state-space modelling of soil-moisture

IntroductionA simple deterministic model

Calibration

State-space modelling of soil-moisture

Jonathan Rougier

Department of Mathematics, University of Bristol, UK

Currently visiting SAMSI

Joint work with Cari Kaufman (SAMSI and NCAR)

Advised by Jim Clark, Sean McMahon (Duke)

Jonathan Rougier State-space modelling of soil-moisture


Calibration

Introduction

This project is part of the Terrestrial Program of the SAMSIyear-long Program Development, Assessment and Utilizationof Complex Computer Models.

The Scalable Landscape Inference and Prediction (SLIP)model is a stochastic model of a forest, treated tree-by-tree,developed by Biologists and Computer Scientists. It can beused to study forest dynamics, and forest response to climaticchange.

A small but important part of this model is soil-moisture,which has a big impact on seed germination.

We would like to develop a model for soil-moisture to beincorporated into SLIP. It should be sensitive to criticalclimatic quantities like temperature and precipitation.



Calibration

Soil moisture

At the plot-scale, soil-moisture is the balance between inflows,mainly precipitation; and outflows, mainly evapotranspirationand net ground-water run-off.

Within a plot, soil-moisture is spatially heterogeneous, beingstrongly affected by local conditions, primarily topography(directly and indirectly), soil-type, soil-depth, and plant-type.

SLIP requires a fairly low-resolution model, operating at theplot-level (∼ 1km), for time-steps of months or years.However, to calibrate this model we have pixel data (∼ 20m)for time-steps of days (this is proxy data on soil-conductivity).

We will develop our model from the top-down, incorporatingjust as much spatial information as is necessary. The morestandard approach is bottom-up, starting with a detailedspecification of the topography.



Calibration

Modelling at the plot-levelDisaggregating to the pixel-level

Plot-level model

We have a single soil-moisture quantity, V (t), and a single otherquantity A(t), which accumulates run-off (R is run-off).

V ′(t + 1)← V (t) + [P − E ](t,t+1] + γA(t)

R ← 0 ∨{V ′(t + 1)− F

}V (t + 1)← V ′(t + 1)− R

A′(t + 1)← (1− γ − δ)A(t) + R

A(t + 1)← H ∧ A′(t + 1)

Model parameters:

F Field capacity, above which soil-moisture runs-off;

H Upper limit on accumulation, above whichsoil-moisture exits the plot;

γ, δ Rates for recirculation and exit.



Calibration


Evapotranspiration

We model evapotranspiration (ET) as a damped function ofpotential evapotranspiration (PET):

ET(V , t) =

{PET×

(1− exp{−κ(V −W )}

)V ≥W

0 otherwise.

W is termed the wilting point: in our model there is no ETwhen V < W .

For PET we start with the Blaney-Criddle model:

BC(t) = daylight(t)×(0.46× temp(t) + 8

)× 365

1000,

units of metres per year. Daylight is the proportionate numberof annual daytime hours, which depends on latitude. Temp. isin degrees centigrade.



Calibration


Evapotranspiration (cont)

To allow for plot-specific characteristics we introduceadditional parameters,

PET(t) = β0 + β1 × BC(t).

When we disaggregate to the pixel level, we will also includeinformation on pixel slope and aspect.

Our model for ET introduces additional parameters:

W The wilting point (metres);β Coefficients in the PET function, initially (0, 1).



Calibration


Coweeta at the plot-level

Precip minus PET, by month

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2000 2001 2002 2003 2004 2005 2006 2007

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Soil−moisture and accumulation

Time in years, labelled at the start of each year

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Calibration


Modelling at the pixel-level

We introduce the spatially-indexed quantity c(s) to denote theconvergence of pixel s. We also introduce a(s), the area of pixel s,and allow F to vary with s according to soil-depth.

V ′(s, t + 1)← V (s, t) + [P − E ](t,t+1] + c(s) γA(t)/a(s)

R(s)← 0 ∨{V ′(s, t + 1)− F (s)

}V (s, t + 1)← V ′(s, t + 1)− R(s)

A′(t + 1)← (1− γ − δ)A(t) +∑

sa(s) R(s)

A(t + 1)← H ∧ A′(t + 1)

Aggregate soil-moisture is V (t) =∑

s a(s) V (s, t).



Calibration


Coweeta by pixels

Location of Coweeta measuring stations

Easting

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Calibration


Coweeta elevations

275400 275500 275600 275700 275800 275900 276000

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Elevation (metres)

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Calibration


Coweeta convergence

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Total Convergence Index

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Calibration


Coweeta soil-depth

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Soil depth (cm)

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Calibration


Coweeta by super-pixels


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Calibration


Coweeta by pixels


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Calibration


Findings so far . . .

Soil-moisture is an example of a fairly simple process subjectto complicated forcing.

Our simple model does a good job of showing how spatialheterogeneity causes variations in soil-moisture evolutionacross a plot, subject to common forcing. It is easy for us tointroduce further spatial factors, e.g., aspect and slope inevapotranspiration.

It’s a puzzle, though, why modelling at different resolutionsdoes not cause more variation in the aggregate quantitiesV (t) and A(t): possibly due to Coweeta being very wet?

If we want to widen the time-stepping, we need a carefultreatment of evapotranspiration and soil-moisture (the wiltingpoint, W , not discussed).



Calibration

Looking ahead to calibration

We introduce a stochastic term accounting for modelimperfection, as part of the forcing:

V ′(s, t+1)← V (s, t)+[P−E ](t,t+1]+ω(s, t+1)+c(s) γA(t)/a(s)

Putting the uncertainty inside the model means that it doesnot contradict the physics. ω(s, t + 1) accounts mainly for(i) local differences in forcing; and (ii) local effects of run-off.

We can simulate our model forward. With some judiciouschoices we can calibrate using sequential Monte Carlomethods (aka particle filters).



Calibration

Sequential calibration

Denote the uncertain model parameters as θ: this mightinclude quantities in the measurement equation.

In the simplest case we choose to have ω(s, t)⊥⊥ ω(s ′, t ′) forall s,′ s where t 6= t ′, and the system state vector is

St ={V (s1, t), . . . ,V (sN , t),A(t)

}Then we have the required DAG

Zt−1 Zt Zt+1

//___ St−1//

OO

St//

OO

St+1

OO

//___

θ

ffNNNNNNN

^^<<<<<<<<<<OO

DD

88ppppppp

AA��

for which we learn sequentially about {S1, θ}, {S2, θ}, . . . .



Calibration

More generality

In the more general case where ω(s, t) 6⊥⊥ω(s ′, t ′) we wouldhave to include ω(t) in St , treating ω(t) as approximatelyMarkov.

To account for meter-specific measurement biases (likely to becommon and substantial in our case), we would have toinclude a vector of biases in θ.

For efficient calibration in these more general cases, it isimportant that we keep the number of pixels down (whichmeans we require fewer particles), and that we exploit largetime-steps where possible.


state-space modelling of soil-moisture

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