| Home > Newsevents > Training > Rcourse_notes > GENERAL_CIRCULATION > GENERAL_CIRCULATION > |
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
| GENERAL CIRCULATION | |||||
|
Table of contents >>
Next Section >>
Previous Section >>
2 . The Liouville equation
The evolution equations in a climate or weather prediction model are conventionally treated as deterministic. These (N dimensional) equations, based on spatially-truncated momentum, energy, mass and composition conservation equations will be written schematically as
|
(1) |
where
describes an instantaneous state of the climate system in 
-dimensional phase space. Eq.
(1) is fundamentally nonlinear and deterministic in the sense that,
for any initial state
, the equation determines a unique forecast state 
. (As described in Section
3 below, information from meteorological observations are combined with
a prior background state through a process called data analysis and assimilation.
In meteorology, the initial state is often referred to as the initial `analysis'
- hence the subscript `a'.) The meteorological and oceanic observing network is sparse over many parts of the world, and the observations themselves are obviously subject to measurement error. The resulting uncertainty in the initial state can be represented by the pdf
; given a volume 
of phase space, then 
is the probability that the true initial state 
at time 
lies in 
. If 
is bounded by an isopleth of 
(i.e. co-moving in phase space), then, from the determinism of Eq.
(1), the probability that 
lies in 
is time invariant. Hence, (similar to the mass continuity equation in physical
space), the evolution of 
is given by the Liouville conservation equation (introduced in a meteorological
context by Gleeson, 1966,
and Epstein, 1969)
|
(2) |
where
is given by Eq. (1). In
the second term of Eq. (2),
there is an implied summation over all the components of 
. Fig. 1 illustrates schematically the evolution of an isopleth of
. For simplicity we assume the initial pdf is isotropic (e.g. by applying
a suitable coordinate transformation). In the early part of the forecast,
the isopleth evolves in a way consistent with linearised dynamics; the N-ball
at initial time has evolved to an N-ellipsoid at forecast time 
. For weather scales of 0(103) km, this linear phase lasts for
about 1-2 days into the forecast. Beyond this time, the isopleth starts
to deform nonlinearly. The third schematic shows the isopleth at a forecast
range in which errors are growing nonlinearly. Predictability is finally
lost when the forecast pdf 
has evolved irreversibly to the invariant distribution ?inv of the attractor.
This is shown schematically in Fig.
1 using the Lorenz
(1963) attractor - a `toy-model' surrogate of the real climate attractor
(Palmer, 1993a).
-dimensional phase space. (a) At initial time, (b) during the linearised
stage of evolution. A (singular) vector pointing along the major axis
of the pdf ellipsoid is shown in (b), and its pre-image at initial time
is shown in (a). (c) The evolution of the isopleth during the nonlinear
phase is shown in (c); there is still predictability, though the pdf is
no longer Gaussian. (d) Total loss of predictability, occurring when the
forecast pdf is indistinguishable from the attractor's invariant pdf.
As mentioned in the introduction, the growth of the pdf through the forecast range is a function of the initial state. This can be seen by considering a small perturbation
to the initial state 
. From Eq. (1), the evolution
equation for 
is given by
|
(3) |
where the Jacobian is defined as
|
(4) |
Since
is at least quadratic in 
, then 
is at least linearly dependent on 
. This dependency is illustrated in Fig.
2 showing the growth of an initial isopleth of an idealised pdf at
three different positions on the Lorenz
(1963) attractor. In the first position, there is little growth, and hence
large local predictability. In the second position there is some growth
as the pdf evolves towards the lower middle half of the attractor. In the
third position, initial growth is large, and the resulting predictability
is correspondingly small. 
The nonlinear phase of pdf evolution can be much longer than the linear phase. For example, Smith et al. (1999) have studied the evolution of an initial pdf on the Lorenz (1963) attractor using a Monte Carlo process. The initial pdf was obtained by adding some notional prescribed 'observation' error to points on the attractor. The initial pdf is sharp, consistent with a small 'observation' error, and initially spreads out in a way consistent with linear theory. The pdf resharpens as it enters the region of phase space where small perturbations decay with time (cf Fig. 2 ), and then bifurcates, leading to a highly non-normal distribution. The existence of such bimodal behaviour indicates that it may not be sufficient to describe forecast uncertainty in terms of a simple `error bar'.
