A practical method to assess model sensitivity and
parameter uncertainty in C-cycle models
Sylvain Delahaies, Ian Roulstone, Nancy Nichols
[email protected], [email protected], [email protected]
Context
Variational data assimilation
We consider a simple model for the carbon
budget allocation for terrestrial ecosystems, the
Data Assimilation Linked Ecosystem model
(DALEC). We use variational methods to perform sensitivity analysis and to diagnose the illposedness of model data-fusion problems. Then
we use ecological common sense and constrained
optimization techniques to reduce parameter uncertainties.
Variational data assimilation aims at best combining observations, with prior knowledge, ie model
and initial state. This is achieved by minimizing the cost function
1
1
T −1
T −1
J(x) = J0 (x) + Jobs (x) = (x − x0 ) B (x − x0 ) + (h(x) − y) R (h(x) − y),
2
2
where h is a generalized observation operator, y is a generalized observation vector and R is the
observation error covariance matrix, x0 is the background term and B its error covariance matrix.
The cost function is minimized using a gradient based method where the gradient ∇J(x) is given by
∇J(x) = B
C-cycle model
DALEC
links the carbon pools (C) via allocation uxes (green), litterfall uxes (red),
decompostion (black).
Respiration is represented by the blue arrows. The orange
arrow represents the feedback of foliar carbon to gross primary production (GPP).
−1
T
(x − x0 ) + H R
−1
(h(x) − y),
∂h
H=
,
∂x
where H T is the adjoint operator of the tangent linear operator H.
Linear analysis: resolution matrices
Studying the linear inverse problem Hz = d using resolution matrices and unit covariance matrix
R = HH
−g
,
N =H
−g
H,
C=H
−g
(H
−g T
) ,
allows us to understand the nature of the ill-posedness and to nd strategies to regularize the problem.
The gures show the model and data
resolution matrices (left and centre) for
NEE ux, and parameter uncertainties
(right) using TSVD regularization.
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10p11p12p13p14p15p16p17 CL Cf Cr Cw Cl Cs
6
p1
p2
100
p3
4
p4
200
p5
p6
300
p7
2
p8
400
p9
p10
0
500
p11
p12
p13
600
−2
p14
p15
700
p16
p17
−4
800
CL
Cf
900
Cr
−6
Cw
1000
Cl
Cs
100
200
300
400
500
600
700
800
900
1000
−8
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10p11p12p13p14p15p16p17CL Cf Cr Cw Cl Cs
Ensemble 4DVAR and ecological common sense
The net ecosystem exchange (NEE) is given by
(1)
N EE = GP P − Ra − Rh.
The model depends on 23 variables: 6 initial
carbon stocks and 17 parameters controlling the
photosynthesis and allocation processes.
Model sensitivity
In [2] ecological common sense constraints were dened to restrict the parameter space for DALEC.
We use them in the form of inequality constraints together with barrier function techniques to solve
the optimization problem
argmin Jobs (x), g(x) ≤ 0, l ≤ x ≤ u.
This formulation converges quickly and multiple runs can be performed to produce ensembles of
solutions from which non gaussian parameter distributions can be drawn.
LAI
7
The mean normalized sensitivity dened in [3]
by

6
5
4
3
2

. X ∂h σ
∂h
σ
i
j 

si = E
,
∂xi σh
∂x
σ
j
h
j
1
0
0
100
200
300
400
500
600
700
800
900
1000
600
700
800
900
1000
NEE
10
is a dimensionless quantity that allows to assess
the importance of the parameter xi in controlling the ux h, where σi and σh are error covariances and E denotes the mean over the time
window.
5
0
−5
0
100
200
300
400
500
LAI normalized sensitivity
0.6
0.5
Conclusion
0.4
• We use variational methods to investigate qualitative aspects of the model-data fusion problem
0.3
for a simple but yet non trivial C-cycle model, DALEC.
0.2
• Simple linear analysis allows us to study the eect of various regularization methods.
0.1
0
• The rapid convergence of the constrained nonlinear problem allows us to produce non gaussian
−0.1
p17 CL p15 p5 p13 Cf
p11 p12 p3 p14 p2 p16 p1 p4 p6
NEE normalized sensitivity
p7
p8
p9 p10 Cr Cw
Cl
Cs
0.45
parameter distributions.
0.4
References
0.35
0.3
[1] S. Delahaies, I. Roulstone, and N. Nichols (2013). Regularisation of a carbon cycle model-data fusion problem.
0.25
University of Reading preprint series.
0.2
0.15
[2] Bloom, A.A. M. Williams (2014). Constraining ecosystem carbon dynamics in a data-limited world: integrating
0.1
ecological common sense in a model-data-fusion framework. Biogeosciences Discussions 11, 12733-12772.
0.05
[3] Zhu, Q., and Q. Zhuang (2014), Parameterization and sensitivity analysis of a process-based terrestrial ecosystem
0
−0.05
p15 p17 p9
p5 p11 p1
Cs CL p8 p13 Cl
Cf
p2 p12 Cr p16 p7
p3 p14 p4 p10 Cw p6
model using adjoint method, J. Adv. Model. Earth Syst., 6, doi:10.1002/2013MS000241.