A Model of Sunspot Number with Modified Logistic Function

TL;DR

This paper introduces a modified logistic model to predict sunspot numbers and solar cycle features with high accuracy, using minimal initial data, and demonstrates its effectiveness on historical and recent solar cycles.

Contribution

The paper develops a two-parameter sunspot prediction model based on a modified logistic differential equation, enabling early and accurate forecasts of solar cycle maxima, minima, and lengths.

Findings

Average maximum prediction error of 8.8%

Cycle length prediction error of 9.5%

Accurate prediction of cycle 24 sunspot variations

Abstract

Solar cycles are studied with the Version 2 monthly smoothed international sunspot number, the variations of which are found to be well represented by the modified logistic differential equation with four parameters: maximum cumulative sunspot number or total sunspot number $x_m$, initial cumulative sunspot number $x_0$, maximum emergence rate $r_0$, and asymmetry $\alpha$. A two-parameter function is obtained by taking $\alpha$ and $r_0$ as fixed value. In addition, it is found that $x_m$ and $x_0$ can be well determined at the start of a cycle. Therefore, a prediction model of sunspot number is established based on the two-parameter function. The prediction for cycles $4-23$ shows that the solar maximum can be predicted with average relative error being 8.8\% and maximum relative error being 22\% in cycle 15 at the start of solar cycles if solar minima are already known. The quasi-online method for determining solar minimum moment shows that we can obtain the solar minimum 14 months after the start of a cycle. Besides, our model can predict the cycle length with the average relative error being 9.5\% and maximum relative error being 22\% in cycle 4. Furthermore, we predict the sunspot number variations of cycle 24 with the relative errors of the solar maximum and ascent time being 1.4\% and 12\%, respectively, and the predicted cycle length is 11.0 (95\% confidence interval is 8.3$-$12.9) years. The comparison to the observation of cycle 24 shows that our prediction model has good effectiveness.