## 3 Inversion Results and Stability  
### Electrical Conductivity Model  
The C-responses (Figure 2) of the 50 stations in the study area were inverted using a 3D \(L_{1}\)-norm GDS method in which the data misfit and model roughness were measured by \(L_{1}\)- and \(L_{2}\)-norm measurements, respectively. Stations close to the research region were also considered in our inversion in order to reduce the influence of the electrical structures of the surrounding mantle. The data misfit in inversion is normalized using the observed data error derived from the estimated responses, which will be helpful to lessen the weight of these data in inversion. In traditional \(L_{2}\)-norm inversion, data with large uncertainties are usually excluded, but in \(L_{1}\)-norm inversion, the contributions of unstable data are suppressed by \(L_{1}\)-norm measurement and the large data error in normalization. As a result, it is anticipated that we will be able to determine a reliable 3D electrical structure of the mantle beneath North Africa and Southern Europe. A homogeneous grid is densified as fine as \(5^{\circ}\times 5^{\circ}\) horizontally in order to discretize the conductive Earth, and the Earth is divided into 12 spherical layers vertically. The conductivity of the initial model for our 3D \(L_{1}\)-norm inversion adopts the global average 1D model obtained from the geomagnetic data observed in satellites [39] and depicted in Figure 4. Considering the mineral phase transition interfaces in the mantle, conductivities are allowed to jump at 410 km, 520 km, and 670 km. Figure 4: The one-dimensional (1D) initial model used in this paper. The initial regularization parameter of the 3D inversion is set at 1.0. During iterations in which the deviation of the data misfit is below the given threshold value, the regularization parameter is cut in half. The inversion process continues iteratively until the given target data misfit is arrived at or the regularization parameter is rather small. The inversion was conducted with several different regularization parameters in order to better access the stability of the detected anomalies. The results of these inversions demonstrate that the distribution of anomaly is almost similar, except that the sharpness at boundaries is different. After 66 iterations, the inversion terminates when the regularization parameter is smaller than \(10^{-4}\). The root-mean-square (RMS) error of the data misfit is 1.96 (Figure 5), which is rather larger than the expected value of 1.0. We suggest that this large value is caused by some unreliable responses included in the inversion. This indicates that most of the responses calculated from the inverted model fit the observed data well. The curves in Figure 5 changed slightly when the inversion came to an end, indicating that the inversion result was reliable and that the process was stable. In Figure 6, we show the data fitting curves of the observed data and inverted responses in some stations. The curves show a good consistency for most of the periods of responses, indicating a convincing result of our inversion. Figure 7 displays the preferred 3D electrical conductivity model near the MTZ (250-900 km) beneath North Africa and Southern Europe. The most attractive feature (anomaly A) is the obviously reduced conductivity area in the lower MTZ and the topmost lower mantle. The electrical structure is similar to that in previous studies [3; 17; 21] but with a better resolution, and the western portion of the anomaly coincides with the results of [PERSON] et al. [22]. The western portion of the anomaly banding the area extending from the Mediterranean Sea to the Red Sea is named Anomaly A, and the eastern portion spreading from the Red Sea to the Caspian Sea is named Anomaly B. It should be noted that Anomaly A only started from the Eastern Mediterranean in the topmost lower mantle. The strongest variation of anomalies is exhibited beneath Greece in the lower mantle, with conductivity falling to around 1/3 of the global average (Figure 8).