orthogonal decomposition methods for inclusion of climatic data into environmental studies
TRANSCRIPT
Od
VIN
a
A
P
K
E
S
C
O
P
a
A
s
1
Trat
0d
e c o l o g i c a l m o d e l l i n g 2 1 7 ( 2 0 0 8 ) 279–291
avai lab le at www.sc iencedi rec t .com
journa l homepage: www.e lsev ier .com/ locate /eco lmodel
rthogonal decomposition methods for inclusion of climaticata into environmental studies
ladimir Penenko ∗, Elena Tsvetovanstitute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of the Russian Academy of Sciences,ovosibirsk 630090, Russia
r t i c l e i n f o
rticle history:
ublished on line 9 July 2008
eywords:
nvironmental modeling
cenario approach
limatic data
rthogonal decomposition
rincipal component and factor
nalysis
ctivity centers in the climatic
ystem
a b s t r a c t
Scenario approach is widely used in numerical modeling for the assessment of current
and prospective states of environmental and ecological systems. The results of modeling
essentially depend on the representation of the underlying hydrodynamics. We suggest a
methodology for a quantitative description of the behavior of a dynamic system for a long
time interval in a compressed, generalized form. Following it, the necessary information,
intended to be used for the construction of scenarios, is extracted from a database containing
measured and/or calculated data on hydrodynamic state functions.
The proposed methodology is a development of ideas of principal component analysis
and factor analysis. Calculations are made with the help of a method of orthogonal decom-
position of functional spaces formed by multi-variate, multi-dimensional state functions
from the database. A two-level data structuring with allowance for given goal criteria is
firstly produced. This provides an efficient realization of the methodology practically with-
out restrictions upon the amount of data and component content of the database. The
targeted structuring is just the element which differs the methodology proposed from the
traditional approaches to data decomposition.
The NCEP/NCAR reanalysis database for 56 years (1950–2005) is used to demonstrate the
possibilities of the methodology. The method of orthogonal decomposition results in the
subspaces which correspond to the processes on different scales: from global climatic pro-
cesses to weather noises. These subspaces serve as informative bases for analysis of the
climatic system behavior. Moreover, the subspaces are key elements for the construction
of deterministic–stochastic scenarios to obtain an atmospheric background for problems of
environment protection and design, ecological risk/vulnerability assessment and control,
etc.
ronmental protection problems considering not only the direct
. Introduction
he idea of this study originates from the analysis of envi-
onmental forecasting and projecting problems, in which thessessment of possible nature changes for a long period ofime has to be made.∗ Corresponding author. Tel.: +7 383 3306152; fax: +7 383 3308783.E-mail address: [email protected] (V. Penenko).
304-3800/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2008.06.004
© 2008 Elsevier B.V. All rights reserved.
A way to solve such problems is the development ofscenario approach. In this case it is natural to formulate envi-
impacts of the specific objects, but also the indirect effectsarising additionally due to climate change. As a matter of fact,the goal of our work is formulated in the following way. A
i n g
280 e c o l o g i c a l m o d e l lmethodology should be created for incorporation of climaticdata and some knowledge about assumed climate changesinto environmental studies. To provide this, the climatic datashould be presented in a compressed and generalized form.A deterministic “main” part extracted from the data under agiven criterion of information completeness is implied here.To separate the scales of physical processes presented inthe data, we propose a version of orthogonal decomposi-tion method suited to the treatment of multi-dimensional,multi-variate fields of data. The ideas of principal componentanalysis (PCA) and factor analysis (FA) are essentially usedhere.
The fundamental and applied aspects of PCA and FA areof interest for many researchers from the different areas ofknowledge. The origins of the methods are related to theend of the 19th and the beginning of the 20th centuries. Thedifferent ways of PCA and FA are actively used in meteorol-ogy and oceanography from the middle of the last century.The systematic description of the main statements and someapplications can be found, for instance, in the monographs ofHarman (1976), Mescherskaya et al. (1970), and Preisendorfer(1988). The voluminous bibliography containing more than athousand of papers with surveys of the history, theoreticalbackground and practical applications are presented there,too.
The limiting parameter for these methods is the dimen-sion of the data analyzed. In practice, these algorithms arevery sensitive to the increase of the dimensions of eigen-vector problems (EVPs). Some existing approaches reducingthe dimensions are discussed in detail by Mescherskaya et al.(1970) and Preisendorfer (1988). Among different algorithms,the eigenvector-partition methods should be mentioned. Insuch algorithms, the data are divided by parts and the totalEVP is replaced by several EVPs of low dimensions. Then thesolution is obtained by constructing the basis from the sepa-rate parts.
The problem of the informative bases construction holdsa priority for many years. Lorenz (1965) used singular vec-tors of the linearized operator of a simple model to study theflow-dependent predictability. An application of biorthogonaldecomposition method to the meteorological fields with theuse of EVP of the linearized forward and adjoint operatorsof atmospheric hydrodynamics was considered by Marchuk(1968). Recently the other approaches for optimal representa-tion of the perturbations have been intensively developed forthe ensemble weather prediction. The singular vectors of thetangent linear operators associated with forecasting modelswere used by Molteni and Palmer (1992), Mureau et al. (1992),Ehrendorfer and Tribbia (1997), Gelaro et al. (1998), and Kim etal. (2004). Toth and Kalnay (1993) applied the bred vectors forgeneration of perturbations.
Our previous experience in these directions was acquired inthe construction and use of orthogonal and biorthogonal basesfor implementation of variational principles in the problemsof atmospheric dynamics (Penenko, 1974, 1975, 1981). To con-struct such bases, a computational methodology was built. In
this methodology, the eigenvectors and singular vectors of thelinearized operators of the main and adjoint problems werecalculated. Lanczos’s algorithms as well as the other itera-tive procedures were applied there. The subspaces obtained2 1 7 ( 2 0 0 8 ) 279–291
by means of PCA and FA were also introduced. The develop-ment of a concept and methods of adjoint sensitivity analysisand variational data assimilation with adjoint problems werethe most important fundamental results of that period.
In this paper we have been developing a methodology oforthogonal decomposition for analysis of the large databaseswhich describe the behavior of the multi-dimensionaldynamic systems. The kernel element of the methodology isthe solution of EVP. It should be noted that our methodol-ogy differs from the above mentioned eigenvector-partitionmethods (Preisendorfer, 1988; Mescherskaya et al., 1970). Weorganize the data in such a way to avoid splitting the EVP andto solve the problem of searching for the main componentswithout loss of information quality. A distinctive elementof our approach is a two-level structuring of the analyzeddatabase. This allows us to solve the decomposition problempractically without limitations on amount of data.
