56-1 Departures from Spatial Homogeneity as Complementary Statistical Measures of Spatial Heterogeneity in Point Patterns of Sandstone Thin Sections

Abstract:

Modal composition of a sandstone describes constituent proportions in a thin section, so does grain-size distribution for size frequencies; neither records spatial distribution of the studied objects. Sandstones with similar summary statistics may have different spatial arrangements, reflecting different depositional and/or diagenetic processes. Quantification of spatial heterogeneity requires a homogeneity reference and a departure measure from it.

We define spatial heterogeneity as the departure of the observed pattern of a sandstone attribute in a thin section from an analysis-specific probabilistic homogeneity reference. Using its centroid to represent an object, we compare Clark–Evans R of the observed pattern of an attribute with the mean value of empirical Monte Carlo null distribution and define Heterogeneity Index (HI) as the unstandardized departure [R_obs − mean (R_null)], and ZHI as standardized departure [HI/SD(R_null)], where SD(R_null) is the standard deviation of R under the null distribution. Negative HI and ZHI values indicate clustering, zero randomness, and positive values regularity or dispersive. The Monte Carlo p-value measures tail extremeness. At a chosen significance threshold, a small p-value indicates statistically detectable heterogeneity relative to the homogeneity reference.

Using a mapped thin section, we demonstrated that HI and ZHI are complementary, not interchangeable, by comparing HI and ZHI values for the patterns of pore space and coarse-silt, very-fine-sand, and lithic grains. For all objects, HI and ZHI are negative, indicating departures toward clustering, relative to their respective homogeneity references. The HI values of coarse-silt (HI = −0.2150; ZHI = −21.84) and very-fine sand grains (HI = −0.2164; ZHI = −31.61) are only 0.7% different, but ZHI values are 45% different. In contrast, HI values of pore space (HI = −0.1735; ZHI = −38.58) and lithic grains (HI = −0.2287; ZHI = −38.28) are 32% different, but ZHI values are 0.8% different. Thus, similar HI values may correspond to substantially different ZHI magnitudes, and vice versa. For the pattern of a specific attribute, HI shows the magnitude and direction of departure from its homogeneity reference on the original R scale, whereas ZHI expresses the same departure relative to the standard deviation of the null distribution. Spatial heterogeneity should be an important aspect in sandstone petrography and be considered in sandstone classification.

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