facies metamorphic rocks featuring abundant almandine-rich porphyroblasts.

Specimens PM1, PM2, and PM4 are garnetiferous quartzites from a subunit (informally R₆) of the Rinconada Formation in the Picuris Range in north-central New Mexico, United States, featuring euhedral dodecahedral garnet in a nonfoliated to weakly foliated quartz-biotite matrix with minor plagioclase and muscovite. Garnet growth took place during a single postkinematic episode (Chernoff and Carlson, 1997; Holdaway, 1978). Carlson et al. (1995) estimate the onset of garnet nucleation at ∼435 °C, and Holdaway (1978) infers peak metamorphic conditions to be 532 ± 20 °C and 3.7 kbar based on mineral equilibria corrected for fluid compositions.

Specimens WR1 and WR3 are garnet amphibolites from the Whitt Ranch mafic body from the Llano Uplift of central Texas, United States. Both contain abundant, mostly anhedral porphyroblasts of manganiferous almandine in a fine-grained matrix of symplectitic hornblende, plagioclase, and augitic clinopyroxene replacing an original equigranular mosaic of sodic clinopyroxene. Garnet porphyroblasts in each WR specimen have undergone limited resorption (Carlson and Johnson, 1991); the more resorbed specimen, WR1, is estimated to have lost on average 150 μm of material radially from each garnet, based on the measurement of multiple coronas in a single thin section. Many porphyroblasts are also cut by alteration zones. The timing of garnet formation is uncertain, but the geological setting suggests a single episode of nucleation and growth; deformation during this metamorphic episode was partitioned into the more ductile rocks surrounding the Whitt Ranch mafic body, so these specimens show very weak or no penetrative fabrics (Carlson, 1998). Carlson et al. (1995) estimate the onset of nucleation as occurring at ∼500 °C.

Specimen MD is a schist from the Mica Creek locality near Mica Dam, British Columbia, in the Shuswap complex, featuring subhedral to anhedral garnet in a foliated, coarse-grained matrix of muscovite, biotite, kyanite, plagioclase, and quartz. The locality underwent three episodes of folding (F_1–3), each with an associated axial planar foliation (S_1–3); large-scale isoclinal F₃ folding postdated metamorphism (Simony et al., 1980). Recent work (Crowley et al., 2000) has revealed a complex history for garnet crystallization in this area. Oriented inclusion trails in garnet discordant with the surrounding foliation indicate that at least some growth predated S₁₊₂ development. Sensitive high-resolution ion microprobe (SHRIMP) dating of monazites included in garnet, kyanite, and matrix indicates that garnet predated kyanite formation and peak metamorphism by at least 30 m.y. (perhaps as much as 100 m.y.) and that there may have been multiple episodes of garnet growth. Garnet compositional profiles are weakly zoned, probably indicating partial diffusional homogenization of major elements. Garnet-biotite geothermometry and geoba rom etry from phase equilibria for garnet core compositions in the area are in the range 655– 695 °C and 5.6–8.4 kbar, with each range excluding one outlier (Ghent et al. 1982). These temperatures are generally corroborated by oxygen isotopic analyses (Ghent and Valley, 1998).

To allow acquisition of higher-resolution HRXCT data for this study, subsamples were extracted from many of the specimens. Cores 25 mm in diameter were extracted from specimens PM1, WR1, and WR3; for PM4, a 45 mm core was cut; for PM2 a rectangular prism roughly 1.9 × 2.5 × 4.6 cm was used. Specimen MD was scanned in its original state (i.e., an irregular hand sample with one cut face). In previous work some parts of specimen MD were omitted from scanning, whereas for this study imagery was acquired and processed for the entire specimen volume.

The CT data of Carlson and Denison (1992) and Denison and Carlson (1997) were acquired with a commercial industrial scanner, imaging roughly fist-sized and larger samples to ensure adequate numbers of crystals for statistical analysis. In the intervening decade, advancements in HRXCT technology and scanning techniques have improved the quality of imagery considerably, and new statistical methods have been formulated that provide robust results with fewer crystals, allowing smaller samples to be used.

