^{1}

^{1}

The results of kinematic modeling of the 24th solar cycle (SC) are discussed. Time series of solar and cosmic ray monthly mean indices were received from web resources of international data centers. The previous prediction of the solar cycle shape using kinematic modeling technique demonstrated rather close agreement with the final phase of the SC23 and relatively large magnitude of SC24. The forecast of 2005 is updated with respect to the modern observation of monthly mean data. The study allows concluding that the SC24 magnitude will probably near the previous cycle. Predicted moments of the maximum monthly mean values are expected in July–September 2012. The uncertainty of this interval is about

As it was concluded by the special solar cycle prediction panel, the sample of different forecasts was separated in two main predictions of reasonably large or small SC24 magnitudes at 2007 (

All the predictions were classified according to the used technique as physical, statistical, and mixed. It is remarkable that the physical predictions have lower magnitudes than statistical and mixed ones, with the exception of Dikpati researches [

One of the statistical techniques called as kinematical approach previously allowed to predict the long final stage of the 23rd cycle and high amplitude of the 24th one (

The time series of the monthly mean values of solar indices were used in the analyses. The international sunspot numbers (Ri) were received from the website (

The observation data were processed using the numerical technique called as the sequential analysis of dominated harmonics method. The method was used firstly for the Caspian Sea level change prediction [

The results of repeated observation

In the absence of a priory information about a number and a contribution level of periodical harmonics in the whole character of variations, the iteration step by step procedure seems to be efficient in the analysis.

At the first stage, it is natural to determine and remove a trend component from the series using the equation

This task can be solved under condition of

For the common case, we can have a priory inverse weight matrix

At the second step, removing the modeled linear component from the initial time-series, subtracting (

The model for the residual time series will be presented as

The iteration process will be written as

Estimation of the vector-function

Then, the iteration process (

Hence, the results of the last iteration will be accepted as final estimations of the model (

Empirical standards

Then, we will compute normalized parameter estimations

Having a reason to consider a harmonic to be statistically confident, we receive an approximation function (

The technique considered in this work is used in a statistical modeling of time variations in observed characteristics. This method can be considered among the periodogram analysis methods the general theory of which was described, for example, in [

The above-described method was used in the modeling and prediction of the 23rd solar cycle form using Ri and 10.7 cm radio flux data [

The practice proved that the best predicting results were received in a case of relatively short solar time series usage. Predictions usually became confidently lower than reality in the use of hundreds years long-time series of monthly mean indices. This feature can be explained by the less accurate ancient observation and a possible miss of some cycles as it was discussed in publications.

It is necessary to remark that a quantity of a sample of determined periodically components is normally about several tenses for a 50-year monthly mean solar time series. In these cases, a statistical degree of freedom is more than 400, that demonstrates a level of an effectiveness of the used approximation method. The time spans of series of relative sunspot numbers and sunspot areas were spread from 1950.0. Used series of 10.7 cm solar radio flux and cosmic ray indices had begun from 1947.0 and 1958, respectively. The final moments of all-time series were limited by the beginning date for the last and the current predictions.

The last SC predictions using the sequential analysis of dominated harmonics were made in 2004-2005 (see Figures

Model, prediction, and epignosis of the monthly mean Ri pattern. Dotted line demonstrates the real data. Dashes show the one sigma intervals. Blue dots are principal epignosis control marks. Abscissa axis fix years of the last modeled and predicted intervals. Ordinate axis shows international sunspot number values.

Model, prediction, and epignosis of the monthly mean indices of 10.7 cm radio flux (S.F.U) pattern. Dotted line demonstrates the real data. Dashes show the one sigma intervals. Blue dots are principal epignosis control marks. Abscissa axis fix years of the last modeled and predicted intervals. Ordinate axis shows S.F.U. values.

