4. Time coordination is such that the angle readings for the location 

 of the submarine position, as provided by the sensors, correspond 

 to the observed submarine position at the time considered. 



5. The observed directions furnished by the two sensors are statisti- 

 cally independent. 



6. The observed direction provided by a sensor, at the time considered, 

 has a normal (Gaussian) probability distribution with mean equal 



to the true direction of the submarine. 



7. The sensors are positioned so that true submarine direction from 

 any sensor is siobstantially different from the direction of the 

 other sensor (with respect to this sensor) . 



8. The standard deviations of the angular probability distributions 

 for the sensors are known and small (say, at most 0.03 radians) . 

 This combined with assumption 7, is considered to imply that second 

 and higher order terms in angular errors (deviations from true 

 direction) can be neglected in mathematical expressions. 



Notation and Relationships . 



For standardization purposes, and ease of analysis, one sensor is con- 

 sidered to be located at the origin of the (x,y) rectangular coordinate 

 system that is used to represent the ocean surface. The second sensor is 

 considered to be located at (D,0) , where D > is the distance between the 

 sensors. Also, the submarine is considered to be located in the first 

 quadrant. These standardizations do not represent any loss of generality 

 for the type of analysis that is made. 



The true location of the submarine is denoted by (A,B) , while the 

 position estimated on the basis of the directional observations is (X,y) . 

 With respect to the first sensor, the true angular direction of the sub- 

 marine is ai radians, while the observed angle ai+£i radians. For the 

 second sensor, the true direction is a.2 ^radians, and the observed direction 

 is a2+Ji2 radians. Figiore 1 contains a schematic diagram that illustrates 

 the standardization and notation used. 



Now, consider some relationships that occur for this situation. First, 

 ai and a2 are directly determined by A, B, and D. That is, tan ai=B/A 

 and tan a2=B/(A-D) , where A-D can be negative. Second, 



X = d{1 + cot(a2 + £2)/tcot(ai + li) - cot(a2 + i-z)]) 



= D cot(cti + £i)/[cot(ai + £i) - cot(a2 ■*" ^2^ ^ 



Y = D/[cot(ai + li) - cot(a2 + £2) 1 • 



In addition, the following relations are useful in the derivations: 



cot (a + e) = (1 - tan a tan e)/(tan a + tan e) , 



and, for e reasonably small (e in radians) , tan e = e , so that 



451 



