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  • View in gallery

    An example from experiment 1: (a)–(i) maps show the spatial patterns of (left) input modes, (middle) modes derived by EOF analysis and (right) modes derived by REOF analysis for (top to bottom) modes 1–3. Contour is 0.25 unit with negative values shaded. The pair of numbers on top of the maps in (a)–(i) denotes the SCF of each map and the CCI with the input map. (j)–(l) The PCs of the corresponding modes with the solid line for the input, the dashed line for EOF analysis, and the dotted line for REOF analysis. The numbers on top of denote the CCI of the EOF and REOF time series with that of the input, respectively.

  • View in gallery

    As in Fig. 1, but for a case from experiment 2.

  • View in gallery

    The percentage of cases where CCI exceeds 0.95 in experiment 3, as a function of the number of input modes: mode (a) 1, (b) 2, (c) 3, and (d) all. The solid curves are for EOF analysis, the dotted curves for REOF analysis, and the dashed–dotted curves for the cases where the criterion is met by both. The vertical dotted lines correspond to the values shown in Table 2.

  • View in gallery

    As in Fig. 2, but for the percentage of cases where CCI exceeds 0.95 in experiment 4, as a function of the maximum radius of the input patterns.

  • View in gallery

    As in Fig. 2, but for the percentage of cases where CCI exceeds 0.95 in experiment 5, as a function of the rotation strength.

  • View in gallery

    (a),(c) The first two EOFs and (b),(d) their time evolutions of the tropical Pacific SST anomaly, calculated from the HadISST dataset for the period of 1948–2007. The contour interval in (a),(c) is 0.1°C with negative values shaded, and the numbers on top of the panels show the SCF of each mode.

  • View in gallery

    As in Fig. 6, but with the linear trend removed.

  • View in gallery

    As in Fig. 6, but for REOF analysis.

  • View in gallery

    (a) The linear trend and (b) the SST difference between the averages of 1978–2007 and 1948–77 in the HadISST dataset. The unit is °C (60 yr)−1 for (a) and °C for (b). The contour interval is 0.25 for both. Heavy black curves denote zeros, and values >0.2 are shaded.

  • View in gallery

    The spatial patterns of (a),(b) input modes, (c),(d) EOF modes, and (e),(f) REOF modes of (top) idealized ENSO and (bottom) long-term trend. Contour interval is 0.1, and negative values are dashed. The numbers on top of each map in (a)–(f) denote the SCF of each mode and the CCI with the input pattern. (g),(h) The PCs of the corresponding modes with the solid curve for input, the dashed curve for EOF analysis, and the dotted curve for REOF analysis. The numbers on top of (g),(h) denote the CCI of the two PCs with the input time series.

  • View in gallery

    As in Fig. 10, but for idealized ENSO and WPEN modes.

  • View in gallery

    As in Fig. 10, but for ENSO, WPEN, and trend combined.

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An Evaluation of Rotated EOF Analysis and Its Application to Tropical Pacific SST Variability

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  • 1 Department of Ocean Science and Engineering, Zhejiang University, and State Key Lab of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Hangzhou, China
  • 2 Department of Ocean Science and Engineering, Zhejiang University, and State Key Lab of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Hangzhou, China, and Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York
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Abstract

As an effective eigen method for phenomenon identification and space reduction, empirical orthogonal function (EOF) analysis is widely used in climate research. However, because of its orthorgonality constraint, EOF analysis has a tendency to produce unphysical modes. Previous studies have shown that the drawbacks of EOF analysis could be partly alleviated by rotated EOF (REOF) analysis, but such studies are always based on specific cases. This paper provides a thorough statistical evaluation of REOF analysis by comparing its ability with that of EOF analysis in reproducing a large number of randomly selected stationary modes of variability. The synthetic experiments indicate that REOF analysis is overwhelmingly a better choice in terms of accuracy and effectiveness, especially for picking up localized patterns. When applied to the tropical Pacific sea surface temperature variability, REOF and EOF analyses show obvious discrepancies, with the former making much better physical sense. This challenges the validity of the so-called sea surface temperature cooling mode and the spatial structure of “El Niño Modoki,” both of which are recently identified by EOF analysis. At any rate, one has to be cautious when claiming new discoveries of climate modes based on EOF analysis alone.

