Edge detection is a critical component of SAR image understanding. When SAR images include serious speckle noises, the core problem of their edge detection is how to keep the constant false alarm rate and how to achieve the unbiased localization ability for detecting the edges. In this paper, a novel edge detection method is proposed to solve this core problem. The capital idea of the proposed method is to combine the difference operation with the ratio operation. The proposed method consists of a 2D separable edge detection filter to compute two local weighted averages and an estimation algorithm to estimate the parameter related to the scattering coefficient. And then, the proposed method performs a difference operation and follows a ratio operation on the computed averages and the estimated scattering parameter, respectively. Theoretical analysis shows that the proposed detector not only has unbiased localization ability for edges, but also keeps constant false alarm rate under the influence of speckle. Theoretical analysis also shows that simply replacing the estimated scattering parameter with computed local weighted average does not affect the properties of the proposed detector, but largely reduces the computational complexity. Both quantitative and qualitative evaluations show that the proposed method can provide unbiased edge position and obtain the closed skeleton edges. The detection performance is insensitive to the changes of edge contrast, transition zone width and noise intensity. Benefitting from the high localization precision and good anti-noise ability, the performance of our method is further improved in the true positive detection rate.