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

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Optimal forecasting of atmospheric quality in industrial regions: risk and uncertainty assessment

Vladimir Penenko

Institute of Computational Mathematics and Mathematical Geophysics SD RAS

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Goal

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

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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

CONCEPT OF ENVIRONMENTAL MODELING

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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

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Desired!

Mathematical model

) , ( = − − + ∂ ∂ r f Y G t B r r r r r ϕ ϕ

,

ς ξ ϕ ϕ r r r r r r + = + = , Y Y

;

) (

t

D ℑ ∈ ϕ r

is the state function ,

) (

t

D Y ℜ ∈ r

is the parameter vector.

G is the “space” operator of the model

A set of measured data

m

ϕ r ,

m

Ψ r

n

m t

D η ϕ r r r + = Ψ

m m

H )] ( [

,

m

H )] ( [ ϕ r

is the model of observations.

η

ς ξ r r r r , , , r are the terms describing uncertainties and errors of the corresponding objects.

Statement of the problem

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i

f source term,

} ; { D x t t Dt ∈ ≤ ≤ = r

,

( ) ( )

( ) ( ) ( )

i i i i

S ϕ ≡ Γ − Π ϕ ϕ ϕ

transformation operators,

( ) ( )

( ) 0, ( ) 0,

i i i

ϕ Γ ≥ Π ≥ ≥ ϕ ϕ ϕ

Граничные и начальные условия

) (x, , ) (

t i i

t q R Ω ∈ = ϕ (x) ) (x, ϕ ϕ =

.

grad div = − − + − + ∂ ∂ ≡

i i i i c i i

r t f S c c t c L

i

) (x, ) ( ) u ( ϕ ϕ µ π π

} , {

,

m i ci 1 = = ϕ

state function

div u t π π ∂ + = ∂

continuity equation

Transport and transformation model

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General form of functionals

( )

( ) ( ) ( , ) , , 1,...,

t

k k k k k D

F x t dDdt F k K ϕ ϕ χ χ Φ = ≡ =

∫

r r r

k

F are the Lipschitz's functions of the given form, differentiable, bounded dDdt

k

χ

are Radon’s or Dirac’s measures on

t

D

,

) (

* t k

D ℑ ∈ χ

.

Quality functionals

( ) ( ( )) ( ( )) ( , ) ,

t

T k m m k D

H M H x t dDdt Φ ϕ = Ψ− ϕ Ψ− ϕ χ

∫

r r r r

“Measurement” functionals

[ ]

1

( ) ( ) ( ) ,

t

K m mk mk t mk k D

H x x dDdt x D

=

Φ ϕ = ϕ δ − ∈

∑ ∫

r r r r r

Restriction functionals ) , ( ( , ) , ( ≤ ≤ t x N t x

k

r r r r ϕ ϑ ϕ

distributive constraints

dDdt t x

k k D k k

t

) , ( ) ) ( ) ( ( ) ( r r r r χ ϕ ϑ ϕ ϑ ϕ + = Φ

∫

Differentiability in extended sense

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{

1 1 2 2

( ) ( ) 0.5 ( ) ( )

m h t t

h h T T k D D

M r M r ϕ ϕ α η η α Φ = Φ + + r r r r r r %

}

3 3 4 4 ( )

( ) ( )

h h h t

T T h D R D

M M α ξ ξ α ζ ζ + + r r r r

( , , )

h t

h D

Y ϕ ϕ∗   +  I r r r

) 4 , 1 ( , = i Mi

are the weight matrices, 1 ,

4 1

= ≥

∑

= i i i

α α are the weight coefficients,

∗

ϕ ϕ r r, are the solutions of the direct and adjoint problems generated from

) , , ( =

∗

ϕ ϕ r r r Y

h

I

Extended functional for construction

f optimal algorithms and uncertainties assessment

Additive aggregation of the functionals for decomposition

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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

