Quantitative ultrasound (QUS) holds promise in enhancing diagnostic accuracy. For attenuation imaging, the regularized spectral log difference (RSLD) can generate accurate local attenuation maps. However, the performance of the method degrades when significant changes in backscatter amplitude occur. Variations in the technique were introduced involving a weighted approach to backscatter regularization, which, however, is not effective when changes in both attenuation and backscatter are present. This study introduces a novel approach that incorporates an L1-norm for backscatter regularization and spatially varying weights for both fidelity and regularization terms. The weights are calculated from an initial estimation of backscatter changes. Comparative analyses with simulated, phantom, and clinical data were performed. When changes in backscatter and attenuation occur, the proposed approach reduced the lowest root mean square error by up to 73%. It also improved the contrast-to-noise ratio (CNR) by a factor of 4.4 on average compared with previously available methods, considering the simulated and phantom data. In vivo results from healthy livers, thyroid nodules, and a breast tumor further confirm its effectiveness. In the liver, it is shown to be effective at reducing artifacts of attenuation images. In thyroid and breast tumors, the method demonstrated an enhanced CNR and better consistency of the attenuation measurements with the posterior acoustic enhancement. Overall, this approach offers promise for enhancing ultrasound attenuation imaging by helping differentiate tissue characteristics that may indicate pathology.