# The Accuracy Assessment of Lithospheric Density Models
## 2 Numerical Model
### Sub-Lithospheric Mantle Geoid
#### 2.2.2 Crust-Stripped Disturbing Potential  
The gravitational potential of consolidated-crust density contrast \(V^{C}\) and the gravitational potential of sediment density contrast \(V^{S}\) are subtracted from the Bouguer disturbing potential \(T^{B}\) in order to remove the gravitational contribution of density heterogeneities within the whole crust. This procedure yields the crust-stripped disturbing potential \(T^{CS}\) (cf. [PERSON] et al. [48]):  
\[T^{CS}=T^{B}-V^{S}-V^{C}. \tag{23}\]  
The potentials \(V^{S}\) and \(V^{C}\) in Equation (23) are computed according to a method developed by [PERSON] et al. [82] that utilizes the expression for a gravitational potential \(V\) generated by an arbitrary volumetric mass layer with a variable depth and thickness while having laterally distributed vertical mass density variations. It reads as follows:  
\[V\big{(}r_{g},\Omega\big{)}=\frac{\rm GM}{R}\sum_{n=0}^{\pi}\sum_{m=-n}^{n}V_ {n,m}\,Y_{n,m}(\Omega). \tag{24}\]  
The potential coefficients \(\rm V_{n,m}\) in Equation (24) are defined by the following  
\[\rm V_{n,m}=\frac{3}{2n+1}\frac{1}{\overline{p}^{Earth}}\sum_{i=0}^{l}\left( Fl_{n,m}^{(i)}-Fu_{n,m}^{(i)}\right), \tag{25}\]  
where the coefficients \(\rm\{Fl_{n,m}^{(i)},Fu_{n,m}^{(i)}:\ i=0,\ 1,\ \ldots,\ I\}\) are computed from the following:  
\[\rm Fl_{n,m}^{(i)}=\sum_{k=0}^{n+2}\binom{n+2}{k}\frac{(-1)^{k}}{k+1+i}\frac{ L_{n,m}^{(k+1+i)}}{R^{k+1}},\]  
\[\rm Fu_{n,m}^{(i)}=\sum_{k=0}^{n+2}\binom{n+2}{k}\frac{(-1)^{k}}{k+1+i}\frac{ U_{n,m}^{(k+1+i)}}{R^{k+1}}. \tag{26}\]The terms \(\sum\limits_{\rm m=-n}^{\rm n}{\rm L_{n,m}\,Y_{n,m}}\) and \(\sum\limits_{\rm m=-n}^{\rm n}{\rm U_{n,m}\,Y_{n,m}}\) in Equation (26) define the spherical lower-bound and upper-bound laterally distributed radial density variation functions \({\rm L_{n}}\) and \({\rm U_{n}}\) of degree \(n\). These numerical coefficients combine information on the geometry and mass density (or density contrast) distribution within a volumetric layer. The computation of these coefficients is realized to a certain degree in spherical harmonics from discrete data of the spatial mass density distribution (typically provided by means of density, depth, and thickness data) of a particular structural component of the Earth's interior. The spherical functions \({\rm L_{n}}\) and \({\rm U_{n}}\), including their higher-order terms \(\{{\rm L_{n}^{(k+1+i)}},{\rm U_{n}^{(k+1+i)}}:\)\(k=0,\,1,\,\ldots\,;\,\,i=1,\,2,\,\,\ldots\,\,I\}\) in Equation (26), are defined by the following:  
\[{\rm L_{n}^{(k+1+i)}}(\Omega)=\left\{ \begin{array}{l}\frac{2n+1}{4\pi} \iint\limits_{\rm\Phi}\rho(D_{U},\Omega^{\prime})\,D_{\rm L}^{k+1}(\Omega^{ \prime}){\rm P_{n}}(t)\ d\Omega^{\prime}\\ =\sum\limits_{\rm m=-n}^{\rm n}{\rm L_{n,m}^{(k+1)}Y_{n,m}}(\Omega)\qquad i=0 \\ \\ \frac{2n+1}{4\pi}\iint\limits_{\rm\Phi}\beta(\Omega^{\prime})\ \alpha_{i}(\Omega^{ \prime})\,D_{\rm L}^{k+1+i}(\Omega^{\prime}){\rm P_{n}}(t)\ d\Omega^{\prime}\\ =\sum\limits_{\rm m=-n}^{\rm n}{\rm L_{n,m}^{(k+1+i)}Y_{n,m}}(\Omega)\qquad i= 1,\,2,\,\ldots,\,I\end{array} \right. \tag{27}\]  
and  
\[{\rm U_{n}^{(k+1+i)}}(\Omega)=\left\{ \begin{array}{l}\frac{2n+1}{4\pi} \iint\limits_{\rm\Phi}\rho(D_{U},\Omega^{\prime})\,D_{U}^{k+1}(\Omega^{\prime }){\rm P_{n}}(t)\ d\Omega^{\prime}\\ =\sum\limits_{\rm m=-n}^{\rm n}{\rm U_{n,m}^{(k+1)}Y_{n,m}}(\Omega)\qquad i=0 \\ \\ \frac{2n+1}{4\pi}\iint\limits_{\rm\Phi}\beta(\Omega^{\prime})\ \alpha_{i}(\Omega^{ \prime})\,D_{U}^{k+1+i}(\Omega^{\prime}){\rm P_{n}}(t)\ d\Omega^{\prime}\\ =\sum\limits_{\rm m=-n}^{\rm n}{\rm U_{n,m}^{(k+1+i)}Y_{n,m}}(\Omega)\qquad i= 1,\,2,\,\ldots,\,I\end{array} \right.