a study of relations between activity centers of the climatic system and high-risk regions
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A study of relations between activity centers of the climatic system and high-risk regions. Vladimir Penenko & Elena Tsvetova. Goal. Development of theoretical background and computational technology for: revealing and identification of activity centers of the climatic system; - PowerPoint PPT PresentationTRANSCRIPT
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A study of relations between activity centers of the
climatic system and high-risk regions
Vladimir Penenko & Elena Tsvetova
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Goal
Development of theoretical background and computational technology for:
• revealing and identification of activity centers of the climatic system;
• assessment of risk/vulnerability domains;• study of relations between activity centers
and risk domains;• applications to ecology and climate.
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Mathematical background
• Analysis of multi-dimensional vector spaces with the help of orthogonal decomposition;
• Variational principles for joint use of measured data and models;
• Sensitivity theory.
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Multicomponent spatiotemporal bases and factor spaces in orthogonal
decomposition: focuses and applications• data compression ( principle components and
factor bases);
• typification of situations for analysis and modeling ;
• revealing the key factors in data;
• variability studies;
• classifying the processes with respect to informativity of basic functions: climatic scale, interannual scale, weather noises
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• efficient reconstruction of meteo-fields on the base of observation;
• development of a few component models;
• construction of leading phase spaces for deterministic- stochastic models;
• formation of subspaces for long-term climatic and ecological scenarios;
• focus on “ activity centers”, ”hot spots” , and “risk/vulnerability” studies
Multicomponent spatiotemporal bases and factor spaces in orthogonal decomposition:
focuses and applications
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Primary concept and data bases
M e asu redd a ta
R e a n a lys is M o d e lin g re su ts S e n s it iv ityfu n ctio ns
D a ta ba se
Sensitivity studies:forward modeling,inverse modeling,adjoint problems
Computational technology and tools
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Basic idea: Representation of multi-component and multi-dimensional data base
as a set of orthogonal spaces
S eto f p rin c ip a l co m p o ne n ts
S eto f o rtho g on a l sp aces
D a ta ba se
Internal structure of decompositionState 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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Basic algorithm of orthogonal decomposition of linear vector spaces
Main stages of decomposition:
• extraction of principle components;
• construction of main factors;
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Basic algorithm of orthogonal decomposition of linear vector spaces
Realization:• constructing inner scalar product;• generating Gram matrix (GM) with respect to
principle variable;• creating GM elements for inner structure of
decomposition by means of scalar product;• solving eigenvalue and eigenvectors problem for
GM;• assembling large units of factor spaces
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Mathematical model for general outlook and creation of algorithm constructions
0
rfY),(G
tB ,
000
0 YY, ;
)( tD is the state function ,
)(Y tD is the parameter vector. G is the “space” operator of the model A set of measured data m , m on m
tD ,
mm H )]([
is a model of observations. ,,r, are the terms describing
uncertainties and errors of the corresponding objects.
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F u n c t i o n a l s o f g e n e r a l f o r mf o r c o n s t r u c t i o n o f s e n s i t i v i t y a n a l y s i s a n d r i s k a s s e s s m e n t
KkdDdttFtD
kkk ,...,,)(x,)()( 1
kF g i v e n f u n c t i o n s
dDdtk R a d o n ’ s m e a s u r e d o n tD , )(*tk D .
V a r i a t i o n a l d e s c r i p 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 n n e r p r o d u c t i n ],[ tDD t 0 .
0)Y,,( I e n e r g y b a l a n c e c o n d i t i o n
C o n s t r u c t i o n o f 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,,( e x t e n d e d f u n c t i o n a l
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0)Y,Y,(Y),()(
kh
khk I
for the sensitivity functions
Inner products for basic constructions
{ }* * * * 2 *( ) 0( , ) ( ( )/ ) ...