Quantitative information of atmospheric and ocean circu-lation for long time periods can be found in the high capacitydatabases such as reanalysis. For example, information onthe basic state variables of the climatic system and someother characteristics are collected in the NCEP/NCAR reanal-ysis database (Kalnay et al., 1996). In this study, we haveconsidered this database as the description of the concreterealization of the climatic system evolution. And a data sub-set for 56 years has been used here as the climatic background.In the presented example on decomposition of the reanalysisdatabase, a two-level structuring with respect to time scaleshas been made. This gives the possibility of producing thetime-dependent dynamical bases that is convenient for prog-nostic goals. In frames of our approach, the other variants ofstructuring are also possible. To realize the methodology, anefficient computational technology has been developed to beused for climate studies and for organizing the scenarios forenvironment protection and design.
2. Mathematical models, functionals, andvariational principles
To solve the above-mentioned problems, we need to formulatesome goal functionals for short description of the generalizedcharacteristics of the state variables of complicated dynamicsystems. The form of such functionals is conveniently chosenfrom the statements of mathematical models of the processesunder discussion. Without going into details, let us writea model in which dynamics of the atmosphere as well astransport and transformation of impurities are presented. Thestructure of such a model can be given in the operator form:
L(�) ≡ B∂�
∂t+ G(�, Y) − f − r = 0, �0 = �0
0 + �, Y = Y0 + �,
(1)
where �(x,t) is the state vector that belongs to the real vectorspace Q(D ), B is a diagonal matrix, G(�,Y) is a nonlinear matrix
tdifferential operator, f is a source function, Dt = D × [0, t̄], D is adomain of spatial coordinates x, t ∈ [0, t̄] is a time interval. Thevector Y describes the model parameters belonging to a rangeof admissible values of R(Dt), �0 is an initial state, �0
0, Y0 are
g 2 1
aridi
fmtgtgt
˚
HdrctFDttc
ardbtTtsta
˚
I
Hsmovg
(tTitm
e c o l o g i c a l m o d e l l i n
priori estimates of the corresponding objects. The functions, �, and � describe the errors and uncertainties of the model,nitial data, and parameters, respectively. The boundary con-itions depend on the problem formulation. They participate
n the definition of the class of functions Q(Dt).The problems of environment quality diagnosis and
orecasting are of multi-goal character. Therefore, suchethodology of modeling is necessary which can be adapted
o the divers statements and criteria. To provide this, a set ofeneralized characteristics is introduced into the model sys-em. The characteristics for quantitative estimations will beiven as the functionals defined on the set of the state func-ions and parameters:
k(�) =∫
Dt
Fk(�)�k(x, t)dD dt ≡ (Fk, �k),
�k ⊂ Q∗(Dt), k = 1, Kc. (2)
ere Fk(�) are the estimated functions, which are bounded,ifferentiable, and satisfying Lipschitz’s condition withespect to �; �k(x,t) ≥ 0, �k(x,t)dDdt are weight functions andorresponding Radon’s measures (if the values of the func-ions Fk(�) are distributed in space) or Dirac’s measures (if
k(�) are determined on a set of discrete points in the domain
t) (Schwartz, 1967); Q*(Dt) is the space of functions adjointo Q(Dt). The domains of nonzero values of the weight func-ion carriers can be determined as receptor areas in Dt whoseonfigurations are prescribed as input parameters in Eq. (2).
The required estimates of the functionals and their vari-tions are obtained with the help of the adjoint sensitivityelations for the functionals (2). The spatial and temporalynamics of these relations in Q(Dt) and R(Dt) is expressedy means of the sensitivity functions (SFs) of analyzed func-ionals with respect to variations of the model parameters.o jointly analyze data, models, and SFs, the agreed defini-ions of “energy” inner products in corresponding functionalpaces have to be used. The appropriate algorithmic construc-ions can be built on the base of a variational principle forssessment of functionals and models (Penenko, 1981):
˜ hk(�) ≡ ˚h
k(�) + [Ih(�, Y, �∗)]Dh
t, (3)
(�, Y, �∗) ≡(
B∂�
∂t+ G(�, Y) − f − r, �∗
)= 0. (4)
ere �* ∈ Q*(Dt) are auxiliary functions determined by thepecific features of the variational principle; ˜̊ h
k(�) are the aug-
ented functionals taking into account the goal functionalsf the form (2) and the description the mathematical model inariational form; the superscript h marks the discrete analo-ous of corresponding objects.
The integral identity (4) is a variational form of the model1). The functional is chosen so that the relation (4) becomeshe equation of the total energy balance of the system if �* = �.
he required algorithmic structures and the definitions ofnner products are obtained on the basis of the integral iden-ity (4) and the functional (3). For instance, if the quasi-static
odel of hydrodynamics is taken, the energy inner product
7 ( 2 0 0 8 ) 279–291 281
for the state functions can be chosen in the form:
(�, �∗)Q(Dt ) =∫
Dt
{uu∗ + vv∗ + �0
[TT∗ +
((�(p)R2
)HH∗
]}dD dt
+nf∑i=1
∫Dt
ˇiϕiϕ∗i dD dt, (5)
where � = (u, v, T, H, (ϕi, i = 1, nf )) is the vector-function of thestate variables, u and v are the horizontal components of thevelocity vector, T is temperature, H is geopotential height, �i
are the components of hydrolodical cycle, admixtures in thegaseous state, and aerosols, nf is the total number of sub-stances, R is the universal gas constant, and �0, �(p) and ˇi,are weight factors. The detailed description of the variationalform of models by means of the integral identity can be foundin Penenko (1981) and Penenko and Tsvetova (1999).
To study sensitivity functions, let us introduce an innerproduct agreed with Eqs. (2)–(5) and generated by the right sideof the sensitivity relations for the functionals ˚k(�) (Penenko,1981):
ı˚hk(�) ≡ (gradY˚h
k(�), ıY) ≡ ∂
∂˛Ih(�, Y + ˛ıY, �∗
k)|˛=0, (6)
ıY = � gradY˚hk(�), k = 1, Kc. (7)
Here ˛ and � are real parameters; ıY are variations of the modelparameter vector that are chosen to be proportional to SFs; �
is the solution to the direct problem (1) with prescribed val-ues of the set of parameters Y; �∗
kare solutions to the adjoint
problems generated by variational principles to estimate thevariations of the augmented functionals ˜̊ h
k(�) (3). The deriva-
tive in Eq. (6) is considered in Gateaux sense (Lévy, 1951). Forconvenience, the initial data and uncertainty functions areincluded into the number of components of the parametervector Y. The structure of the phase space of SFs defined bythe right side of Eq. (7), is determined by the structure anddimensions of the components of parameter vector and bythe form of integral identity functional.
Thus the functionals of four main types participate in thevariational principle. They are defined as the inner products(2)–(7). For the aims of orthogonal decomposition, the func-tionals of energy type (5) and (6) are directly used. The first ofthem defines the metrics and norm in the space of state vari-ables. The second one introduces the metrics and norm in thephase spaces of SFs.
