propagation conditions, and operation error. Because of the random nature of 

 these events they must be averaged over a period of time and treated statistically , 

 In this manner it is possible to state the probability tliat the error of a given po- 

 sitional fix Lies between zero and some upper limit. Much of the more recent 

 navigation literature makes use of the very important root -mean -square error of 

 repeatability which is discussed below. 



In Figure III -9 point P is the apparent location of a navigational fix. 

 The individual position line errors are indicated as x and y. 



FIGURE III-9 



LINE -OF -POSITION GEOMETRY 



The radius d is taken as the radial error or the straight -line deviation of the 

 probable position from the true position. The root -mean -square error dj- is 

 often used to describe the random errors occurring in a navigation fix formed 

 by two intersecting lines of position. Measurement errors are calculated along 

 each of the two orthogonal axes, assuming they are uncorrelated and have a nor- 

 mal distribution. The error probability is found by first computing the standard 

 deviation a from the errors of a large number of observations . With equal 

 standard deviations of cj^ ~ ""^v ^ circle may be drawn with its center at the 

 mean value of its position coordinates . A circle of dj- =^ o sjl is then defined 

 as the root -mean -square error. The circle defines the locus of points having 

 a constant probability density and encloses an area where a single measurement 

 will fall 63% of the time . For a circle of 2dj. the probability increases to . 982 . 



For the case a^ 7^ o a parallelogram is formed which defines an 

 error ellipse . However, by a proper transformation of the axis the parameter d 



26 



artlixir 3i.lLittlc.3inr. 



