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Emergence of patterns in the geologic record and what those patterns can tell us about Earth surface processes. Hydrologic Synthesis Reverse Site Visit – August 20, 2009. Rina Schumer Desert Research Institute, Reno NV, USA. Water Cycle Dynamics. Hydrosphere/ Biosphere. Hillslopes. - PowerPoint PPT Presentation
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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.