Figure 2 shows examples of the image data used in earlier studies with corresponding imagery used here, and ⁰¹ provides some comparative scanning parameters. For the PM and WR specimens, much of the improvement stems from scanning a smaller volume which allowed the use of the ultra-high-resolution system featuring more focused X-rays and finer detector spacing. In addition, scanning cylindrical specimens permitted the use of a “wedge” calibration, which counteracts beam-hardening artifacts and allows the use of higher X-ray intensity, diminishing data noise (Ketcham and Carlson, 2001). The improvement for specimen MD is from the use of second-generation, or translate-rotate scanning, which utilizes extensive oversampling to achieve better resolution.

The improvement in data quality has a substantial influence on how accurately textural measurements can be made. For specimen PM4, the noisiness of the earlier data obscured many small garnet porphyroblasts and often obscured cases of impingement among crystals (Figs. 2A–2C). Although the crystals in specimen WR1 are larger than in specimen PM4, the imagery for the current study features superior detail, allowing for improved interpretation of original crystal size, shape, and impingement, despite the overprint of resorption and alteration (Figs. 2D–2F). The noise and beam-hardening artifacts in the old image data precluded their analysis using BLOB3D.

⁰² lists the Segment operations used for each data set during BLOB3D processing (see Ketcham, this volume, for details and parameter descriptions). The garnet in the PM specimens was sufficiently distinct from other phases that a simple threshold was sufficient, after noise reduction. In specimen MD the seeded expanding threshold was necessary to exclude other phases with overlapping grayscale ranges. The WR specimens contain oxides, predominantly magnetite, that are more attenuating than garnet and thus had to be segmented first to prevent contamination of the garnet data. To test for the textural effects of resorption, specimen WR1 was processed twice, first by segmenting for the garnet material only (“normal”) and a second time by expanding all segmented garnet material by three voxels in all directions, roughly equivalent to the 150 μm average corona width (“expanded”). This expansion was visually verified to coincide with fine magnetite rims marking the outer edges of the coronas where they are visible in the scan data. This procedure is at best only a partial correction, however, as it cannot recover small crystals that may have been fully resorbed, and large crystal boundaries remain irregular.

All data sets in this study were processed using Extract both with and without primitive fitting (Ketcham, this volume). Primitive- fitting methods for each specimen are given in ⁰³ . In general, primitive fitting was most successful for the PM specimens, in which the crystals are nearly spherical and impingement is the principal factor impacting their shapes. Primitive fitting for specimen MD was also reasonably successful, although the crystals are elongated, necessitating the use of an ellipsoid primitive. Primitive fitting was problematic for many WR porphyroblasts, due to their oblong shapes and irregular boundaries stemming from resorption and alteration.

All statistical analyses of data output from Extract were done using REDUCE3D (Hirsch et al., 2000; Hirsch, 2000). In the tables below it may be noted that the numbers of crystals and sample volumes may change slightly depending on whether and how primitive fitting was done. This is a consequence of the way in which REDUCE3D makes these measurements. The volume reported corresponds to the spatial region subtended by the smallest convex hull surrounding all crystals, and all crystals on the convex hull are excluded from the analysis under the assumption that they are likely to be incomplete. The inconsistencies are thus caused by small changes in the crystal center positions owing to primitive fitting that in turn subtly shift the position of the convex hull.

The results of the BLOB3D analysis for each specimen suite are provided in ⁰⁴ . Corresponding data obtained by Denison and Carlson (1997) are also listed, though it should be borne in mind that the data are not directly comparable because different subvolumes were analyzed.

All of the data sets were inspected using commercial three-dimensional visualization software, to seek for evidence of large-scale trends or inhomogeneities and any other textural feature that could impact the statistical analysis. Figure 3 shows that the WR suite and specimen MD all feature significant layering, as revealed in visualizations and reflected in the BLOB3D results. To control for the effects of the observed layering, statistical analyses were carried out on both the entire specimens and relatively homogeneous subsets as listed in ⁰³. Animations through the data sets and various three-dimensional renderings are also available online at http://www.ctlab.geo.utexas.edu/pubs/ketcham_et_al/ketcham_et_al.htm.

