Total Energy Singular Vectors for Atmospheric Chemical Transport Models

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1 Total Energy Singular Vectors for Atmospheric Chemical Transport Models Wenyuan Liao and Adrian Sandu Department of Computer Science, Virginia Polytechnic Institute and State University, Blacksburg, VA {liao, Abstract. The aim of this paper is to address computational aspects of the total energy singular vector analysis of atmospheric chemical transport models. We discuss the symmetry of the tangent-linear/adjoint operator for stiff systems. Results for a 3D simulation with real data reveal that the total energy singular vectors depend on the target domain, simulation window, chemical reactions, and meteorological data. Keywords: Adjoint models, sensitivity analysis, data assimilation, total energy singular vectors. 1 Introduction Improvements of air quality require accurate and timely predictions of atmospheric pollutant concentrations. A critical element for accurate simulations is the use of observational data to constrain model predictions. Widely used data assimilation techniques include 3D-Var, 4D-Var, Kalman filter and ensemble nonlinear filters. Kalman filter techniques provide a stochastic approach to the data assimilation problem. The filter theory is described by Jazwinski [8] and the applicability to atmospheric modeling is discussed by Daley [4]. As explained by Fisher [6], the Kalman filter is too expensive to be a practical assimilation method for large-scale systems. The ensemble Kalman filter [7] is a feasible approach which approximates the Kalman filter covariance matrix by a Monte- Carlo-type technique. In ensemble Kalman filters the random errors in the statistically-estimated covariance decrease only with the square-root of the ensemble size. Furthermore, the subspace spanned by the random vectors is not optimal for explaining the forecast error. For good statistical approximations with small size ensembles it is essential to properly place the initial ensemble to span the directions of maximum error growth. These directions are the total energy singular vectors as explained below. In this paper we study some of the challenges encountered when computing singular vectors for large transport-chemistry models. The paper is organized as follows. In Section 2 we introduce the total energy singular vectors in the context of data assimilation. Computational aspects are discussed in Section 3, and numerical results are presented in Section 4. Conclusions and future directions are given in Section 5. V.S. Sunderam et al. (Eds.): ICCS 2005, LNCS 3515, pp , c Springer-Verlag Berlin Heidelberg 2005

2 Total Energy Singular Vectors for Atmospheric Chemical Transport Models Ensembles and Singular Vectors An atmospheric model propagates the model state (from x b (t 0 )tox f (T )) and its covariance matrix (from P b (t 0 )top f (T )) using: x f = M t0 T (x b ), P f = M t0 T P b M T t 0 + Q. (1) Here x b and x f represent the background and the forecast state, while P b, P f, and Q represent the covariance matrices of the errors in the background state, forecast state, and of the model errors respectively. The model solution operator is denoted by M t0 T, M t0 T is the solution operator of the tangent linear model and MT t 0 the solution operator of its adjoint. Consider a set of observables y (assumed, for simplicity, to be a linear function of model state, y = Hx). The extended Kalman filter uses the forecast state and its covariance (x f (T ),P f (T )) and the observations and their covariance (y, R) to produce an optimal ( analyzed ) estimation of the model state and its covariance (x a (T ),P a (T )): x a = x f + P f H T (R + HP f H T ) 1 (y Hx f ) (2) P a = P f P f H T (R + HP f H T ) 1 HP f The computational expense of the Kalman filter (2) is extremely large because one needs to invert the matrix R + HP f H T and apply the tangent linear model to each column and the adjoint model to each row of the covariance matrix. The commonly used method to reduce the computational cost is to propagate (only) the projection of the covariance matrix onto a low-dimensional subspace span{s 1,,s k }. The subspace (at the analysis time T ) should contain the directions s k (T ) along which the error has the maximal growth. Singular vector analysis was introduced in meteorology in the 60 s by Lorenz [10] to compute the largest error growth rates. At the beginning of 90 s, adjoint technique was introduced by Molteni [13] and Mureau [14] to compute singular vectors in meteorology problems, then singular vector analysis become viable with sophisticated atmospheric general circulation models (see e.g., Navon et. al. [15]). We define the energy of an error vector at time t 0 as the Euclidean inner product s k (t 0 ),As k (t 0 ), and the energy at the final time T as s k (T ),Bs k (T ). A is a symmetric positive definite matrice and B is a symmetric positive semidefinite matrice. The errors evolve in time according to the dynamics of the tangent linear model, s k (T )=M t0 T s k (t 0 ). The ratio between error energies at t 0 and T offers a measure of error growth: λ = s k(t ),Bs k (T ) s k (t 0 ),As k (t 0 ) = s k(t 0 ),MT t 0 BM t0 T s k (t 0 ) s k (t 0 ),As k (t 0 ) (3) The total energy singular vectors (TESV) are defined as the directions of maximal error growth, i.e. the vectors s k (t 0 ) that maximize the ratio λ in Eq.(3). These directions are the solutions of the generalized eigenvalue problem M T t 0 BM t0 T s k (t 0 )=λas k (t 0 ) (4)

