## 3 Lessening Fluctuation Method
### Fluctuations in Cascade Development  
To determine the primary energy \(E\) based on the energy release at the observation level, usually the following dependence is used: \(q=aE^{b}\), where \(a\), \(b\) are parameters depending on the penetration depth \(d\) and the mass of the primary particle. The equation is statistically correct. However, \(q\) (at the observation level) strongly fluctuates in an individual event. Therefore, in order to solve the problem of large fluctuations in the development of a cascade, it is necessary to start with an analysis of fluctuations in individual events. For this, we first considered individual cascades with significantly different cascade curves. As the analyzed value of the cascade size, we used the logarithm of the energy release, log \(q\), at the measuring layer. The measuring levels of the PAMELA calorimeter were equidistant. Therefore, as the cascade curved, we analyzed the dependence of the cascade size on the layer number, \(L\). Figure 1: Proton spectra of various experiments [11; 12; 13; 14; 15; 16; 17; 18; 19]. The cascade, the development of which starts at the beginning of the calorimeter, is called a fast cascade. The cascade, the significant development of which begins in the second half of the calorimeter, is called a slow cascade. Figure 2a shows the cascade curves for three cascades of 10 TeV protons with fast, medium, and slow character of the development of the cascade process. As can be seen in Figure 2a, the total energy release in the calorimeter for these three showers differs significantly. The total energy release by the fast shower is several times higher than that released by the slow shower. Therefore, methods using this parameter to reconstruct the primary energy will determine that the energy of these showers is significantly different, while the primary energy of these showers is the same. The situation is similar for methods that use the energy release on the last calorimeter layer as a calculated value, since this value also differs significantly for these three showers. Determination of the energy from the maximum of the cascade curve is possible only for fast showers. Slow and medium showers cannot be analyzed because they do not reach the maximum of the cascade curve. Thus, the statistics of the analyzed events are significantly reduced. Moreover, the maximum point shifts towards greater depth with increasing energy. Therefore, the higher the primary energy, the thicker the calorimeter should be. To understand how to solve the problem of large fluctuations in the development of a cascade, it is necessary to understand what their causes are. The behavior of the cascade curve depends on the features of the interaction of the primary nucleus with the nuclei of the calorimetric substance. Different approaches and methods are used in order to study features in the multi-particle production [22; 23; 24; 25; 26]. First, the fluctuations of the penetration depth before the first interaction are very important. The earlier the primary particle interacts, the faster the cascade begins to develop. Second, fluctuations in the multiplicity of the first interaction are important. If the first interaction is central, then many particles are produced and the cascade develops rapidly. If the first interaction is peripheral, then the cascade develops slowly. For example, the proton that initiated the middle cascade (Figure 2a) interacted at the beginning of the calorimeter, but the rapid development of the cascade began only after layer 9. Figure 2: (**a**) Cascade curves for three showers of 10 TeV protons with fast, medium, and slow character of the development of the cascade process; (**b**) The size–rate curves for the same showers. After the first interaction, each secondary particle can interact, also producing secondary particles. The parameters (the penetration depth before the first interaction, the number of secondary particles, etc.) of each subsequent interaction also fluctuate.