Techniques involve obtaining acoustic data from an acoustic logging tool, where the acoustic data includes waves reflected from the casing, the annular fill material, the formation, and/or interfaces between any of the casing, the annular fill material, and the formation. A crude casing thickness, tool position (e.g., eccentering), mud sound velocity may be estimated using the acoustic data. Techniques also involve computing a model spectra and an estimated casing thickness using a forward model and based on a crude casing thickness, an initial mud acoustic impedance, and an initial annular acoustic impedance, estimating a specular signal using the model spectra and the acoustic data in a first time window, computing a calibrated model signal using the estimated specular signal and computed model spectra, computing a misfit of the computed calibrated model signal and acoustic data in a second time window comprising the initial time window, and computing a correction update to one or more of the estimated casing thickness an estimated apparent annular acoustic impedance and an estimated mud acoustic impedance, based on the misfit. Techniques involve iteratively estimating the model spectra and the Jacobian curve, computing the specular signal, computing the misfit, and computing the update until the update is below a threshold. Outputs may include one or more of a casing thickness, an apparent acoustic impedance of the annular fill material, and the acoustic impedance of mud.