a. Overview

Many different methods have been applied in the past to diagnose equatorial Kelvin wave signals. Some authors have projected data onto meridional structure functions for the waves (e.g., Boulanger and Menkes 1995), others have forced ocean models with wind data (e.g., Bergman et al. 2001; S07), while others have applied simple time filtering analogous to the method discussed above, assuming that Kelvin waves dominate the selected filter bands (e.g., Johnson and McPhaden 1993). Since the intent of this reconstruction is to estimate amplitudes of Kelvin waves that may be coupled to atmospheric wind stress patterns (e.g., RK06) and since such stress may distort waves from their expected structures, we made no assumptions concerning meridional structure. Ocean-model-based reconstructions depend on wind data, which might not be reliable during the 1970s through early 1980s. Model-based reconstructions would also depend on the choice of model.

As in RK06, we used time filtering and regression to diagnose and reconstruct Kelvin wave signals. Regression models were constructed to fill in all values missing from the interpolated and filtered daily equatorial Kelvin wave dynamic height dataset between 1 January 1974 and 31 December 2005. Selected data from all available equatorial TAO buoys and sea level observations from the sites listed in Table 1 were utilized simultaneously to estimate the values of the missing data except that data from the buoy at 137°E were not used because they contributed to loss of skill across the western basin (see section 4b). Two additional reconstructions starting with smaller subsets of the predictors were made for analyzing the skill of the method as described below. Equation (1) gives the form of the models used:

View Expanded

where y is filtered TAO dynamic height data at a given equatorial buoy, I_n is a sea level or dynamic height time series from the nth station or buoy, and the a_x coefficients are regression parameters calculated by matrix inversion (e.g., Wilks 2006; Strang 2005). For convenience, the first 11 subscripts on I represent the station numbers listed in Table 1. Power terms diagnose relationships that cross frequency bands, allowing modeling of relationships between time series of different shapes (e.g., Wilks 2006). These are important both because the frequency of the Kelvin waves tends to decrease toward the east (RK06) and because signals measured at stations farther from the equator tend to be dominated by frequencies lower than those of related processes on the equator. A dot over I represents the first finite time difference. These derivative terms, in combination with the others, help the model to diagnose the phase difference between the sea level data at a given station and the dynamic height y at a buoy. This matches the phases of waves in the reconstructed dynamic height data to those in TAO dynamic height. These are important both because the signal at each location is shifted in phase from that at every other location and because dynamic height and sea level are not generally in phase at a given location. The OLR term is the east-central Pacific OLR index described above and is included because no sea level data are available in the region and because RK06 showed that Kelvin waves are occasionally coupled to atmospheric convection across the region during El Niño. Cross-product terms help the model resolve spatial scales and phase speeds of waves crossing portions of the east Pacific where sea level data are not available. Here ε represents the model residuals.

As described so far, the model assumes that the current state of a missing data point is determined by the state at the same time of all of its available neighbors. Since the state also depends on available data occurring before and after the missing point, we include four additional terms for each sea level and dynamic height predictor series. The first two are shifted 10 days backward and 10 days forward, respectively, and the second two are shifted 20 days backward and forward, respectively. These terms improve the resolution of the reconstruction, especially during periods when many predictors are simultaneously missing but data are available within a short time before or after the missing point.

Several predictors are significantly correlated with each other, leading to overdetermination or overfitting. Such colinearities make the model coefficients difficult to resolve and prevent unique solutions from being found. Prior to optimization (see below), the total number of predictors used for training the model exceeded 200. We eliminated colinearity and reduced the number of predictors through the following process. Each predictor was first normalized by subtracting the mean and dividing by its standard deviation. An orthogonal basis for the set of normalized predictors was then found by the Gram Schmidt process (Lay 1994) as described by Roundy and Frank (2004) and RK06. Next, the projections of each predictor onto the selected dynamic height time series were found by taking the vector dot product. These projections were sorted from minimum to maximum absolute values, and a cumulative sum of these absolute values was taken. The series of sums was then divided by the maximum sum, resulting in values increasing from zero to one. We then eliminated predictors associated with cumulative sum values less than 0.2. This cutoff was determined by optimizing the skill of the reconstruction across the basin. This method selects the subset of available predictors in Eq. (1) that produces the best overall fit to observations outside of the training set, following the methods in section 3b. Thus the number of predictor terms assigned is not arbitrary.