Corresponding author address: Tao Lian, Department of Ocean Science and Engineering, Zhejiang University, 866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China. E-mail: liantao@sio.org.cn

Abstract

As an effective eigen method for phenomenon identification and space reduction, empirical orthogonal function (EOF) analysis is widely used in climate research. However, because of its orthorgonality constraint, EOF analysis has a tendency to produce unphysical modes. Previous studies have shown that the drawbacks of EOF analysis could be partly alleviated by rotated EOF (REOF) analysis, but such studies are always based on specific cases. This paper provides a thorough statistical evaluation of REOF analysis by comparing its ability with that of EOF analysis in reproducing a large number of randomly selected stationary modes of variability. The synthetic experiments indicate that REOF analysis is overwhelmingly a better choice in terms of accuracy and effectiveness, especially for picking up localized patterns. When applied to the tropical Pacific sea surface temperature variability, REOF and EOF analyses show obvious discrepancies, with the former making much better physical sense. This challenges the validity of the so-called sea surface temperature cooling mode and the spatial structure of “El Niño Modoki,” both of which are recently identified by EOF analysis. At any rate, one has to be cautious when claiming new discoveries of climate modes based on EOF analysis alone.

Corresponding author address: Tao Lian, Department of Ocean Science and Engineering, Zhejiang University, 866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China. E-mail: liantao@sio.org.cn

1. Introduction

Since the introduction of empirical orthogonal function (EOF) analysis in atmospheric science by Lorenz (1956), this simple yet effective method has been used extensively in atmospheric, oceanic, and climatic research. The essence of EOF analysis is to identify and extract the spatiotemporal modes that are ordered in terms of their representations of data variance. Because a small number of leading modes usually account for most of the total variance, EOF analysis enables one to pick out the dominant and sometimes physically meaningful modes of variability while greatly reducing the data space. By construction, the spatial patterns (EOFs) and the temporal coefficients [principal components (PCs)] of these modes are both orthogonal. Such orthogonality has the advantage of isolating uncorrelated modes, but sometimes it also leads complexity of spatial structures and difficulty in physical interpretations (Hannachi 2007); thus, it may cause misunderstanding and confusion.

Rotated EOF (REOF) analysis was then introduced to alleviate these problems of EOF analysis. There are many types of REOF schemes (Richman 1986), but the most popular is the Varimax REOF analysis (Kaiser 1958), which linearly transforms the spatial patterns derived by EOF analysis into a rotated basis (REOFs) based on the fact that the energy at a given point, which is represented by variance in a sense, is always affected only by points around it but not by all grid points of the whole data domain. Varimax updates the alignment of the eigenfunctions with the actual data and improves the relationship between their spatiotemporal patterns and known physical mechanisms. Previous studies have shown that REOF analysis is able to avoid the unphysical dipolelike EOF analysis pattern that often appears when the known dominant mode has the same sign across the domain (Houghton and Tourre 1992; Dommenget and Latif 2002) and to simplify spatial structures while retaining the robust patterns (Hannachi et al. 2006; Cheng et al. 1995). Despite the obvious advantages of REOF analysis, however, it has not been as widely accepted as the traditional EOF analysis. One of the possible reasons is the lack of a comprehensive comparison of the two methods in a statistically convincing manner.

In the present study, we carried out a thorough statistical comparison of REOF and EOF analyses by evaluating their ability to reproduce a large number of randomly selected stationary modes of variability. This is done by constructing a set of random signals that covers a large range of possible temporal evolutions and spatial distributions in a fixed time–space domain, and then applying REOF and EOF analyses to reconstruct these known spatiotemporal structures. The rest of this paper is organized as follows. The basic algorithms of our synthetic experiments are described in section 2, followed by the results of five different experiments in section 3 to compare the performance of REOF and EOF analyses. In section 4, the two methods are applied to an analysis of the tropical Pacific sea surface temperature (SST) variability, which shows large discrepancies between the two and thus raises questions about some newly identified modes based on EOF analysis, including the so-called SST cooling mode (Zhang et al. 2010, hereafter Z10) and the “El Niño Modoki” (Ashok et al. 2007, hereafter A07). Summary and discussion are given in section 5.

2. Design of synthetic experiments

A suite of synthetic experiments was designed for the statistical comparison of REOF and EOF analyses. The basic idea was to evaluate the ability of these analyses to reproduce synthetic modes of variability with predetermined temporal evolutions and spatial distributions. As mentioned earlier, both temporal and spatial matrices of EOF analysis are orthogonal, but only one of them can remain orthogonal after rotation (Jolliffe 1995). Without losing generality, we chose to make the temporal evolutions of input modes orthogonal (uncorrelated) in all synthetic experiments of this study, with the spatial distributions of these modes either orthogonal or nonorthogonal. Furthermore, since our focus is to compare REOF and EOF analyses, both of which are best used for stationary modes, the patterns of our synthetic modes were assumed to be stationary. The nonstationary and propagating variations, which are better represented by either extended EOF analysis (Weare and Nasstrom 1982) or complex EOF analysis (Wallace and Dickinson 1972; Horel 1984; Barnett 1983, 1985), were not considered in our synthetic experiments.

