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;

• assessment of risk/vulnerability domains;• study of relations between activity centers

and risk domains;• applications to ecology and climate.

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.

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

• 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

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

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

Basic algorithm of orthogonal decomposition of linear vector spaces

Main stages of decomposition:

• extraction of principle components;

• construction of main factors;

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

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.

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

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

jimkDATA

Principle components and Factor analysis

11 nknk ,,

),,( , kmijjimk t

functionsysensitivitqTpHvu ,,,,,,,

Data set

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

)( jimk

norm

SS

11

),,( normnormrR 2121

),( 2S

Gram Matrix

nk,, 121

Data preprocessing

nknknfFa p

fn

pp

norm ,,, 11

nfpaVnk

pk ,, 11

2

,

fn

kkkaar

12121

with restrictions

nk,12,1

Successive minimization

Principle components and EOF

R

nk ,,, 21

nk ,,,diag 21

nknkjj ,,,, 11

nk 21

Spectral problem

nfknkjaank

jjk

jkkjk

kk

kkk ,,,,

),( / 11

1

221

nfkVa kk

nk

jjk ,, 1

1

2

Principle components

normnk

p

pp

aF

1

nkmfm

kkm

,)(1

nkfnk )(

Informative function

Empirical orthogonal functions (EOF)

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).

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%

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

The main basis vector (EOF N1) for 1950-2002 , HGT500mb, January

The main activity centers in the global atmosphere

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

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

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

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.

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.

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