Speckle noise is an inherent property of medical ultrasound imaging, and it generally tends to reduce the image resolution and contrast, thereby reducing the diagnostic value of this imaging modality. As a result, speckle noise reduction is an important prerequisite, whenever ultrasound imaging is used for tissue characterization. Among the many methods that have been proposed to perform this task, there exists a class of approaches that use a multiplicative model of speckled image formation and take advantage of the logarithmical transformation in order to convert multiplicative speckle noise into additive noise. The common assumption made in a dominant number of such studies is that the samples of the additive noise are mutually uncorrelated and obey a Gaussian distribution. The present study shows conceptually and experimentally that this assumption is oversimplified and unnatural. Moreover, it may lead to inadequate performance of the speckle reduction methods. The study introduces a simple preprocessing procedure, which modifies the acquired radio-frequency images (without affecting the anatomical information they contain), so that the noise in the log-transformation domain becomes very close in its behavior to a white Gaussian noise. As a result, the preprocessing allows filtering methods based on assuming the noise to be white and Gaussian, to perform in nearly optimal conditions. The study evaluates performances of three different, nonlinear filters--wavelet denoising, total variation filtering, and anisotropic diffusion--and demonstrates that, in all these cases, the proposed preprocessing significantly improves the quality of resultant images. Our numerical tests include a series of computer-simulated and in vivo experiments.