## 2 Recent Advances in the Evaluation of Rockburst Characteristics  
One of the significant parameters that allows for the sufficient description of the rockburst mechanism as a function of time is the motion's velocity (\(v_{e}\)), which may be assessed from the following relationship based on the empirical back-calculation approach [16]:  
\[v_{e}=d\sqrt{\frac{g}{2 hcos^{2}\theta+dsin^{2}\theta}} \tag{1}\]  
where \(\theta\)--initial ejection angle measured upwards from the horizontal; \(h\)--1/2 height of excavation; \(d\)--observed horizontal range of the motion; \(g\)--gravity. This equation will be later used for validation of the usefulness of the below-proposed approach for the rockburst hazard estimation. Appropriate calculations proved that the value of the above velocity reaches about 5 m/s in the case of serious events. For the cases in which the ejected rock mass is smaller, this velocity is greater and reaches a value of about 10 m/s [17; 18]. As pointed out by [PERSON] [19], in the case of good-quality rocks, one may assume that under high-stress conditions, multiple joints may develop, and their spatial orientations should fulfill the condition of the minimum value of a safety index. The issue mentioned herein is a typical problem occurring during underground excavation and should be taken into consideration during the planning of mining works. Thus, it is recommended to develop small-scale models at the first stage for a better understanding of the considered processes rather than large models that are theoretically able to give an absolutely accurate answer, but at the same time, they use a lot of parameters whose values are characterized by unknown reliability levels [20]. Therefore, the implementation of an analytical approach that permits the evaluation of rockburst safety indexes around the excavation is highly recommended at the preliminary risk evaluation stage. Such an analytical approach may be based on the modified limit equilibrium method (LEM), which has been used for years, particularly for obtaining approximate solutions for different stability problems in rock mechanics and geotechnical engineering [21; 22; 23]. It should be kept in mind that LEM also requires some assumptions in terms of the stress distribution along the failure/wedge surface, which allows for providing a general equation of mechanical equilibrium addressed to different structures. If more complicated mechanisms of failure need to be considered, such as translation integrated with rotation, the literature proposes different more complex algorithms, e.g., Discontinuous Deformation Analysis (DDA) [24; 25; 26] or Block Stability 3D (BS3D) [27; 28], which also permits considering the dynamic events. However, these numerical methods require reliable knowledge of joint systems, which unfortunately is not always available, especially near the mining fronts. Therefore, in most cases, these methods are simplified and limited just to translation mechanisms [29] without any severe losses in accuracy [30; 31] and may be utilized in typical geologic conditions in underground mines. In turn, as highlighted by [PERSON] et al. [10], from numerical approaches and codes in rock mechanics applied for rockburst prediction in the last 20 years, continuous methods such as the finite element method (FEM) and finite difference method (FDM) are still the most common. These methods involved over 77% of the published works, whereas the discontinuous methods (e.g., DEM--discrete element method) were utilized in more than 20% of publications. The first successful attempts at numerical rockburst predictions were presented in 1972 when [PERSON] determined possible locations of rockburst occurrence [32]. Then rapid development of rockburst analyses with dynamic FEM was observed which resulted in the improvement of reliable constitutive models for underground mining conditions [33; 34; 35]. Today, FEM, FDM, and DEM seem to be the most suitable methods for determining the shape and characteristics of rockburst formation under high-stress levels.