For simplicity, the construction of the orthogonal temporal evolution matrix was based on Fourier decomposition, since any smooth time series can be represented by the sum of a set of Fourier basic functions that are orthogonal with one other. In our experiments, the time duration of the synthetic modes was fixed to be 256 steps. Thus, we could produce any smooth time series to represent the temporal evolution of these modes with periods from 4 to 256 steps by using the first 128 Fourier basic functions (64 for the sine part and 64 for the cosine part). To form a time series in the orthogonal matrix, we first picked up a random subset of available Fourier basic functions and gave the functions random coefficients, and then we took the functions’ sum and normalized it. The procedure was repeated using the rest of the Fourier basic functions until all dimensions of the orthogonal matrix were filled (default is three modes).

Each time series formed above was associated with a spatial distribution, which was randomly chosen within a square domain of 30 × 30 grid points. Each distribution was allowed to have up to three Gaussian-shaped patterns, with the number of patterns and the location, amplitude, and radius of each pattern randomly determined. The radius was set to be no more than but at least half of a given scale (default is eight grid points). The product of the temporal and spatial matrices constitutes an input signal, and a large number of such signals could be generated according to the randomness in both temporal and spatial components as detailed above.

Five experiments with 1000 random signals each were then designed to provide a comprehensive statistical comparison between EOF and Varimax REOF analyses. In the first experiment, the modes of each input signal were not allowed to overlap spatially, in which case both EOF and REOF analyses should be able to reproduce the input signal since it is orthogonal in space as well as in time. In the second experiment, the restriction of spatial orthogonality was relaxed by overlapping at least two modes of each input signal, in which case REOF analysis was expected to have a superior performance. In the last three experiments, a series of sensitivity tests were conducted on the number of input modes, the spatial scale of input signal, and the magnitude of rotation, respectively, to ensure the statistical robustness of our results. A brief summary of these experiments is given in Table 1.

Table 1.

Summary of synthetic experiments.

Table 1.

It is somewhat arbitrary to decide how many EOFs should be chosen for rotation (Jolliffe 1989; Hannachi et al. 2007). In our synthetic experiments, we rotate all EOFs to obtain REOFs with the goal of simplicity and efficiency (Hawkins 1973). Because the rotated PC matrix is orthogonal, we can project the data onto the PC matrix to obtain the variance of each rotated mode. For the sake of ordering modes in terms of their explained variances, we need to set a minimum distance (in variance) between two adjacent modes. Following the “rule of thumb” proposed by North et al. (1982), this distance should be at least , where λ is the squared covariance fraction (SCF) and N is the length of sample (256 here). To make sure that modes are completely separated, we doubled this minimum distance in our experiments. An index called correlation of congruence (CCI) (Cheng et al. 1995) was used to measure the matching of two series (or maps) a and b as follows:
eq1
Because our input time series and the PCs derived by EOF and REOF analyses have zero means, this index is equivalent to the traditional anomaly correlation in the time domain. In contrast, when a and b denote grid maps, CCI gives a good estimation of similarity in spatial distributions (Richman 1986).

3. Results of synthetic experiments

a. Experiment 1

This experiment was really just a test of algorithm. With the input signals being orthogonal in both space and time, we never found a case where either EOF or REOF analysis failed to reproduce the input. While Varimax rotation was applied, the maximum element in the pure transform matrix (defined as the difference between transform matrix and identity matrix) was very close to zero, indicating that the rotation was almost negligible. In other words, EOF and REOF analyses were equivalent on such occasions, both being able to correctly pick up the input modes.

As an example, Fig. 1 shows a typical case taken from experiment 1. The three randomly chosen input distributions (left column) do not overlap with one another, and their corresponding time series (last column, solid lines) include both low-frequency (mode 2) and high-frequency oscillations (modes 1 and 3). Calculated by spatial variance, the SCF of these three modes is 72.12%, 19.28%, and 9.60%, respectively. It is clear that either EOF or REOF analysis is able to reproduce the three input modes almost completely in terms of both spatial distribution and temporal evolution, with perfect scores (CCI = 1.0) throughout.

Fig. 1.
Fig. 1.