Different combinations of predictors were available at different times, so regression relationships were obtained for all combinations that occur. The normalization and orthogonalization procedure was performed for each subset of predictors. After applying the orthogonalization procedure, each missing value at each buoy longitude was filled by finding the appropriate set of regression coefficients relating the selected subset of available predictors to the data from that buoy. The quantity returned by Eq. (1) after substituting predictor values appropriate for the missing value was then substituted for that value.

It is important to point out that a few of the selected sea level stations are located more than one equatorial Rossby radius of deformation off the equator and, thus, probably do not include much Kelvin wave signal. However, we found that including them increases the skill of the reconstruction in approximating observed wave anomalies, especially during periods when little other data are available, presumably because the wind stress patterns linked to Kelvin waves disturb sea surface height over regions broader than the oceanic Rossby radius, producing signals that correlate locally with Kelvin waves developing near the equator. As discussed above, these and all predictors are used only when they project significantly onto the signals to be reconstructed in a manner that improves the fit of the reconstruction to the observed dynamic height anomalies, as discussed below in section 3b.

b. Skill score tests

Three reconstructions were generated. The first is the reconstruction product including filtered TAO dynamic height whenever it is available for the period 1974–2004. All missing data were filled in the first reconstruction, based on all available predictors. The second and third reconstructions were generated to help assess the skill of the method in approximating the observed dynamic height anomalies. Missing data were artificially created in the second reconstruction in consecutive 100-day segments over the entire dataset at all buoys, and these missing segments were reconstructed by using all available data except that from the buoy being reconstructed. The third set is the same as the second, except that TAO data were excluded from the predictor set and sea level data and the east-central Pacific OLR index were applied to predict the artificially missing segments. Skill of the second and third reconstructions was determined by applying a Brier skill score test, as in Eq. (2):

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where MSE = is the mean square error in which K_recon is the time series of reconstructed dynamic height and K_TAO is the time series of TAO dynamic height; and MSE_clim, = represents the mean square error of the climatology, with the climatology assumed to be the zero anomaly everywhere. These tests are most often applied for probabilistic forecasts but they are also applicable in the same manner to predictions of data fields (Wilks 2006). A prediction with a skill score greater than zero is on average a better estimate of the actual values than climatology. This method allows us to objectively compare two different predictions of the same quantity relative to an observed value and it incorporates information about correlations and amplitudes (Wilks 2006). The result for the second reconstruction suggests how well the method fills in missing data when TAO data are available. The result for the third reconstruction shows how the method performs when TAO data are not available, such as before the mid-1980s. A comparison between the third reconstruction and dynamic height during recent periods with plentiful TAO data thus measures the skill of the method using sea level data and other proxies. This is important because, as shown below, the variance of the reconstruction is likely to be artificially reduced early in the dataset when TAO data are not available even though high-amplitude reconstructed waves appear to occur with the correct phasing.

We also applied the same skill test for those time periods when the amplitudes of either the verification or the reconstruction were large. Variance indices were made by smoothing time series of the squares of the reconstructions and verification series at each buoy by 121-day centered moving averages (see RK06). Then Eq. (2) was applied only for those time indices when the local variances of either the reconstruction or the verification were greater than the variances of the entire time series.

c. Wave activity and ENSO indices

An index of basinwide Kelvin wave activity was calculated from the first set of reconstructed data by averaging the squares of the dynamic height estimates over all buoys from 147°E to 125°W and smoothing the result with a 121-day centered moving average (as in RK06). Although this index probably is a good indicator of the timing of periods of enhanced activity, actual variance was probably higher than the index suggests prior to the mid-1980s. To help address this issue, a second activity index was calculated from a reconstruction based only on sea level observations and the east Pacific OLR index. Comparison of the two activity indices helps to estimate how much variance might be artificially reduced in the reconstruction prior to the mid-1980s.

