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.
