Optimal forecasting of atmospheric quality in industrial regions: - PowerPoint PPT Presentation

Optimal forecasting of atmospheric quality in industrial regions: risk and uncertainty assessment Vladimir Penenko Institute of Computational Mathematics and Mathematical Geophysics SD RAS

Goal Development of theoretical background and computational technology for environmental and ecological applications

CONCEPT OF ENVIRONMENTAL MODELING The methodology is based on: • control theory, • sensitivity theory, • risk and vulnerability theory, •variational principles, •combined use of models and observed data, • forward and inverse modeling procedures, • methodology for description of links between regional and global processes ( including climatic changes) by means of orthogonal decomposition of functional spaces for analysis of data bases and phase spaces of dynamical systems

Basic elements for concept implementation: • models of processes • data and models of measurements • adjoint problems • constraints on parameters and state functions • functionals: objective, quality, control, restrictions etc. • sensitivity relations for target functionals and constraints • feedback equations for inverse problems

Statement of the problem • Mathematical model r ∂ ϕ r r r r + ϕ − − = B G ( , Y ) f r 0 ∂ , t r r r r r r ϕ = ϕ + ξ = + ς 0 0 , Y Y ; 0 0 r ϕ ∈ ℑ ( D ) is the state function , r t ∈ ℜ ( ) Y D is the parameter vector . t G is the “space” operator of the model r r , ϕ Ψ • A set of measured data m D on m m t r r r Ψ = ϕ + η [ H ( )] , m m r ϕ [ H ( )] is the model of observations . m r r r r ξ ς η • , , , r are the terms describing Desired! uncertainties and errors of the corresponding objects.

Transport and transformation model ∂ π c ϕ ≡ + π − µ + ϕ − − = i ( u ) ( ) (x, ) L div c grad c S f t r 0 ∂ i c i i i i t i ∂ π + π = div u 0 continuity equation ∂ t ϕ = = state function { , } c i i 1 m , r = ≤ ≤ ∈ f source term, , D t { 0 t t ; x D } i ( ) ( ) ϕ ≡ Γ ϕ ϕ − Π ϕ transformation operators, ( S ) ( ) ( ) i i i i ( ) ( ) Γ ϕ ϕ ≥ Π ϕ ≥ ϕ ≥ 0 ( ) 0, ( ) 0, i i i Граничные и начальные условия ϕ = ∈ Ω ( R ) q , (x, t ) i i t ϕ = ϕ 0 . (x, ) (x) 0

General form of functionals r r r ( ) ∫ Φ ϕ = ϕ χ ≡ χ = ( ) F ( ) ( , ) x t dDdt F , , k 1,..., K k k k k k D t F are the Lipschitz's functions of the given form, differentiable, bounded k χ ∈ ℑ χ * ( D ) dDdt D are Radon’s or Dirac’s measures on , . k t k t Quality functionals r r r r ∫ Φ ϕ = Ψ− ϕ Ψ− ϕ χ T ( ) ( ( )) ( ( )) ( , ) , H M H x t dDdt k m m k D t “Measurement” functionals K r ∑ ∫ [ r ] r r r Φ ϕ = ϕ δ − ∈ m ( ) H ( ) ( x x ) dDdt , x D mk mk t mk = 1 k D t Restriction functionals r r r r ϕ ≤ ϑ ϕ ≤ Differentiability ( x , t ) N , ( ( x , t ) 0 distributive constraints k r r r r ∫ in extended sense Φ ϕ = ϑ ϕ + ϑ ϕ χ ( ) ( ( ) ( ) ) ( , ) x t dDdt k k k k D t

Extended functional for construction of optimal algorithms and uncertainties assessment { r r r r r r Φ ϕ = Φ ϕ + α η η + α % h h T T ( ) ( ) 0.5 ( M ) ( r M r ) 1 1 m 2 2 h k D D t t r } r r r r r r   ϕ ∗ + ϕ h + α ξ ξ + α ζ ζ I ( , , Y ) T T h  ( M ) ( M ) 3 3 h 4 4 h h h D R ( D ) D t t , = M i ( i 1 , 4 ) are the weight matrices, 4 ∑ α ≥ α = 0 , 1 are the weight coefficients, i i = i 1 ϕ r r , ϕ ∗ are the solutions of the direct and adjoint problems generated from r r Y r ∗ ϕ ϕ = h I ( , , ) 0 Additive aggregation of the functionals for decomposition

Optimal forecasting and design Optimality is meant in the sense that estimations of the goal functionals do not depend on the variations : • of the sought functions in the phase spaces of the dynamics of the physical system under study • of the solutions of corresponding adjoint problems that generated by variational principles • of the uncertainty functions of different kinds which explicitly included into the extended functionals

