**Xuebo Zhang 1,2, Cheng Tan 3,\* and Wenwei Ying <sup>4</sup>**


Received: 20 February 2019; Accepted: 17 March 2019; Published: 20 March 2019

**Abstract:** For the multireceiver synthetic aperture sonar (SAS), the point target reference spectrum (PTRS) in the two-dimensional (2D) frequency domain and azimuth modulation in the range Doppler domain were first deduced based on a numerical evaluation method and accurate time delay. Then, the difference between the PTRS and azimuth modulation generated the coupling term in the 2D frequency domain. Compared with traditional methods, the PTRS, azimuth modulation and coupling term was better at avoiding approximations. Based on three functions, an imaging algorithm is presented in this paper. Considering the fact that the coupling term is characterized by range variance, the range-dependent sub-block processing method was exploited to perform the decoupling. Simulation results showed that the presented method improved the imaging performance across the whole swath in comparison with existing multireceiver SAS processor. Furthermore, real data was used to validate the presented method.

**Keywords:** synthetic aperture sonar (SAS); multireceiver; numerical evaluation; numerical transfer function; imaging algorithm

#### **1. Introduction**

Synthetic aperture sonar (SAS) [1] provides high resolution images via the coherent processing of successive echo data along the virtual aperture. This makes SAS a suitable technique for applications such as searching for small objects [2], imaging of wrecks [3], underwater archaeology [4] and pipeline inspection [5]. Additionally, it improves the classification and detection of objects based on SAS images [6]. Multireceiver SAS [7], as opposed to monostatic SAS constellation, offers a fast mapping rate at a given resolution. However, it does this at the cost of complicated signal processor.

For the multireceiver SAS, the point target reference spectrum (PTRS) [8] is a prerequisite of fast imaging algorithms. The two-way slant range of the multireceiver SAS consists of two hyperbolic range histories, which include the instantaneous range between the moving transmitter and target and that between the target and moving receiver. The two hyperbolic range histories make it difficult to deduce the point of stationary phase (PSP) and PTRS using the method of stationary phase [9]. In order to solve this problem, approximations are often exploited. In [10–12], the phase center approximation (PCA) was used to model a transducer located at the midpoint between the transmitter and receiver. With this, the echo data of multiple receivers is converted into the monostatic format. However, the preprocessing includes the compensation of phase errors [13]. Due to the space variance of approximation errors [13], it is difficult to compensate phase errors completely. Loffeld et al. [14] have presented an analytic PTRS. Their method was based on the approximation that the transmitter

and receiver contribute equally to the Doppler frequency. Based on the method of stationary phase [9], two PSPs corresponding to the transmitter's phase and receiver's phase were deduced. The phase history of the transmitter and that of the receiver were expanded into a power series at their individual PSPs. Both phases were then combined to generate a quadratic function. It can be found that two approximations are exploited by this method. One is the equal Doppler contribution of the transmitter and receiver, and the other is the Taylor approximation of the transmitter's phase and receiver's phase. In general, this method only applies to the narrow beam case [14,15]. There are still some other methods deducing the analytic PTRS. The basic idea relies on series approximation. In [16], the quadratic approximation of the two-way range was exploited. This introduced a large residual error, which degraded the imaging performance at close range. Moreover, this method did not consider the compensation of the stop-and-hop error [12]. A single target suffers from the coordinate deviation in azimuth [17]. However, a distributed target suffers from the distortion. In [18,19], the two-way range was expanded into a power series with respect to the slow time. Additionally, the PSP was expanded into a power series based on the series reversion method. The accuracy of the two-way range and PSP was limited by the number of terms in the polynomial. With this method, the series approximation was used twice. The approximation error increased with the slow time. The accumulative error would be large when the SAS system works with the wide beam case. In [20,21], the instantaneous Doppler wavenumber was exploited to deduce the analytic PTRS. The two-way slant range was formulated as a function of equivalent bistatic squint angle and half bistatic angle [20,21]. Based on the method of stationary phase [9], the azimuth wavenumber can also be expressed as a function of equivalent bistatic squint angle and half bistatic angle. Then, the PTRS was expressed as a function of half bistatic angle, which should be analytically calculated. Considering that the triangle whose vertices were the transmitter, receiver and target, the fourth order equation with respect to half bistatic angle was obtained based on the theorem of sine and some basic algebra skills. In [21], this equation was solved using the series reversion method [18,19]. The accuracy of the PTRS is also limited by the number of terms in the power series.

