**4. Corrections to gravimetric data**

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

.

Assuming that all observation errors are stochastic with expectation zero, an error propagation of the least squares solution in Eq. (21) yields that the covariance

<sup>0</sup> **A**<sup>T</sup>**Q**�<sup>1</sup>

Note that there is no denominator in Eq. (26), because in the present adjustment example with 3 observations and 2 unknowns per pixel there is only 1 degree of

<sup>0</sup> is the variance of unit weight, which can be unbiasedly estimated by

**A** � ��<sup>1</sup> (25)

*<sup>x</sup>* of the solution *x* for the MD or

� � (28)

*<sup>y</sup>* <sup>þ</sup> *E x*<sup>2</sup> � *<sup>y</sup>*<sup>2</sup> � �*:* (27)

*=* 2*σxσ<sup>y</sup>*

<sup>q</sup> (29)

*x* ¼ *x* � *ex* ¼ *y* � *ey*, (30)

*<sup>y</sup>* . If both

<sup>0</sup> <sup>¼</sup> **<sup>L</sup>**<sup>T</sup>**Q**�<sup>1</sup> **<sup>L</sup>** � **<sup>Α</sup>X**^ � �*:* (26)

**3.2 The uncertainties in VMM Moho depth and density contrast**

*Geodetic Sciences - Theory, Applications and Recent Developments*

**QXX** <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

*s* 2

First, we will find an estimate of the variance *σ*<sup>2</sup>

*<sup>k</sup>* <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

MDC by assuming that we know another solution *y* with variance *σ*<sup>2</sup>

solutions have vanishing expected errors, the solution becomes

*σ*2 *<sup>x</sup>* <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

The correlation coefficient between *x* and *y* follows from

*<sup>x</sup>* <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>

where *ex* and *ey* are random errors with zero-expectations.

*<sup>y</sup>* � *E x*ð Þ � *y* <sup>2</sup> h i n o

One can also plot the t-test parameter of the normalized (and unitless) differ-

*<sup>T</sup>* <sup>¼</sup> *<sup>x</sup>* � *<sup>y</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *σ*2 *<sup>x</sup>* þ *σ*<sup>2</sup>

To verify Eqs. (27)–(29), one may start from the substitutions that the true value

In practice, *x* and *y* are the Moho quantities at a pixel estimated from two models, and the expectation operator should be replaced by the (weighted) mean value over the central and surrounding pixels. Note that the solution in Eq. (27) is independent on whether *x* and *y* are correlated or not. Eq. (29) can be used in an

*<sup>y</sup>* � 2*kσxσ<sup>y</sup>*

RMSE value becomes 1.17 � <sup>10</sup><sup>4</sup> kg/m<sup>2</sup>

matrix of **X**^ becomes

**3.3 Verification of the solutions**

ence between *x* and *y*:

for *x* and *y* is given by

**30**

to study the expected difference.

where *σ*<sup>2</sup>

freedom.

Nowadays, the Earth's gravity field has been recognized as an important source of information about the Earth's structure. Such data contain both short- and longwavelength features, i.e., signals from the topographic and bathymetry geometries and density heterogeneities in the topography, ice caps, sediment basins and also in the mantle and core/mantle topography variations.

The long-wavelength contribution to the gravity field, say to spherical harmonic degree and order 10, may be assumed to be related to the mantle and below located heterogeneities.

To isolate the gravity data caused only by the geometry and density contrast of the Moho interface, all aforementioned signal contributors to the gravity data must be removed by applying the so-called stripping corrections and NIEs [34] and NIEs (see section 4.2). Another gravity correction corresponds to the gravimetric effect of filling-up all oceans with masses to a standard density of 2670 kg/m<sup>3</sup> . Finally, by removing also normal gravity from the resulting stripped free-air gravity observation, one obtains the refined Bouguer gravity disturbance. As a result, the ideal stripped Bouguer gravity disturbance can be explained as caused by a spherical Earth without solid Earth topography and mass anomalies below the crust.
