This paper presents a new approach for describing the shape of 11-year sunspot cycles by considering the monthly averaged values. This paper also brings out a prediction model based on the analysis of 22 sunspot cycles from the year 1749 onward. It is found that the shape of the sunspot cycles with monthly averaged values can be described by a functional form of modified binary mixture of Laplace density functions, modified suitably by introducing two additional parameters in the standard functional form. The six parameters, namely two locations, two scales, and two area parameters, characterize this model. The nature of the estimated parameters for the sunspot cycles from 1749 onward has been analyzed and finally we arrived at a sufficient set of the parameters for the proposed model. It is seen that this model picks up the sunspot peaks more closely than any other model without losing the match at other places at the same time. The goodness of fit for the proposed model is also computed with the Hathaway -Wilson -Reichmann χ measure, which shows, on average, that the fitted model passes within 0.47 standard deviations of the actual averaged monthly sunspot numbers.
The evolution of the space debris scenario consisting of a very large number of fragments is described using the propagation of the characteristics of equivalent fragments without propagating each and every individual debris fragment. This is similar to characterizing a fluid in terms of the average density, pressure, and temperature without considering the velocities of individual molecules in a fluid element. The space debris fragments are assigned to a three-dimensional bin of semimajor axis, eccentricity, and ballistic coefficient. A suitably defined representative semimajor axis, eccentricity, and an equivalent ballistic coefficient (a, e, B) are defined for the equivalent fragments in each of the bins. A constant gain Kalman filtering technique based on 1) propagating the above characteristics, and 2) updating them as and when further measurements become available, has been proposed. Further the assimilation of the information from other breakups with the passage of time is also possible. The robustness of the constant Kalman gain approach instead of using the Kalman filter statistics helps to handle better the unmodeled or unmodelable errors due to the finite bin size and the environmental perturbations. This methodology is also suggested to handle massive atmospheric data assimilation problems.
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