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Chapter 15: Multivariate Statistical Modeling:
the other n-m modes. However, in the case m > n, m - n of the m POPs
cannot be interpreted, since they normally do not fit to the chosen process
and are considered to be useless. Since n is unknown, there is no strict way
to determine which m of n POPs are useless.
Although there is no complete solution to this problem, a rule-of-thumb is
often used in practice. Since all processes are approximated by (15.1), the
spectrum of the estimated POP coefficient must have the form of (15.16)
and (15.17), which is a function of the estimated eigenvalue. In the case of
a complex POP, the real and imaginary part of the POP coefficient should
additionally be significantly coherent and 90 o -out-of phase around the estimated POP period. We therefore select only those mo des whose coefficients
satisfy these conditions.
15.5 POPs as Normal Modes
Equation (15.1) represents also the discrete form of a first order differential
equation with A being the discrete version of a linear operator. The minimization in (15.25) suggests that the estimated system matrix A describes
optimally the approximated linear dynamics of the considered system in a
least squares sense. Thus, the POPs 'are also referred to as the estimated
normal mo des of the system.
The ability of a POP analysis in identifying normal mo des of a system
is demonstrated by the example of medium-scale synoptic waves in the atmosphere (Schnur et al., 1993). In this study, the linear system matrix A
has been obtained in two conceptually different ways. In one case, it was
estimated from observational data via (15.25) and in the other it was derived
theoretically from the quasigeostropic equations, assuming small disturbances
(linear stability analysis). Since the theoretical normal modes are known in
this example, a comparison between the POPs provides insights into the
interpretation of the POPs.
In the study of Schnur et al. (1993), the state vector X was the same in
both empirical and theoritical approaches and consists of Fourier expansion
coefficients along each latitude and at each pressure level. This construction
of x allows one to study the meridional-height structure of the waves. No
zonal dependence of the waves was considered. The theoretical A was a
function of a zonally symmetric basic state.
Schnur et al.'s theoretical A describes the linear mechanism of baroclinic
instability which generates the synoptic waves, whereas their empirical A
contains in general linear approximations of all possible processes. Although
they limited them by using filtered data, the empirical A contains still more
information than the theoretical A. One candidate which operates on the
same time scales is the one which causes nonlinear decaying of the waves.
Thus, the empirical A may contain information about processes which generate and damp the waves. At this point, it is unclear whether a linearization
Chapter 15: Multivariate Statistical Modeling:
the other n-m modes. However, in the case m > n, m - n of the m POPs
cannot be interpreted, since they normally do not fit to the chosen process
and are considered to be useless. Since n is unknown, there is no strict way
to determine which m of n POPs are useless.
Although there is no complete solution to this problem, a rule-of-thumb is
often used in practice. Since all processes are approximated by (15.1), the
spectrum of the estimated POP coefficient must have the form of (15.16)
and (15.17), which is a function of the estimated eigenvalue. In the case of
a complex POP, the real and imaginary part of the POP coefficient should
additionally be significantly coherent and 90 o -out-of phase around the estimated POP period. We therefore select only those mo des whose coefficients
satisfy these conditions.
15.5 POPs as Normal Modes
Equation (15.1) represents also the discrete form of a first order differential
equation with A being the discrete version of a linear operator. The minimization in (15.25) suggests that the estimated system matrix A describes
optimally the approximated linear dynamics of the considered system in a
least squares sense. Thus, the POPs 'are also referred to as the estimated
normal mo des of the system.
The ability of a POP analysis in identifying normal mo des of a system
is demonstrated by the example of medium-scale synoptic waves in the atmosphere (Schnur et al., 1993). In this study, the linear system matrix A
has been obtained in two conceptually different ways. In one case, it was
estimated from observational data via (15.25) and in the other it was derived
theoretically from the quasigeostropic equations, assuming small disturbances
(linear stability analysis). Since the theoretical normal modes are known in
this example, a comparison between the POPs provides insights into the
interpretation of the POPs.
In the study of Schnur et al. (1993), the state vector X was the same in
both empirical and theoritical approaches and consists of Fourier expansion
coefficients along each latitude and at each pressure level. This construction
of x allows one to study the meridional-height structure of the waves. No
zonal dependence of the waves was considered. The theoretical A was a
function of a zonally symmetric basic state.
Schnur et al.'s theoretical A describes the linear mechanism of baroclinic
instability which generates the synoptic waves, whereas their empirical A
contains in general linear approximations of all possible processes. Although
they limited them by using filtered data, the empirical A contains still more
information than the theoretical A. One candidate which operates on the
same time scales is the one which causes nonlinear decaying of the waves.
Thus, the empirical A may contain information about processes which generate and damp the waves. At this point, it is unclear whether a linearization