1. Introduction

An accurate prediction of the Asian monsoon precipitation on a monthly or longer time scale is very important for water management, economy, and human life in the Asian monsoon region. In particular, over 80% of precipitation occurs during the five summer months (May–September), and its variation often causes serious flooding or drought (Hidore and Oliver 1993). Thus, a reliable forecast of its variation is extremely important for the quality of life in the area. Long-range climate forecasts of monsoon precipitation, however, remain a big challenge. A typical long-range forecast is limited to either “above normal” or “below normal” accuracy on a coarse (e.g., monthly) temporal resolution (Barnston et al. 1999; Hwang et al. 2001). Typical long-range numerical prediction models produce quantitative estimates of monsoon precipitation. The accuracy of the forecasts, however, is not significantly better than the so-called “probabilistic” forecast (above normal or below normal).

There are, in general, two types of approaches for long-range monsoon prediction—dynamical methods and statistical methods. A major reason for the difficulty of dynamical prediction methods is that a dynamical model is so sensitive to given initial conditions of sea surface temperature, wind, moisture, etc. These variables are difficult to determine accurately a few months ahead of time; slight error leads to a significant discrepancy in the ensuing prediction (Fan et al. 2000). Further, model physics is not sufficiently accurate to warrant a long integration from a given observational data of some model variables. Tracton et al. (1989), Brankovic et al. (1990), and Van den Dool and Rukhovets (1994) address that the dynamical extended-range forecasts may not necessarily be a good approach due to several problems intrinsic to general circulation models. Thus, the general consensus of the scientific community is that the development of statistical prediction methods should parallel the improvement of dynamical methods (Anderson et al. 1999; DelSole and Shukla 2002).

Statistical methods, on the other hand, show moderate success in some applications (Barnston et al. 1994, 1999; Webster et al. 1998; Waliser et al. 1999a, b; Lo and Hendon 2000; Mo 2001). Such statistical methods as multiple regression, analog, canonical correlation, and neurological network have been attempted to perform seasonal forecasts (Barnston et al. 1999). Statistical methods, however, have a serious shortcoming in that little physical mechanism is built into them (Vautard et al. 1999). This tends to make long-lead predictions inaccurate, especially for precipitation which is an important but difficult variable to predict. Furthermore, most long-range predictions are limited to a coarse temporal resolution (e.g., monthly or seasonal) specifically due to the lack of physical understanding of variability on a fine temporal scale. For this reason, climate predictions still remain in a “crawling” stage.

According to Vautard et al. (1999), physical understanding of observational data, namely identification of physical components in them, is essential for improving the long-range predictive skill. Waliser et al. (2003), for example, employed extended EOFs in their MJO prediction method in an attempt to resolve physical evolution in their data. The present study is consistent in motivation and spirit with the idea presented by Vautard et al. (1999). In the present study, observational data are decomposed via cyclostationary EOF (CSEOF) analysis into the evolution of individual climate signals in the dataset. The present approach is different from that based on EOFs since a faithful representation of climate signals is not possible in terms of stationary EOFs.

Section 2 explains the conceptual framework of the prediction algorithm based on the separation of variability into “regularly occurring” deterministic climate signals. Results and diagnostics of 3-yr forecasts are given in section 3 for pentad precipitation data. In section 4, results and diagnostics are given for the 30–60-day bandpass-filtered precipitation data in order to make inferences on the predictability of the intraseasonal oscillations. Discussion and concluding remarks are given in section 5.

2. The conceptual framework of the prediction method

The concept of the new statistical prediction method is best illustrated as follows. Let us consider an artificial time series in the form

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where deterministic components, cos2πωt and sinπωt, are slowly modulated by stochastic, or equivalently, nondeterministic amplitude time series, A(t) and B(t), respectively. At the outset, it should be pointed out that actual climate signals may not necessarily be cos2πωt or sinπωt. Simple artificial signals have been used here to facilitate our discussion. A crucial motivation for viewing observational data in this way is that the evolution features of climate signals are (nearly) deterministic, whereas the overall strengths of climate signals change stochastically in time. For example, the physical evolution of El Niño as manifested in sea surface temperature over the tropical Pacific appears to be nearly deterministic (Philander 1990), as is also suggested by some theoretical and modeling studies. The occurrence (timing) of El Niño as seen in the sea surface temperature record, however, is stochastic (not deterministic) and is subject to prediction.

Figure 1 shows an example of time series (1) with ω = 1 yr⁻¹, and A(t) and B(t) are given in Fig. 2 (solid lines). Once the deterministic components are identified in P(t), only the stochastic components (amplitude variation of climate signals) are predicted since the deterministic components are already known. Since the stochastic components have much longer time scales than the deterministic components, the former are easier to predict than raw data. As contrasted in Figs. 1 and 2, A(t) and B(t) would be much easier to predict than the raw time series P(t). This new paradigm of prediction contrasts with the conventional prediction methods in which raw data are directly predicted without separating deterministic components from stochastic components.