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( , )

h h k t

B G Y f r ϕ ϕ ϕ ∗ ∂Φ ≡ Λ + − − = ∂ % r r r r r r

( ) ( , ) 0,

h T T k t k k k

B A Y d ϕ ϕ ϕ ϕ

∗ ∗

∂Φ ≡ Λ + + = ∂ % r r r r r

)), ( 5 . ) ( (

1 1

η η α ϕ ϕ r r r M d

T h k k

+ Φ ∂ ∂ =

( )

k t t

x ϕ∗

=

= r r

), , (

1 3

= + =

∗ −

t x M

k a

r r r r ϕ ϕ ϕ

), , ( ) , (

* 1 2

t x M t x r

k

r r r ϕ

−

=

k a

M Y Y Γ − =

− r

r r

1 4

) , , (

∗

∂ ∂ = Γ

k h k

Y I Y ϕ ϕ r r r r r

[ ]

) , ( ) , (

=

+ ∂ ∂ ≡

α

ϕ δ α ϕ α ϕ δ ϕ Y G Y A

h

r r r r r r ϕ

t

Λ is the approximation of time derivatives Initial guess:

a a

Y Y r r r r r r = = =

) ( ) ( ) (

, , ϕ ϕ

The universal algorithm

f forward & inverse modeling

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( ) ( , Y) ( ,Y Y, )

h h k k k

I

α

δ δ α δ α

∗ =

∂ Φ ≡ ≡ + ∂ ϕ Γ ϕ ϕ

( ,Y Y, ) Y

h k k

I

α

α δ δ α

∗ =

∂ ∂   = +   ∂ ∂   Γ ϕ ϕ

The main sensitivity relations Algorithm for calculation

f sensitivity functions

} { ki

k

Γ = Γ

are the sensitivity functions

} { Y

i

Y δ δ =

are the parameter variations

N i K k , 1 , , 1 = = N N N dt dY

k

≤ = Γ − =

α α α α α

α η , , 1 ,

The feed-back relations

Some elements

f optimal forecasting and design

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Real time equations of back relations

( , ) ( ) ( )

k ks kp

Φ = Φ + Φ Y Y ϕ ϕ

( ) ( )

( )

2 2 (1) (2) 1 2 1

( ) 0.5 grad

t

N kp ip i i ip i i i D

Y Y Y Y dDdt γ γ

=

  Φ = Γ − + Γ −    

∑ ∫

Y % %

( ) ( )

* (1) (2) 1 2

( , , ) div grad

h i ip i i ip i i i

Y I Y Y Y Y t Y κ γ γ   ∂ ∂ = − − Γ − + Γ −   ∂ ∂   Y ϕ ϕ % % Y % - a priori parameter values

1, 2

γ γ ≥

weight coefficients

( ) ip α

Γ

matrices of scale coefficients and weights

( , ) , 1, ; ( , )/ ,

i k k k k i

Y i N t Y κ κ ∂ ∂Φ ∂Φ ∂Φ   = − = ≅ Φ   ∂ ∂ ∂ ∂   Y Y Y Y ϕ ϕ

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1 2 * k

r( x,t ) M ( x,t ),

−

= ϕ r r r

1 3 k

M ( x, ), t

− ∗

ξ = ϕ = r r r

1 1 4 4 h k k

M M I ( ,Y , ) Y

− − ∗

∂ ζ = Γ = ϕ ϕ ∂ r r r r r r

Algorithms of uncertainty calculation based on sensitivity analysis and data assimilation:

in model in initial state in model parameters and sources

2 4 ,( , )

i

M i =

are the weight matrices

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Threshold of safety intervals safe ecological conditions Estimations for deterministic case

Risk assessment with the help of sensitivity functions

K k

s k

, 1 , = ∆

s k k

∆ ≤ Φ δ

∑

= Γ

≤ Φ

N i i ki k

Y

1

δ δ

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E( E E( ( 1, )

i i

Y i N δ δ = ≡ = Y) { ), }

1

E( E(

N i i i

Y δ δ

=

Φ = Γ

∑

) ) D( (D( ) , ) δ δ Φ = Γ Γ ) Y ( ) ( ) , P f x dx x δ δ

∆

Φ∈∆ = ≡ Φ

∫

2

( E( )) 2D( )

