Emergence of patterns in the geologic record and what those patterns can tell us about Earth surface processes Rina Schumer Desert Research Institute, Reno NV, USA Hydrologic Synthesis Reverse Site Visit – August 20, 2009

Stochastic Transport and Emergent Scaling in Earth-Surface Processes (STRESS) Hydrosphere/ Biosphere Water Cycle Dynamics Hillslope s How can we improve predictability? Transport of water/sediment/biota over heterogeneous surfaces Synthesis subgroup #5

Synthesis (Carpenter et al., BioScience) Sustained, intense interaction among individuals with ready access to data: mine existing data from new perspectives that allow novel analyses develop and use new analytical/computation/modeling tools that may lead to greater insights bring theoreticians, empiricists, modelers, practitioners together to formulate new approaches to existing questions integrate science with education and real-world problems

solute transport in groundwater flow systems 1990’s solute transport in streams ~2000 STRESS working group flow through heterogeneous hillslopes bedform deformation gravel transport slope- dependent soil transport non-local transport on hillslopes sediment transport in sand bed rivers sediment accumulation rates landslide geometry and debris mobilization hillslope evolution depositional fluvial profiles transport on river networks Timeline showing use of heavy-tailed stochastic processes in modeling Earth surface systems Results of Synthesis “acceleration of innovation”

Introduction Geology records the “noisiness" of sediment transport, as seen in wide range of sizes of sedimentary bodies  intermittency at many scales Describe nature and pace of landscape evolution by separating random transport from forcing mechanisms (glacial cycles,tectonics,etc) Need to estimate deposition rate

Modified from Sadler 1999 hiatus Influence of transport fluctuations on stratigraphy

“Sadler Effect” accumulation rate = thickness/time 1,000 yr. hiatus 50 yr. hiatus 2,000 yr. hiatus 40,000 yr. hiatus 1,000 yr. hiatus 10 yr. hiatus 1,000 yr. hiatus 500 yr. hiatus 100 yr. hiatus -3/ LOG (Time interval, t ) [yr] LOG (Accumulation rate) [mm/yr] -1/5 Shoreline Shelf Delta Continental Rise Abyssal Plain measured deposition rate depends on measurement interval

LOG (Time interval, t ) [yr] LOG (Accumulation rate) [mm/yr] -3/4 accumulation rate = thickness/time 1,000 yr. hiatus 50 yr. hiatus 2,000 yr. hiatus 40,000 yr. hiatus 1,000 yr. hiatus 10 yr. hiatus 1,000 yr. hiatus 500 yr. hiatus 100 yr. hiatus -1/5 Shoreline Shelf Delta Continental Rise Abyssal Plain “Sadler Effect” measured deposition rate depends on measurement interval

LOG (Time interval, t ) [yr] LOG (Accumulation rate) [mm/yr] -3/4 accumulation rate = thickness/time 1,000 yr. hiatus 50 yr. hiatus 2,000 yr. hiatus 40,000 yr. hiatus 1,000 yr. hiatus 10 yr. hiatus 1,000 yr. hiatus 500 yr. hiatus 100 yr. hiatus -1/5 Shoreline Shelf Delta Continental Rise Abyssal Plain “Sadler Effect” measured deposition rate depends on measurement interval

LOG (Time interval, t ) [yr] LOG (Accumulation rate) [mm/yr] -3/4 accumulation rate = thickness/time 1,000 yr. hiatus 50 yr. hiatus 2,000 yr. hiatus 40,000 yr. hiatus 1,000 yr. hiatus 10 yr. hiatus 1,000 yr. hiatus 500 yr. hiatus 100 yr. hiatus >350 references to Sadler(1981) ! -1/5 Shoreline Shelf Delta Continental Rise Abyssal Plain “Sadler Effect” measured deposition rate depends on measurement interval

“Sadler Effect” 1.Strong correlation between sample age and measurement interval Young samples  small interval Old samples  long intervals No constant sampling intervals 2.Greater probability of encountering a long hiatus in a longer interval: -3/ LOG (Time interval, t ) [yr] LOG (Accumulation rate) [mm/yr] -1/5 Shoreline Shelf Delta Continental Rise Abyssal Plain Attributed to (Sadler, 1981) const

