## 2 Materials and Methods
### [PERSON]'s Model and the Values of Its Parameters  
This article examines the efficiency of the most popular model ([PERSON]'s formula) used in forecasting continuous deformations of a land surface induced by underground mining in the Upper Silesian Coal Basin in Poland and on examples of land inclinations and where linear discontinuous deformations also occurred on the land surface. [PERSON]'s model allows for the calculation of the subsidence values (\(W_{B}\)) in the mining area and its derivatives (first derivative--the inclinations (\(T_{B}\)) of the land surface; second derivative--the curvatures (\(K_{B}\)) of the land surface) induced by underground extraction of hard coal from exploitation fields, as follows:  
* Of any shape and any number of vertices;
* Of any height;
* Located at various depths;
* Located in one or several hard coal seams lying horizontally or inclined at any angle;
* With a longwall front moving at any velocity;
* Exploited at any time;
* Located in old goaves;
* Containing mining margins;
* Causing distant influences;
* Causing asymmetry of the profile of the subsidence trough on its slopes. Figure 7: Surveys of the ground points by use of the following: (a) GNSS receiver–situational measurements; (b) precise leveler with tripod and surveying patch-altitude measurements; (c) electronic meter-length distance measurements (source: own study). [PERSON]'s formula for the subsidence of a mining area is based on the geometric integral theory of influences by [PERSON] [9; 10]. [PERSON]'s theory describes the course of subsidence in a mining area using a two-parameter Gaussian function. Assuming that the exploitation field has the shape of a half-plane and the subsidence trough, created as a result of its excavation, is stabilized (static), the formula for the \(W_{K}\) subsidence has the following form:  
\[W_{K}(x;a,\ g,\ r)=-\frac{a\cdot g}{r}\int_{x}^{\infty}e^{-\pi\frac{x^{2}}{r^{ 2}}}dx\ [\mathrm{m}], \tag{1}\]  
where the variables mean the following:  
* \(a\)--Exploitation coefficient;
* \(g\)--Thickness of hard coal seam (a height of longwall) [m];
* \(r=\frac{h}{tan\beta}\)--Range radii of the main influences [m];
* \(h\)--Depth of the exploitation [m];
* \(\beta\)--Range angle of the main influences [\({}^{\circ}\)];
* \(x\)--Distance from the longwall front [m]. The profiles of the real subsidence troughs are characterized by certain deviations from the profiles calculated using [PERSON]'s formula. Because they are difficult to describe using a linear theory of influences, [PERSON] significantly modified this formula by introducing additional parameters that allow for [PERSON]'s function to be more flexible. These modifications led to an improvement in the description of the asymmetry of the profile of the subsidence trough in relation to the edge of exploitation and the exploitation's influences in the extreme parts of this trough. [PERSON] developed a very general methodology for describing deformations, allowing for summing of the influences of an exploitation distributed in various seams. This methodology includes, in addition to lateral activation (influence of old goaves located in the same seam), the activation of old goaves in neighboring seams. The operator method (\(A_{n}\) operators) of modifying the description of the subsidence trough profile used by [PERSON] is related to the presence of the \(d\) mining margin. It refers to the issue of the increase in the volume of the rock mass as a result of vertical strains caused by mining exploitation. The essence of this description is the assumption that the \(W_{B}\) subsidence can be calculated as the sum of the \(W_{K}\) linear influences calculated using [PERSON]'s formula and the \(\Delta W_{K}\) nonlinear correction, as follows:  
\[W_{B}(x)=W_{K}(x)+f(\tau,W_{K})\ [\mathrm{m}], \tag{2}\]  
where the following holds:  
* \(\gamma\)--A simplified form of the \(\gamma_{oct}\) octahedral strain [\(\frac{\mathrm{mm}}{\mathrm{m}}\)].