^{1}

^{2}

^{3}

^{*}

^{2}

^{1}

Chenghai Wang, Lanzhou University, China

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In contrast to the historical forecast test which is temporally successive with a near-steady forecast skill, the real-time forecast made at any one moment produces a forecast time-series whose skill rapidly decreases as the forecast lead time increases; thus, only the leading section of the latter is adopted in practical applications. As compared with the intraseasonal filtered historical forecast, the real-time extended-range forecast has a lower skill in the absence of filtering. In addition, it is difficult to estimate the intraseasonal phases near the end of the real-time forecast time-series due to missed information afterward. The current work developed a simple but useful method to improve the real-time forecast skill from the above two aspects for an empirical extended-range forecast scheme. The scheme is devoted to predict the intraseasonal variabilities of Indian summer monsoon precipitation, in which the boreal summer intraseasonal oscillation acts as the precursor. The intraseasonal signals in the previous observations, the better forecast skills of shorter lead times, the implicit information regarding the intraseasonal phases in the forecast of longer lead times, and the data-adaptive intraseasonal filter are adopted in the improving method, so as to extract intraseasonal signals as much as possible from the currently available information at each forecast moment. A practical comparison shows that the forecast skills of the real-time forecast improved by this method are close to or even better than the intraseasonal filtered historical forecast. Even at the longest acceptable forecast lead time that the forecast after which is considered to be worthless, it helps extract useful information regarding the intraseasonal phases.

Each summer, the Indian summer monsoon (ISM) brings about 80% of annual rainfall over Peninsular India and affects more than one billion people living there. While the precipitation experiences a smooth climatological seasonal cycle that onsets in late May, maximizes during July, and slowly decreases through September, in any one year it comprises several wet episodes (“active” phases) and dry episodes (“break” phases), each lasting 10–30 days (

The boreal summer intraseasonal oscillation (BSISO,

Numerical models’ performance in predicting BSISO has been significantly improved in recent years, particularly with coupled models (

On the other hand, empirical models through statistical approaches have a long history in predicting the seasonal mean ISM precipitation (

The empirical schemes for extend-range forecast of ISM precipitation introduced above achieved great success; however, we argue that there is still room for improvement. As a quantitative forecast, considerable high anomaly correlations were attained in

As mentioned above, in contrast to a successive time-series obtained from a long-time historical forecast test (namely, each point is predicted at a fixed lead time), the real-time forecast made at any one moment produces a time-series with a rapidly decreasing skill as the forecast lead time increases. Our question is: can we get information regarding the phases at the end of such a time-series even if the predicted magnitudes have large deviations? It could still be very useful since it tells whether a peak or a valley is coming or decaying and when it will arrive. Furthermore, for extended-range forecast, there is no doubt that a significantly higher skill would be obtained after applying an intraseasonal filter to the historical forecast result. In the real-time forecast, however, the terminal effect of filtering would possibly make it even worse. Can we improve the real-time forecast quality by extracting its intraseasonal component as much as possible? The current work is devoted to improve the real-time forecast skill from the above two aspects for an empirical extended-range forecast scheme.

The rest of the article is organized as follows. The “Data and Procedures” section introduces the data and procedures. The “Empirical Prediction of Intraseasonal Variabilities of ISM Precipitation Based on Precursory BSISO Signals” section describes the empirical forecast scheme for predicting the intraseasonal variabilities of the ISM precipitation based on its relationship with the BSISO. The “Improving the Real-Time Forecast” section discusses how to improve the real-time forecast. A summary is given in the “Summary” section.

The datasets used in this study include data on the observed daily outgoing longwave radiation (OLR) from the National Oceanic and Atmospheric Administration polar-orbiting satellites (

The empirical forecast scheme used in the current work is based on multiple linear regression, and the predictors are selected according to the lagged maximum covariance analysis [MCA, also known as singular value decomposition analysis (

With smoothed daily anomalies, a lagged MCA is applied to the precipitation over the India region and the OLR over the tropical Indian Ocean. The predictors are obtained by projecting the OLR anomalies onto the heterogeneous covariance maps of the corresponding MCA modes. These predictors are employed in constructing the multiple linear regression model, whose predictand is the precipitation and the forecast lead time agree with those in the lagged MCA. Each year for historical forecast tests, a model is constructed with the data of the previous 25 years, so that the interannual and inter-decadal variations of the precipitation–OLR relationships are taken into consideration, if they exist.

