# The Accuracy Assessment of Lithospheric Density Models
## 5 Error Analysis
Adopting the above error estimates of \(\varepsilon_{\delta\rho^{L/\Lambda}}\approx\pm\)100 kg\(\cdot\)m\({}^{-3}\), \(\varepsilon_{\Delta\rho^{c/\mathrm{m}}}\approx\pm\)15 kg\(\cdot\)m\({}^{-3}\), and \(\varepsilon_{\Delta\rho^{c/\mathrm{m}}}\approx\pm\)50 kg\(\cdot\)m\({}^{-3}\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Differences** & **Min [km]** & **Max [km]** & **Mean [km]** & **RMS [km]** \\ \hline SLNAAFSA–SL2013 sv & \(-\)237 & 303 & 3 & 11 \\ SLNAAFSA–LITHO1.0 & \(-\)196 & 289 & 1 & 77 \\ SLNAAFSA–CAM2016 & \(-\)160 & 307 & 4 & 26 \\ SLNAAFSA–3D2015-07 Sv & \(-\)638 & 309 & 5 & 26 \\ SL2013 sv–LITHO1.0 & \(-\)195 & 298 & \(-\)1 & 74 \\ SL2013 sv–CAM2016 & \(-\)160 & 306 & 1 & 22 \\ SL2013 sv–3D2015-07 Sv & \(-\)637 & 303 & 3 & 24 \\ LITHO1.0–CAM2016 & \(-\)184 & 206 & 2 & 71 \\ LITHO1.0–3D2015-07 Sv & \(-\)603 & 206 & 4 & 74 \\ CAM2016-3D2015-07 Sv & \(-\)561 & 195 & 2 & 18 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistics of differences between the lithospheric thickness models. and considering the lithospheric thickness \(L\)\(\sim\)100 km and the Moho depth \(M\)\(\sim\)30 km, the geoid modeling error \(\varepsilon_{\rm{\it{NSM}}}\) could, according to Equation (51), reach or even exceed \(\pm\)0.5 km. To confirm the above theoretical estimates, we rearranged the error propagation in Equation (51) into a form that directly incorporates density uncertainties within the uppermost asthenosphere and the lowermost crust. Inserting for \(\Delta\rho^{\rm{c/m}}=\rho^{\rm{LM}}-\rho^{\rm{c}}\), \(\delta\rho^{\rm{LM}}=\rho^{\rm{LM}}-\rho^{\rm{LM}}\), and \(\Delta\rho^{\rm{L/A}}=\rho^{\rm{A}}-\rho^{\rm{LM}}\) in Equation (51), we obtain the following:  
\[\varepsilon_{\rm{\it{NSM}}}\approx-\frac{3}{4\pi\overline{\rho}^{\rm{ Earth}}}\Big{[}\Big{(}\varepsilon_{\rho^{\rm{LM}}}-\varepsilon_{\rho^{\rm{c}}} \Big{)}M+\Big{(}\varepsilon_{\rho^{\rm{LM}}}-\varepsilon_{\rho^{\rm{LM}}} \Big{)}(L-M)+\Big{(}\varepsilon_{\rho^{\rm{A}}}-\varepsilon_{\rho^{\rm{LM}}} \Big{)}L\Big{]}\]  
\[\approx-\frac{3}{4\pi\overline{\rho}^{\rm{Earth}}}\Big{[}\varepsilon_{\rho^{ \rm{LM}}}M-\varepsilon_{\rho^{\rm{c}}}M+\varepsilon_{\rho^{\rm{LM}}}L- \varepsilon_{\rho^{\rm{LM}}}M-\varepsilon_{\rho^{\rm{LM}}}(L-M)+\varepsilon _{\rho^{\rm{A}}}L-\varepsilon_{\rho^{\rm{LM}}}L\Big{]}\]  
\[\approx\frac{3}{4\pi\overline{\rho}^{\rm{Earth}}}\Big{[}\varepsilon_{\rho^{ \rm{c}}}M+\varepsilon_{\rho^{\rm{LM}}}(L-M)+\varepsilon_{\rho^{\rm{A}}}L\Big{]}, \tag{52}\]  
where \(\varepsilon_{\rho^{\rm{c}}}\) and \(\varepsilon_{\rho^{\rm{A}}}\) are errors in the average crustal \(\rho^{\rm{c}}\) and asthenospheric \(\rho^{\rm{A}}\) densities, respectively. As seen in Equation (52), the modeling error \(\varepsilon_{\rm{\it{NSM}}}\) depends on the accuracy of the lithospheric mantle density distribution as well as the adopted average densities of the crust and the asthenosphere, while errors in the adopted value of the average lithospheric mantle density \(\rho^{\rm{LM}}\) cancel. [PERSON] and [PERSON] [111] reported an average value of 2835 kg\(\cdot\)m\({}^{-3}\) for the continental crustal density. [PERSON] et al. [50] provided a very similar value of 2790 kg\(\cdot\)m\({}^{-3}\). The oceanic crust, composed primarily of mafic rocks, is typically heavier than the continental crust [118]. [PERSON] et al. [50] estimated that the average density of the oceanic crust (without marine sediments) is 2860 kg\(\cdot\)m\({}^{-3}\). [PERSON] and [PERSON] [119] provided an estimate of 2890 kg\(\cdot\)m\({}^{-3}\). [PERSON] and [PERSON] [92] confirmed a similar value (when disregarding marine sediments) of 2900 kg\(\cdot\)m\({}^{-3}\) based on the analysis of global samples of marine bedrock densities. If we focus only on the lowermost crustal density uncertainties, the density according to CRUST1.0 varies from 2850 kg\(\cdot\)m\({}^{-3}\) under the Himalayas to 3050 kg\(\cdot\)m\({}^{-3}\) under the oceans, with a standard deviation of \(\sim\)70 kg\(\cdot\)m\({}^{-3}\).