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Mathematical models for ecological prognosis, design and monitoring V.V. Penenko. Institute of Computational Mathematics and Mathematical Geophysics SD RAS, Novosibirsk. What is the role of atmospheric chemistry in amplifying or damping climate change?. - PowerPoint PPT Presentation
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Institute of Computational Mathematics and Mathematical Geophysics SD RAS,Novosibirsk
Mathematical modelsfor ecological prognosis,
design and monitoringV.V. Penenko
What is the role of atmospheric chemistry in amplifying or damping climate change?
How will human activities transform the dynamical and chemical properties of the future atmosphere?
How will quality of life change?
System organizationof environmental modeling
Models of processes•hydrodynamics•transport and transformation of pollutants
Data basesModels of observations
FunctionalsQuality, observations,restrictions, control, cost,etc.
extended:functional +model as integral identity
Solution of forward problems Solution of adjoint problems
Calculation of sensitivity functionsand variations of functionals
Analysis of sensitivity relationsrisk/vulnerability, observability, sources
System of decision making, design
Identification of parameters,decrease of uncertainties,data assimilation, monitoring
Analysis of the climatic system for constructionof long-term scenarios:
• Extraction of multi- dimentional and multi-component factors from data bases
• Classification of typical situations with respectto main factors
• Investigation of variability
• Formation of “leading” spaces
Approaches and tools
Scenario approach
Models of hydrodynamics
Models of transport and transformation of pollutants (gases and aerosols)
Sensitivity and observability algorithms
Combination of forward and inverse techniques
Joint use of models and data
Nested models and domains
M a t h e m a t i c a l m o d e l
0
rfY),( G
tB ,
000
0 YY, ;
)( tD i s t h e s t a t e f u n c t i o n ,
)(Y tD i s t h e p a r a m e t e r v e c t o r . G i s t h e “ s p a c e ” o p e r a t o r o f t h e m o d e l
A s e t o f m e a s u r e d d a t a m , m o n m
tD ,
mm H )]([
i s t h e m o d e l o f o b s e r v a t i o n s .
,,r, a r e t h e t e r m s d e s c r i b i n gu n c e r t a i n t i e s a n d e r r o r s o f t h ec o r r e s p o n d i n g o b j e c t s .
,01
u
i
i
Fxx
mlvdtdu
,01
v
i
i
Fyy
nludtdv
,0
i
iRT
.0)(
ii Lt
TTBTB ppppp ;/)(;/)(
SBBSB pppppp );1/()(;/)(
)( 1D
)( 2D
Model of hydrodynamics
ya
myxa
nxmn
F ayiaxi
i
sa
a
F aii
va
1
amav
ynau
xmnaL i
iii
)(
Transformation of moisture and pollutants
Gases and aerosols
•interaction with underlying surface •dry and wet deposition•condensation and evaporation•coagulation
Model of atmospheric chemistryModel of aerosol dynamicsModel of moisture transformation•water vapour •cloud water•rain water
Model of transport and transformation
if i s t h e s o u r c e t e r m
iS )( i s t h e t r a n s f o r m a t i o n o p e r a t o r ,};0{ DxttD t .
Boundary and initial conditions
)(x,,)( tii tqR (x))(x, 00 .
0)(x,)(
)gradu(div
iii
iii
rtfSt
Li
{ , , , , , 1, }v c r iq q q c m is the function of pressure
Model of aerosol dynamics
1 1 1 11 10
( ) 1( ) ( , ) ( )
2
q N Ni
ik k km mk m
qq K q q q q q dq
t
)()(),()( qrq
dqqqqKq ii
q N
kkiki
M
1
0 111
),()()( tqQqRqq iiiii 2
2
11 NNi ;,
},),,({ Nitqi 1 -concentration of particles in volume ;, qqq
),( 1qqK - coagulation kernel;),( tqri - rates of condensation and evaporation;),( tqi - coefficients of diffusive change of particles;),( tqRi - removal parameter;),( tqQi -source term;
},,,{,,, Nmkiikikik 1-parameters of collective interaction of particles
Hydrological cycle of atmospheric circulation for studying aerosols
If supersaturation -->condensation
rcv qqq ,, - content of water vapor, cloud water and rain water in respectively
Notations:
RCP - autoconversion of cloud water to rain water (*dt)
RAP - accretion of cloud droplets by rain drops (*dt)
REP - evaporation of rain water(*dt)
CONP - condensation (evaporation)(*dt)
3gm
?0cq ?0cc qq 0RCP
0
0
RA
RC
P
P
Hydrological cycle
RC
calculate
P
REcalculate P ?0rq ?0rq RAcalculateP
0RAP
?cRARC qPP c
RC RA
redistribute q
between P and P?rRE qP
rRE qP 0REP
no no
no
yes
no
yes yes
yes
yes
yes
output
input
no
yes
G e n e r a l f o r m o f f u n c t i o n a l s
KkdDdttF
tD
kkk ,...,,)(x,)()( 1
kF a r e t h e f u n c t i o n s o f t h e g i v e n f o r m ,
dDdtk a r e R a d o n ’ s m e a s u r e s o n tD , )(*
tk D .