An ENSO index for Niño-3.4 dynamic height was reconstructed as for the Kelvin waves using Eq. (1) except that 200-day low-pass filtered dynamic height and sea level heights were used and the squared and cubic terms were omitted because the associated coefficients were not well resolved, probably because the dataset is too short relative to the time scales of the index. The Niño-3.4 dynamic height index of RK06 was substituted for y, coefficients were calculated, and the 200-day low-pass filtered sea level and dynamic height data were substituted to reconstruct the index. The result is dominated by the Christmas Island sea level time series.

a. Comparison of reconstructed and actual dynamic height

Figure 3a shows the observed dynamic height and Figs. 3b,c show the second and third sets of reconstructed dynamic height, discussed in section 3, during the 2002–03 El Niño (cf. Fig. 1 of RK06). Some phase shifts are evident between the three panels, but the general wave patterns seen in the TAO data are maintained by the reconstructions, even in the third reconstruction, which does not utilized TAO data. Although the amplitudes of anomalies in the third reconstruction are generally smaller than those of corresponding anomalies in the first, a few low-amplitude anomalies in TAO data correspond to higher-amplitude reconstructed anomalies (e.g., February 2003). In general, the highest-amplitude observed waves tend to be well described by both reconstructions. The decreasing phase speeds of successive waves documented by RK06 (especially early in the period) are evident in both reconstructions. RK06 provides a list of possible sources of this variability including changes in the preferred vertical modes of the waves, background currents, and variations in the effects of coupling between the waves in the ocean and convection and winds in the atmosphere. S07 suggest that even comparatively weak wind stress anomalies across the east Pacific dramatically influence the phase speeds of sea surface height anomalies linked to Kelvin waves.

b. Skill score tests

Figure 4 shows the skill score estimates calculated as described in section 3b. The heavy solid curve in Fig. 4 is the skill of the second set of reconstructed data. The thin solid curve shows the corresponding skill during periods when amplitudes are greater than average according to the local wave activity indices discussed above. The heavy dashed curve shows the skill of the third reconstruction. The thin dashed curve is the corresponding skill during periods when the local wave activity indices exceeded their long-term mean.

All skill values in Fig. 4 are positive except those for the second reconstruction at 137°E. Skill values approach 0.8 near the date line in the second reconstruction. Figure 4 shows that, except at 137°E, the reconstruction is on average a better estimate of the actual dynamic height than the climatology. Except at 137°E, thin lines representing the skill during periods of above average activity are higher everywhere than the corresponding heavy lines representing skill over all available data, suggesting that the reconstruction is more skillful during periods of high wave activity than during periods of low activity. After studying the time series at 137°E, we concluded that a few inappropriately high amplitude values there cause the negative skill. This observation is consistent with the reduced skill at 137°E during periods of enhanced activity. Large sections of data are missing at this buoy since it first began collecting, so the filtered signals are not sufficiently well fit. Data at 137°E should therefore be used with caution. In spite of this problem, the occurrence of Kelvin wave signals at 137°E is generally consistent with those of the neighboring buoys.

Since the skill is everywhere less than one, the reconstruction cannot be considered a perfect representation of the data. Skill of regression-based reconstructions can be reduced by loss of either variance or correlation between the reconstruction and the observations. All regression-based reconstructions show lower variance than the data that they are designed to approximate. Examination of the reconstructed data suggests that it is well correlated with the observed dynamic height, especially during periods of enhanced wave activity, but that the amplitudes are often reduced, especially on the east and west sides of the basin (e.g., Fig. 3). The general improvement in skill when only high-amplitude anomalies are considered suggests that low-amplitude reconstructed anomalies are usually poorer approximations of observed anomalies than are high-amplitude anomalies.

