THE I NVERSE PROBLEM FOR ESTI MATI ON OF AEROSOL FALLI NG FI ELDS - - PowerPoint PPT Presentation

THE I NVERSE PROBLEM FOR ESTI MATI ON OF AEROSOL FALLI NG FI ELDS FROM AREA SOURCES

RAPUTA V.F. Institute of Computational Mathematics and Mathematical Geophysics

• f SB of RAS, Novosibirsk

Problem setting

( )

( )

z

C C C VC w K K C t z z z ∂ ∂ ∂ ∂ + ∇ − = + ∇ ∇ ∂ ∂ ∂ ∂ u r

s z z

C C K wC z t

=

∂ ∂   + =   ∂ ∂  

s g s s z

C V C C t α

=

∂ = − ∂

( , , ) , ( , )

s s t t

C C x y z C C x y

= =

= =

( 1 ) ( 2 ) ( 3 ) ( 4 )

1

• I. Estimation of concentration fields of

local gauge

( ) ( ) ( )

, q q q q u z w k z v z x z z z y y ∂ ∂ ∂ ∂ ∂ ∂ − = + ∂ ∂ ∂ ∂ ∂ ∂

( ) ( )

0, 0, ,

x x z

q k wq q q M y z H z δ δ

→∞ = =

∂ + = → = − ∂

r

( ) ( ) ( ) ( )

1 1 1 1

, ,

n m

z z u z u k z k v z k u z z z     = = =        

( )

3 2 2 max max max

3 , ,0 exp 1 , 2 4 x x y q x y q x x k x       = ⋅ − −              

Point source

( 5 ) ( 6 ) ( 7 ) ( 8 )

2

3 2 3 2 1 max max 2 max 3

3 1 , , , 2 4 e q x x k θ θ θ = ⋅ = =

( )

2 3 1 2 3/ 2

, exp . y q x x x x θ θ θ θ   = − −       r r

( )

( )

4 4

2 3 1 2 2 3/2 1 4

, exp , 1

i i

w K i w w i i

p y q x x x x Г w x

θ θ

θ θ θ θ θ θ

=

  = − − ⋅     +   ∑ r r

( )

4 1

1 . 1 k n θ = +

3

Light impurity Polydisperse aerosol

( 9 ) ( 1 0 ) ( 1 1 )

Area source

( )

( ) ( ) ( )

ϕ     ⋅    

∑

2 N 2 i 2 1 i 2 2 i i=1 i

y-y θ q x,y,θ =θ exp -

• x-x

x-x 2 x-x r

( ) ( )

ϕ

1+n 1 1 2 2 1 1

M u H θ = , θ = 1+n k 2π 1+n k

( )

[ ]

2

, , 0 , , , 1, .

k k k n k j kj k

r q x E E k j N θ ξ ξ ξ ξ δ σ = +   = = =   r r

( ) ( )

2 2 1

, .

N N k k k k

J r q x θ σ θ

− =

  = −  

∑

r r r LS method – estimate of parameters

4

( 1 2 ) ( 1 3 ) ( 1 4 ) ( 1 5 )

• II. Numerical modeling of processes of distribution

sulphate aerosol in a neighbourhood of lake Selitrennoe

5

• Fig. 1. The plan of air samples

and the recovered field of countable concentration (thousands of particles per litre) of 0.3 – 0.4 microns fraction of sulphate aerosol

• Fig. 2. The calculated field of

countable concentration of 0.3-0.4 microns light fraction, formed with south wind

6

The analysis of the observation data of countable concentration

• Fig. 3. The measured and calculated values of countable concentration of

an aerosol for fraction 0.3-0.4 microns

Размер частиц

1997 г. 2004 г.

R max, км / R max, км /

0,3 – 0,4 мкм 1,25 4,24 0,7 1,01 0,4 – 0,5 мкм 1,15 3,51 0,5 0,81 0,5 – 1 мкм 0,6 0,91 0,4 0,25

1

θ

6

10 ⋅

6

10 ⋅

Estimates of parameters of the model (12) according to observation data

1

θ

7

8

The analysis of the observation data

• f mass concentration

(A linear source model)

• Fig. 4. The measured and calculated values of sulphate

aerosol mass concentration

• III. Interpretation of density data of dropouts
• f plant pollen

9

а) Leban

• Fig. 5. The measured and recovered density values of

dropout of leban pollen

b) Birch

10

• Fig. 6. The calculated density curve of dropout of birch

pollen, recovered on the points had on distances

• IV. Methods of estimation of regional

pollution of territories

( ) ( , ) , 2 M g q r u h r ϕ ϕ π ⋅ = ⋅ ⋅ ⋅

( )

( ) , g r r θ ϕ ϕ ⋅ Φ =

Point source

( ) ( )

( , ) ( ) , 2 Mg B u h g r d r u h r λ ϕ θ ϕ ϕ π

Ω

⋅ ′ ′ ′⋅ Φ = Ω = ′ ′ ⋅

∫∫

( , ) 2 M B u h d u h λ θ π

Ω

′ ′ ⋅ ′ = Ω ′ ′ ⋅

∫∫

( , ) 1 1 ( , ) B u h d B u h d u h u h u h

Ω Ω

′ ′ ′ ′ Ω = Ω = ′ ′ ⋅ ⋅ ⋅

∫∫ ∫∫

/(2

),

M u h θ λ π = ⋅ ⋅ ⋅

( 1 7 ) ( 1 6 ) ( 1 8 ) ( 1 9 )

11

Замечание 1.

