atmosphere to emit fluorescence and Cherenkov light which intensity and spectrum depend on the electron state at the time of emission. Thus, when measured at many times while the shower traverses the atmosphere, the light characteristics enable a derivation of the total energy deposit of the shower particles changing with the atmospheric depth. This provides the best estimation of the shower primary energy i.e. the energy of the cosmic ray particle $E_0$. It also allows one to calibrate the much more frequent showers, registered by the ground detectors all the day round, not only at dark nights necessary for the light detectors measurement, as this was done for the first time at the Pierre Auger Observatory [@pao].\
However, an exact determination of the shower atmospheric profile, from which $E_0$ and $X_{max}$, the depth of the shower maximum, can be derived, demands a detailed knowledge of the state of shower most abundant particles, electrons, at any stage of its development. Our present study shows that it is possible to obtain it, in contrast to the common belief that extensive air showers are a highly fluctuating phenomenon. Indeed, the atmospheric depths of the cosmic particle interactions, the energies transferred to the secondary 