# Successful Tests on Detecting Pre-Earthquake Magnetic Field Signals from Space
## 3 Results and Discussion
### Magnitude-Energy correlation  
Another significant parameter obtained from the results is the identified anomaly energy, En. This \"energy\" is represented here by the amplitude of the anomaly divided by the average amplitude of the whole half-orbit. As explained earlier, the amplitude of the anomaly is related to the earthquake's magnitude. As a result, the anomaly energy is also related to the magnitude. According to Figure S5, where the y-axis is the natural logarithm energy and the x-axis is the magnitude, there is a natural logarithmic relationship between the two quantities. In Figure S5a, which is for all data, there is the following relationship between energy \(En\) and magnitude \(M\):  
\[M=(1.469\pm 0.877)\times\ \ln(En)+(-0.032\pm 0.758) \tag{3}\]  
The R-square of the system is equal to 0.96, which shows the high capability of the model to predict the results, and the error value is equal to 0.167. Here, we also experiment with the obtained relationship using geomagnetic indices (Figure S5b) and a signal-to-noise ratio greater than two and greater than five (Figure S5c,d). In both cases, it can be  
Figure 3: Magnitude-duration correlation. The plot illustrates the relationship between the natural logarithm of anomalies duration (\(y\)-axis) and the magnitude of earthquakes (\(x\)-axis) under different conditions. (**a**) All the data have been considered without applying any specific conditions. (**b**) The data are filtered based on quiet geomagnetic conditions, with criteria such as Kp \(\leq\) 3, \(\mid\)Dist \(\mid\)\(\leq\) 20 nT, and F10.7 \(\leq\) 150. (**c**) The data are filtered based on a signal-to-noise ratio greater than 2. (**d**) The data are filtered based on a signal-to-noise ratio greater than 5.  
observed that the relationship maintains its stability well, and there is not much change in the relationship. **Magnitude-(Duration \(\times\) Distance) correlation**  
The results of the magnitude-duration correlation indicate a clear pattern: for each distinct magnitude, there exists a range of magnetic anomaly durations. The values within this range are influenced by the distance from the orbit to the earthquake epicenter. In essence, a shorter distance from the orbit to the epicenter corresponds to a longer duration of the observed anomaly, and conversely, a greater distance results in a shorter duration (refer to Figures S9 and S10 for illustrative examples). In Figure S6a, the correlation between the natural logarithm duration of the detected anomaly at a certain distance from the earthquake epicenter (y-axis) and the magnitude \(M\) (x-axis) can be clearly demonstrated. This relationship is given as:  
\[M=(0.587\pm 1.034)\times\ln(Du\times Dist)+(1.022\pm 1.168) \tag{4}\]  
Here \(Dist\) is the distance, i.e., the horizontal projected distance of the satellite anomaly with respect to the earthquake epicenter (in km). The R-square of the system here is equal to 0.99, which shows how robust is the relationship (3) and the consequence of distance on it. The error value in \(M\) is equal to 0.074, so it is practically negligible. In Figure S5b, the effect of geomagnetic indices on the system is investigated. Another test was performed on the system (Figure S6c,d), in which the signal-to-noise ratio for values higher than two (Figure S6c) and values higher than five (Figure S6d) is applied to the data. Compared to the general state of the data (Figure S6a), it can be observed that the system has maintained its stability. The error and stability values have changed very little (R-square = 0.98 and R-mse = 0.108). **Magnitude-(Duration \(\times\) Time) correlation**  
In the discussion of earthquake precursors, in addition to magnitude and distance, another important parameter is the time with respect to the earthquake occurrence. According to the obtained results, the anomalies that are close to the time of the earthquake are clearer and longer.