I. Introduction
Modal interferences are evident in acoustic signals propagated over distances [1] . For a coherent broadband signal, the signal energy travels at the modal group velocities. Modal arrivals and dispersion can be studied by displaying the signal as a function of delay time and frequency for broadband signals [1]. The time delay between modal intensity maxima at a given frequency is proportional to the source range [2]. For random (noise-like) broadband signals (having no well-defined waveform), individual mode arrivals cannot be identified in the received signal. However, the received intensity spectrum plotted as a function of frequency and range shows a well-defined pattern due to modal interference. One finds striated bands of intensity maxima and minima as a function of frequency and range, corresponding to constructive and destructive interference of normal modes. For many types of sound-speed profiles (SSPs), the slope of the striations is invariant to the details of the waveguide and can be characterized by a scalar parameter , referred to as the waveguide invariant [3]–[5]. Given that is known for a given environment, one can estimate the (approximate) range between the source and the receiver using a single receiver by measuring the slope of the striations from the data at a given frequency. The power of this ranging method lies in its simplicity as compared with more sophisticated source localization methods such as matched-field processing [6]. As pointed out before, the modal interference distance, measured by the shifts of (modal) intensity maxima at a given frequency, is expected to increase with the source range [7]. The waveguide invariant has also been used for time-reversal focusing by frequency shifting [8], source localization by range shifting (in a range-dependent environment) [9], source localization by the sidelobes of matched-field processing [10], and target motion compensation by frequency shifting [11], [12].