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.
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