Accounting for bias in horizontal wavenumber estimates due to source motion

Modal-based techniques for geoacoustic inversion require estimates of discrete horizontal wavenumbers corresponding to the propagating modes in a shallow-water waveguide. Horizontal wavenumber estimates can be obtained from the pressure field of a monochromatic point source measured as a function of range at a fixed depth. Previous experimental efforts have employed either fixed source/moving receiver or fixed receiver/moving source geometries to measure the field with range. Often, due to the slowly moving source/receiver, the problem is assumed to be static and that reciprocity holds. However, source motion and receiver motion impact the time-domain solution to the depth-dependent acoustic wave equation in different ways. One particular effect of source motion is a Doppler frequency shift in the depth-dependent Green's function, or integration kernel, used in the inverse Fourier transform to get the time-domain solution for the field. The Doppler shift in the Green's function equates to a Doppler shift in the horizontal wavenumbers corresponding to the individual propagating modes. Therefore, modal wavenumber estimates made for a source moving at constant velocity will be biased due to the Doppler shifts. In order for the estimates to properly represent modes consistent with the waveguide boundary conditions as required for modal inverse techniques, the bias should be removed. In this paper, a simple method is presented for effectively removing the bias in wavenumber estimates due to the Doppler effects induced by a moving source. An example is provided showing the Doppler shift measured from real experimental data. The impact on inversion results due to not accounting for moving source effects is also presented. Finally, a method for directly measuring modal group velocity from Doppler shifted wavenumber estimates is proposed.