Thus, the first step prior to prediction is to identify deterministic components in a dataset and separate them from their respective stochastic modulation time series. This separation can be accomplished by performing CSEOF analysis (Kim and North 1997). The method, in essence, identifies from a space–time dataset, P(r, t), repeating deterministic components of variability, B_n(r, t), such that they are mutually orthogonal in space and time and that their amplitude time series, S_n(t), are uncorrelated,

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Unlike EOF modes, CSEOF modes, B_n(r, t), have both spatial and temporal structures. The deterministic signals repeat in time, that is,

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where the so-called nested period d is assumed to be one year in this study. Namely, each climate signal repeats every year; only the strength of each climate signal varies from one time to another. CSEOF analysis on the time series in Fig. 1 identifies reasonably the two deterministic components (Fig. 3) and their modulation time series (dashed lines in Fig. 2). As should be obvious in (3) and Fig. 3, the nested period d should be a multiple of physical periods; otherwise, deterministic signals cannot be rendered accurately. This is an important caveat of the CSEOF technique.

Each amplitude time series is predicted separately with the assumption that each principal component (PC) time series is independent of others; this is nearly true for the PC time series of CSEOFs. The prediction technique employed here was developed in Kim and North (1998). Kim (2000) showed that this prediction method is equivalent to the prediction normal equation (Newton 1988). The performance of the employed method is similar to other linear statistical prediction methods based on the minimization of prediction error variance. The details of the prediction method are beyond the scope of this study and can be found in Kim and North (1998, 1999). Once the amplitude of each climate signal is predicted, the forecast field, P̂(r, t), can be obtained via

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where Ŝ_n(t) are the forecasted amplitude of individual signals.

3. Prediction of the ASM precipitation

To extract climate signals, CSEOF analysis was first conducted on the 20-yr (1979–98) Xie–Arkin pentad (5-day mean) precipitation data (Xie and Arkin 1996) over the Asian monsoon domain, (0°–60°N) × (30°E–180°). The precipitation data, P(r, t), were decomposed into a number of deterministic climate signals and their stochastic undulation time series as described in section 2. The analysis identified such deterministic components as the annual cycle (Lim et al. 2002; Wang and LinHo 2002), the ENSO signal (Kim et al. 2003), the Intraseasonal Oscillation (ISO) (Madden and Julian 1972, 1994; Knutson and Weickmann 1987; Annamalai and Slingo 2001; Seo and Kim 2003) in the 40–50-day period band, and other shorter intraseasonal oscillations (Krishnamurti and Ardanuy 1980; Hartmann et al. 1992; Hsu and Weng 2001).

The first 16 dominant modes were taken for climate prediction, which explain about 67% of the total variability of the pentad precipitation data. The remaining several hundred modes account individually for less than 1% of the total variability (see Table 1) and represent the “sporadic” or “noisy” components of variability. They were not used for prediction not only because they individually explain a small fraction of the total variability but because they essentially have no predictability. To assess the error incurred by taking only the first 16 modes, the reconstructed data (1979–2001) based on the 16 modes were compared with the observational data over the five regions (Fig. 4) of prominent Asian summer monsoon (ASM) precipitation. They are India, Bangladesh and the Bay of Bengal, the Indochina peninsula, China (Yangtze River area), and northeast Asia (Korea and Japan). It is evident in Fig. 5 that significant active/break phases as well as the year-to-year variation of rainfall were reasonably captured in the first 16 modes.

Each amplitude time series, S_n(t), of the 16 CSEOF modes was forecasted from 1 January 1999 to 31 December 2001 using the linear statistical prediction model addressed in section 2 (Kim and North 1998). The training period and, henceforth, the prediction filter were updated every six pentads (∼1 month) including new training data for the last six pentads. It should be noted that data in the forecast period have not been used for constructing a prediction model. Forecast fields, then, were generated according to (4).

Figure 6 shows the forecasts of the ASM precipitation 2 months ahead of time for the five regions shown in Fig. 4. As can be seen in Fig. 6, predictions (dashed lines) at 2-month lead reasonably reproduce fluctuations in the observational data (solid lines). It is encouraging that the active and break phases during the ASM period are, in general, predicted faithfully. Also obvious is some phase lagging, common to most linear prediction methods. Regional comparisons indicate good predictive skills for the Indian monsoon area, including the Bay of Bengal and Bangladesh, the heaviest rainfall region over the ASM domain (Figs. 6a,b; see also Fig. 7). The East Asian monsoon regions (China, Japan, and Korea) also exhibit nontrivial predictability. In particular, prediction is more accurate over the China area, where the intraseasonal peaks are relatively small in magnitude compared with other areas. The only region among the test sites where prediction is relatively poor is the Indochina peninsula.