Crystal size distributions (CSDs) are an important petrological tool (e.g., Marsh, 1988), and recent work has sought to use details of CSDs to obtain information about various aspects of crystal growth, including the underlying rate laws (Eberl et al., 2002). The data produced by HRXCT and BLOB3D are ideal for such inquiries, as the data are fully three-dimensional and do not require the stereological models or assumptions necessary when attempting to extrapolate them from two- dimensional (i.e., thin-section) data.

Figure 4 shows various CSDs for the PM suite of specimens, as obtained in this study and by Denison and Carlson (1997). It is clear that the superior resolution of the newer scan imagery has greatly increased the number of small porphyroblasts measured. For specimen PM1, the result has been to shift the crystal size distribution from nearly symmetrical to strongly negatively skewed (Figs. 4A and 4B). BLOB3D-based CSDs for PM2 and PM4 also show negative skewness. With the exception of Figure 4C, all of the CSDs are based on extended volumes; i.e., the radius of the best-fit sphere after correcting for impingement. An alternative is to compute the CSD based on the actual volume (i.e., not corrected for impingement) by using the radius of the equivalent-volume sphere. As can be seen by comparing Figures 4B and 4C, this change of convention can have a noticeable effect on CSD shape in cases where impingement is heavy, as would be expected when more than a third of the sample volume is porphyroblasts.

The CSDs for the WR suite (Fig. 5 ) and specimen MD (Fig. 6 ) also indicate that the improved data quality has led to detection of more porphyroblasts per unit volume and extended the size range to a smaller minimum. However, in direct contrast to PM, here the CSD skewness is largely unchanged and strongly positive. Because the size of specimen MD necessitated using a lower-resolution HRXCT system, the minimum resolvable size in the current study is somewhat larger than for the PM and WR specimens, and it is likely that some small crystals were not measured. However, it appears unlikely this omission can have grossly changed the CSD shape.

Some of the BLOB3D-based CSDs show smaller maximum sizes than the earlier data, and proportionally fewer large crystals. The reasons for this are twofold. First, for all samples except MD the HRXCT data were only acquired for smaller pieces cut or drilled from the original samples. Areas of the samples with larger porphyroblasts may thus have been excluded, and the smaller sample size also makes it proportionally more likely that larger porphyroblasts intersect the side of the sample. Second, the better-resolution data and improved processing made it more likely that garnet masses appearing as one large porphyroblast in the original data were distinguished as two or more impinging crystals in the new data. Also, in Denison and Carlson (1997), only the data covering the garnet-rich “cap” of sample PM2 were presented; the cut chip examined in this study did not include this region.

It is clear from Figure 2 that the different specimen suites vary markedly in their potential for detecting and measuring strongly impinged porphyroblasts. The Picuris suite features extremely euhedral porphyroblasts, so any appreciable departure from near-sphericity can be reliably identified as an instance of impingement during separation using BLOB3D. In contrast, the irregular, oblong, and partially resorbed crystals of the Whitt Ranch suite greatly complicate the interpretive task. The Mica Dam specimen probably lies somewhere in between these two end members in terms of garnet shape, but the data are also lower resolution.

Figures 7A and 7B show the parameters from ² and ¹ calculated for the PM data sets and compared to corresponding data from Denison and Carlson (1997). Results for all contacting porphyroblast pairs in each data set are plotted as cumulative distribution functions (CDFs), and the region containing the lowermost 3% of pairs—the most severely interpenetrating cases—is shown separately in an enlarged view. On both plots, the degree of impingement increases from right to left. As described previously, l/r_s (Fig. 7A) is primarily sensitive (reaches its lowest values) in cases where impinging porphyroblasts are similar in size, while d/d_I (Fig. 7B) is most sensitive when contacting porphyroblasts have very disparate sizes. When comparing curves, a rightward displacement of the overall CDFs indicates that impingements are on average less severe according to that measure, and a rightward displacement in the lowermost 3% indicates that the maximum amount of impingement detected is less severe. The maximum possible value for d/d_I is 2.0, but the maximum for l/r_s is arbitrarily high, with higher values indicating progressively smaller crystals contacting larger ones. There is a clear difference between the old data and the new in the CDF positions, indicating that the BLOB3D analysis detected on average a greater degree of impingement among differently sized porphyroblasts and more severe maximum impingements regardless of relative size.

