Problems of Global Change through Natural Systems … · Prigogine’s chaos/ordertheory...

Discrete Dynamics in Nature and Society, Vol. 1, pp. 243-253

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Predictability Problems of Global Change as Seenthrough Natural Systems Complexity Description

1. General Statements

VLADIMIR V. KOZODEROVa’*, VICTOR A. SADOVNICHII b, SERGEY A. USHAKOVb

and OLEG A. TIMOSHINb

alnstitute of Computational Mathematics Russian Academy of Sciences, Gubkine Street 8,117333 Moscow, Russia," bM. V. Lomonosov Moscow State University, 119899 Moscow, Vorobiovy Gory, MSU Russia

(Received in finalform 2 June 1997)

The overall problem of global change is considered as the mathematical discrete dynamicsdiscipline that deals with the sets, measures and metrics (SMM) categories in informationsub-spaces. The SMM conception enables to unify techniques of data interpretation andanalysis and to explain how effectively the giant amounts of information from multispectralsatellite radiometers and ground-based instruments are to be processed. It is shown thatPrigogine’s chaos/order theory and Kolmogorov’s probability space are two milestones inunderstanding the predictability problems of global change. The essence of the problems ismaintained to be in filtering out a "useful signal" that would spread from key regions of theglobe as compared to their background. Global analysis, interpretation and modellingissues are outlined in the framework of incorrect mathematical problems and of the SMMcategories, which contribute to solving the comparability problem for different sets ofobservations.

Keywords." Order and chaos, Imagery interpretation, Modelling

INTRODUCTION

Understanding of predictability problems of globalchange is considered by the world scientific com-

munity as having one of the highest priority in thecontext of international cooperation. Possiblechanges in climate, land use and composition ofthe atmosphere due to natural and anthropogenicinfluences are usually implied under the overall

*Corresponding author.

243

term of global change. Separate projects of theInternational Geosphere-Biosphere Programme/IGBP (Global Change, 1992), World ClimateResearch Programme/WCRP (Morel, 1990), Inter-national Human Dimensions Programme onGlobal Environmental Change/IHDP (Moss,1992) and other Programmes have been designedto solve the problems. The listed three Programmes(IGBP, WCRP, IHDP) are developed under the

244 V.V. KOZODEROV et al.

International Council of Scientific Unions (ICSU),World Meteorological Organization (WMO) andInternational Council of Social Sciences (ICSS),respectively. Opportunities to study components ofglobal change issues using satellite imagery, simula-tion techniques and data bases of specially arrangedfield campaigns have been realized within theProgrammes. They reflect the-state-of-the-art inglobal change research.

Research objectives of the Programmes are tocharacterize and document variability and trendsof the Earth’s system through operational globalobservations, to understand the nature of keyregulating parameters to identify the causal factorsof observed variations and feedback processes,and to assess the predictable aspects of long-termvariability and changes, including regional impacts,through the combined application of observationsand models. The comprehensive research pro-grammes that addresses these objectives shouldencompass the capability to measure and monitorkey parameters, a strong field observation anddata analysis stage to improve knowledge of keyprocesses, a modelling activity to develop predic-tive capability for regional change, and an inte-grative analysis phase to evolve methods forimpact assessments.To achieve the ultimate goal of the Programmes,

that is predictability of global change, is incon-ceivable without having in mind progress in math-ematical disciplines ofdiscrete dynamics for systemsencountered in the natural and social sciences. Butcommon sense and traditional principles are seen

prevailing in the current projects realization. Math-ematical aspects of information content assess-

ments, discrete dynamics techniques and theevolutionary processes description are usually con-

sidered as something too abstract to be taken intoaccount in the international projects. The concep-tual endeavours, based on the mathematical aspectsand their on-going improvements, seem to be not so

attractive for researchers within the Programmes.The reasons of a certain kind of ignoring themathematical frameworks to the detriment of

common sense may be due to by the complication,connected with the general assertion of globalchange problems. Such conventional attitude topriorities in global change research gives way toallegedly more understandable results of simulationtechniques. In fact, in the projects follow-up, theultimate goal is somewhat forgotten, and numericalmodels (Global Circulation Model- GCM, Soil-Vegetation-Atmosphere Transfer- SVAT, Dy-namic Global Vegetation Model- DGVM, etc.)are putting forward in the related studies (Asrar,1989; Running et al., 1994). Satellite observationsand remote sensing techniques are anticipated asserving mainly to run the models. These are modelsupon which main results of decision making arepractically based, and it is clear that these models arenecessary but not sufficient to solve problems ofglobal change.

