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:

(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.) a1 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.