Advanced scenario approach for assessment of environmental changes - - PowerPoint PPT Presentation

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Advanced scenario approach for assessment of environmental changes

Vladimir Penenko & Elena Tsvetova

Institute of Computational Mathematics and Mathematical Geophysics SD RAS Novosibirsk

SLIDE 2

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

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x y

100 200 300 50 100 150 August 15

Leading OBV-1 for 500-hPa geopotential height for 56 years (1950-2005), August

Main energy active regions in the global atmosphere

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Функция чувствительности для оценки областей риска/уязвимости для озера Байкал

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Siberian Federal District. Lake Baikal region. Angarsk as aggregated source of pollution

Animation

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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)

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Scenarios construction and adaptive monitoring with SV

[ ]

( ,t ) L ( , ), x D, t ,t δϕ = δϕ ∈ ∈ x x r

L

A t ∂δϕ + δϕ = ∂

Tangent linearization about ( ,t )

ϕ x %

∂ϕ + ϕ = ⇒ ∂ A( ) t

[ ] [ ]

Σ

ψ = δϕ ∈ → x x

t

* D

( , ) L ( ,t ) ,t t

( , ) ( a priori ) δϕ = x

( , ) L t x

• forward tangent propagator about ϕ x

% ( ,t )

SLIDE 8

Basic relations and patterns for SVs

( ) ( ) (

)

( )

Σ

δϕ = δϕ δϕ = = δϕ δϕ = δϕ δϕ = = δϕ ψ

t

*

( t ) ( t ), ( t ) L ( ),L ( ) ( ),L L ( ) ( ), ( )

t

D ∈

∑

evaluation domain at t

t = D ∈

∑

target area at t=0

[ ]

0,t “optimal” time interval ( ≤ 48 h)

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Partial eigenproblem for SVs

* 2 , (

)

i i i

L LV V i K σ = ∈

i i

,V σ

singular values and vectors of L ( SEVs, SVs)

• Lanzosh algorithm
• Ortogonal decomposition of perturbation spaces
• Optimal construction of perturbations

with respect to rapidly growing SVs

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Target area

Weighted sum of SVs’ energy

1 1 M i m i i i

F ( x ) f (V ( x )), x D

=

  σ = ∈   σ  

∑

r r r

i i

f (V )

full energy of

i

SV

• x

r

maximum point of

M

F ( x ) r

M M

if F 0 5F x , ( x ) . ( x ) ∈Σ ≥ r r

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System organization

• f environmental modeling

Models of processes:

• hydrodynamics
• transport and transformation
• f pollutants

Data bases Models of observations Functionals

goal functionals: quality, observations, restrictions, control, cost,etc. augmented functionals: goal functionals + integral identities ( models)

Forward problems Adjoint problems Sensitivity, observability, controllability, risk/vulnerability Revealing sources and control System of decision making, design Identification of parameters, decrease of uncertainties, data assimilation, targeted monitoring

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Analysis of data for construction

• f 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

Approaches

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Basic idea: Presentation of multi-component and multi-dimensional data base as a product of orthogonal spaces

Set

• f principal components

Set

• f orthogonal spaces

Data base

Internal structure of decomposition State vector functions ( space, time): temperature, wind velocity components, geopotential, humidity, gas phase and aerosols substances, etc Principle variable for general (external) structure decomposition: year number

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• Mathematical model for general outlook

and creation of algorithm construction

= − − + ∂ ∂ r f Y) , (ϕ ϕ G t B

,

ς ξ ϕ ϕ + = + = Y Y ,

;

) (

t

D ℑ ∈ ϕ

is the state function ,

) ( Y

t

D ℜ ∈

is the parameter vector.

G is the “space” operator of the model

• A set of measured data

m

ϕ ,

m

Ψ on

m t

D , η ϕ Ψ + =

m m

H )] ( [ is a model of observations.

• η

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

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i,k( x,t )

Γ r r

Sensitivity and uncertainty functions

{ }

1 k i,k i,q i,q q

B ( x,t ) supp ,k K

=

= Γ ≥ε ≤ r

U

Observability functions Localization functions

{ } { }

1 1 k i,q i,q q i,k k i,q i,q q

supp L ( x,t ) ,

= =

 Γ ≥ε   =   χ Γ ≥ε  ∑ r

I

1 , a (a) , a  ≥ε  χ =  <ε  

Observability, sensitivity and uncertainty

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Factor subspaces for deterministic-stochastic scenarios

• Factor subspaces

X     r r

is a linear subset of the vector space

X  r

arbitrary element of

X

X  

!! is invariant at algebraic transformations in X While modeling, is leading phase space,

 r

• generated disturbances

X

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1 1

, , 0 max

d

N i i i d i

X c N N c                      



r

Construction of

X

Calculation of

 r

• 1. In deterministic case:

with the help of models of processes

• 2. In deterministic-stochastic case:

spectral methods for generation of random processes

• f fractal type with dispersion

2 2 1

1 ,

H q q

H λ   σ = ≤ ≤   λ   H is “fractal” parameter,

1

q

  are eigenvalues of Gram matrix “Weather noise” part of subspaces is used for randomisation

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Feedback relations

( ) ( )

( )

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

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

i k k k i k i

Y i N t Y κ κ

−

∂ ∂Φ ∂Φ ∂Φ   = − Γ = ≅ Φ   ∂ ∂ ∂ ∂   Y Y Y Y ϕ ϕ

( ) ( )

* 1 (1) (2) 1 2

( , , ) div grad

h i i ip i i ip i i i

Y I Y Y Y Y t Y κ γ γ

− 

 ∂ ∂ = − Γ − Γ − + Γ −   ∂ ∂   Y ϕ ϕ % %

Goal functional

_ _

( , ) ( ) ( )

k k state k parameters

Φ = Φ + Φ Y Y ϕ ϕ

Feedback equations

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Forming the guiding phase space with allowance for

• bservation data on the subdomain

( ) ( ) ( )

2 1

a m p m

n m m m p p a p D

a D n n

τ

τ

τ τ τ

=

− ∈ ≤

∑

a

min Z x, x, , x, , Ψ

( ) ( ) ( )

1

a

n m p p t a p

t a t t D n n

=

= ∈ ≤

∑

Z x, x, , x, , Ψ

measured data;

m

τ Z (x, )

{ }

1

m p a

a p n = = a , ,

( )

1 m m

Γ

−

= a F

( )

{ }

1

m

m m m pq p q a D

W p q n

τ

Γ Γ = = = , , , , Ψ Ψ

( )

1

a m

n m m m p D p

W

τ

=

= ∑ F , Z Ψ

If

t τ <

then

t Z(x, ) is forecast !

( )

p

t x, Ψ

basis

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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

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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)

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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.

• European Commission

contract No 013427