The misleading aspect of the statistical analyses used in Dommenget and Latif, which raises concerns on some of the reported climate modes, is demonstrated. Adopting simple statistical techniques, the physical existence of the Indian Ocean dipole mode is shown and then the limitations of varimax and regression analyses in capturing the climate mode are discussed.
In a recent article, Dommenget and Latif (2002, hereafter DL) commented on the limitations of widely used statistical methods such as EOF and varimax analyses in identifying physical modes related to climate variability. They used specifically designed artificial examples to verify their findings. One of the targeted examples in their study was the Indian Ocean dipole (IOD) mode; DL questioned the existence of the IOD as a physical mode. We find that the concern raised by DL on the IOD issue is superficial. This is because the detection of the phenomenon was based not on the EOF analysis but on the physical and dynamical understanding of various ocean–atmosphere parameters (Saji et al. 1999; Webster et al. 1999; Behera et al. 1999; Rao et al. 2002a,b; Feng et al. 2001), on simulation studies using ocean general circulation models (Vinayachandran et al. 1999, 2002; Murtugudde et al. 2000), and coupled general circulation models (e.g., Iizuka et al. 2000).
The construction of DL's artificial example is debatable not only in a statistical sense (Jolliffe 2003) but also in a physical sense with regard to IOD. We shall first demonstrate the physical existence of the IOD using multiple datasets, and then discuss why DL failed to capture the dipole mode using the varimax and regression analyses. Although the present discussion is simple, we believe that it is instructive to the climate research community. The statistical analyses and physical analyses should proceed hand in hand in order to deepen the understanding of the physics of our climate system.
2. IOD as a climate mode
As noted in previous studies, a basinwide monopole mode is the dominant EOF mode for interannual SST variability in the Indian Ocean (Cadet 1985; Klein et al. 1999; Wallace et al. 1998; Venzke et al. 2000). We note that the principal component of this mode is highly correlated to the SST variation in the eastern Pacific. The correlation coefficient peaks at 0.75 when the Niño-3 (5°N–5°S, 90°–150°W) index leads the EOF mode by 4 months. Changes in surface heat fluxes associated with the Pacific El Niño–Southern Oscillation (ENSO) events explain the SST anomalies related to the monopole mode (e.g., Venzke et al. 2000); the flux induced heating/cooling of the upper ocean takes about a season to give rise to the SST anomaly.
Recently, the tropical dipole mode in SST anomalies, called the IOD, was documented. Several studies, using ocean surface and subsurface data in addition to atmospheric data, suggest that it is a coupled ocean–atmosphere phenomenon inherent in the Indian Ocean (Saji et al. 1999; Webster et al. 1999; Behera et al. 1999; Iizuka et al. 2000; Murtugudde et al. 2000; Rao et al. 2002a). This mode appears as the EOF-2 in the EOF analysis of SST anomalies from the tropical Indian Ocean. We note here that finding the dipole mode as the second dominant mode in the EOF analysis does not necessarily imply that the associated SST variability is of second order; the associated signal explains close to 50% of the total variance when the analysis is limited to boreal summer and fall season (as discussed later in this section). Lesser persistence and spatial coverage as compared to the monopole mode, however, compel the dipole mode to take the lower seat in the contribution to the whole variance in a statistical sense when considered for all 12 months. In reality, the major IOD signal in the Indian Ocean is clearly seen during 1961, 1967, 1972, 1982, 1994, and 1997 in the raw SST data (cf. Meyers 1996; Vinayachandran et al. 2002). We clarify that identification of the dipole mode in Saji et al. (1999) was based on these prominent SST anomaly signals and not on EOF analysis, which was used only to substantiate the findings.