Model, prediction, and epignosis of the monthly mean sunspot areas pattern. Dotted line demonstrates the real data. Dashes show the one sigma intervals. Blue dots are principal epignosis control marks. Abscissa axis fix years of the last modeled and predicted intervals. Ordinate axis shows sunspot area values in units of millionths of a hemisphere.

It is necessary to explain that the one sigma tendencies were previously determined like mean square values of true errors in five retrospective forecasts of the way of behavior of Ri indices.

The analysis of the correlation between the different pair of indices was allowed to determine the correspondent linear regression formulas which were used in one sigma interval determination for the S.F.U and sunspot area values.

The tests of the previous SC forecasts demonstrate that the best prediction is present in the 5-year interval as it was expected from the earliest experiments. One can see that the true errors of the forecast are near to one sigma margins. Actual solar indices are systematically lower than the forecasted positions for the Ri and 10.7 radio flux variations (see Figures

The general conclusions of the former SC behavior forecast were the next [

The minimal monthly mean indices were expected for the 2006.5–2008.

The final stage of the 23rd solar cycle was expected to be reasonable long and have some local maxima.

The 24th solar cycle monthly mean index was expected to be near

The final phase was estimated especially for the main solar indices together with galactic cosmic ray data [

In a reference to the intercycle minimum phase, we have to give a passing mention to some principal studies. The nature of the minimal phase between 23rd and 24th solar cycles was carefully analyzed using 10.7 cm solar radio flux indices in the paper [

Today, it is a propitious moment for updating the published previous prediction because the amplitude and the time of maximum activity are not obvious yet.

The same data type sets updated by the modern observed values were used for the correction of the former SC24 prediction. Analyzed time series were consisted of monthly mean indices. Data archive addresses are denoted above. The main task for the moment is improving the estimations of the amplitude and the length of the SC24.

New data processing using the sequential analysis of dominated harmonics method was performed for the sunspot numbers, 10.7 cm radio flux indices, and sunspot area time series. Cosmic ray observations from the Moscow station were used too. The results of new SC24 predictions are illustrated in Figures

SC24 shape prediction for monthly mean sunspot number variation. Dotted line demonstrates the real data. Dashes show the one sigma intervals. Solid is the modeled curve.

SC24 shape prediction of monthly mean indices for 10.7 cm radio flux variation. Dotted line demonstrates the real data. Dashes show the one sigma intervals. Solid is the modeled curve.

SC24 shape prediction for cosmic ray monthly index variation. Dotted line demonstrates the real data. Dashes show the one sigma intervals. Solid is the modeled curve.

SC24 shape prediction for monthly mean sunspot areas variation. Dotted line demonstrates the real data. Dashes show the one sigma intervals. Solid is the modeled curve.

As we can see from the Figures

The interesting fact is that the previous high magnitude SC24 forecast of the SC prediction panel coincides with the presented prediction in general in a reference to the magnitude and relatively narrow shape. It can be seen in comparison with Figure

The first international consensus in SC24 prediction dated April 7, 2008 (

The updated international prediction of SC24 dated April 9, 2009 (

Three types of solar data were used in the kinematic modeling for the previous prediction [

As it is shown by Jan Janssens, the medium 90–140 amplitude SC24 forecasts predominate for today. At present, some researchers conclude that the maximal SC phase has already started. Our modeling allows presuming that researchers have at least a half of a year before monthly mean sunspot maximum will occur and experts have an opportunity to check and update current SC forecasts.

The results obtained from the simulation described above support the relatively high (about 130 Ri) or moderate magnitude variants of the 24th SC predictions for the monthly mean value case. Forecast capabilities of the modeling method were verified by several retrospective simulations. In the case of the forecast coming true, the entry into a super centurial solar minimum would be shifted in decades to come.

The author would like to thank his closest collaborator Michail Krainev for his significant impact on cosmic ray variation modeling and to express gratitude to heads of the experiments who placed the data in the Internet. Especial acknowledgment has to be expressed to the anonymous reviewer for his valuable comments and recommendations.