1 ( ) , 2 D( )

x x x

f x e x π

− −

= ( )

s s

R P δ = Φ ≤ ∆

Risk estimates for deterministic-stochastic case

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Safe range Risk domain

( ) ( E( ))/ D(x)

s s

R x λ λ = = ∆ −

2 2

( E( )) 2D( ) 2

1 2 ( ) , 2 D( ) 2

s

x x t s x

R e dx e dt x

λ

λ π π

− ∆ − −

= = = Ψ

∫ ∫

1 { }

r s s

R R P δ = − ≡ Φ > ∆ E( ) D( )

s

δ λ δ ∆ − Φ = Φ

Probability risk assessment

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Fundamental role

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

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From risk assessment to design of sustainable development strategy

Risk/vulnerability

assessment Models of processes & data bases + environment quality functionals

Strategy of sustainable

development Models of processes & data bases + superposition of different multi-criteria functionals:

environment quality,

bjective, control, restrictions,

etc.

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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

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Idea and basic approximations

( )

1 1 1 1

* * * * *

, ( ) ( )

i i i i i i i i

x x x x x x x x

L f dx L dx A f x x dx ϕ ϕ ϕ ϕ ϕ ϕ ϕ

− − − −

= − = + − =

∫ ∫ ∫

Differential operators of common kind in the models

* *

0, Lϕ =

( )

1 1

* *

, ( ) ( ) 0, 2,

i i i i

x x x x x

A f x x dx i n ϕ ϕ ϕ

− −

− = =

∫

If then

*( ) 1

( ), , 1,2

i i

x x x x

α

ϕ α

− ≤

≤ =

integrating multipliers

{ } { }

*(1) *(1) *(2) *(2) 1 1

1, 0 , 0, 1 , 1, 1

i i i i x

i n ϕ ϕ ϕ ϕ

+ +

= = = = = −

Fundamental analytical solutions of local adjoint problems

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Variational principle for successive schemes

( ) ( )

{

( )

}

( ) ( ) ( )

* 2 1 1 * 1 1 * 1 * * 1 1 1 2

( , ) 1

J r j j j j j j j j j j j j j j j r j j j j j j j j j r j

Y L f t t

α α α α α α α α α α α α α α α α α α

ϕ ψ ϕ τ ψ ψ ψ σ ϕ σ ϕ ϕ δ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ δ

= = − − − − − =

   Φ + − + ∆ −       + − + −      + − + − + −      

∑ ∑ ∑

( )

1

j j j α α α α α

ψ σ ϕ σ ϕ − = + − 0.5 1,

j j

t

α α α

σ τ σ ≤ ≤ ∆ = ∆

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Variational principle for parallel schemes

( ) ( )

{

( )

}

1 * 2 1 1 * * 1 2 1

1 1 ( , )

J r j j j j j j j j j j j j j r J r j j j j j k j

L f t t Y r

α α α α α α α α α α α α α α α α α

ψ ϕ τ ψ ψ ψ σ ϕ σ ϕ ϕ δ ϕ ϕ ϕ δ ϕ

− = = − = = =

   − + ∆ −       + − + −      + − + Φ      

∑ ∑ ∑ ∑∑

r r

2 1 2 1

( , ) ( , ) ( , )

J r J r h j j j k k k k j j j D

Y F Y x t dD t

α α α α α

ϕ ϕ χ δ

= = = =

  Φ = Φ =    

∑∑ ∑∑ ∫

r r r r r

1

( , ) x t ϕ r

given

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Long-term forecast of environmental risk for Lake Baikal region

0.1 0.1 0.1 0.2 0.2

0.2 0.2 0.2

0.3 . 3 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 . 6

0.6

. 6 . 7 0.7 0.7 0.8 0.8 0.9 . 9

95 100 105 110 115 50 55 60

Baikalsk Irkutsk Ulan-Ude Nijhne-Angarsk Cheremkhovo Bratsk Angarsk

Surface layer, October

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Conclusion

Algorithms for optimal environmental forecasting and design are proposed:

uncertainty calculations
risk assessment
feed-back relations

The fundamental role of uncertainty is highlighted

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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.

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Thank you for your time!