## 2 Approach and Associated Methods
### Addressing Offsets in Empirical Equation Results  
By substituting the TSP concentrations retrieved from the datasets into the equations in Table 1, it becomes evident that the resulting visibilities (Figure 2) generally agree with the recorded values in the distribution form, but suffer from considerable offsets. Furthermore, performing RMSE analysis (Table 3) revealed that there was a significant performance difference among the evaluated models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Area of Sampling** & **Measured TSP Concentration** & **Record Visibility** & **Number of** & **Step Type** & **Reported Measurement Errors** \\  & **Range, w/gm3** & **Range, km** & **Data Points** & **TSP, w/gm3** & **Visibility, km** \\ \hline KSA [17; 18] & 269.86–7639.09 & 0.2–10 & 16 * & Urban & \(\pm 0.1\) & \(\pm 10\)\% * \\  & 50–395 & 4.83–56.31 & 14 & Urban & 848.185\({}^{\circ}\)/-0.843 & Not specified \\ USA [3; 19] & & & & & Max * \\  & & & & & 128.687\({}^{\circ}\)/-0.841 & \\  & & & & & 128.687\({}^{\circ}\)/-0.841 & \\  & 2000–440,000 & 0.06–8 & 13 & Rural & Min 105007\({}^{\circ}\)/-1.16 & Not specified \\  & 43.95–1401.5 & 1.41–12.41 & 23 & Rural & Not specified & \(\pm 10\)\% * \\ North-East Asia [13; 20] & 101.5–3071.4 & 0.97–24.17 & 30 * & Various sites & Not specified & \(\pm 10\)\% * \\ Australia [15] & 24.99–2050.55 & 0.2–40.31 & 47 * & Rural & Not specified & \(\pm 10\)\% * \\ \hline \hline \end{tabular}  
* data point measurements were extracted from images using the digitization [21] method (TSP: Figure 7 in [17]/Figure 9 in [18] and visibility: Figure 8 in [18]; Figure 2 in [14]; Figure 5 in [15]). b minimum (R\({}^{2}\) = 0.7321) and maximum (R\({}^{2}\) = 0.7634) errors are fitted with a trendline for conciseness. Raw data may be inspected individually in its respective reference. c an assessment [22] found that modern visibility sensors, despite their accuracy claims, usually operate within an accuracy of \(\pm 10\)%.  
\end{table} 
Table 2: Measured total suspended particles (TSPs) concentration ranges and their corresponding recorded visibilities from different sources/regions. Figure 1: Recorded visibility (linear scale) vs. recorded TSP concentration (logarithmic scale) ranges; data point measurement ranges from 6 different sources. Figure 2: Logarithmically plotted results of listed equations (refer to legend) in Table 1 and TSP concentration from Australia (**a**), USA (**b**) and (**e**), North-East Asia (**c**) and (**d**), and KSA (**f**). It can be observed that empirical formulas over- or under-estimate the originally recorded visibilities (\(\bullet\)). However, the variation in performances in the form demonstrated in Figure 2 implies the tendency to either over- or under-estimate visibilities based on given TSP concentrations. Therefore, it indicates a chance that the closest values may reside within the average of two equations. To develop an algorithm addressing the noted offsets in the calculated visibilities, we first defined a layout for the retrieved data of calculated and recorded visibilities. By breaking the layout into specific segments, the data became hospitable for transitioning into a programming environment. Figure 3 highlights the breakdowns, with brief descriptions, of the tabulated data assigned to the arrayed variables. Three arrays were defined to hold the necessary data: namely, OriVal[_Var_] to store the recorded visibilities, EqResults[_Var_] [5] to store the results of the equations in Table 1, and EqNum [5] to store the alphanumeric characters to track alphanumerically the final, 2-averaged equations, with the lowest error percentage. Additionally, the _Var_ variable was substituted by the number of rows until Row \(n\), thus setting a boundary for loops and associated counters. As demonstrated in Figure 4, the program comprises a few sequential steps, starting with initializing the necessary variables and data-holding arrays.