3. Methodology of database analysis
To analyze a high dimension database describing the evo-lution of the processes examined, we apply the methods oforthogonal decomposition. The starting point to this aim isthe definition of the database structure. The form of the totalenergy inner product is taken into account. This gives the met-
rics and type of the functionals in the space of analyzed fieldswhich, in terms of Eq. (5) and (7), are multi-dimensional andmany-component aggregates of high dimension. The meteo-rological fields can be the results of observations of the nature
i n g
282 e c o l o g i c a l m o d e l lprocesses, as well as the results of model runs. Although themodel might be nonlinear, we assume that the calculatedfields belong to the linear vector spaces.
3.1. Database structuring
So, let these fields be the vector-matrix objects in the reallinear spaces. The set of vectors is defined as
{�(x, t, Y) ∈ Q(Dt); (x, t) ∈ Dt; Y(x, t) ∈ R(Dt)}. (8)
For organizing the algorithms, we arrange the database in theform of a block matrix. To describe the blocks, two groups ofindependent variables-indices are introduced. The first groupdescribes the governing (external) structure of the data: thenumber of blocks and their order in the overall hierarchy. Thesecond group defines the numeration and location of the com-ponents inside each block. Thus, the vectors can be presentedin the block form as
� = {�i(k)}, �i(k) ∈ RN, i = 1, n, n ≥ 1, k ∈ K, (9)
where n is the number of blocks in the external structure; Kis the set of values of the multi-indices k of the componentsin the internal structure of each block. The total number ofelements in the internal structure is denoted by N. Hereinafter,all operations are performed in the real vector spaces RN andRn with corresponding inner products.
It should be noted that all algorithms are universal, and thespecific features of each problem are determined by the struc-ture of the data matrix and by the form of the inner products.Let the energy inner product be
(�i, �j) = 〈�i(k), C�j(k)〉= 〈C1/2�i(k), C1/2�j(k)〉, i, j = 1, n, k ∈ K. (10)
Here C is the diagonal N × N-matrix whose elements containthe scaling factors and the volume metrics in the discrete rep-resentation of the functionals (5)–(7). It should be stressed thatthe calculation of inner product is the key element which isresponsible for the accuracy of solution to the decompositionproblem.
Now we introduce the transformation of the state variablesof the form Zi = C1/2�i so that the components of vectors inthe new form should have the identical physical dimensionand energy property. The dimensions of the inner product (10)and norms should remain unchanged. Finally, the databasefor solving the problem can be presented as (n × N)-matrixZ = [Zi], i = 1, n, where n is the number of vector columns.Each column contains the complete internal structure withtotal number of components being equal to N. The quanti-ties n, N, and the content of the set of multi-indices K are theinput parameters for structuring the database Z and formingthe inner product (10). The obtained vectors are centered withrespect to a given center vector which is introduced as input
parameter of the algorithm. In particular, the vector of thesampling mean can be used as the center. Then the vectorsare reduced to the unit length in the norm defined by the innerproduct.2 1 7 ( 2 0 0 8 ) 279–291
The data matrix Z can be considered in two ways: as theset of n vector-columns {Zi, i = 1, n} from RN, and as theset of N vector-rows {Vq, q = 1, N} from Rn. Hence, we canuse two Gram matrices: the (n × n)-matrix � = ZTZ = {�ij =(Zi, Zj), i, j = 1, n} and the (N × N)-matrix M = ZZT = {Mij =(Vi, Vj), i, j = 1, N} (the superscript T indicates transposition).The Gram matrices are real, symmetric, and positive semi-definite. It is known that r ≡ rank (� ) = rank (M) ≤ min (n,N). Tomake the algorithm efficient, we organize the source databaseso that n should be much less than N. And all the calculationsare made with the (n × n)-matrix � . In this case, the value ofthe parameter n can be limited only by the capabilities of theprocedures used for solving the EVP for the matrix � . The sizeN of the vectors belonging to the internal structure can be aslarge as desired.
3.2. Quadratic forms and decomposition methods
Construction of multi-dimensional orthogonal spaces is a fun-damental problem of functional analysis and linear algebra.In addition to Gram matrices, the effective tools for studyingthe linear transformations of vector fields and the corre-sponding databases are bilinear and quadratic forms (Courantand Hilbert, 1953; Gantmacher, 1959). Let us introduce thequadratic form joining the matrices Z and � :
S(V, Z) ≡ (ZV)T(ZV) = VTZTZV = VT� V. (11)
This form is defined on the space of vectors V ∈ Rn, whichsatisfy the condition VTV = 1. In essence, the quadratic formcharacterizes the scattering of the sample with respect to thegiven center vector (Wilks, 1962). The Gram matrix � is alsoreferred to as scatter matrix.
Then using the methods for studying the extreme prop-erties of the form S(V,Z), we perform the orthogonaldecomposition of the space of vectors forming the matrix Zand belonging to RN × Rn. Omitting the description of inter-mediate stages, we obtain the solution to the problem as a setof orthogonal subspaces
{p, vp ∈ Rn, �p ∈ RN,
vTp vq = pıpq, �T
p �q = ıpq, p, q = 1, n
}. (12)
Here p ≥ 0; vp are the eigenvalues and eigenvectors of the(n × n) Gram matrix � ; ıpq is Kronecker delta function; �p arethe orthogonal basis vectors-subspaces (OBV) obtained by pro-jection of the vector-rows {Vi, i = 1, N} of the matrix Z onto thebasis of principal components (PCs) {vp, p = 1, n}. The internalstructure of the vectors �p is the same as that of the vector-columns {Zi, i = 1, n} of the matrix Z in the space RN.
The value of p can be taken as an information measure ofthe pair {vp, �p}. An information index of the pair is definedas p = (p/
∑n
i=1i). The smallest eigenvalue n ≥ 0 is usuallyreferred to as the measure of linear independence of the vectorsystem (9). It does not grow if the number of vectors n in the
system increases. The largest eigenvalue 1 characterizing themaximum scale of perturbations, does not decrease with theincrease of n. Detailed studies of the vector systems and thecorresponding Gram matrices can be found in (Taldykin, 1982).
g 2 1
3
Ls
3Tss
�
Tidsom
3Tv
�
Hp
3To
�
weoit
3oTst
Z
Uc
e c o l o g i c a l m o d e l l i n
.3. Basic algorithms
et us take a brief look at a set of algorithms for analysis andynthesis of databases.
.3.1. Calculation of the PC-basis in the space Rn
he eigenproblem is solved for the (n × n) Gram matrix withpecific normalization of the eigenvectors to account for thecales of perturbations
v = v, vTp vq = ıpqp, vp = {vp(ˇ), ˇ = 1, n},
tr(� )=n∑
p=1
�pp=n∑
p=1
p, p, q=1, n, 1 > 2 . . . > n ≥ 0.
(13)
he eigenvectors vp are normalized and arranged in descend-ng order as the corresponding eigenvalues p. In theseecompositions, only the vectors for which p > 0 are con-idered. The eigenvectors vp constitute the orthogonal basisf the PCs in the space Rn containing the vector-rows of theatrix Z.
.3.2. Formation of the basis subspaces in RN
he structure of the subspaces is similar to that of the originalectors Zj. The algorithm for calculation the OBVs is follows:
p(k) = −1p
n∑j=1
vp(j)Zj(k), (�p, �q)=ıpq, p, q=1, n, k ∈ K.
