1 Introduction

The value of 600 kg/m3 was traditionally assumed for the crust-mantle density contrast (see e.g., Heiskanen and Moritz, 1967, p. 135). Martinec (1994) claimed that this value corresponds better with the Moho interface density contrast within the ocean crust, while he estimated the value of 280 kg/m3 for the continental crust by minimising the external gravitational potential induced by the Earth’s topographic masses and the Moho discontinuity under the assumption that the Moho density contrast is constant. The continental Moho density contrast of 200 kg/m3 was reported by Goodacre (1972) for Canada. The density contrast across the Moho boundary was determined regionally from seismological studies using the wave receiver functions (e.g., Niu and James, 2002 and Jordi, 2007). The results of these studies indicate that the density contrast across Moho may regionally vary as much as from 160 kg/m3 (for the mafic lower crust) to 440 kg/m3 (for the felsic lower crust), with an apparently typical value for the craton of about 440 kg/m3. Dziewonski and Anderson ( 1981 ) (Table 1) adopted the value of 480 kg/m3 for the global crust-mantle density contrast in the definition of the Preliminary Reference Earth Model (PREM). This value of the crust-mantle density contrast was derived from the seismic reflection data. Tenzer et al. (2009a) estimated that the average value of the crust-mantle density contrast is about 520 kg/m3. In the most recent study based on solving Moritz’s generalisation of the Vening-Meinesz inverse problem of isostasy, Sjöberg and Bagherbandi (2011) estimated that the Moho density contrast varies globally from 81.5 kg/m3 in the Pacific region to 988 kg/m3 in Tibet, with the average values of 678 ± 78 and 334 ± 108 kg/m3 for the continental and oceanic areas, respectively. They also estimated that the global average of the Moho density contrast is 448 ± 187 kg/m3. This estimated value is about 7% smaller than the adopted value (480 kg/m3) for the global crust-mantle density contrast in PREM.

Table 1 Statistics of the topographic and crust-stripping corrections to gravity disturbances
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The presented study is an update of the article by Tenzer et al. (2009a) based on using more recent data and more accurate numerical models. The global average value of the crust-mantle density contrast is estimated based on minimising the correlation between the refined gravity and crust-thickness data by adding the crust-mantle density contrast to the adopted reference crustal density. There are two major reasons for estimating the average value of the crust-mantle density contrast. In the context of global studies for a recovery of the Moho density interface from gravimetric data, this value is a required parameter of the functional model formulated between the (known) refined gravity and (unknown) crustal-thickness data. Moreover, when the objective is to study the sub-crustal density distribution anomalies, then the density contrast of the crustal components is usually taken relative to the average density of the lithospheric mantle (upper mantle) estimated based on minimising the correlation between the refined gravity and crust-thickness data. When speaking of the Moho interface density contrast or the crust-mantle density contrast, we have to distinguish between the contrast of the lower-most crust with respect to the encompassing upper-most mantle, and the average crust as a whole with respect to the mantle. The consolidated crust-stripped gravity field (used to estimate the crust-mantle density contrast) describes the refined gravity field generated by the regularised Earth of which all the masses outside the Earth’s ellipsoid are removed and the known crust density distribution is replaced by the homogenous reference crustal density.

We adopt the reference crustal density of 2,670 kg/m3. This value is often assumed for the upper continental crust in geological and gravity surveys, geophysical exploration, gravimetric geoid modelling, compilation of regional gravity maps, and other applications. Although this density value is widely used, its origin remains partially obscure. Woollard (1966) suggested that this density was used for the first time by Hayford and Bowie (1912). In reviewing several studies seeking a representative average density from various rock type formations, Hinze (2003) argued that this value was used earlier by Hayford (1909) for gravity reduction. Hayford (1909) referred to Harkness (1891) who averaged five published values of surface rock density. Harkness’s (1891) value of 2,670 kg/m3 was confirmed later, for instance, by Gibb (1968) who estimated an average density for the surface rocks in a significant portion of the Canadian Precambrian shield from over 2,000 individual measurements. Woollard (1962) examined more than 1,000 rock samples and estimated that the average basement (crystalline) rock density is about 2,740 kg/m3. Subrahmanyam and Verma (1981) determined that crystalline rocks in low-grade metamorphic terranes in India have an average density of 2,750 kg/m3, while 2,850 kg/m3 in high-grade metamorphic terranes. We note here that the choice of the reference crustal density is optional depending on a particular purpose of the numerical study.

