In this paper, we introduce a new spatially adaptive homomorphic Bayesian wavelet-based method for despeckling synthetic aperture radar (SAR) images. The wavelet coefficients of the logarithmically transformed reflectance image and the speckle noise image are modeled using a symmetric normal inverse Gaussian prior and an additive white Gaussian noise distribution, respectively. These models are then exploited to develop a Bayesian maximum a posteriori estimator. A method is proposed for estimating the parameters of the assumed prior. The noise-free variance of a wavelet coefficient is locally estimated, and used in a minimum mean square error estimator to obtain the corresponding noise-free coefficient. Experiments are carried out on two synthetically speckled images and a real SAR image. The results show that the proposed method has a performance that is superior to that of the other existing methods in terms of the peak signal-to-noise ratio, ability to suppress the speckle in the homogeneous regions.