Performance of the developed prediction method was investigated via correlation and relative error variance of prediction as a function of lead time for each test site (see Fig. 7). Here, comparisons are with the observational data truncated at the 16 CSEOF modes as well as with the ±1 pentad moving-averaged observational data.

Additional comparison with the 16-mode truncated data examines the prediction skill for the 16 modes retained for predictions only; in a comparison with the raw data, the effect of truncation obscures the actual performance of the prediction method. Note in the computation of correlation and relative error variance that the climatological (mean) seasonal cycle, which represents a significant fraction of the precipitation variability, was removed from both the observational data and the prediction for a stringent comparison of the prediction results. In a typical operational prediction, the climatological seasonal cycle is removed as a repetitive signal. With the seasonal cycle, the correlation of prediction with the 16-mode reconstructed data is close to 0.9 or higher for a forecast at 6-month lead.

As should be expected, correlation of prediction, in general, decreases with lead time, and relative error variance generally increases with lead time. At all stations except for the Indochina peninsula, forecasts sustain correlation >0.5 and relative prediction error <0.75 (Brier skill score >0.25) with the 16-mode reconstruction data up to the lead time of ∼25–30 days. This should be compared with the persistence skill in Fig. 7; persistence reaches the same threshold within the lead time of ∼10 days. This is a significant improvement; predictability of precipitation was generally tripled from the persistence at each station (figure not shown). A comparison with the raw data indicates that a large percentage of variability still remains to be explained. Nonetheless, prediction results are nontrivial in the sense that the persistence barrier is less than 5 days. Some stations exhibit correlation >0.3 and relative error variance <1.0 with the raw data at the lead time of ∼30 days. As noted above, higher modes have essentially no predictability; addition of higher modes generally deteriorates the prediction results.

As shown in the results, forecasts of pentad precipitation seem plausible about a month ahead of time over many regions of the ASM based on the new paradigm of prediction. It is remarkable that some fine temporal structures of regional precipitation have been captured in a long-term climate prediction. The present results compare favorably with previous attempts (Barnston et al. 1994, 1999; Webster et al. 1998; Anderson et al. 1999; Waliser et al. 1999a, b; Hwang et al. 2001; Lo and Hendon 2000; Mo 2001; Webster and Hoyos 2004). Unfortunately, a direct comparison is difficult because the temporal resolution of the present prediction is 5 days in contrast to the monthly or seasonal mean prediction in many earlier studies. Note that the 16 CSEOF modes together explain almost the same amount of variance in monthly precipitation over the ASM domain.

To highlight the conceptual difference of the present method, prediction results of a conventional method were compared with that of the new method. Figure 8 shows the 15-day prediction (dotted lines) and the Indian precipitation (solid lines) rendered via 16 CSEOFs (upper panel) and 24 EOFs (lower panel). The leading 24 EOFs together explain approximately the same amount of precipitation variability as 16 CSEOFs do. The prediction algorithm is identical with that used for forecasting CSEOF time series. As can be seen in the figure, the performance of the new method (ρ = 0.76) is superior to a conventional method (ρ = 0.42). While the former reproduced reasonable peaks, the latter underestimates the peaks seriously, which is a typical characteristic of a conventional linear prediction method. The conventional method barely beats the persistence. A similar conclusion is derived for a 30-day prediction with ρ = 0.65 for the new method and ρ = 0.41 for the conventional method (figure not shown). As addressed in Kim and North (1999), many linear prediction methods based on the minimization of error variance yield similar performances.

4. Prediction of the propagation of ISO

Prediction of ISO is an integral component of forecasting the ASM precipitation. The propagation of the alternating phases of ISO is a crucial factor determining the intraseasonal variability of precipitation locally. As already seen in Fig. 5, local precipitation records exhibit strong signs of the ISO activity. In this section, therefore, predictability of ISO is specifically addressed.

Shown in Fig. 9 are the ISO signals predicted 60 days ahead of time in comparison with the observations in 2000, the year of the worst ISO prediction in the 1999–2001 forecast. In both the prediction and the observation, ISO signals were extracted by conducting a 30–60-day bandpass filtering using the Parzen window (Newton 1988). It should be emphasized that filtering was conducted after, not before, the prediction. As shown, the predicted ISO signals closely resemble those in the observation, although the prediction slightly lags the observation. Even at a considerable lead, the evolution of the ISO signals was faithfully reproduced. The present result is very encouraging since the prediction of ISO at 20–30-day lead is considered a serious challenge (Waliser et al. 2003).

