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Space-time models for soil moisture dynamics

Valerie Isham

Department of Statistical ScienceUniversity College London

valerie@stats.ucl.ac.uk, 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