The estimation of the crust-mantle density contrast is done using two types of the refined gravity data, namely the consolidated crust-stripped gravity disturbances and the corresponding gravity anomalies. Whereas the gravity disturbance is defined as a vertical derivative of the disturbing potential, the formulation of the gravity anomaly is based on the fundamental gravimetric equation (cf. Vaníček et al., 2004, Eq. 9). The second term of the fundamental gravimetric equation represents the distinction between the gravity anomaly and the gravity disturbance. This also has implications in applying the gravity corrections, giving a theoretical foundation for the secondary indirect effects (cf. Vajda et al., 2006, 2007; and Tenzer et al., 2008a, b). As we demonstrate in Sect. 3, the consolidated crust-stripped gravity disturbances differ significantly from the corresponding gravity anomalies due to large contributions of the secondary indirect effects.

The methodology of computing the consolidated crust-stripped gravity data is reviewed in Sect. 2. The results are demonstrated in Sect. 3. The correlation between the refined gravity and crust-thickness data is investigated and the global crust-mantle density contrast is estimated in Sect. 4. The summary of results and conclusions are given in Sect. 5.

2 Methodology

The computation of the consolidated crust-stripped gravity disturbances \( \delta g^{c} \) and the corresponding gravity anomalies \( \Updelta g^{c} \) is done using the following expressions

$$ \delta g^{c} = \delta g - g^{t} + g^{b} + g^{i} + g^{s} + g^{c} , $$
(1)

and

$$ \Updelta g^{c} = \delta g^{c} - \frac{2}{r}\left( {T - V^{t} + V^{b} + V^{i} + V^{s} + V^{c} } \right), $$
(2)

where r is the geocentric radius of the computation point; T is the disturbing gravity potential; \( \delta g \cong \partial T/\partial r \) is the gravity disturbance and the gravity anomaly \( \Updelta g \) is defined by the fundamental gravimetric equation in the following form (e.g.; Heiskanen and Moritz, 1967): \( \Updelta g = \delta g - 2r^{ - 1} T \); \( V^{t} \), \( V^{b} \), \( V^{i} \), \( V^{s} \), and \( V^{c} \) are, respectively, the gravitational potentials generated by the topography and density contrasts due to the ocean, ice, sediments, and remaining anomalous density structures within the Earth’s crust. The respective gravitational attractions in Eq. 1 are denoted as \( g^{t} \), \( g^{b} \), \( g^{i} \), \( g^{s} \), and \( g^{c} \). The coefficients of the Earth Gravitational Model 2008 (EGM2008) complete to spherical harmonic degree 180 (Pavlis et al., 2008) were used to compute the gravity disturbances and gravity anomalies.

The coefficients \( E_{n,m} \) of the global topographic/bathymetric model DTM2006.0 and the coefficients \( N_{n,m} \) of the EGM2008 global geoid model were used to generate the global elevation model (GEM) coefficients \( H_{n,m} \) using the following expression (Novák, 2010)

$$ H_{n,m} = E_{n,m} - N_{n,m} . $$
(3)

The DTM2006.0 coefficients \( E_{n,m} \) describe the global geometry of the heights above mean sea level (MSL) which are reckoned positive, and the depths below MSL which are reckoned negative. The global topographic/bathymetric model DTM2006.0 was publicly released together with EGM2008 by the US National Geospatial-Intelligence Agency EGM development team. The coefficients \( N_{n,m} \) were generated from the numerical coefficients \( T_{n,m} \) of the disturbing potential (derived from EGM2008) as follows

$$ N_{n,m} = \frac{{T_{n,m} }}{{\gamma_{0} }}, $$
(4)

where \( \gamma_{0} \) is the normal gravity at the surface of the reference ellipsoid GRS-80 (Moritz, 1980). The GEM coefficients complete to degree and order 180 were used to compute the topographic corrections (for a homogenous density of 2,670 kg/m3) to gravity data according to expressions derived by Novák (2010).