An example from experiment 1: (a)–(i) maps show the spatial patterns of (left) input modes, (middle) modes derived by EOF analysis and (right) modes derived by REOF analysis for (top to bottom) modes 1–3. Contour is 0.25 unit with negative values shaded. The pair of numbers on top of the maps in (a)–(i) denotes the SCF of each map and the CCI with the input map. (j)–(l) The PCs of the corresponding modes with the solid line for the input, the dashed line for EOF analysis, and the dotted line for REOF analysis. The numbers on top of denote the CCI of the EOF and REOF time series with that of the input, respectively.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

b. Experiment 2

Now we proceed to a more interesting and revealing experiment by relaxing the orthogonality in spatial dimension. Natural modes of variability in the earth’s climate system are often overlapped in space and thus not to be readily separated by EOF analysis, whose modes are orthogonal by construction. Presumably the transform matrix of REOF allows rotated modes to overlap, which would make them a better representation of the physical modes in nature. This experiment is a test for the difference of EOF and REOF analyses in dealing with spatially nonorthogonal signals. To make the comparison more meaningful, in this test and the following two experiments, we only consider the cases where rotation is significant, that is, the maximum element in the pure transform matrix (rotation parameter) falls in the range [0.3, 0.4]. Note that the rotation parameter is not predetermined; we choose the subset from all randomly built synthetic cases. The sensitivity to the degree of rotation will be evaluated in experiment 5.

Figure 2 shows a typical case taken from experiment 2. The first and the second modes of the input signal are partly overlapped near the center of the grid map, and the third and the second modes are also slightly overlapped. It is obvious that EOF analysis has difficulty reproducing the input signal mode by mode. Because of the constraints of orthogonality and variance maximization, the first and the third EOF modes show strong imprints of the second input mode, and the second EOF mode is impacted by both the first and the third input modes. The spatial CCIs of the three EOF modes with the corresponding input modes are 0.91, 0.74, and 0.83, respectively, while the corresponding temporal CCIs are 0.87, 0.76, and 0.89, respectively. These numbers may not seem too bad, but for our idealized experiments they do represent serious biases, as evident in Fig. 2. In contrast, the REOF analysis captures the three input modes almost perfectly in both space and time, with all CCIs very close to 1.0.

Fig. 2.
Fig. 2.

As in Fig. 1, but for a case from experiment 2.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

Using the minimum of spatial and temporal CCI exceeding 0.95 as a criterion for acceptable match between reproduced and original modes, Table 2 shows the statistical differences between EOF and REOF analyses in terms of the percentage of matches for 1000 randomly chosen input signals. Since the evaluation metrics shown in Table 2 are used in all the following experiments, a few words of explanation are in order. Aside from the match percentages for each mode, two more sets of percentages are calculated. One set is for the cases where the match criterion is met for all three modes simultaneously (right column in Table 2), which evaluates the ability of EOF and REOF analyses to reproduce the whole input signal. The other set is for the cases where both EOF and REOF modes meet the criterion (bottom row), which helps to explain the statistical difference between the EOF and REOF analyses.

Table 2.

Percentage of cases where CCI exceeds 0.95 in experiment 2. The rightmost column is for cases where the criterion is met for all three modes. The bottom row is for cases where the criterion is met for both EOF and REOF analyses.

Table 2.

The following is evident in Table 2. First, the percentages of cases for the three EOF modes to meet the CCI criterion are 48.4%, 9.6%, and 42.0%, respectively, while those for the REOF modes are 77.2%, 68.8%, and 70.8%, respectively, indicating that REOF analysis is indeed a much better performer in a statistical sense. The particularly low score of the second EOF mode is due to the large distortion of this mode being orthogonal to the leading EOF mode, which contains the largest variance. Second, when all three modes are considered simultaneously, the match percentage of EOF outputs is dramatically reduced to 0.4%, while that of REOF modes remains at 55.6%, suggesting that EOF analysis has little skill to completely reproduce spatially nonorthogonal input signals. Finally, the percentage of cases where both EOF and REOF analyses meet the match criterion is very close to that for EOF analysis alone, indicating that when EOF analysis works fine, so does REOF analysis, but not the other way around. There is little chance for EOF analysis to reproduce an input signal better than REOF analysis; the latter outperforms the former almost universally.

c. Experiment 3

In most of our experiments, we used three input modes with up to three spatial patterns each, which could give fairly complex structures for input signals. Here we further explore the sensitivity of EOF and REOF analyses to the complexity of input signals by increasing the number of modes. Figure 3 shows the evaluation metrics used in Table 2 as a function of input modes up to seven. It seems that EOF analysis is not very sensitive to the increasing number of modes, because three overlapping modes are already too many for EOF analysis to handle. REOF analysis has a stronger sensitivity, with a degraded performance as the complexity of input signals increases, especially when all modes are accounted for at the same time. This is because with a large number of modes, the limited grid map becomes so crowded that multimode overlapping is everywhere, which makes it difficult to distinguish different modes no matter how EOFs are rotated. When the number of input modes is larger than six, REOF analysis becomes as unskillful as EOF analysis in reproducing the whole signal; otherwise, REOF analysis always outperforms EOF analysis.

