Time scales and gaps, Haar fluctuations and multifractal geochronologies

Abstract Much of our knowledge of the Earth’s past is gleaned by interpreting varying layers in outcrops, cores drilled through lake and ocean sediments, and sheets of rock and ice. These were laid down under highly variable conditions and their stratigraphy generally include gaps of all sizes1. Since erosion, sedimentation, aggradation, denudation and progradation rates of sediment sources and sinks operate over wide ranges of scale2, the simplest hypothesis is that they are scaling in accord with scaling Holocene and Quaternary climate records 3,4,5,6,7,8,9,10,11,12,13 . In this study, we analyze chronologies and proxies from records spanning the Holocene, Quaternary, and Phanerozoic, focusing on the measurement density, the number of measurements per unit time, that we quantitatively characterize in a multifractal framework. This density is a new and distinct multi-scale proxy that can be used to understand the evolution of Earth’s system history. Implicitly, it includes information on the absence of evidence, thus – paradoxically - providing positive information about the structure of non-recorded history, a significant phenomenon that can be quantified in its own right. We also analyze the scaling of the probability distributions of the inter - measurement time intervals and show that completeness of the records decreases with the number of measurements, thus quantitatively explaining the Sadler effect 14,15,16,2,17 .