Epileptic seizure detection using scalp electroencephalogram (sEEG) and intracranial electroencephalogram (iEEG) has attracted widespread attention in recent two decades. The accurate and rapid detection of seizures not only reflects the efficiency of the algorithm, but also greatly reduces the burden of manual detection during long-term electroencephalogram (EEG) recording. In this work, a stacked one-dimensional convolutional neural network (1D-CNN) model combined with a random selection and data augmentation (RS-DA) strategy is proposed for seizure onset detection. Firstly, we segmented the long-term EEG signals using 2-s sliding windows. Then, the 2-s interictal and ictal segments were classified by the stacked 1D-CNN model. During model training, a RS-DA strategy was applied to solve the problem of sample imbalance, and the patient-specific model was trained with event-based K-fold (K is the number of seizures per patient) cross validation for detecting all seizures of each patient. Finally, we evaluated the performances of the proposed approach in the two levels: the segment-based level and the event-based level. The proposed method was tested on two long-term EEG datasets: the CHB-MIT sEEG dataset and the SWEC-ETHZ iEEG dataset. For the CHB-MIT sEEG dataset, we achieved 88.14% sensitivity, 99.62% specificity and 99.54% accuracy in the segment-based level. From the perspective of the event-based level, 99.31% sensitivity, 0.2/h false detection rate (FDR) and mean 8.1-s latency were achieved. For the SWEC-ETHZ iEEG dataset, in the segment-based level, 90.09% sensitivity, 99.81% specificity and 99.73% accuracy were obtained. In the event-based level, 97.52% sensitivity, 0.07/h FDR and mean 13.2-s latency were attained. From these results, we can see that our method can effectively use both sEEG and iEEG data to detect epileptic seizures, and this may provide a reference for the clinical application of seizure onset detection.