Geometric interpretation of time-scale dependent sedimentation rates

Empirical rate data show that aggradation and progradation rates are negatively correlated with the duration they are averaged over. This means that we cannot understand deep time sedimentation rates in terms of the abundant rate data we have from modern environments. We are addressing this by investigating how growth of a sediment volume in multiple directions affects time-averaged accumulation rates that do not necessarily measure all directions of growth. We derive a mathematical model from mass-balance considerations of a sedimentary deposit that grows without erosion. We show that the non-linear relationship between the lengths, areas and volumes of growing objects require that time-averaged sedimentation rates are time-scale dependent when a deposit grows in multiple directions. We generate rate compilations from simple synthetic stratigraphies, which show that we can reproduce the empirical rate compilations by requiring only that deposition occur in a dynamic mosaic of sequential patches in which the areal extent of deposition increases with time-scale as described by our mathematical model. Our model has two implications with application in stratigraphic analysis. The first is that it provides a theoretical model of why we cannot expect high stratigraphic completeness if we measure a deposit in fewer directions than it has grown. The second is that it provides a framework by which to attribute anomalies in the stratigraphic completeness of a one-dimensional or two-dimensional sample to scaling effects or erosional hiatuses.