Meanwhile, temporal data sets of satelliteobservations would give an impetus for solvingthe predictability problems based on geoinforma-tion models (Kozoderov, 1994), i.e. derived as aresult of data sets processing using GeographicalInformation Systems (GIS), but not from taken inadvance equations and numerical models con-structed from these equations. This is a new class,called geoinformation models, that is wider inapplications than pure simulation techniques. Inclimate research the predictability is usuallyassociated with finding global responses of theEarth’s climate system to anomalies of relatedprocesses in key regions of the World ocean andland surface. These anomalies, characterized by acertain kind of variability of thermodynamicalparameters (atmospheric pressure, temperature,humidity) of the global system, are analyzed usingtemporal data sets of the Earth Radiation Budget(ERB) components from satellite observations

(Marchuk et al., 1990).The essence of the predictability problems is

maintained in these studies to be in filtering outmathematically a "useful signal" that would spreadfrom the key regions in the system as compared totheir background. The concept of teleconnections,

PREDICTABILITY PROBLEMS: PART 245

i.e. signal-to-noise ratios extraction from temporaldata sets, is revealed in cross-correlations of theobserved anomalies between the regions and otherparts of the globe on the scales of the ERB inter-annual variability.Another problem to be solved as a part of the

overall predictability problem is in the necessity torealize that interpretation of remote sensing datausing results of modelling of subsequent processesin the global system is not a mathematicallycorrect procedure (Tikhonov and Arsenin, 1979)in the sense that even small disturbances of initialdata could result in large changes in solutions ofthe inverse problem. A minimization procedure fora smoothing functional, that is equal to the sum ofa stabilizing functional for the problem and a

deviation measure between the operator of a

hypothesized model and initial data of observa-tions, is needed to make it possible to comparemodelling and measurements data. This procedureis called as a regularization of approximate solu-tions finding for the corresponding functionalequations, coupling modelling and measurementresults (Tikhonov, 1982). Otherwise, the consis-tency of theoretical models application, whileemploying data of observations as their inputs,looks ambiguous without such procedures. Thismeans in mathematical assertion of the compara-bility problem that data of modelling and observa-tions could be in different sub-spaces, and takinginto account stringent terms of sets, measures andmetrics (SMM) (Kolmogorov and Fomin, 1976)with applied mathematical formalism may be theonly way to understand to what extent sets ofmodelling and observation data are useful to solvethe predictability problems.

In addition to the remarks concerning theconceptual SMM models application, one can

say that besides a new sounding of such modellingtechniques in data interpretation and analysis(Kozoderov, 1995), the SMM categories are meantto be a profitable tool in assessing state parametersof natural ecosystems using remotely sensed andground-based data. On the eve of the Earth

Observing System (EOS) era (see Earth ObservingSystem, 1993), including GCOS, GOOS and GTOSas the Global Climate, Ocean and TerrestrialObserving Systems, respectively, the scientificcommunity mostly relies on traditional applica-tions of the giant amounts of data to be acquiredfrom the Systems. These applications are usuallyforeseen in diagnostic studies and numerical simu-lation techniques, which allegedly are to be fed bydata of the observations.

