Objective.　Monitoring　the　depth　of　anaesthesia　(DOA)　during　surgery　is　of　critical　importance.　However,　during　surgery　electroencephalography　(EEG)　is　usually　subject　to　various　disturbances　that　affect　the　accuracy　of　DOA.　Therefore,　accurately　estimating　noise　in　EEG　and　reliably　assessing　DOA　remains　an　important　challenge.　In　this　paper,　we　proposed　a　signal　quality　index　(SQI)　network　(SQINet)　for　assessing　the　EEG　signal　quality　and　a　DOA　network　(DOANet)　for　analyzing　EEG　signals　to　precisely　estimate　DOA.　The　two　networks　are　termed　SQI-DOANet.　Approach.　The　SQINet　contained　a　shallow　convolutional　neural　network　to　quickly　determine　the　quality　of　the　EEG　signal.　The　DOANet　comprised　a　feature　extraction　module　for　extracting　features,　a　dual　attention　module　for　fusing　multi-channel　and　multi-scale　information,　and　a　gated　multilayer　perceptron　module　for　extracting　temporal　information.　The　performance　of　the　SQI-DOANet　model　was　validated　by　training　and　testing　the　model　on　the　large　VitalDB　database,　with　the　bispectral　index　(BIS)　as　the　reference　standard.　Main　results.　The　proposed　DOANet　yielded　a　Pearson　correlation　coefficient　with　the　BIS　score　of　0.88　in　the　five-fold　cross-validation,　with　a　mean　absolute　error　(MAE)　of　4.81.　The　mean　Pearson　correlation　coefficient　of　SQI-DOANet　with　the　BIS　score　in　the　five-fold　cross-validation　was　0.82,　with　an　MAE　of　5.66.　Significance.　The　SQI-DOANet　model　outperformed　three　compared　methods.　The　proposed　SQI-DOANet　may　be　used　as　a　new　deep　learning　method　for　DOA　estimation.　The　code　of　the　SQI-DOANet　will　be　made　available　publicly　at　https://github.com/YuRui8879/SQI-DOANet.