t
t
Q DD
uu vv TT p R HH dDdtj j s aé ù= + + + +ë ûòr r
for the state functions
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jimkDATA
Principle components and Factor analysis
11 nknk ,,
),,( , kmijjimk t
functionsysensitivitqTpHvu ,,,,,,,
Data set
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jimk
j
imk
jk
j
ik
i
mk
m
kk
k
k
D
1 1 1 1 12121 ,),(
jimkjimk tDD
kimimimk SD
Multicomponent inner product
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)( jimk
norm
SS
11
),,( normnormrR 2121
),( 2S
Gram Matrix
nk,, 121
Data preprocessing
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nknknfFa p
fn
pp
norm ,,, 11
nfpaVnk
pk ,, 11
2
,
fn
kkkaar
12121
with restrictions
nk,12,1
Successive minimization
Principle components and EOF
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R
nk ,,, 21
nk ,,,diag 21
nknkjj ,,,, 11
nk 21
Spectral problem
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nfknkjaank
jjk
jkkjk
kk
kkk ,,,,
),( / 11
1
221
nfkVa kk
nk
jjk ,, 1
1
2
Principle components
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normnk
p
pp
aF
1
nkmfm
kkm
,)(1
nkfnk )(
Informative function
Empirical orthogonal functions (EOF)
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Caution !
• Due to huge dimensions of vector spaces and individual vectors of this spaces it is recommended to conserve informative quality of calculations of GM elements
• It needs to provide exact orthogonality of principle components vectors and multi-blocks factor spaces (especially important).
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1960 1970 1980 1990
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
Eigenvector N1 (20,3%), November, global scale
The principle component ( eigenvector N1), November, 1960-1999
20,3%
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1960 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
17%
The principle component ( eigenvector N1), June, 1960-1999
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The main basis vector (EOF N1) for 1950-2002 , HGT500mb, January
The main activity centers in the global atmosphere
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x
y
0 100 200 3000
50
100
150
January 15January 15January 15January 15January 15
x
y
0 100 200 3000
50
100
150
January 1January 1January 1January 1January 1
The main basis vector (EOF N1, 16,06%) for 1950-2002Horizontal velocities at 500 mb
number of eigenvalue
eigenvalue
10 20 30 40 500
1
2
3
4
5
6
7
8
9
10
Informativityof orthogonal spaces
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Revealing the areas of ecological risk Sensitivity function of the atmospheric quality
functional of the zone-receptor is taken as a measure of ecological risk for the receptor to be polluted by the sources distributed on the Earth’s surface in the Northern Hemisphere.
Here are four scenarios. In each scenario the same configuration of the zone-receptor was taken. But each receptor was placed in the different parts of the Northern Hemisphere: in the Far East, Central Asia, North America, and Western Europe ( in some activity centers).
Quality functionals were estimated in the interval 14-24 of April, 1999. Inverse modeling was carried out within the interval 24.04- 23.03.1999 in back time
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1
1
1
5
11E-05
30.0001
50.001
70.01
90.1
111
1
1
1
3
3
3
5
57
9
11E-05
30.0001
50.001
70.01
90.1
111
1
1
1
1
3
3
33
5
5
5
5
7
7
9
9
11E-05
30.0001
50.001
70.01
90.1
111
1
1
1
1
1
3
3
3
3
3
5
5
57
11E-05
30.0001
50.001
70.01
90.1
111
USA Central Asia
Western Europe
Far East
The risk functions for the receptors
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Comparative analysis of the sensitivity functions shows that there are the areas of high potential vulnerability with respect to the pollution from the sources which can be distributed over the Northern Hemisphere .
It is seen, that Far East region and West Europe are examples of such areas of high vulnerability.
In the contrary, the receptors located in North America has got relatively favorable conditions.
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Conclusion•The set of numerical algorithms for multicomponent 4D factor analysis and sensitivity studies is developed for climate and ecology applications
• The orthogonal bases ( principle components and EOFs) are constructed as a result of decomposition of Reanalysis data for 53 years
• The main activity centers in the global atmosphere are revealed.
•The structure of risk domains is demonstrated in dependence on the position of receptors with respect to activity centers.
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Acknowledgements
The work is supported by•RFBR
Grant 04-05-64562•Russian Ministry of Science and Education
Contract № 37.011.11.0009• Russian Academy of Sciences
Program 13Program 14Program 1.3.2
•Siberian Division of Russian Academy of SciencesInterdisciplinary projects NN 130, 131, 137, 138