# A 2D Lithospheric Magnetic Anomaly Field over Egypt Using Gradient Data of Swarm Mission
###### Abstract  
The current work makes use of the geometrical configuration of the two lower-altitude Swarm satellites (Swarm A and C), moving side by side with a longitudinal distance of \(1.4^{\circ}\), to estimate a two-dimensional (2D) model of the lithospheric magnetic anomaly field over Egypt using gradient data. The gradient in both the north-south and the east-west directions have been inverted using the weighted damping least-squares fit technique to estimate the best model coefficients of the 2D model. The best model coefficients have been obtained under the expansion of the Legendre polynomial from degree \(n=7\) to \(n=56\). Results showed that the gradient of the field in the north-south direction is always much smoother than that in the east-west direction. The noise in the east-west direction is attributed to the different environmental conditions surrounding both satellites. The modeled field always showed smoother variations than the observed data, even for the horizontal components (\(B_{x}\) and \(B_{y}\)). Swarm satellites; inverse theory; lithospheric magnetic anomaly field; Legendre polynomial; Egypt +
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a spatial resolution of 300 km [18]. The CM model is derived from data collected from ground-based observatories and the Orsted, SAC-C, CHAMP, and Swarm satellites. The CM model calculates oceanic tidal magnetic sources, ionospheric quiet-time fields, and the induction fields of both the ionosphere and the magnetosphere, in addition to internal and external fields [9; 12; 13]. The lithospheric field is expanded up to spherical harmonic degree 120, corresponding to the resolution of 333 km [13]. The magnetic anomaly field with short-scale wavelength is not applicable to the global models because it requires a high degree of expansion that makes the computation intensive. Therefore, regional approaches have been used for modelling the lithospheric field, such as spherical cap harmonic analysis (SCHA) [19], 2D models, and rectangular harmonic analysis (RHA) [20; 21]. However, these models give accurate anomaly field data in a short-scale wavelength with a lower order than global models. There are shortcomings regarding the use of these local field models. The first problem with the SCHA model is the difficulty in separating the internal and external fields from each other [22] because both are expanded in terms of the potential function at the surface of the cap. Second, in order to reach a high spatial resolution, we must use a small cap (dense grid); subsequently, a large expansion must be implemented. Therefore the computation is very intense and requires a long time to reach convergence [23]. Third, the edge effect problem is considered a common issue in all local models that could be overcome by enlarging the area under study [24]. Additionally, RHAs produce unrealistic values when expanding non-cyclic data [25; 26]. This problem could be overcome through expanding the lithospheric field data to a polynomial [26]. The current work makes use of 8 years of magnetic data recorded by Swarm A and C to model the lithospheric magnetic anomaly field over Egypt and its surroundings, using a 2D model built on Legendre polynomials. Unlike [27], we use gradient magnetic data to obtain an accurate model because gradient data is less affected by external noise [17]. The results presented by [28] neglect the analysis of the lithospheric anomaly of the \(\mathbf{B}_{y}\) component because this component was very noisy, more than the \(\mathbf{B}_{x}\) component. Therefore, in the current work, we have tried to model the whole components using the gradient data from the lower-altitude Swarm satellites. The structure of the current work is as follows: Section 2 presents the data set and the selection criteria in addition to a theoretical discussion of the modelling techniques.