Bayesian Range Weighting for Sonar
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Active sonars are often plagued by false contacts from fish, which are much more numerous than the submarine targets that the sonar is actually trying to find. Even though submarines are much larger and therefore produce much larger echoes than do fish located at similar ranges, fish located near the sonar can produce echoes that confuse the sonar operator. Bayes's theorem provides a convenient framework for defining a range weighting technique to help ignore unwanted sonar contacts from nearby fish.
Sonars resolve targets in range and bearing. A sonar records the arrival time and bearing of the echo from a target. When an echo is detected at time measured relative to the time of pulse transmission, the range to the target is , where is the speed of sound in water. The detection of an echo implies that there is a target in a cell with area , where is the angular resolving capability of the sonar and is the range extent of the sonar pulse. The length in time of the sonar pulse is denoted by . Modern sonars use short pulses and have good angular resolving capability (small ). Thus the sonar detection cell is small enough in area to contain only one target.
Consider a sonar operating in a region in which there are only two types of targets: fish () and an enemy submarine (). Both the fish and the submarine are capable of producing an echo that the sonar can detect. Suppose that a sonar detection occurs at a particular range . A question of great practical significance: what is the probability that this observed detection at range is caused by a fish? If the echo is caused by a fish, it is a false alarm and should be ignored. However, if the echo comes from a submarine it is a very important piece of information.
In the situation that we are considering there are only two types of targets present: fish and submarines. When an echo is detected, we have two pieces of information: 1) There was a target in the sonar detection cell. Otherwise there would have been no echo. 2) The echo from the target was detectable. Since we are only considering two types of targets, the echo must come from one or the other. Let and respectively denote the events of fish in the cell and submarines in the cell, given that there is a target in the cell. Since the cell is small enough in area that it can only contain one target, the events and are disjoint and it is the case that . If there are 9999 fish in the area of sonar operation and one submarine, then and .
In order to address our question, we proceed in the following fashion. Let denote the probability that there is a fish in the cell, given that an echo is observed at range , and let |r) denote the probability that there is a submarine in the cell, given that an echo is observed at range . The probabilities and depend on two types of information: the sonar's performance capability against fish and submarines, and the likelihood of encountering fish and submarines. In most practical situations the sonar is more likely to encounter a fish than a submarine, since fish are much more numerous. On the other hand, submarines are easier to detect because they produce larger echoes. Bayes's theorem provides a convenient means of combining these different types of information. Bayes's theorem tells us that
where and denote the sonar's detection capability against fish and submarines, and the probabilities and are the a priori probabilities of encountering fish and submarines. Their values are based upon the relative concentrations of fish and submarines in the area in which the sonar is operating.
Use the upper two sliders on the graph to control the selection of the reflectivity (TS is target strength) of the submarine and the fish. Use the lower slider to control the a priori density of fish targets. Blue and red are used to denote performance against fish and submarines, respectively. The dashed curves show sonar probability of detection as a function of range against a given type of contact, either fish or submarine. The solid curves are the probabilities and
 P. Gregory, Bayesian Logical Data Analysis for the Physical Sciences, New York: Cambridge University Press, 2005.
 E. T. Jaynes, Probability Theory: The Logic of Science (G. Larry Bretthorst, ed.), New York: Cambridge University Press, 2003.
 R. J. Urick, Principles of Underwater Sound, New York: McGraw–Hill, 1983.
 A. D. Whalen, Detection of Signals in Noise, San Diego: Academic Press, 1971.