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

,

,

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

Eliminate v:

Similarly, eliminate u:

-Blows up as 0 or 180°

-Not an optimal solution for signals with error.

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 .

The simple elimination solution takes the form

.

Expanding and , rewrite the "optimal" version of (e.g.) a₁ as

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