Visual and automatic analyses using synthetic aperture radar (SAR) images are challenging because ofinherently formed speckle noise. Thus, reducing speckle noise in SAR images is an important research area for SARimage analysis. During speckle noise reduction, homogeneous regions should be smoothed while details such as edgesand point scatterers need to be preserved. General speckle noise model contains gamma distributed multiplicative partwhich is dominant and Gaussian distributed additive part which is in low amount and mostly neglected in literature. Inthis study, a novel sparsity-driven speckle reduction method is proposed that takes both multiplicative noise model andadditive noise model into consideration. The proposed speckle reduction method uses a cost function with multiplicativeand additive data terms besides the total variation smoothness term. Also, an efficient and stable numerical minimizationscheme is proposed for the proposed cost function that deals with multiplicative and additive noise. Speckle reductionperformance of the proposed method is shown on synthetically generated SAR images and real-world SAR images.