Figures 7C and 7D show the same plots for the computer models generated by Denison and Carlson (1997) to match as closely as possible the measured textures for these specimens using interface-controlled (after Carlson, 1991) and diffusion-controlled (after Carlson et al., 1995) growth laws. On the overall CDFs the data plot between the two end- member models. In the l/r_s graph (7C), the principal difference is at the high end of the curve, where the interface-controlled models feature a much larger incidence of small crystals contacting large ones. Here the revised data (7A) are closer to the diffusion-controlled prediction. In Figure 7D, the interface- controlled cases are characterized by their CDFs being nearly linear between d/d_I values of 1.2 and 2.0, whereas the diffusion- controlled models are more curved. In this case the main body of the BLOB3D data plot squarely between the end-member predictions.

In the enlarged graphs it is apparent that the BLOB3D data feature a small number of garnet pairs that are even more severely interpenetrating than predicted by either model. Inspection of these instances shows them to be confined to cases in which crystals were severely impinged on multiple faces, which decreased somewhat the accuracy of the sphere primitive fit. Even excluding these cases, however, it is apparent that the BLOB3D data feature porphyroblasts that impinge as severely as those in the idealized computer models, and thus for these specimens we assume that all impingements are observed. Consequently, we do not apply any observability criteria to these new data sets.

Figures 8A and 8B show CDF plots for the current and Denison and Carlson (1997) WR data sets. As with PM, there are a few instances in which d/d_I reflects an extent of impingement unseen even in models, again because of inaccurate primitive fits. Overall, however, the level of detected impingement is far below that in the PM suite. The plots for sample MD (Figs. 8C and 8D) show an inverse relationship from the others: although overall impingement is similar, the level of severe impingement in the Denison and Carlson (1997) data set exceeds that obtained with BLOB3D analysis. The full reason behind this switch is uncertain, but it should be noted that the large apparent shift is caused by the relatively small number of impingements. Only 212 intersections occur in the old data set, as opposed to 610 in the new, and thousands in the other suites. Of these, only four exceed the maximum d/d_I observed in the BLOB3D data.

The appropriate cutoff values for the spatial analysis, a₁ and a₂, are to some extent a matter of interpretation. Although the observability criteria are implemented as firm limits, it is almost certain that in the most extreme cases of impingement, detection is a matter of probability: some may be measured and some missed. In other words, the criteria use a step function whereas reality is more likely a continuum. It is thus not mandatory that the cutoffs include every intersection found in the data. For the WR data, the values selected by Hirsch et al. (2000), a₁ = 3.0 and a₂ = 0.85, are reasonable, and we employ them again here for consistency. For specimen MD, for which less severe impingement is observed, we compare the Hirsch et al. (2000) values to more conservative values derived from the data: a₁ = 3.25 and a₂ = 1.04.

Results for the single-valued statistics are listed in ⁰⁴, and correlation-function plots are shown in Figure 9 . Analysis of the original Denison and Carlson (1997) data for PM1 (Fig. 9A) strongly indicates ordering at scales equal to and less than the mean nearest-neighbor distance. At longer length scales, the PCF indicates clustering of porphyroblast centers and the MCF suggests some spatial ordering with respect to size.

To examine the effects of the primitive- fitting procedure on the results, we first analyzed the BLOB3D data for specimen PM1 without primitive fitting (Fig. 9B). Again, the data are broadly and strongly indicative of ordering at length scales up to the mean nearest-neighbor distance. When the proper, primitive-fit data are analyzed, however, the ordering signal is distinctly weaker (Fig. 9C). Although the PCF and MCF still indicate ordering at length scales close to and slightly below the nearest-neighbor distance, at shorter-length scales the results fall within the null- hypothesis envelopes for a 95% confidence level.

