# Regional Seismicity of the Northeastern Tibetan Plateau Revealed by Crustal Magnetic Anomalies
## 4 Methodology  
### Data Sources  
In this study, crustal magnetic anomaly data were extracted from the EMM2017 model. The model was compiled from satellite, marine, aeromagnetic, and ground magnetic surveys. EMM2017 is a hybrid model where the core field is represented by spherical harmonics up to degree 15, while the crustal field is represented by spherical harmonics from degrees 16 to 790. The core field was based on the Swarm satellite magnetic data. The crustal field was derived from the latest (2017) release of EMAG2-v3, the Earth Magnetic Anomaly Grid at 2-arcminute resolution, version 3. By relying only on observations rather than using prior geological structures or ocean age information for constructing the underlying EMAG2-v3, EMM2017 can obtain more accurate information of magnetic anomalies (EMM2015, 2016). It is of a higher spatial resolution (51 km) than previous models (e.g., EMM2015, 56 km). Conventional spherical harmonic (SH) modeling approaches exhibit two principal constraints in achieving enhanced spatial resolution: (i) inherent limitations in terrestrial station distribution density and spatial homogeneity; (ii) prohibitive computational demands associated with high-degree expansions. Theoretical studies demonstrate that SH models truncated at degree 720 (N_max = 720) yield a theoretical spatial resolution threshold of 56 km (equivalent to \(\pi\)a/N_max, where a denotes Earth's radius), requiring the inversion of 519,840 independent Gauss coefficients. Extending the resolution to 5 km scales mandates N_max \(\approx\) 8000, yielding coefficient matrices comprising approximately 6.4 \(\times\) 10\({}^{7}\) elements--a configuration inducing prohibitively high computational overhead. In response to these constraints, regional modeling techniques (such as rectangular harmonic analysis, spherical cap harmonics, and equivalent source dipole inversion) have been developed. Notwithstanding these inherent limitations, a global SH model (e.g., EMM2017) maintains critical scientific utility through their mature algorithmic framework for assimilating satellite, marine, aeromagnetic, and ground magnetic surveys, while it can be evaluated at any desired location to provide the magnetic field vector, its direction, and its magnitude. Figure 3: Focal depth of earthquakes in the study areas of Ms \(\geq\) 5.0 since 1900: (a) proportion of earthquake occurrence in different focal depth sections; (b) distribution of earthquakes with different magnitudes in focal depth. Given that the EMM2017 model was established using spherical harmonic analysis theory, a convenient way of representing geomagnetic fields is to expand the scalar magnetic potential into spherical harmonic functions [7]:  
\[U(r,\theta,\lambda)=a\sum_{n=1}^{\infty}\sum_{m=0}^{n}{\left(\frac{a}{r}\right)^ {(n+1)}}(g_{n}^{m}\cos m\lambda+h_{n}^{m}\sin m\lambda)P_{n}^{m}(\cos\theta), \tag{1}\]  
where \(\lambda\) and \(\theta\) are the longitude and colatitude, respectively; \(a\) is the Earth's radius (6371.2 km); \(P_{n}^{m}(\cos\theta)\) is the Schmidt quasi-normalized associated Legendre function of degree \(n\) and order \(m;g_{n}^{m}\) and \(h_{n}^{m}\) are spherical harmonic coefficients; and \(N\) is the truncation level. In this study, \(N\) was set to degree 790, and the grid resolution was 0.1\({}^{\circ}\)\(\times\) 0.1\({}^{\circ}\). Three components of the magnetic field in rectangular coordinates (i.e., northward \(X\), eastward \(Y\), and vertical \(Z\)) can be obtained through differentiation (\(X=\partial U/r\theta\theta\), \(Y=-\partial U/r\sin\theta\partial\lambda\), and \(Z=\partial U/\partial r\)).