Solving multistatic sonar location problems with 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 reflected sound waves. Knowing the speed of sound in water, the time when the sound was sent from a source, and the arrival time of the sound at one or more receivers, it is possible to determine the location of surrounding objects. The propagation of underwater sound is a complex phenomenon that depends on various attributes of the water (density, pressure, temperature, and salinity) and the emitted sound (pulse length and volume), as well as the reflection properties of the water’s surface. These effects can be approximated by nonlinear equations. Furthermore, natural obstacles in the water, such as the coastline, need to be taken into consideration. Given an area of the ocean that should be endowed with a sonar system for surveillance, this paper formulates two natural sensor placement problems. In the first, the goal is to maximize the area covered by a fixed number of sources and receivers. In the second, the goal is to cover the entire area with a minimum-cost set of equipment. For each problem, this paper considers two different sensor models: definite range (“cookie-cutter”) and probabilistic. It thus addresses four problem variants using integer nonlinear formulations. Each variant can be reformulated as an integer linear program in one of several ways; this paper discusses these reformulations, then compares them numerically using a test bed from coastlines around the world and a state-of-the-art mixed-integer program solver (IBM ILOG CPLEX).