Data assimilation in terrestrial biosphere models: exploiting mutual constraints among the

carbon, water and energy cycles

Michael Raupach, Cathy Trudinger, Peter Briggs, Luigi Renzullo, Damian Barrett, Peter Rayner

Thanks: colleagues in the Australian Water Availability Project (CSIRO, BRS, BoM); participants in OptIC project

CarbonFusion (Edinburgh, 9-11 May 2006)

Outline

Data assimilation challenges for carbon and water

Multiple-constraint data assimilation

Using water fluxes (especially streamflow) to constrain carbon fluxes

Observation models for streamflow (with more general thoughts on scale)

Example: Murrumbidgee basin

Model-data fusion: comparison of two methods

Carbon DA

Challenges for carbon cycle science (including data assimilation)

• Science: finding state, evolution, vulnerabilities in C cycle and CCH system

• Policy: supporting role: IPCC-SBSTA-UNFCCC, national policy

• Management: trend detection, source attribution ("natural", anthropogenic)

Terrestrial carbon balance

Required characteristics of an observation system

• pools (Ci(t)), fluxes (GPP, NPP, NBP, respiration, disturbance)

• Long time scales (to detect trends)

• Fine space scales (to resolve management and attribute sources)

• Good process resolution (to detect vulnerabilities, eg respiration, nutrients)

• Demonstrated consistency from plot to globe

fire harvest herbivory

heterotrophicChange of C allocated disturbance flux out of poolrespirationin pool NPPpartitioned NBP

i i NPP i i i i i

ii

dC dt a F k C F F F

Water DA

Challenges for hydrology (including water data assimilation)

• Science: state, evolution, vulnerabilities in water as a limiting resource

• Policy: supporting role at national and regional level

• Management: providing tools (forecasting, allocation, trading)

Terrestrial water balance (without snow)

Required characteristics of an observation system

• W(t) and fluxes for soil water balance (also rivers, groundwater, reservoirs)

• Accurately enough to support regulation, trading, warning (flood, drought)

• With forecast ability from days to seasons

precipitation interception transpiration runoff to drainage tosoilChange of soil

rivers groundwaterevaporationwater store

P I T S R DdW dt Q Q Q Q Q Q

Coupled terrestrial cycles of energy, water, carbon and nutrients

C flowN flow

P flow

Water flow

PLANTLeaves, Wood, Roots

ORGANIC MATTERLitter: Leafy, WoodySoil: Active (microbial)

Slow (humic)Passive (inert)

Photosynthesis

Respiration

ATMOSPHERE

Disturbance

Leaching Fluvial, aeolian transport

Fertiliser inputs

N fixation,N deposition,N volatilisation

N,P Cycles

C Cycle

Rain

Transpiration

Runoff

WaterCycle

Soil evap

SOILSoil water

Mineral N, P

Energy

Confluences of carbon, water, energy, nutrient cycles

Carbon and water:

• (Photosynthesis, transpiration) involve diffusion of (CO2, H2O) through stomata

• => (leaf scale): (CO2 flux) / (water flux) = (CsCi) / (leaf surface deficit)

• => (canopy scale): Transpiration of water ~ GPP ~ NPP

Carbon and energy:

• Quantum flux of photosynthetically active radiation (PAR) regulates photosynthesis (provided water and nutrients are abundant)

Water and energy:

• Evaporation is controlled by (energy, water) supply in (moist, dry) conditions

• Priestley and Taylor (1972): evaporation = 1.26 [available energy][Conditions: moist surface, quasi-equilibrium boundary layer]

Carbon and nutrients:

• P:N:C ratios in biomass (and soil organic matter pools) are tightly constrained

• 500 PgC of increased biomass requires ~ (5 to 15) PgN

• Estimated available N (2000 to 2100) ~ (1 to 6) PgN (Hungate et al 2003)

The carbon-water linkage

Terrestrial water balance (without snow):

Residence time of water in soil column ~ (10 to 100) days, so over averaging times much longer than this, dW/dt is small compared with fluxes

In an "unimpaired" catchment with constant water store: [streamflow] = [runoff] + [drainage]

Chain of constraints:

• Streamflow (constrains (total) evaporation

• Total evaporation (= transpiration + interception loss + soil evaporation) constrains transpiration

• Transpiration constrains GPP and NPP

• GPP, NPP control the rest of the terrestrial carbon cycle

precipitation interception transpiration runoff to drainage tosoilChange of soil

rivers groundwaterevaporationwater store

P I T S R DdW dt Q Q Q Q Q Q

Streamflow: observation model

Basic principle

• In an unimpaired catchment,

• d[water store]/dt = [runoff] + [drainage] [streamflow]

• If d[water store]/dt can be neglected (small store or long averaging time):

• [streamflow] = [runoff] + [drainage]

• [water store] includes groundwater within catchment, rivers, ponds ...

