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The broadband noise field of a ship of opportunity often exhibits environment dependent striation structure in the frequency-range plane. For the soft-layered sediment environment studied in this paper, the striation structure is critically determined by sub-bottom sound speed (Cbot), sediment thickness (H) and sediment sound speed (Csed). Numerical simulations demonstrate that striations in different frequency bands have different sensitivities to the three critical parameters. The sensitivity differences are used here to progressively estimate the Cbot, H and Csed. We first use low-frequency striation structure to estimate the Cbot, then obtain a preliminary estimation of the H and Csed with a set of low-frequency striations, and finally find the best-fit solutions from previous estimates using high frequency striation structure. We processed passive ship run data collected in Mediterranean Sea in 2007. The good agreement between our results with active inversion methods demonstrates the accuracy of the method for ocean bottom geoacoustic characterization.