PROBABILITY OF LOCATING A SUBMARINE WITHIN A STATED DISTANCE 

 ON THE BASIS OF TWO DIRECTIONAL SENSORS 



John E. Walsh* 

 Southern Methodist University 



ABSTRACT 



The general problem is to estimate the surface position under which a 

 siibmarine is located. The information for this estimation is provided by 

 two directional sensors whose locations are known. The observed directions, 

 in combination with the sensor locations , are combined to vi eld the estimated 

 position. The specific problem is to determine approximately the proba- 

 bility that the true submarine position is within a stated distance (on the 

 ocean surface) of the estimated position. This article identifies the para- 

 meters involved and, in terms of these parameters, develops an approximate 

 expression for the probability value. 



Introduction . 



The general problem is to estimate the surface position (on the ocean) 

 under which a submarine is located. Here, all positions considered are 

 on the ocean surface and, for brevity, the word "surface" is deleted in 

 referring to positions or locations of the submarine or sensors. 



The information for estimating the submarine position is provided by 

 two directional sensors whose locations are known. That is, at a single 

 fixed time, each sensor furnishes an observed direction (along the ocean 

 surface) for the submarine location. These observed directions, in combi- 

 nation with the known locations of the sensors, yield an estimated location 

 for the sxobmarine at the time considered. 



Let a circle with given radius be centered at the estimated submarine 

 location. The specific problem is to determine approximately the probability 

 that, at the time considered, the true sxobmarine location is contained in 

 this circle. This probability depends on the locations of the sensors, the 

 radius of the circle, and on the probability distributions of the angular 

 errors for the sensors. The purpose of this article is to identify the para- 

 meters involved and, in terms of these parameters, to develop an approximate 

 expression for the probability that the circle contains the true (but unknown) 

 location of the submarine. 



Assumptions and Simplifications . 



1. The ocean surface is considered to be flat (a geometric plane) . 



2. The submarine and the sensors can be represented as points. 



3. The locations of the sensors are exactly known at the single time 

 that is considered. 



*Based on work done while the author was with Lockheed Aircraft Corporation 

 and with System Development Corporation. Written in association with the 

 Office of Naval Research Contract No. N00014-68-A-0515. 



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