Pure and Applied Geophysics

, Volume 169, Issue 3, pp 447–465 | Cite as

Direct and Inverse Problems in a Variational Concept of Environmental Modeling

  • Vladimir Penenko
  • Alexander Baklanov
  • Elena Tsvetova
  • Alexander Mahura


A concept of environmental forecasting based on a variational approach is discussed. The basic idea is to augment the existing technology of modeling by a combination of direct and inverse methods. By this means, the scope of environmental studies can be substantially enlarged. In the concept, mathematical models of processes and observation data subject to some uncertainties are considered. The modeling system is derived from a specially formulated weak-constraint variational principle. A set of algorithms for implementing the concept is presented. These are: algorithms for the solution of direct, adjoint, and inverse problems; adjoint sensitivity algorithms; data assimilation procedures; etc. Methods of quantitative estimations of uncertainty are of particular interest since uncertainty functions play a fundamental role for data assimilation, assessment of model quality, and inverse problem solving. A scenario approach is an essential part of the concept. Some methods of orthogonal decomposition of multi-dimensional phase spaces are used to reconstruct the hydrodynamic background fields from available data and to include climatic data into long-term prognostic scenarios. Subspaces with informative bases are constructed to use in deterministic or stochastic-deterministic scenarios for forecasting air quality and risk assessment. The results of implementing example scenarios for the Siberian regions are presented.


Variational principle weak-constraint formulation sensitivity uncertainty 4D-Var data assimilation uncertainty assessment environmental risk inverse problems 



We thank the anonymous reviewers for their constructive criticism and useful comments. This study was supported by the 6th and 7th Framework Programmes of the European Commission (CA-project Enviro-RISKS: Man-induced Environmental Risks: Monitoring, Management and Remediation of Man-made Changes in Siberia,, INCO-CT-2005-013427; MEGAPOLI:; grant no: 212520; and TRANSPHORM: projects), the Danish Council for Strategic Research (Center for Energy, Environment and Health—CEEH;, and the Russian Foundation for Basic Research (grants RFBR 10-05-01083-a, 11-01-00187-a). We are also grateful to of the Presidium of the Russian Academy of Sciences (program nos. 4 and 20) and the Department of Mathematical Sciences of RAS (program no 3). The DMI supercomputing facilities have been used extensively in this study. We also thank Jens H. Sørensen (DMI) for active participation in the Enviro-RISKS project.


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Vladimir Penenko
    • 1
  • Alexander Baklanov
    • 2
  • Elena Tsvetova
    • 1
  • Alexander Mahura
    • 2
  1. 1.Institute of Computational Mathematics and Mathematical Geophysics (ICM&MG)Siberian Branch of the Russian Academy of SciencesNovosibirskRussia
  2. 2.Research DepartmentDanish Meteorological Institute (DMI)CopenhagenDenmark

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