In practice, tens of different instruments of theSystems (with various spectral and spatial resolu-tions) will give many Gigabytes of information,and no rigorous approach has confirmed that allthese sets of data will be beneficial for employingin numerical models and other interpretations. Inmathematical terminology this situation is under-stood that sets of observations and modelling arein non-overlapping sub-spaces, and additionalsteps are to be undertaken to give proofs concern-ing the effectiveness of the above Systems. Prac-tical purposes of the Systems may look problematicunless the comparability problem of different datasets is elucidated by a proven scientific approach.Evolving the information and mathematical basisof the defined global change science, using theSMM categories in Kolmogorov’s probabilityspace (see part 2 of the publication), might bethe only way to link different sets of measurementswith their practical applications. This is analternative to pure technological means in utilizing"the three-dimensional memory" while collectingand storing many Gigabytes of data within theEOS plans. The information content assessmentand studies of redundance of sets of data will givea scientific tool for data processing, interpretationand analysis in the newly defined domain addi-tionally to routine pattern recognition and sceneanalysis techniques.

Being adherent to Prigogine’s ideas of dynami-cal systems description by equations of the evolu-tionary type (Prigogine and Stengers, 1986; 1994),we shall outline below major statements of therelated theory to be known in global change,


subject incentives of some IGBP projects tocriticism because of their unreadiness to use thetheoretical statements, and consider the naturaltargets complexity description problem as well asthese targets variability and predictability of theiralteration in the context of the global changescience. In our attempts to give an explanation ofthe theoretical approach in terms of axiomatic pre-requisites and rigorous mathematical definitions,we shall follow the proposed data sets concept inKolmogorov’s probability space along with theinformation measures and entropy metrics repre-sentation. These attempts are intended to linkPrigogine’s chaos/order theory and Kolmogorov’sprobability space as two milestones in under-standing predictability problems of global changehaving in mind EOS data sets to be interpreted interms of the above SMM categories.

Part of the publication presents a generalscope of mathematical aspects of the problems tobe solved to understand what is the essence of theglobal change predictability: the specificity ofremote sensing Earth sciences techniques to solvethe inverse, mathematically incorrect, problems tounify data both of modelling and measurements;order and chaos as the main categories, theproposed conceptual models to be pertained to;global analysis, interpretation and modellingaspects of the overall predictability problems touse the mathematical procedures of multispectralsatellite imagery processing as an example of newgeoinformation models application.

Part 2 will embrace a description of our

approach: Kolmogorov’s probability space; prop-erties of information measures that would enableto operate with information sub-spaces; imageryvisualization procedures and the SMM categories;Markov’s chains in the sub-spaces to find mea-sures of any form of ordering on multispectralimagery; natural objects complexity and structureanalysis techniques on sets of regular configura-tions of elements given by the informationmeasures; the predictability problems in cross-correlation techniques for anomalies of processesto be picked up by temporal sets of the imagery.

BACKGROUND

We have developed an information/mathematicalapproach that enables to solve the comparabilityproblem of different sets of measurements relatingto a particular natural object in terms of theSMMs (Sadovnichii, 1996). The approach is basedon information measures and entropy metrics forany set of data using a mathematical expectanceoperator from specially interpreted probabilitydistribution functions. The well-known Shannon’sformula for the amount of information in any setof data gives a particular number to be calculated("expected") as the sum, taken with the minus sign,of the products of a known probability distribu-tion function from elements of the set and thelogarithm (usually on base 2) of the function. Thetrue sense of the information measure gets widerthan simply the number calculated, if the formulais considered as a functional defined by the scalarproduct in the probability space that includesprobability vectors, given by the distribution ofthe elements of the set, and other vectors, given bythe non-linear logarithmic function from the dis-tribution. The latter vectors can be called as theuncertainty vectors implying that no uncertaintyexists in an event when the probability function isconstant and equal to (the certain event). Thus,additionally to the SMM categories we have todeal with representation of data in spaces, amongwhich the distinction of Kolmogorov’s space is notonly in its containing sets of widely-used elemen-tary events in mathematical sciences, but in using a

specially defined algebra, i.e. this space gives "arule of actions" with its elements along with itsparticular probability measures (Kolmogorov andFomin, 1976). This space can be the n-measureEuclidean space for groups ofnumbers or elements,the linear normalized Hilbert’s space with the scalarproduct on it, etc.To outline scientifically the construction of the