Here, using the Global Sea Ice and Sea Surface Temperature (GISST) data for the period from January 1958 to December 1999 and the zonal wind anomalies derived from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis dataset (Kalnay et al. 1996), we demonstrate the existence of the dipole mode in a somewhat different way. The low-frequency variability comprising the periodicity of greater than 7 yr is removed from all the datasets. Further, a 5-month running mean is applied to highlight the interannual variabilities. A western box (10°S–10°N, 50°–70°E) and an eastern box (10°S–0°, 90°–110°E), as defined in Saji et al. (1999), are considered for this analysis. When SST anomalies show opposite polarities in these two boxes, it is considered to be the manifestation of the IOD. In the analysis period, 178 months out of 504 months are found to be associated with the dipole mode. The zonal anomaly difference in SST associated with the IOD is closely coupled to the zonal wind anomalies in the central equatorial Indian Ocean (5°S–5°N, 70°–90°E). Figure 1a shows the linear relationship between them very clearly. The dynamical response of the ocean to these winds during IOD events is well documented in several ocean-modeling studies (Vinayachandran et al. 1999, 2002; Behera et al. 1999; Murtugudde et al. 2000; Rao et al. 2002a). This paradigm is demonstrated in Fig. 1b where a strong relationship is found between the wind anomaly and the anomalous thermocline tilt [realized from sea surface height anomalies; Carton et al. (2000)]. Further, Figs. 1c and 1d clearly show the coupling among the anomalous variability in SST, wind, and the atmospheric convective activity during IOD events.
In Fig. 2a, we show the monthly percentage distribution of the SST dipole events during the 42-yr period. As seen in the figure, about 50% of dipole events occur during August–November. Thus, the dipole is the dominant mode during this part of the year. It is obvious that we will find an east–west negative correlation if we consider only this season when the dipole is dominant (see the discussion in section 4): an underlying fact overlooked in DL's analysis. Most importantly, the dipole mode is found to be associated with a similar structure at the subsurface prominently during boreal summer and fall (Fig. 2b).
Figures 1 and 2 confirm not only the physical existence of the IOD but also the associated ocean–atmosphere interaction and the role of ocean dynamics in the evolution of dipole events as discussed by several recent studies (Saji et al. 1999; Webster et al. 1999; Murtugudde et al. 2000; Iizuka et al. 2000; Rao et al. 2002a; Vinayachandran et al. 2002). In particular, using satellite altimeter data, ocean general circulation model results, and upper-ocean temperature data, Rao et al. (2002a) have shown that the subsurface variability is dominated by the IOD, which is in contrast to the leading monopole mode in the SST anomalies linked to ENSO. Since we do not repeat those analyses of dynamics, readers may refer to the above references. The present analysis therefore demonstrates the power of simple statistical analysis to capture physical climate signals, when used appropriately.
3. Limitations of varimax and EOF analyses
Though simple statistical analyses as shown in the previous section captured the physical mode, more sophisticated methods could sometimes be misleading. The basic limitations of statistical methods such as the EOF and varimax are well pointed out by DL, as already discussed in numerous references on this topic (e.g., Richman 1986, 1987; Jolliffe 1987, 2003; Kim and Wu 1999). Here we explain the misconstruction inherent in DL's analysis that fails to identify the IOD. The EOF analysis is performed here using the singular value decomposition method. The spatial vectors are then multiplied by the singular values (S) and the associated time series are divided by S, this being the synthesis formulation of the principal component model (Mestas-Nunez 2000). For the varimax analysis, we rotate the first 10 normalized eigenvectors using a varimax rotation.
Dommenget and Latif anticipated that the presence of the dipole mode should lead to a negative correlation between the western and eastern poles (see also discussions in the previous section) and that the varimax rotation should retain the dipole pattern of EOF2. However, these assumptions are not always valid as demonstrated in the following experiment. In this experiment, the basin corresponding to the tropical Indian Ocean has two well-defined modes of variability: a monopole mode and a dipole mode (Fig. 3a). Both the modes are defined in such a way that variance explained by the monopole mode clearly dominates that of the dipole mode as in the observations. The spatial structures of the modes that are constructed using the trigonometric cosine and sine functions are well defined and spatially orthogonal. The time series associated with both modes are uncorrelated and characterized by random white noises. To make the experiment more realistic, we have added higher modes having as much variance as the two synthetic modes put together. These higher modes are realized by taking the residual SST anomalies that are obtained after removing the first 10 EOF modes in the observed data.
The two synthetic modes being significantly separated from each other in their explained variances are very well resolved by the EOF analysis (Fig. 3b). The explained variances are almost identical to the prescribed values. Now, the data are subjected to varimax rotation. We note that varimax combines the defined modes to form artificial localized modes here (Fig. 3c). Furthermore, the east and west poles of the dipole are separately correlated with the data in the whole domain (Fig. 3d). The correlation analysis does not capture the dipole mode. Three important points that emerge from the current exercise are as follows:
EOF analysis succeeds in resolving these modes realistically, since the synthetic modes are well separated;
being global modes relative to the domain under consideration, a varimax rotation gives rise to artificial localized modes;
since the correlation analysis captures the dominant mode of variability, which is the monopole mode in the present case as well as in the observations, we do not see a negative correlation between east and west.
