Emergence of patterns in the geologic recordand what those patterns can tell us about Earth surface processes

Rina SchumerRina SchumerDesert Research Institute, Reno NV, USADesert 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

DynamicsHillslopes

How can we improve predictability?

Transport of water/sediment/biota over heterogeneous surfaces

Synthesis subgroup #5

Synthesis (Carpenter et al., 2009 - 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

STRESSworking group

2007-2009

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

thicknesstime intervalobsR

1( )S t

2( )S t

“Sadler Effect”

accumulation rate = thickness/time

1,000 yr. hiatus

1,000 yr. hiatus

50 yr. hiatus2,000 yr. hiatus

40,000 yr. hiatus

1,000 yr. hiatus10 yr. hiatus

1,000 yr. hiatus

500 yr. hiatus100 yr. hiatus

-3/4

0

1

2

3

4

5

6

7

-1

-2

-3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8LOG (Time interval, t ) [yr]

LOG

(Accum

ulation rate) [m

m/yr]

-1/5

ShorelineShelfDeltaContinental RiseAbyssal Plain

measured deposition rate depends on measurement interval

0

1

2

3

4

5

6

7

-1

-2

-3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8LOG (Time interval, t ) [yr]

LOG

(Accum

ulation rate) [m

m/yr]

-3/4

accumulation rate = thickness/time

1,000 yr. hiatus

1,000 yr. hiatus

50 yr. hiatus2,000 yr. hiatus

40,000 yr. hiatus

1,000 yr. hiatus10 yr. hiatus

1,000 yr. hiatus

500 yr. hiatus100 yr. hiatus

-1/5

ShorelineShelfDeltaContinental RiseAbyssal Plain

“Sadler Effect” measured deposition rate depends on measurement interval

0

1

2

3

4

5

6

7

-1

-2

-3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8LOG (Time interval, t ) [yr]

LOG

(Accum

ulation rate) [m

m/yr]

-3/4

accumulation rate = thickness/time

1,000 yr. hiatus

1,000 yr. hiatus

50 yr. hiatus2,000 yr. hiatus

40,000 yr. hiatus

1,000 yr. hiatus10 yr. hiatus

1,000 yr. hiatus

500 yr. hiatus100 yr. hiatus

-1/5

ShorelineShelfDeltaContinental RiseAbyssal Plain

“Sadler Effect” measured deposition rate depends on measurement interval

0

1

2

3

4

5

6

7

-1

-2

-3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8LOG (Time interval, t ) [yr]

LOG

(Accum

ulation rate) [m

m/yr]

-3/4

accumulation rate = thickness/time

1,000 yr. hiatus

1,000 yr. hiatus

50 yr. hiatus2,000 yr. hiatus

40,000 yr. hiatus

1,000 yr. hiatus10 yr. hiatus

1,000 yr. hiatus

500 yr. hiatus100 yr. hiatus

>350 references to Sadler(1981) !

-1/5

ShorelineShelfDeltaContinental RiseAbyssal 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 intervalsNo constant sampling intervals

2. Greater probability of encountering a long hiatus in a longer interval:

-3/4

0

1

2

3

4

5

6

7

-1

-2

-3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 LOG (Time interval, t ) [yr]

LOG

(Accum

ulation rate) [m

m/yr]

-1/5

ShorelineShelfDeltaContinental RiseAbyssal Plain

Attributed to (Sadler, 1981)

thicknesstime intervalobsR

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, 2007

2. Log-log slope of the Sadler plot is directly related to the tail of the hiatus

length density

-3/4

0

1

2

3

4

5

6

7

-1

-2

-3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 LOG (Time interval, t ) [yr]

LOG

(Accum

ulation rate) [m

m/yr]

-1/5

ShorelineShelfDeltaContinental RiseAbyssal Plain

Infinite-mean probability density

Exponential (mean=10) Pareto (tail parameter=0.8)

number of random variables number of random variables

runn

ing

mea

n

runn

ing

mea

n

( ) ( )E t tP t dt

What if there is no average size hiatus because there is always a finite probability

of intersecting a larger hiatus?

Measured accumulation rate Robs

S(t1)

S(t2)

accumulation rate Rretardation coeff.obsR

Incorporate avg. fraction of time with no deposition

2 1

2 1

thicknesstime intervalobs

S t S tR R

t t

No hiatus periods

CTRW – discrete stochastic model

Y1

Y3

Y4

Y6

Y7

Y5

J

1

t

t

N

N ii

S Y

location of Sediment surface

random # of events by time t is a function of

the hiatus lengths

sediment accumulationevent length

J

J

J

J

J

J

Y2

T1 T2 T3 T4 T5 T6 T7

max :t nN n T t

Governing equations for scaling limits of CTRW

S R St x

, 0< 1S R St x

Advection equation with retardation

Time-fractional advection equation

CTRW Governing Equation

Constant jump lengthRandom hiatus length with thin tails

Constant jump lengthRandom hiatus length with heavy tails

scaling limit

S(t)= surface location with time, R=deposition rate, =retardation coeff.

sedi

men

t sur

face

ele

vatio

n (m

m)

time (yr)

sedi

men

t sur

face

ele

vatio

n (m

m)

time (yr)

1

heavy tails in hiatus density

NO heavy tails in hiatus

density

Expected location of sediment surface with timeanalytical and numerical modelling: CTRW with

constant (small) depostional periods, random hiatus length

( )t t

time (yr) time (yr)

obse

rved

dep

ositi

on ra

te (m

m/y

r)

1

Obs

erve

d de

posi

tion

rate

(mm

)

convergence to constant

Sadler effect arises from heavy tailed hiatus distributionanalytical and numerical modelling: CTRW with

constant (small) depostional periods, random hiatus length

heavy tails in hiatus density

NO heavy tails in hiatus

density

( )t t

Measured accumulation rate Robs

S(t1)

S(t2)

accumulation rate Rretardation coeff.obsR

Incorporate avg. fraction of time with no deposition

2 1

2 1

thicknesstime intervalobs

S t S tR R

t t

No hiatus periods

1R

1obstR

Incorporate heavy tailed hiatuses

1

A power-law function of time

( )t t

Implications:

0.1 1 10 1001000

10000

100000Erosion rate [km

3/M

yr]Age [Ma]

Age [Ma]0 10 20 30 40 50 60

5

10

15

20

25

30

Sedim

ent mass [x 1018

kg]

Global values for terrigenous sediment accumulation

(after Hay 1988 and Molnar 2004)

0 2 4 6 8 10 120

5000

10000

15000

20000

25000

f(x) = 14483.8560927064 x^-0.279776793890249R² = 0.939373061026276

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), 14933-14940.

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, 67-89.

Pelletier, J.D. (2007) Cantor set model of eolian dust deposits on desert alluvial fan terraces, Geology, 35, 439-442.

Plotnick, R.E. (1986) A fractal model for the distribution of stratigraphic hiatuses, J. Geology, 94(6), 885-890.

Sadler, P.M. (1981) Sediment accumulation rates and the completeness of stratigraphic sections, J. Geology, 89(5), 569-584.

Sadler, P.M. (1999) The influence of hiatuses on sediment accumulation rates, GeoRes. Forum, 5, 15-40.