Extending Graphic Correlation to Many Dimensions
William G. Kemple, Peter M. Sadler, David Strauss · SEPM (Society for Sedimentary Geology) eBooks · 1995
Abstract Stratigraphic correlation involves three distinct tasks: establishing the temporal sequence of marker events (sequencing task), determining the relative sizes of the intervals between those events (spacing task), and locating the horizons that correspond in age with each event in every section (locating task). Stratigraphic sections do not yield enough information to solve this problem exactly. Instead, stratigraphers must search for the approximation that “best” fits all local stratigraphic observations. The concept of economy of fit, as used in graphic correlation, provides a rigorous definition of “best” and implies the existence of a penalty function that can be used to rank possible solutions. Unfortunately, traditional graphic correlation requires severe simplifying assumptions about accumulation rates because it attempts to solve all three tasks at the same time, and yet it incorporates the local sections into the solution one at a time. Using a penalty function based on economy of fit, an alternative solution technique naturally emerges in the form of constrained optimization. This technique is J dimensional, in the sense that it treats the observations in all J sections simultaneously. It can complete the sequencing task before making assumptions necessary to the spacing task. Constrained optimization eliminates impossible solutions (constraint) and then searches for the best of all the possible ones (optimization). For realistic instances of the problem, it is not feasible to calculate the penalty function for all possible solutions. Instead, we use a probabilistic search procedure termed “simulated annealing” to find very good solutions without an exhaustive search. Simulated annealing does not maintain any memory of the search path or search exhaustively at the local scale. We reject an alternative procedure that incorporates these features because it proved difficult to tallor to produce satisfactory solutions. Our J-dimensional procedure quickly finds solutions for Palmer’s (1954) classical data set (/= 7 sections from the Cambrian of Texas) that are comparable to the solution Shaw (1964) achieved by traditional graphic correlation. The expert judgements that the stratigrapher uses to draw the traditional lines of correlation and which give the appearance of excessive subjectivity, in fact, derive primarily from a knowledge of all the sections. By treating all sections at once, constrained optimization eliminates much of the apparent subjectivity associated with traditional graphic correlation. Constrained optimization still allows the user to explore the consequences of different and genuinely subjective judgements about the relative reliability of different taxa and sections.