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Chapter 15: Multivariate Statistical Modeling:
15.3.5 Example
In the tropics, there are two phenomena with similar time scales. One is
the EI Niiio Southern Oscillation (ENSO) which is the dominant interannual
signal in the tropical atmosphere and ocean. The other is the Quasi-Biennial
Oscillation (QBO) in the lower stratosphere. Theoretically, it is shown (an
overview is given by Holton, 1983) that the QBO is driven by the inter action
between the stratospheric mean fiow and vertically propagating equatorial
waves generated in the troposphere. It is commonly assumed that equatorial
convective disturbances which display large changes during an ENSO event
might e~cite these waves in the upper troposphere. The fact that the wave
sources probably have substantial interannual variability related to the ENSO
suggests a possible link between the QBO and the SO. Such a link would be
of importance for the ENSO forecast.
In order to investigate a possible link, (15.1) was fitted to time series which
contained both surface data [gridded equatorial zonal wind at surface and sea
surface temperature (SST)] and stratospheric data (zonal wind at 6 levels
between 70 hPa to 15 hPa) (Xu, 1992). Surface wind, SST and stratospheric
wind were normalized to have the same variance in the combined time series.
The first 10 EOFs ofthe combined times series were used to form the "signal"
subspace. The system matrix A and its eigenvectors (POPs) were estimated
(see Section 15.4) from this combined time series.
Figure 15.1 shows real (thin lines) and imaginary parts (thick line) of two
POPs of the estimated A. POPI (Figure 15.1a) has large amplitude in stratospheric winds whereas POP2 (Figure 15.1b) has large amplitude in surface
winds and SST. As discussed in Sections 15.3.2 and 15.3.3, a complex POP
picks up two patterns in the "signal" subspace which oscillate coherently
around frequency f = /:; given by the corresponding eigenvalue A = IA le"' .
The estimated frequency for POPI is 11 ~ 1/(28 months) and for POP2
12 ~ 1/(45 months). Since the real and imaginary part of each complex
POP are 90 0 out of phase around f;, the stratospheric wind patterns in
Figure 15.1a describe propagation of easterly anomalies from about 20 hPa
(imaginary part) to ab out 40 hPa (real part) Td4 = 1/(411) = 7 months
later. A propagation is also seen in Figure 15.1b for the surface winds with
an eastward propagation of westerly anomalies from Indian Ocean to Indonesian regions within about T2/4 = 1/(412) = 11 months. The extreme values
in SST of POP2 are in nearly the same locations in the real and imaginary
part of POP2 (Figure 15.1b). This behavior suggests that the SST signal is
characterized by a strengthening of standing anomalies, especially over the
central and eastern Pacific, rather than by a propagation of the anomalies.
The autospectrum and cross-spectrum between the corresponding POP
coefficients are shown in Figure 15.2. In these spectra, a clockwise rotation
(from real to imaginary part) is defined as positive. The POP frequencies
11 and 12 are therefore negative. As indicated by (15.17), the autospectrum
of a complex POP coefficient is not symmetric about zero frequency, that
Chapter 15: Multivariate Statistical Modeling:
15.3.5 Example
In the tropics, there are two phenomena with similar time scales. One is
the EI Niiio Southern Oscillation (ENSO) which is the dominant interannual
signal in the tropical atmosphere and ocean. The other is the Quasi-Biennial
Oscillation (QBO) in the lower stratosphere. Theoretically, it is shown (an
overview is given by Holton, 1983) that the QBO is driven by the inter action
between the stratospheric mean fiow and vertically propagating equatorial
waves generated in the troposphere. It is commonly assumed that equatorial
convective disturbances which display large changes during an ENSO event
might e~cite these waves in the upper troposphere. The fact that the wave
sources probably have substantial interannual variability related to the ENSO
suggests a possible link between the QBO and the SO. Such a link would be
of importance for the ENSO forecast.
In order to investigate a possible link, (15.1) was fitted to time series which
contained both surface data [gridded equatorial zonal wind at surface and sea
surface temperature (SST)] and stratospheric data (zonal wind at 6 levels
between 70 hPa to 15 hPa) (Xu, 1992). Surface wind, SST and stratospheric
wind were normalized to have the same variance in the combined time series.
The first 10 EOFs ofthe combined times series were used to form the "signal"
subspace. The system matrix A and its eigenvectors (POPs) were estimated
(see Section 15.4) from this combined time series.
Figure 15.1 shows real (thin lines) and imaginary parts (thick line) of two
POPs of the estimated A. POPI (Figure 15.1a) has large amplitude in stratospheric winds whereas POP2 (Figure 15.1b) has large amplitude in surface
winds and SST. As discussed in Sections 15.3.2 and 15.3.3, a complex POP
picks up two patterns in the "signal" subspace which oscillate coherently
around frequency f = /:; given by the corresponding eigenvalue A = IA le"' .
The estimated frequency for POPI is 11 ~ 1/(28 months) and for POP2
12 ~ 1/(45 months). Since the real and imaginary part of each complex
POP are 90 0 out of phase around f;, the stratospheric wind patterns in
Figure 15.1a describe propagation of easterly anomalies from about 20 hPa
(imaginary part) to ab out 40 hPa (real part) Td4 = 1/(411) = 7 months
later. A propagation is also seen in Figure 15.1b for the surface winds with
an eastward propagation of westerly anomalies from Indian Ocean to Indonesian regions within about T2/4 = 1/(412) = 11 months. The extreme values
in SST of POP2 are in nearly the same locations in the real and imaginary
part of POP2 (Figure 15.1b). This behavior suggests that the SST signal is
characterized by a strengthening of standing anomalies, especially over the
central and eastern Pacific, rather than by a propagation of the anomalies.
The autospectrum and cross-spectrum between the corresponding POP
coefficients are shown in Figure 15.2. In these spectra, a clockwise rotation
(from real to imaginary part) is defined as positive. The POP frequencies
11 and 12 are therefore negative. As indicated by (15.17), the autospectrum
of a complex POP coefficient is not symmetric about zero frequency, that