Fig. 3.
Fig. 3.

The percentage of cases where CCI exceeds 0.95 in experiment 3, as a function of the number of input modes: mode (a) 1, (b) 2, (c) 3, and (d) all. The solid curves are for EOF analysis, the dotted curves for REOF analysis, and the dashed–dotted curves for the cases where the criterion is met by both. The vertical dotted lines correspond to the values shown in Table 2.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

d. Experiment 4

The maximum radius of input spatial patterns in most of our experiments was set to eight grid points (in a 30 × 30 grid map). Figure 4 shows the sensitivity of our evaluation metrics to this parameter with a range from 3 to 13 grid points. A nearly linear trend against the maximum radius is evident in the performances of both EOF and REOF analyses. When the radius is smaller than the default, there is a greater chance for input spatial patterns to be independent of one another and thus a higher skill for EOF and REOF analyses. When the radius is larger than the default, there is a greater possibility for input spatial patterns to overlap and thus a worse performance of EOF and REOF analyses. It is worth noting that, despite the dependence of both EOF and REOF analyses on the spatial scale of input signals, the superiority of REOF analysis over EOF analysis is not sensitive to this parameter.

Fig. 4.
Fig. 4.

As in Fig. 2, but for the percentage of cases where CCI exceeds 0.95 in experiment 4, as a function of the maximum radius of the input patterns.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

e. Experiment 5

The REOF analysis in the previous four experiments was limited to the cases with an effective rotation, where the rotation parameter falls in the range [0.3, 0.4]. Here we examine the sensitivity of our results to the degree of rotation, with the rotation parameter ranging from [0.0, 0.1] to [0.5, 0.6], as shown in Fig. 5. When the rotation parameters are in the range [0.0, 0.1], which corresponds to the cases where input modes are mostly orthogonal, the results of EOF and REOF analyses are very close and both exhibit high skills. As the rotation parameter increases, the skill of REOF analysis remains acceptable, while that of EOF analysis shows a sharp descending trend, especially for the second mode and for all modes combined. When the rotation parameter goes beyond the range [0.4, 0.5], the skill of REOF analysis tends to degrade more rapidly.

Fig. 5.
Fig. 5.

As in Fig. 2, but for the percentage of cases where CCI exceeds 0.95 in experiment 5, as a function of the rotation strength.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

This kind of behavior can be briefly explained as follows. When the rotation parameters are very small, the rotation matrix is close to be the identify matrix and EOF analysis itself satisfies the Varimax criterion. In a situation like this, the traditional EOF analysis is as reliable as the rotated one, as shown in experiment 1. However, when the rotation parameters are very large, the energy distribution of the input signal must be scattered. Consequently, the spatial patterns given by EOF analysis are full of small-scale variations and form a poor basis for rotation. Thus, the accuracy of REOF outputs drops rapidly toward the end of strong rotation. For the cases when the rotation parameters are in the midrange, REOF analysis is most effective as its superiority over EOF analysis is most pronounced.

From the five synthetic experiments presented above, it is clear that for spatially overlapped, stationary modes of variability, Varimax REOF analysis is an overwhelmingly better tool for mode identification as compared to EOF analysis. Except for situations with extremely complicated, heavily overlapped pattern distributions, REOF analysis does a decent job in reproducing the original signals mode by mode. Next we examine a realistic application of REOF analysis to the tropical Pacific climate variability, hoping to clarify the meaning of some newly identified “modes” based on EOF analysis.

4. A case study of tropical Pacific SST

As the most important indicator of El Niño–Southern Oscillation (ENSO), the tropical Pacific SST variability has been studied extensively. There is no doubt that the dominant mode of variability is associated with ENSO, with the largest SST anomalies in the eastern and central equatorial Pacific. This physically meaningful mode is always well captured by the first EOF mode regardless of the datasets being used for analysis. However, the interpretation of the second and higher modes of tropical Pacific SST variability is still a subject of considerable debate. Here we examine a couple of newly identified modes based on EOF analysis, both of which have been claimed to be the second largest mode of tropical Pacific SST variability and have been given distinctive physical meanings.