The Hilbert transform is utilized in identifying the instantaneous phase of the time-series. It is defined by

The Hilbert transform

With the property of the phase shifting, the complex function constructed with the original function as the real part and its Hilbert transform as the imaginary part can be written in the following form according to Euler’s formula:

In this study, the forecast instantaneous phase

It is worth pointing out that the instantaneous frequency defined by

In order to obtain a smoothed signal and to extract the intraseasonal component as well, a preprocessing that is partially equivalent to a frequency filtering based on the variational mode decomposition (VMD) is applied to the time-series before calculating its instantaneous phases. The VMD (

We also compared other filtering algorithms, such as the Fast Fourier Transform-based filter. Since the VMD is data adaptive, it performs better in most cases, particularly for short time-series whose length is comparable with the timescale of filtering. This is especially useful for real-time forecast.

The intraseasonal standard deviations during boreal summer for daily OLR over the Indo-western Pacific region, and precipitation over South Asia are shown in

Standard deviations of 20- to 80-day filtered daily anomalies of

The precipitation over South Asia and the large-scale tropical OLR anomalies share some common features in the power spectrum distributions. For instance, for the raw pentad anomalies of precipitation over northern Peninsular India and OLR over southwestern equatorial Indian Ocean (

As revealed in the lagged MCA (

Heterogeneous correlation maps of the first

Essentially, these two leading MCA modes captured the northeastward-propagating BSISO1 mode and its lagged impact on precipitation (

With the knowledge of the precursor signals revealed by the MCA, a multiple linear regression model is constructed, in which the predictand is the precipitation anomalies, and the predictors are the projections of the OLR anomalies onto the corresponding heterogeneous covariance maps of the MCA. The lead time of the OLR anomalies agrees with that in the lagged MCA, and so is the period for estimating the regression coefficients. For instance, based on the MCA results exhibited in

The forecast skill of the empirical scheme described above varies with the lead time and the number of MCA modes employed in the regression model. Based on 16 years of historical forecast tests from 2004 to 2019, we found that as long as the two leading MCA modes are included, introducing the other modes into the regression model does not significantly improve the forecast skill. In some cases, it could be even worse. For simplicity, only the forecast results based on the two leading MCA modes are analyzed below.

Such a simple mode with two predictors captures the dominant intraseasonal variabilities over Peninsular India 2–3 weeks in advance.

Anomaly correlations of the observed and forecast precipitation anomalies on the intraseasonal timescale (20- to 80-day band-pass filtered) during 2004–2009 JJA at different forecast lead times (indicated at the top-right corners). Dotted areas passed the significance test at the 95% level (Student’s

The anomaly correlations shown in

Time-series of 10-day moving-averaged daily precipitation over northern Peninsular India (72.5–87.5°E, 17.5–22.5°N) during 2004–2019 JJA for raw observational anomalies (thin-gray lines), 20- to 80-day VMD-filtered observational anomalies (thick-black lines), and 20- to 80-day VMD-filtered forecast anomalies made at a lead time of 15 days in advance (thick-red lines). Points marked by circles indicate that the 20- to 80-day observational and forecast anomalies are instantaneously in phase with each other. The ratio of in-phase is 66.7% over the entire period, and their anomaly correlation is 0.65 (denoted on the top-right corner by “In-phase” and “ACC,” respectively). The forecast time-series has been multiplied by a fixed factor of 2.0 to compensate the systematic underestimation of the amplitude. The discontinuities between 2 years should be ignored.

As discussed in the

An example for regular

An analogous real-time forecast is shown in

Same as in

From another point of view, the significant improvement of the intraseasonal filtered historical forecast implies that the intraseasonal signal does exist in the unfiltered and interrupted real-time forecast result. It is thus speculated that for the real-time forecast, extracting intraseasonal signals as much as possible from the currently available information would be an approach to improve the forecast skill. The following facts or assumptions are thus utilized in achieving this purpose:

The previous observations contain intraseasonal signals.

The forecast of a shorter lead time has better skills.

Even if the forecast anomalies have large deviations as the lead time increases, there could still be information regarding the intraseasonal phases.

An intraseasonal filter is favorable for extracting the intraseasonal signal.

Based on the above premises, a simple but useful method is proposed for improving the real-time forecast.

The improved real-time forecast based on the method described above is shown in

Same as in

The sensitivity of the forecast skill to different KBs and KFs in the improved real-time forecast is tested and shown in

Ratio of in-phase and anomaly correlation (ACC) of improved real-time forecast at different KBs and KFs.

Lead | KB | KF | In-phase (Raw/VMD) (%) | ACC (Raw/VMD) |
---|---|---|---|---|

20 | N/A | N/A | 42.9/57.8 | 0.51/0.63 |

20 | 0 | 0 | 25.1/26.0 | 0.51/0.53 |

20 | 0 | 5 | 47.6/47.3 | 0.51/0.50 |

20 | 0 | 15 | 49.3/49.9 | 0.51/0.51 |

20 | 0 | 30 | 49.4/49.9 | 0.51/0.51 |

20 | 15 | 0 | 37.8/38.6 | 0.51/0.51 |

20 | 15 | 5 | 52.7/54.1 | 0.51/0.54 |

20 | 15 | 15 | 54.0/56.6 | 0.51/0.55 |

20 | 15 | 30 | 52.5/55.3 | 0.51/0.53 |

The first line is for the result in the regular real-time forecast and intraseasonal filtered historical forecast. The latter and the best improved results are marked by red.

