The Multistatic Sonar Location Problem and Mixed-Integer Programming

Abstract

A multistatic sonar system consists of one or more sources that are able to emit underwater sound, and receivers that listen to the direct sound as well as its reflected sound waves. From the differences in the arrival times of these sounds, it is possible to determine the loca- tion of surrounding objects. The propagation of underwater sound is a complex issue that involves several factors, such as the density and pres- sure of the water, the salinity and temperature level, as well as the pulse length and volume and the reflection properties of the surface. These ef- fects can be approximated by nonlinear equations. Furthermore, natural obstacles in the water, such as the coastline, need to be taken into consid- eration. Given a certain area of the ocean that should be endowed with a sonar system for surveillance, we consider the task of determining how many sources and receivers need to be deployed, and where they should be located. We give an integer nonlinear formulation for this problem, and several ways to derive an integer linear formulation from it. These formulations are numerically compared using a test bed from coastlines around the world and a state-of-the-art MIP solver (CPLEX).