One of them is the so-called SST cooling mode recently identified by Z10. Based on various SST products that cover the past one and a half centuries, they found an apparent cooling strip in the eastern and central equatorial Pacific as the second EOF mode. This mode explains about 10% of the total variance and is highly correlated with the long-term trend. Moreover, when the twentieth-century simulations of a set of global climate models were analyzed, the cooling mode also showed up as the second EOF mode. Their interpretation of this mode is linked to global warming. A positive feedback mechanism similar to the “thermostat theory” (Clement et al. 1996; Cane et al. 1997) was invoked. Basically, because of ocean dynamics, global warming would strengthen the zonal SST gradient in the equatorial Pacific, which in turn enhances the trade wind, and the enhanced wind then induces stronger upwelling in the east that brings more cold water from subsurface, thereby further reinforcing the zonal SST gradient.

The other mode of interest here is the El Niño Modoki (A07), and some variants of it have also been referred to as the warm pool El Niño (WPEN) (Kug et al. 2009) or the central Pacific El Niño (Kao and Yu 2009). The mode is characterized by a dipole pattern with warm SST anomalies around the date line and comparable cold anomalies in the eastern equatorial Pacific. A07 pointed out that for the period of 1979–2004, El Niño Modoki was the second EOF mode of tropical Pacific SST variability, while a weak cooling strip in the eastern equatorial Pacific appeared as the third mode. The obvious contradiction between Z10 and A07 cannot be simply attributed to their different periods for analysis. As shown in Z10, the zonal SST gradient was at its maximum during the recent three decades, and thus by their argument the cooling mode should be stronger rather than weaker for this period. What is the cause of the discrepancy? Is there a way to reconcile these different opinions? Are these two modes of tropical Pacific SST variability physically real?

To address these questions, we first reproduced the EOF modes of Z10 using the same SST dataset for the period of 1948–2007. As shown in Fig. 6, the first and dominant mode is clearly ENSO, and the second mode is a narrow strip in the eastern equatorial Pacific, with its evolution being a long-term trend superimposed by interannual fluctuations. Then we removed the linear trend from the SST data and recalculated the EOF modes (Fig. 7). While the first mode remains the same, the second mode of detrended data becomes totally different, with the cooling strip replaced by a dipolelike mode that bears a strong resemblance to the El Niño Modoki of A07. Furthermore, we applied Varimax rotation to the first 10 EOFs to see what happens when the spatial orthogonality constraint is relaxed. As evident in Fig. 8, the first REOF mode is again the ENSO mode, but the second REOF mode is neither the cooling mode of Z10 nor the El Niño Modoki of A07. The mode has its center of action in the western-central equatorial Pacific, and it stands out as the second REOF mode no matter if the SST data are detrended or not (figure not shown).

Fig. 6.
Fig. 6.

(a),(c) The first two EOFs and (b),(d) their time evolutions of the tropical Pacific SST anomaly, calculated from the HadISST dataset for the period of 1948–2007. The contour interval in (a),(c) is 0.1°C with negative values shaded, and the numbers on top of the panels show the SCF of each mode.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

Fig. 7.
Fig. 7.

As in Fig. 6, but with the linear trend removed.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

Fig. 8.
Fig. 8.

As in Fig. 6, but for REOF analysis.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

The results shown in Figs. 68 are quite revealing, and a few points are worth noting here. First, the appearance of the SST cooling mode found by EOF analysis is partly a consequence of the long-term trend in the data. However, the spatial pattern of this mode does not look anything like that of the actual SST trend (Fig. 9), which shows warming over most of the tropical Pacific though the magnitude is somewhat less in the region where the cooling mode is located. Second, the dipole pattern of El Niño Modoki seems to be a result of the orthogonality requirement of EOF analysis, since previous studies and our synthetic experiments all show that the second mode tends to be a dipole when the first mode is a strong basinwide mode. Third, because of the better ability of REOF analysis to separate spatially overlapped modes, the rotated second mode in Fig. 8 is probably a good representation of real SST variability. In fact, it looks quite similar to the composite SST anomaly pattern of WPEN in Kug et al. (2009). They showed that, except for the 1990/91 event, no strong cooling was found in the eastern equatorial Pacific during the peak phase of WPEN.

Fig. 9.
Fig. 9.

(a) The linear trend and (b) the SST difference between the averages of 1978–2007 and 1948–77 in the HadISST dataset. The unit is °C (60 yr)−1 for (a) and °C for (b). The contour interval is 0.25 for both. Heavy black curves denote zeros, and values >0.2 are shaded.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

It is therefore reasonable to postulate that neither of the SST cooling modes of Z10 and the El Niño Modoki dipole of A07 are real physical modes; they are artificially created by EOF analysis from overlapped SST variations related to ENSO, WPEN, and a long-term warming trend. To confirm our postulation, we designed an additional set of idealized experiments in which EOF and REOF analyses were used to reconstruct input signals that mimic observed modes of variability. The idealized ENSO mode has a period of 4 yr and covers a large portion of the tropical Pacific with the maximum variance in the east, while the WPEN mode is a 2-yr oscillation centered just east of the date line. The idealized long-term trend has a spatial pattern with three warming maxima located in the equatorial western Pacific and off-equatorial eastern Pacific, consistent with the patterns of the linear SST trend during 1948–2007 (top panel of Fig. 9) as well as the SST difference between the 30-yr averages of 1978–2007 and 1948–77 (bottom panel of Fig. 9).