The universal algorithm of forward & inverse modeling ∂Φ ≡ % r r h r r r Λ ϕ + ϕ − − = h k B G ( , ) Y f r 0 r ϕ ∗ ∂ t ∂Φ ≡ % r r r h r r r ϕ ∗ = ∗ ∗ Λ ϕ + ϕ ϕ + = T T ( ) x 0 k ( B ) A ( , ) Y d 0, r = ∂ ϕ t k k k k t t ∂ r r r r − ∗ r r r ϕ = ϕ + ϕ = 0 0 1 = Φ ϕ + α η η M ( x , 0 ), t 0 h T d ( ( ) 0 . 5 ( M )), a 3 k ∂ ϕ k k 1 1 r r − r r r r − = ϕ = − Γ 1 * 1 r ( x , t ) M ( x , t ), Y Y M 2 k a 4 k ∂ r r r r ∗ Γ = ϕ ϕ h r I ( , Y , ) ∂ k k Y ∂ [ ] r r r r r r ϕ δ ϕ ≡ ϕ + α δ ϕ h A ( , Y ) G ( , Y ) α = ∂ α 0 Λ ϕ is the approximation of time derivatives t r r r r r = ϕ = ϕ = ( 0 ) 0 ( 0 ) 0 ( 0 ) Initial guess: r 0 , , Y Y a a

Some elements of optimal forecasting and design The main sensitivity relations ∂ ∗ δ Φ ϕ ≡ Γ δ ≡ ϕ + α δ ϕ h h ( ) ( , Y) ( ,Y Y, ) I α = ∂ α 0 k k k Algorithm for calculation of sensitivity functions ∂ ∂   ∗ Γ = ϕ + α δ ϕ h   I ( ,Y Y, ) α = ∂ δ ∂ α k  k 0  Y The feed-back relations dY α = − η Γ α = ≤ , 1 , N , N N α α α α k dt Γ = Γ { ki } are the sensitivity functions k δ = δ are the parameter variations Y { Y } i = = k 1 , K , i 1 , N

Real time equations of back relations Φ ϕ = Φ ϕ + Φ ( , Y ) ( ) ( Y ) k ks kp ( )   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   = i 1 D t ∂ ∂Φ ϕ ∂Φ ∂Φ   Y ( , Y ) , = − κ = κ ≅ Φ ϕ i k  k k  i 1, N ; ( , Y )/ , ∂ ∂ ∂ ∂ k   t Y Y Y i   ∂ ∂ ϕ ϕ h * ( ) ( ) Y I ( , Y , ) = − κ − γ Γ − + γ Γ − % % (1) (2)   i div grad Y Y Y Y ∂ ∂ 1 2 ip i i ip i i   t Y i % - a priori parameter values Y γ γ ≥ 0 - weight coefficients 1, 2 α Γ ( ) - matrices of scale coefficients and weights ip

Algorithms of uncertainty calculation based on sensitivity analysis and data assimilation: in model r r r − = ϕ 1 * r( x,t ) M ( x,t ), 2 k in initial state r r r − ∗ ξ = ϕ = 1 M ( x, ), 0 t 0 3 k in model parameters and sources ∂ r r r r r − − ∗ ζ = Γ = ϕ ϕ 1 1 h r M M I ( ,Y , ) ∂ 4 k 4 k Y i = ,( 2 4 , ) M are the weight matrices i

Risk assessment with the help of sensitivity functions Threshold of safety intervals ∆ = s , k 1 , K k safe ecological conditions δ Φ ≤ ∆ s k k Estimations for deterministic case N ∑ δ Φ ≤ = Γ δ Y k ki i i 1

Risk estimates for deterministic-stochastic case δ = ≡ δ = E( Y) { E E( Y ), ( i 1, N ) } i i N ∑ δ Φ = Γ δ E( ) E( Y ) i i = 1 i δ Φ = δ Γ Γ D( ) (D( Y ) , ) ∫ δ Φ∈∆ = ≡ Φ δ P ( ) f x dx ( ) , x ∆ − 2 ( x E( )) x − 1 = 2D( ) x f x ( ) e , π 2 D( ) x = P δ Φ ≤ ∆ s s R ( )

Probability risk assessment − 2 ∆ λ s ( x E( )) x 2 t − − 1 2 ∫ ∫ = = = Ψ λ s 2D( ) x 2 R e dx e dt ( ) , π π 2 D( ) x 2 0 0 λ = λ = ∆ − s s ( R ) ( E( ))/ x D(x) Risk domain = − ≡ P δ Φ > ∆ r s s R 1 R { } Safe range ∆ − δ Φ = λ δ Φ s E( ) D( )

Fundamental role of uncertainty functions • integration of all technology components • bringing control into the system • regularization of inverse methods • targeting of adaptive monitoring • cost effective data assimilation

From risk assessment to design of sustainable development strategy • Risk/vulnerability • Strategy of sustainable assessment development Models of processes & Models of processes & data bases data bases + + environment quality superposition of different functionals multi-criteria functionals: environment quality, objective, control, restrictions, etc.

Advantage of the approach • Consistency of all technology elements • Optimality of numerical schemes based on discrete-analytical approximations(without flux-correction procedures ) • Cost-effectiveness of computational technology