Ultrasonic inverse problems such as spike train deconvolution, synthetic aperture focusing or tomography aim to reconstruct spatial properties of an object (discontinuities, delaminations, flaws, \textit{etc.}) from noisy and incomplete measurements. They require an accurate description of the data acquisition process. Dealing with frequency-dependent attenuation and dispersion is therefore crucial since both phenomena modify the wave shape as the travel distance increases. In an inversion context, this paper proposes to exploit a linear model of ultrasonic data taking into account attenuation and dispersion. The propagation distance is discretized in order to build a finite set of radiation impulse responses. Attenuation is modeled with a frequency power law and then dispersion is computed in order to yield physically consistent responses. Using experimental data acquired from attenuative materials, this model outperforms the standard attenuation-free model and other models of the literature. Due to model linearity, robust estimation methods can be implemented. When matched filtering is employed for single echo detection, the model that we propose yields precise estimation of the attenuation coefficient and of the sound velocity. A thickness estimation problem is also addressed through spike deconvolution where the proposed model also achieves accurate results.