In the first experiment, the input signal is considered a combination of ENSO and the linear trend (Fig. 10). ENSO is largely reproduced by both EOF and REOF analyses as the first mode, but the spatial pattern of the trend is much better captured by the latter than the former. There is apparent mode mixing for both EOF and REOF analyses, as indicated by the shift of variance from the second to the first mode, the trend in the PCs of the first mode, and the 4-yr oscillations in the PCs of the second mode. However, the mode mixing is much less severe for REOF analysis than for EOF analysis. The most notable difference is that, while the second REOF mode still has its largest variances in the three areas corresponding to the input trend, the second EOF mode has the strongest variance in a narrow strip of opposite sign in the eastern equatorial Pacific. In other words, a “SST cooling mode” similar to that of Z10 is artificially generated by the spatial orthogonality requirement of EOF analysis.

Fig. 10.
Fig. 10.

The spatial patterns of (a),(b) input modes, (c),(d) EOF modes, and (e),(f) REOF modes of (top) idealized ENSO and (bottom) long-term trend. Contour interval is 0.1, and negative values are dashed. The numbers on top of each map in (a)–(f) denote the SCF of each mode and the CCI with the input pattern. (g),(h) The PCs of the corresponding modes with the solid curve for input, the dashed curve for EOF analysis, and the dotted curve for REOF analysis. The numbers on top of (g),(h) denote the CCI of the two PCs with the input time series.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

In our second experiment, the input signal is considered a combination of ENSO and WPEN (Fig. 11). Again, ENSO is well reproduced by both EOF and REOF analyses as the first mode, but the spatial pattern of WPEN is much better captured by REOF analysis than EOF analysis. There is still obvious mode mixing for both EOF and REOF analyses, as indicated by the shift of variance from the second to the first mode (though less than that in the first experiment) and the harmonic modulations of the PC time series, but such mixing is much less severe for REOF analysis than for EOF analysis. The second mode REOF is able to capture the basic structure of WPEN, while its amplitude is somewhat smaller than that of the input. In contrast, the second mode EOF not only has a distorted and much weaker WPEN but also shows an out-of-phase pattern in the eastern equatorial Pacific with comparable amplitude. Clearly, a dipole structure similar to the El Niño Modoki of A07 is artificially produced by EOF analysis.

Fig. 11.
Fig. 11.

As in Fig. 10, but for idealized ENSO and WPEN modes.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

In our final experiment, all three modes, including ENSO, WPEN, and the linear trend, are considered simultaneously (Fig. 12). For the input signal, the variances explained by WPEN and the trend are the same, while that by ENSO is 4 times larger, in accordance to the observed energy partition among these modes. The EOF and REOF modes in this case are more or less a rearranged linear combination of those from the last two experiments. Some of the previous findings, such as the good skill of both EOF and REOF analyses in reproducing ENSO, the shift of energy toward higher modes, and the mixing among overlapped modes, are still obvious here. In general, REOF analysis does a much more credible job in the reconstruction of all three input modes, especially for the second and third modes, as indicated by the CCIs in both space and time. It is interesting to note that in the second EOF mode, the unphysical cooling strip in the eastern equatorial Pacific is even stronger than that in the first experiment, obviously due to combined interplays of WPEN and the trend with ENSO.

Fig. 12.
Fig. 12.

As in Fig. 10, but for ENSO, WPEN, and trend combined.

Citation: Journal of Climate 25, 15; 10.1175/JCLI-D-11-00663.1

5. Summary and discussion

This study consists of two main parts. First, a statistical evaluation of REOF analysis against EOF analysis was carried out in terms of their ability to identify stationary modes of variability from data, based on a series of synthetic experiments in which a vast number of randomly chosen signals were used as input. Second, the leading modes of tropical Pacific SST variability from EOF analysis were reexamined using REOF analysis, and the reasons for the large discrepancies between the two analyses were elucidated using an additional set of idealized experiments. We found that REOF analysis outperforms EOF analysis almost certainly in reconstructing spatially overlapped modes, and that this superiority is not sensitive to parameters such as the number of modes, the spatial scale of the signal, and the degree of rotation (figures not shown). We also found that REOF analysis gives a more realistic representation of the natural modes of tropical Pacific climate variability, and that some recently identified EOF modes, such as the SST cooling mode and the El Niño Modoki dipole, are not physically meaningful.