GCOS, GOOS and GTOS systems, our approachenables to merge the following interrelated blocks:Climate/Biosphere Models, Observation Systems,Geoinformation Systems and Predictability


Problems. The opportunity to unify these blocks inthe general context of global/regional changestudies are determined by the proposed applicationof the information and thermodynamic propertiesof the entropy category. It reveals its dualism inthe information content assessments for data ofmodelling/monitoring and in state parametersretrievals for natural targets using multispectralremote sensing images and ground measurementson test sites of the Earth.The Climate/Biosphere block is designed to

highlight how global and regional change prob-lems are drawn up in terms of models, data andprocesses for natural media (the atmosphere,hydrosphere, land surface, biosphere). The Ob-servation Systems block, linking modelling andmonitoring quantities on various scales, serves toaccount for recommendations to optimizationtechniques based on certain criteria, value func-tions and information content metrics. The re-commendations., are due to the necessity toimprove or make cost-effective the systems whichare planned within the EOS and similar otherprogrammes. The Geoinformation Systems blockgives an explanation to what extent GIS-technol-ogies, databases and information systems can beunified with general principles of data processing,related algorithms and procedures. The Predict-ability Problems block enables one to understandwhat can be determined from temporal data setanalyses and interannual variability of multispec-tral satellite products and samples of groundmeasurements.

These our findings, which are designated tocouple basic and applied research relating todiscrete dynamics in nature and society, open upan opportunity to reduce uncertainties aboutcurrent knowledge of natural processes by explain-ing differences between "pure random events"(chaotic distributions) and "an excessive ordering"to be characterized by uniform distributions(Prigogine and Stengers, 1986). As a result, tack-ling with the entropy functionals (conditionalentropy, information divergence, mutual informa-tion, etc.), the effectiveness measures of global

observing systems are derived in specific subjectareas. Metrics is fitted in the relevant probabilityspace to consolidate the unified description of datasets from different instruments. Besides that, we

have shown that probabilistic grammatic rules andrelated computer languages on sets of regular con-

figurations, extracted from sets of data using GIS,do contribute to a proper description of the com-

plexity problem for structures of natural targets.Thus, global change is a particular sphere of

discrete dynamics application to study processes innature and society using the proposed unificationof scientific investigations in the SMM form fordata analysis and interpretation in the relatedinformation sub-spaces. The conception of thesesub-spaces as a composite part of widely usedmetrical spaces must become such a universalmeans to assess the state of the objects understudy while processing data sets that would carryinformation about objects using computer workstations. Though not terminate in applications, theterminology of the functional analysis discipline(for example, linear shells or compactness of setsas well as metrical spaces or their completeness)are routinely coming into software of existingwork stations for data sets processing. A pair ofquantities is understood as usual under themetrical space, that consists in a set of elements(pixels for imagery) of a particular structure andin a distance, i.e. a single, non-negative and realfunction to be defined for two optional elements ofthe given set and satisfied the known axioms of"the zero element existence, symmetry and trian-

gle" (Kolmogorov and Fomin, 1976).The idea to employ the probability space

representations is vital for us to help unifyprocessing of different data sets: of measurementsin remote sensing Earth sciences, of questionnairesconcerning regional data on the health care andinfrastructure, of indicators of possible changeusing ground observations, etc. In any case,understanding of the relevant probability distribu-tions is needed in the form of order and chaosmeasures to be extracted from the specific sets ofdata. In this respect we can say about applications


of our approach as a new step in global changescience operating with discrete sets of data ininformation sub-spaces.

ORDER AND CHAOS CATEGORIES

Irreversibility of processes in the natural world isone of its fundamental properties. The importantattainment of Prigogine’s theory (see Prigogineand Stengers, 1986; 1994) is that the theory hasdemonstrated by non-contradictory evidence theuniversal entropy increase for spontaneous naturalprocesses as a reflection of objective properties ofinstability in dynamic systems (the upper limit ofwhich are chaotic systems). We do not dwell hereon thermodynamic and information properties ofthe entropy (Kozoderov, 1995), which are knownto be formally equivalent (Volkenstein, 1986). Ourmain interest is concerned the information entropyand the consequent measures of uncertaintydescribed by it.