c. Consistency with winds during 1974–88

Although reconstructed dynamic height prior to about 1988 cannot be effectively compared with TAO dynamic height at most buoys, they can be checked for consistency with surface wind data because most intraseasonal Kelvin waves in the Pacific are forced by wind stress (e.g., Cravatte et al. 2003; Boulanger et al. 2003). Figure 5 shows the NCEP–NCAR reanalysis 1000-hPa u wind anomalies overlaid with contours of reconstructed dynamic height for the period September 1981 through mid-February 1983. In general, positive anomalies of reconstructed dynamic height (red contours) follow periods of westerly anomalies, and negative dynamic height anomalies follow periods of easterly anomalies. The timing of wave development relative to the wind bursts appears to be generally consistent with RK06. The December 1981 Kelvin wave had a particularly high amplitude (e.g., Weickmann 1991), but this reconstruction suggests that the intraseasonal Kelvin waves that developed March–April, June–July, and November 1982 might have been more instrumental to the development of the warm event. Although the timing of the wind bursts appears to be consistent with the reconstructed Kelvin waves, the wind data contain substantial biases and might not accurately portray intensity or duration of wind bursts during this period.

Figure 6 shows the winds and reconstructed dynamic height for 27 April 1986 through 9 September 1987. All downwelling Kelvin waves, suggested by positive (red) contours of reconstructed dynamic height, are preceded by westerly wind bursts of varying amplitudes, and upwelling waves (blue contours) tend to be associated with easterly anomalies near their origination points. Higher amplitude dynamic height anomalies are usually associated with higher amplitude wind anomalies.

Figures 5 and 6 show Kelvin wave anomalies responding to changes in surface winds. Westerly wind bursts tend to amplify downwelling waves and are associated with eastward propagation, whereas enhanced trade winds appear to attenuate downwelling waves and occasionally result in slower eastward propagation, as shown by S07. Away from downwelling waves, easterly wind anomalies are associated with the development of upwelling waves. These results suggest that the reconstruction is consistent with the expected location and timing of Kelvin waves during the periods shown. Further analysis suggests similarly consistent relationships during other periods before 1988 as well (such as an active period during 1979), though the highest amplitude waves during the period 1974–87 occurred during the periods shown in Figs. 5 and 6.

d. Activity index

Figure 7 shows the wave activity index discussed in section 3c (dashed–dotted line) and the supplemental wave activity index (based only on sea level observations and the east Pacific OLR index: gray line), plotted with the reconstructed ENSO dynamic height index, comparable with Fig. 3b in RK06. Where the two activity indices nearly overlap, only the supplemental index is shown. The differences between the wave activity index and the supplemental wave activity index during the period 1986–92 suggest that peak wave activity during the period 1974–85 may be reduced by up to 25% from actual values.

High levels of variance tend to occur when the ENSO index is high, often when it is increasing with time. However, the ENSO index peaked before the wave activity index during the 1976 and 1982–83 El Niño events. The tendency for waves to be most active after these Niño-3.4 dynamic height maxima does not necessarily suggest that Kelvin waves did not modulate the development of the El Niño events. To illustrate, Fig. 5 suggests that the highest amplitude waves during 1982 occurred during December 1981–January 1982, March–April, June–July, and November–December. Figure 5 shows a weak general transition in the zonal wind from anomalous easterly to anomalous westerly across January 1982, suggesting that the initial trend toward El Niño conditions was first associated with the December 1981–January 1982 wave. This high amplitude wave was not a part of a longer period of enhanced wave activity, so the activity index is not very sensitive to it. Weickmann (1991) gives a detailed discussion of the wind burst associated with this event, and Fig. 5 suggests that the resulting wave propagated across the entire basin. However, later wave events might have been more relevant to the development of the El Niño. Figure 8 shows OLR and reconstructed dynamic height perturbations for the period late March through December 1982. This figure shows that the wave during June–July 1982 was followed by gradual eastward progression of anomalously active convection and the development of mature El Niño conditions. This wave and convection were directly preceded by a negative OLR anomaly moving eastward across the Indian Ocean into the west Pacific, consistent with an active convective phase of the MJO. The solid gray line in Fig. 9 shows unfiltered sea level height at Christmas Island (the long-term mean has been subtracted), and the dashed–dotted black line shows the same quantity filtered for the 20–120-day band. The region in Fig. 9 enclosed by the solid rectangle highlights the June–July Kelvin wave. The filtered sea surface height shows that the wave passed Christmas Island during July, consistent with the reconstruction. The amplitude of this wave suggested in the reconstruction and the filtered sea level data were not extraordinary, but the unfiltered sea level height jumped about 25 cm with the wave passage and remained high through December. These results suggest that the June–July Kelvin wave event may have largely determined the ultimate amplitude and timing of the 1982–83 El Niño (consistent with the timing noted by Harrison and Schopf 1984). Chu and Frederick (1990) describe an intense westerly wind burst during mid-May 1982. The reconstruction suggests, however, that this event produced a low-amplitude Kelvin wave. Figures 5 and 8 together suggest that westerly anomalies during June were more relevant to the development of the El Niño. Figure 8 shows that the active MJO OLR anomaly slowed down as the oceanic Kelvin wave developed. Figure 8 thus indicates that the Kelvin wave was forced in a manner consistent with the coupling process discussed by RK06. Thus, the MJO was likely more relevant to the development of the 1982–83 El Niño than suggested by Bergman et al. (2001) and others. The wind burst and Kelvin wave of November 1982 were more intense than the June–July event, but the El Niño was already at its peak. Figure 5 suggests that Kelvin waves observed during the 1982/83 El Niño probably were characterized amplitudes higher than those suggested in the wind-forced model-based wave reconstruction of Bergman et al. (2001), especially since the present reconstruction likely underestimates actual wave amplitude.

