risk and uncertainty assessment

Transcription

1 Optmal forecastng of atmospherc qualty n ndustral regons: rsk and uncertanty assessment Vladmr Penenko Insttute of Computatonal Mathematcs and Mathematcal Geophyscs SD RAS

2 Goal Development of theoretcal background and computatonal technology for envronmental and ecologcal applcatons

3 CONCEPT OF ENVIRONMENTAL MODELING The methodology s based on: control theory, senstvty theory, rsk k and vulnerablty theory, varatonal prncples, combned use of models and observed data, forward and nverse modelng procedures, methodology for descrpton of lnks between regonal and global processes ( ncludng clmatc changes) by means of orthogonal decomposton of functonal spaces for analyss of data bases and phase spaces of fdynamcal systems

4 Basc elements for concept mplementaton: models of processes data and models of measurements adjont problems constrants on parameters and state functons functonals: objectve, qualty, control, restrctons t etc. senstvty relatons for target functonals and constrants feedback equatons for nverse problems

5 Statement of the problem Mathematcal model B G(, Y ) f r t 0 0, Y Y 0 0 ; 0, ( Dt ) s the state functon, Y ( Dt ) s the parameter vector. G s the space operator of the model A set of measured data m, m H ( )] [ m, m on [ H ( )] m s the model of observatons. m D t r,,, are the terms descrbng uncertantes and errors of the correspondng objects. Desred!

6 L Transport and transformaton model c t dv ( c u grad c ) ( S ) dv u 0 contnuty equaton t c { c, 1, m} state functon f source term, D t { 0 t t; x D}, S transformaton operators, ( ) ( ) ( ) ( ) 0, ( ) 0, Граничные и начальные условия ( R) q, (x, t t ) 0 (x, 0) (x). f (x, (, t ) r 0

7 General form of functonals ( k ) F ( k ) k( xtdddt, ) Fk, k, k 1,..., K D t F are the Lpschtz's functons of the gven form, dfferentable, bounded k * are Radon s or Drac s measures on D, t ( ). kdddt k D t Qualty functonals T k( ) ( H( )) mm( H( )) mk( x, t) dddt, K D t Measurement functonals m mk mk t ( ) H ( ) ( x x ) dddt, x D k 1 D t mk Restrcton functonals ( x, t) N, k ( ( x, t) 0 dstrbutve constrants ( ) ( ( ) ( ) ) ( x, t dddt k k k k ) D Dfferentablty n extended sense

8 Extended functonal for constructon of optmal algorthms and uncertantes t assessment h h T T ( ) ( ) 0.5 M r M r 1( 1) m 2( 2 ) k D h t D t h I (,, Y ) h T T h 3 ( M 3 ) 4( M4 ) ) D h R h ( D h t M, ( 1,4) are the weght matrces, 4 0, 1 are the weght coeffcents, 1, are the solutons of the drect and adjont problems generated from h I (, Y, ) Addtve aggregaton of the functonals for decomposton 0 D t

9 Optmal forecastng and desgn Optmalty s meant n the sense that t estmatons of the goal functonals do not depend on the varatons : of the sought functons n the phase spaces of the dynamcs of the physcal system under study of the solutons of correspondng adjont problems that generated by varatonal prncples p of the uncertanty functons of dfferent knds whch explctly ncluded nto the extended functonals

10 The unversal algorthm of forward & nverse modelng h k h B t G (, Y ) f r 0 h k T T ( B t) k A (, Y) k dk 0, k( x) tt 0 h T dk ( k ( ) 0.5 1( M1 )), a M 3 k ( x,0), t 0 r ( x, t) 1 * M 2 k ( x, t), k Y I h Y a M 1 4 k Y (, Y, h A(, Y ) G (, Y ) 0 t s the approxmaton of tme dervatves (0) 0(0) 0 (0) Intal guess: r 0,, Y Y a k ) a

11 Some elements of optmal forecastng and desgn The man senstvty relatons h h k( ) ( k, Y) I (,Y Y, k) Algorthm for calculaton of senstvty functons h k I (,Y Y, k) 0 Y The feed-back relatons dy k, 1, N, N N dt k { k } are the senstvty functons Y { Y Y } are the parameter varatons k 1, K, 1, N 0

