## 1 Introduction  
In modern agriculture, there is a tendency to increase efficiency, which has undeniable advantages but may also have negative effects. In order to achieve high efficiency, it is necessary to provide large working widths of machines, which allows for processing a large area with fewer work passes [1]. However, machines with large widths have a high energy requirement, which in turn requires the use of larger tractors [2]. These are characterized by large sizes and weights, which have an impact on the arable soil. In this case, the main negative phenomenon will be soil compaction caused by elements of the chassis of tractors and agricultural machines. The soil compaction is most often determined as a reduction in the volume of soil pores and the resulting closer proximity of soil particles [3; 4]. The consequences of compaction on arable soils include the disruption of water-air relations in the soil, increased mechanical resistance for plant roots, and the need to use more energy to loosen the soil [5; 6; 7]. In order to reduce the compaction of arable soils, methods are used based primarily on increasing the contact surface of the running gear elements with the ground. For this purpose, the following methods are used, among others: the reduction in air pressure in the tires, the use of twin tires, the use of tires with a wider width than standard [8; 9; 10], and in extreme cases, the installation of tracked systems instead of wheeled systems [11]. Soil compaction is the subject of much scientific research. Research directions include determining changes in parameters describing compaction (density, compactness, maximum shearing stress), describing the geometric parameters of the rut or footprint [2; 12; 13; 14], and formulating mathematical equations describing the impact of operational parameters on the intensity of compaction. The most frequently described changes are those in the compactness or parameters of the rut as a function of the changes in the vertical wheel load or tire inflation pressure [15; 16; 17; 18; 19]. Research on compaction can be divided into laboratory and field research. In the first case, soil bins or soil boxes are used. An example of laboratory research is the work of [PERSON] et al. [20], which showed differences in the values of the density and compactness of \"artificial\" soil at two levels of vertical load and three levels of inflation pressure. Similar results are also described in the work of [PERSON] et al. [21] (where footprints obtained in a soil bin were modeled) or in the work of [PERSON] and [PERSON] [22], where the contact surfaces of wheels with the ground in a soil bin were examined. In turn, field tests are performed in real conditions, most often as a result of the impact of full-size agricultural vehicles and machines, and sometimes using special test stands. The works [23; 24; 25; 26] can be taken as examples. Both in the case of laboratory and field tests, the formulation of mathematical models requires knowledge of many parameters, so it is very important to ensure correct measurement techniques. While there are no major technical problems when measuring tire deformations, they may arise when trying to record what is happening under the soil surface as a result of compaction [15]. There are many studies that attempted to determine stresses in soil and their distribution [27; 28; 29; 30; 31; 32]. Due to the mentioned technical difficulties related to measuring stresses inside the soil, innovative simulation techniques such as the Finite Elements Method (FEM) are increasingly used--this method was used in the works [33; 34; 35]. Another innovative technique is the method of scanning a tire nut or footprint to obtain a 3D computer image. This technique was used in the works [36; 37; 25; 38; 14; 39]. In many studies on the impact of wheels on the ground, the analyzed parameter is the contact area of the wheel with the ground.