274
PIERRE BRASSEUR
to simulate the transition of the state vector up to time t i+1 ,
x
f
i+1 = M(t i , t i+1 )x
a
i
(1)
where x
f
i+1 is the vector describing the “forecast” state of the system.
At instant t i+1 , another piece of useful information is available about
the state, collected in the observation vector y i+1 of dimension p. From
the two independent pieces of information x
f
i+1 and y i+1 , how can the
true state of the system x
t
i+1 be best estimated at time t i+1 ? To answer
this question, we need to know more about the precision of the dierent
pieces of information.
2.2
Uncertainties and PDFs
The precision of the forecast x
f
i+1 can be quantified in terms of errors
on the initial guess and numerical model errors. The dierence between
the initial guess x a
i and the true state at time t i is the error vector noted
a
i = x a
i x t
i . Of course, its value is unknown but we can make a number
of assumptions about its statistical properties: we will assume that the
estimation x a
i is unbiased ( a
i = 0, where the overbar represents the
expected value), and its error a
i is distributed as a gaussian, multivariate
random variable. The corresponding probability density function (pdf)
is
a
i $ N (0, P
a
i ) exp
1
2
a T
i P
a 31
i
a
i
(2)
where P
a
i = a
i a T
i is the n × n error covariance matrix associated with
x
a
i and
T denotes the transpose. Error covariances are formally obtained
by multiplying an error vector by its transpose and averaging over many
realizations, leading to symmetric and positive definite matrices. Morrison [1988] contains excellent background information on multivariate
statistical methods. Similarly, the model operator M(t i , t i+1 ) is imperfect and the simulation error is noted as follows:
= M(t i , t i+1 )x
t
i x
t
i+1
(3)
Again, the individual realization of this error is unknown (otherwise a
perfect model operator could be run) but its statistical distribution is
assumed to be gaussian and centered (= 0):
$ N(0, Q) exp
1
2
T Q
31
(4)
where Q = T is the n × n model error covariance matrix. We assume
in addition that
a
i and are uncorrelated: a
i T = 0. In general, these
PIERRE BRASSEUR
to simulate the transition of the state vector up to time t i+1 ,
x
f
i+1 = M(t i , t i+1 )x
a
i
(1)
where x
f
i+1 is the vector describing the “forecast” state of the system.
At instant t i+1 , another piece of useful information is available about
the state, collected in the observation vector y i+1 of dimension p. From
the two independent pieces of information x
f
i+1 and y i+1 , how can the
true state of the system x
t
i+1 be best estimated at time t i+1 ? To answer
this question, we need to know more about the precision of the dierent
pieces of information.
2.2
Uncertainties and PDFs
The precision of the forecast x
f
i+1 can be quantified in terms of errors
on the initial guess and numerical model errors. The dierence between
the initial guess x a
i and the true state at time t i is the error vector noted
a
i = x a
i x t
i . Of course, its value is unknown but we can make a number
of assumptions about its statistical properties: we will assume that the
estimation x a
i is unbiased ( a
i = 0, where the overbar represents the
expected value), and its error a
i is distributed as a gaussian, multivariate
random variable. The corresponding probability density function (pdf)
is
a
i $ N (0, P
a
i ) exp
1
2
a T
i P
a 31
i
a
i
(2)
where P
a
i = a
i a T
i is the n × n error covariance matrix associated with
x
a
i and
T denotes the transpose. Error covariances are formally obtained
by multiplying an error vector by its transpose and averaging over many
realizations, leading to symmetric and positive definite matrices. Morrison [1988] contains excellent background information on multivariate
statistical methods. Similarly, the model operator M(t i , t i+1 ) is imperfect and the simulation error is noted as follows:
= M(t i , t i+1 )x
t
i x
t
i+1
(3)
Again, the individual realization of this error is unknown (otherwise a
perfect model operator could be run) but its statistical distribution is
assumed to be gaussian and centered (= 0):
$ N(0, Q) exp
1
2
T Q
31
(4)
where Q = T is the n × n model error covariance matrix. We assume
in addition that
a
i and are uncorrelated: a
i T = 0. In general, these