Correlation-function results from the Denison and Carlson (1997) data for specimen PM2 are also supportive of ordering, although the signal is weaker than for PM1 (Hirsch et al., 2000, their Fig. 15). However, the new data acquired in this study show only weak evidence of ordering (Fig. 9D). Of the single-value statistics the CI shows weak clustering and the others are within the null-hypothesis envelopes, while at short-length scales the correlation functions occasionally brush the lower limits of the null-hypothesis envelopes or dip below them. Results for specimen PM4 are similar (Fig. 9E). Specimens PM2 and PM4 both have a much lower crystal volume fraction than PM1 (6%–8% versus 37%); this feature may be linked to the divergence in their ordering signals. It is also interesting that all three PM specimens have MCF values above 1.0 at the shortest length scales. The strength of this signal is inversely correlated with crystal volume fraction, although in no case is it strong enough to disprove the null hypothesis at the 95% confidence level.

Results for the WR specimens are given in ⁰⁵ and Figure 10 . In both cases, the Denison and Carlson (1997) data are indicative of ordering (Fig. 10A and 10G). In the analyses of the entire WR1 core without compensation for resorption (⁰⁵), the single-valued statistics give inconsistent readings. The OI indicates a random distribution, the CI indicates clustering, and the II indicates ordering. The correlation functions have distinctive patterns (Fig. 10B) in which the PCF reflects clustering of nucleation sites and the MCF indicates an arrangement of crystal sizes consistent with competition. When the different sections of the volume are considered separately, the top section (featuring larger and sparser porphyroblasts) has a clear ordering signal across all statistics (Fig. 10C). The results for the bottom section are suggestive also of ordering, although the departures from the 95% confidence intervals among the single-value statistics are comparatively modest, and the PCF indicates only randomness and clustering (Fig. 10D). Expanding the regions segmented as garnet to compensate for resorption (⁰⁵) produces no significant change in interpretation based on any of the single-valued statistics, although the departure from the 95% confidence limits is in all cases reduced. Expansion substantially reduces the ordering signal in the PCF for the top section, giving it only a single point below the null-hypothesis envelope (Figs. 10E and 10F).

In specimen WR3, the single-valued statistics are largely similar between the current study and Denison and Carlson (1997) and are consistent with ordering (Figs. 10G–10J, ⁰⁵). The one exception is the CI, which indicates clustering. When the different zones of the specimen volume are considered separately, however, the CI for each is consistent with ordering. The PCF diagrams suggest ordering at length scales less than the mean nearest-neighbor distance and clustering at greater length scales. The ordering signal is stronger than observed in WR1, and it is unlikely that compensating for resorption during segmentation would eliminate the ordering signal.

In the earlier analysis of specimen MD, the OI and II indicated ordering, although the CI indicated clustering (Carlson and Denison, 1992). Correlation functions calculated from the earlier data are consistent with ordering at short-length scales (Fig. 11A ). The new data display a weaker ordering signal, with the OI only slightly above the envelopes of expected values for a random distribution (⁰⁶) and the PCF just below the null-hypothesis envelope when the same observability parameters a₁ and a₂ are used as for the WR data (Fig. 11B). As with the WR specimens, the II and MCF are both below the random envelope, which is the expected result if there has been local competition for nutrients. When observability parameters are increased to reflect the low incidence level of observed impingement in the data, the null-hypothesis envelopes are shifted such that the PCF result falls just inside the lower limit of the null-hypothesis envelope; the MCF still strongly indicates ordering (Fig. 11C).

