Advanced scenario approach for assessment of environmental changes Vladimir Penenko & Elena Tsvetova Institute of Computational Mathematics and Mathematical Geophysics SD RAS Novosibirsk
Goal Methodology for prognosis of long-term environmental changes ( air, water): • background hydrodynamics • pollutants’ transport • pollutants’ transformation • risk/vulnerability assessment Scenario approach as a way of goal achievement
Main energy active regions in the global atmosphere August 15 150 100 y 50 0 0 100 200 300 x Leading OBV-1 for 500-hPa geopotential height for 56 years (1950-2005), August
Функция чувствительности для оценки областей риска / уязвимости для озера Байкал
Siberian Federal District. Lake Baikal region. Angarsk as aggregated source of pollution Animation
State-of-the-art in scenario approach • Singular vectors (SV) for forward tangent operator of dynamical models and the use of SV-decomposition for scenario construction and errors analysis ( uncertainty ruducing); • ensembles of prognostic scenarios with generation of perturbations (“breeding cycle”); Monte-Carlo methods for scenario construction • • Stochastic-dynamic moment equations and Liouville equations ICMMG technology • Orthogonal decomposition of multi-dimesional databases for formation of informative subspaces Minimization of uncertainties with respect to given criteria of • prognosis quality ( + data assimilation if any)
Scenarios construction and adaptive monitoring with SV ∂ϕ + ϕ = ⇒ A( ) 0 ∂ t ϕ x % Tangent linearization about ( ,t ) ∂δϕ + δϕ = δϕ = ( , ) x 0 ( a priori ) A 0 ∂ L t [ ] r δϕ = δϕ ∈ ∈ ( ,t ) x L ( , ), x x 0 D, t 0 ,t - forward tangent propagator about ϕ x % x ( ,t ) L ( , ) t [ ] [ ] ψ = δϕ ∈ → * ( , ) x 0 L ( ,t ) x ,t t 0 Σ D t
Basic relations and patterns for SVs ( ) δϕ = δϕ δϕ = ( t ) ( t ), ( t ) Σ t ) ( ) ( = δϕ δϕ = δϕ δϕ = * L ( ),L ( ) ( ),L L ( ) 0 0 0 0 ( ) = δϕ ψ ( 0 ), ( 0 ) ∑ ∈ = D evaluation domain at t t t ∑ target area at t=0 ∈ D 0 [ ] 0 ,t “ optimal” time interval ( ≤ 48 h)
Partial eigenproblem for SVs 2 , ( = σ ∈ * L LV V i K ) i i i σ ,V singular values and vectors of L ( SEVs, SVs) i i •Lanzosh algorithm •Ortogonal decomposition of perturbation spaces •Optimal construction of perturbations with respect to rapidly growing SVs
Target area Weighted sum of SVs’ energy σ M r r r ∑ = ∈ i F ( x ) f (V ( x )), x D σ m i i = i 1 1 f (V ) full energy of SV i i i r r ∈Σ ≥ x , if F ( x ) 0 5F . ( x ) 0 M M 0 r r x F ( x ) maximum point of o M
System organization of environmental modeling Models of processes: Data bases •hydrodynamics Models of observations •transport and transformation of pollutants Functionals goal functionals: augmented functionals: quality, observations, goal functionals + restrictions, control, cost,etc. integral identities ( models) Forward problems Adjoint problems Identification of parameters, Sensitivity, observability, decrease of uncertainties, controllability, risk/vulnerability data assimilation, targeted monitoring Revealing sources and control System of decision making, design
Approaches Analysis of data for construction of long-term scenarios: • Extraction of multi- dimentional and multi-component factors from data bases (main part and noise) • Classification of typical situations with respect to main factors • Investigation of variability • Formation of “leading” spaces
Basic idea: Presentation of multi-component and multi-dimensional data base as a product of orthogonal spaces Internal structure of decomposition Principle variable State vector functions ( space, time): for general (external) temperature, wind velocity components, structure decomposition: geopotential, humidity, gas phase and year number aerosols substances, etc Data base Set Set of principal components of orthogonal spaces
• Mathematical model for general outlook and creation of algorithm construction ∂ ϕ + ( ϕ − − = , Y) f r B G 0 ∂ , t ϕ = ϕ + ξ = + ς 0 0 , Y Y ; 0 0 ϕ ∈ ℑ ( ) D is the state function , t ∈ ℜ Y ( D ) is the parameter vector. t G is the “space” operator of the model • A set of measured data ϕ , Ψ on m D , m m t Ψ = ϕ + η [ ( )] H m m is a model of observations . • ξ ς η r, , , are the terms describing uncertainties and errors of the corresponding objects.
