## 1 Introduction  
### Physical Background of the Method  
Automating the measurement process of the magnetic properties of macro-objects in motion is a task in various industries where ferrous scrap is a common raw material. The capability to directly control the metallic raw material composition in moving transport without unloading and sampling is a scientific and technical problem which is gaining much attention [1; 2]. Operation monitoring is mainly required to control the various forms of the hidden mass fraction of non-magnetic substances present in ferromagnetic raw materials. Non-magnetic inclusions add technologically useless weight to the freight, which is documented during weighing and included in the subsequent financial calculations. The damage caused by non-magnetic inclusions in railway freight per annum adds up to more than seven billion rubles of direct financial losses in Russia. At present, there is no system, device, or unit that enables the express control of these inclusions without stopping or unloading the train or automobile. The closest analogues are X-ray television (TV) units, which are being positioned by some developers as instruments for freight truck express control [3,4]. In addition to high technical and maintenance costs, the usage of high-powered sources of ionizing radiation for express control has a series of restrictions related to work safety standards. X-ray measurements are vulnerable to video signal distortion via special freight stowing and X-ray radiation shielding. X-ray measurements offer no ability to evaluate the non-magnetic material mass fraction. Various metal detectors [3, 4, 5, 6] offer no solution to the problem of non-magnetic substance quantity control in ferrous material, since a metal detector's purpose is to reveal insignificant quantities of metal in non-magnetic or weakly magnetic media, which is an inverse problem to the one considered here. Numerous works on magnetic induction tomography [7, 8, 9, 10, 11] were devoted to magnetic scanning of biological objects, which solves the same problem as the metal detectors--measuring a weak magnetic signal in a non-magnetic medium. However, the difference in magnetic induction tomography compared to metal detectors is that it uses mathematical signal processing, for example, a three-dimensional (3D) finite-element method (FEM) [12, 13, 14]. Therefore, tomography can produce a 3D picture of the distribution of magnetic centers by scanned volume. Mathematically, this is achieved by the region being divided into a finite number of subdomains (elements), where the solutions of differential equations are sought. In each of the elements, the type of the approximating function is chosen arbitrarily. The coefficients of the approximating functions are usually sought from the condition of the equality of the value of neighboring functions at the boundaries between the elements (at the nodes). These coefficients are then calculated for the functions of the elements in the nodes. A system of linear algebraic equations is compiled and solved. However, this method is not applicable to the task of searching for a signal of non-magnetic inclusions in a ferromagnetic medium at low frequencies. The task of detecting weakly magnetic signals in a non-magnetic matrix is solved using high frequencies to scan an object (at hundreds and thousands of megahertz [7]). This approach is fundamentally not applicable to solving our inverse problem of searching for non-magnetic inclusions in a strongly magnetic matrix. Scanning a ferromagnetic material with a high-frequency field will lead to a strong decrease in the amplitude of the electromagnetic waves as they penetrate deep into the conducting medium [15]. As a result of this effect, a high-frequency alternating current, when flowing through a conductor, is distributed mainly in the surface layer. This shields the signals from the depths of the object. In our case, a railway car will screen the signal from the load.