Area source

( ) ( ) ( )

, , 2

S

m g Q x y d d uh d ξ η ϕ λ ξ η π =

∫∫

( ) ( ) ( )

2 2 1

, , , , y x y arctg d M M x y x η ϕ ξ η ξ η ξ   − = = = − + −   −  

2

1 1 1 2 d r α αµ = + −

1 1 1

, , , cos r r OM r OM r α µ θ = = = = uuuu r

( )

1 1

n n n

P d r α µ

∞ =

=

∑

( 2 0 ) ( 2 1 ) ( 2 2 )

12

( ) ( ) ( ) ( ) ( ) ( )

1 2 3 2 1 1 1 2 2 3

, , , , , ... ( )

n n n S S S S

c Q x y P m g x y d d Q Q Q r c c c mgP d d mgrP d d mgr P d d r r r α µ ξ η ξ η ξ η µ ξ η µ ξ η µ ξ η

∞ =

= = + + + = = + + +

∑∫∫ ∫∫ ∫∫ ∫∫

K

( ) ( ) ( )

g g g ϕ ϕ ϕ ψ ′ ≅ +

, y arctg x ϕ ψ ϕ ϕ = = −

( ) ( )

2

cos x x y y r x y rd rd ξ η ξ η ψ − + − − − = =

( ) ( ) ( )

2 2

1 2 x y g g g r r π ϕ ϕ ϕ ξ η   ′ ≅ + − − −    

( 2 3 ) ( 2 4 ) ( 2 5 )

13

( ) ( ) ( ) ( ) ( )

1 2 2

, , 1 2

S

c x y Q x y m g g g d d r r r π ξ η ϕ ϕ ϕ ξ η ξ η       ′ ′ = + − − + =            

∫∫

( ) ( ) ( ) ( )

1 2 3 3 3

1 2 g g g x g y r r r π ϕ ϕ ϕ ϕ θ θ θ   ′ + −   ′ ′   = + + ( ) ( ) ( )

1 2 3

, , , , , .

S S S

c m d d c m d d c m d d θ ξ η ξ η θ ξ ξ η ξ η θ η ξ η ξ η = =− =−

∫∫ ∫∫ ∫∫

Замечание 2.

( )

g ϕ ′ =

( 2 6 ) ( 2 7 )

14

• Fig. 7. The plan of snow samples selection in

neighbourhoods of Novosibirsk

15

• V. Experimental researches

• VI. The Numerical analysis of the
• bservation data

ФЛУОРЕН

10 20 30 40 50 60 10 20 30 40 50 60 70 км нг/л

ФЕНАНТРЕН

20 40 60 80 100 120 10 20 30 40 50 60 70 км нг/л

НАФТАЛИН

5 10 15 20 25 30 35 40 45 10 20 30 40 50 60 70 км нг/л

БЕНЗ(а)ПИРЕН

5 10 15 20 25 30 35 40 45 10 20 30 40 50 60 70 км нг/л

16

• Fig. 8. PAH concentration in a direction on northeast from

Novosibirsk (2006)

Zn

5 10 15 20 25 10 20 30 40 50 60 70 км мкг/л

Cr

0,5 1 1,5 2 2,5 3 10 20 30 40 50 60 70 км мкг/л

Fe

50 100 150 200 250 10 20 30 40 50 60 70 км мкг/л

Осадок пыли

10 20 30 40 50 60 70 10 20 30 40 50 60 70 км мг/л

17

• Fig. 9. Concentration of heavy metals and dust in northeast

direction from Novosibirsk (2007)

• Fig. 10. The field of benzapyrene aerosol dropouts recovered by

model (11) in neighbourhoods of Novosibirsk (ng/l)

18

• Fig. 11. The plan of samples selection in neighbourhoods
• f Irkutsk

19

• Fig. 12. Levels of beryllium

fallout on a route Irkutsk – Listvyanka for a winter continuance

20

• Fig. 13. Levels of

beryllium fallout on a route Irkutsk – Bayandai for a winter continuance 1994-1995 гг. (а), 1995-1996 гг. (б).

21

• Fig. 14. The plan of samples selection in neighbourhoods
• f Tomsk:

1 – 1974 г., 2 – 1976 г., 1 – 1979 г., 4 – country standing facility

22

Conclusion

• 1. Few-parametric models of reconstruction of concentration

fields of impurity in a neighbourhood area sources are constructed within the limits of statements of inverse problems settings of conduction of impurity in ground and boundary layers of an atmosphere.

• 2. On the basis of these models and in-situ data the

quantitative patterns of creation of local and regional dropouts

• f

aerosols fields for some concrete natural and anthropogenous sources are erected.

• 3. The select of reference points should be spent by methods
• f experiment planning mathematical theory.

23

• 4. Possibility of economical monitoring systems making,
• btaining of state estimates of the long-term of a city air

contamination and definition

• f

emission

• f

the characteristic impurities from its territory is shown.

24

Спасибо за внимание