# Chinese Geomagnetic Reference Field 2020 by the Revised Surface Spline Method
## 2 Materials and Methods  
### Revised Surface Spline Method  
The surface spline method was first applied in 1987 [34]. It can be expressed by Equation (1).  
\[\left\{\begin{array}{c}P(x,y)=a_{0}+a_{1}x+a_{2}y+\sum\limits_{i=1}^{N}A_{i }r_{i}^{2}\ln\left(r_{i}^{2}+\varepsilon\right)\\ r_{i}^{2}=\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}\\ \sum\limits_{i=1}^{N}A_{i}=\sum\limits_{i=1}^{N}x_{i}A_{i}=\sum\limits_{i=1}^{ N}y_{i}A_{i}=0\end{array}\right. \tag{1}\]  
Here, \(x\) and \(y\) are the geographical longitude and latitude of the calculated position, respectively; \(x_{i}\) and \(y_{i}\) are the geographic longitude and latitude of the \(i\)th spatial position, respectively; \(P(x,y)\) is the value of an element of the geomagnetic field at position \((x,y)\); \(a_{0}\), \(a_{1}\), \(a_{2}\), and \(A_{i}\) are the fitting coefficients of the surface spline function; \(N\) is the amount of original data; \(r_{i}\) is the distance between the calculated position and the \(i\)th spatial position; and \(\varepsilon\) is the curvature factor and is usually \(10^{-7}\).  
From Equation (1), the traditional surface spline fitting formula consists of two parts. The first part is 1-order polynomials, and the second part includes some U-shaped waves centered on the base points. Because the base points are the field repeat stations in the traditional surface spline method, the geographical longitudes and latitudes of field repeat stations should be input when calculating geomagnetic data. Therefore, in practical use, we should not only know the fitting coefficients but also know the longitudes and latitudes of the repeat stations. In addition, the temporal characteristics of the geomagnetic field cannot be obtained by analyzing the time series of coefficients, due to the different centers of the basis function corresponding to the coefficients. This makes it inconvenient to extend and apply the surface spline method.  
So, we redefine the centers of the U-shaped waves as a grid lattice distribution position in the second part. In addition, we increase the truncation order of polynomials in the first part to describe a more complex trend distribution of the geomagnetic field. Then the revised surface spline method can be expressed as:  
\[\left\{\begin{array}{c}P(x,y)=\sum\limits_{i=0}^{K}\sum\limits_{j=0}^{i}a_{ ij}x^{i-j}y^{j}+\sum\limits_{i=1}^{N}A_{i}r_{i}^{2}\ln\left(r_{i}^{2}+\epsilon\right) \\ r_{i}^{2}=\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}\\ \sum\limits_{i=1}^{N}A_{i}=\sum\limits_{i=1}^{N}x_{i}A_{i}=\sum\limits_{i=1}^{ N}y_{i}A_{i}=\ldots=\sum\limits_{i=1}^{N}x_{i}y_{i}^{K-1}A_{i}=\sum\limits_{i=1}^{N}y_{ i}^{K}A_{i}=0\end{array}\right. \tag{2}\]  
Here, \(K\) is the truncation order of polynomials; \(a_{ij}\) and \(A_{i}\) are the fitting coefficients of the revised surface spline function; \(x\) and \(y\) are the geographical longitude and latitude of the calculated position, respectively; \(x_{i}\) and \(y_{i}\) are the geographic longitude and latitude of the \(i\)th grid lattice distribution position, respectively; \(P(x,y)\) is the value of an element of the geomagnetic field at position \((x,y)\); \(N\) is the number of the grid lattice distribution positions; \(r_{i}\) is the distance between the calculated position and the \(i\)th grid position; and \(\epsilon\) is the curvature factor and usually is also \(10^{-7}\).