The coefficients \( E_{n,m} \) and \( N_{n,m} \) were further used to generate the global bathymetric model (GBM) coefficients \( D_{n,m} \) according to the following expression (Novák, 2010)

$$ D_{n,m} = N_{n,m} - E_{n,m} . $$
(5)

The GBM coefficients complete to degree and order 180 were used to compute the bathymetric stripping gravity corrections. Tenzer et al. (2011) utilised a depth-dependent seawater density model in deriving expressions for computing these corrections in order to reduce large errors otherwise presented in results when using only a constant seawater density. The depth-dependent seawater density model was derived by Gladkikh and Tenzer (2011) based on the analysis of the global data of pressure/depth, salinity, and temperature from the World Ocean Atlas 2009 (provided by NOAA’s National Oceanographic Data Center; products description can be found in Antonov et al., 2010, Johnson et al. 2009, and Locarnini et al., 2010) and the World Ocean Circulation Experiment 2004 (provided by the German Federal Maritime and Hydrographic Agency; see Gouretski and Koltermann, 2004). Since the topographic and bathymetric stripping corrections computed using the GEM and GBM coefficients (\( H_{n,m} \) and \( D_{n,m} \)) according to Eqs. 3–5 are referred relative to the EGM2008 geoid surface, we applied additional corrections due to the gravitational contributions of the topographic and ocean density contrast masses enclosed between the surfaces of the EGM2008 geoid model and the GRS-80 Earth’s ellipsoid in order to obtain the ellipsoid-referenced topographic and bathymetric stripping corrections. For more details we refer readers to Tenzer et al. (2009a); see also Vajda et al. (2008).

The 10 × 10 arc-min mean heights computed by spatial averaging of the 30 × 30 arc-sec global elevation data from GTOPO30 (provided by the US Geological Survey’s EROS Data Center) and the 10 × 10 arc-min continental ice-thickness data from ICE-5G (VM2 L90) made available by Peltier (2004) were used to generate the global ice-thickness model (GIM) coefficients. The GEM and GIM coefficients complete to spherical harmonic degree 180 were used to compute the ice density contrast stripping corrections to gravity data according to expressions derived in Tenzer et al. (2010b). The density of glacial ice 917 kg/m3 (cf. Cutnell and Kenneth, 1995) was adopted for a definition of the ice density contrast. The accuracy of computed ice density contrast stripping gravity corrections is discussed in Tenzer et al. (2010b).

For global studies the best currently available global crustal model is CRUST2.0 (Bassin et al., 2000), which is an upgrade of CRUST5.1 (Mooney et al., 1998). The CRUST2.0 model contains information on the crustal thickness and the subsurface spatial distribution and density of the following global components: ice; soft and hard sediments; upper, middle, and lower (consolidated) crust. We note here that the global crust-thickness models CUB2 (Shapiro and Ritzwoller, 2002) and MDN (Meier et al., 2007) do not provide additional information on the crust density structure. The stripping corrections due to sediments and remaining anomalous crustal density structures were computed using the forward modelling techniques which facilitate the spatial representation of Newton’s volume integral. In particular, we applied two integration approaches. The semi-analytical approach was used for computing the far-zone contributions, utilising the Newton–Cotes numerical integration scheme for the surface component of Newton’s volume integral, while the vertical component was defined by the closed analytical expression according to Gradshteyn and Ryzhik (1980), see also Martinec (1998). To reduce the inaccuracy due to using the semi-analytical integration approach, the rectangular prism approach (see Nagy et al., 2000) was utilised for the analytical integration within the near zone up to the distance of 100 km around the observation point. The 2 × 2 arc-deg CRUST2.0 data of density, depth, and thickness of the (soft and hard) sediments and (upper, middle, and lower) crustal components were used to compute globally the stripping gravity corrections due to sediments and consolidated crust components.

