# Virtual Source Array-Based Multiple Time-Reversal Focusing
## 1 Introduction  
Over the last several decades, time reversal (TR) processing has been extensively studied in various fields [1; 2; 3]. In TR processing, a transmitted probe signal is received at an array of source-receive elements, which is referred to as a time-reversal mirror (TRM), and the received signal is time reversed to be backpropagated into the medium. If the propagation medium is static, TR processing results in coherent acoustic focusing (or pulse compression) at the probe source (PS) location where the signal was generated. One of the main advantages of TR processing is generating a focused field without a-priori knowledge about the propagation medium (i.e., self-adaptive) because of the TR invariance of the wave equation.  
The adaptive time-reversal mirror (ATRM), proposed by [PERSON] et al. [4], backpropagates the weight vector obtained as the optimal solution of an objective function with a single imposed constraint. It can focus distortionless response at a single location while simultaneously forming nulls by minimizing the total reception power at different arbitrary locations. The ATRM can be applied directly to selective focusing on a weak target [4] and to underwater multiuser communications to mitigate cross talk [5]. By extension, multiple TR focusing based on adaptive methods was proposed for the long range underwater communication [6] and stable focusing in a fluctuating ocean environment [7].  
However, a practical limitation of TR processing is the requirement of a PS to obtain a spatio-temporal focus at the PS location; this is the primary weakness of TR processing. Virtual source array (VSA)-based TR focusing, which was proposed by [PERSON] et al. [8], was designed to perform TR focusing without the requirement of a PS and provide a resolution comparable to that of conventional TR focusing. The basic idea is that a-priori sampled (or measured) transfer functions between the TRM and VSA are appropriately time delayed prior to backpropagation from the TRM. These functions are referred to as synchronized transfer functions. Consequently, the backpropagated field is steered over the VSA to a selected location for which the transfer function is unknown.  
The objective of this study is to extend the VSA-based single TR focusing to simultaneous multiple focusing at arbitrarily selected locations. To achieve this, the multiple constraints method (MCM) [9] is applied by imposing a set of constraints in the formulation of the weight vector (i.e., backpropagation vector). Then, the detailed response of VSA-based multiple TR focusing to sound speed mismatch is examined. Further, singular value decomposition (SVD) [7; 10] is applied to achieve robust VSA-based multiple TR focusing in the sound speed mismatch environment. It was found that a wavefield is created to pass through multiple locations, but not necessarily at the same time. As multiple TR focusing without the VSA is applicable only when two-way propagation is involved (i.e., at least one physical PS is needed), which extends its applicability to underwater communications or focal-spot broadening to achieve stable focusing, our approach can be more attractive in these practical applications because one-way propagation can only be used in implementing multiple focusing at arbitrarily selected locations beyond the VSA, if a-priori measured transfer functions between the TRM and VSA are updated at regular intervals, e.g., a week [3].  
The remainder of the paper is organized as follows. Section 2 reviews VSA-based single TR focusing. In Section 3, the approach is extended to simultaneous multiple focusing using the adaptive method, where the backpropagation vector is obtained with multiple constraints. In Section 4, the performance of VSA-based multiple TR focusing is compared based on two different profiles (isovelocity and downward-refracting) via numerical simulations in an ocean waveguide. The robustness of the method is investigated in the presence of sound speed mismatch in Section 5, followed by a summary in Section 6.