## 3 Lessening Fluctuation Method
### The Analysis Procedure: The Size-Rate Function  
The analysis procedure consisted of several main stages.  
First stage: simulation of cascades with fixed energies.  
We simulated 100 cascades initiated by iron nuclei, 100 carbon cascades and 100 proton cascades with fixed energies of 1 TeV and 10 TeV in the PAMELA calorimeter.  
The mean cascade curves for these cascades are presented in Figure 3a. Error bars show statistical errors. As can be seen in Figure 3a, the development of the proton and Fe cascades is significantly different. At \(L<8\) Fe showers with energy of 1 TeV have \(\log\,q\) higher than proton showers with energy of 10 TeV. The most significant factors determining the observed differences are the penetration depth before the first interaction and the number of secondary particles formed.  
Second stage: smoothing the cascade curve.  
To reduce fluctuations from layer to layer, a signal accumulation method along the spectrum was used. This method allows for a softer minimization of fluctuations from layer to layer, in contrast, for example, to fitting with a polynomial function [29]. Smoothing was carried out at three points in accordance with the formula:  
\[S_{L}=\frac{1}{3}\sum_{i=L-1}^{L+1}\log q_{i}\]where \(\log q_{\rm i}\) is the measured value of the shower size, and \(S_{L}\) is the accumulated value of the shower size.  
Third stage: searching for the beginning of the cascade development.  
As can be seen in Figure 3a, the avalanche-like process of the cascade development does not always begin immediately after the interaction of the primary particle with the calorimeter substance. If the first interaction is peripheral, then the energy release from layer to layer can not only increase, but also decrease. We considered such cascades as not having started. In order to separate the part of the measurements in which the cascade had not yet begun, we compared the rate of development of the cascade in adjacent layers. If the rate of development of a cascade increased at three adjacent observation levels in a row, then such a cascade was considered to have begun. For example, the fast cascade shown in Figure 3a was considered to have started at the first level, the middle at layer 9, and the slow cascade only at layer 15.  
Fourth stage: plotting \(SR\) distributions.  
Figure 3b shows the average size-rate dependences for the same proton, carbon and Fe cascades as in Figure 3a.  
As can be seen in Figure 3b the size-rate dependences are an ordered structure depending on the primary energy and are practically independent of the type of the primary nucleus. This fact can also be attributed to the advantages of the presented approach.  
Fifth stage: creating the size-rate function.  
To create the \(SR\) function, we fitted the \(SR\) curves of third-order polynomial functions for each fixed energy:  
\[S\left(R\right)=a_{0}+a_{1}R+a_{2}R^{2}+a_{3}R^{3} \tag{1}\]  
Then the coefficients \(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{3}\) were fitted depending on the energy. The size-rate function for reconstructing primary energy was in the following form:  
\[S(R,E)=a_{0}(E)+a_{1}(E)R+a_{2}(E)R^{2}+a_{3}(E)R^{3} \tag{2}\]  
Using the size-rate function (2), an analysis of test cascades was performed.  
Figure 3: (**a**) Cascade curves for showers initiated by protons, C and Fe nuclei with energies of 1 TeV and 10 TeV in the PAMELA calorimeter; (**b**) The size–rate curves for the same showers.