ERROR PROPAGATION VIA AREA BALANCING IN BALANCED CROSS SECTIONS

Balanced cross sections are models that are drawn to fit incomplete data; as such, a key question is “how well do they fit the data?” Despite the ubiquity of balanced sections in structural and tectonic studies during the past five decades, there remains no general, rigorous, quantitative method for evaluating uncertainties in balanced sections. Line length balanced sections, by far the most common type, are usually cited as “minimum shortening estimates” because of the uncertainty in the position of eroded hanging wall cutoffs in emergent thrust belts. Other sources of uncertainty -- the depth to the decollement, internal structural geometry, shortening at scales below the resolution of the cross section, and initial stratigraphic thickness -- are seldom explicitly considered. To estimate the magnitude or percent horizontal shortening, we require a method that encompasses all possible sources of error and all possible structural/kinematic fold-fault models.

Area balancing meets all of these requirements and has an additional advantage: because area can be calculated analytically, the errors on the input parameters can be propagated formally through the calculation to the provide a robust estimate of error in the magnitude of shortening. The inputs are the vertices of a polygon that envelopes the deformed region and the initial stratigraphic wedge of pre-growth strata for the initial state. Each are assigned errors based on geological criteria. We increase the number of vertices in the enveloping polygon until the magnitude of the shortening solution and its error stabilizes, usually at < 20 vertices in the case of the Subandean cross sections we have analyzed. Formal error propagation clearly shows how each of the input errors contributes to the overall error. An under-appreciated factor is stratigraphic thickness: just 10% error in initial thickness estimate can contribute half of the overall uncertainty in shortening. Our error analysis complements line-length sections: because area balance must encompass all possible viable and admissible line length solutions, it immediately shows the author where their solution lies with the solution space.