This paper proposes a deep separate source-channel coding (DSSCC) scheme for the semantic-aware image transmission, where image is lossily compressed and transmitted to receiver for recovery and processing certain semantic tasks. To improve the compression efficiency, the forward adaption (FA) method is incorporated into the DSSCC scheme to capture the density information of compressed features as side information. For a typical application of image classification task, we derive a novel rate-distortion optimization problem by analyzing the Bayesian model of the FA-based DSSCC framework. Then, a variational autoencoder approach is proposed to effectively compress image for semantic-aware transmission by minimizing the proposed rate-distortion problem. Simulation results reveal that the proposed FA-based DSSCC scheme achieves better image recovery and classification performance in most scenarios, compared to the classical compression schemes and the emerging deep joint source-channel schemes.