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The acoustic attenuation coefficient of most soft biological tissues has an approximately linear-with-frequency attenuation characteristic. The slope of the attenuation coefficient with frequency, denoted by β, has been observed to vary with the severity of liver disease. Two approaches for estimating the β value from reflected ultrasound signals are examined: the spectral-difference approach, which estimates β from the slope of the difference between log-spectra from two locations within the liver, and the spectral-shift approach, which estimates β from the downward shift experienced by the propagating pulse spectrum with penetration into the liver. This paper considers signals reflected from a small tissue region, defined by a cell measuring D by D centimeters in the plane of the sonogram, thus determining the feasibility of producing attenuation images. Lower bounds on the variance of the two β estimators are calculated by deriving maximum-likelihood estimators and by locating the tissue cell in the focal plane of the transducer. If W denotes the usable bandwidth in the reflected signals, the bounds are shown to be proportional to (WD)-3for β estimates calculated from individual reflected signals, and (WD)-4for estimates from all the signals reflected from the tissue region. With currently available technology, clinically useful results can be obtained for cell sizes measuring approximately 2.0 cm on a side.