84
P. Courtier
In the next section we shall introduce linear estimation theory. Section 4.3 briefly describes the
Optimal Interpolation algorithm and then concentrates on the variational approach. Section
4.4 deals with the time dimension; it theoretically introduces the Kalman filter then the fourdimensional variational assimilation algorithm is described, together with some simplifications
which allow an operational implementation in the foreseeable future. Finally, an algorithm is
presented as an attempt to provide error bars from the variational algorithms and to improve
the treatment of the time dimension.
4.2 Introduction to Linear Estimation
In the following the random variables are underlined whereas realizations of these random
variables are not.
4.2.1 A simple illustration
Let us assume that we are interested in an estimation of the temperature of the ECMWF
Lecture Theatre. We consider the room temperature as a random variable 'L of expectation
<'L> and variance «'L- <'L»2> which are not supposed to be known.
We have a thermometer of known precision ao and we read To. The measurement To is considered as one realization of a random variable 'La of expectation < 'L > (the observation is
assumed unbiased) and variance
In the absence of any other information, the Best Linear Unbiased Estimator (BLUE) of the
room temperature is 'La = 'La and for the particular realization To of the measurement we have
Ta = To.
In other words, we have been able to produce a rather pedantic presentation of what you should
do when you want to have an idea of the room temperature - simply read the thermometer; at
least it is not against common sense!
However, if you look carefully at the ECMWF attendees of a presentation in the Lecture
Theatre, most of them are wearing a jumper: they have a priori (background) information Tb
on the room temperature (it's gonna be cold!). Tb is considered as one realization of a random
variable L of expectation <'L> and variance < (L- <'L»2 >= al
Intuitively, one realizes that the background information, combined with the observation, should
lead to a better estimate than the observation alone. We then look for the BLUE, a linear
estimate
'La = aL + /3' La
(4.1 )
which is unbiased and best (of minimum error variance). As the background information and
the observations are assumed unbiased, we get
<'L>= a <'L> +/3 <'L> .
and then a + /3 = 1.
In meterological practice (4.1) is usually rewritten as
'La = L + /3('La - L)·
P. Courtier
In the next section we shall introduce linear estimation theory. Section 4.3 briefly describes the
Optimal Interpolation algorithm and then concentrates on the variational approach. Section
4.4 deals with the time dimension; it theoretically introduces the Kalman filter then the fourdimensional variational assimilation algorithm is described, together with some simplifications
which allow an operational implementation in the foreseeable future. Finally, an algorithm is
presented as an attempt to provide error bars from the variational algorithms and to improve
the treatment of the time dimension.
4.2 Introduction to Linear Estimation
In the following the random variables are underlined whereas realizations of these random
variables are not.
4.2.1 A simple illustration
Let us assume that we are interested in an estimation of the temperature of the ECMWF
Lecture Theatre. We consider the room temperature as a random variable 'L of expectation
<'L> and variance «'L- <'L»2> which are not supposed to be known.
We have a thermometer of known precision ao and we read To. The measurement To is considered as one realization of a random variable 'La of expectation < 'L > (the observation is
assumed unbiased) and variance
In the absence of any other information, the Best Linear Unbiased Estimator (BLUE) of the
room temperature is 'La = 'La and for the particular realization To of the measurement we have
Ta = To.
In other words, we have been able to produce a rather pedantic presentation of what you should
do when you want to have an idea of the room temperature - simply read the thermometer; at
least it is not against common sense!
However, if you look carefully at the ECMWF attendees of a presentation in the Lecture
Theatre, most of them are wearing a jumper: they have a priori (background) information Tb
on the room temperature (it's gonna be cold!). Tb is considered as one realization of a random
variable L of expectation <'L> and variance < (L- <'L»2 >= al
Intuitively, one realizes that the background information, combined with the observation, should
lead to a better estimate than the observation alone. We then look for the BLUE, a linear
estimate
'La = aL + /3' La
(4.1 )
which is unbiased and best (of minimum error variance). As the background information and
the observations are assumed unbiased, we get
<'L>= a <'L> +/3 <'L> .
and then a + /3 = 1.
In meterological practice (4.1) is usually rewritten as
'La = L + /3('La - L)·