## 3aUW15. Coherent localization of an unknown cw source.

### Session: Wednesday Morning, December 4

### Time: 11:20

**Author: L. Neil Frazer**

**Location: Dept. Geology and Geophys., School of Ocean and Earth Sci., Univ. of Hawaii, Honolulu, HI 96822**

**Abstract:**

Let D[sup j]=D[sup j](f) be the data spectrum from the jth element of a
hydrophone array of any shape, all of whose phones have the same unknown
transfer function with unknown gains. Then D[sup j]=B[inf d]G[sup j](m[inf d]),
where B[inf d] is the product of the unknown source spectrum and the unknown
hydrophone response, G[sup j] is the Green's function for the jth hydrophone
location, and m[inf d] is the unknown true value of the vector m containing
source location and environmental parameters. Similarly, let S[sup j]=B[inf
s]G[sup j](m[inf s]) be a candidate synthetic spectrum for comparison with D[sup
j]; here B[inf s] is a guess at B[inf d] and m[inf s] is a candidate parameter
vector. Conventional coherent localization searches for an m[inf s] that
maximizes the agreement of D[sup j] and S[sup j], an agreement clearly dependent
on the guess B[inf s]. To remove the problem of the unknown source define F[sup
ij]=D[sup i]G[sup j](m[inf s]) and search for the m[inf s] that maximizes
agreement of F[sup ij] with F[sup ji]. To see why this works note that F[sup
ij]=B[inf d]G[sup i](m[inf d])G[sup j](m[inf s]), whereas F[sup ji]=B[inf
d]G[sup j](m[inf d])G[sup i](m[inf s]). Thus F[sup ij] and F[sup ji] have the
same source function B[inf d], and F[sup ij] equals F[sup ji] only if m[inf
s]=m[inf d]. Note that knowledge of the source is not required; in particular it
need not be compact in time, hence cw data can be used to localize with (greater
than or equal to) two hydrophones. The localizer for hydrophones with equal
gains is (phi)[inf 1]=|(summation)[inf i(not equal to)j](xi)[inf i](xi)[inf
j](summation)[inf f](zeta)[inf f]F[sup ij]F[sup ji*]|[sup 2]/((summation)[inf
i(not equal to)j](xi)[inf i](xi)[inf j](summation)[inf f](zeta)[inf f]F[sup
ij]F[sup ij*])[sup 2], in which optional real (xi)[inf i] give array shading,
and optional real (zeta)[inf f] give spectral shading. A localizer allowing
hydrophones to have different unknown gains is (phi)[inf 2]=|(summation)[inf
i(not equal to)j](xi)[inf i](xi)[inf j](summation)[inf f](zeta)[inf f]F[sup
ij]F[sup ji*]|[sup 2]/((summation)[inf i(not equal to)j](xi)[inf i](xi)[inf
j])[sup 2], where F[sup ij]=F[sup ij]/[radical (summation)[inf f](zeta)[inf
f]F[sup ij]F[sup ij*][radical .

ASA 132nd meeting - Hawaii, December 1996