Acoustic noise interferometry offers new, cost-effective, and environmentally friendly approaches to monitoring dynamics of the ocean and atmosphere. It has been established theoretically and confirmed experimentally that two-point cross-correlations function of diffuse acoustic noise approximates the Green's function, which describes deterministic sound propagation between the two measurement points. The similarity between noise cross-correlations and Green's functions motivates application to acoustic noise interferometry of the techniques that were originally developed for remote sensing using broadband, coherent compact sources. In this paper, time reversal is applied to cross-correlation functions of the ambient and shipping noise measured in 100 meter-deep water in the Straits of Florida. Noise was recorded continuously for about six days at three points near the seafloor by pairs of hydrophones separated by 5.0, 9.8, and 14.8 km. In numerical simulations, a strong focusing occurs in the vicinity of one hydrophone when the Green's function is back-propagated from the other hydrophone, with the position and strength of the focus being sensitive to density, sound speed, and attenuation coefficient in the bottom. When sound propagates in a shallow-water waveguide over ranges large compared to the ocean depth, performance of a time-reversal mirror is enhanced by multiple image sources corresponding to various orders of surface and bottom reflections. A single virtual point source, i.e., a one-element time-reversal mirror, is found to provide distinct, well-defined foci of the back-propagated acoustic field under conditions of the experiment. Values of the geoacoustic parameters of the seafloor at the experimental site are estimated by optimizing focusing of the back-propagated noise cross-correlation functions. The results are consistent with the values of the seafloor parameters evaluated independently by other means.