Attenuation of ultrasound waves varies with tissue composition, hence its estimation offers great potential for tissue characterization and diagnosis and staging of pathology. We recently proposed a method that allows to spatially reconstruct the distribution of the overall ultrasound attenuation in tissue based on computed tomography, using reflections from a passive acoustic reflector. This requires a standard ultrasound transducer operating in pulse-echo mode and a calibration protocol using water measurements, thus it can be implemented on conventional ultrasound systems with minor adaptations. Herein, we extend this method by additionally estimating and imaging the frequency-dependent nature of local ultrasound attenuation for the first time. Spatial distributions of attenuation coefficient and exponent are reconstructed, enabling an elaborate and expressive tissue-specific characterization. With simulations, we demonstrate that our proposed method yields a low reconstruction error of 0.04dB/cm at 1MHz for attenuation coefficient and 0.08 for the frequency exponent. With tissue-mimicking phantoms and ex-vivo bovine muscle samples, a high reconstruction contrast as well as reproducibility are demonstrated. Attenuation exponents of a gelatin-cellulose mixture and an ex-vivo bovine muscle sample were found to be, respectively, 1.4 and 0.5 on average, consistently from different images of their heterogeneous compositions. Such frequency-dependent parametrization could enable novel imaging and diagnostic techniques, as well as facilitate attenuation compensation of other ultrasound-based imaging techniques.