Bayesian Estimation and Tracking: A Practical Guide

4.4 The DIFAR Likelihood Function

If we assume that the DIFAR sensor likelihood functions are independent from sensor to sensor, then the likelihood function for the observation vector conditioned on the target ship position is given by the joint density

(4.32)

where with

(4.34)

Although it is an interesting problem in itself, the full derivation of the likelihood function for a DIFAR buoy is not the main emphasis of this book, so any interested reader can find the derivation in the appendix of Ref. [2], available online. Here, because it is needed in Part III of this book, we present just the resulting likelihood density and its properties.

If we ignore the time index for clarity and let z_m tan θ_m, then from Appendix A in Ref. [2] we can write

(4.35)

with

(4.35)

Here

(4.36)

(4.37)