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As a newly emerging seismic acquisition technology, distributed acoustic sensing (DAS) has attracted widespread attention due to its remarkable attributes of dense sampling, high sensitivity, efficiency, and cost-effectiveness. Nevertheless, a new challenge lies in the low signal-to-noise ratio (SNR) of DAS data, particularly for the downhole microseismic data in the hydraulic fracturing monitoring. Therefore, enhancing the SNR of DAS data is important for accurate data processing and interpretation. We have developed a new DAS microseismic data denoising method using dual domain generative adversarial networks with attention mechanism. The proposed denoising network is built upon the robust framework of a generative adversarial network. In addition to the standard GAN network operating in the spatial domain, we incorporate an additional GAN network in the spectral domain. It can capture and account for the spectral differences between clean and noisy data in the spectral domain, providing complementary information to the conventional spatial GAN. As a result, both the spatial and spectral GANs learn from both spatial and spectral information during the deep learning training process. This combined learning approach leads to better denoising outcomes. Within the generator, we introduce channel and spatial/spectral attention mechanisms to enhance its feature representation capabilities. Numerical examples have demonstrated the effectiveness and robustness of our proposed method. It achieves better denoising results for DAS microseismic data than than the traditional GAN method.