OCEAN DATA ASSIMILATION
273
several reasons why most assimilation methods used today in operational
forecasting systems are inspired by the statistical approach. In the context of GODAE (Global Ocean Data Assimilation Experiment), one objective will be to test and compare the bunch of algorithms implemented
in operational systems, in order to better understand the importance of
the various possible approximations made in each of these.
The objectives of this chapter are to review the fundamentals of sequential data assimilation for ocean state estimation and to expose the
basic ingredients of practical assimilation algorithms developed for applied ocean research and operational systems, focusing mainly on highresolution applications. Section 2 is dedicated to the fundamentals of
applied estimation methods leading to the KF equations. Numerical representations of the mathematical objects introduced by the theory will
then be illustrated using oceanographic examples in Section 3. Section
4 will provide a brief description of traditional simplifications of the KF.
In Section 5, we discuss various approaches to reduce the size of the estimation problem and, in Section 6, we derive the framework of low-rank
Kalman filters. The important question of the verification of consistency
will be addressed in Section 7, where the concept of adaptivity is also
mentioned. Finally, a number of advanced implementation issues such
as the transition to incremental/smoothing algorithms will be discussed
in Section 8, before concluding the chapter.
2.
Kalman filtering: Fundamentals
2.1
Problem definition
In this section, we introduce the basic assimilation problem in the
state space using the conventional notations proposed by Ide et al.
[1997]. The goal here is not to present a rigorous and comprehensive
derivation of the Kalman filter, which can be found elsewhere in dedicated text books (e.g., Gelb [1974]), but rather to introduce a simplified
framework that still contains the essential characteristics needed to illustrate the more advanced concepts and implementation issues discussed
in the following sections.
To start, let us assume that some a priori knowledge about the state
of the ocean is available at time t i , represented by vector x
a
i . A physical
model is also available to describe the transition of the state vector
from time t i to time t i+1 , which is represented by a numerical (matrix)
operator M(t i , t i+1 ). A linear model will be considered at this stage.
The dimension of the state space is noted as n. The state vector x
contains the minimum set of independent variables needed to perfectly
characterize the state of the system at any time. The model can be used
273
several reasons why most assimilation methods used today in operational
forecasting systems are inspired by the statistical approach. In the context of GODAE (Global Ocean Data Assimilation Experiment), one objective will be to test and compare the bunch of algorithms implemented
in operational systems, in order to better understand the importance of
the various possible approximations made in each of these.
The objectives of this chapter are to review the fundamentals of sequential data assimilation for ocean state estimation and to expose the
basic ingredients of practical assimilation algorithms developed for applied ocean research and operational systems, focusing mainly on highresolution applications. Section 2 is dedicated to the fundamentals of
applied estimation methods leading to the KF equations. Numerical representations of the mathematical objects introduced by the theory will
then be illustrated using oceanographic examples in Section 3. Section
4 will provide a brief description of traditional simplifications of the KF.
In Section 5, we discuss various approaches to reduce the size of the estimation problem and, in Section 6, we derive the framework of low-rank
Kalman filters. The important question of the verification of consistency
will be addressed in Section 7, where the concept of adaptivity is also
mentioned. Finally, a number of advanced implementation issues such
as the transition to incremental/smoothing algorithms will be discussed
in Section 8, before concluding the chapter.
2.
Kalman filtering: Fundamentals
2.1
Problem definition
In this section, we introduce the basic assimilation problem in the
state space using the conventional notations proposed by Ide et al.
[1997]. The goal here is not to present a rigorous and comprehensive
derivation of the Kalman filter, which can be found elsewhere in dedicated text books (e.g., Gelb [1974]), but rather to introduce a simplified
framework that still contains the essential characteristics needed to illustrate the more advanced concepts and implementation issues discussed
in the following sections.
To start, let us assume that some a priori knowledge about the state
of the ocean is available at time t i , represented by vector x
a
i . A physical
model is also available to describe the transition of the state vector
from time t i to time t i+1 , which is represented by a numerical (matrix)
operator M(t i , t i+1 ). A linear model will be considered at this stage.
The dimension of the state space is noted as n. The state vector x
contains the minimum set of independent variables needed to perfectly
characterize the state of the system at any time. The model can be used