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This paper presents an algorithm for speckle reduction of synthetic aperture radar (SAR) images within a framework of multiscale curvelet analysis. First, we introduce a novel method to investigate the presence of 2-D heteroscedasticity based on Lagrange multiplier procedure. Employing this test confirms the heteroscedasticity of SAR image curvelet coefficients. Therefore, we employ a generalization of 2-D generalized autoregressive conditional heteroscedastic (2-D GARCH) model, called 2-D GARCH generalized Gaussian (2-D GARCH-GG), to these coefficients. This model preserves the appropriate properties of 2-D GARCH for modeling the curvelet coefficients while extending the dynamic formulation of 2-D GARCH model. Then, we design a novel Bayesian processor based on employing 2-D GARCH-GG model to estimate the noise-free curvelet coefficients. Experiments carried out on synthetic SAR images, as well as on true SAR images, verify the performance improvement in utilizing the new strategy compared with other established despeckle algorithms.