In contrast to the historical forecast test which is temporally successive with a near-steady forecast skill, the real-time forecast made at any one moment produces a forecast time-series whose skill rapidly decreases as the forecast lead time increases. Generally, such a real-time forecast is interrupted and only the leading section with a relatively higher skill is adopted in practical applications. The interruption brings two problems. Due to the terminal effect, the real-time filtering is unworkable. Hence, as compared with the intraseasonal filtered historical forecast, the real-time extended-range forecast has a significantly lower skill in the absence of filtering, even though the intraseasonal signal does exist as proved by such a comparison. In addition, it is difficult to estimate the intraseasonal phases near the end of the interrupted forecast time-series due to missed information afterward, particularly when the forecast amplitudes have a large deviation. The current work developed a simple but useful method to improve the real-time forecast skill from the above two aspects for an empirical extended-range forecast scheme.

The empirical scheme used in the current work is devoted to predict the intraseasonal variabilities of the ISM precipitation, in which the BSISO acts as the precursor. A lagged MCA is applied to the 10-day moving-averaged daily anomalies of precipitation over the Indian region and the OLR over the tropical Indian Ocean for JJA. The projections of the OLR anomalies onto the heterogeneous covariance maps of the two leading MCA modes are employed in constructing the multiple linear regression model for precipitation, whose forecast lead time agrees with that in the lagged MCA.

The anomaly correlation and the ratio of in-phase on the intraseasonal timescale are applied in evaluating the forecast skill. The instantaneous phase is estimated by the phase angle of the complex function whose real part is the original signal and the imaginary part is its Hilbert transform. A VMD filter on intraseasonal timescale is applied on the time-series before estimating its instantaneous phase. As long as the phase difference between the forecast and observational anomalies is less than or equal to

For areas of large intraseasonal variabilities of ISM precipitation, the empirical scheme used in the current work has an acceptable forecast skill when the forecast lead time is <20 days. However, relative to those in the intraseasonal filtered historical forecast, both the anomaly correlation and the ratio of in-phase have a significant fall in the real-time forecast, as expected. The ratio of in-phase drops more sharply.

A method for improving the real-time forecast is developed. For each forecast moment, a successive time-series is constructed by linking up the observational anomalies before the initial time of making the forecast, and the forecast made at the initial time for every day after it. The latter is 10–20 days longer than the originally demanded forecast lead time. Next, a 20-day low-pass VMD filter is applied to such a time-series, and the result is used to estimate the instantaneous phase and the amplitude at the time of forecast. The intraseasonal signals in the previous observations, the better forecast skills of shorter lead times, the implicit information regarding the intraseasonal phases in the forecast of longer lead times, and the data-adaptive intraseasonal filter are adopted in this method so as to extract the intraseasonal signal as much as possible from the currently available information at each forecast moment.

A practical test for the method described above shows that for the forecast lead time of 15 days, both the anomaly correlation and the ratio of in-phase in the improved real-time forecast are even slightly higher than those in the intraseasonal filtered historical forecast. For the lead time of 20 days, which is normally the longest acceptable lead time of our forecast scheme and is close to the interrupting point in regular applications, the improving method increases the ratio of in-phase so that it is close to that in the intraseasonal filtered historical forecast. The anomaly correlation has also increased for the lead time of 20 days, although it is still lower than the filtered one. Hence, this method does improve the real-time forecast skill. Even at the longest acceptable forecast lead time that the forecast after which is considered to be worthless, it helps extract useful information regarding the intraseasonal phases.

The current work is not intended to design a better empirical forecast scheme than before (such as

Publicly available datasets were analyzed in this study. These data can be found here: NOAA Interpolated Outgoing Longwave Radiation (OLR) [

All authors contributed to the article and approved the submitted version.

This work was jointly supported by the National Key Research and Development Program of China (Grant Nos. 2018YFC1505803 and 2018YFC1505903) and the National Natural Science Foundation of China (Grant Nos. 41505059 and 41775074). The authors are thankful for the support of the Jiangsu Provincial Innovation Center for Climate Change. The SOEST contribution number is 11154 and the IPRC contribution number is 1478. CPC Global Unified Precipitation data and OLR data were provided by the NOAA/OAR/ESRL PSL, Boulder, Colorado, USA, from their website at https://psl.noaa.gov/.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.