# The Accuracy Assessment of Lithospheric Density Models
## 2 Numerical Model
### Sub-Lithospheric Mantle Geoid
#### 2.2.1 Bouguer Disturbing Potential  
The Bouguer disturbing potential \(T^{\text{B}}\) is obtained from the disturbing potential \(T\) after subtracting the gravitational potentials of topography \(V^{T}\), bathymetry \(V^{B}\), and polar glaciers \(V^{I}\). We then write the following (cf. [PERSON] et al. [48; 50]):  
\[T^{\text{B}}=T-V^{T}-V^{B}-V^{I}. \tag{4}\]  
The gravitational potentials of atmosphere [80] and lakes [81] are completely negligible when compared to values of the gravitational potentials \(V^{T}\), \(V^{B}\), and \(V^{I}\). The topographic potential in Equation (4) is often computed individually for a uniform and anomalous topographic density. The topographic potential \(V^{T,\rho^{T}}\) for a uniform topographic density reads as follows [12]:  
\[V^{T,\rho^{T}}\big{(}r_{g},\Omega\big{)}=\frac{\mathrm{GM}}{\mathrm{R}}\sum_{n=0 }^{\overline{n}}\sum_{m=-n}^{n}\mathrm{V}_{n,m}^{T,\rho^{T}}\mathrm{Y}_{n,m}( \Omega). \tag{5}\]  
The potential coefficients \(\mathrm{V}_{n,m}^{T,\rho^{T}}\) in Equation (5) are given by the following:  
\[\mathrm{V}_{n,m}^{T,\rho^{T}}=\frac{3}{2n+1}\ \frac{\rho^{T}}{\mathrm{p}^{ Earth}}\sum_{k=0}^{n+2}\binom{n+2}{k}\frac{1}{k+1}\frac{\mathrm{H}_{n,m}^{ (k+1)}}{\mathrm{R}^{k+1}}, \tag{6}\]  
where \(\overline{\rho}^{\mathrm{Earth}}=5500\) kg\(\cdot\)m\({}^{-3}\) is the Earth's mean mass density, and \(\rho^{T}\) is the average (constant) topographic density. The Laplace harmonics \(\mathrm{H}_{n}\) of topographic heights \(H\) are defined by the following integral convolution:  
\[H(\Omega)=\sum_{n=0}^{\infty}\mathrm{H}_{n}(\Omega),\]  
\[\mathrm{H}_{n}(\Omega)=\frac{2n+1}{4\pi}\iint\limits_{\Phi}\rho^{T}H^{\prime} \,\mathrm{P}_{n}(t)\ d\Omega^{\prime}=\sum_{m=-n}^{n}\mathrm{H}_{n,m}\, \mathrm{Y}_{n,m}(\Omega), \tag{7}\]  
where \(\mathrm{H}_{n,m}\) are the topographic coefficients, \(\mathrm{P}_{n}\) is the Legendre polynomial for the argument \(t\) of cosine of the spherical angle \(\psi\) between two points \((r,\Omega)\) and \((r^{\prime},\Omega^{\prime})\); i.e., \(t=\cos\psi\). The infinitesimal surface element on the unit sphere is denoted as \(d\Omega^{\prime}=\cos\varphi^{\prime}\,d\varphi^{\prime}\,d\lambda^{\prime}\), and \(\Phi=\{\,\Omega^{\prime}=(\varphi^{\prime},\lambda^{\prime}):\varphi^{\prime} \in[-\pi/2,\,\pi/2]\wedge\lambda^{\prime}\in[0,\,2\pi)\ \}\) is the full spatial angle. The corresponding higher-order terms \(\{\mathrm{H}_{n,m}^{(k)}:\ k=2,\,3,\ldots\}\) read as follows:  
\[\mathrm{H}_{n}^{(k)}(\Omega)=\frac{2n+1}{4\pi}\iint\limits_{\Phi}H^{\prime k} \,\mathrm{P}_{n}(t)\ d\Omega^{\prime}=\sum_{m=-n}^{n}\mathrm{H}_{n,m}^{(k)} \,\mathrm{Y}_{n,m}(\Omega). \tag{8}\]  
The topographic potential \(V_{g}^{T,\delta\rho^{T}}\) for an anomalous lateral topographic density is computed from the following:  
\[V^{T,\delta\rho^{T}}\big{(}r_{g},\Omega\big{)}=\frac{\mathrm{GM}}{\mathrm{R}} \sum_{n=0}^{\overline{n}}\ \sum_{m=-n}^{n}\ \mathrm{V}_{n,m}^{T,\delta\rho^{T}}\mathrm{Y}_{n,m}( \Omega), \tag{9}\]  
where  
\[\mathrm{V}_{n,m}^{T,\delta\rho^{T}}=\frac{3}{2n+1}\frac{1}{\overline{\rho}^{ \mathrm{Earth}}}\sum_{k=0}^{n+2}\binom{n+2}{k}\frac{1}{k+1}\frac{\delta\rho^{T }\mathrm{K}_{n,m}^{(k+1)}}{\mathrm{R}^{k+1}}. \tag{10}\]  
The topographic density-heights coefficients \(\{\delta\rho^{T}\mathrm{K}_{n}^{(k)}:\ k=1,2,\,3,\ldots\}\) in Equation (10) are given by the following:  
\[\delta\rho^{T}\mathrm{K}_{n}^{(k)}(\Omega)=\frac{2n+1}{4\pi}\iint\limits_{ \Phi}\delta\rho^{T}\big{(}\Omega^{\prime}\big{)}\,K^{\prime k}\,\mathrm{P}_{ n}(t)\ d\Omega^{\prime}=\sum_{m=-n}^{n}\delta\rho^{T}\mathrm{K}_{n,m}^{(k)}\, \mathrm{Y}_{n,m}(\Omega). \tag{11}\]  
The height \(K\) in Equation (11) equals the topographic height \(H\), except for areas covered by polar glaciers. In this case, the height \(K\) is defined as the topographic height minus the ice thickness. The anomalous lateral topographic density \(\delta\rho^{T}\) in Equation (11) is taken with respect to the average topographic density \(\rho^{T}\), so that \(\delta\rho^{T}(\Omega)=\rho^{T}-\rho^{T}(\Omega)\).