**3. Uncertainty estimations**

First, the result of the least-squares procedure depends on the quality and weighting of the gravity and seismic observations. The weights should be selected as proportional to the inverse standard errors (STEs) of the observations squared. The STEs of seismic data is, hopefully, provided along with the data files. For the gravity data we derive the global mean STE in Section 3.1. In Section 3.2 we propagate the data errors to error estimates in the VMM least-squares results of Moho constituents. Finally, in Section 3.3 a method for validating the modeled Moho undulations is presented.

### **3.1 The uncertainty in the gravimetric-isostatic observation equation**

Assuming that there are no systematic errors and disregarding 2nd –order terms in Eq. (8), one obtains the error in *χ* by simple error propagation from Eq. (12):

$$
\varepsilon\_{\chi} = \frac{1}{4\pi} \sum\_{n=n\_1}^{n\_2} \frac{2n+1}{n+1} df\_n,\tag{23}
$$

where *df <sup>n</sup>* is the error in *f <sup>n</sup>*. Then it follows that the global Root Mean Square Error (RMSE) of *χ* becomes

$$RMSE(\chi) = \frac{1}{4\pi} E\left\{ \iint \left( \varepsilon\_{\chi}^{2} \right) d\sigma \right\} = \frac{1}{4\pi} \sqrt{\sum\_{n\_{1}}^{n\_{2}} \left( \frac{2n+1}{n+1} \right)^{2}} d\varepsilon\_{n},\tag{24}$$

where *E*fg denotes the statistical expectation of the term in the bracket, and *dcn* are the error degree variances of the gravity disturbances. Using this formula with harmonics between 10 and 180 of the XGM2019e gravity field model (see [33]), the RMSE value becomes 1.17 � <sup>10</sup><sup>4</sup> kg/m<sup>2</sup> .
