Toy model description

Authors: P. Peylin and N. MacBean

1. Model of soil carbon decomposition based on two pools and first order kinetics

\[ \frac{dC_{pool}}{dt} = Input-C_{pool} \frac{\Delta t}{\tau_{pool}} f(T) f(W) \]

where \( C \) is the carbon content the pool, \( \tau \) is the turnover time and \( M_e \) is the microbial assimilation efficiency The microbial assimilation efficiency governs fraction that partitions decomposition into input to the (passive) pool and respiration \( (1-M_e) \).

Two pools: Active (\( C_{active}\)) and Passive (\( C_{passive}\)) pools

Input for the Active pool: litter input (\(L\)) (prescribed)

Input to the passive pool:

\[ C_{pool1 \to pool2} = C_{pool1} \frac{\Delta t}{\tau_{pool1}} f(T) f(W) M_{e.pool1} \]

\( f(t) \): Temperature function cotroling the decomposition (based on Q10 parameter)

\( f(W) \): Soil water stress function (inverse parabolic function) depending on soil water (based on two parameters)

2. Model set up

Period of simulation: nb years variable

Daily time step and daily data

Input (litter, Temp, soil moisture) taken from a specific site (HESSE forest in North-Eastern France)

3. Parameters to optimize

• the initial C pool contents — \( C_{litter,t0} \) and \( C_{soil,t0} \)
• the turnover time of both pools — \( \tau_{litter} \) and \( \tau_{soil} \)
• the microbial efficiency of both pools — \( M_{e,litter} \) and \( M_{e,soil} \)
• the parameter in the \( f(T) \) temperature stree function — Q10
• two parameters controlling the shape of the inverse parabola in the \( f(W) \) moisture stress function — \( wf_{x0} \) and \( wf_m \).

Note that the Q10 and \( wf \) parameters are the same for both pools, though the temperture and moisture are different.

4. Data streams

Two data streams are considered:

• Daily flux measurements corresponding to heterotrophic respirations (Rh): possibly split between litter and soil organic pools contributions
• Total Soil carbon content at the beginning of the assimilation window: possibly split between litter and soil organic pools

Type of data:

• Pseudo-data: created with one set of parameters (standard values) + added random noise
• Real-Obs: Respiration taken as 50% of the site measured total respiration (50% to account for Heterotrophic Resp only; Measurements correspond to fluxnet data partitioning); Soil initial carbon pools taken as the total soil carbon from ORCHIDEE after a spin up.

5. Bayesian optimisation — equations and approach (step-wise vs simultaneous)

• Non linear model; Use an iterative solution based on a linearization (Matrix approach); Iterative solution based on a fixed point algorithm (Tarantola, 1987, p197)

\[ x_{i+1} = x_b + [H^T R^{-1} H + B^{-1}]^{-1} H^T R^{-1} (y - H(x) - H(x_i - x_b)) \]

• Test the problem of potential local minima by having multiple tests starting from different first guess values.

Step-wise approach

Step 1: Assimilation of data-stream 1. The prior parameters, including their values and error covariance (\(x_0\) and \(P_0\)), are optimised to produce a first set of posterior optimised parameters \(x_1\) with error covariance \(P_1\).

Step 2: Assimilation data-stream 2. The parameters, \(x_1\), and their covariances, \(P_1\) (estimated using linear assumption), are used as a prior to the optimisation system and further optimised to produce the second (and final) set of posterior optimised parameters, \(x_2\), and the associated error covariance \(P_2\). The \(x_1\) and \(P_1\) prior parameter vector and error covariance matrix are augmented to include parameters that were not optimised by data stream 1. For the additional parameters, the prior value and uncertainty is the default value of ORCHIDEE.

Simultaneous approach

Both data-stream 1 and 2 are included in the optimisation and all parameters are optimised at the same time. The prior parameters, including their values and error covariance (\(x_{prior}\) and \(P_{prior}\)) are optimised to produce the posterior parameter vector (\(x_{post}\)) and associated uncertainties \(P_{post}\).

Appendix. Model description

NOTATIONS

\(t\): current time step
\(t_+\): next time step
\(C_{active}\): quantity of soil carbon in the active pool
\(C_{passive}\): quantity of soil carbon in the passive pool
\(C_{active,t0}\), \(C_{passive,t0}\): quantity at \(t_0\)
\(M_e\): Microbial efficiency (\(1-M_e\): fraction of the active pool decomposition that is respired)
Rh: Heterotrophic respiration flux
Q10: value of the so-call Q10 for the temperature dependance
\(\tau_{active}\) and \(\tau_{passive}\): turnover rate for active and passive pools
\(L_{input}\): input of \(C\) to the active pool from the litter (provided from ORC simulation)
Temp: soil temperature (Input provided from ORC)
\(W\): soil water content (input provided from ORC)
\(W_w\): \(W\) at wilting point
\(W_f\): \(W\) at field capacity

MODEL EQUATIONS

\[ \frac{dC_{active}}{dt} = L_{input} - C_{active} \frac{\Delta t}{\tau_{active}} f(T) f(W) + C_{passive \to active} \] \[ C_{active \to passive} = C_{active} \frac{\Delta t}{\tau_{active}} f(T) f(W) M_{e.active} \] \[ \frac{dC_{passive}}{dt} = C_{active \to passive} - C_{passive} \frac{\Delta t}{\tau_{passive}} f(T) f(W) \] \[ C_{passive \to active} = C_{passive} \frac{\Delta t}{\tau_{passive}} f(T) f(W) M_{e.passive} \] \[ Rh = C_{active} \frac{\Delta t}{\tau_{active}} f(T) f(W) (1-M_{e.active}) + C_{passive} \frac{\Delta t}{\tau_{passive}} f(T) f(W) (1-M_{e.passive}) \] \[ f(T) = Q10^{((T_{air}-30)/10)} \] \( f(W) \): soil water stress function: increases from 0 to 1 depending on \(W\) (0 up to \(W_w\), and then linear increase up to a value \(W_f\))