Effect of sea-bottom elasticity on the propagation of acoustic–gravity waves from impacting objects

Recent analysis of data, recorded on March 8th 2014 at the Comprehensive Nuclear-Test-Ban Treaty Organisation’s hydroacoustic stations off Cape Leeuwin Western Australia, and at Diego Garcia, has led to the development of an inverse model for locating impacting objects on the sea surface. The model employs the phase velocity of acoustic–gravity waves that radiate during the impact, and only considers their propagation in the water layer. Here, we address a significant characteristic of acoustic–gravity waves: the ability to penetrate through the sea-bottom, which modifies the propagation speed and thus the arrival time of signals at the hydrophone station. Therefore, we revisit some signals that are associated with the missing Malaysian Aeroplane MH370, and illustrate the role of sea-bottom elasticity on determining impact locations.

that can travel long distances before dissipating. We therefore filtered low frequencies (below 5 Hz) with a high pass Butterworth IIR filter. Since the signal randomness measure will change when the signal's nature changes, transient signals over a noisy background can be identified by calculating a windowed entropy value. Peaks in the entropy trace are present where transient signals are detected. These peaks were considered for the subsequent bearing calculation. After separating the signals, the bearing is calculated using time of arrival based triangulation, see ref. 1 for detailed bearing calculations.

HA01.
We address two possible impact events that were identified by ref. 1  Transects along the bearings of E1 and E2 are given in Figs 2 and 3. Acoustic-gravity waves can transmit between layers at the highlighted regions T 1j (j = 1, …, 8) in the case of bearing 301°, and T 2k (k = 1, …, 6) in the case of bearing 234°. One possibility, in both E1 and E2, is that the recorded acoustic-gravity waves travelled only in the water layer, which results in the locations originally identified by ref. 1 . Another possibility is that acoustic-gravity waves couple with the elastic layer shortly after the impact and all along the way until the signals are received at the station. This scenario dictates farest location distance that is c s /c l (almost as twice) the distance originally calculated by ref. 1 . The second scenario is more likely when the water depth at impact is critical. Between these two marginal possibilities there are a number possible transmission between layers that result in different locations, as summarised in Tables 1 and 2.   HA08s. Analyses of signals recorded at station HA08s (−65.5445°, 32.4730°) were more challenging, partially due to disturbances in the recordings that are believed to be caused by military action in the region. A summary of identified signals of interest (see Fig. 4) is given in Table 3. Note that bearings if signals HA_30 and HA_32 fall within the military action bearings, so it is also possible that the signals are associated with the military action. Among the rest of the signals, it is remarkable that three have a bearing of 170.9°, and one 173°. The first occurred at 12:11 UTC whereas the other three followed about three hours later, all after 3.30. Last but not least, a fifth signal appears at 3:07 (see Fig. 5). This signal probably indicates restarting the system after it was shutdown for 25 minutes, i.e. there is a missing data in these specific CTBTO recordings.

Discussion
Acoustic-gravity waves can travel at speeds near the speed of sound in water, yet they can double their speed when coupling with the elastic layer. As they propagate they carry information on their source and thus can be used, among others, for locating impacting objects at the sea surface by applying a proper inverse model 1 . However, since the location directly relies on the propagation speed, and the later depends on the medium, it is important to know the route travelled by acoustic-gravity waves. For example, signals E1 and E2 travel through different routs, as given in Tables 1 and 2, which can results in different calculated locations. In the case of multiple transmissions one expects teh signals to be composed of a number of smaller signals, which is not the case here unless if other signals are buried in the ambient noise. Since only one complete signal has been identified for each of E1 and E2, it is more likely that the signals either did not transmit at all into the elastic layer, or transmitted only once at the initial stage and coupled with the elastic layer all a long the way, i.e. the first and last possibilities of each table.
The locations of signals found on HA08s are with high uncertainty or unknown and require further analysis. Though, if related to MH370 that might suggest a location in the northern part of the Indian Ocean. Due to the sensitivity of the recorded data, it is unlikely that the three hydrophones on HA08s had a simultaneous technical failure and the reason behind the shut down is to-date unknown. The missing data might be related to the military action in the area (during or after the impact), but another argument is that a violent nearby activity (including impact, explosion) could have resulted in a shutdown of the system. Both the signal HA_30 of bearing 247° recorded at 11:57 on March 7th, and the missing data if related to MH370 could (independently) suggest that the impact location is closer to Diego Garcia's station, as opposed to Cape Leeuwin's station. With the absence of the recordings, there is currently no scientific evidence that an impact occurred during this time window. However, it might be possible to extract more information after processing hidden signals in the ambient noise. To study this possibility and to further assess the effects of elasticity and transmissions due to sea-bottom topography we intend to carry out a set of field experiments, while in parallel develop a depth-integrated, see 10 , sea-bottom elastic model for the radiation of acoustic-gravity waves from impacting objects.

Methods
Dispersion relation. The solution for the propagation of acoustic-gravity waves in compressible water under the effects of gravity, and an elastic half space was treated by ref. 7 , who derived the dispersion relation, ( 2 ) where r is the eigenvalue, h is the water depth, g is the acceleration due to gravity, k is the wavenumber, ω is the frequency, λ and μ are Lame's elasticity constants, ρ l is the water density, and q and s are separation constants in the solid sea-bottom,  Table 3).