## 2 Materials and Methods
### Micro-Scale Spherical Surface Modeling via Genetic Algorithm
Thus, the initial parameters (\(h\), \(k\), \(l\)) are determined by means of the next expressions  
\[k =\frac{\mathcal{Z}_{1}[\mathcal{Y}_{4}+\mathcal{Z}_{4}]-\mathcal{Z}_{ 3}[\mathcal{Y}_{2}+\mathcal{Z}_{2}]}{2[\mathcal{Y}_{3}\mathcal{Z}_{1}- \mathcal{Y}_{1}\mathcal{Z}_{3}]} \tag{4}\] \[l =\frac{\mathcal{Y}_{1}[\mathcal{Y}_{4}+\mathcal{Z}_{4}]-\mathcal{Y }_{3}[\mathcal{Y}_{2}+\mathcal{Z}_{2}]}{2[\mathcal{Z}_{3}\mathcal{Y}_{1}- \mathcal{Z}_{1}\mathcal{Y}_{3}]}\] (5) \[h =\frac{\mathcal{Z}_{5}[\mathcal{X}_{8}+\mathcal{Z}_{8}]-\mathcal{Z }_{7}[\mathcal{X}_{6}+\mathcal{Z}_{6}]}{2[\mathcal{X}_{7}\mathcal{Z}_{5}- \mathcal{X}_{5}\mathcal{Z}_{7}]} \tag{6}\]  
where, \(\mathcal{Z}_{1}=(z_{i,j+\Delta})-(z_{ij})\), \(\mathcal{Z}_{2}=(z_{i,j+\Delta})^{2}-(z_{ij})^{2}\), \(\mathcal{Z}_{3}=(z_{i+\delta,j+\Delta})-(z_{i+\delta,j})\), \(\mathcal{Z}_{4}=(z_{i+\delta,j+\Delta})^{2}-(z_{i+\delta,j})^{2}\), \(\mathcal{Z}_{5}=(z_{i+\delta,j})-(z_{ij})\), \(\mathcal{Z}_{6}=(z_{i+\delta,j})^{2}-(z_{ij})^{2}\), \(\mathcal{Z}_{7}=(z_{i+\delta,j+\Delta})-(z_{i,j+\Delta})\), \(\mathcal{Z}_{8}=(z_{i+\delta,j+\Delta})^{2}-(z_{ij,\mu})^{2}\), \(\mathcal{Y}_{1}=(y_{i,j+\Delta})-(y_{ij})\), \(\mathcal{Y}_{2}=(y_{i,j+\Delta})^{2}-(y_{ij})^{2}\), \(\mathcal{Y}_{3}=(y_{i+\delta,j+\Delta})-(y_{i+\delta,j})\), \(\mathcal{Y}_{4}=(y_{i+\delta,j+\Delta})^{2}-(y_{i+\delta,j})^{2}\), \(\mathcal{X}_{5}=(x_{i+\delta,j})-(x_{i,j})\), \(\mathcal{X}_{6}=(x_{i+\delta,j})^{2}-(x_{ij})^{2}\), \(\mathcal{X}_{7}=(x_{i+\delta,j+\Delta})-(x_{i,j+\Delta})\), and \(\mathcal{X}_{8}=(x_{i+\delta,j+\Delta})^{2}-(x_{ij+\Delta})^{2}\). By computing Equations (4)-(6), four parents (P\({}_{1,P}\), \({}_{2,P}\), \({}_{3,P}\), \({}_{4,P}\)) are determined for each parameter, where, the sub - index \(t\) indicates the generation number. In this way, the parent \(P_{1,1}\) is determined by computing Equations (4)-(6) via sub-indices (\(i=0\), \(j=0\), \(\delta=n/2\), \(\Delta=m/2\)). The parent \(P_{2,1}\) is determined by computing Equations (4)-(6) via sub-indices (\(i=n/2\), \(j=n/2\), \(\delta=n/2\), \(\Delta=m/2\)). The parent \(P_{3,1}\) is determined by computing Equations (4)-(6) via sub-indices (\(i=n/2\), \(j=0\), \(\delta=n/2\), \(\Delta=m/2\)). The parent \(P_{4,1}\) is determined by computing Equations (4)-(6) via sub-indices (\(i=n/2\), \(j=m/2\), \(\delta=n/2\), \(\Delta=m/2\)). Thus, four coordinates (\(h\), \(k\), \(l\)) are obtained, and they are defined as the initial population. Additionally, the ratio \(\bar{R}\) is determined by the average distance between the origin coordinates and the surface points. Then, the genetic algorithm determines the search space based on the maximum and minimum value of each parameter. In this way, the minimum and maximum of the parameter \(h\) are defined as \(x_{0,0}\) and \(x_{n,m}\), respectively. The minimum and maximum of the parameter \(k\) are defined as \(y_{0,0}\) and \(y_{n,m}\), respectively. The minimum and maximum of the parameter \(l\) are defined as (\(l+\bar{R}/2\)) and (\(l-\bar{R}/2\)), respectively. Thus, the initial population is completed. The second step is to perform the crossover to create the current children. This procedure is carried out via exploration and exploitation [46]. In this procedure, two children are created inside parents and two children are created outside parents. Thus, the children (\(C_{1+4^{q},l}\), \(C_{2+4^{q},l}\), \(C_{3+4^{q},l}\),\(C_{4+4^{r}q,l}\)) are created via parents (\(P_{1+2^{q},l}\), \(P_{2+2^{q},l}\)) for \(q=0\) and \(q=1\). These children are computed by means of the next equations  
\[C_{1+4*q,l}=\left\{ \begin{array}{l}P_{1+2*q,l}-0.5\beta\big{|}P_{1+2*q,l}- \mathrm{minimum}\big{|},\text{ if }P_{1+2*q,l}<P_{2+2*q,l}\\ P_{2+2*q,l}-0.5\beta\big{|}P_{2+2*q,l}-\mathrm{minimum}\big{|},\text{ if }P_{2+2*q,l}<P_{1+2*q,l}\end{array} \right.