(14)
ere Zj(k) ∈ RN are the vector-columns of the matrix Z, thearameter k ∈ K is the multi-index stated above.
.3.3. Composite subspaces formed by the basis OBVshese objects are calculated as weighted linear combinationsf the OBVs (14) in the form:
m(k) =m∑
i=1
�̃i
(i
1
)�i(k), 1 ≤ m ≤ n, k ∈ K, (15)
here �̃i, 1 ≤ i ≤ m are some deterministic or stochastic param-ters. The factor subspaces �m(k) are constructed on the basef the leading OBVs, in accordance with the scales of their
nformation significance with respect to the dominating vec-or.
.3.4. Synthesis of subspaces with filtering the processesf chosen scaleshe composition of the state vectors for the specific modelingcenarios is made with the use of OBVs with filtering underhe given information completeness criteria
(˛)∑̨
i(k) =
p=1
�p(k)vp(i), k ∈ K. (16)
sually the filtering is fulfilled at ˛ < n by excluding theomponents corresponding to the small eigenvalues. The
7 ( 2 0 0 8 ) 279–291 283
vectors of the initial set Zi from Eq. (9) are obtained if˛ = n.
3.3.5. Energy criteria for revealing the areas of theprescribed level of significanceThe criteria are useful for revealing the regions of increasedrisk and vulnerability, as well as for detection of the centersof energy activity in the climatic system. Following the defini-tions from Buizza and Montani (1999), let us denote the energyof the vector � in the subdomain identified by the multi-indexk ∈ K in Eq. (9) as �c(�,k) = [�,�]k and define
A ≡ �c(�, k0) = maxk ∈ K
�c(�, k). (17)
Here � is one of the vectors from Eqs. (8), (9), and (14) or fromtheir linear combinations of the type (15) and (16) and k0 isthe multi-index of localization of the maximum value of thefunction. Using these definitions, the region in which
�c(�, k) ≥ ˛A, k ∈ K˛0(�) ⊂ K (18)
is calculated. Here ˛, (0 < ˛ < 1) is a parameter chosen forextracting the region having the given relative level of energysignificance. If the inner product (5) is chosen, the analysis ofthe region pattern K˛
0(�) in dependence on the parameter ˛ isused for assessment of the energy active regions. If the innerproduct (6) and (7) is taken, the regions of increased sensi-tivity/risk/vulnerability in the global climatic system or in itsparts can be revealed.
4. Information content and physicalmeaning of orthogonal basis spaces
The decomposition algorithms and their results are used invarious practical applications. The specific interpretation ofthe triplets {q, vq, �q, q = 1, n} obtained from the system oforthogonal decomposition (12) depends on many factors suchas: the goal of the study, the content of the database, theway of data structuring, and the concrete form of the energyinner product. Following the terms accepted in PCA and FA(Harman, 1976; Preisendorfer, 1988), the set of the eigenvec-tors vq ∈ Rn can be conditionally related to the PCs, while theOBVs, �q ∈ RN, can be considered as the main factors which area generalized version of the empirical orthogonal functions(EOFs) for multi-dimensional and multi-component spaces ofthe complicated inner structure which is introduced by thegiven energy inner product.
The trace of Gram matrix is equal to the value of the totalvariance �2 of the source set Z. Owing to normalization ofthe vector-deviations to the unit length, we have: �qq = 1, andtr(� ) = n. The part of the total variance equal to 1 falls on eachvector block Zi(x,t) of the initial set.
The physical dimensions of the elements of � coincide withthose of the eigenvalues q. They are defined by the dimensionof “energy” which we imply in the functionals of the inner
product. The dimensions of the components of the eigenvec-tors vq follow the relations (12) and (13) and are equal to thoseof the square root of energy or, in terms of standard deviation,�q =√q.
i n g
284 e c o l o g i c a l m o d e l lThe value of q is equal to the fraction of the total vari-ance relating to the vector vq. As
∑n
ˇ=1v2q(ˇ) = q from Eq. (13),
each component vq(ˇ) is equal to the standard deviation thatdefines the fraction of the input of Zˇ(x,t) into the OBVs �q(x,t)from Eq. (14).
According to Eq. (16), the components vq(ˇ) of the vector vq
define the values of the projections of each vector-column ofthe matrix onto the OBV-basis. In other words, the vectors vq
characterize the variability of the initial sampling with respectto the constructed OBV-system. Owing to Eqs. (12) and (14),the basis functions �q(x,t) are dimensionless. They describethe time–space distribution of the total variance q in frac-tions of the standard deviation �q over the region Dh
t and,hence, they characterize the relative measure of variability ofthe processes at each point of the grid domain.
5. An example of constructing orthogonalsubspaces with the use of the reanalysisdatabase
The methodology described above is applied for studying thelong-term behavior of the global climatic system in terms oforthogonal decomposition. To this end we use the NCEP/NCARreanalysis database (Kalnay et al., 1996) which is a well-structured universal-purpose information system containingthe basic set of atmospheric characteristics. This database isactively used for different goals. As for orthogonal decomposi-tion, for instance, the monthly mean EOFs have been recentlycomputed for studies of multiple regimes and low-frequencyoscillations in the Northern Hemisphere’s zonal-mean flow(Kravtsov et al., 2006).
In our study the data for the period from 1950 to 2005 (56years) are considered. The chosen time interval is greater thanthe period of 30 years normally used in climatic estimates(IPCC, 2001). By example of these data, we make the problemformulation more definite and identify the main elements ofthe technique used.
To organize the study and to ensure efficient operation ofthe algorithms, we form a database as a subset of the reanal-ysis database and define the matrix Z and the quadratic form(11) in a suitable manner. The physical content of the com-ponents is selected in accordance with the goal functionals(2), inner products (5)–(7), and functionals (10) for calculationof the Gram matrix elements. Having in mind the acceptedtwo-level structuring (8)–(10), we introduce two scales in time:external and internal. The external scale has the total durationof 56 years (n = 56) with time step of one year. As for inter-nal scale, there are many variants beginning with a few daysto a year. From the formal point of view the annual intervalis the simplest one. But the loss of information quality mayarise in the calculation of inner products (10) due to sum-ming the huge number of data. Taking this fact into accountand realizing our wishing to study the seasonal behavior ofthe climatic system, we choose a variant in which the annualinterval is divided into 12 parts of the month’s length (Jan-
uary, February, etc.) with 12-h time step within a month. Thus,the data for the same name month for 56 year is separatelyconsidered. Splitting data in time by monthly intervals pro-vides a compromise between information comprehensiveness2 1 7 ( 2 0 0 8 ) 279–291
and efficiency of computations. Moreover, it is convenient forthe content analysis and organization of the modeling sce-narios. Retention of time dependence in the inner structureof the input vectors and, hence, in the basis subspaces, givesa new facility and allows the time-dependent scenarios to bedesigned.
