Some Vector and Arithmetic Operations on Two-Dimensional Orientation Variates, with Applications to Geological Data

In part I we compare vector (ā) and arithmetic (x̄) means of circular-type variates by showing how each changes with addition of one measurement (an) to (n - 1) measurements. Differentiating, σ x/σ aₙ = 1/n (no dispersion term appears here); σ a/σ aₙ = (R₁ cos aₙ + 1)/(R²₁ + 2R₁ cos aₙ + 1), where R1 the vector resultant of the first (n - 1) measurements, is a measure of dispersion. As aₙ → 0 and as R₁ → (n - 1) (i.e., dispersion of the [n - 1] measurements → 0), then σ a/σ aₙ → σ x̄/σ aₙ. Thus we obtain requirements for approximate equivalence. In part II, using trigonometric symmetry to simplify computations, a short method for computing vector mean and vector resultant is presented. We then examine ten sets of earth-science data; to nine of these we attempt to fit linear normal and circular-type normal density functions, testing goodness of fit with x2. The former (i.e., linear) fits satisfactorily in five out of nine comparisons. Circular-type frequencies are computed in four of nine examples; two of the four fit satisfactorily. In part III, we define the dispersion measure ß' as the angle enclosing a fictitious uniform distribution having the same vector strength (ā ≡ R/n) as the observed data. After showing a high linear correlation between β' and s (standard deviation) of our earth-science data, we derive the basis for this relationship in terms of equivalent uniform distributions. Further, the corollary relation between ā and s is shown to be ā = (sin √3 s) / √3 s. Concluding remarks deal chiefly with this study's shortcomings.