problem, the presentation of the 
acquisition-loop portion of MULCAP 
will be omitted. Furthermore, for 
the sake of brevity, only MULCAP's 
passive mode will herein be presented 
in detail. 
Let there be known ann xl 
vector dg whose n elements are the n 
ranges to the vehicle from the n 
hydrophones each diminished by their 
common mean. Let the squares of these 
n elements constitute the n elements 
of bo. Matrices Do and dog contain 
the information derivable from time- 
of-arrival data of the sort assumed 
for passive-mode MULCAP. The initial 
lack of knowledge of the vehicle's 
true n ranges is reflected in the 
initia], quite arbitrary, assumption 
that their mean value is zero. 
Let N be ann x 3 matrix whose 
elements are the doubles, in (xyz) 
order, of the n hydrophone coordinates. 
Let the 3 x m matrix M denote the 
conditional inverse of ZN, where Z is 
a systematic differencing matrix of 
order m xn, where m = $n(n - 1). 
Let the n elements of the n x 1 matrix 
r denote the n distances, squared, 
from the origin to each of the n 
hydrophones. 
It is assumed that the field 
characteristic n xn matrix F defined 
as follows: 
(1) F = 272 — 2?zn(ntz?zn)-1NTz?z, 
is stored in the computer. 
The mean range Pg to the vehicle 
is computed by means of the formula: 
(2) Py = do.Flr ~ b9) 
2dolFdo 
Let the 3 x 1 matrix p, be com- 
puted by means of the formuta: 
(3) po = MZ(r- bo) . 
Pc,a 3 x 1 matrix whose three 
elements are respectively the best 
estimates available of the Cartesian 
coordinates of the vosition of the 
sound source, is, finally, computed 
as follows: 
(4) Pe = Po - (2PA¢)MZdo 
CONCLUSION 
When the locations of underwater 
sound sources near the surface are 
computed utilizing a MULCAP routine 
based upon equations (1), (2), (3), 
and (4) above, from simulated data 
30 
obtained with a velocity profile 
representing the real-ocean medium at 
depths of 6000 feet and with bottom- 
mounted hydrophones with base lines 
up to 10,000 feet, in hexagonal con- 
figuration, the precisions attpined 
are better than one part in 10” . 
REFERENCE 
1. Morrissey, J. H., MULCAP, Philco 
Tech Rep Division Bulletin, Vol. 11, 
No. 2, March-April, 1961. 