3 808 W. Liao and A. Sandu Eq.(4) can be solved efficiently using software packages like ARPACK [9] (or its parallel version PARPACK). The left side of Eq.(4) involves one integration with the tangent linear model followed by one integration with the adjoint model. A special set of energy norms is provided by the choice B = I and A =(P b ) 1. In this case the resulting Hessian singular vectors s k (t 0 ) evolve into the leading eigenvectors s k (T ) of the forecast error covariance matrix P f (T ). 3 Computation of Chemical Singular Vectors The numerical eigenvalue solver applied to (4) requires a symmetric matrix M BM in order to successfully employ Lanczos iterations, and guarantee that the numerical eigenvalues are real. The symmetry requirement imposes to use the discrete adjoint M of the tangent linear operator M in (4). The computation of discrete adjoints for stiff systems is a nontrivial task [17]. In addition, computational errors (which can destroy symmetry) have to be small. For a given model a symmetry indicator is constructed based on two random perturbation vectors u(t 0 )andv(t 0 ) which are propagated forward in time, u(τ) =M t0 τ u(t 0 )andv(τ) =M t0 τ v(t 0 ). The symmetry residual is the difference r(τ) = u(τ),mt τ M τ T v(τ) v(τ),mt τ M τ T u(τ). Clearly if M is exactly the discrete adjoint of M then r(τ) = 0 for all τ. However, both M and M are evaluated numerically and in practice we expect the symmetry residual r(τ) to have small (but nonzero) values. As an example we consider the SAPRC-99 atmospheric gas-phase reaction mechanism [2] with 93 species and 235 reactions. The forward, tangent linear, and adjoint models are implemented using the automatic code generator KPP [3, 5, 17]. Several numerical experiments revealed that the magnitude of the symmetry residual depends on the choice of numerical integrator. Among the Rosenbrock integrators available in KPP Rodas4 [17] performs best. The variation of r(τ) with time for Rodas4 is shown in Fig. 1 (solid line). Surprisingly, the symmetry is lost for a short transient at the beginning of the time integration interval, where the symmetry residual jumps from to This behavior is due to the stiffness of the chemical terms. Consider a singular perturbation model for the chemical system y = f(y, z), ɛz = g(y, z). Here ɛ 1, y is the slow component, and z is the fast component. For ɛ 0, the perturbation vectors that are propagated through the tangent linear model are of the form δz = gz 1 (y, z)g y (y, z)δy (5) During the numerical computation of the eigenvectors ARPACK (or any solver package) generates vectors [δy, δz] T which do not satisfy Eq.(5). To correct this we apply the tangent linear model on the initial perturbation for a short time, which is equivalent to projecting the initial perturbation onto the slow evolution manifold described by (5). The result is then used to initialize the subsequent tangent linear model run. In order to preserve operator symmetry,

4 Total Energy Singular Vectors for Atmospheric Chemical Transport Models 809 another projection using the adjoint model needs to be performed at the end of the adjoint integration. Consequently the operator is computed as w = P M T t 0 M t0 T Pu, (6) where P and P denote the projection operations performed with the tangent linear and the adjoint models respectively. Numerical tests revealed that a small number of projection steps ( 7) is sufficient in practice to substantially enhance symmetry. Fig.1 (dashed) presents the evolution of the symmetry residual when 6 projection steps are performed with the very small stepsize of 10 9 seconds. The symmetry error during the transient is only Fig. 1. Symmetry residual vs. time. Projection improves symmetry considerably These results can be extended to 3D chemistry-transport models, which solve the advection-diffusion-reaction equations in the atmosphere. A detailed description of such models and the corresponding tangent linear and adjoint models is given in [16]. 4 Numerical Results The numerical tests use the state-of-the-art regional atmospheric chemical transport model STEM [1]. The simulation covers a region of 7200 Km 4800 Km in East Asia and uses a computational grid with a horizontal resolution of 240 Km 240 Km. The chemical mechanism is SAPRC-99 [2] which considers the gas-phase atmospheric reactions of volatile organic and nitrogen oxides in urban and regional settings. Both the forward and adjoint chemical models are implemented using KPP [3, 5, 17]. The simulated conditions correspond to March More details about the forward model simulation conditions and comparison with observations are available in [1]. The forward and adjoint models are parallelized using PAQMSG [12]. PARPACK [9] was used to solve the symmetric generalized eigenvalue problems. To visualize the four-dimensional eigenvectors in (4) we consider separately the vector sections corresponding to different chemical species. Two-dimensional top views are obtained by adding the values in each vertical column.