BLOB3D analysis provides additional information that can aid in the interpretation of metamorphic textures such as shape, orientation, and contact relationships. For example, the garnet porphyroblasts in the WR and MD specimens were found to be oblate. The mean aspect ratio (maximum axis/minimum axis) of the ellipsoid primitives describing these was 1.59 for WR1, 1.61 for WR3, and 1.52 for MD. This obliquity may impact their spatial statistics, as the null-hypothesis envelopes are based on pseudorandom placement of spheres. Stereo plots of the long-axis orientations of the MD porphyroblasts reveal a preferred axis orientation aligned with the principal foliation direction, and garnet-garnet contact normals are preferentially aligned to one side of this great circle, possibly indicating a shear sense (Ketcham, 2005). These observations strongly suggest that the structural history of the MD specimen impacted the spatial distribution of porphyroblasts, possibly rearranging them during shear and/or influencing late-stage growth patterns.

The principal changes brought about by the improvements in data quality and processing techniques are in the detection of small crystals and the recognition and appropriate measurement of impingement. The former is probably not influential for the statistical analyses used here, as tests by Denison et al. (1997) and for the PM1 and WR3 data sets in this study have indicated that the smallest 10%– 20% of crystals can be deleted without changing any inferences. The latter, on the other hand, is very influential. Whether a mass of garnet material is interpreted as one crystal or two strongly impinged crystals can greatly affect measurements of degree of impingement and nearest-neighbor distance, as well as spatial correlations.

As shown above, successful detection and measurement of impingement in three dimensions requires high-resolution imagery, sensitive data processing, and favorable specimen characteristics. In the data sets with the best instances of all three, the PM suite, it is thus noteworthy that the ordering signal is significantly altered from earlier results. Only specimen PM1, with a higher volume fraction of porphyroblasts, shows a strong degree of ordering and competition, and even then not on the shortest length scales, where diffusional effects are in theory strongest. The other two specimens, which have much smaller volume fractions of garnet, show tendencies toward ordering, but those signals fail to reach statistical significance at the 95% confidence level. Because normalized radius-rate analysis of compositional zoning of PM garnet is most clearly interpretable as indicative of crystal growth controlled by thermally accelerated intergranular diffusion (Carlson, 1989), the implication of the correlation-function results is that low volume fractions of porphyroblasts may reduce the effects of diffusional suppression of nucleation and growth to levels difficult to detect by these techniques. It should be noted that failure to depart from the null- hypothesis envelope does not verify the null hypothesis of interface control with random nucleation sites but only fails to disprove it at the 95% confidence level. If a slightly less ambitious degree of confidence were chosen (90%, 80%), most of the PCF results at length scales below the nearest neighbor distance would lie below the envelopes, indicating ordering.

One interpretation of these results is that they reflect a number of simultaneous factors that impact the texture in ways that cancel each other to some extent, in a manner similar to that described for the single-valued statistics by Denison et al. (1997, p. 38–41). Ordering at the mean–nearest neighbor length scale is probably indicative of diffusional control, but this signal may be overprinted at various length scales by clustering of potential nucleation sites. Compositional layering would produce such effects at long-length scales. At short-length scales, nucleation sites might be localized on the grains of one or more precursor phases that are reacting to provide the chemical components needed for the growth of garnet. If nucleation sites are concentrated on such precursor phases, and if overgrowth is gradual and these phases persist through early stages of garnet nucleation and growth, then many favorable sites for nucleation will lie in close proximity to already-growing porphyroblasts, leading eventually to short-length–scale clustering and excessive impingement. Because the composition of the intergranular medium will likely be buffered to high values of the chemical affinity for the reaction when in direct contact with the precursor phase, the diffusionally depleted zone surrounding a growing porphyroblast (theorized to suppress nucleation events) will be limited to regions in which the precursor has been exhausted by prior reaction.

This portrayal of diffusion-controlled crystallization is more complex than that depicted by the Carlson et al. (1995) model, which simplified the calculation by assuming that all nutrient material is homogeneously distributed and entirely available for diffusional redistribution in the intergranular medium. New modeling efforts are under way (Ketcham and Carlson, 2004) that provide a more realistic picture by simulating persistent precursor phases and local fluid buffering in a diffusional system.