Although the results in Fig. 7 help clarify our points about the reconstruction, the above analysis alone is not sufficient to explain the general relationship between ENSO and intraseasonal Kelvin waves, nor is this analysis as comprehensive as that of others with respect to the relationship among the MJO, Kelvin waves, and ENSO (e.g., Zhang and Gottschalck 2002; RK06; and many others). Further analysis of the reconstruction to more thoroughly assess relationships between intraseasonal variability in the atmosphere and ENSO is a major focus of continued research. The regression analysis discussed below in section 4d was designed to diagnose differences between Kelvin wave behaviors during ENSO events prior to the period of plentiful TAO data with more recent waves and is one step in many potential future applications of the reconstruction.

e. Regression comparison

The regression composites of RK06 suggest patterns of coupling between intraseasonal Kelvin waves in the ocean and convection and wind stress in the atmosphere, but RK06 was limited to the period 1988–2005. This short dataset could not be easily applied to generalize their results to earlier periods. To demonstrate one application of the reconstruction, we apply a similar regression analysis here, reproducing their analysis for the period 1988–2005 and developing a second regression model for the period 1974–87. We then test for the significant differences between the composite wind stress to discern whether the patterns differed between the two periods. Figure 10 shows the regression composite applying Eq. (1) of RK06 for the period 1974–87, and Fig. 11 shows the composite for the period 1988–2005 (comparable to Fig. 6 of RK06 using TAO data). Figures 10a and 11a show regressed OLR and reconstructed dynamic height, and Figs. 10b and 11b show regressed OLR and NCEP–NCAR reanalysis wind stress. Both figures show negative OLR and westerly wind stress preceding positive dynamic height anomalies of downwelling Kelvin waves. Both figures also suggest that subsequent Kelvin waves during a trend toward El Niño are characterized by successively slower phase speeds. The wind stress patterns (Figs. 10b and 11b) are similar because the strongest westerly wind bursts occur with the active MJO convection near lag zero. The westerly wind burst west of the date line near lag zero appears to be more intense and to last longer in the later period, but the differences are not statistically significant above the 95% level (see appendix), although they are significant at the 90% level. The earlier period shows significantly more westerly anomalies in the central and east Pacific near lag zero than the later period. The Indian Ocean OLR pattern of the MJO appears to repeat more than one full cycle during the period 1974–87, while only one cycle appears during the period 1988–2005. Figure 7 suggests that during the earlier period, there was a tendency for the highest amplitude Kelvin waves to occur after the corresponding peak in the ENSO index, whereas during the later period, the greatest Kelvin wave activity tended to occur at the same time as or before maxima in the ENSO index.

The overall comparison of Figs. 10 and 11 suggests that the wind stress and convective patterns of the two periods are broadly similar. The appendix describes a simple resampling test that demonstrates that the two wind stress composites (Figs. 10 –11b) cannot be assumed to be different across most of their time and space domains, with the main exception of the significantly stronger westerly stress across the east Pacific near lag 0 during the earlier period.