Ekelund ranging

One of the best known TMA techniques is Ekelund ranging.^[1] It is a method that is specifically designed for a 2leg-1zig scenario. This method works by first estimating the bearing rates during the first ${\displaystyle BR^{1}}$ and second leg ${\displaystyle BR^{2}}$. Secondly, one calculates the speed of advance along the line of sight with the target on the first ${\displaystyle SOA^{1}}$and second leg ${\displaystyle SOA^{2}}$. The rule then states that the range of the target at the moment of maneuver^[2] is given by:

${\displaystyle r_{2}=\textstyle {\frac {SOA^{2}-SOA^{1}}{BR^{1}-BR^{2}}}}$

To check the solution of an Ekelund Ranging solution there is also an iPhone app available.^[3]

Maximum Likelihood Estimator (MLE)

The MLE method tries to fit the directional measurements (bearings) to a theoretical linear motion model of the target. The bearing function to be fitted is:

${\displaystyle \beta (t)=arctan2(\textstyle {\frac {V_{y}*t+y_{0}-y^{O}BS}{V_{x}*t+x_{0}-x^{O}}})}$

If ${\displaystyle N}$ measurements of ${\displaystyle \beta _{i}}$ have been collected, the problem reduces to an overdetermined system of ${\displaystyle N}$ non-linear equations. The state vector associated is

${\displaystyle X=[x_{0}y_{0}V_{x}V_{y}]}$

and can be solved by numerical estimation procedures like Gauss-Newton.