## 3 Methodology
### Ant-Colony Optimization  
ACO is a probabilistic method that is capable of resolving computational altercations that are constrained to finding paths in several ways [72]. It monitors the behavior of ants in the pursuit of food and fetching back to their nest by analyzing their paths [73]. The assemblage of the food is a unified effort of a group of ants for fetching the food, and they diffuse pheromones while making their way back; this diffusion appears as a fracture in a seismic volume. The diffusing pheromone can be recognized as an odor, and the scouting ants hunt the food by smelling this pheromone. The diffused pheromone is very volatile and, hence, evaporates. Thus, the solitary route remains, which is of shorter length and is followed by the majority of the ants to detect the noisy trends in the seismic data [11]. The scouting group of ants shows intelligence to extract the discontinuities that lead to colony optimization. The scouting ants detect their ideal track to food based on their intelligence and pheromone strategy [74], which is termed as swarm intelligence [75]. Swarm intelligence augments the discontinuous features in an edge-detection seismic volume since it tacitly tracks those trends that are continuous and are likely to be fault or fracture.  
The algorithm supposes that the number of ants is proportional to the amount of diffused pheromone on a branch [15]. The pheromone on a branch at an intensity of path (_i_, _j_) at time \(t\) is denoted by \(\tau_{ij}\), while the branch tracking \(\rho\) (\(0\leq\rho\leq 1\)) and the pheromone amount per unit length of ant \(k\) at a path (_i_. _j_) is denoted by \(\Delta\tau_{ij}^{k}\).  
\[\tau_{ij}(t+1)=\rho\tau_{ij}(t)+\sum\Delta\tau_{ij}^{k}.(t) \tag{1}\]  
The branch length of ant \(k\) in a circle is represented by \(Z_{k}\), and \(\Delta\tau_{ij}^{k}=Q/Z_{k}\), where \(Q\) is a constant. The factor \(\eta_{ij}\) shows the visibility of the edge path \((i,j)\), \(\beta(\beta\geq 0)\) shows the relative importance of the path visibility, \(\alpha(\alpha\geq 0)\) shows the relative importance of the branch, \(U\) represents the possible vertex set, and \(\rho_{ij}^{k}(t)\) gives the transition probability of the ant \(k\) at time \(t\). The probability of electing the branch was used in this study by using the 3D seismic volume to identify and sharpen the tracked faults and fractures [76].  
\[\rho_{ij}^{k}(t)=\left\{\begin{array}{c}\frac{[\tau_{ij}(t)]^{a}[\eta_{ij}] ^{\beta}}{\sum_{i\in U}[\tau_{ij}(t)]^{a}[\eta_{ij}]^{\beta}},\ j\in U\\ 0,\ \textit{otherwise}\end{array}\right. \tag{2}\]  
The application of ant tracking entails several factors that extract fault information in a seismic amplitude volume. These factors incorporate an initial ant boundary, ant track deviation, ant step size, permissible illegal steps, obligatory legal steps, and stopping criteria [77]. The initial ant boundary governs that how narrowly the preliminary ant agents may be kept within a seismic volume. The ant track deviation illustrates the examining tolerance of the ants along the side of its tracking direction. A higher value of ant track deviation gives higher connections of the ant agents. The step size shows how far an ant can progress for every increment of its search. The permissible illegal steps showcase the ability of ants to image discontinuities further than the existing position. The obligatory legal steps factor works with the illegal step feature by acquiring the designated number of legal steps after an illegal step. The stop criteria also work with the illegal steps factor and do not permit the ants' movement when plenty of illegal steps are involved.