Thus, we have seven parameters to characterize each ele-ment of the database (8) and (9). The first three of them arethe ordinal numbers of the year, the month, the data field interms of the physical content. The last four parameters definethe space–time coordinates of data on the globe or its part:latitude, longitude, height, and time.
In terms of Eq. (9), the year number is the index-parameteri = 1, n, where n is the number of years in the database. Thelast five parameters set the internal structure and arrange themulti-index k in terms of Eqs. (9) and (10). The name (number)of the month is used as an input (dumb) parameter of thealgorithm. It should be noted that the efficient computationsmight be organized for all the months’ names in parallel.
Now the internal structure will be described. The size ofthe vector-columns of the matrix Z is defined by the followingparameters. In terms of time, it is 2m (m is the number of daysin a month, the fields are taken twice a day). The number ofgrid points in space, 144 × 72 × l, presents the global domain inspherical coordinates with resolution of 2.5◦ × 2.5◦ for the hor-izontal variables multiplied by the number of vertical levels,l ≥ 1, parametrically given. The number of different physicalcomponents (≥1) is also introduced parametrically.
As the result of decomposition, the basis set of 56 elements(subspaces) is built. In all, there are 12 such sets of the factorspaces in accordance with the number of months in a year.
Thus, the long-term variability of the elements of the cli-matic system is presented by the structure which has twoscales in time and uses the orthogonal spaces of two types.The first one is presented by the PCs, i.e. by the vectors vq thatcharacterize the year-to-year variability in the range of theexternal time scale. Their components vq(ˇ) are the functionsof the current number of the year ˇ in the aggregate (9). Thesecond type consists of the multi-variate OBVs �q(x,t) depen-dent on the variables (x, t) ∈ Dh
t , where t is a local time withinthe monthly interval.
First of all, to analyze the long-term climatic tendencies,it is worth considering the predominant PCs and OBVs cor-responding to q > 1. The information content of the PCs andOBVs corresponding to q ≈ 1 is equal to that of the normalizeddeviation vectors of the input aggregate (9). The PCs and OBVscorresponding to q < 1 can be considered as climatic and/orweather noises in the total information flow. They might befiltered out using Eq. (16) and excluding the terms in the sumwhich are associated with the small values of q. In essence,each OBV is a multi-component subspace, the elements ofwhich depend on the space coordinates and time within themonth.
The merits of representation of the functions (9) as theorthogonal subspaces set are obvious. This is the usability ofprojecting the state functions onto OBVs and PCs and con-
structing the new subspaces with desired properties by meansof linear combinations (15) and (16). As the spaces are orthog-onal, the total variance of the entire set is the sum of variancesof all subspaces without interactions among them. Therefore,
g 2 1 7 ( 2 0 0 8 ) 279–291 285
ar
da{dctso
adna
6
SictTaaisttbZ
aatcpe
qassP
mmt
yt�
nirftO
13 informative PCs and OBVs accumulate more than 50% ofthe total variance. Exactly these components can serve as thebase for the climatic scale behavior description. The horizon-tal line in Fig. 2 separates them from the other eigenvalues
e c o l o g i c a l m o d e l l i n
s it was mentioned above, the designed subspaces can beanged with respect to the level of their significance.
The simultaneous analysis of the (PC, OBV)-pairs forifferent months shows the year-to-year and seasonal vari-bility within the long-term interval. Analyzing the systemvq, �q(x, t), q = 1, n} jointly, one can notice that, in accor-ance with Eqs. (17) and (18), there are domains in which theomponents of OBVs have the maximal modules. They seemo pretend being the centers of energy activity in the climaticystem. Remember, that here the energy is meant in the sensef the inner product (10).
Thus, using the results of orthogonal decomposition, bothquantitative and qualitative analyses of the complicated
ynamic system behavior can be performed. Besides, the sce-arios based on the prescribed criteria for solving diagnosticnd prognostic problems can be deliberately formed.
. Climatic data analysis
ince the middle of the last century, the research has beenntensively performed on revealing and studying the climaticenters of activity in the atmosphere and ocean to definehe most significant patterns having the prediction character.hese regions on the globe where the relatively stable char-cteristics of the general circulation of the atmosphere exist,re usually referred to as such activity centers. They are, fornstance, the quasi-stationary zones of high and low atmo-pheric pressure. Firstly Teisserence de Bort (1884) definedhe concept on activity centers of the atmosphere. Sincehen, some areas of the sufficiently stable localizations haveeen described in the synoptic meteorology (see, for example,verev, 1968).
In the 80th years of the last century G. Marchuk consideredconcept of the energetically active zones of the ocean whichre characterized by increased energy releases from the oceano the atmosphere (Marchuk et al., 1984). In frames of the con-ept, the studies of the activity zones for the goals of weatherrediction were fulfilled with the use of the theory of adjointquations (Marchuk, 1974).
The methodology proposed here gives the possibility touantitatively examine the hypotheses on existence of energyctive zones in the atmosphere and ocean and to study theirpatial–temporal variability as manifestation of the dynamicystem evolution in the phase spaces with the basis ofCs–OBVs.
Now we shortly present some results of analysis of the cli-atic system behavior for 56 years (1950–2005). For the longestonths of 31 days, as many as 3472 fields for each state func-
ion have been used.The time behavior of the leading PC-1 for January of many
ears is shown in Fig. 1. It is calculated for the 500-hPa geopo-ential height. Circles show the values (standard deviation,
1 = √1) of the PC-1 components corresponding to the year
umber. The scale of the vertical axis is defined by normal-zation from Eq. (13). The components of the PC-1 reflect the
elative input of each generalized January field into the leadingactor space OBV-1. The years in which the absolute values ofhe component PC-1 are greater, give the greater input in theBV-1.Fig. 1 – Time variation of the leading PC-1 for the 500-hPageopotential height for January (1950–2005).
In Fig. 2, the eigenvalues (13) of the Gram matrix � areshown in descending order. The vertical scale is defined bythe value of the Gram matrix trace which presents the totalvariance equal to 56. The logarithmic scale of the verticalaxis is chosen in Fig. 2. The first eigenvalue is seen to bestrictly dominated. It gives the input of 26.34% into the totalvariance of the whole set. The leading PSs and OBVs withnumbers from 1 to 13 hold the information in the climaticsense: the corresponding eigenvalues are greater than 1. These
Fig. 2 – Eigenvalues of the Gram matrix for the 500-hPageopotential height for January (1950–2005).
286 e c o l o g i c a l m o d e l l i n g 2 1 7 ( 2 0 0 8 ) 279–291
asisto 00
Fig. 3 – One of the 62 fragments of the leading OBV-1. The bgeopotential height for 56 years. The fragment corresponds
mentioned above as presenting the climatic and /or weathernoises.
