EXAMPLES AND RANGE ESTIMATION

 

Two_short_MUSIC
Figure 2 The MUSIC central frequency estimate of the location of two targets in random media with different strength of the fluctuations of the sound speed. The exact location of the targets is denoted by green stars (in the color version). The standard deviation s and maximum fluctuations (M.F.) are indicated on the top of each view. The horizontal axis is the range in mm and the vertical axis is the cross-range in mm.

Examples for frequency-domain MUSIC with two targets are displayed in Fig.2. It is clear from this Figure that no range information is obtained from frequency-domain objective functionals, and even the cross-range information is often quite haphazard in random media. Lack of statistical stability prevents these imaging approaches from being useful in random media with significant multipathing as considered here. When the realization of the random medium is changed, the images obtained typically change also -- which is what we mean by the phrase ``lack of statistical stability'' for these methods. Note that this approach works well for homogeneous media, but quickly breaks down when randomness of the velocity field is important.

 

Two_short_DOA
Figure 3 The DOA estimate (5) of the location of two target in random media with different strength of the fluctuations of the sound speed. The exact location of the target is denoted by the green star (in the color version). The standard deviation s and maximum fluctuations (M.F.) are indicated on the top of each view. The horizontal axis is the range in mm and the vertical axis is the cross-range in mm.

Examples for time-domain MUSIC with two targets are displayed in Fig.3. The cross-range results show dramatic improvement over results using other methods (Berryman et al., 2002). Range information is still not to be found here, due to loss of coherence in the random medium; we cannot get exact cancellation at the targets in this situation whereas coherent refocusing is possible in homogeneous media. But the statistical stability of the universal ``comet tails'' -- which was also anticipated by recent theoretical analyses (Blomgren et al., 2002) -- is now easily observed. The images are necessarily shown for specific realizations, but the results do not change significantly when the underlying realization of the random medium is changed. This fact has been repeatedly shown in our simulations, and is the main operational characteristic of statistically stable methods.

Target localization requires an estimate of the range. In the far field, only the arrival time information is useful for this purpose. Arrival time information is present in the singular vectors and can also be averaged (for the same random medium) using the multiple copies available in the array response matrix for random media -- see Borcea et al. (2002) -- to obtain very stable estimates of arrival times. We will now combine this approach with the time-domain methods to obtain well-localized images of the targets.

For each search point ${\bf y}_s$, we compute the objective functional  

$${\cal R}_{\mbox{SAT}}({\bf y}^s) = {\displaystyle \sum_{j=1}... ...^{(j)}({\bf y}^s) } { {\cal G}_{\mbox{SAT}}^{(j)}({\bf y}^s)},$$ (8)

where  

$${\cal G}_{\mbox{SAT}}^{(j)}({\bf y}^s) = {\displaystyle \sum... ... \right\vert^2 \left[ \tau^{(j)}_p - t_p({\bf y}^s) \right]^2.$$ (9)

Here $\hbox{\boldmath{${\cal F}$}}^{(j)}({\bf y}^s,t)$ is defined by (Fj), $t_p({\bf y}^s)$, for $p = 1,\ldots, N$, are the deterministic arrival times given by (det-tr-time) and $\tau^{(j)}_p$, for $p = 1,\ldots, N$, and $j=1,\ldots,M$, are the computed arrival times. We call (Rsat) the Subspace Arrival Time (SAT) estimator.

 

Two_short_SAT-1
Figure 4 The SAT estimate for two targets.

Examples of SAT (or time-domain MUSIC with arrival time estimates from the averaged singular vectors) for two targets are displayed in Fig.4. This method is statistically stable and gives good estimates of the target locations. These localization results have degraded the least of all those considered (Borcea et al., 2002; Berryman et al., 2002) at the highest values of the random fluctuations.