process yields, by itself, the pro- 
jection of the sound source's location 
on whatever sub-space is effectively 
spanned. For example, if the hydro- 
phones constituting the “front end" 
of a sonar surveillance system must, 
for reasons of economy, be mounted 
on a flat portion of the ocean botton, 
then MULCAP yields by itself merely 
the projection on that bottom of the 
gound source's location. In this 
case, however, a supplementary routine 
involving n square-root operations is 
employed to obtain the necessary 
depth (z) coordinate from the already- 
known (xy) coordinates; the supple- 
mentary routine does not involve the 
simultaneous solution of quadratic 
equations. MULCAP abolishes the 
latter process from all multi-static 
location systems; in so doing it opens 
the way to much more rapid computation 
of vehicle location. It also elimi- 
nates entirely the need for rejection 
of false roots of quadratic equations. 
By virtue of the greater speed 
of a MULCAP system (resulting from 
4ts exclusive use of linear equations), 
and of the related convenience and 
ease with which such a system can be 
established, augmented, and calibrated, 
data from many hydrophones can be 
processed almost as easily as that 
from a minimum number. Precision of 
vehicle location is directly dependent 
on the number of hydrophones in the 
field. Thus any desired accuracy can, 
in principle, be obtained, provided 
the economics of the situation permit, 
merely by increasing the number of 
hydrophones to a suitable value. 
Five non-co-planar hydrophones con- 
stitute a useful minimum for three- 
space applications; four non-co-linear 
hydrophones constitute a useful mini- 
mum for planar applications. MULCAP 
is, however, a location system based 
in a fundamental sense on the exploi- 
tation of redundancy; minimal config- 
urations are, therefore, not to be 
encouraged. Because of the impor- 
tance of redundancy to optimum util- 
4zation of MULCAP, the principle of 
least squares plays a central role 
therein; ten or more linear equations 
(equations of planes) must be utilized 
in order to determine three Cartesian 
spatial coordinates; six or more 
linear equations (equations of lines) 
must be utilized in order to determine 
two Cartesian planar coordinates. 
29 
GEOMETRIC INTERPRETATION 
The "families" of planes or lines 
(more generally, of linear loci) which 
are utilized in this method of sound- 
source location are, each of then, 
associated with a distinct transducer- 
hydrophone or hydrophone-hydrophone 
pair, depending on whether the active 
mode or the passive mode is under 
discussion. If a field of p trans- 
ducers and q hydrophones is utilized 
to locate a reflecting vehicle, there 
are pq families of planes. Similarly 
if a field of n hydrophones only is 
used to locate a transmitting vehicle, 
there are n(n - 1) families of planes. 
Each member of a family of lines, or 
planes, is parallel to every other 
member of the same family; thus each 
member has the same set of direction 
cosines as every other member, that 
set determined by a line joining the 
two members of a pair of field elements. 
A vehicle, transmitting or reflecting, 
in the surveillance space, is located 
by the common intersection of a set 
of planes, a "locating cluster", 
consisting of one plane from each of 
the families. 
Only in a completely noise-free 
situation will the intersection of the 
planes, or lines, in the locating 
cluster be "clean", i.e., only then 
will all these planes pags through a 
single point. A residual sunm-of- 
squares measure of the departure of 
the planes in the cluster from a 
unique common intersection provides a 
measure of the precision of vehicle 
location. Ina noise-free situation, 
this measure is zero. 
A crucial step in the series of 
arithmetic operations which consti- 
tutes MULCAP, is the computation of 
the critical, or true, value of "mean 
range" (the sum, divided by n, of the 
n ranges from the vehicle to the n 
hydrophones). This quantity may be 
thought of as a variable parameter; 
its critical value is that value which 
minimizes the residual sum-of-squares 
value referred to above. 
ALGEBRAIC FORMULATION 
Since, in the development of a 
satisfactory precision underwater 
tracking system, toward which this 
presentation of MULCAP is oriented, 
vehicle acquisition is not a major 
