IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 35, NO. 4, OCTOBER 2010 877 Peer-Reviewed Technical Communication Modeling Bottom Reverberation for Sonar Sensor Motion Jinyun Ren, Student Member, IEEE, and Rodney G. Vaughan, Fellow, IEEE Abstract—Simplified signal models of the bottom reverberation are widely used in the sonar literature, but the simplifications themselves are seldom discussed. In this paper, the formulation of the joint modeling of signal propagation, bottom scattering, and reception is reviewed, and an emphasis is placed on the assump- tions which lead to model simplification. The interest extends to modeling for sensor motion compensation algorithm development, and it is in this context that an in-depth understanding of the assumptions and simplifications becomes essential. The formula- tions help pave the way for the derivation of motion compensation algorithms. One such algorithm is presented in the companion paper. Index Terms—Bottom reverberation, motion compensation, scattering and propagation modeling, sonar sensor motion. I. INTRODUCTION A major problem in active sonar systems is reverberation, which is volume scattering from the water body and sur- face scattering from the water surface or the water bottom (al- ternatively referred to as the seafloor) [1], [2]. In this paper, we address the scattering contributions from the water bottom. The bottom reverberation is an undesired signal in some applications such as target detection in shallow water, while it is the wanted signal in seafloor mapping. In conventional applications such as sonar depth sounders, the amplitude or power of bottom reverberation is of interest. However, with the introduction of sonar arrays, both the phase and amplitude of the received bottom reverberation become im- portant to the corresponding algorithms. For example, swath bathymetric sidescan sonars calculate depth by using the phase difference between signals received from different elements of a sonar array [3]. Here, the time of flight gives range, and the receiving array gives an angle of arrival, and these parameters allow the depth to be calculated. In these applications, there is seldom ground truth available, so the raw and processed data must be interpreted carefully. Consequently, recent theoretical performance analysis of this kind of sonar relates to character- Manuscript received April 04, 2009; revised July 02, 2010; accepted September 15, 2010. Date of publication November 11, 2010; date of current version November 30, 2010. Associate Editor: D. Knobles. The authors are with the School of Engineering Science, Simon Fraser University, Burnaby, BC V5A 1S6 Canada (e-mail: jren@alumni.sfu.ca; rodney_vaughan@sfu.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JOE.2010.2079591 izing estimation accuracy, e.g., [4] and [5]. Similarly, for syn- thetic aperture sonar (SAS), the signal processing algorithms (including vehicle motion compensation and bottom imaging) and corresponding performance analysis rely on the phase as well as the amplitude of the received signal [6]–[9]. To develop sonar signal processing algorithms and conduct theoretical performance analysis, the modeling of physical mechanisms of bottom reverberation should be thoroughly examined. A simplified signal model of bottom reverberation, defined from a plane (i.e., 2-D only) and assuming the bottom to be like a line (i.e., 1-D), is adopted by many researchers [4], [5], [10]. For example, in [4] and [5], this model acts as a start point for deriving the correlation function of reverberation sig- nals received by different elements of a sonar sensor array, and this correlation function is further used to analyze the accuracy of swath bathymetric sonars. A more general bottom reverber- ation signal model (considering the bottom as a 2-D surface) is adopted in deriving the space-time correlation function of signals scattered from the bottom for correlation sonars in [11]. This resulting correlation is also the key for estimating the displacement of a 3-D sonar array. However, these references do not elaborate the simplifications or their justification; or the interplay between the sonar mechanisms, their modeling, their assumptions, and the algorithms used. The modeling of bottom reverberation becomes important when discussing the sonar signal itself, in particular its phase, because the algorithm performance is typically derived from a model of the complex transmission characteristics. The bottom reverberation physics is modeled using rough sur- face scattering theory (e.g., [12]–[15]), pioneered in electro- magnetic wave research (e.g., [16]–[21]). These texts mainly discuss scattered power, rather than complex amplitudes, in- cluding the phase features of scattered signals. To the authors’ knowledge, there is no concise discussion of the joint modeling of signal propagation, bottom scattering, and reception. In this paper, the bottom reverberation modeling for a stationary sonar is discussed in detail. The model derives from the principles of propagation, and includes all the steps to the known formula in the sonar literature. The important assumptions and simpli- fications, which are not previously discussed in the literature, are significant for algorithms and performance analysis. The assumptions are particularly important for developing motion compensation algorithms. It is shown how it is possible to esti- mate the sensor motion by comparing two signals received from the same sensor element but at different times. Using the known, simplified model for monostatic sonar, an analysis of the effects of small sensor motion is presented. The assumptions which 0364-9059/$26.00 © 2010 IEEE