Using an accurate time delay of the transmitted signal [12], the back projection (BP) algorithm [22] can provide high resolution results. In this paper, we present an imaging algorithm, which was also based on the accurate time delay of the transmitted signal. With the numerical evaluation method [23], we first calculated the PSP and azimuth PSP. The PTRS and azimuth modulation can be easily obtained based on their respective numerical PSPs. Then, we obtained the coupling term in the two-dimensional (2D) frequency domain using the phase difference between the numerical PTRS and azimuth modulation. The PTRS, coupling term and azimuth modulation avoiding approximations were further exploited to develop the imaging processor, which compensated the coupling phase based on the sub-block processing method.

This paper is organized as follows. In Section 2, the imaging geometry and signal model are introduced. Section 3 introduces the PTRS, azimuth modulation and coupling term based on the numerical evaluation method. Section 4 presents the imaging algorithm based on three functions. Section 5 compares the presented method with traditional methods, and highlights the advantages of the presented method. In Section 6, processing results of the simulated data and real data are used to validate the presented method. Finally, some conclusions are reported in the last section.

#### **2. Imaging Geometry and Signal Model**

The imaging geometry of the multireceiver SAS is shown in Figure 1. The linear array consists of a transmitter and receiver array including *M* uniformly spaced receivers. In Figure 1, the black rectangle denotes the transmitter. Each receiver has an integer index *i* ∈ [1, *M*]. For the *i*-th receiver, the distance between the receiver and transmitter is denoted by *di*. The linear array is aligned in the sonar moving direction, which is called the azimuth dimension. The horizontal direction represents the range dimension. Since the SAS configuration shown in Figure 1 is characterized by azimuth invariance, an ideal point target located at coordinates (*r*, 0) is used. *t* denotes the slow time in the

azimuth dimension. The fast time in the range dimension is represented by *τ*. *c* is the sound speed in water. *v* represents the velocity of the sonar platform.

**Figure 1.** Imaging geometry of the multireceiver synthetic aperture sonar (SAS).

In Figure 1, the two-way slant range is from the transmitter to target and then back to the *i*-th receiver. When the transmitter moves to the position *v* · *t*, a chirp signal *p*(*τ*) is transmitted. The accurate time delay [12] of the echo signal corresponding to the *i*-th receiver is given by:

$$\tau\_{l} = \frac{v(\mathbf{v} \cdot \mathbf{t} + d\_{l}) + c\sqrt{v^{2}t^{2} + r^{2}}}{c^{2} - v^{2}} + \frac{\sqrt{\left[v(\mathbf{v} \cdot \mathbf{t} + d\_{l}) + c\sqrt{v^{2}t^{2} + r^{2}}\right]^{2} + (c^{2} - v^{2})(2\mathbf{v} \cdot \mathbf{t} \cdot d\_{l} + d\_{l}^{2})}}{c^{2} - v^{2}} \tag{1}$$

In (1), we consider the forward distance of the *i*-th receiver during the signal reception, because the sonar is continuously travelling along the azimuth dimension [17]. After demodulation, the echo signal corresponding to the *i*-th receiver is expressed as:

$$\text{ss}\_{i}(\tau, t) = p(\tau - \tau\_{i})\omega\_{a}(t)\exp\{-j2\pi f\_{c}\tau\_{i}\} \tag{2}$$

where *fc* is the center frequency. The composite beam pattern corresponding to the *i*-th receiver and transmitter is represented by *ωa*(·). For simplicity, we neglect this beam pattern to concentrate on the phase processing.

#### **3. PTRS, Azimuth Modulation and Coupling Term**

The BP algorithm [22] should compute the instantaneous range from the moving transmitter to target and then back to moving receiver. Therefore, (1) is very suitable for the BP algorithm. Since (1) considers the influence of stop-and-hop assumption, BP algorithm can provide high resolution image. However, (1) cannot be used by traditional fast algorithms. For simplicity, the forward distance of the *i*-th receiver during the signal reception is approximated by *vτ<sup>i</sup>* ≈ 2*vr*/*c*. This approximation degrades the imaging performance when the SAS system works with the wide beam case. In this paper, the accurate time delay shown as (1) is extended to fast imaging algorithms. The PTRS, azimuth modulation and coupling term play an important role in developing fast imaging algorithms. We start from the deduction of the PTRS, azimuth modulation and coupling term in this section.
