Combining radial velocities from 2 directions.

Given radial velocities from two directions (sonars at 2 locations):

,

,

we want to reproduce estimates of the orthogonal components *u* and *v*.

(1) Simple elimination (least squares):

Eliminate *v*:

Similarly, eliminate *u*:

Defects:

-Blows up as _{}0 or 180°

-Not an optimal solution for signals with error.

(2) Optimal combination:

Given two (or more) signals *Sj* with errors _{}

we want

such that = minimum. Following *Bretherton et al* [1976], set

Defining

we can write the solution in vector and matrix form (summing over *j*):

which, for just two radial estimates, simplifies to

.

Now we need to evaluate the correlations represented by all the quantities in braces "<>". For a general-purpose solution (this is the key), we set

Then we obtain (after a little math)

,

where _{}.

Comparison with simple elimination:

The simple elimination solution takes the form

.

Expanding _{} and _{}, rewrite the "optimal" version of (e.g.) *a*_{1} as

We see that this goes asymptotically to the simple elimination solution as the noise goes to zero.

Reference:

Bretherton, F. P., R. E. Davis, and C. B. Fandry, A technique for objective analysis and design of oceanographic experiments applied to Mode-73, *Deep Sea Res.*, *23*, 559-582, 1976.