The decomposition of the time-reversal operator (DORT) is a detection and focusing technique using an array of transmit receive transducers. It can extract Green's functions of scatterers in a medium. A variant consists in transmitting focused beams (FDORT). It is shown here that the FDORT method can be interpreted as the decomposition of a time-reversal operator between an array of virtual transducers located at the transmit beams' foci and the physical array. The receive singular vectors correspond to scatterers' Green's functions expressed in the physical array while the transmit singular vectors correspond to Green's functions expressed in the virtual array. The position of the virtual array can be changed by varying the position of the foci, thus offering different points of view. Parameters and performance of some transmit schemes are discussed. Appropriately positioning the virtual transducers can simplify some problems. One application is measuring and correcting aberration in the case of a far-field phase screen model. Placing the virtual transducers near the phase screen transforms the problem in a simpler near-field phase screen problem.