Handling of Some Classes of Inverse Problems Part A: Adjoint Methods and their Application in Earth Sciences

The subjects of this chapter are the adjoint methods, which are widely used in environmental sciences. The adjoint methods are based on contemporary results of mathematical analysis, like variational calculus and functional analysis. The first nonmathematician users of the technique were the great generation of nuclear physicists in the 20th century. Actually the method was first transferred to Earth sciences by them, and has been used in this field successfully since the 1970's. The earliest Earth science applications appeared in meteorology, but for today it is a widespread technique in all branches of Earth science like oceanography or geophysics. In the 21st century its use widened to the field of almost all natural sciences, like chemistry, biology, etc. It is basically an inverse method, which utilizes the notion of adjoint operator of the considered model operator. The adjoint operator provides a duality between a model inputs and outputs, this way it is an efficient tool for sensitivity studies or for optimization problems. Probably the greatest success of the method in Earth sciences, at least in meteorology, is the basis of the so called ensemble forecasting, which is considered as the numerical forecasting method of the future. The adjoint functions act like backward signal transmitters, they can reveal the sensitive or unstable parts of a considered dynamical system. Following from this feature they have definite physical meaning, and give an insight how the given dynamical system is functioning. In this paper the most important mathematical formulations of the method are described and also the most important applications are introduced like sensitivity analysis, variational data assimilation, and finally the use of singular vectors in ensemble forecasting and in the method of targeted observations.

Keywords: Banach space, Hilbert space, non-linear dynamical system/operator equations, initial-boundary value problems, sensitivity analysis, direct method, adjoint (inverse)method, response functional, sensitivity of a chosen response, sensitivity to initial/boundary condition and parameter perturbations, tangent linear operator/equations, adjoint operator/equations, nuclear engineering, G.I. Marchuk, D.G. Cacuci, Earth sciences, meteorology, numerical weather prediction, climatology, oceanology, ensemble forecasting, singular vectors, targeted observations, 3D and 4D variational data assimilation, physical meaning and analysis of adjoints and singular vectors, signal transmission mechanism in non-linear systems.