# The Accuracy Assessment of Lithospheric Density Models
## 5 Error Analysis
[PERSON] and [PERSON] [111] provided an uncertainty of \(\pm\)50 kg\(\cdot\)m\({}^{-3}\) for the crystalline crust that could be reduced to \(\sim\)\(\pm\)30 kg\(\cdot\)m\({}^{-3}\) when considering only the lowermost crust. The error \(\varepsilon_{\rho^{\rm{c}}}\) of \(\pm\)50 kg\(\cdot\)m\({}^{-3}\) could then be a realistic expectation for an average density within the lowermost crust. Density variations within the asthenosphere are probably controlled by a mantle flow pattern. Large density variations attributed to mantle plumes, for instance, were detected under Iceland, Hawaii, the Azores, and elsewhere [120]. Seismic tomography results also indicate relatively large density heterogeneities and density discontinuities in the mantle transition zone at depths approximately between 410 and 660 km [121; 122]. Based on petrological evidence, [PERSON] et al. [123] reported density variations of 3300-3400 kg\(\cdot\)m\({}^{-3}\) in the asthenosphere. From these estimates, we could assume that the error \(\varepsilon_{\rho^{\rm{A}}}\) in the average asthenosphere density is roughly \(\pm\)30 kg\(\cdot\)m\({}^{-3}\). Assigning these density uncertainties to Equation (52), the geoid modeling errors \(\varepsilon_{\rm{\it{NSM}}}\) within \(\pm\)0.6 km closely correspond with the estimated value of \(\pm\)0.5 km according to Equation (51). To check the correctness of our initial assumption that the lithospheric mantle density uncertainties \(\varepsilon_{\rho^{\rm{LM}}}\) are mostly within \(\pm\)100 kg\(\cdot\)m\({}^{-3}\), we used differences between the lithospheric thickness models (in Table 4) to reversely estimate errors of lithospheric mantle density. The computation was performed approximately by using a simple relation for the gravitational potential \(V\) of an infinite planar plate that is computed from the density \(\rho\) and thickness \(s\) values as follows:  
\[V\approx 2\pi\,GR\rho\,s. \tag{53}\]The error propagation model for Equation (53) reads as follows:  
\[\varepsilon_{V}\approx 2\pi\,GR\left(s\,\varepsilon_{\rho}+\rho\,\varepsilon_{s} \right), \tag{54}\]  
where \(\varepsilon_{\rho}\) is a density uncertainty, and \(\varepsilon_{s}\) is a thickness uncertainty. Since we here inspect the propagation only between errors \(\varepsilon_{\beta\rho\text{{{LM}}}}\) and \(\varepsilon_{\text{{NSM}}}\) (or equivalently \(\varepsilon_{V}\)), the error \(\varepsilon_{s}\) in Equation (54) is disregarded. We then write the following:  
\[\varepsilon_{V}\approx 2\pi\,GR\,\,s\,\varepsilon_{\rho}. \tag{55}\]  
Defining the potential errors \(\varepsilon_{V}\) in Equation (55) in terms of geoid uncertainties \(\varepsilon_{N}\), we arrive at the following:  
\[\varepsilon_{N}=\frac{\varepsilon_{V}}{\gamma_{o}}\varepsilon_{V}\approx \frac{2\pi}{\gamma_{o}}\,GR\,\,s\,\varepsilon_{\rho}\approx 2\pi\frac{GR}{\text{ GM}/R^{2}}\,\,\,s\,\varepsilon_{\rho}\approx 2\pi\frac{R^{3}}{\text{ M}}\,\,s\,\varepsilon_{\rho}, \tag{56}\]  
where M = 5.9722 \(\times\) 10\({}^{24}\) kg is the Earth's total mass. According to Table 4, the RMS of differences between the lithospheric thickness models is on average roughly \(\pm\)0.4 km. This RMS corresponds to a density error \(\varepsilon_{\rho}\)-\(\pm\)160 kg\(\cdot\)m\({}^{-3}\). As seen, this error estimate agrees quite closely with the error analysis according to Equation (51), where we assumed density errors within a similar range (i.e., \(\varepsilon_{\beta\rho\text{{{LM}}}}\approx\pm\)100 kg\(\cdot\)m\({}^{-3}\), \(\varepsilon_{\Delta\rho\text{{{L}}}/\Lambda}\approx\pm\)15 kg\(\cdot\)m\({}^{-3}\), and \(\varepsilon_{\Delta\rho\text{{{L}}}/\Lambda}\approx\pm\)50 kg\(\cdot\)m\({}^{-3}\)). As confirmed in this study, sub-lithospheric mantle geoid modeling is affected mainly by errors in lithospheric thickness and lithospheric mantle density. Since we disregarded density variations within the asthenosphere, additional errors in our result are possibly attributed to temperature anomalies and small-scale convection patterns within the asthenosphere.