## 3 Results and Discussion
### Approximative Expressions and Simulations  
Nowadays, more complicated methods and formulas exist and can be used in this field of science [27; 33; 34]. They are mostly related to calm conditions in the ionosphere, but still they are also used for perturbed conditions. However, the question arises of how easy some of them are for use and whether they are applicable when speed of analysis and obtaining products are important. The idea is to make the formula easy to use as well as to enable rapid calculations and analysis, such as for example the IRI model--the international standard empirical model for the terrestrial ionosphere [35; 36]. In order to create an easier and more adequate use of the results and data, here we introduce the electron-density expression:  
\[\log Ne(h,Ix(t),t+\Delta t)=\left\{ \begin{array}{cc}\sum\limits_{i=0}^{2}a_ {i}(h)\cdot(\log(I_{x}^{\max}))^{i},&t\leq t_{Ix^{\max}}\quad\text{a})\\ y0+(2*A/pi)*(w/(4*(t-xc)^{2}+w^{2})),&t>t_{Ix^{\max}}\quad\text{b})\end{array} \right. \tag{4}\]  
where, \(t_{\max}\) is time of maximum of X-ray flux (Wm\({}^{-2}\)) and \(h\) is height (km), \(\Delta t\) is time delay defined earlier in the text and \(a_{i}(h)\) represents height-dependent dimensionless coefficients. Here, \(y0+(2*A/pi)*(w/(4*(t-xc)^{2}+w^{2}))\) is Lorentzian function. Equation (4) gives more than satisfactory results when simplified (which was our intention to make it easy to use), i.e., when (4a) is valid for the whole time interval. Consequently, instead of whole Equation (4) we can use a simple logarithmic second-degree polynomial expression for theentire time interval, with height-dependent coefficients:  
\[\log Ne(h,Ix(t),t+\Delta t)=\sum\limits_{i=0}^{2}a_{i}(h)\cdot(\log(I_{x}^{ \max}))^{i} \tag{5}\]  
Figure 10: Variations in the effective reflection height \(H^{\prime}\), sharpness \(\beta\) and X-ray irradiance during the occurrence of SF on 11 June 2014 are shown in panels from lower to upper, respectively. Height-dependent coefficients \(a_{i}(h)\) can be found in the Github project. Compared to paper [37], here for the first time we introduce \(\Delta t\) and improve the expression by taking into account the delay time of the ionosphere response, making it more physical. Time delay \(\Delta t\) in min can be presented (we use it further in the investigation) by linear dependence on the logarithm of X-ray flux (see, e.g., Figure 11)  
\[\Delta t=\sum_{i=0}^{1}c_{i}\cdot(\log(I_{x}^{\max}))^{i}\ [\min] \tag{6}\]  
where dimensionless quantities take values \(c_{0}=0.45385\) and \(c_{1}=-0.44863\) and \(I_{x}^{\max}\) is X-ray flux at peak time in [Wm\({}^{-2}\)]. Further in the investigation we will use expression (6), which can be additionally corrected by inserting various influences. Of course, a more sophisticated formula might have been developed, but such precision was unnecessary since the values of the D-region parameters are known to fluctuate by an order of magnitude. The goal was to make the formula simple to use while also allowing for quick calculations and analysis. In Figure 11a, we present values of obtained time delay \(\Delta t\) between the X-ray flux and the VLF amplitude as a function of peak flux. Black circles are data from present work and red stars are data from [27], obtained under the same conditions, i.e., with the same equipment and for the same transmitter-receiver path. Additionally, in Figure 11b we provide the distribution of \(\Delta t\) data points--a rug plot. The time delay between the X-ray flux and the VLF amplitude maximum \(\Delta t\) has a mean value of about 3 min. For C-class SFs, \(\Delta t\) is more scattered and has the highest variation, and for the larger values of flux \(I_{x}\) (M- and X-class SFs), the increase is the smallest. From Figure 6 of the paper by [PERSON] et al. [38], one can see that presented data are scattered, even more so than in our case, with \(\Delta t\) up to 10 min and with a mean value of about two minutes, which is in accordance with our results.