3 Step-Wise Consolidated Crust-Stripped Gravity Data

We computed and subsequently applied the topographic and crust-stripping corrections to gravity disturbances and gravity anomalies. All computations were realised globally on a 1×1 arc-deg geographical grid at the Earth’s surface. The density contrasts were taken relative to the reference crustal density of 2,670 kg/m3. The statistics of the topographic and crust-stripping corrections to the gravity disturbances and gravity anomalies are summarised in Tables 1 and 2. The complete corrections to gravity anomalies comprise the combined contribution of the direct and secondary indirect effects. The statistics of the applied secondary indirect effects are given in Table 3. The statistics of the step-wise consolidated crust-stripped gravity disturbances and gravity anomalies are summarized in Tables 4 and 5.

Table 2 Statistics of the topographic and crust-stripping corrections to gravity anomalies
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Table 3 Statistics of the secondary indirect topographic and crust-components stripping effects
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Table 4 Statistics of the step-wise consolidated crust-stripped gravity disturbances
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Table 5 Statistics of the step-wise consolidated crust-stripped gravity anomalies
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Tenzer et al. (2009b) demonstrated that the atmospheric correction to gravity disturbances globally varies between −0.18 and 0.03 mGal, and the complete atmospheric correction to gravity anomalies varies from 1.13 to 1.76 mGal. Since these values are very small compared to the topographic and crust-stripping gravity corrections (see Tables 1, 2), the gravitational effect of the atmosphere is negligible in the context of this study.

The gravity disturbances and gravity anomalies computed using the EGM2008 coefficients complete to degree 180 of spherical harmonics are shown in Fig. 1. The topography-corrected gravity disturbances and gravity anomalies are shown in Fig. 2. The application of the topographic corrections to observed gravity data exhibits the isostatic compensation in mountainous regions, where the topography-corrected gravity data have the largest negative values. The largest positive values are mainly located over oceanic areas. The topography-corrected and bathymetry-stripped gravity disturbances and gravity anomalies are shown in Fig. 3. The application of the bathymetric stripping corrections significantly changed the gravity field over oceanic areas and revealed main structures of the ocean floor relief and the global pattern of the tectonic plates more likely due to the different density and thickness of the continental and oceanic lithospheric plates. Moreover, some features in the gravity signal computed at locations of the world oceans uncovered more detailed features of the density and thickness within the oceanic lithospheric plates (for example along the Mid-Atlantic Ridge). Since the bathymetric stripping corrections to gravity data have mostly a long-wavelength character over continental areas, the higher-frequency spectrum of the topography-corrected gravity data remains almost unchanged. The convergent ocean-to-continent tectonic plate boundaries represent the regions with the largest gravity signal spatial variations. The gravity disturbances and gravity anomalies obtained after applying the ice density contrast stripping corrections are shown in Fig. 4. The ice density contrast stripping corrections significantly changed the gravity field quantities over the regions with a large thickness of the continental ice sheet in Greenland and Antarctica. The application of the ice stripping corrections also exhibited more realistically the possible presence of isostasy in the gravity signal over the polar continental regions to a large extent magnified after removing the gravitational effect of the topographical masses of density 2,670 kg/m3. The gravity disturbances and gravity anomalies obtained after applying the sediment density contrast stripping corrections are shown in Fig. 5. The application of the stripping corrections exhibited slightly more the contrast between the continental and oceanic crust in some parts of the world and also slightly changed the pattern of gravity field at the continental shelf regions with the largest sediment deposits. These changes are, however, not as significant as the corresponding changes in the gravity signal after applying the topographic, bathymetric, and ice stripping corrections due to a relatively small gravitational contribution of the sediment density contrast (globally mainly below 100 mGal). The consolidated crust-stripped gravity disturbances and gravity anomalies are shown in Fig. 6. The consolidated crust-stripped gravity disturbances globally vary from −1,319 to 508 mGal. The corresponding gravity anomalies vary between −853 and 392 mGal. As seen in Figs. 1 and 6, the differences between the EGM2008 gravity disturbances and gravity anomalies are mostly less than 20 mGal, while these differences reach up to several hundreds of milligals in computed values of the consolidated crust-stripped gravity data. This is due to large contributions of the secondary indirect (topographic and crust density contrasts) effects.

