Meteorological Data Assimilation -
Local Analysis and Efficient Numerical Techniques
for Constrained Differential Optimization
Abstract
The
aim of this thesis is to present a new
local analysis tool for analyzing and solving the
meteorological
data assimilation problem. This analysis is based on the
representation
in form of wavepackets and extents the standard Fourier analysis
for differential optimization problems governed by elliptic
partial differential equations to non-elliptic problems.
In
our study the data assimilation which is an
inverse problem is formulated as a constrained optimization problem,
where
the cost functional consists of a distance function and the constraint
is given by a model problem. An accurate model for the data
assimilation
involves the Navier-Stokes equations and in certain cases the Euler
equations.
In order to understand diverse intricate aspects of the problem
we
have focused on the Euler case and formulated three model problems
which
aim at tackling some of the basic difficulties faced in the real-life
problem.
We study the effect of dissipation and dispersion in different
discretization
schemes on the identifiability in the problem. Basically, dissipation
will
result in bad estimation far from the measurement locations due to loss
of information as waves propagate.
We
demonstrate results for several model cases,
including the advection equation and a wave equation. For this purpose,
we use different types of measurement in terms of location of the
measurements
and the amount of data.
Keywords: Constrained Optimization problem, Data Assimilation, Differential Optimization, Dissipation, Local Fourier Analysis
