Echo Time Spreading and the Definition of Transmission Loss for Broadband Active Sonar

Paper Number 97, Proceedings of ACOUSTICS 2011 2-4 November 2011, Gold Coast, Australia Echo time spreading and the definition of transmission loss for broadband active sonar Z. Y. Zhang Defence Science and Technology Organisation P.O. Box 1500, Edinburgh, SA 5111, Australia ABSTRACT In active sonar, the echo from a target is the convolution of the source waveform with the impulse responses of the target and the propagation channel. The transmitted source waveform is generally known and replica-correlation is used to increase the signal gain. This process is also called pulse compression because for broadband pulses the re- sulting correlation functions are impulse-like. The echo after replica-correlation may be regarded as equivalent to that received by transmitting the impulsive auto-correlation function of the source waveform. The echo energy is spread out in time due to different time-delays of the multipath propagation and from the scattering process from the target. Time spreading leads to a reduction in the peak power of the echo in comparison with that which would be obtained had all multipaths overlapped in time-delay. Therefore, in contrast to passive sonar where all energies from all significant multipaths are included, the concept of transmission losses in the active sonar equation needs to be handled with care when performing sonar performance modelling. We show modelled examples of echo time spreading for a baseline case from an international benchmarking workshop. INTRODUCTION ECHO MODELLING ISSUES The performance of an active sonar system is often illustrated In a sound channel, the transmitted pulse generally splits into by the power (or energy) budget afforded by a separable form multipath arrivals with different propagation angles and of the sonar equation, such as the following, travel times that ensonify the target, which scatters and re- radiates each arrival to more multipath arrivals at the re- SE = EL – [(NL – AGN) ⊕ RL] + MG – DT (1) ceiver. Hence the received echo is a multiple convolution of the transmitted source pulse with the following three impulse where SE is the signal excess, EL is the sound pressure levels responses: (1) propagation from source to target; (2) target of the echo, (NL – AGN) is the in-beam noise level, i.e., noise scattering; (3) propagation from target to receiver. level NL in the receiver bandwidth reduced by the receiver array gain AGN against noise, RL is the in-beam reverbera- Whilst Eqs.(1) and (2) are instructive to help understand the tion level from scatterers on the source and receiver’s beam power (or energy) budget of the sonar process, rigorous mod- pattern, ⊕ represents intensity summation. MG is the gain in elling of broadband coherent active sonar is more complex signal to noise ratio (SNR) from matched filtering. The detec- than computing the seemingly simple sonar equations be- tion threshold, DT, is the SNR required at the output of the cause of the dependence of the sonar quantities on frequency, matched filter to achieve a certain probability of detection angle, and time. and false alarm for given signal and noise statistics. In general, due to the presence of multipath with different The DT in Eq.(1) is defined at the output of the matched propagation angles, the EL can not be computed from sepa- filter, which is the convention in Ainslie (2010), Lurton rable terms as in Eq.(2). Target scattering generally depends (2002), Bradley (1996), and the Radar literature (Barton on incident and outward angles. Source beam patterns, 2005; DiFranco & Rubin 1968). When detection threshold is propagation multipaths or multi-modes, and target scattering defined at the input to the temporal processor, as in Urick become coupled in angle domain and there is no separable (1983), Burdic (1991), Waite (2002), and Etter (2003), the equation to calculate EL from SL, TS, and TL. It is only matched filtering gain is included in the detection threshold. when the target scattering cross section varies little across the angular domain of the significant multipaths, that the terms Traditionally, the echo level is often written as (Urick 1983; become separable (Ratilal et al 2002). In addition, effects of Burdic 1991; Waite 2002; Etter 2003), multiple scattering can be significant when the target is near an ocean boundary (Giddings & Shirron 2008; Hackman & EL = SL – TLS→T + TS – TLT→R (2) Sammelmann 1988). Furthermore, in a multipath environ- ment, the array signal gain also degrades from that computed where SL is the source power level, TLS→T is the one-way for a single plane wave in free field (Zhang 2009). transmission loss from source to target, TS is the target strength, TLT→R is the one-way transmission loss from target Another complication arises from the dependence of the so- to receiver. nar quantities on frequency. The source level, propagation, target strength, noise and reverberation, array beam patterns and array gain, generally vary with frequency. Integration Acoustics 2011 1 2-4 November 2011, Gold Coast, Australia Proceedings of