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1
Space-time models for soil moisture dynamics
Valerie Isham
Department of Statistical ScienceUniversity College London
[email protected], http://www.ucl.ac.uk/stats/
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Collaborators:
David Cox
Nuffield College, Oxford
Ignacio Rodriguez-Iturbe
Civil and Environmental Engineering, Princeton
Amilcare Porporato
Civil and Environmental Engineering, Duke
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Introduction
Temporal models of soil moisture at a single-sitepoint rainfall (ie concentrated at discrete time points)
Spatial-temporal models of soil moisturespatially-distributed rainfall (at a point or temporally
distributed in time)variable vegetationproperties at a point and averaged over space-time
Coupled dynamics of biomass and soil moisturetemporal process at a single site
Summary and future directions
Overview
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Fundamental problem of hydrological interest…
Soil moisture (and its spatial and temporal variability)
is the dynamic link between climate, soil and vegetation,
and impacts processes at a range of spatial scales.
Point scale: infiltration, plant dynamics, biogeochemical cycle
Hillslope: controlling factor for slope instability and land slides
Basin: drought assessment, flood forecasting
Region/continent: interaction with atmospheric phenomena
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R u n o f f
IN P U T : R A IN F A L L( in te rm it te n t-
s to c h a s t ic )
t
h
E v a p o -tr a n s p ira t io n
T r o u g h fa l l
Z r
E f fe c t iv e p o ro s i ty , n
Z r
E f fe c t iv e p o ro s i ty , n
L e a k a g e
R u n o f f
Soil moisture……• increases due to precipitation• decreases due to evapotranspiration
and leakage and is dependent on • soil properties• vegetation
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We consider dynamics
• at a daily time scale (no effects of diurnal fluctuations in temperature on evapotranspiration)
• within a single season
• on relatively small spatial scales (no feedback between soil moisture and rainfall).
The impact on the vegetation as well as of the vegetation is of interest.
La Copita, Texas;courtesy of Amilcare Porporato/ Steve Archer
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0.05
0.15
0.25
150 200 250 300 350
Julian Day
q (%
)interspace
canopy
0
5
10
15
20
Pre
cip
itat
ion
(mm
day
)
Sevilleta, New Mexicocourtesy of Amilcare Porporato/Eric Small
Precipitation and soil moisture
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Temporal process of soil moisture
Modelling approach
We use
piecewise deterministic Markov processes (Davis
1984) in continuous time: sample paths have
• periods of deterministic change governed by a
differential equation
• random jumps occurring at random times
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S(t)
X2 X3
X4
X1
T1 T2 T3 T4 t
S(t): the Takács virtual waiting time process for a M/G/1 queue
ie the service requirement of all the customers in the system at t,
Alternatively: S(t) is the content of a store (reservoir)
* replenished by random amounts at random times
* subject to depletion at a constant rate when non-empty
A very simple such process ……
Times: a Poisson process, rate
Jumps: iid, density gDecay: constant rate
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Let
and let S have density for s > 0
Forward equation:
Many properties of the process can be determined
Special case: Xi ~ exp( )
Equilibrium: if
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Other properties and extensions (Cox and Isham, 1986)
• transient solution: Laplace transform (wrt to t) of the moment generating function
• expansions determining convergence to equilibrium
• autocovariance function, in equilibrium
• slowly varying arrival rate
(small )
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For soil moisture• state-dependent decay
losses depend on current soil moisture level
• boundedness of soil moisture
excess rainfall runs off saturated soil
state-dependent jumps, density g(x,s)
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Soil moisture balance equation:
n soil porosity
Zr depth of root zone
I (random) rate of infiltration (dependent on ground cover)
E rate of evapotranspiration (dependent on vegetation)
L rate of leakage (dependent on soil properties)
Standardise I, E, L
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Losses…approximated by
0 s* s1 1.0
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distribution of infiltration…Assume that standardised infiltration I*(s,t) has an
exponential ( ) distribution, truncated at 1- s
The excess rainfall is lost as surface run-off.
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Forward equation… for density of S(t)
(no atom at origin since ).
Equilibrium distribution (Rodriguez-Iturbe et al 1999) has the form
Use piecewise linear form of continuity of p(s) at s* and s1.
Normalise to 1 to find c.
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Note: the atom of probability at 1-s in the state dependent jumps is not used explicitly in the derivation. Soil saturation only affects the restricted range over which p(s) is normalised – an effect of the Markov nature of the soil moisture process.
properties, impact of parameters on properties etc
Note: Equilibrium distribution is for linear evapotranspiration
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Impact of climate, soil and vegetation on equilibrium distribution
Parameters chosen to represent
a) tropical climate and vegetation, frequent moderate rainfall, deep soil;
b) hot arid region, shallow sandy soil, mixture of trees and grasses;
c) cold arid region; d) forested temperate
region.