Figure 10 is the time–longitude and time–latitude plots showing the propagation of the ISO signals over the equatorial region (Fig. 10a), the Indian monsoon region (Fig. 10b), and the South China Sea region (Fig. 10c) in 1999–2001. Contours and shading denote, respectively, the prediction and the observation. It appears that the direction and the phase of ISO propagation are, in general, reasonably predicted. This suggests that subseasonal active/break monsoon periods caused by the propagation of the ISO can be predicted reasonably. Some inaccuracy, however, is apparent; the ISO signals in the prediction lag those in the observation by 10 days or so on several occasions. Specifically, the sign of the predicted eastward equatorial ISO is opposite to that in the observational data in early 2000 (Fig. 10a).

Figure 11 shows that the ISO was predicted well over India, central China, and northeast Asia; even at a 60-day lead, the predicted local ISO signals compare fairly well with those in the 16-mode reconstruction data with correlation >0.6 and relative error variance <0.8. The two other locations, however, exhibit limited predictability of the ISO signal; in particular, the amplitude was predicted more poorly than the phase of the ISO. A comparison with the ISO signals in the raw data reveals that the ISO signal is somewhat underrepresented in the 16-mode reconstruction data. Nonetheless, the new method extends the predictability of the ISO well beyond that of persistence. The present result also compares favorably with that of Waliser et al. (2003), who obtained a faithful MJO prediction using extended EOFs and a dynamical method. In their results, however, the potential predictability of MJO drops drastically at around 15-day lead. This threshold of predictability comes from the semiperiodic nature of MJO. That is, the MJO is easily predictable at or near its peak position. Between two peaks, however, the MJO is difficult to predict. The present method does not exhibit such a behavior since it is the strength (amplitude) of MJO that is predicted and not the evolution of MJO itself.

5. Summary and concluding remarks

This study aimed at developing a new prediction method based on a sound physical understanding of the ASM variability. In the new method, ASM variability is decomposed into a number of computational modes with corresponding amplitude time series using the CSEOF technique. Each mode describes a regularly occurring “deterministic” component of variability, and the corresponding amplitude (PC) time series describes how the strength of the deterministic component varies on an interannual time scale. Once this decomposition is completed, the amplitude time series of each mode is predicted independently.

The new prediction method was applied to the pentad ASM precipitation. Results indicate that the ASM precipitation as represented by the first 16 modes (67% of the total variance) is reasonably predictable ∼30 days in advance. At the lead time of 30 days, the prediction anomaly and the observed precipitation anomaly from the climatological (mean) seasonal cycle are correlated at above 0.5 and the relative error variance is less than 1.0 at all of the selected test stations except for the Indochina peninsula. It should be noted that higher modes, each of which explains a small fraction (<1%) of the total precipitation variance, are essentially unpredictable; the addition of higher modes did not improve the accuracy of prediction.

Specifically, ISO signals were extracted from the prediction results by conducting a 30–60-day bandpass filtering and were compared with the observed counterparts similarly extracted from the 16-mode reconstruction data. Whether ISO can be forecasted reasonably bears an important implication on a successful prediction of ASM precipitation since local precipitation records are populated with ISO activity. It appears that the ISO forecast at 60-day lead reasonably reproduced the observed ISO signals for the three ASM years studied (1999, 2000, and 2001). Although some significant delays are apparent in 2000, prediction results are generally encouraging with correlation greater than 0.6 and relative error variance less than 0.8 at three of the five stations investigated. Thus, the subseasonal oscillations and henceforth active/break periods of precipitation seem to be predictable, which has been a serious obstacle for successful prediction of monsoon precipitation on a relatively short time scale (Waliser et al. 2003).

While prediction products on a fine temporal resolution are in much demand, few prediction studies were conducted on a 5-day resolution (e.g., Webster and Hoyos 2004). Most previous climate predictions have been performed on a monthly or a seasonal time scale; even many of these are probabilistic forecasts and not quantitative forecasts. Although a direct quantitative comparison cannot be made, it is clear that the performance of the new prediction method excels most of the methods currently used in the operational forecasts of the Asian monsoon precipitation. It is worthwhile to point out that the performance of the method is significantly better than persistence. The improved performance of the present method comes clearly from the separation of deterministic and stochastic components of variability, which was confirmed through a limited test using a conventional prediction method. On the other hand, a prediction method directly employing the raw data may not be so successful, barely excelling persistence. This improved performance is due to the fact that the deterministic components (CSEOFs) in the observational data have much shorter time scales than the stochastic components (PC time series); only the stochastic component is predicted in the new method since the deterministic components are already known. Based on the sensible approach and reasonable performance, the new prediction method is reported as a viable approach to climatic predictions of highly noisy variables.