Section 3.0.3 provides background information about MRA in general, and intro- duces the class of deterministic and stochastic regression models used. More de- tailed information is necessary in order to accurately interprete the MRA results. To provide the information, this section discusses the combined deterministic/- stochastic regression model more deeply, and explains important characteristics of the ozone predictors.
3.3.1
Combined regression model
Together, Equations 3.1, 3.2, and 3.4 define the combined regression model. The dependent variableYtrepresents zonal monthly mean total-column ozone in Dob-
son Units, given by a threefold ensemble mean of independent REF realisations. The three Yt time series of interest (Figure 3.3) represent the following latitudi- nal means: SH mid-latitudes (35-60◦S), NH mid-latitudes (35-60◦N), and tropics
(15◦N-15◦S). In fact, the MRA involves the full non-deseasonalised ozone data,
but, with regards to interannual variability, graphing deseasonalised values is more informative. Finally, separate sets of ozone predictors apply to each of the three latitude bands (Table 3.1), but the associated MRAs refer to the same regression model mentioned above.
Seasonal expansion of the ozone predictors
The ozone predictorsβk,t (Figure 3.1) represent typical ozone-modulating bound- ary conditions of the REF scenario, such as the QBO and ENSO. An expansion of the deterministic regression parameters xk (Equation 3.1) with sinusoidal terms (Equation 3.4) allows the relationship between ozone and any of the predictors to depend on the seasonal cycle; for instance, the impact of the QBO on mid-latitude total ozone has a large seasonal component (Section 3.0.2).
Which and how many of the sinusoidal terms in Equation 3.4 contribute to the regression equation depends on both the dependent variable Yt, as well as on the individual ozone predictors βk,t. It is decided on the basis of several diagnostics (Section 3.4.2). It can happen that an individual ozone predictor is chosen to contain, for instance, the semi-annual term, but not the lower-frequency annual oscillation. Finally, owing to aliasing effects, the monthly mean approach largely precludes sinusoidal terms of higher than quarterly order.
xk =xk,0+xk,1cos µ 2π 12 t+xk,2 ¶ | {z } annual +xk,3cos µ 2π 6 t+xk,4 ¶ | {z } semi-annual +xk,5cos µ 2π 4 t+xk,6 ¶ | {z } four-monthly +xk,7cos µ 2π 3 t+xk,8 ¶ | {z } quarterly (3.4)
The particular method of sinusoidal expansion (Equation 3.4) conforms with the regression approach byPandit and Wu (1983), and is nonlinear as it contains products of regression parameters. More common in stratospheric ozone research is a linear sinusoidal expansion of the form
xk=xk,0 +xk,1cos µ 2π 12 ¶ +xk,2sin µ 2π 12 ¶ +. . . (3.5)
but the two different approaches have both their respective advantage and draw- back. The nonlinear approach is more parsimonious as it displays a weaker
collinearity (not shown), probably because for each sinusoidal term the first pa- rameter is devoted exclusively to the amplitude and the second parameter to the phase. The linear expansion is easier to handle.
Regression initial values
FollowingPandit and Wu (1983), initial parameter values for the nonlinear com- bined model (Equations 3.1, 3.4, and 3.2) are obtained from separate fits, first of the deterministic model (Equations 3.1 and 3.4) and then of the stochastic model (Equation 3.2) using the respective deterministic residuals.
Without considering the stochastic parts, the deterministic model contains nonlinear seasonal terms (Equation 3.4) that need to be initialised. Picking op- timum initialisation values is accomplished by choosing the minimum RSS from several model fits with a range of different initialisation values for the seasonal parameters. However, it is convenient that the deterministic model is not very sensitive to initialisation (not shown). In contrast to the deterministic model
mentioned above, an ARMA(n,0) stochastic model without a moving average
part is linear and does not need to be initialised. Pandit and Wu (1983) provide a routine that yields initial parameters for a full ARMA(n, m) nonlinear model, but the implementation is beyond the scope of this study and ARMA(n, m) mod- els are ignored.
3.3.2
Design of the ozone predictors
To clarify and quantify causal relationships of total column ozone variability, the MRA adopts as ozone predictors the most important ozone-modulating boundary conditions of the E39/C scenario REF (Section 3). In order to increase the MRA deterministic response, for each ozone predictor a particular design is chosen (Figure 3.1). This section provides information how the choice is made, and briefly covers the issue of time lags.
QBO
Compared to other predictors such as ODS, ENSO, or SULF, the design of QBO response predictors is more complex because the QBO involves the tropical wind profile throughout a range of lower-stratospheric levels. For each of the three lat- itude bands of interest, a single time-lagged QBO response predictor is used, and, following Bodeker et al. (1998), accounts for the QBO seasonal synchronisation by a seasonal expansion (Equation 3.4). The MRA adopts a nonlinear expansion and, in this respect, adopts an approach which differs from Bodeker et al.(1998).
The choice of a particular time series as QBO response predictor is not straightforward and there exist different approaches. Most of the studies either incorporate a single time-lagged QBO response predictor (i.a.Weatherhead et al., 2000; Connor et al., 1999; Logan, 1994), or two QBO proxies without time lag (i.a.Fleming et al., 2007;M¨ader et al., 2007;Ziemke et al., 1997;Randel and Wu, 1996).