In Figs. 3–6, the fragments of the leading OBVs correspond-ing to the maximum eigenvalues of the Gram matrix for the500-hPa geopotential height for 56 years are presented for fourmonths, one for each season. Each vector OBV presents thesubspace that consists from 2m such fields in time, where mis the number of days in a month. Remember that the OBVsare normalized as in Eq. (14). The fragments are taken at 0000UTC on 15 January (Fig. 3), 15 April (Fig. 4), 15 July (Fig. 5), and15 October (Fig. 6), respectively. In Figs. 3–7, the vertical axisshows the function of latitude so that the South Pole is 0◦
and the North Pole is 180◦. The horizontal axis is the longi-
tude starting from the Greenwich meridian. The continents’configurations are marked by the thick lines. Regions of localmaxima and minima can be interpreted as centers of activ-ity or energy active regions. Analyzing the temporal behaviorFig. 4 – The same as in Fig. 3 b
is constructed on reanalysis data of the 500-hPa00 UTC 15 January.
of the OBVs-1 for different components in different months,we can conclude the following. The space structures in theOBVs-1 are of the global scale. Their behavior can be classi-fied as quasi-stationary because their geographical locationschange insignificantly within the month. In the high latitudesof the Northern Hemisphere, in January, and in the SouthHemisphere, in June, they are more stable than in the tran-sition periods. The intensity of perturbations and the scalesof domains of their carriers decrease in autumn and spring.The contrasts in the character of the circulation mechanismsin the high latitudes of the both hemispheres become less pro-nounced. With the increase of OBV’s number, the amplitudesof perturbations decrease and the patterns of the small-scale
structure manifest themselves (not shown here).In Fig. 7, one of the 62 fragments in time of OBV-1 for Jan-uary (1950–2005) for the velocity component is demonstrated.It corresponds to the fields of the horizontal components of
ut for 0000 UTC 15 April.
e c o l o g i c a l m o d e l l i n g 2 1 7 ( 2 0 0 8 ) 279–291 287
ig. 3
tlcOrtcOih
tottot
Fig. 5 – The same as in F
he wind velocity at the level of 500 hPa. In January, the circu-ation above the continents in the Northern Hemisphere areaused by the competition between the Pacific and Atlanticceans. The circulation structures over the Eurasia are sepa-
ated near the meridian of 100◦E (compare with Fig. 3). In Julyhe summer type of circulation is characterized by latitudinalharacter (Fig. 5). It should be noted that the leading spaceBV-1 for velocities is quasi-stationary during the month as
t is for the geopotential height. The main circulation systemsave the sufficiently stable localization in the space as well.
Fig. 8 shows the seasonal behavior of the information index
1 (%) of the first leading pair (PC-1 and OBV-1) with respecto the total variance of the systems presenting the subsetsf reanalysis data for each of 12 months. In this case it is
he energy of perturbations in the sense of Eq. (5). It is seenhat the seasonal behavior has two local maxima. The greaterne is related to the winter month, the lesser one is relatedo the summer month. As the seasons in the Southern andFig. 6 – The same as in Fig. 3 bu
but for 0000 UTC 15 July.
the Northern hemispheres are opposite, the difference in theintensity of maxima can be explained by the fact that theland–ocean contrasts manifest themselves more brightly inthe Northern hemisphere.
The same characteristics of decomposition were early cal-culated for the period of 40 years (1960–1999) (Penenko andTsvetova, 2002, Penenko and Tsvetova, 2003). Comparing theconfigurations of the first leading vectors for the periods of40 years with those of 56 years, we can conclude that thenumber of activity centers keeps to be the same. Their geo-graphical locations did not change, either. Thus, there is noessential reconstruction in the leading basis vector when thetotal number of the input vectors is significantly increased.
Summarizing, we can repeat that the leading OBVs have
a relatively high information significance and they are quasi-stationary. More over, 10–15 of them contain more than 50%of the total variance of the system. Hence, it can be concludedthat the system possesses a long-term deterministic mem-t for 0000 UTC 15 October.
288 e c o l o g i c a l m o d e l l i n g 2 1 7 ( 2 0 0 8 ) 279–291
asisent
Fig. 7 – One of the 62 fragments of the leading OBV-1. The bcomponents of wind at 500-hPa level for 56 years. The fragm
ory. This fact may serve as an argument for the validity ofthe hypothesis of relative stability of the climatic backgroundwhich is described by the leading PCs and OBVs. It shouldbe mentioned that this conclusion is made when the con-crete database is used and the concrete structuring is chosen.Nevertheless, the designed bases seem to be useful for theconstruction of guiding phase spaces for the goals of long-term environmental prediction and ecology studies.
7. Principles of prognostic scenariosorganization for environmental studies
We propose that the climatic information should be taken intoaccount for environmental forecasting by means of the specialscenarios. In them, the results of orthogonal decompositionare involved for calculation of the hydrodynamic background
along which the processes of transport and transformationof substances flow. The analysis of the main factors in theglobal and regional scales fulfilled jointly with the sensitivityanalysis of the environment quality functionals, shows thatFig. 8 – Seasonal variability of the information index �1 (%)of the leading pair (PC-1 and OBV-1).
constructed on reanalysis data for the horizontalcorresponds to 0000 UTC 15 January.
the adequate simulation of the regional processes is impos-sible without their interconnection with global processes. Forinstance, this fact becomes strictly apparent while assessingthe areas of ecological risk/ vulnerability for the protectedregion with respect to harmful impact (Penenko and Tsvetova,2005, 2007a,b). Hence, the modeling technology is organizedon the principles of combining the different scales models andof decomposing the state function on the background and per-turbations. The essence of the technology is as follows. Newelements named the guiding phase spaces are introduced intothe model. These are the multi-component fields of the space-time structure. They provide the description of the backgroundstructure with desired degree of information completenesswith respect to the global climatic system state. The com-position of the components depends on the form and thecontent of the state function of the model. These fields areused in the technology by means of the assimilation proce-dures. The current state, generated by the model, is calculatedwith allowance for the background processes prescribed by theleading spaces.
7.1. Deterministic and deterministic–stochasticguiding spaces
As stated above, for construction of the guiding phase spaceswhich participate in the formation of the hydrodynamicsbackground, the method of orthogonal decomposition is used.Let us define the guiding spaces as the sum of two constructiveelements:
�d(x, t) = �0d(x, t) + �1
d(x, t), (x, t) ∈ Dhdt . (19)
Here �0d(x, t) is a large-scale part expressed by the linear com-
bination (16) of the leading basis subspaces in frames oforthogonal decomposition (12); �1(x, t) is the space built on the
d
components of the smaller scales; Dhdt is the given set of points
in Dt. The constituent �1d(x, t) may be deterministic–stochastic
in the range of variability of the corresponding parametersfrom the database. The relations (15) and (16) are intended for
g 2 1
smtt
tfsrsi
7f
Lgs
Z
Tf
m
HD
c
F
Htist
bcb
mwaobttestl
e c o l o g i c a l m o d e l l i n
uch constructions. For calculation of the stochastic part oneay apply the probabilistic density functions of the state vec-
ors defined in the phase spaces spanned by the leading set ofhe PCs and OBVs.