Classical thermodynamics is unable to explainprocesses of energy and mass exchange in openbiological and other systems. The rate of theentropy production in stationary positions to bereached as a result of internal irreversible processesin such systems is equal to its minimal positivemeaning. So, the entropy is not only unlimited"slipping down" to the stationary position withoutconstructing any organized structure, but it servesas a source of a certain measure of order emergingin the systems. The classical description of thesystems behaviour in terms of trajectories in thephase space, containing spatial coordinates andtime, turns out to be not sufficient for chaoticsystems, for which behaviour cannot be predictedsimilarly to the classical case. The constructive roleof chaos, that leads to the rise of local orderingand to decrease of natural systems entropy whilstits overall tendency is to be increased, is amongbasic principles in Prigogine’s theory.

Overall findings of the theory are in gainingproofs that producing the chaos is an objectiveproperty of dynamic systems. Such systems would

characterize the evolution of nature and society.The properties of the systems can be analyzed byan irreducible probabilistic description of thesesystems in generalized spaces, where complex eigenvalues of mathematical operators in the equationsof the evolutionary type give information aboutprocesses of decay, damping and irreversibility inaddition to classical descriptions of the dynamicprocesses. These findings of Prigogine’s theorywould entail a necessity of utilizing mathematicalaxiomatics of probability theory, informationcontent assessments and entropy characterizationto employ the basic ideas of dynamical systemsbehaviour for representing sets of remote sensingdata as units of the relevant metrical spaces. Thesetreatments, in turn, would unify the process of thenatural objects complexity description, compara-bility of their images and finding their variabilityusing temporal data sets of their sub-sequentimagery.

Saying about chaotic systems as having theupper level of instability, when a phase trajectoryloses its sense through a characteristic intervalcalled "Ljapounov’s time", we are able to observechanges on multispectral imagery with time byrepresenting sets of data in metrical spaces. Thisour opportunity to visualize structures of naturalobjects on their imagery would enable to interpretthe emergence "of particular points" of possiblebifurcation as they belong to "rules of theirsystematization" using special computer languageswhile processing sets of the imagery. Chaoticsystems are known to "forget" their initial condi-tions in their evolution after the period ofLjapounov’s time. This definition has allowedscientists to solve many problems of naturaldevelopment and evolution while satisfying thefollowing main requirements of chaos origin:

-the existence of irreversible processes that isexpressed in the symmetry violation betweenthe past and future events;the necessity to introduce "the event" categoryas permitting descriptions only in terms ofprobabilities; this category cannot be derived


from any deterministic law, reversible in time or

not;some events must have the ability to change thecourse of evolution, i.e. the evolution is to bedefinitely "unstable" and characterized by mech-anisms that make some events initial points ofnew developments and a new global order.

Contrary to closed or isolated (without interac-tion with their environment) deterministic systems,characterized by classes of stable solutions in therelevant differential equations, new possibilitiesenter into consideration in "the open world"dealing with the irreducible probabilistic laws.The irreducibility is the direct consequence of theinstability of the chaotic systems. There is athreshold for open systems in hydrodynamics,chemical kinetics and other sciences, from whichthe systems behaviour gets radically changed. Theresult is in emerging "dissipative structures"(Kurdyumov, 1982). These are understood as

having "an organized behaviour" resulting frominteractions between two opposite processes in thenon-equilibrium thermodynamics: dissipation andorder. Even no new ordering force is induced intothe system, far from the equilibrium position a

spontaneous violation of the spatial symmetry can

emerge in the system under coherent collectivemotions because of non-linear properties of thestrongly non-equilibrial situations. These are

situations that result in such effects as a sensitivityof the solutions to initial conditions, instabilityand bifurcations, i.e. giving rise to many ways of a

particular system developing near these points ofits phase trajectories.The thermodynamic equilibrium state is called

