TECHNICAL APPROACH

In our simulations, the array response matrix 262#262[see definition in (Pt)] in the frequency domain is symmetric but not Hermitian. In general (as for array elements with nonisotropic radiation patterns), it is neither Hermitian nor symmetric, but with slight modifications our methods apply to this case as well. The eigenvectors of 263#263 having unit norm are denoted by 264#264, for 265#265.The eigenvalues of 263#263 are 266#266, with 267#267 being the singular values of 262#262.The significant singular vectors 264#264 [i.e., those in the range of 262#262] have singular values 268#268 for 269#269, where M is either the number of targets, or the size of the array (N) -- whichever is smaller. We assume that the number of targets is smaller than the array size N, so that M is in fact the number of distinguishable targets; this assumption is required by the imaging methods we employ (such as MUSIC) as will become clear while presenting the method.

The notation used here is the same as in Borcea et al. (2002). We denote by 270#270 the deterministic source vector observed at the array for a source located at 271#271. Then, 270#270 is given by  

 272#272 (111)

where 273#273 is the deterministic two-point Green's function, and 274#274 is the location of the j-th array element.

We also define the projection 275#275 of 276#276 onto the null-space of 277#277 by

 

 278#278 (112)

for each frequency in the support of the probing pulse 279#279.

The method we describe here is a time domain variant of MUSIC (Schmidt, 1979; 1986; Cheney, 2001; Devaney, 2002) which we label DOA, because it gives very stable estimates of the direction of arrival. Frequency domain MUSIC takes a replica (or trial) vector, which is the impulse response or Green's function for a point source at some point in the space, and dots this vector into an observed singular vector at the array. With appropriate normalization, this dot product acts like a direction cosine of the angle between the replica vector and the data vector. If the sum of the squares of these direction cosines is very close to unity, then it is correct to presume that the source point of that replica vector is in fact a target location since it lies wholely in the range of the array response matrix. Crudely speaking, imaging is accomplished by plotting 280#280, which will have a strong peak when the replica source point is close to the target location.

We form the sum  

 281#281 (113)

with  

 282#282 (114)

and display the objective functional  

 283#283 (115)

for points 271#271 in the target domain.

The arrival time 284#284 is the deterministic travel time from the p-th transducer to the search point,  

 285#285 (116)