Coherent Matched-Filter Reflection Loss From a Moving Rough Surface as a Function of Pulse Duration and Ensonified Extent

In many applications of active sonar, the propagation paths connecting a source, an object of interest, and a receiver include reflection from the ocean surface. A classic result describes the coherent surface reflection loss (SRL) of an acoustic pulse solely as a function of the Rayleigh roughness parameter by requiring a large ensonified area (LEA) and a pulse short in duration relative to the surface wave period. At the other extreme are results accounting for motion of the surface throughout reflection of the pulse but requiring a small ensonified area (SEA) where, in effect, less than a correlation length of the surface contributes to the reflection. The pulse compression achieved by matched filtering broadband pulses enables the SEA while using a pulse that may be long in duration relative to the surface wave period. The coherent SRL for a medium ensonified area (MEA) is derived in this paper by extending the SEA results to account for the combination of multiple correlated components of the surface contributing to specular reflection. The derivation adds a dependence on both the ensonified area and the spatial correlation function of the surface. The model is seen to simplify to the classic LEA result when the ensonified area is large relative to the correlation length. A low-roughness approximation allows simple evaluation of the MEA SRL that, for long pulses, is a mixture of the SEA and LEA results. An example evaluation of SRL using a Pierson-Moskowitz wave spectrum is presented as a function of the duration and bandwidth of linear-frequency-modulated pulses to illustrate how the loss is near the LEA result for low bandwidths and tends to the SEA result as bandwidth increases.