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 .