12 Real tme equatons of back relatons (, Y) ( ) ( Y) k ks kp N 2 (1) (2) 2 Y 0.5 grad Y Y Y Y kp( ) 1 p 2 p dddt D 1 t Y k(, ), 1, ; (, )/ k, k Y N k Y t Y Y Y Y t h * I (, Y, ) dv grad Y - a pror parameter values Y 1, weght coeffcents ( ) p - matrces of scale coeffcents and weghts Y Y Y Y (1) (2) 1 p 2 p

13 Algorthms of uncertanty t calculaton l based on senstvty analyss and data assmlaton: n model r( x,t ) M ( x,t ), 1 2 * k n ntal state M (x, ), t 1 3 k 0 0 n model parameters and sources 1 1 h M4 k M 4 I (,Y, k ) Y M,( 24, ) are the weght matrces

14 Rsk assessment wth the help of senstvty functons Threshold of safety ntervals s, k 1, k K safe ecologcal condtons s k k Et Estmatons for determnstc dt t case k N 1 k Y Y

15 Rsk estmates for determnstc-stochastc case E( Y) { E E( Y ), ( 1, N) } N E( ) E( Y ) 1 D( ) (D( Y), ) P( ) f( x) dx, x ( x E( x )) 2D( x) 1 f( x) e, 2 D( x) 2 R s s P( )

16 Probablty rsk assessment s ( x E( x)) 2 2 t 2D( x) 2 s 1 2 R e dx e dt ( ), 2D( x) s s ( R ) ( E( x ))/ D(x) Rsk doman R r 1 R s P{ s } Safe range s E( ) D( )

17 Fundamental role of uncertanty functons t ntegraton t of all lltechnology components brngng control nto the system regularzaton of nverse methods targetng g of adaptve montorng cost effectve data assmlaton

18 From rsk assessment to desgn of sustanable development strategy Rsk/vulnerablty assessment Models of processes & data bases + envronment qualty functonals Strategy of sustanable development Models of processes & data bases + superposton of dfferent mult-crtera functonals: envronment qualty, objectve, control, restrctons, etc.

19 Advantage of the approach Consstency of all technology elements Optmalty of numercal schemes based on dscrete-analytcal approxmatons(wthout flux-correcton procedures ) Cost-effectveness of computatonal technology

20 Idea and basc approxmatons Dfferental operators of common knd n the models *( ) 0 x x * * * f L f dx L dx x 1 1 x * x *, x ( ) ( ) 0 1 x A f x x dx x x 1 If 1 * * L 0, x * *, ( ) ( ) 0, 2, 1 x 1 A f x x dx n 1 x then ( x), x x x, 1,2 ntegratng multplers Fundamental analytcal solutons of local adjont problems *(1) *(1) *(2) *(2) 1 1 1, 0, 0, 1, 1, n 1 x x

21 Varatonal prncple for successve schemes r (, Y) L f J j j j j j j j * j j j2 1 1 * 1 t j j j j j r j1 j1 * j j1 j * j j j * j r t j j j j 0.5 1, j t j

22 J j 2 1 Varatonal prncple for parallel schemes r 1 * j L f j j j j j j 1 * 1 t j j j j j r J r j 1 j * j j tj k(, Y) r 1 j21 J r J r h j j j k k(, Y) Fk (, Y) k ( x, t) dd tj j21 j21 D 1 ( xt, 0) gven

23 0.6 Long-term forecast of envronmental rsk for Lake Bakal regon Bratsk Njhne-Angarsk Cheremkhovo Angarsk 0.6 Irkutsk Bakalsk Ulan-Ude Surface layer, October

24 Concluson Algorthms for optmal envronmental forecastng and desgn are proposed: uncertanty calculatons rsk kassessment feed-back relatons The fundamental role of uncertanty s hghlghted

25 Acknowledgements The work s supported by RFBR Grant Presdum of the Russan Academy of Scences Program 16 Department of Mathematcal Scence of RAS Program 1.3.

26 Thank you for your tme!