# A 2D Lithospheric Magnetic Anomaly Field over Egypt Using Gradient Data of Swarm Mission
## 3 Results and Discussions  
Figure 5 shows the standard deviation of the N-S gradient \(\
abla\mathbf{B}_{t}\) field components. The standard deviation data binned to regions of the surface area of 2.5 degrees. For each binned area, the standard deviation has been calculated as shown in Figure 5 for the \(\
abla\mathbf{B}_{t}\) component. It shows regions of higher standard deviation corresponding to large errors in the data. The same calculations have been performed for the other vector field components, \(\
abla\mathbf{B}_{x}\), \(\
abla\mathbf{B}_{y}\), and \(\
abla\mathbf{B}_{z}\), respectively. According to Equation (4), each data point located within a specific binned region has been divided by its corresponding standard deviation. This process has been implemented by several users to reduce the effect of outlier points [34; 35]. Figure 4: The power spectrum of the model with respect to the degree (\(n\)), (**a**) in a linear scale and (**b**) in a logarithmic scale. The non-gradient lithospheric magnetic field data presented by [27] is more affected by external sources/noise. It exhibited N-S stripes in the X-component, especially at higher altitudes. To reduce this noise, we used gradient data from a Swarm satellite [36]. Figure 6 shows the E-W gradient data calculated from Swarm A and C. Rows from top to bottom correspond to the \(\
abla\mathbf{B}_{x}\), \(\
abla\mathbf{B}_{y}\), \(\
abla\mathbf{B}_{z}\), and \(\
abla\mathbf{B}_{t}\) components, respectively. Columns from left to right correspond to observed, modeled, and weighted modeled gradient data, respectively. The observed \(\
abla\mathbf{B}_{x}\) and \(\
abla\mathbf{B}_{y}\) are noisier than \(\
abla\mathbf{B}_{z}\) and \(\
abla\mathbf{B}_{t}\). The noise in the horizontal components (\(\
abla\mathbf{B}_{x}\) and \(\
abla\mathbf{B}_{y}\)) could be attributed to external sources. In addition, we can interpret it in terms of the sensitivity of the data sets onboard the Swarm A and Swarm C satellites. In order to reduce this noise, we have implemented the weighted damped least-squares technique as shown in the most right column of Figure 6. It is worth noting that the magnetic anomaly in the \(\mathbf{B}_{y}\) component was excluded from [27] because the non-gradient data are strongly affected by external and lithospheric activities (e.g., Earthquakes, ionospheric dynamos, etc.). However, using the gradient data is a powerful technique in removing external noise in comparison with [27], which is strongly affected by external ionospheric and magnetospheric noise. The weighted damping least-squares fit technique of the gradient data has the capability to emphasize a lithospheric magnetic anomaly in a much smoother form, especially in the \(\mathbf{B}_{x}\) and \(\mathbf{B}_{y}\) components. The weighted damped modeled data in the \(\mathbf{B}_{x}\) and \(\mathbf{B}_{y}\) components illustrated magnetic anomalies that could not clearly be seen in the observed gradient data in most left panels. The most clearly enhanced anomalies under the weighted damping technique are shown in the northwest and in the middle of the map of the \(\mathbf{B}_{y}\) component. The field strength of these regions is enhanced under the action of the damping technique in comparison with the observed lithospheric gradient data. Figure 3 shows the diagonal elements of the resolution matrix R with respect to the parameter numbers of \(\
abla\mathbf{B}_{t}\) of the N-S gradient data at the best damping parameter (\(10^{-4}\)). It shows that the diagonal element of the resolution matrix approaches unity, which demonstrates the great description of the model for the data. The increase of R at higher orders could be interpreted in terms of the dominance of the noise at higher orders of expansion [27; 33]. Figure 5: The standard deviation of the N–S gradient \(\
abla\mathbf{B}_{t}\) field component binned to regions of the surface area of 2.5 degrees.