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Predicting
uncertainty in forecasts of weather and climate
(Also published as ECMWF Technical Memorandum No. 294)
By T.N. Palmer
Research Department, ECMWF
November 1999
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3 . The probability density function of initial error
In order to discuss how the pdf of initial error can be estimated in weather and climate prediction, it is necessary to outline the method by which observations are used to determine the initial conditions for a deterministic weather or climate forecast.
In meteorology and oceanography, data assimilation is a means of obtaining a forecast initial state which in some well-defined sense optimally combines the available observations for a particular time with an independent background state (Daley, 1991). This background state is usually a short-range forecast (e.g. 6 hour) from an estimate of the initial state valid at an earlier time, and this carries forward information from observations from earlier times. A very simple example of the basic notion can be illustrated by considering two different independent estimates,
and 
, of a scalar 
. Suppose that the errors associated with these two estimates are random,
unbiased and normally distributed, with standard deviations 
and 
, respectively. Then the maximum-likelihood estimate of 
is the state 
which minimises the cost function
|
(10) |
The least-squares solution
|
(11) |
is easily found. The error associated with
is normally distributed with variance given by
|
(12) |
The data assimilation technique used in weather prediction (e.g. at ECMWF) is a multi-dimensional generalisation of this technique (Courtier et al., 1994, 1998). The analysed state
of the atmospheric state vector is found by minimising the cost function
|
(13) |
where
is the background state, 
and 
are covariance matrices for the pdfs of background error and observation
error respectively, 
is the so-called observation operator, and 
denotes the vector of available observations. For example, if 
includes a radiance measurement taken by an infrared radiometer onboard
a satellite orbiting the earth then 
includes an estimate of the infrared radiance that would be emitted by a
model atmosphere as represented by the state vector 
. Similarly, if 
includes a surface pressure measurement taken at some point 
on the earth's surface, then 
includes the surface pressure at 
given 
. Since 
is finite dimensional, the operator 
inevitably involves an interpolation to 
. Similar to Eq. (12),
the Hessian of 
is given by (Fisher and Courtier,
1995)
|
(14) |
We refer to
as the analysis error covariance matrix. In the current ECMWF operational data assimilation system, the background error covariance matrix
is not dependent on the present state of the atmospheric circulation. This
is believed to introduce considerable imprecision in the estimate of the
initial pdf as given by (14).
This estimate can be improved; within the linearised regime (cf. Fig.
1 ), the forecast error covariance matrix F implied by Eqs.
(6) and (8) can be
written
|
(15) |
where
is the tangent propagator along the trajectory between the initial state
and the forecast state 
at time 
. Since the time between consecutive analyses (typically 6 hours) is broadly
within this linearised regime, then a flow-dependent estimate of the background
error covariance matrix at time 
can be obtained by propagating the analysis error covariance matrix from
the earlier analysis time 
, i.e.
|
(16) |
The propagator
, and its transpose 
are essential components of 4-dimensional data assimilation (Courtier
et al., 1994) where observations are assimilated over a time window.
Using 
, a perturbation 
can be evaluated at the same time that an observation is taken. Given the
dimension of comprehensive weather prediction models, 
is not known in matrix form, and is represented in operator form (cf. Eq.
(8)). Similarly the transpose 
is also represented in operator form 
(see Eq. (18) below) and
is known as the adjoint (tangent) propagator. However, Eq. (16) is computationally intractable for numerical weather prediction, requiring O(1014) individual linearised integrations of
for a complete specification of the propagated matrix 
. Three possible solutions have been proposed. The first is essentially
a Monte Carlo solution, whereby a random sampling of 
is evolved using 
(Evensen, 1994; Andersson
and Fisher, 1999). The second proposal involves solving the propagation
Eq. (16) with an intermediate
complexity model (Ehrendorfer,
1999). The final proposal (the so-called reduced-rank Kalman filter; Fisher,
1998) is to propagate 
explicitly only in the appropriate unstable subspace defined by the dominant
flow-dependent local instabilities of the attractor. Broadly, speaking,
the proposal is to have the best possible knowledge of the initial state
in that part of phase space from which forecast errors are most likely to
grow. At present these three different proposals are being evaluated.
Since the notion of local flow-dependent instability features strongly in later sections of this paper, it is worth outlining some more detail on how these instabilities can be estimated. First consider a Euclidean inner product <..,..> so that for any perturbations
, 
,
|
(17) |
In terms of <..,..> the adjoint tangent propagator
is defined by
|
(18) |
where
|
(19) |
for an arbitrary pair of perturbations
, 
. The analysis error covariance matrix
defines a secondary inner product
|
(20) |
Here
is the covariant form of an analysis error covariance metric, 
(Palmer et al., 1998).
Hence the perturbation 
, which has maximum Euclidean amplitude at 
and unit 
norm at initial time 
is given by
|
(21) |
This is equivalent to finding the dominant eigenvector of the generalised eigenvector equation
|
(22) |
Formally, by taking the square root of
, Eq. (22) can be transformed
to a singular vector equation which can be solved using a Lanczos algorithm
(Strang, 1986). More
generally, Eq. (22) is
solved using a generalised Davidson algorithm (Barkmeijer
et al., 1998). We refer to the solutions 
in Eq. (22) as 
-singular vectors of 
. The set of dominant
singular vectors of 
(with largest singular values 
) defines an unstable subspace in the tangent space at 
. It comprises the set of most rapidly-growing directions defined locally
on phase space, relative to a basic-state trajectory between 
and 
, subject to the constraint that the initial perturbations are normalised
with respect to the initial pdf. At forecast time 
, these singular vectors have evolved into the major axes of the forecast
error ellipsoid, or, equivalently, into the eigenvectors of the forecast
error covariance matrix. The first two parts of Fig.
1 show schematically a dominant 
singular vector at initial and forecast time. Further discussion of these
singular vectors, and their relation to more familiar forms of perturbation
growth (such as normal mode and Lyapunov exponent growth) are discussed
in Section 5.
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