The use of the orthogonal decomposition results allowshe scenarios for the typical and extreme situations to beormed. The quantitative measure of the basis informationignificance gives the possibility to classify the processes withespect to the scales of perturbations as well as to identify thepatial–temporal domains of perturbations with respect to thentensity of their development.
.2. Forming the leading phase space with allowanceor observation data on the subdomain
et a set of the measured values Zm(x,�) of the state vector beiven in a subdomain Dm
� ⊂ Dt. Then for approximate recon-truction of the vector Z in Dt, we use the algorithm
(x, t) =na∑
p=1
amp �p(x, t), (x, t) ∈ Dt, na ≤ n. (20)
he coefficients of expansion a = {amp , p = 1, na} can be found
rom the condition
in〈am
p 〉
∥∥∥∥∥Zm(x, �) −na∑
p=1
amp �p(x, �)
∥∥∥∥∥2
Dm�
, (x, �) ∈ Dm� , na ≤ n. (21)
ere || · ||Dm�
denotes the Euclidean norm of vectors, given onm� , with a positive diagonal weight matrix Wm. Finally, theoefficients are obtained as
a = (� m)−1 Fm, � m = {� mpq = (�p, Wm�q)Dm
�, p, q = 1, na},
m =na∑
p=1
(�p, WmZm)Dm�
.
ere (., .)Dm�
is the inner product of vectors defined on Dm� with
he weight Wm. It introduces the norm for Eq. (21), and � m
s a positively definite symmetric na × na Gram matrix for theystem of vectors obtained by projection or interpolation ofhe values of basis vectors �p onto the set Dm
� .If the time interval � is less than the interval t, on which the
asis �p is defined, then the relation (20) has the prognosticharacter for calculation of the state variables by means of theasis in Dt.
The problem of reconstruction of meteorological fields withissing data are typical everywhere and especially in Siberiahere the monitoring net is rare and nonregular in space. The
dvantage of using the informative bases for reconstructionf the system state by means of the observed data shoulde stressed. In this case, to solve the problem, we need onlyhe number of observations which is commensurable withhe number of the basis functions. This amount of data is
nough for estimation of the expansion coefficients of theought fields onto the informative basis. The combination ofhis approach and data assimilation procedures gives effectiveow-cost computational algorithms. Note that the availabil-7 ( 2 0 0 8 ) 279–291 289
ity of reconstruction of the field structure on insufficientdata by means of data assimilation was firstly demonstratedat the Joint IUTAM/IUGG International Symposium on Mon-soon Dynamics, New Dheli, 5–9 December, 1977 (Marchuk andPenenko, 1981).
7.3. Analysis of the scenarios ensemble
In this section, some possibilities of application of the pro-posed methodology for the analysis of scenarios results arediscussed. It is implied that the results are collected in adatabase in a unified form.
Let us consider a set of scenarios simultaneously andjointly. For this purpose we define a functional space of thescenarios results (FSS) and an energy inner product on it. Thenwe form the matrix Z, the scattering function S(V,Z), and theGram matrix � . The ordinal numbers of scenarios are iden-tified with the numbers of vector-columns in the matrix Z.These numbers is used as the variable of external structure.The content of the column is the result of the correspondingscenario.
Further all operations are produced in accordance with thegeneral scheme of decomposition algorithm.
According to Eq. (14), the PCs show the relative input of eachscenario in the formation of the OBV’s basis. As it is seen fromEq. (16), the fraction of each OBV is reflected in the result ofeach scenario. The pairs (PCs, OBVs) present the general char-acteristics of the ensemble ranged by the degree of significancequantified by the eigenvalues of the Gram matrix.
The proposed scheme of collective analysis of the scenar-ios can be applied to the generalized studies of the quality ofdifferent models which are developing for the solution of thesame class of problems. The interesting point might be theassessment of the generality of the different model results. Itshould be noted that the appropriate metrics has to be chosen.As a result, the PCs give a quantitative estimate of the relativerole of each model in the formation of the general descrip-tion of the processes under study. Moreover, they show howthe constructed set of the OBVs is reproduced in the results ofeach model.
The potential applicability areas of such analysis could bethe ensemble weather prediction projects and the climaticprojects, like the Atmospheric Model Intercomparison Project(AMIP) (Philips, 1994) and the Coupled Model IntercomparisonProject (CMIP) (Meehl et al., 2000).
8. Conclusion
The methodology for the efficient analysis of information fromthe large databases in terms of orthogonal subspaces hasbeen built. It has been designed with allowance for the goalcriteria. It can be applied to the problems of environmentalprotection, ecology, and climate. The main elements and thealgorithms of the methodology are universal. The orthogonaldecomposition method agrees with the structure of the vari-
ational principles for the models of processes under study bymeans of corresponding set of the functionals. The reductionof the dimensions by the two-level data structuring providesthe high efficiency of the algorithms for construction of the
i n g
r
290 e c o l o g i c a l m o d e l l
informative subspaces. The computational cost of the pro-posed algorithms is significantly lower than that of the othermethods demanding the solution of spectral problem for thelinearized operators of the dynamic system models.
The developed approach supplements the methodologyof the solution to the direct (forward) and inverse problemsof environment protection and design. It is intended for thepreparation of the scenarios, as well as for the analysis of themodel results.
The data decomposition in the phase spaces is importantas it allows us to take the entire picture of many years in aglance, to correlate and interpret the results of modeling andthe observed features of the atmosphere behavior.
An optimistic perspective for studying interrelationsbetween the energy active climatic zones and the regionsof increased risk/vulnerability has appeared. It is due toa joint use of the functionals of two types. The first oneis defined on the spaces of the state functions. The sec-ond one is the generalized estimations of the environmentquality defined on the spaces of the sensitivity functions.Taking these interrelations into account is useful for thelong-term prediction in environmental studies and ecologicalmodeling.
Acknowledgements
This research was supported by Program 16 of RAS, Program 3of the Mathematical Sciences Department of RAS, RFBR grant07-05-00673, and European Commission contract no. 013427.
e f e r e n c e s
Buizza, R., Montani, A., 1999. Targeting observations usingsingular vectors. J. Atmos. Sci. 56, 2965–2985.
Courant, R., Hilbert, D., 1953. Methods of Mathematical Physics,vol. 1. Interscience Publishers, NY, 476 pp.
Ehrendorfer, M., Tribbia, J.J., 1997. Optimal prediction of forecasterror covariances through singular vectors. J. Atmos. Sci. 54,286–313.
Gantmacher, F.R., 1959. The Theory of Matrices. Volumes One andTwo. Chelsea Publishing Company, 576 pp.
Gelaro, R., Buizza, R., Palmer, T.N., Klinker, E., 1998. Sensitivityanalysis of forecast errors and the construction of optimalperturbations using singular vectors. J. Atmos. Sci. 55,1012–1037.
Harman, H.H., 1976. Modern Factor Analysis, 3rd ed. University ofChicago Press, Chicago, 486 pp.