Observability, sensitivity and uncertainty Sensitivity and uncertainty functions r r Γ i,k ( x,t ) Observability functions { } k r = Γ ≥ε ≤ U B ( x,t ) supp ,k K i,k i,q i,q = q 1 Localization functions { } k Γ ≥ε I supp i,q i,q r = = q 1 L ( x,t ) ≥ε 1 , a i,k { } k ∑ χ = χ Γ ≥ε (a) , <ε i,q i,q , a 0 = q 1
Factor subspaces for deterministic-stochastic scenarios • Factor subspaces r r X 0 X is a linear subset of the vector space r X arbitrary element of X !! is invariant at algebraic transformations in X 0 X is leading phase space, While modeling, 0 r - generated disturbances
X Construction of 0 N r d i X c , N N , 0 c max 0 i i d i 1 r 1 Calculation of 1. In deterministic case: with the help of models of processes 2. In deterministic-stochastic case: spectral methods for generation of random processes of fractal type with dispersion 2 H λ σ = ≤ ≤ q 2 , 0 H 1 λ q 1 1 H is “fractal” parameter, are eigenvalues of Gram matrix q “Weather noise” part of subspaces is used for randomisation
Feedback relations Goal functional Φ ϕ = Φ ϕ + Φ ( , Y ) ( ) ( ) Y k k _ state k _ parameters ( ) N ( ) ( ) ∑ 2 ∫ 2 Φ = γ Γ − + γ Γ − % % (1) (2) ( ) Y 0.5 grad Y Y Y Y dDdt kp 1 ip i i 2 ip i i = 1 i D t Feedback equations ∂ ∂Φ ϕ ∂Φ ∂Φ Y ( , Y ) , − = − Γ κ = κ ≅ Φ ϕ 1 i k k k i 1, N ; ( , Y )/ , ∂ ∂ ∂ ∂ i k t Y Y Y i − ∂ ∂ ϕ ϕ h * ( ) ( ) Y I ( , Y , ) = − Γ κ − γ Γ − + γ Γ − % % 1 (1) (2) i div grad Y Y Y Y ∂ ∂ 1 2 i ip i i ip i i t Y i
Forming the guiding phase space with allowance for observation data on the subdomain ( ) τ Ψ m measured data; Z (x, ) x, t basis p n ( ) ∑ ( ) ( ) a = Ψ ∈ ≤ m Z x, t a x, t , x, t D , n n p p t a = p 1 2 n ∑ ( ) ( ) ( ) a τ − Ψ τ τ ∈ ≤ m m m min Z x, x, , x, , a D n n τ p p a m a = p 1 p m D τ { } ( ) − = = 1 m = Γ m m a a , p 1 , n a F p a { } ( ) = ∑ n ( ) a Γ = Γ = Ψ Ψ = Ψ m m m m m m , W , p q , 1 , n F , W Z pq p q a p m m D D τ τ = p 1 τ < If then Z(x, ) is forecast ! t t
Summary of scenario approach • Hydrodynamic background for environmental needs: – informative basis extracted from databases – leading spaces – dynamical model+ assimilation of data of guiding subspaces and accessible monitoring data – extrapolation on basis subspaces
Summary of scenario approach for environmental studies • Comprehensive models; • optimal numerical schemes ( variational technique); • universal algorithm of forward and inverse modeling + sensitivity and uncertainty analysis; • functional space of phase trajectories of the system over prognostic interval • basis in this functional space • numerical models with a limited number of degrees of freedom ( projecting on basis) • return to physical space ( adjoint to projection)
Acknowledgements The work is supported by •RFBR Grant 07-05-00673 • Presidium of the Russian Academy of Sciences Program 16 •Department of Mathematical Science of RAS Program 1.3. •European Commission contract No 013427
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