Section 14.4: Climatic Applications of SSA
269
having enormous covariance matrices to diagonalize, a prior EOF expansion
is performed on the data, and the first 10 principal components, the "channels" , are retained for the analysis. This reduction has the advantage also of
filtering synoptic motions which we are not interested in anyway. With these
L = 10 "channels", a window length m* of 200 days and a sampling rate Ll
of 5 days, the size of the covariance matrix is 400.
Since no time filter is applied beforehand, the first two ST-EOFs sticking out represent the seasonal cyde. They explain about 50% of the total
variance. Next, in the eigenvalue spectrum, one finds an osci11atory pair (satisfying the phase quadrature relationships) for eigenmodes 7 and 8, with a
period of about 70 days. Figure 14.1 displays ST-EOFs 7 and 8 in eaeh of
the 10 channels. Eaeh pair of curves represents the value of the ST-EOFs as
a function of the lag, each channel being associated to one spatial EOF. First
of a11, one remarks that the amplitude varies a lot from a ehannel to another.
The channel where the amplitude is the largest is the second, meaning that
the associated oscillation projeets mostlyon the second EOF, which is the
famous NAO pattern of Wall ace and Gutzler (1981). If its components on
other channels were exactly zero, then the oscillation would be a standing
oscillation of the second EOF. This is not the case, and we see partieularly
that the phase differs from a channel to another. Therefore we expect a
propagative spaee-time pattern.
In Figure 14.1 we displayed the ST-EOFs 7 and 8 in the lag spaee, i.e., for
each channel, as a function of the lag. ST-EOFs ean also be represented in
physical space as a sequence of maps. In order to be slightly more general,
we display next on Figure 14.2 the composite half-eyde of the oseillation as
a function of its phase. One could have used ST-PCs in the complex plane as
explained in Section 13.3.2. In fact, the phase used to key the composites of
Figure 14.2 are based on a slightly more accurate index, the reconstruction
filter, ealculated also from the two ST-PCs 7 and 8 [see Plaut and Vautard
(1994) for more details]. From this figure one sees that the oscillation has
a poleward-propagating component in the region of the Atlantic jet, with
a standing component over Siberia. Its amplitude is maximal, as expected,
when it is in phase with the NAO (Figure 14.2d).
As explained above, SSA or MSSA allow the study of the explained variance in the frequency domain. Figure 14.3 shows the Maximum Entropy
speetra of several ST-PCs, divided by the total variance. Each curve represents therefore the fr action of variance explained by one ST-PC as a function
of frequency. In particular, we recover the two ST-EOFs 7 and 8 peaking
near 70 days, explaining about 30% of the varianee at this period. Another
remarkable oscillatory pair is the pair 12 and 13 peaking at aperiod of 32
days. ST-PCs 1 and 2 peak out of the graph exaetly at aperiod of 1 year.
ST-PCs 3 to 6 explain most of the varianee between 60 days and 1 year.
Another interesting feature is the fact that the dominant oseillations have
periods of, roughly speaking, 70 and 35 days. The study of the phase index
269
having enormous covariance matrices to diagonalize, a prior EOF expansion
is performed on the data, and the first 10 principal components, the "channels" , are retained for the analysis. This reduction has the advantage also of
filtering synoptic motions which we are not interested in anyway. With these
L = 10 "channels", a window length m* of 200 days and a sampling rate Ll
of 5 days, the size of the covariance matrix is 400.
Since no time filter is applied beforehand, the first two ST-EOFs sticking out represent the seasonal cyde. They explain about 50% of the total
variance. Next, in the eigenvalue spectrum, one finds an osci11atory pair (satisfying the phase quadrature relationships) for eigenmodes 7 and 8, with a
period of about 70 days. Figure 14.1 displays ST-EOFs 7 and 8 in eaeh of
the 10 channels. Eaeh pair of curves represents the value of the ST-EOFs as
a function of the lag, each channel being associated to one spatial EOF. First
of a11, one remarks that the amplitude varies a lot from a ehannel to another.
The channel where the amplitude is the largest is the second, meaning that
the associated oscillation projeets mostlyon the second EOF, which is the
famous NAO pattern of Wall ace and Gutzler (1981). If its components on
other channels were exactly zero, then the oscillation would be a standing
oscillation of the second EOF. This is not the case, and we see partieularly
that the phase differs from a channel to another. Therefore we expect a
propagative spaee-time pattern.
In Figure 14.1 we displayed the ST-EOFs 7 and 8 in the lag spaee, i.e., for
each channel, as a function of the lag. ST-EOFs ean also be represented in
physical space as a sequence of maps. In order to be slightly more general,
we display next on Figure 14.2 the composite half-eyde of the oseillation as
a function of its phase. One could have used ST-PCs in the complex plane as
explained in Section 13.3.2. In fact, the phase used to key the composites of
Figure 14.2 are based on a slightly more accurate index, the reconstruction
filter, ealculated also from the two ST-PCs 7 and 8 [see Plaut and Vautard
(1994) for more details]. From this figure one sees that the oscillation has
a poleward-propagating component in the region of the Atlantic jet, with
a standing component over Siberia. Its amplitude is maximal, as expected,
when it is in phase with the NAO (Figure 14.2d).
As explained above, SSA or MSSA allow the study of the explained variance in the frequency domain. Figure 14.3 shows the Maximum Entropy
speetra of several ST-PCs, divided by the total variance. Each curve represents therefore the fr action of variance explained by one ST-PC as a function
of frequency. In particular, we recover the two ST-EOFs 7 and 8 peaking
near 70 days, explaining about 30% of the varianee at this period. Another
remarkable oscillatory pair is the pair 12 and 13 peaking at aperiod of 32
days. ST-PCs 1 and 2 peak out of the graph exaetly at aperiod of 1 year.
ST-PCs 3 to 6 explain most of the varianee between 60 days and 1 year.
Another interesting feature is the fact that the dominant oseillations have
periods of, roughly speaking, 70 and 35 days. The study of the phase index