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U.S. GEOLOGICAL SURVEY
Scientific Investigations Report 2008–5167

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Statistical Analysis of Sedimentary Interbed Thickness

Nonparametric statistical tests of spatial and temporal stationarity of sediment interbed thickness distributions were performed and are discussed in the following sections.

Tests of Spatial Stationarity: Composite Units 2–7

To test for spatial stationarity, interbed thickness distributions from two geographically grouped sets of five coreholes were compared (Northeast Hole Group and Southwest Hole Group, fig. 5). Middle 1823 was excluded from this analysis because lithologic information was not available for much of composite units 2–7 in this corehole. All tests of similarity showed no statistically significant difference between these sample groups at the 95 percent confidence level, thereby supporting the hypothesis of spatial stationarity of distributions, medians, and variances of interbed thicknesses (fig. 6; table 4) within the area studied. P-values greater than 0.05 indicate a less than 5 percent probability that the tested populations actually differ in the property being tested and therefore violate the hypothesis of stationarity.

Tests of Temporal Stationarity

The hypothesis of temporal stationarity was tested by comparing sedimentary interbed thickness distributions between groups of composite units. A general grouping scenario using the three groups shown in table 5 (composite unit 1, units 2–7, and units 8–14) was used to test if particular stratigraphic intervals differ from others with respect to interbed thickness distributions (fig. 4, table 5). The K-W test of similarity of multiple medians and the M-W test of similarity of two medians shows thickness distributions of composite units 2–7 are statistically different from composite unit 1 and units 8–14 (table 5). These results corroborate the conclusions of Welhan and others (2006) regarding composite unit 1. They concluded that sediment abundances in composite unit 1 were higher than in other composite units. The data obtained in this study also indicate that interbeds in composite unit 1 tend to be thicker than those in composite units 2–7 (fig. 4, table 5), possibly because of relative volcanic quiescence during the past 200 Ka (Anderson and others, 1997; Champion and others, 2002).

Although the results of this study corroborate the conclusions of Welhan and others (2006) regarding temporal stationarity among composite units 2–7, the new data obtained in this study on interbed thicknesses in composite units 8-14 do not support their conclusion that median sediment content in composite units 2–14 does not statistically vary. The low p-values indicate that median interbed thicknesses in composite units 2–7 are statistically different than those in composite units 8–14 (table 5). Sample size, however, is small; more deep stratigraphic data would be needed to confirm this result.

Welhan and others (2006) inferred temporal stationarity among two groupings of composite units 2-7 (composite units 2-5 and 6-7), not among individual composite units. The K-W test was used to analyze the similarity of medians among three grouping scenarios (fig. 7, table 6). Results indicate that the interbed distribution of composite units 2–3 is significantly different at the 95 percent confidence level, whereas the differences among other groups are not statistically significant (table 6). With the exception of composite units 2–3, interbed thicknesses among different groupings support the hypothesis of temporal stationarity across composite units 4–7.

The apparent difference between composite units 2–3 and 4–7 may be due to random variations arising from small sample size (Welhan and others, 2006) or it may reflect a real difference. More data are needed to determine if composite units 2 and 3, like composite unit 1, tend to have significantly thicker interbeds than older units.

Interbed Thicknesses Inferred from Natural-Gamma Logs

Welhan and others (2006) used the data on Quaternary interbed thicknesses that Anderson and Liszewski (1997) interpreted solely from natural-gamma logs. Those data were compared with the interbed thickness data compiled in this study from a combination of lithologic and natural-gamma logs. The statistical distributions of interbed thicknesses in the eight USGS series coreholes (USGS 127 through 134), Middle 2050A, and Middle 2051 (fig. 2) were compared with the statistical distributions of interbed thicknesses in 10 nearby boreholes analyzed by Anderson and Liszewski (1997) (table 7), using the same statistical tests described above (K-S, M-W, and L). Data from only 10 boreholes of Anderson and Liszewski (1997) were used so the number of borings from their analysis was equal to the number of new coreholes studied here. Only composite unit group 1–7 was analyzed because sample size from composite unit group 8–14 was small.

Distributions of interbed thicknesses from composite units 1-7 were quite different from those of the Anderson and Liszewski (1997) database, consistently showing a statistically significant difference in both the medians (M-W test) and the shapes (K-S test) of the distributions (fig. 8; table 8). These differences are due to the presence of relatively thin (~1-3 ft) interbeds that were distinguished in the new data set from lithologic logs, but not in the Anderson and Liszewski (1997) data set, and probably reflects the difference in data collection methods used in the two studies. Because both the medians and distribution shapes are statistically distinguishable, the similarity of variances demonstrated by the L test has no relevance to the overall similarity of these two samples.

By using lithologic logs in conjunction with geophysical logs to quantify interbed thicknesses, many more thin interbeds could be identified than could be inferred from natural-gamma signatures alone. The physical sensing limitation of natural-gamma sonde censors small features such as thin beds (Keys, 1997). Because the Anderson and Liszewski (1997) lithologic data were compiled using only natural-gamma logs, it is likely that their data are censored of these thin interbeds.

To test the hypothesis that the Anderson and Liszewski (1997) lithologic data are censored of thin interbeds, the interbed thickness data obtained in this study were artificially censored to varying degrees and compared to the data of Anderson and Liszewski (1997) (table 9). By excluding interbeds of thicknesses less than 2 ft, the statistical similarities between the two datasets increased such that they were not statistically different (table 9). By evaluating a range of censoring thresholds, it is possible to conclude that the thickness data of Anderson and Liszewski (1997) are censored at a threshold of between 1 and 2 feet with a 95 percent confidence level, and therefore are not representative of the statistical distribution of all interbed thicknesses.

Conversely, statistical comparisons of the new corehole data based solely on natural-gamma logs showed no statistically significant difference with the Anderson and Liszewski (1997) data (fig. 9; table 10). A few relatively thin (2–3 ft) interbeds were still observed with the new geophysical data, indicating that an unintentional bias may have been introduced because of pre-existing knowledge of the lithologic log information.

The analyses conducted here document the censoring effect of natural-gamma logs and indicate that detailed lithologic logs are essential for accurately interpreting subsurface interbed occurrences. Lithologic logs do not require significantly more effort to interpret than geophysical logs, and in fact may entail less effort overall. Drawbacks include potentially poor knowledge of the actual sediment recovery when coreholes are logged, together with the expense and infrastructural overhead involved with coring, logging, and core storage.

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