Fixed time versus fixed range reverberation calculation: Analytical solution

Abstract

Reverberation is commonly calculated by estimating the propagation loss to and from an elementary area, defined by transmitted pulse length and beam width, and treating the resulting backscatter from the area as a function of its range. In reality reverberation is strictly a function of time and contributions for a given time come from many ranges. Closed-form solutions are given for reverberation calculated both at fixed range and at fixed time isovelocity water and some variants of Lambert's law and linear reflection loss with an abrupt critical angle. These are derived by considering the shape of the two-way scattered multipath pulse envelope from a point scatterer. The ratio of these two solutions is shown to depend on the dominant propagation angle spread for the particular range or time. The ratio is largest at intermediate ranges (though typically less than 1 dB) and depends explicitly on the critical angle. At longer ranges mode-stripping reduces the propagation angle spread and the ratio reduces ultimately to unity. At short range the ratio is also close to unity although interpreting it requires care.

Target time smearing with short transmissions and multipath propagation

“…However it is not particularly a flux, mode, or eigenray approach since the same formulation can be derived from all three by assuming either a continuum of modes or a continuum of eigenrays (see Appendix B of Harrison and Ainslie, 2010). In that respect the formulation is very robust; it will break down, however, if there is either a small number of eigenrays, or a small number of modes, or if there is significant convergence or focusing.…”

“…If the transmission is very short it essentially picks the value for each travel time and multiplies by the transmitted pulse duration. This was developed further for the short transmission case (actually an impulse) to include refraction and to consider the fine differences between reverberation as a function of time rather than range (Harrison and Ainslie, 2010). Also this one-way and two-way multipath pulse shape has been a) Author to whom correspondence should be addressed.…”

Echo and reverberation in a Pekeris waveguide by convolution and by the product rule

“…24, an exact solution to the integral of Eq. (18) is possible using the special form for the reflection coefficient (HA-15)…”

“…This convolution method was extended to reverberation by Harrison and Ainslie. 24 The present work builds on the ideas of Harrison and co-workers 23,24 by removing the limitation to near horizontal ray paths in Ref. 23.…”

“…A simple correction to the product rule is obtained by recasting the results of Ref. 24 in a form that permits direct comparison between the integrands of the two methods. The convolution method is applied to the target echo in Sec.…”