Requirements for unimpaired catchment

• All runoff and drainage finds its way to the river (no farm dams)

• No other water fluxes from the river (eg irrigation, urban water use)

• No major dams (otherwise d[store]/dt dominates streamflow)

• Groundwater does not leak horizontally through catchment boundaries

Snow

• needs a separate balance

Streamflow (and other) data issues

Requirements on catchments

• Unimpaired, gauged at outlet

• Catchment boundary must be known

Requirements on measurement record

• Well maintained gauge

• The water agency must be prepared to give you the data

Requirements on other data

• Need spatial distribution of met forcing (precip, radiation, temperature, humidity)

• Need spatial distribution of soil properties (depth, water holding capacity ...)

• Catchments are hilly:

• Downside: everything varies

• Upside: exploit covariation of met and soil properties with elevation

(eg: d(Precipitation)/d(elevation) ~ 1 to 2 mm/y per metre

• ANUSplin package (Mike Hutchinson, ANU)

Modelling at multiple scales

We often have to predict large-scale behaviour from given small-scale laws:

Small-scale dynamics Large-scale dynamics

Four generic ways of approaching this problem:

1. Full solution: Forget about F, integrate dx/dt = f(x,u) directly

2. Bulk model: Forget about f, find F directly from data or theory

3. Upscaling: Find a probabilistic relationship between small scales (f) and large scales (F), for example by:

4. Stochastic-dynamic modelling: Solve a stochastic differential equation for PDF of x (small scale), and thence find large-scale F:

,state vectorexternal forcing

d dttt

x f x uxu

, , withd dt

X xX F X U

U u

, , , d d xuF X U f x u x u x u

Raupach, Barrett, Briggs, Kirby (2006)

Steady-state water balance: bulk approach

Steady state water balance:

Dependent variables: E = total evaporation, R = runoff

Independent variables: P = precipitation, EP = potential evaporation

Similarity assumptions (Fu 1981, Zhang et al 2004)

Solution finds E and R (with parameter a)(Fu 1981, Zhang et al 2004)

1

1

,

,

aa aP P P

aa aP P P

E P E P E P E

R P E P E E

1 1

2 2

, with 0 0 (wet limit)

, with 0 0 (dry limit)

P

P P

E P f E E P f

E E f P E E f

0 P T S R DdW dt Q Q Q Q Q

P E R

Fu (1981)Zhang et al (2004)

Normalise with potential evap EP:plot E/EP against P/EP

Normalise with precipitation P:plot E/EP against EP/P

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3

P/Q

E/Q

NECoastSECoastTasAgricNtropicsArid2345

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3

Q/P

E/P

NECoastSECoastTasAgricNtropicsArid2345

Steady water balance: bulk approach

dry wet

wet dry

a=2,3,4,5

a=2,3,4,5

Fu (1981)Zhang et al (2004)

P/EP

E/E

P

EP/P

Stochastic-dynamic modelling

Forcing (u) on small-scale process (x(t)) is often random, and dynamics (dx/dt = f(x,u)) often has strong nonlinearities such as thresholds

Examples: soil moisture, dust uplift, fire, many other BGC processes

If we can find x(x), the PDF of x, we can find any average (large-scale) property

Equation for state (x) Equation for PDF of state [px(x)]

Deterministic system

Deterministic dynamic equation Liouville equation

d dt x f x,u xx

d

dt

x f

Stochastic-dynamic modelling

Forcing (u) on small-scale process (x(t)) is often random, and dynamics (dx/dt = f(x,u)) often has strong nonlinearities such as thresholds

Examples: soil moisture, dust uplift, fire, many other BGC processes

If we can find x(x), the PDF of x, we can find any average (large-scale) property

Equation for state (x) Equation for PDF of state [x(x)]

Deterministic system

Deterministic dynamic equation Liouville equation

Stochastic system

(deterministic system with

random perturbations)

Stochastic dynamic equation

u(t) is a Markov process, with transition prob obeying CK eq

Stochastic Liouville equation

-------------- and then --------------

d dt x f x,u

u u uT t L T

xx

d

dt

x f

xuxu u xu

dL

dt

x f

, ,d dt t tx f x u

d d xuF f f x u

Steady-state water balance: stochastic-dynamic approach

Dynamic water balance for a single water store w(t):

Then:

• Let precipitation p(t) be a random forcing variable with known statistical properties (Poisson process in time, exponential distribution for p in a storm)

• Find and solve the stochastic Liouville equation for w(w), the PDF of w

• Thence find time-averages: <w>, E = <e(w)>, R = <r(w)>

dw dt p t e w t r w t

Rodriguez-Iturbe et al (1999)Porporato et al (2004)

0.2 0.4 0.6 0.8 1w

0.5

1

1.5

2

2.5

3

3.5

PDF PDF of w

w(w)

w = relative soil water

increasing precipitation

event frequency

<w>

1 2 3 4PQ

0.2

0.4

0.6

0.8

w parameterbb: zzrrQ

dry wet

P/EP

increasing precipitation

event frequency

Water and carbon balances: dynamic model

Dynamic model is of general form dx/dt = f(x, u, p)

All fluxes (fi) are functions fi(state vector, met forcing, params)

Governing equations for state vector x = (W, Ci):

Soil water W:

Carbon pools Ci:

Simple (and conventional) phenomenological equations specify all f(x, u, p)

Carbon allocation (ai) specified by an analytic solution to optimisation of NPP

precipitation transpiration runoff to drainage tosoilChange of soil

rivers groundwaterevaporationwater store

P T S R DdW dt Q Q Q Q Q

heterotrophicChange of C allocated

respirationin pool NPPpartitioned NBP

i i NPP i i

i

dC dt a F k C

11.051.11.151.21.251.31.351.41.451.51.551.61.651.71.751.81.851.91.9522.052.12.152.22.252.32.352.42.452.52.552.62.652.72.752.82.852.92.9533.053.13.153.23.253.33.353.43.453.53.553.63.653.73.753.83.853.93.9544.054.14.154.24.254.34.354.44.454.54.554.64.654.74.754.84.854.94.9555.055.15.155.25.255.35.355.45.455.55.555.65.655.75.755.85.855.95.9566.056.16.156.26.256.36.356.46.456.56.556.66.656.76.756.86.856.96.9577.057.17.157.27.257.37.357.47.457.57.557.67.657.77.757.87.857.97.958

U rban

H orticu lture

C ropping

FertilisedG razing

W oodland &R angeland

Forest

W ater

D ra inageBasins

R oads

R ivers

Melbourne

Adelaide CanberraNarrandera

Hay

Shepparton

Ballarat

Albury

Renmark

100 100 200 km0

W agga

Test area: Murrumbidgee basin

Murrumbidgee basin

Murrumbidgee: relative soil moisture

Jan 1981 to Dec 2005

0

0 . 1

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J F M A M J J A S O N D

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Murrumbidgee Relative Soil Moisture (0 to 1)

J F M A M J J A S O N D

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MurrumbidgeeTotal Evaporation(mm d-1)

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Tumbarumba

Wagga Wagga

Southern MDB: "unimpaired" gauged catchments

Predicted and observed discharge 11 unimpaired catchments in Murrumbidgee basin

25-year mean: Jan 1981 to December 2005Prior model parameters set roughly for Adelong, no spatial variation

0

50

100

150

200

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0 100 200 300 400

ZDisCM = Predicted Discharge [mm/y]

AD

isC

M =

Ob

serv

ed D

isch

arg

e [m

m/y

]

Adelong:410061

Goobarragandra:410057

Both ZDisCM and ADisCM are conditioned on ADisCM>=0 (discharge data avai;able)[m/mth] [m/mth] [mm/y] [mm/y]ZDisCM ADisCM ZDisCM ADisCM

410044:ZDisCM 0.001445 0.00398 17.34367 47.76458 410044 MuttamaCreek@Coolac410038:ZDisCM 0.012694 0.015991 152.3273 191.8946 410038 AdjungbillyCreek@Darbalara410047:ZDisCM 0.003045 0.008373 36.53617 100.4813 410047 TarcuttaCreek@OldBorambola410048:ZDisCM 0.002893 0.00471 34.71892 56.52 410048 KyeambaCreek@Ladysmith410057:ZDisCM 0.015951 0.032463 191.4125 389.5508 410057 GoobarragandraRiver@Lacmalac410061:ZDisCM 0.016078 0.018988 192.932 227.8604 410061 AdelongCreek@BatlowRoad410059:ZDisCM 0.03018 0.03469 362.1584 416.2745 410059 GilmoreCreek@Gilmore410097:ZDisCM 0.001358 0.004217 16.29116 50.60565 410097 BillabongCreek@Aberfeldy410033:ZDisCM 0.009444 0.005276 113.3247 63.31652 410033 MurrumbidgeeRiver@MittagangCrossing410141:ZDisCM 0.007159 0.003185 85.90418 38.21652 410041 AdjungbillyCreek@Darbalara222007:ZDisCM 0.000616 0.001175 7.395969 14.10417 222007 WullwyeRiver@Woolway

Predicted and observed discharge 11 unimpaired catchments in Murrumbidgee basin

25-year time series: Jan 1981 to December 2005

0

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Out

flow

(m

/mth

)

410044:ZDisCM410044:ADisCM

0

0.020.04

0.060.08

0.10.12

0.140.16

0.18

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Out

flow

(m

/mth

)

410038:ZDisCM410038:ADisCM

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flow

(m

/mth

)

410047:ZDisCM410047:ADisCM

0

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0.040.05

0.06

0.070.08

0.09

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Out

flow

(m

/mth

)

410048:ZDisCM410048:ADisCM

0

0.05

0.1

0.15

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Out

flow

(m

/mth

)

410057:ZDisCM410057:ADisCM

0

0.05

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0.15

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0.25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

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flow

(m

/mth

)

410061:ZDisCM410061:ADisCM

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Out

flow

(m

/mth

)

410059:ZDisCM410059:ADisCM

0

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Out

flow

(m

/mth

)

410097:ZDisCM410097:ADisCM

0

0.010.02

0.03

0.040.05

0.06

0.070.08

0.09

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Out

flow

(m

/mth

)

410033:ZDisCM410033:ADisCM

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0.14

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Out

flow

(m

/mth

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410141:ZDisCM410141:ADisCM

0

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(m

/mth

)

222007:ZDisCM222007:ADisCM

Model-data fusion

Basic components• Model: containing adjustable "target variables" (y)• Data: observations (z) and/or prior constraints on the model• Cost function: to quantify the model-data mismatch z – h(y)• Search strategy: to minimise cost function and find "best" target variables

Quadratic cost function:

1 1T TJ z yy z h y C z h y y y C y y

Cost function

MeasurementsPrior information

about target variables

Target variables

Model prediction of observations

Covariance matrix of observation error

Covariance matrix of prior information error

Observations Prior information

Estimates the time-evolving hidden state of a system governed by known but noisy dynamical laws, using data with a known but noisy relationship with the state.

Dynamic model:

• Evolves hidden system state (x) from one step to the next

• Dynamics depend also on forcing (u) and parameters (p)

Observation model:

• Relates observations (z) to state (x)

Target variables (y): might be any of state (x), parameters (p) or forcings (u)

Kalman filter steps through time, using prediction followed by analysis

• Prediction: obtain prior estimates at step n from posterior estimates at step n-1

• Analysis: Correct prior estimates, using model-data mismatch z – h(y)

Kalman Filter

1 , , noise with covariancen n n Q x φ x u p

, noise with covariance R z h x u

Parameter estimation with the Kalman Filter

Dynamic model includes parameters p = pk (k=1,…K) which may be poorly known:

Include parameters in the state vector, to produce an "augmented state vector"

The dynamic model for the augmented state vector is

1

1

, state variables: 1,...,

parameters: 1,...,

n n nj

n nj j

X j N

X X j N N K

φ X u

11, , , with ,...n n n n n n

Mx x x φ x u p x

1 1,... , ,... lengthn n n n nM Kx x p p M K X

Parameter estimation from runoff data

Compare 2 estimation methods

• EnKF with augmented state vector (sequential: estimates of p and Cov(p) are functions of time)

• Levenberg-Marquardt (PEST)(non-sequntial: yields just one estimate of p and Cov(p))

Model runoff predictions with parameter estimates from EnKF

Final thoughts

Applications of "Multiple constraints"

• Data sense: atmospheric CO2, remote sensing, flux towers, C inventories ...

• Process sense: measuring one cycle (eg water) to learn about another (eg C)

Requirement for multiple constraints (in process sense)

• "Confluence of cycles"

• Fluxes: cycles share a process pathway controlled by similar parameters

• Pools: cycles have constrained ratios among pools (eg C:N:P)

Streamflow as a constraint on water cycle, thence carbon cycle

• Strength: Independent constraint on water-carbon (and energy-water) cycles (strongest in temperate environments with P/EP ~ 1)

• Limitation 1: obs model = full hydrological model (sometimes can be simplified)

• Limitation 2: streamflow data (availability, quality, access)

Model-data fusion

• Several methods work (focus on EnKF in parameter estimation mode)

• OptIC (Optimisation InterComparison) project: see poster by Trudinger et al.

Hilary Talbot