Fig. 1
figure 1

The gravity disturbances (a) and gravity anomalies (b) computed globally on a 1 × 1 arc-deg grid at the Earth’s surface using the EGM2008 coefficients complete to spherical harmonic degree 180

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Fig. 2
figure 2

The topography-corrected gravity disturbances (a) and gravity anomalies (b) computed globally on a 1 × 1 arc-deg grid at the Earth’s surface using the EGM2008 and DTM2006.0 coefficients complete to spherical harmonic degree 180

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Fig. 3
figure 3

The topography-corrected and bathymetry-stripped gravity disturbances (a) and gravity anomalies (b) computed globally on a 1×1 arc-deg grid at the Earth’s surface using the EGM2008 and DTM2006.0 coefficients complete to spherical harmonic degree 180

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Fig. 4
figure 4

The topography-corrected and bathymetry-, and ice-stripped gravity disturbances (a) and gravity anomalies (b) computed globally on a 1 × 1 arc-deg grid at the Earth’s surface using the EGM2008, DTM2006.0, and ICE-5G-derived coefficients complete to spherical harmonic degree 180

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Fig. 5
figure 5

The topography-corrected and bathymetry-, ice-, and sediment-stripped gravity disturbances (a) and gravity anomalies (b) computed globally on a 1 × 1 arc-deg grid at the Earth’s surface using the EGM2008, DTM2006.0, and ICE-5G-derived coefficients complete to spherical harmonic degree 180 and the 2 × 2 arc-deg CRUST2.0 data of (soft and hard) sediments density and thickness

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Fig. 6
figure 6

The consolidated crust-stripped gravity disturbances (a) and gravity anomalies (b) computed globally on a 1 × 1 arc-deg grid at the Earth’s surface using the EGM2008, DTM2006.0, and ICE-5G-derived coefficients complete to spherical harmonic degree 180, and the 2 × 2 arc-deg CRUST2.0 data of (soft and hard) sediments and (upper, middle, and lower) crust components density and thickness

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4 Interference of the Crust-Mantle Density Contrast

The correlations between the step-wise refined gravity and crust-thickness data (defined by means of Pearson’s linear correlation coefficients) are summarised in Tables 6 and 7. As seen from the results, the consolidated crust-stripped gravity data have the largest correlation with the Moho density interface among all gravity data computed in Sect. 3. We used the 2 × 2 arc-deg global data of the crust thickness taken from CRUST2.0. It is worth mentioning that the differences between the correlations of the gravity field data with the Moho depths (beneath the Earth’s ellipsoid) and with the entire crust thickness are less than 0.05. Therefore, there would not be any substantial differences in results of the correlation analysis done for the Moho depths instead of using the crust-thickness data. The relation between the crust-thickness and step-wise refined gravity data is shown in Fig. 7. The observed gravity data (computed based on EGM2008) are not correlated with the crust thickness more likely due to the isostatic balance of the Earth’s lithosphere. The application of the topographic and bathymetric stripping corrections substantially increased the correlation (in absolute sense) between the gravity and crust-thickness data. The application of the stripping corrections due to ice, sediments, and remaining CRUST2.0 crustal components further increased the absolute correlation between the refined gravity and crust-thickness data up to 0.96 (for the consolidated crust-stripped gravity disturbances) and 0.80 (for the consolidated crust-stripped gravity anomalies).

Table 6 Pearson’s linear correlation between the CRUST2.0 crust-thickness data and the step-wise refined gravity disturbances: observed δg, topography-corrected δg t, topography-corrected and bathymetry-stripped δg bt, and consolidated crust-stripped δg c
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Table 7 Pearson’s linear correlation between the CRUST2.0 crust-thickness data and the step-wise refined gravity anomalies: observed Δg, topography-corrected Δg t, topography-corrected and bathymetry-stripped Δg bt, and consolidated crust-stripped Δg c
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Fig. 7
figure 7

The relation between the CRUST 2.0 crust-thickness d and the step-wise refined gravity disturbances (left panels) and gravity anomalies (right panels): a observed δg and Δg, b topography-corrected δg t and Δg t, c topography-corrected and bathymetry-stripped δg bt and Δg bt, and d consolidated crust-stripped δg c and Δg c