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Spatial-temporal soil moisture
Soil moisture is spatially dependent, because of• correlated rainfall input• ground topology causing run-off from one location to
affect nearby locations• correlated vegetation cover We assume • a stochastic process of rain cells with random spatial
extents• a flat landscape to avoid run-off problems, eg savannah • a) a homogenous vegetation, or b) a stochastic process of trees with random canopies in a
grassy landscape
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The simplest model…• temporally instantaneous rainfall (ie daily timescale) at random times Tk
• linear losses (hot arid region, cf Fig (b)) ( will be vegetation and soil-dependent)
• ignore bound on soil moisture
In this case
• proportional interceptionstandardised infiltration for rainfall
• heterogeneous soil and vegetation and depend on location
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Equilibrium distribution for….
hot arid region, shallow sandy soil, mixture of trees and
grasses; s* = 0.45
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S(t)
t
Shot-noise process
Linear losses ( ) and no saturation
exponential decay:
if there is no input in (0,t). In this case
and S(t) has no atom at 0.
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Rainfall process…
• Poisson process of rain cell origins, rate in space-time
• circular cells, random radii (iid)
• rainfall is instantaneous in time over the cell, depths Y (iid)
• at a fixed location, A say, rain events occur in a temporal Poisson process of rate
• events occur at locations A and B, d apart, in a temporal Poisson process of rate
Here
is the area of overlap of two unit discs, centres u apart.
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Marginal distribution …
• Transient distribution and its properties
• Equilibrium distribution
where is the mgf of the rain depth Y, with
• If Y ~ exp( ), S ~
• For general infiltration, replace integrand by
where is the mgf of infiltration from a rain depth Y
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Joint distribution: sites A and B, d apart…• SA (t) - rain events before t that only affect A, rate
- rain events before t that affect both A and B, rate
• SB (t+h) - events before t+h that only affect B, rate
- events in (t, t+h) that affect both A and B, rate
- events before t that affect both A and B, rate • Properties of transient distribution• Equilibrium distribution
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In particular
For general infiltration
Joint equilibrium mgf:
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As before, assume • Poisson process of rain cell origins, rate in space-time,
and circular cells, random radii (iid) Assume • rain cell duration D, with constant intensity V (iid)
Observe their superposition where
Soil moisture
(assuming, as before, linear losses, proportional interception and ignoring bound on soil moisture)
Alternative model: rain cells with exponential durations…
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formal solution…
In particular, the covariance properties of
(assuming D ~ exp( ) ) imply those of S (via Campbell’s Th)
The corresponding covariance for the pulse rainfall model is
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Properties for homogeneous vegetation
• Correlation as a function of the spatial and temporal lags
• Effect of spatial averaging (different spatial scales). Analytic results can be obtained by using a Gaussian approximation to
• Effect of spatial and temporal averaging (different scales)
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Correlation as a function of spatial and temporal lags
(rainfall parameters fitted to data from 17 gauges in Southern Italy, two values for soil porosity-root depth factor)
nZr=100mm nZr=500mm
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Standard deviation of spatially averaged field relative to standard deviation at a point
Here is the mean rain cell radius.The ratio depends only on and the spatial area
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Standard deviation of spatially and temporally averaged fields
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Heterogeneous vegetation – trees in a grassy landscape
A model for tree crowns…..
• Poisson process of tree locations, rate in space
• Circular canopies, random radii (iid)
• No. of trees covering location, A say,
• No. of trees covering A and B, d apart,
• P(neither A nor B covered)
• P(A is covered, B is not)
• P(both A and B are covered)
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a realisation of the vegetation process…
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Use probabilities to remove conditioning of previous results on vegetation cover, and determine corresponding properties with random vegetation
eg variance of spatially integrated soil moisture
10-6
10-4
10-2
100
102
104
106
10-4
10-3
10-2
10-1
Area (km2)
Var
ian
ce
HeterogeneousAll TreeAll Grass
Slope -0.915
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Biomass and soil moisture…temporal process
For water-limited ecosystems, a simple model for the coupled system of biomass B and soil moisture S is
Assume (within a growing season)
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transient solution…
moments, eg
Equilibrium:
(deterministic)
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Summary and scope for further work• Single-site, temporal models of soil moisture
• Spatial-temporal models of soil moisture *simplifications - flat landscape
- linear evapotranspiration - ignore bound at s = 1
*spatially-distributed rainfall instantaneous distributional results temporally distributed second order results
(proportional interception only)variable vegetationareally-averaged properties
• Coupled dynamics of biomass and soil moisture *single site, temporal process