As described above in Sect. 2.1.7, post-processing of tomographic signals detected by our sensor must take into account the mutual motion of the transceiver and the receiver. This motion was mostly due to the drift of the vessel. Sometimes it was significant (about 1 m/s). To extract a correct pulse response of the waveguide we must remove this effect in processing of our data. But the motion of our sensor gives an unexpected advantage. The Doppler shift is different for the different rays, it is proportional to the cosine of the grazing angle of the ray. Using the coherent post-processing of signals we can find different optimal Doppler shifts for different rays and in this way we can evaluate these angles. To say it in other words, our receiver synthesized a horizontal aperture of a length L equal to , where v - the average speed of vessel motion and is the duration of signal. Sometimes L was up to 200 meters. Estimating the Doppler effect and travel time is within the scope of estimation theory . This particular problem has been extensively studied in the RADAR or even SONAR literature since it is the common signal model for active target identification . There is a set of papers dealing with such a problems [38, 39], etc. We also considered this effect and carried out some estimates.
The removal of Doppler effect includes two procedures. The first one is substitution of demodulation frequency by the frequency , where is:
where c - sound speed at the receiver location. The second step consists in extending the duration of signal by a value :
that is changing the time scale by Mach number v/c. Under this procedure we must reinterpolate our ADC samples to the new time scale. After these steps we can average all cycles of the M-sequence at a single reception taking into account their phase (coherent averaging). If we take the exact value of averaged speed of vessel drift multiplied by a cosine of grazing angle of a separate ray we will receive a maximum amplitude of this ray at the correlation function with respect to other values of the Doppler shift. Such amplitude must be equal to the value of the correlation function under the noncoherent averaging (or little less, which is due to the little disturbances of speed of receiver motion, e.g., swing of the sensor).
The dependence of maximum of the correlation function for the separate arrival versus error of evaluation of averaged speed of vessel drift ( ) has a typical view of , where . The separation between the adjacent minima of this function is , and its main maximum (which corresponds to the true speed) has the double width with respect to the others. At Fig. 2.46 (b) one can see the typical view of this function, calculated for one of experimental records. Scanning the correlation function over the interesting range of values of speed of R/V motion we receive a 3D picture. At such plot the coherently averaged correlation function reachs maxima at different time delays and different Doppler shifts. Thus, we can evaluate the grazing angles corresponding to those time delays (i.e., different incident rays) and enrich the information about the ray pattern. The regular dependence of maxima versus the Doppler shift (as shown at Fig. 2.46 (b)) can serve as an indicator of arrival presence. Dashed curve at this figure corresponds to the same function for the time delay without any arrival.
Figure 2.46: Recording of the tomographic signal at a synthesized aperture create by the R/V drift:
a) - Dependence of the maximum of correlation function versus error of evaluation of averaged velocity of vessel drift (resolution of two rays).
b) - The same dependencies as at a) for a single ray (solid line) and without signal arrival (dashed line).
c) - Grazing angles of eigenrays which formed the acoustic field at the receiver.
d) - Result of noncoherent averaging of the reception (solid line),
numerical prediction of this function by the ray theory (dashed line).
To determine the absolute values of grazing angles of rays we must identify some arrival with an appropriate ray (it may be an axial group of rays with very small grazing angles and close arrival times). After that we can evaluate the true velocity of the R/V drift and then find the modules of the grazing angles for the other rays. According to the estimate of the array resolution , the error of finding the grazing angle of a separate ray by our synthesized aperture array equals
(here is an aperture of our array) for rays with small grazing angles. If we have a good signal-to-noise ratio in the correlation function and clearly see the sidelobes such as at the slices presented at Fig. 2.46 (b) we can increase an accuracy of angle evaluation using these sidelobes. It can be done by finding the more accurate position of main maximum.
One more advantage of application of the synthesized aperture consists in resolution of two or more rays which have the same arrival times, but substantially different grazing angles, if such situation takes place in experiment. At Fig. 2.46 (a) one can see this effect. We can clearly see the composition of two functions of type and, hence, at this time moment two different rays came to our receiver.
Figure 2.47: 3D picture of correlation function level versus time delay and velocity of the R/V drift. Markers correspond to theoretically calculated rays, red digits show their grazing angles
Finally, at Fig. 2.47 a color plot of the correlation function is presented. The peaks are located along one line. A small slope of this line with respect to the vertical line ( ) is explained by the time scale variation as described above. The results of synthesized aperture processing of this reception are summarized at Fig. 2.46 (c-d). The following curves are presented at the lowest plot: the result of noncoherent averaging of the reception (solid line) and prediction of this function by the ray theory (dashed line). The grazing angles of the rays which formed the acoustic field at the receiver are shown at Fig. 2.46 (c). Markers at Fig. 2.47 correspond to the time delays and grazing angles of these rays. The signal was received from the transceiver S. The receiver-transceiver separation was about 125 km, the speed of R/V drift - about 0.786 m/s, time duration of the signal - 204 s (40 cycles), the carrier frequency - 400 Hz. So, the expected angular resolution is about . In computer modeling the medium between source and receiver was defined by two vertical profiles of the sound speed - one at the transceiver location and the other at the reception point, with linear interpolation between them. One can see good coincidence between theory predictions and experimental results. Certainly, the agreement will be improved after the tomographic reconstruction of propagation conditions.