Q u a l i t y f u n c t i o n a l s
KkdDdttMtD
kmT
mk ,...,,)(x,)()()( 1
“ M e a s u r e m e n t ” f u n c t i o n a l s
mtmkmk
D
K
k
DdDdttt
x,)x(x)(x,)( 1
R e s t r i c t i o n f u n c t i o n a l s
0 )(x,(,)(x, tNt k d i s t r i b u t i v e c o n s t r a i n t s
dDdttkk
D
kk
t
)(x,))()(()(
Functionals of measurements
dDdttxHMH mm
T
Dmm
t
),()()()( 111
dDdttxM mm
T
D
mm
t
),()( 222
2 1( ) ( ) ( , ) , 1,t
m c o
D
x t dDdt nr
2 2( ) ( ) ( , ) , 1,t
m c o
D
x t dDdt nr
, if 0( )
0, otherwise
A AA
The structure of the source term
K
kkkk kXxtQf
1
1),,()(
)(tQksource power
),( kk Xx source shape
kX reference point of the source
Particular case
))((),( tXxXx kkk
Functionals for assessment of source parameters
tD
aT
a dtdDffMffJ 11111 )()(
dtMQQJ ak
K
k
t
k 122
1 0
2 )(
dtMXXJK
k
takk 13
1 0
2
3
dtMXXdtd
JK
k
takk 14
1 0
2
4 )(
0)(
,)0(
:0
dtXXd
XX
takka
kk
T h e v a r i a t i o n a l f o r m u l a t i o n o f t h e p r o b l e mI n t e g r a l i d e n t i t y
0 )f,Y),(()Y,,( G
tBI
)(),( tt DD ),( ba i s i n n e r p r o d u c t o n ],0[ tDD t .
0)Y,,( I ( e n e r g y b a l a n c e e q u a t i o n )
C o n s t r u c t i o n o f t h e d i s c r e t e a p p r o x i m a t i o n s
)()(~ hk
h htD
hI )Y,,( i s e x t e n d e d f u n c t i o n a l .
S t a t i o n a r y c o n d i t i o n s
)(allfor,~
* ht
hhh
DI
0 ( d i r e c t p r o b l e m )
)(allfor,~
ht
hh
D
0 ( a d j o i n t p r o b l e m s )
E x t e n d e d f u n c t i o n a l f o r t h e m a i n a l g o r i t h m o f i n v e r s e m o d e l i n g
ht
mt D
TD
Thk
h MM r)(r)(.)()(~221150
hDR
TD
Tht
hh MM)(
)()( 4433
htD
hI )Y,,(
)4,1(, iM i a r e t h e w e i g h t m a t r i c e s ,
1,04
1
i
ii a r e t h e w e i g h t c o e f f i c i e n t s ,
, a r e t h e s o l u t i o n s o f t h e d i r e c t a n d a d j o i n t p r o b l e m s
0 )Y,,( hI
The basic algorithm of inverse modeling
0
rfY),(~
ht
hk GB
0
kkT
kT
t
hk dAB
Y),()(~
)),(.)(( 1150 Md Th
kk
0
ttk (x)001
300 tM ka ),(x,
),(x,)r(x, * tMt k1
2 ka M 14
YY
)Y,,(Y
k
hk I
0
Y),(Y),( hGA
t i s t h e a p p r o x i m a t i o n o f t i m e d e r i v a t i v e s
I n i t i a l g u e s s : aa YY,,r )()()( 00000 0
0
)Y,Y,(Y),()( kh
khk I
0
)Y,Y,(
Y kh
k I
The main sensitivity relations
The algorithm for calculation of sensitivity functions
}{ kik are the sensitivity functions}{Y iY are the parameter variations
NiKk ,1,,1
NNNdtdY
k ,,1,
The feed-back relations
Factor analysis ( global scale). Reanalysis 1960-1999
hgt 500, june
West Siberia region
60-105 E, 45-65 N
June, 1960-1999
1960 1970 1980 1990
0.1555
0.156
0.1565
0.157
0.1575
0.158
0.1585
0.159
Eigenvector N1 (97%), June, Western Siberia1960 1970 1980 1990
0.156
0.1565
0.157
0.1575
0.158
0.1585
0.159
0.1595
Eigenvector N1 (97%), June, Eastern Siberia
West Siberia, 97%
Global, 17%
Eigenfunction N1, June, 1960-1999
Novosibirsk
East Siberia Region
90-140 E, 45-65 N
June,1960-1999
1960 1970 1980 1990
0.156
0.1565
0.157
0.1575
0.158
0.1585
0.159
0.1595
Eigenvector N1 (97%), June, Eastern Siberia1960 1970 1980 1990
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
Eigenvector N1 (17%), June, global scale
Global, 17%
East Siberia, 97%
Eigenvectors N1, June, 1960-1999
Irkutsk
. June 16 2003 г
18.06.2003
n
i D
ikiikiy
ikix
t
ccyc
yc
xc
xc
1
[{
dtdcnc
dDdtcfcS iki
nikiiki
t
])(
}]))(([ dtdcqRccudDcc iki
D
iikintiki
t
00
Kk ,1
Simbol denotes variations of corresponding functions,and coefficients at are the sensitivity functions )(k
Y)),(()( hkY
hk grad
Sensitivity relation for estimation of risk/vulnerability and observability
Sensitivity function for estimation of risk/vulnerabilitydomains for Lake Baikal
Conclusion
Combination of• forward and inverse modeling• factor and principle component analysis• sensitivity theory on the base of variational principles
gives the possibility for coordinated solutionof the variety of environmental problems, such as
• diagnosis• prognosis• monitoring (mathematical background)• design