Depending on the study, a single lagged QBO time series may refer to the QBO-dominated tropical zonal wind velocity at a particular height level or refer to a composite of several levels; similar to the single time series, the two unlagged time series may each also refer to a single level or a level composite. The latter approach without time lags takes into account that there are two QBO ozone response patterns in the real stratosphere which are out of phase at tropical latitudes. One of the response patterns maximises in the lower stratosphere, the other one maximises in the middle stratosphere (Randel and Wu, 1996); the out-of-phase behaviour is not important in case of E39/C modelled ozone data because the CCM neglects parts of the middle stratosphere. Also, the unlagged approach facilitates the implementation of automatic stepwise regression (M¨ader et al., 2007), but goes hand in hand with a stronger collinearity (not shown). For both of the approaches mentioned above, methods akin to empirical orthogonal decomposition (EOD) may yield a QBO response predictor in form of a linear composite of several height levels that is more effective in capturing the ozone
QBO than a less sophisticated predictor (Randel and Wu, 1996). The present
analysis does not apply EOD, but a more subjective approach instead. The MRA is done for all QBO heights and some linear height combinations, finally using the most effective QBO response predictor.
A difficulty is that the optimum QBO response predictor varies with the latitude at which the ozone response is considered (Yang and Tung, 1995), and the ozone response at a given latitude is probably not purely linearly proportional to a linear composite of equatorial zonal wind velocity (Lee and Smith, 2003). Hence, the mentioned QBO proxies are unlikely to remove the whole ozone QBO signal, particularly in case of latitudinal averages.
Finally, Steinbrecht et al. (2004); Newchurch et al. (2003) apply a range of sinusoidal terms without zonal wind velocity to remove the QBO signal in strato- spheric ozone concentrations, thus treat the QBO signal as a nuisance parameter since the approach is likely to impede a causal attribution. The method does well if the focus is on trend detection, but is obviously unsuitable if the causal attribution of interannual variability itself is of interest.
ODS
It is common practice among MRA studies of total column ozone to adopt as ozone trend predictor the evolution of equivalent effective stratospheric chlorine, a
weighted integral over all ozone-depleting substances in the stratosphere (WMO, 2007). E39/C does not account for the second most important ozone-deplet- ing element bromine (Appendix A.1); Global chlorine concentrations therefore represent a suitable predictor for ODS-related trends.
SSC
Figure 3.1b shows that the SSC time series displays a significant monthly vari- ability, compared to the amplitude of the 11-year oscillation. Every linear or nonlinear MRA involves the time derivative of the ozone predictors and, hence, the monthly SSC variability tends to unwontedly bias the 11-year response to- wards the monthly fluctuations, resulting in a too weak 11-year response (not shown). To overcome this bias, a yearly running mean SSC response predictor is implemented (Figure 3.1b).
E39/C accounts for the ozone chemistry above its model domain by the pre- scription of NOy contents in the two uppermost model layers (see Appendix A.2). The NOy time series contains a SSC signal, but using NOy concentrations as SSC response predictor does not reduce the RSS more strongly than using the solar-flux-based predictor in Figure 3.1b (not shown).
ENSO
The ENSO response predictor is based on the mean SST associated with those twenty model boxes that cover the area defined by the longitude section 215.625◦-
256.875◦ and the latitude section 3.715◦N-3.715◦S. In this part of the tropical Pa-
cific, the SST interannual variability is dominated by ENSO (i.a. Rayner et al., 2003) and hence gives a suitable ENSO response predictor. To avoid correlations with the time series of both ODS and annual cycle, the ENSO time series is de- trended and deseasonalised. Finally, the resulting monthly mean ENSO response predictor is superior to multi-monthly running means (not shown).
SULF
Sulfate aerosol surfaces exclusively cause heterogeneous ozone destruction in the
presence of significant ODS concentrations (i.a. Dameris et al., 2005). ODS
weighted SULF response predictors (Figure 3.1c) are taken since these more strongly reduce the RSS than the pure SULF response predictors without weight- ing (not shown). In addition to incorporating sulfate aerosol surfaces, the REF scenario prescribes eruption-induced heating rates (Dameris et al., 2005;Kirchner et al., 1999), but these are not considered here.
Annual cycle
Many of the published MRA studies, that explicitely account for the annual cycle in monthly mean ozone data, apply a range of sinusoidal terms in the regression equation to remove the annual cycle (i.a. Fleming et al., 2007; Vyushin et al., 2007;Brunner et al., 2006; Newchurch et al., 2003;Fioletov et al., 2002;SPARC, 1998). Here, however, this approach results in an excessively high yearly residual autocorrelation of the deterministic model (not shown), signifying that sinusoidal terms do not fit the annual cycle well.
To overcome the yearly autocorrelation, this Chapter adopts an approach
similar to that in Soukharev and Hood (2006). A long-term mean annual cycle
is determined from the E39/C ozone data prior to the MRA, using the full data set length of 60 years. Then, the resulting annual cycle is implemented into the regression equation in terms of an ordinary ozone predictor without the sinusoidal expansion given by Equation 3.4. In case of the SH mid-latitudes, which display a robust trend, the annual cycle is calculated from detrended data.
Time lags
Time lags not only apply in case of the QBO, but also in case of the other monthly varying predicors ENSO and SULF. Following the approach by, i.a.,Bodeker et al.
(1998), optimum time lags (Table 3.1) are determined by fitting the full combined regression model for a range of monthly lags up to 24 months, finally adopting the lag which is associated with a minimum RSS. The remaining ozone predictors, ODS and SSC, vary on the scale of several years or more slowly and in this case time lags only negligibly reduce the RSS.