**5. Reconstruction of electron beam energy spectra**

A necessary condition for successful irradiation with electron beams is complete information about the spatial distribution of the absorbed dose in the irradiated object, which is determined both by the properties of the irradiated object (i.e., geometry, elemental composition, and density) and by the source parameters, primarily the energy beam spectrum [13, 14]. Modern approaches to determine the energy spectra of accelerators are based on direct measurement of electron energy using special equipment [12] and on indirect methods based on the reconstruction of the spectra using experimentally measured data [9–11].

The indirect method of reconstructing the energy spectrum is based on the solution of the Fredholm integral equation of the first kind, which can be formulated as follows:

$$D(\mathbf{x}) = \int\_0^{E\_{\text{max}}} f(E)d(\mathbf{x}, E)dE,\tag{6}$$

where *D x*ð Þ is the distribution of some parameters, such as the absorbed charge, dose, fluence, and flux density, along the parameter *x* (depth, angle, etc.); *d x*ð Þ , *E* is the distribution of the same parameters for a monoenergetic beam with the energy E; *f E*ð Þ is the energy spectrum.

Usually, this equation is reduced to a system of linear algebraic equations by approximating a continuous spectrum *f E*ð Þ linear combination of basic functions *Fj*ð Þ *E* with *<sup>a</sup><sup>j</sup>* acting as weights and by approximating distribution *D x*ð Þ with a discrete set of values *D<sup>i</sup>* at points *xi*. The corresponding system of linear algebraic equations takes the form:

$$D^i = \sum\_{j=1}^N d^{ij} d^j. \tag{7}$$

Here *d<sup>i</sup>*,*<sup>j</sup>* is the parameter at the point *xi* created by an electron beam with an energy spectrum *Fj*ð Þ *E* .

A common method for solving the system (7) is the least squares method. Fredholm equation of the first kind in the general case is an incorrectly posed problem, that is, the solution of the equation may not exist or there may be several of them. Also, the solution of the system (7) can change greatly with small changes of *D x*ð Þ. These properties of the integral equation are transferred to its discrete counterpart, which leads to non-physical sharply oscillating solutions that have little to do with the true spectrum of the beam, as demonstrated in **Figure 5a**.

To address this phenomenon, the regularization procedure is used [11, 58] that involves the modification of the original problem, which turns an incorrectly posed problem into a correctly posed one. There are two types of regularization: the first type modifies the equation; the second type modifies allowed solutions. The simplest