Here we note that these conclusions are not significantly affected by the introduction of the higher modes in our study. Even if we consider an exact design, as in DL, we arrive at the same conclusions. For example, let us consider only two modes of variability that are a monopole mode z1 (5.0, 4.0) and a dipole mode z2 (3.5, −3.0), and that these are acting on two spatial points x1 and x2. Note that these two modes explain variances of 65.86% and 34.14%, respectively. As in DL, we also assume that the standard deviations of the time series of these modes amount to unity. The following expressions relate those two spatial locations to the two defined modes:
In matrix form we can express them as X = 𝗩Z, where column matrices X and Z are vectors of two variables and 𝗩 is the matrix (2 × 2) of loadings. The covariance structure and therefore the EOF and varimax as well as linear regression can be calculated exactly (Table 1) by using matrix 𝗩. As seen in the Table, the EOF decomposition reproduces the monopole and dipole modes. However, the varimax rotation and linear regression completely fail to recognize the dipole mode and produce degenerated monopole modes instead.
The conclusions from both the simplified and general experiments are thus the same. In some special situations, in which the physical modes could be localized and might have nearly equal variances, EOFs may be misleading as suggested by DL's study. However, in general, especially over tropical oceans where coupled ocean–atmospheric interactions potentially produce large-scale spatial structures, these assumptions do not hold well and there is no guarantee that we can always find the physical modes by using varimax and regression methods (see also Jolliffe 2003).
4. The dipole mode in EOF and varimax
The synthetic example shown in the previous section demonstrates the inherent incapability of the varimax and regression methods in capturing the IOD. It may be noted that the EOFs are unstable in DL's example because the defined variances for the prescribed artificial modes are not significantly different from each other (e.g., Krzanowski 1984; Jolliffe 1989). That those artificial modes in DL are mostly localized complicates the analysis further. Hence, it will be misleading to make conclusions about the nonexistence of a basin-scale physical mode simply because varimax and regression methods fail to capture it.
The question, which arises here, is: is it possible to identify the dipole mode in the real SST data using these two methods? This can be achieved by filtering out the monopole mode related to ENSO (Fig. 4). By this procedure, it is assured that this mode does not co-occupy spatial locations with the dipole mode (Jolliffe 2003). The dominant mode in the EOF analysis on the reduced data is a dipole (Fig. 4a). Since the dipole is now the dominant mode, a linear regression succeeds in reproducing the negative correlation across the basin (Fig. 4b). The varimax method also succeeds in identifying the dipole mode (Fig. 4c). The spatial patterns that emerged from all three methods agree with Saji et al.'s (1999) conclusions. Further, the time series of the EOF and varimax modes are in excellent agreement (Fig. 4d) with the original dipole mode index proposed by Saji et al. (1999).
We have reconfirmed that at least two types of basin-scale interannual variability exist in the SST anomalies of the tropical Indian Ocean: the monopole mode and the dipole mode (IOD). As shown in the literature cited earlier and as seen in the analyses of multiple datasets here, the latter mode reflects the ocean–atmosphere interaction in the Indian Ocean. Several examples are introduced in this short note in order to demonstrate how simple statistical analyses can lead to a deeper insight on the physical mode. On the other hand, improper use of statistical methods without proper understanding of the climate system may lead to misleading results.
The best way to judge whether the statistical patterns represent physical modes is to look at the raw data carefully and to recognize the real phenomenon. The next step may be to prepare composite maps to describe the evolution of the phenomenon, not in one climate parameter like SST but in several other available parameters. Comparison between those composite maps and statistical patterns may provide clues on its relevance to the physical mode as done in Saji et al. (1999). All these are just clues and, most importantly, must be supported by our physical understanding of the climate system.
The critical comments provided by Dr. H. Hendon and an anonymous reviewer were useful in revising the manuscript. We thank Dr. I. Jolliffe for providing helpful suggestions.
Corresponding author address: Dr. Toshio Yamagata, Institute for Global Change Research, Frontier Research System for Global Change, 3173-25 Showamachi, Kanazawa-ku, Yokohama 236-0001, Japan. Email: firstname.lastname@example.org