As shown in Ehrendorfer (1994a), the Liouville equation can be formally solved to give the value of
at a given point 
in phase space at forecast time 
. Specifically
|
(5) |
where '
' denotes the trace operation. The point in this equation corresponds to
that initial point, which, under the action of Eq.
(1) evolves to the given point 
at time 
. 
Using the identity
, then Eq. (5), can be
written as
|
(6) |
where
|
(7) |
is the so-called forward tangent propagator, mapping a perturbation
, along the nonlinear trajectory from 
to 
to
|
(8) |
A simple example which illustrates this solution to the Liouville equation is given in Fig. 3 , for a 1 dimensional Riccati equation (Ehrendorfer, 1994a)
|
(9) |
where
, based on an initial Gaussian pdf. The pdf evolves away from the unstable
equilibrium point at 
and therefore reflects the dynamical properties of Eq.
(9). Within the integration period, this pdf has evolved to the nonlinear
phase. The forward tangent propagator plays an important role in meteorological data assimilation systems; see Section 3 below. However, even though the forward tangent propagator may exist as a piece of computer code, this does not mean that the Liouville equation can be readily solved for the weather prediction problem. Firstly, the determinant of the forward tangent propagator is determined by the product of all its singular values (see Section 5). For a comprehensive weather prediction model, a determination of the full set of O(107) singular values is currently impossible. Secondly, the inversion of Eq. (1) to find an initial state
, given a forecast state 
, is itself problematic. Even on timescales of a day or so, decaying phase-space
directions (as determined by the existence of small singular values of the
propagator, see Section 5)
will lead to the inversion being poorly conditioned (Reynolds
and Palmer, 1998). Thirdly, a particular type of weather at a particular
location is not related 1-1 with a state 
of the climate system. For example, to estimate the probability of it raining
in London two days from now, we would have to apply Eq.
(6) and the inversion to find 
, to each state on the climate attractor, for which it is raining in London.
An alternative to using the solution form (6) is to integrate the partial differential equation (2) by randomly sampling the initial pdf, and integrating each sampled point using (1); the Monte-Carlo solution. However, the problem of dimensionality continues to be a significant issue. If phasespace is
dimensional, then, even in the linear phase, O(N2) integrations
will be needed to determine the forecast error covariance matrix. In the
nonlinear phase, many more integrations are needed to determine the pdf,
as it begins to wrap itself around the attractor. Ehrendorfer (1994b) has
shown that even for a 3-dimensional dynamical system, a Monte-Carlo sampling
of O(102) points can be insufficient to determine the pdf within
the nonlinear range. Yet another method of solution of the Liouville equation is possible, writing Eq. (2) in terms of an infinite hierarchy of equations for the moments of
, and applying some closure to this set of moments (Epstein,
1969). This method is certainly useful for evolving the pdf within the linearised
phase, and indeed forms the basis of the so-called Kalman filter approaches
to data assimilation (see Section
3 below). Ehrendorfer
(1994b) has shown that in the nonlinear phase, substantial errors in estimating
the first and second moments of 
can arise from neglecting third and higher order moments. A more sophisticated
approach to closure is to use arguments from turbulence theory (see Section
5) to seek scaling relations between moments (Frisch,
1995). Nicolis and Nicolis
(1998) have studied an approach in which high order moments are expressed
as time-independent functionals of low-order moments, based on a study of
dynamical systems which showed that subsets of moments vary on a timescale
given by the dominant eigenvalues of the Liouville operator 
, defined in Eq. (2).
In general, however, this method of moment decomposition has not yet been
studied in the context of realistic weather and climate systems.
In conclusion, whilst a formal analytic solution can be found to the problem of predicting the forecast pdf, there are practical problems associated with the dimension of the underlying dynamical system. However, the issue of dimensionality affects the problem in other, more insidious, ways. These are discussed in the next two sections.
Training Course Notes Front Page >>
Table of contents >>
Next Section >>
Previous Section >>
Copyright © 2003, ECMWF. All rights reserved.
 |
13.05.2003 |
 |
© ECMWF |
