Gravity Spectra from the Density Distribution of Earth’s Uppermost 435 km

Abstract

The Earth masses reside in a near-hydrostatic equilibrium, while the deviations are, for example, manifested in the geoid, which is nowadays well determined by satellite gravimetry. Recent progress in estimating the density distribution of the Earth allows us to examine individual Earth layers and to directly see how the sum approaches the observed anomalous gravitational field. This study evaluates contributions from the crust and the upper mantle taken from the LITHO1.0 model and quantifies the gravitational spectra of the density structure to the depth of 435 km. This is done without isostatic adjustments to see what can be revealed with models like LITHO1.0 alone. At the resolution of 290 km (spherical harmonic degree 70), the crustal contribution starts to dominate over the upper mantle and at about 150 km (degree 130) the upper mantle contribution is nearly negligible. At the spatial resolution $<150\,\hbox {km},$ the spectra behavior is driven by the crust, the mantle lid and the asthenosphere. The LITHO1.0 model was furthermore referenced by adding deeper Earth layers from ak135, and the gravity signal of the merged model was then compared with the observed satellite-only model GOCO05s. The largest differences are found over the tectonothermal cold and old (such as cratonic), and over warm and young areas (such as oceanic ridges). The misfit encountered comes from the mantle lid where a velocity–density relation helped to reduce the RMS error by 40%. