data assimilation in terrestrial biosphere models: exploiting mutual constraints among the carbon,...
TRANSCRIPT
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
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1
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1.4
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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
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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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212028
212021
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210091210088210082
210048
210042210040
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
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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
0.01
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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
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
)
410047:ZDisCM410047: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
)
410048:ZDisCM410048:ADisCM
0
0.05
0.1
0.15
0.2
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
0.1
0.15
0.2
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
Out
flow
(m
/mth
)
410061:ZDisCM410061:ADisCM
0
0.05
0.1
0.15
0.2
0.25
0.3
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
)
410059:ZDisCM410059:ADisCM
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
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
)
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
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
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
)
410141:ZDisCM410141:ADisCM
0
0.01
0.02
0.03
0.04
0.05
0.06
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
)
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