Schematic of the SSR method using TR. Sources and receivers are denoted by and , respectively. Subplots (a) and (b), respectively, display the forward propagation and back propagation from a typical TR experiment. The additional forward propagation and back propagation steps required by SSR are depicted in subplots (c) and (d), respectively.
Photographs of the two samples used in the experiments described in this paper. Subplot (a) displays the silica glass cube sample and subplot (b) displays the doped glass block sample.
Subplots (a)–(i) display typical temporal signals used throughout the proposed SSR method. (a) Voltage signal sent to transducers. (b) Signal received at from transducers . (c) Signal received at from transducers . (d) Effective TRM signal for (for first TR step). (e) Focal signal at (after TR step). (f) Windowed focal signal at (after first TR step). (g) Effective TRM signal for (for second TR step). (h) Subtraction result for . (i) Focal signal at after SSR.
Plots of the instantaneous spatial distribution of the (normalized) out of plane velocity at the focal time after standard implementation of the TR process [subplot (a)], and after employing the SSR method [subplot (b)]. A different source is identified in each of the subplots.
Plots of temporal signals to demonstrate the quality of the source reconstruction of TR and of the SSR method. The original source signal is displayed in blue. The focal signal at after standard TR is displayed in red. The focal signal at after using the SSR method is displayed in black.
Spatial dependence of the time reversed focal spots which result from progressively decreasing the distance between two sources (virtual sources were used to generate these plots). In subplot (a) the sources are placed 8 mm apart, in subplot (b) they are 6 mm apart, and in subplot (c) they are 4 mm apart (to be compared to the dominant signal wavelength of 11.7 mm). Note the progressive merging of the two focal spots with decreasing separation of sources.
Spatial dependence of the time reversed focal spots generated by two virtual sources separated by 4 mm. Subplot (a) displays the unmodified, merged focal spot. Subplot (b) displays the result of performing SSR on the data in subplot (a) assuming the source is located at the center of the focal spot. Subplot (c) displays the result of performing SSR on the data in subplot (a) with knowledge of the actual location of the upper source: the result is that the lower source location is determined [note that the center of the focal spot shifted vertically from 12 mm down to 10 mm when comparing subplots (a) and (c)].
TR focal signals at (red) and (black), after using standard TR [subplot (a)] and after using the SSR method [subplot (b)]. The envelope of each signal is displayed.
Plots of the instantaneous spatial distribution of the out of plane velocity at the times corresponding to the peaks of the signal at after employing the SSR method, before [subplot (a)] and after [subplot (b)] the time of focus.
Subplot (a): time reversed focus at after source reduction with original ratio (red line), and time reversed focus at after source reduction with (blue line). Subplot (b): time reversed focus at after subtraction of the data represented by the blue line from that represented by the red line in subplot (a) (blue line). The time reversed focus detected at with is also displayed (red line). The correlation coefficient of the two signals displayed in the subplot (b) is 91%. The data are normalized with respect to the peak amplitude in subplot (b).
Schematic of the aluminum slab. and represent sources and receivers, respectively.
Metrics to demonstrate the performance of the SSR method using TR. Subplot (a): amplitude ratio of the peak of time reversed focus to the highest side lobe amplitude for the two sources before and after reduction in the dominant source vs the source ratio . The blue and red lines represent the metric calculated at and , respectively. The solid and dashed lines refer to before and after source reduction on . Subplot (b): ratio between the metrics calculated before and after the source reduction in vs the source ratio . The blue solid line represents the relative decrease in amplitude at after performing SSR. The red dashed line represents the relative increase in amplitude at as a result of SSR.
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