14-3] SYSTEM CONSIDERATIONS 735 



is further assumed to consist of the error in doppler velocity (or the along- 

 heading velocity component) av and a computer error contribution to the 

 range errorac^j. The cross-track or transverse error or is assumed to consist 

 of the error in doppler drift angle (or cross-heading velocity component) an, 

 the error of the heading reference an, and a computer error contribution to 

 the transverse error acT- Based on measured results, these component 

 errors, and hence also the range and transverse errors, exhibit roughly 

 normal or Gaussian probability distributions. 



The resulting position error therefore exhibits a two-dimensional normal 

 or elliptical probability distribution. If the standard deviations of the 

 component errors av, o-cr, <^d-> ctcti and an are known, the standard devia- 

 tions of the range and transverse errors can be determined from the 

 following expressions: 



<JH = i<rv-' + <7§^ (14-8a) 



CTT = i<TD~ + <ThT + <Th'' (14-8b) 



where all symbols are standard deviations of the quantities which were 

 defined in the text above. 



The major and minor axes of the ellipse — namely, the range and 

 transverse error, aR and or — having thus been determined, one can obtain 

 the radius of the position error circle for a desired probability of occurrence 

 as a function of the transverse error ctt and the ratio o-r/o-t — b and a 

 multiplying factor a, by consulting "Tables of Integrated Probability 

 within a Circle Centered at the Origin of Normal Elliptical Distributions."^ 



Expressed in general mathematical terms, if the ratio b is known, where 



b = ^ (14-9) 



(It 



a multiplying factor a can be found from the elliptical probability tables for 

 the determination of the radius of the position error £p for any desired 

 probability, such that 



E, = auT. (14-10) 



For the special case for which (tr and ot are equal, the distribution 

 becomes circular (rather than elliptical) and is known as a Rayleigh 

 distribution. As will be demonstrated later by means of measured data, in 

 typical doppler navigation systems, <jt and ur are most frequently not 

 equal; rather, their ratio is often found to be 2 or greater {<JTl<yR > 2), 

 which is usually attributable to the effect of the heading reference error. 

 From the probability tables mentioned above, one finds, for instance, that 



2Such tables have been calculated — for instance, in yet unpublished work by R. H. Bacon 

 and others at GPL Division, General Precision, Inc., Pleasantville, New York. 