"an attractor", the isolated system is directed to,that consists of many billions of particles, butmight be described by only a small number ofparameters such as temperature and pressure. Theattractors are synonymous with the stability andreproducibility categories. However, attractorswith a fractional dimension ("strange attractors")cannot be described in terms of simple geometricalobjects and thus require that the cause and

consequence categories, known in deteministicrelations, be replaced by their new properties.The dissipative chaos is a phenomenon that

arises in unstable dissipative structures for greatlynon-equilibrial processes when the regime of thestrange attractor gets chaotically unstable, i.e. whenthe distance between two points of the system,initially very small, is exponentially enhanced withtime. The possibility of predicting the behaviourof such systems is vague, differing from smallfluctuations in the case of "the deterministicchaos," which can be described in terms ofexpected correlations in temporal data sets, con-

trary to "a completely random process." The latteris characterized by infinitely large dimensions, andits temporal evolution is governed by internalrandom phenomena, but not global dynamics oran attractor.

These major statements of Prigogine’s theoryenable us to formulate the predictability problemsof global change in a linear cross-correlationapproach when the possibility of predicting thedynamic behaviour of the Earth’s geophysical/biogeochemical/climate system remains real. Weoriginate our research from the fact that theultimate goal of understanding the predictabilityproblems in the system can be achieved by multi-disciplinary studies using global monitoring sys-tems, regional networks and models mergingscientific and technological issues. Therefore, thepredictability theory is outlined in part 2 of this

publication based on temporal data sets analysisfor remote sensing imagery of various spatialresolutions including global and regional scales.In fact, these goals can be reached provided thatrules of information measures application are

determined to represent sets of data in metricalsub-spaces. Only for the sub-spaces, mathemati-cally formalized by definitions of sets and a

distance between two points in them, it is possibleto use common measures of comparability, order-ing and calculations.Some extractions from the information theory in

part 2 are assumed to use the particular form of themathematical expectance operator for matching


sets of data in information sub-spaces instead ofHamilton’s operator used in Prigogine’s publica-tions for the classical approach. Our conviction toyield a new quality in understanding of predict-ability problems of global change using theinformation amount assessments is strengthenedby the fact that evolving imagery visualizationprocedures in terms of the entropy measures, wecan motivate what is the "most interesting" innatural observable structures being exposed tonatural and antliropogenic changes. Analyzingthe commensurate "contextually coherent struc-tures" of natural targets, we are able to find what isessential in a particular event, given by a sequenceof symbols on multispectral imagery. This is a

possible way to remove the major vulnerability ofthe algorithmic theory of information, which isdue to by the supposition that the structure underconsideration is not infinite. Thus, an opportunityis opened up to elucidate relations between theinitial sense of the structure formation and meansto describe its variability. Our approach enables tomake the complexity description of natural targetsbehaviour using contextual computer languages.

Before going to information measures consid-eration, let us shortly outline main tendencies indata analysis, interpretation and modelling. Theproposed conceptual framework of global changehas turned out not yet to be of particularimportance in global change science due to lackof its proper mathematical influence on existingstudies.

GLOBAL ANALYSIS, INTERPRETATIONAND MODELLING IN INTERNATIONALPROGRAMMES

Possibilities to solve global change problems are togreat extent associated with global system model-ling, parameterizations of natural processes in therelevant numerical models and integration ofconsequent sub-system models. In particular, theIGBP Task Force for Global Analysis, Interpreta-tion and Modelling (GAIM) is conceived by its

initiators as directed "toward establishing theresearch capabilities necessary for developing themodels and understanding needed for constructionof integrated biogeochemical Earth system mod-els" (Research GAIM, 1997). This is only one partof the multidisciplinary problem to understand thepredictive capabilities of global change. Its secondpart is concerned the role of regular observationsand current measurements in the studies. How can

these parts be coupled? This question is behind theGAIM and similar other projects consideration,though in light of the EOS era the second partseems to be more actual to date than the first one.An impression can be gathered that the GAIM

