Total variation is a hot point of research on speckle reduction. The nonconvex regularizer can better preserve or even enhance the information about the edges of an image. Spatially adaptive regularization parameters can reasonably control the level of speckle reduction according to the region in which the pixels are, and improve the speckle reduction effect. In this paper, we present a new total variation model for speckle reduction by integrating nonconvex regularizer and spatially adaptive regularization parameters. In order to solve the model, a new algorithm is designed based on Newton's method, augmented Lagrange multiplier, alternating direction method of multipliers, and iteratively reweighted method. The numerical examples demonstrate that the proposed model can obtain the better speckle reduction effect.