# Combination Analysis of Future Polar-Type Gravity Mission and GRACE Follow-On
## 2 Numerical Simulation Scheme  
### Mathematical Model for Gravity Field Recovery  
The dynamic approach is adopted in our gravity field model recovery [43], which is widely used in real GRACE data processing by different scientific institutions [7; 8; 44; 45; 46]. In this section, we outline basic formulas of the dynamic approach used in the article, as well as relevant references for more detailed descriptions. The Earth gravity field model can be expressed via a set of Spherical Harmonic Coefficients (SHC) \(\mathbf{u}=\left\{\overline{C}_{lm},\ \overline{S}_{lm}\right\}\) with following Spherical Harmonic (SH) expansion,  
\[V(\rho,\varphi,\lambda;\mathbf{u})=\frac{GM}{\rho}\left[1+\sum_{l=2}^{L_{MAX}}\sum_ {m=0}^{l}\left(\frac{R_{E}}{\rho}\right)^{l}\left(\overline{C}_{lm}\cos m \lambda+\overline{S}_{lm}\sin m\lambda\right)\cdot\overline{P}_{lm}(\sin\varphi )\right] \tag{1}\]  
where \(V\) is the gravitational potential of an arbitrary point on or outside the Earth with the spherical coordinate of radius \(\rho\), latitude \(\varphi\) and longitude \(\lambda\), in the Earth-fixed frame. \(GM\) is the product of the gravitational constant and the Earth mass, and \(R_{E}\) is the Earth's mean equatorial radius; \(l\) and \(m\) stand for the degree and order of SH expansion, respectively, with \(L_{MAX}\) being the maximum degree of the expansion; \(\overline{P}_{lm}\) is the fully normalized associate Legendre function of degree \(l\) and order \(m\). The estimation of SHC \(\mathbf{u}\) from satellite observations is based on the [PERSON]'s law of motion [47], which is formulated as a three-dimensional second-order nonlinear ordinary differential equation (ODE),  
\[\ddot{\mathbf{r}}(t)=\mathbf{a}(t;\mathbf{r}(t),\dot{\mathbf{r}}(t),\mathbf{p},\mathbf{u}) \tag{2}\]  
where \(\mathbf{r}(t),\dot{\mathbf{r}}(t)\), and \(\ddot{\mathbf{r}}(t)\) denote the vectors of satellite's position, velocity and acceleration at epoch \(t\), respectively. \(\mathbf{a}\) is the force acting on the unit mass of the satellite, where \(\mathbf{p}\) represents the parameters vector for the force models except the Earth gravity field. In the case of GRACE, \(\mathbf{p}\) is referred to accelerometer calibration parameters (e.g., bias and scale parameters) [48]. When the force models are given, the position and velocity of satellite's orbit are determined as follows [49],  
\[\left\{ \begin{array}{l}\mathbf{r}(t)=\mathbf{r}(t_{0})+(t-t_{0})\cdot\dot{\mathbf{r}}(t_ {0})+\int_{t_{0}}^{t}\int_{t_{0}}^{\tau}\mathbf{a}(\tau^{\prime};r(\tau^{\prime}), \mathbf{r}(\tau^{\prime}),\mathbf{p},\mathbf{u})d\tau^{\prime}d\tau\\ \dot{\mathbf{r}}(t)=\dot{\mathbf{r}}(t_{0})+\int_{t_{0}}^{t}\mathbf{a}(\tau;\dot{\mathbf{r}}( \tau),\mathbf{r}(\tau),\mathbf{p},\mathbf{u})d\tau\end{array} \right. \tag{3}\]  
where \(\left(\mathbf{r}^{T}(t_{0}),\dot{\mathbf{r}}^{T}(t_{0})\right)^{T}\) denote the position and velocity vectors at initial epoch \(t_{0}\) of an orbit arc. According to Equation (3), the position and velocity vectors at any epoch \(t\) within the orbit arc are the function of parameter vector \(\mathbf{x}=\left(\mathbf{w}^{T},\mathbf{u}^{T}\right)^{T}\), where \(\mathbf{w}=\left(\mathbf{r}^{T}(t_{0}),\dot{\mathbf{r}}^{T}(t_{0}),\mathbf{p}^{T}\right)^{T}\) is the arc-specific parameters vector. Linearizing the Equation (3) with respect to the approximate vector \(\mathbf{x}_{0}\) of parameters, we have,  
\[\left\{ \begin{array}{l}\mathbf{r}(t)=\mathbf{r}^{REF}(t)+\frac{\partial\mathbf{r}^{REF} (t)}{\partial\mathbf{x}}\delta\mathbf{x}\\ \dot{\mathbf{r}}(t)=\dot{\mathbf{r}}^{REF}(t)+\frac{\partial\mathbf{r}^{REF}(t)}{\partial \mathbf{x}_{0}}\delta\mathbf{x}\end{array} \right. \tag{4}\]  
where the vectors of position \(\mathbf{r}^{REF}(t)\) and velocity \(\dot{\mathbf{r}}^{REF}(t)\), named as the reference orbit, are numerically integrated with Equation (3) by replacing \(\mathbf{x}\) with \(\mathbf{x}_{0}\). \(\delta\mathbf{x}\) is the correction vector to \(\mathbf{x}_{0}\).