and other IGBP projects (for instance, InternationalGlobal Atmospheric Chemistry- IGAC, GlobalChange and Terrestrial Ecosystems- GCTE, Bio-

spheric Aspects ofthe Hydrological Cycle BAHC,Land Use/Cover Change-LUCC, etc.) participantscome consciously away from vital problems con-

centrating mostly their attention on modelling andpartly on analysis problems, but not on that of datainterpretation (see, for example, Natural Distur-

bances 1996). Not denying attempts within theGAIM to create sub-system models of the globalbiogeochemical system, we are saying here aboutconceptual models to describe complex non-linear

phenomena within the global system using mathe-matical formalism of probability spaces, sets ofdata, information measures, entropy metrics, etc.We see our purpose to fill in the gap between

data of modelling and monitoring having in mind

the challenge in gaining the ultimate goal- tounderstand the predictability problems of global/regional change. Giving tools to different data sets

comparability, the conceptual models would en-

able to solve the problems. Data interpretation inthe GAIM Task Force should include mathemat-ically stable retrieving of state parameters fornatural targets within a concurrent class of modelsusing particular solutions of incorrect problems.These are other procedures of remote sensing dataprocessing than are envisioned by the GAIM.The general direction in the GAIM type projects

implementation is in solving direct problems, i.e.


in obtaining results of related hypothesized theo-retical models. Practical criteria of making con-

clusions, concerning global/regional change, aredetermined by solutions of inverse problems thatare designed to account for the quality of one or

another model from sets of the hypothesizedclasses of comparable models. The questions areput while solving the inverse problems: (i) whetheran individual model in an accepted class existsor not, theoretical consequences and particularobservations for this models are within the accu-racy of the observations (the comparability pro-blem is set up for the class of the models used andthe observations for a particular natural target);(ii) if the class of the models is comparable withobservations, then to realize an interpretation ofthe particular set of observations, i.e. to explainthe value of the individual theoretical modelamong the accepted class of models undercomparison.

CORRECT AND INCORRECT PROBLEMS

The final purpose of the global Earth systemmodels construction will look vague without theabove basic statements of simulation techniques. Ifwe denote by z an approximate model of a naturalobject, then it is usually represented by a set ofnumerical parameters, functions, etc. so that allother models of the same type from the classZ {z} of the comparable models are consideredidentical, provided all their characteristics for themodels are completely coincident. Hence, a devia-tion measure between two models z and z2,

denoted as pz(z,z2), can be defined (Tikhonov,1982) being considered as elements of the givenmetrical space (Kolmogorov and Fomin, 1976). Inpractice, it means that we are eligible to separatecorrect and incorrect problems, the first are tosatisfy the requirements of their solutions existenceand uniqueness as well as stability of the solutionrelative to disturbances of initial data, the secondare called when at least one from these threeconditions are not satisfied.

If u is a theoretical value of observational inputsto be found in accordance with the idealized modelz and A is the mathematical operator thatdetermines a relationship between characteristicsof the z model and input data u so that u Az,then the similar deviation measure pu(u, u2) canalso be defined in the class of possible input curvesof observations U={u}. The form of the Aoperator is usually established owing to the theorythat serves as a basis for interpreting data ofobservations. In specific applications while inter-preting remote sensing data, the following integralFredholm equation of the first kind, well-known inmathematical analysis, has to be solved:

b

Az K(x, s)z(s) ds u(x), c <_ x <_ d,

where the equation kernel K(x,s) has to be acontinuous function c <_ x <_ d, a <_ s <_ b; z(s) is thefunction to be found for the object under study;u(x) is a known function given by the particularexperiment.An explicit form of the A operator while pro-

cessing remote sensing data is usually associatedwith the radiation transfer theory (Kondratyevet al., 1992). But the problem of the relatedobservations interpretation is set up as that offinding solutions of the following equation:

Az- u*

that serves for a pattern recognition of the objectunder study using approximate input data ofparticular observations u*. The assertion of theproblem is mathematically inconsistent since u* isa product of a particular measurement for an

object, with inevitable complications due to themeasurement inaccuracy. The latter could result inpossible absence of any solution that would satisfythe equation in the Z class of all permissiblesolutions.

