Direct and Inverse Problems in a Variational Concept of Environmental Modeling

Abstract

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 longterm 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.

Data assimilation in atmospheric chemistry models: current status and future prospects for coupled chemistry meteorology models

“…The (ensemble) filtering approach requires the augmentation of the state variables with the parameters (Ruiz et al, 2013). 4D-Var easily lends itself to data assimilation since the parameter variables can often be accounted for in the cost function (Penenko et al, 2002;Elbern et al, 2007;Bocquet, 2012;Penenko et al, 2012). However, it is often required to derive new adjoint operators corresponding to the gradient of the cost function with respect to these parameters if the driving mechanisms are not external forcings.…”

Development of variational approach for direct and inverse problems of atmospheric hydrodynamics and chemistry

“…The heat equation (6) and the equation for pressure (7) are constructed with respect to the first law of thermodynamics, equations of continuity (8), and state (9). This variant of the model is suitable for studying nonhydrostatic processes in the environmen tal problems with intense forcings.…”

“…To con struct a modeling system with the estimated uncer tainties, we formulate the variational principle with weak constraints [6,7,11,12]. To obtain a compatible representation for complex mathematical models of different scale processes, we follow the recommenda tions of O.A.…”

“…The weight function is assigned different from zero for If the number of observation types is greater than one, it is necessary to form a common functional for the entire system of observations in D t from the functionals of a type (17). To solve direct and inverse problems, we use a varia tional principle in the statement with weak constraints [6,7,11,12]. For this purpose, starting from (14), (15), we calculate an extended functional that unites all elements of the modeling system (3)- (17) in the discrete representation: (18) Here, is the objective functional of the type (16); are the weight coefficients.…”

“…Recent studies at the laboratory develop the con cept of environment oriented prediction [3][4][5][6][7] based on solving direct and inverse problems formulated within the framework of the variational principle. Under this approach, direct modeling offers models of processes of a different degree of complexity with respect to functional content and detalization and the consideration of different factors.…”