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2.2 Singular vectors
Let us try to quantify further the linear stage of evolution of the forecast
PDF between Fig 1a and Fig lb. Let us suppose our basic system is
described by the (m-dimensional) nonlinear evolution equation
dX = M[X]
dt
(2.1)
Consider a small perturbation x of the state vector X. For sufficiently
short time intervals, its evolution can be described by the linearised approximation
dx
-=M1x
dt
(2.2)
of (1). Ml == ~ IX(t) is the linear evolution operator evaluated on the
nonlinear trajectory X(t).
Equation (2.2) can be written in the integral form
x(t) = L(t, to)x(to)
(2.3)
In practice, we estimate Lin (2.3) by splitting up the trajectory into many
short quasi- stationary segments. For each segment we can write
(2.4)
(Because the full trajectory segment is time-varying, we cannot, in general,
write L(t, to) in terms of the operator exponential.)
The operator L(t, to) is referred to as the forward tangent propagator; it
maps small perturbations along the (nonlinear) trajectory from an initial
time to to some future time t. For the application to weather prediction,
if x(to) is the typical error in the initial conditions for a weather forecast,
then (2.2) and (2.3) hold for approximately 2-3 days of integration time.
We now define an inner product (x; y), which in turn defines a metric
on the tangent space. Following the discussion above this inner product
is not arbitrary, it is defined so that the PDF of the initial conditions
is isotropic. As discussed in section 2.7, the atmospheric PDF appears
reasonably isotropic using an inner product based on total perturbation
energy. Using (2.3), the perturbation norm at time t is given by
Ilx(t)112 == (x(t);x(t)) = (x(to);L*Lx(to))
(2.5)