Mathematical Modeling of the Expanding Square Search Pattern of the Lost Submersible

In this article, the Expanding Square Search Pattern is introduced for locating missing submersibles in three-dimensional ocean environments. Initial deployment points are set based on the highest probability density from the positioning model, and the optimal search area is established using anticipated probabilities. Following discretization, an Expanding Square Search Pattern is adopted within the horizontal plane plus vertical scanning to search the space. Indefinite integration is performed over the scanned space using the probability density matrix, ultimately deriving a function that links search probability to time.