# Comparative Analysis of Deep Learning Methods for Real-Time Estimation of Earthquake Magnitude
## 3 Analysis of Results
### Analysis of the Magnitude Estimation Results for Different Time Durations of the First-to-Wave Wake  
The results presented in Figure 4 further confirm that the overall performance of the DCRNN and DCRNNAmp in the magnitude estimation task has a significant advantage over MagNet and Exams. In order to analyze the error distribution characteristics of these models in more depth, we introduce a plot of the predicted magnitude versus the true magnitude during the first 8 s, as well as a histogram of the error distribution. The error distribution histogram reveals that the error data follow a Gaussian distribution.  
When comparing R\({}^{2}\) values, the DCRNN and DCRNNAmp achieve scores of 0.667 and 0.676, respectively, in the first second. These values are significantly better than MagNet's score of 0.580 and Exams' score of 0.383. As time progresses to the eighth second, the Exams model gradually improves and surpasses MagNet, reaching an R\({}^{2}\) value of 0.816. However, it still falls short of the DCRNN and DCRNNAmp, which maintain R2 values of 0.863 and 0.861.  
The error histogram dynamics provide a visual representation of the gradual optimization process of model performance. As the estimation time increases, the \"tail\" of the high error region starts to shorten, while the low error region expands, resulting in a more concentrated overall error distribution. This trend suggests that the model's estimation error decreases, which ultimately enhances the accuracy and reliability of the estimations.  
Figure 4: Plot of the errors of the four models with magnitude in the first 6 s after the arrival of the P-wave.  
Additionally, we assessed the linear correlation between the predicted and true magnitudes by calculating the correlation coefficient. The data presented in Figure 5 indicate that the correlation coefficients for DCRNNAmp and DCRNN are both 0.82 from the start and onward. This performance is significantly better than that of MagNet (0.78) and Exams (0.76). Furthermore, the correlation coefficients for all models show a gradual increase with the extension of the time window, further confirming the continuous improvement in model performance.  
Notably, DCRNNAmp exhibits a faster and more accurate magnitude estimation capability during the first 3 s when compared to DCRNN. The results indicate that DCRNNAmp outperforms DCRNN in terms of both accuracy and speed in this initial period. This finding underscores the potential application of the DCRNNAmp in earthquake early  
Figure 5: Rows 1 and 3 show the predicted magnitude and accurate magnitude distribution of DCRNN and DCRNNAmp models. Rows 2 and 4 show the frequency distribution of the error (predicted magnitude and true magnitude) for the DCRNN and DCRNNAmp models.  
warning systems and serves as an important reference for future model optimization and algorithm enhancement.  
The performance of the Exams model against other models in a real-time streaming input scenario demonstrates the advantages of RNN and its variants (e.g., Bi-LSTM) in fast-magnitude prediction. The sequence and time dependence of events in seismic waveform data are crucial, and RNN effectively handles the long time dependence problem through a gating mechanism that captures subtle features and maintains the memory of historical information. Therefore, RNN and its hybrid models (e.g., DCRNNAmp) provide more stable and accurate magnitude prediction results when fast and accurate prediction is required, especially when facing continuous data streams.