IPCC, 2001. Climate change: The Scientific Basis/Contribution ofWorking Group 1 to the third assessment report of the IPCC.Cambridge: University Press, 994 pp.
Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D.,Gandin, L., Iredell, M., Saha, S., White, G., Woolen, J., Zhu, Y.,Leetmaa, A., Reynolds, B., Cheliah, M., Ebisyzaki, W., Higgins,W., Jonowiak, J., Mo, K.C., Ropelewski, C., Wang, J., Jenne, R.,Joseph, D., 1996. The NCEP/NCAR 40-year reanalysis project.Bull. Am. Meteorol. Soc. 77, 437–471.
Kim, H.M., Morgan, M.C., Morss, E., 2004. Evolution of analysis
error and adjoint-based sensitivities: implications foradaptive observations. J. Atmos. Sci. 61, 795–812.Kravtsov, S., Robertson, A.W., Ghil, M., 2006. Multiple regimes andlow-frequency oscillations in the Northern Hemisphere’szonal-mean flow. J. Atmos. Sci. 63, 840–860.
2 1 7 ( 2 0 0 8 ) 279–291
Lévy, P., 1951. Problèmes concrets d’analyse fonctionnelle.Gauthier-Villars, Paris, 484 pp.
Lorenz, E.N., 1965. A study of the predictability of a 28-variableatmospheric model. Tellus 17, 321–333.
Marchuk, G.I., 1968. K teorii biortogonal’nikh razlojhenii poleimeteoelementov. Doklady Akademii Nauk SSSR 179, 832–835.
Marchuk, G.I., 1974. Chislennoye reshenie zadach dinamikiatmosferi i okeana. Gidrometeoizdat, Leningrad, 303 pp. (inRussian).
Marchuk, G.I., Penenko, V.V., 1981. Application of perturbationtheory to problems of simulation of atmospheric processes.In: Lighthill, J., Pearce, R., (Eds.), Monsoon dynamics,Proceedings of the Joint IUTAM/IUGG InternationalSymposium on Monsoon Dynamics, New Delhi, December5–9, 1977, Cambridge University Press, pp. 639–655.
Marchuk, G.I., Sarkisyan, A.S., Dymnikov, V.P., Kurbatkin, G.P.,1984. The programme ‘sections’ and monitoring of WorldOcean. Environ. Monit. Assess. 7, 25–30.
Meehl, G.A., Boer, G.J., Covey, C., Latif, M., Stouffer, R.J., 2000. Thecoupled models intercomparison project (CMIP). Bull. Am.Meteorol. Soc. 81, 313–318.
Mescherskaya, A.V., Rukhovets, L.V., Yudin, M.I., Yakovleva, N.I.,1970. Estestvennye Sostavlyayuschie MeteorologicheskikhPolei. Gidrometeorologicheskoe izdatelstvo, Leningrad, 199pp. (in Russian).
Molteni, F., Palmer, T.N., 1992. Predictability and finite-timeinstability of the northern winter circulation. Quart. J. Roy.Meteorol. Soc. 119, 269–298.
Mureau, R., Molteni, F., Palmer, T.N., 1992. Ensemble predictionusing dynamically conditioned perturbations. Quart. J. Roy.Meteorol. Soc. 119, 299–323.
Penenko, V.V., 1974. Spectral’niye zadachi gidrodinamiki ibiortogonal’nye razlojheniya meteorologicheskikh elementovna sfere. In: Marchuk, G.I. (Ed.), Proceedings of theInternational Symposium on Finite-difference and SpectralMethods for Solution of the Problems of Atmospheric andOcean Dynamics. Novosibirsk, 1973, Computing Center of theSiberian Division of the USSR Academy of Sciences, pp. 68–87(in Russian).
Penenko, V.V., 1975. Vichislitel’nyie aspecti modelirovanoyadinamiki atmosfernikh protsessov i otsenki vliyaniyarazlichnikh faktorov na dinamiku atmosfery. In: Lavrent’ev,M.M. (Ed.), Nekotoriye problemi vichslitel’noi i prikladnoimatematiki. Nauka, Novosibirsk, pp. 61–77 (in Russian).
Penenko, V.V., 1981. Metodi chislennogo modelirovaniyaatmosphernikh processov. Gidrometeoizdat, Leningrad, 352pp. (in Russian).
Penenko, V.V., Tsvetova, E.A., 1999. Mathematical models forstudies of interactions in system of Lake Baikal and theatmosphere of the region. Appl. Mech. Tech. Phys. 40 (2),137–147.
Penenko, V.V., Tsvetova, E.A., 2002. Construction ofmultidimensional basis vectors for analysis and prognosis. In:Ritchie H. (ed.), Research activities in atmospheric and oceanicmodeling, report no. 32, WMO/TD-No.1105, pp. 01-46–01-47.
Penenko, V.V., Tsvetova, E.A., 2003. Global and regional principalcomponents of the climatic system and their application toenvironmental studies. Atmos. Ocean. Opt. 16, 372–379.
Penenko, V.V., Tsvetova, E.A., 2005. Energy centers of the climatesystem and their connection with ecological risk domains.SPIE 6160, part 2, 61602U.
Penenko, V.V., Tsvetova, E.A., 2007a. Variational technique for
environmental risk/vulnerability assessment and control. In:Ebel, A., Davitashvily, T. (Eds.), Air, Water and Soil QualityModeling for Risk and Impact Assessment. Springer,pp. 15–28.
g 2 1
P
P
P
S
T
e c o l o g i c a l m o d e l l i n
enenko, V.V., Tsvetova, E.A., 2007b. Mathematical models ofenvironmental forecasting. Appl. Mech. Tech. Phys. 48,428–436.
hilips, T.J., 1994. A summary documentation of the AMIPmodels. PCMDI report no. 18, Lawrence Liviermore NathilalLaboratory, CA, 343 pp.
reisendorfer, R.W., 1988. Principal Component Analysis in
Meteorology and Oceanography. Elsevier, Amsterdam, 425 pp.chwartz, L., 1967. Analyse Matematique, vol. 1., Hermann, 824pp.
aldykin, A.T., 1982. Elementi prikladnogo funkzional’nogoanaliza. Vischaya schkola, Moscow, 383 pp. (in Russian).
7 ( 2 0 0 8 ) 279–291 291
Toth, Z., Kalnay, E., 1993. Ensemble forecasting at NMC: thegeneration of perturbations. Bull. Am. Meteorol. Soc. 74,2317–2330.
Teisserence de Bort, J., 1884, Etude sur l’hiver de 1879–1880 etrecherches sur la position des centers d’action del’atmosphere dans les hivers anormaux. Bureau Centr.Meteorologieque de France “Annales” Pt. 4, 17–62.
Wilks, S.S., 1962. Mathematical Statistics. John Wiley and Sons,NY, 632 pp.
Zverev, A.S., 1968. Sinopticheskaya meteorologiya.Gidrometeorologicheskoe izdatel’stvo, Leningrad, 774 pp. (inRussian).