5 810 W. Liao and A. Sandu Fig. 2. The dominant eigenvalues for 12h, 24h and 48h simulations Fig. 3. Dominant eigenvectors for O 3 and NO 2, the 24h simulation The target is the ground level ozone concentration in a 720 Km 960 Km area covering Korea (the gray area in Fig. 3). The target (region, vertical level, and chemical species) defines the matrix B in (4). The largest 12 eigenvalues for 12h, 24h and 48h simulations started at 0 GMT, March 1 st, 2001 are shown in Fig. 2. The rapid decrease of eigenvalue magnitude indicates that one can capture the uncertainty in the target region with only a few total energy singular vectors. The eigenvalues decrease faster for longer simulation windows. The O 3 and NO 2 sections of the first two dominant eigenvectors are shown in Fig. 3. The simulation interval for this test is 24 hours. We notice that the eigen-

6 Total Energy Singular Vectors for Atmospheric Chemical Transport Models 811 Fig. 4. Adjoint O 3 and NO 2 variables, the 24h simulation Fig. 5. Dominant O 3 eigenvectors for the 12h (a) and 48h (b) simulations vectors are localized around the target area. The shapes of the second eigenvector is different from the first, which illustrates the fact that different eigenvectors contain different information. The shapes and the magnitudes of the O 3 and NO 2 sections are also different, illustrating the different influences that these species have on ground level O 3 after 24h. Total energy singular vectors versus adjoints. To illustrate the difference between the information conveyed by the total energy singular vectors and adjoint variables we show the adjoints (for the total ground level O 3 in the target area after 24h) in Fig. 4. The adjoints cover a wider area following the flow pattern, while the singular vectors are more localized. Influence of the simulation interval. The O 3 sections of the dominant eigenvectors for 12h and 48h simulations starting at 0 GMT, March 1, 2001, are shown in Fig. 5. The plots, together with Fig. 3, show the influence of the simulation interval on the singular vectors. For the 12h simulation the pattern is more localized. Influence of meteorological conditions. The O 3 section of the dominant eigenvector for a 24h simulation started at 0GMT, March 26, 2001, is shown Fig. 6 (a). The shape of the TESV is different than for March 1 st. Influence of the target region. The O 3 section of the dominant eigenvector for another 24h, March 1 st simulation is shown in Fig. 6(b). The target is ground

7 812 W. Liao and A. Sandu Fig. 6. Dominant eigenvectors (O 3 section) for: (a) Korea, March 26, show the influence of different meteorological conditions; and (b) China, March 1, show the effect of different target region level ozone in a region of same area, but located in South-East China. Additional numerical tests revealed that the eigenvalues and eigenvectors are heavily effected by the size of target region. Specifically, the eigenvalues decrease is slower for larger regions, and therefore, more eigenvectors are needed to capture the uncertainty. 5 Conclusions In this work we study the computational aspects of total energy singular vector analysis of chemical-transport models. Singular vectors span the directions of maximal error growth in a finite time, as measured by specific energy norms. The required symmetry of the tangent linear-adjoint operator implies the necessity of using discrete adjoints. A projection method is proposed to preserve symmetry for stiff systems associated with chemical models. Numerical results are presented for a full 3D chemistry-transport model with real-life data. The singular values/vectors depend on the simulation interval, meteorological data, location of target region, the size of target region etc. Future work will focus on computing Hessian singular vectors, and on using singular vectors within nonlinear ensemble filters. Acknowledgements This work was supported by the National Science Foundation through the awards NSF CAREER ACI and NSF ITR AP&IM We would like to thank Virginia Tech s laboratory for Advanced Scientific Computing (LASCA) for the use of the Anantham cluster. References 1. Carmichael, G.R. et. al. Regional-Scale Chemical Transport Modeling in Support of the Analysis of Observations obtained During the Trace-P Experiment. Journal of Geophysical Research, 108(D21), Art. No. 8823, 2004.

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