Our Conclusions 1.Sadler effect will arise if the length of hiatus periods follow a probability distribution with infinite mean (aka power-law*, heavy-tailed) *power law suggested previously by Plotnick, 1986 and Pelletier, Log-log slope of the Sadler plot is directly related to the tail of the hiatus length density -3/ LOG (Time interval, t ) [yr] LOG (Accumulation rate) [mm/yr] -1/5 Shoreline Shelf Delta Continental Rise Abyssal Plain

Infinite-mean probability density Exponential (mean=10)Pareto (tail parameter=0.8) number of random variables running mean

What if there is no average size hiatus because there is always a finite probability of intersecting a larger hiatus? Measured accumulation rate R obs S(t 1 ) S(t 2 ) Incorporate avg. fraction of time with no depositionNo hiatus periods

CTRW – discrete stochastic model Y1Y1 Y3Y3 Y4Y4 Y6Y6 Y7Y7 Y5Y5 JJ location of Sediment surface random # of events by time t is a function of the hiatus lengths sediment accumulation event length JJ JJ JJ JJ JJ JJ Y2Y2 T1T1 T2T2 T3T3 T4T4 T5T5 T6T6 T7T7

Governing equations for scaling limits of CTRW Advection equation with retardation Time-fractional advection equation CTRWGoverning Equation Constant jump length Random hiatus length with thin tails Constant jump length Random hiatus length with heavy tails scaling limit S(t)= surface location with time, R=deposition rate,  =retardation coeff.

sediment surface elevation (mm) time (yr) sediment surface elevation (mm) time (yr) heavy tails in hiatus density NO heavy tails in hiatus density Expected location of sediment surface with time analytical and numerical modelling: CTRW with constant (small) depostional periods, random hiatus length

time (yr) observed deposition rate (mm/yr) Observed deposition rate (mm) convergence to constant Sadler effect arises from heavy tailed hiatus distribution analytical and numerical modelling: CTRW with constant (small) depostional periods, random hiatus length heavy tails in hiatus density NO heavy tails in hiatus density

Measured accumulation rate R obs S(t 1 ) S(t 2 ) Incorporate avg. fraction of time with no depositionNo hiatus periods Incorporate heavy tailed hiatuses A power-law function of time

Implications: Erosion rate [km 3 /Myr] Age [Ma] Sediment mass [x kg] Global values for terrigenous sediment accumulation (after Hay 1988 and Molnar 2004) Eastern Alps volumetric erosion rates estimated from surrounding basin accumulation rates (adapted from Kuhlemann et al. 2001) Measurement bias or…..climate change? Same patterns seen in rate measurements for subsidence erosion incision evolution!

Synthesis (Carpenter, et al. BioScience) Sustained, intense interaction among individuals with ready access to data: mine existing data from new perspectives that allow novel analyses develop and use new analytical/computation/modeling tools that may lead to greater insights bring theoreticians, empiricists, modelers, practitioners together to formulate new approaches to existing questions integrate science with education and real-world problems

References Hay, W.W., J.L. Sloan, and C.N. Wold (1988). Mass/Age distribution and composition of sediments on the ocean floor and the global rate of sediment subduction. J. Geophys. Res., 93(B12), Molnar, P. (2004) Late Cenozoic increase in accumulation rates of terrestrial sediment: How might climate change have affected erosion rates?, Annual Review of Earth and Planetary Sciences, 32, Pelletier, J.D. (2007) Cantor set model of eolian dust deposits on desert alluvial fan terraces, Geology, 35, Plotnick, R.E. (1986) A fractal model for the distribution of stratigraphic hiatuses, J. Geology, 94(6), Sadler, P.M. (1981) Sediment accumulation rates and the completeness of stratigraphic sections, J. Geology, 89(5), Sadler, P.M. (1999) The influence of hiatuses on sediment accumulation rates, GeoRes. Forum, 5,