If the A operator is explicitly known and 5 is theinaccuracy amplitude for the u*, so that for the u-"precise" measurements the deviation measure can


be written

pv(u*, u-) <_ ,then the inverse problem is set up as that of thefollowing inequality solution finding:

p(Az, u*) <_ 6 (z Z).

Elements z e, satisfying this inequality, can benominated as formally comparable with the u*values. Denoting via Z the set of all z, compara-ble with these values

Z {z Z: pv(Az6,u*) <_ 6},

one can obtain classical solutions of the inverseproblem.

If the set of Z is empty, i.e. models comparablewith a set of observations in the Z class are absent,this means that these models from Z are too roughand cannot describe the object under study withthe given accuracy. Thus, it is necessary to widenthe Z class of models taking an entire set ofextending classes of the hypothesized modelsZ1, Z2,... ,Zn until the following inequality (thatwould characterize the infimum of the deviationsbetween the theoretical values of Az and data ofmeasurement u* known approximately) is satisfied:

inf pv(Az, u* <_ 6,zEZn

i.e. until a comparable (consistent) model of theobject is found.

In the case, when Z is not empty and containsdifferent models, the formal comparability is notsufficient to identify the related model of theobject using only observations, given with theiraccuracy, without additional information. It meansthat even the above equation Az u* does have a

solution, the solution of the stated inverse problemcan be unstable in the sense that even smallvariations in data of observations will result inlarge variations in the solutions of the wholeproblem. This equation may not have any solution

in its classical sense, and in this case Mean SquareTec.hniques (MST) are usually implied to find ageneralized solution, which is understood as such amodel realization z0 Z that

pz(Azo, u* inf p(Az, u* ).zEZn

The MST improvements are known to solve theproblem concerning the existence of a formallycomparable model, provided the model z0 can befound that would "almost completely minimize"the deviation with the accuracy r/, such as

50 + r/< 6, where

60 inf pv(Az, u*) < 6.zZ

The MST solutions are often used in remotesensing data processing, but this is an example ofunstable solutions in the sense defined above.Stable algorithms of observational data inter-pretation are given by regularization techniques(Tikhonov and Arsenin, 1979).

If the given Z set of the formally comparablesolutions is too wide, then it is necessary toestablish a selection principle how to choose amost valuable model Z from the point of viewof its applicability for interpretation purposes. Thechoises are given by the regularization techniques.The principle of "minimal complexity" of a model,being selected from sets of similar models to becomparable in accordance with data observationaccuracies, is often laid down into the selectionprocess. The complexity functionals W[z] shouldbe non-negative, continuous and satisfying to thecondition that the set Ze {z Z: W[z] <_ c} iscompact in the Z space. The compactness impliesthat from any sequence of objects in the metricalspace a new sub-sequence can be selected thatshould converge to an element in Z. As a result, anormal solution of the problem is given by such anelement Z, which would minimize the com-plexity functional

W[el_< W[z] for allzZ,


i.e. is the least complex model from the class ofall models comparable with observations accord-ing to the accuracy category.

There are theorems of uniqueness of thedetermination under sufficiently general propertiesof the W[z] functional, of stability of the 2 modelselection process relative to small perturbations ofthe observations as well as of convergence of thenormal solution to exact when the observationaccuracy increase is not restricted (Tikhonov andArsenin, 1979). Thus, is a stable regularizedsolution of the equation Az--u*.

There are several techniques of the optimalfunctional W[z] selection. This example of globalanalysis, interpretation and modelling shows thatthe problems to be put in the GAIM are more

complicated than considered by the Task Force.Creating global sub-system models looks liketypical exercises in convenient simulations withoutstudying the complete set of the EOS and othersystems data interpretation for the global pur-poses. Part 2 of the publication will be devoted toinformation and complexity measures, their de-scription and use to understand the predictabilityproblems of global change.

References

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