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The consolidated crust-stripped gravity data and the CRUST2.0 crust-thickness data were used to estimate the global average value of the crust-mantle density contrast. This was done by minimising the correlation between the refined gravity and crust-thickness data by adding the crust-mantle density contrast to the original reference crustal density of 2,670 kg/m3. The estimation was carried out individually for the consolidated crust-stripped gravity disturbances and gravity anomalies. As shown in Fig. 8, the correlation between the complete crust-stripped (relative to the mantle) gravity disturbances and the CRUST2.0 crust-thickness reached the absolute minima for the crust-mantle density contrast of 485 kg/m3. The corresponding lowest absolute correlation for the complete crust-stripped (relative to the mantle) gravity anomalies was reached for the crust-mantle density contrast of 481 kg/m3. These values differ significantly from the value of 520 kg/m3 estimated by Tenzer et al. (2009a). This is due to applying various theoretical and numerical improvements as well as using the latest available global data sets. The most significant theoretical improvement was achieved by adopting the depth-dependent seawater density model in expressions for computing the bathymetric stripping corrections. Tenzer et al. (2011) demonstrated that the consideration of a depth-dependent seawater density can improve the accuracy in computed values of the bathymetric stripping corrections to about 15 mGal particularly over the open oceans (cf. Tenzer et al., 2010a, 2011). The facilitation of the 10 × 10 arc-min continental ice-thickness data from ICE-5G instead of using the 2 × 2 arc-deg CRUST2.0 ice-thickness data improved the accuracy of the computed ice density contrast stripping corrections locally more than 100 mGal. Nevertheless, we anticipate large errors in the computed values of refined gravity field and subsequently in estimated average values of the crust-mantle density contrast. These errors are attributed mainly to the signal of the crustal model uncertainties and the Moho uncertainty (especially under significant orogens). A realistic assessment of these errors is not simple. Kaban et al. (2003) estimated, for instance, that the errors in computed values of the gravitational attraction can reach about 100 mGal over continental regions, while about 40 mGal over the oceanic areas. Moreover, the estimation model would translate also the signal coming from the mantle lithosphere and deeper mantle into false information on the crust-mantle density contrast.

Fig. 8
figure 8

The absolute Pearson’s correlation between the complete crust-stripped (relative to the mantle) gravity data and the CRUST2.0 Moho depths data for different values of the crust-mantle density contrast

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5 Summary and Conclusions

We have computed and applied the topographic and crust-stripping corrections to observed gravity data. The step-wise refined gravity data were used for the analysis of the correlation with the crust thickness and for the estimation of the crust-mantle density contrast.

We have demonstrated that the consolidated crust-stripped gravity data have the largest correlation with the crust thickness. Therefore, these refined gravity data are the most suitable (among the investigated gravity data types) for the recovery of the Moho density interface by means of the gravimetric modelling or inversion. Our results revealed that the absolute correlation between the crust-thickness and refined gravity data reached 0.96 and 0.80 for the consolidated crust-stripping gravity disturbances and the corresponding gravity anomalies, respectively.

We have used the consolidated crust-stripped gravity data and the CRUST2.0 crust-thickness data to estimate the global average value of the crust-mantle density contrast. This was done by minimising the correlation between the refined gravity and crust-thickness data by adding the crust-mantle density contrast to the original reference crustal density of 2,670 kg/m3. We obtained the values of 485 kg/m3 (for the refined gravity disturbances) and 481 kg/m3 (for the refined gravity anomalies). This good agreement between these two estimated values ascertained the correctness of the forward modelling techniques used for a numerical realisation in this study. Our estimated values are very similar to the corresponding theoretical value of 480 kg/m3 adopted in the definition of PREM. These values of the crust-mantle density contrast differ about 7% from the global average of the Moho density contrast of 448 kg/m3 estimated by Sjöberg and Bagherbandi (2011) based on solving the Moritz’s generalization of the Vening-Meinesz inverse problem of isostasy and by the same amount with respect to a previous assessment (of 520 kg/m3) by Tenzer et al. (2009a).

The adopted reference crustal density of 2,670 kg/m3 and the estimated value of the crust-mantle density contrast 485 kg/m3 (and 481 kg/m3) yield the value of 3,155 kg/m3 (and 3,151 kg/m3) for the global average density of the upper-most mantle.