The advantage of REOF analysis derives from its relaxation of the orthogonality constraint imposed by EOF analysis. However, as wonderful as it is, REOF analysis is just a linear analytical tool and should be used with caution (Jolliffe 2003). As seen in our synthetic experiments, the performance of REOF analysis degrades as the complexity of the input signal increases. In reality, physical modes of variability could be overlapped in space and correlated in time, and they could be propagating and nonstationary. Under such circumstances, simple REOF analysis is not likely to work well and more advanced methods have to be adopted (Kim and Wu 1999). Our argument here is that, if one intends to attach physical meaning to EOF modes, one had better take a close look at their rotated counterparts. By relaxing either spatial or temporal orthogonality, REOF modes are often more representative of reality. Rotation is particularly needed when emphasis is placed on the second EOF mode, which is prone to large distortion because it has to be orthogonal to the first mode that dominates the data field, as seen in our synthetic experiments as well as the case study of tropical Pacific SST.

In the tropical Pacific, ENSO is undoubtedly the dominant mode of variability, and any method would be able to capture this mode. The question is how to pick up the second mode that is physically meaningful. We have shown that the second EOF mode in the tropical Pacific, regardless of which dataset it is derived from, is at least partly artificial. The second REOF mode shown in Fig. 8 is likely to be a robust mode of natural variability related to the eastward movement of the western Pacific warm pool, though its physical mechanism is still not clear. It can be considered the western pole of El Niño Modoki, or preferably warm pool El Niño (WPEN). Because of the interplays among ENSO, WPEN and a long-term trend, the second EOF mode may look like either the cooling mode of Z10 or the El Niño Modoki dipole of A07. For instance, in an idealized test similar to that shown in Fig. 12 but with a broad trend pattern in the eastern tropical Pacific representing that observed for the period after 1980, the second EOF mode appears more like a dipole (not shown) rather than a cooling strip, thus reconciling the difference between the results of Z10 and A07.

The observations used for our analyses in section 4 is from the Met Office Hadley Centre Sea Ice and Sea Surface Temperature (HadISST) dataset (Rayner et al. 2003) for the years 1948–2007, but our test experiments show that the main results and general conclusions of this study do not depend on the length and source of the data being used. For example, we have examined Extended Reconstructed SST (ERSST) (Smith and Reynolds 2004) and Kaplan SST (Kaplan et al. 1998), which were used in Z10 along with the HadISST dataset. Although Z10 claims that a cooling mode is present in all three datasets as the second EOF mode, a closer look reveals that the patterns of this mode for the other two datasets are not a strong narrow strip in the eastern equatorial Pacific as for HadISST. It is centered at the tropical central Pacific near 150°W for ERSST and is a much broader pattern from 180° to 120°W for Kaplan SST. This is because the spatial patterns of the long-term trend in these datasets are different, and thus their overlapping and interaction with ENSO are different. When these two datasets are detrended, the second EOF mode becomes a dipole just like that in the case of HadISST; when the trend of HadISST is added to the two detrended datasets, the narrow cooling strip is recovered (figures not shown).

A clear identification of the leading modes of the tropical climate variability and their response to global warming is of great importance for our understanding of the earth’s climate system as well as for the practical purpose of climate prediction (Cane et al. 1997; Chen 2008; Chen and Cane 2008). The idea that the tropical Pacific Ocean serves as a dynamical thermostat to global warming (Clement et al. 1996) is appealing and plausible. However, calling the reduced warming in the eastern equatorial Pacific a cooling mode (Z10) is misleading, since the reason for its standing out as a prominent cooling trend is simply a consequence of the orthorgonality constraint of EOF analysis. By the same token, despite the newly aroused interest in El Niño Modoki, the dipole structure of this mode is not likely to be a truthful representation of the real climate variability. At any rate, one has to be cautious when claiming new discoveries of climate modes based on EOF analysis alone.

Acknowledgments

This research is supported by the Ministry of Science and Technology (Grants 2010DFA21012, 2007CB816005), the State Oceanic Administration (Grant 201105018), and the National Science Foundation (Grant 40730843) of China, and also by the National Oceanic and Atmospheric Administration, the National Aeronautics and Space Administration, and the Department of Energy of the United States. We thank Dr. Zheng Ling for his assistances in programming, Dr. A. Hannachi for providing the Varimax code, and two anonymous reviewers for their helpful comments and suggestions.

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