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

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### 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

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

Coupled terrestrial cycles of energy, water, carbon and nutrients

Water flow

C flow

N flow

P flow

Energy

ATMOSPHERE

Photosynthesis

Soil evap

Rain

WaterCycle

N fixation,N deposition,N volatilisation

C Cycle

Transpiration

PLANTLeaves, Wood, Roots

Disturbance

Respiration

Fertiliser inputs

N,P Cycles

ORGANIC MATTER

Litter: Leafy, Woody

Soil: Active (microbial)

Slow (humic)

Passive (inert)

SOIL

Soil water

Mineral N, P

Fluvial, aeolian transport

Runoff

Leaching

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

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

Raupach, Barrett, Briggs, Kirby (2006)

- 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:

Steady-state water balance: bulk approach

Fu (1981)Zhang et al (2004)

- 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)

Steady water balance: bulk approach

dry wet

wet dry

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

E/EP

a=2,3,4,5

P/EP

a=2,3,4,5

Fu (1981)Zhang et al (2004)

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

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

Steady-state water balance: stochastic-dynamic approach

rw(w)

<w>

increasing precipitation event frequency

increasing precipitation event frequency

dry wet

w = relative soil water

P/EP

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

- 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 rw(w), the PDF of w
- Thence find time-averages: <w>, E = <e(w)>, R = <r(w)>

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

Test area: Murrumbidgee basin

Murrumbidgee basin

Murrumbidgee: relative soil moisture

- Jan 1981 to Dec 2005

F

M

A

M

J

J

A

S

O

N

D

81

82

83

84

Murrumbidgee Relative Soil Moisture (0 to 1)

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

00

01

02

03

04

05

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

Goobarragandra:410057

Adelong:410061

Predicted and observed discharge 11 unimpaired catchments in Murrumbidgee basin

- 25-year time series: Jan 1981 to December 2005

Model-data fusion

Prior information

Observations

Measurements

Prior information about target variables

Cost function

Model prediction of observations

Target variables

Covariance matrix of prior information error

Covariance matrix of observation error

- 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:

Kalman Filter

- 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)

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

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.

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