Edge detection in SAR images remains a difficult task due to the strong multiplicative noise. Many researches have been dedicated to edge detection in SAR images but very few of them try to address the most challenging situation, namely the edge detection in 1-look real SAR images. Motivated by the success of Convolutional Neural Networks (CNNs) in natural images, we study the applicability of the usual CNN edge detector to SAR images, especially for edge detection in 1-look real SAR images. One crucial factor that contributes to the success of CNNs is the training dataset with labeled ground truth. Instead of doing the tedious job of annotating plenty of SAR images, we simulate a SAR dataset leveraging the optical dataset BSDS500 [1] to train CNN models because edges are mainly corresponding to changes in brightness and textures in grayscale images. In order to cope with the differences in the range of pixel values between SAR and optical images, we propose to train CNN models on the gradient magnitude fields of images because the differences in the gradient distribution between speckled optical images and real SAR images are tiny and insignificant. In the gradient feature space, the gradient magnitude fields of homogeneous areas follow exactly the same distribution regardless of their mean intensity values, and the distribution of gradient magnitude fields for two homogeneous areas across boundaries depends only on the ratio of their mean intensity values. The proposed CNN edge detector GRHED achieves excellent performances in all our simulations including several 1-look synthetic edge images with different ratio contrasts, two hundred 1-look optical images, which are simulated from BSDS500, one synthetic SAR image and two 1-look real SAR images. In addition, it exceeds the existing edge detectors in SAR images a lot.