The graphical techniques presently in use in structural petrology are unsatisfactory, except in special cases, due to (1) their limiting assumptions and (2) their inherent imprecision. These difficulties may be overcome by introducing more general assumptions and by treating the data more directly.
The “best” axis may be defined for any given set of fabric data by a least squares criterion, based upon a general conical model. In the limits this model reduces to (1) the point distribution which is analyzed in beta diagrams and centrosymmetric point maxima of lineations, and (2) the great circle distribution, which is analyzed in pi diagrams. As defined by the least squares criterion, the best axis may be interpreted as the “factor” (by analogy with factor analysis) which explains a minimum of the variance in the data. A variance surface may be constructed from spacially distributed trial axes, plotted on a spherical surface, and then projected onto a plane. Such surfaces show how the variance explained by trial axes varies with orientation.
Variance surfaces allow development and testing of models that approximate the structures under consideration. Examples from metamorphic rocks of the Sierra Nevada show how data may be analyzed and models may be developed.