ACOUSTICS 2011 with frequency is needed and the frequency dependence can For example, if the one-way response has a Gaussian enve- only be ignored if the variation is small over the frequency lope, the two-way response from convolution will also have a band of interest. Gaussian envelope with its peak reduced by a factor of 2 , i.e., the extra loss is a fixed value of 1.5 dB. If the one-way The third complication comes from the time dependence or response is an exponential decay function, the peak of the transient nature of pulse propagation, which is the focus of two-way response is reduced by 4.3 dB relative to the one- this paper. way response (Weston 1965). Transmission Loss for Pulse Propagation Convolution in time is equivalent to multiplication in the Traditional transmission loss, such as that for passive sonar, frequency domain. So if the pulses are narrowband, transmis- is defined for a harmonic wave of a single frequency, where sion losses defined for harmonic waves should be additive in all multipath contributions are added together because the dB. signal is treated as of infinite time extent (Urick 1983; Jensen To consider the relationship of one-way and two-way trans- et al 2000). We may call these harmonic transmission losses. mission losses defined in terms of total energy, it can be Coherent or phased additions, where the complex pressures shown using the Fubini theorem (Kudryavtsev, LD 2001) that are added, reveal constructive and destructive interferences the integral of a convolution equals the product of the inte- between different paths. Incoherent addition, where the inten- grals of respective functions. Applying this property to the sities are added, is often used to approximate the smoothing intensity impulse responses in time domain, e.g., Eq.(A3) of effects of averaging, e.g., over frequency, range, depth, (Har- Harrison & Ainslie (2010), we conclude that transmission rison & Harrison 1995) or other statistical fluctuations. Inco- losses defined on total energy basis are additive in dB. herent addition can be over rays or modes where different types of phase information are discarded. The same discussion and principle should apply to convolution with target responses, i.e., peak target strength generally For pulses, one may define sound levels and transmission is not additive in dB with transmission loss, but target losses in terms of total energy integrated in time or frequency strength defined for harmonic waves and total energy should (Ainslie 2010; Marshall 1996). For simplicity we call these be additive in dB. total transmission loss. One may also define transmission losses in terms of difference in energy spectral density Matched Filtering Gain and Correlation Loss (Badiey et al 1997), which correspond to harmonic transmission losses at the different component frequencies. Under the following idealisations: (1) echoes from all multipath arrive at the same time; (2) each multipath suffers no Peak pressures or intensity have also been used to represent distortion during its propagation, the echo would collapse sound levels (Madsen 2005) or to measure transmission into a single, time-delayed perfect replica of the transmitted losses (Hines et al 1997). We may call these peak transmis- pulse with its amplitude associated with total transmission sion losses. For modelling the echo power level of an active loss. The matched filtering gain will be 10log10(BT), where B sonar, after processing such as matched filtering, envelope is the bandwidth and T is the duration of the transmitted detection, and averaging, what is relevant is the peak of the pulse (Barger 1994; Waite 2002; Lurton 2002). intensity sum of multipath arrivals within the sonar resolution cell. Multipath time spreading, time-varying, and dispersive properties of the channel introduce distortions to the pulse shape The distinction between total and peak transmission loss is and consequently a correlation loss, CL (Ainslie 2010; Wes- important for short pulses and can be illustrated by the simple ton 1965) example of propagation in an isovelocity waveguide. Whilst the total transmission loss may behave as cylindrical or MG = 10log10(BT) – CL (3) “three halves law” spreading depending on the reflectivity of the bottom (Weston 1971;Brekhovskikh & Lysanov It is more intuitive to understand correlation loss due to mul- 2003;Harrison 2003), if the pulse is short enough so that all tipath time spreading in terms of the propagation of equiva- multipaths are separated, the peak transmission loss will be lent short pulses. determined by the strongest direct path, leading to spherical spreading irrespective of bottom properties. Equivalent Short Pulse Two-way versus One-way Losses Matched filtering is equivalent to correlation with a scaled replica of the transmitted pulse. The output signal has a It is often regarded as an indisputable fact in active sonar mainlobe whose effective duration is approximately the in- modelling that the two-way transmission loss is double the verse of the signal bandwidth B and an amplitude that is pro- one-way loss.

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