Systematic collection of air samples was made using commercial jet airliners between Tokyo, Japan and Anchorage, Alaska and between Tokyo and Sydney, Australia in 1984 and 1985. The amplitude of the seasonal CO2 cycle in the upper troposphere was found to be largest in high latitudes of the northern hemisphere but decreased and lagged in phase as the equator was approached. The cycle was still clearly observable in the southern hemisphere, due to southward transport of the northern hemispheric air through the upper troposphere by a monsoon circulation. The yearly mean value of the upper tropospheric CO2 was high in the equatorial region and decreased poleward. The seasonal cycle of the lower stratospheric CO2 in northern high latitudes showed a minimum concentration early in May and a maximum concentration early in September, with a peak‐to‐peak amplitude of 2.2 ppmv. Yearly mean CO2 concentrations were lower by 1.4‐2.2 ppmv than those in the upper troposphere. The difference in yearly mean CO2 concentrations between 1984 and 1985 was about 1.1 ppmv at all locations covered by this measurement.

This paper considers a connected Markov chain for sampling 3×3×K contingency tables having fixed two-dimensional marginal totals. Such sampling arises in performing various tests of the hypothesis of no three-factor interactions. A Markov chain algorithm is a valuable tool for evaluating P -values, especially for sparse datasets where large-sample theory does not work well. To construct a connected Markov chain over high-dimensional contingency tables with fixed marginals, algebraic algorithms have been proposed. These algorithms involve computations in polynomial rings using Gröbner bases. However, algorithms based on Gröbner bases do not incorporate symmetry among variables and are very time-consuming when the contingency tables are large. We construct a minimal basis for a connected Markov chain over 3×3×K contingency tables. The minimal basis is unique.Some numerical examples illustrate the practicality of our algorithms.

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