The lack of strong ordering signals in specimens PM2 and PM4 may be traced to their low crystal density. If nutrients and nucleation sites are concentrated in relatively sparse precursor phases, the arrangement of these phases may be the main controlling factor in the resulting spatial distribution of products. Competition effects may also be inhibited by the larger nearest-neighbor distances, requiring diffusion to act over longer length scales to serve as a textural control.

The CDFs used to evaluate the observability of impinging crystals in the scan data also constitute an original means of evaluating spatial arrangements against theoretical models, as they summarize salient aspects of the character of contacting crystal pairs: how disparate their sizes are and the degree of their interpenetration. In one case (Figs. 7A, 7C) overall the PM data for all three samples more closely resembled the diffusion-controlled models, while in the other (Figs. 7B, 7D) the data lie between the two end-member predictions. We believe that these results are also suggestive, if not yet definitively supportive of diffusion control. The augmented model of diffusion-controlled nucleation and growth in an inhomogeneous precursor discussed above would be expected to make the model CDF curves more closely resemble those observed in the data. Conversely, we know of no physically-grounded enhancement in our depiction of random interface-controlled growth that would have an effect on impingements.

The WR specimens are least favorable for textural analysis due to their extensive resorption, alteration, and layering. With compensation for resorption effects, specimen WR1 shows mixed evidence for ordering of crystal centers. The single-valued statistics suggest moderate to weak ordering, the PCF indicates randomness or subtle ordering, and the MCF results are consistent with competition. The resorption compensation was only partial and inexact, yet it diminished the ordering signal as compared to the data acquired without corrective measures. This is expected, because the textural effect of resorption is to change the relationship between crystal size and degree of separation, artificially decreasing the first while maintaining the second. The other probable effect of resorption, making impingements difficult or impossible to detect, is accounted for by the observability criteria but may obscure any signal of short-length–scale clustering such as was inferred for the PM specimens. Specimen WR3 stands out as a clear case in which all statistical tests suggest ordering of both porphyroblast centers and sizes.

Interpretation of specimen MD is complicated by the poorer resolution of the scan data and the slight obliquity of the garnet. The PCF results are consistently near the bottom of, but within, the null-hypothesis envelope at 95% confidence, possibly indicating a weak tendency toward ordering. The MCF test suggests an ordering of crystal sizes indicative of competition for nutrients. However, given the history of this specimen, with multiple probable garnet growth episodes and subsequent recrystallization and deformation, diffusional competition during crystal growth may not be revealed by this texture, or implied by it. One hypothetical possibility is that, during deformation, the garnet porphyroblasts acted as mostly rigid particles surrounded by less competent material; in a similar manner to boudinage or igneous flow segregation by grain-dispersive pressure, the less resistant or more ductile material may have had a tendency to flow between closely spaced crystals, increasing their separation. Alternatively, original growth may have indeed been ordered, and a statistically detectable aspect of this ordering may have been preserved through subsequent bulk deformation.

Reanalysis of six garnetiferous specimens to evaluate textural signals of ordering of nucleation sites and competition for nutrients has resulted in a significantly more nuanced picture than observed in earlier studies. Although textural signals of ordering consistent with diffusion-controlled growth persist, results for the PM suite suggest an overprint from inhomogeneous nucleation that had been obscured by earlier lower-resolution data. This finding in turn inspires caution in the interpretation of specimens such as those in the WR suite, where alteration complicates data processing and may obscure signals for both randomness and clustering on short-length scales. The results for MD may serve as a caution, as some ordering was detected even though the sample's geologic history weakens the connection between conditions during garnet nucleation and growth and the arrangement of porphyroblasts observed today. Overall, the results of this study corroborate earlier findings of ordering and competition indicating diffusion-controlled nucleation and growth in these rocks, but the bar has been raised for the data quality and specimen favorability required to provide unambiguous information.

This research was supported by National Science Foundation (NSF) grants EAR-9902682 and EAR-0113480. The High-Resolution X-ray Computed Tomography Facility at the University of Texas at Austin is supported in part by NSF grant EAR-0345710. The manuscript benefited from thorough reviews by C.T. Foster and an anonymous reviewer, and from editorial handling from M. Williams.