# Search Space Reduction for Localization and Tracking of an Acoustic Source
## 1 Introduction  
Source localization is at the core of Matched Field Processing (MFP) [1], a fundamental technique of Underwater Acoustics that generalizes the concept of plane-wave beamforming [2]. In MFP, a set of candidate predictions of the acoustic field (called replicas) are to be compared with a given set of signal observations acquired at a single array; the replica with the better fit to the observations can be used to infer the position of the source. A rectification condition for MFP to be successful is to have a proper knowledge of the sound speed profile, bottom properties, and local bathymetry and to take advantage of a reliable underwater acoustic model. Generally speaking, there are two different types of source localization: the first one is two-dimensional (2D) source localization, which can be seen as the problem of finding the source coordinates \((r_{s},z_{s})\) within the search space defined by \([r_{\min},r_{\max}]\times[z_{\min},z_{\max}]\); the second case is three-dimensional (3D) source localization, which corresponds to the problem of finding the source coordinates \((r_{s},z_{s},\phi_{s})\) within the search space defined by \([r_{\min},r_{\max}]\times[z_{\min},z_{\max}]\times[\phi_{\min},\phi_{\max}]\), where \(\phi\) stands for source bearing, and in general \(\phi_{\max}-\phi_{\min}=360^{\circ}\) (the choice of cylindrical coordinates is a typical convention). In both cases, the calculation of replicas takes place among all possible combinations of coordinates within the search space; for obvious reasons, the number of combinations is much higher in 3D source localization than in 2D source localization. Strictly speaking, 2D environments do not exist because ocean bathymetries are always 3D; however, there are often configurations of the source and the array in which the sea bottom transect can be approximated as flat. In such cases, one can replace 3D source localization with 2D source localization and rely on a normal mode model to provide an efficient calculation of replicas over a given interval of source depths; this replacement is attractive in order to reduce the size of the search space but is achieved at the cost of losing the source bearing because the calculation of replicas depends on range and depth only. Two-dimensional MFP-based localization had been discussed in detail consideringsimulations and experimental data for both the narrowband and broadband cases [3; 4; 5; 6; 7; 8]; additionally, non-MFP 2D localization with a single hydrophone has been shown to be possible as well [9; 10]. Environmental Signal Processing (ESP) was proposed to overcome the bearing limitations of 2D source localization in [11] by suggesting to take advantage of the bathymetry. To this end, one can consider two different approaches: the first one is the so-called \(N\times 2\)D modeling, in which replicas are to be calculated along 2D transects over \(N\) bearings; in the second approach, replicas are to be calculated taking into account all the variability of the 3D environment. Environmental Signal Processing based on the \(N\times 2\)D approach using the KRAKEN normal mode model [12] was discussed in detail for experimental data in [13], but only for a subset of all possible bearings; under such conditions, the particular bounds on the source bearing certainly allow us to reduce the size of the search space, but this is a specific type of _a priori_ information that cannot be expected to be always available. Current computational resources certainly allow us to handle 3D source localization successfully for all combinations of parameters of the search space. However, by proceeding in this manner, the method seems to be missing one important fact, namely that the calculation of replicas over the entire 3D search space is undesirable because of the corresponding computational burden; besides, such calculation is also inefficient because a large percentage of the replicas will provide a low fit to the observations.