For shot signals outside a certain range, modes can be resolved in time by narrowband filtering, and in incidence angle by beamforming. The travel time of a mode "pulse" is proportional to its average group slowness. For two signals, the travel time difference is observable at the receiver. If the average group slowness difference also was known, there is a simple formula for estimating the range. But this average is not measurable locally. However, the local phase speed difference can be measured by the beamformer. The quantity called "waveguide invariant β" combines group and phase slowness differences. This quantity can for a given range be computed from the bathymetric profile, alternatively from the top bedrock profile between source and receiver, after making some simplifying assumptions. Thereby the influence of the unknown average group slowness can be "taken care of". Locally at the receiver the relation between travel time difference and phase slowness difference can be measured. This relation can be combined with theoretical β values in a simple equation, from which a range estimate can be extracted. In an experiment at sea an 820 m array with 10 hydrophones was deployed at the bottom in 320 m water depth. For two endfire runs in opposite directions, small explosive charges out to 115 km were used as sound sources. Use of the range estimation formula gave typical range estimation errors of 5-10%, somewhat dependent on center frequency and bandwidths.
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