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:


(2) Optimal combination:

Given two (or more) signals Sj with errors

we want

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


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.) a1 as

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


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.