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

1. 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)

2. 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

3. 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

4. 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

5. 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

6. 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) = (CsCi) / (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)

7. 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

8. 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

9. 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)

10. 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:

11. 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)

12. 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

13. 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

14. 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

15. 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)>

16. 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

17. Test area: Murrumbidgee basin Murrumbidgee basin

18. Murrumbidgee: relative soil moisture • Jan 1981 to Dec 2005

19. J 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

20. J F M A M J J A S O N D 81 82 83 84 Murrumbidgee Total Evaporation (mm d-1) 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05

21. Southern MDB: "unimpaired" gauged catchments

22. 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

23. Predicted and observed discharge 11 unimpaired catchments in Murrumbidgee basin • 25-year time series: Jan 1981 to December 2005

24. 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:

25. 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)

26. 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

27. 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

28. 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.

29. Hilary Talbot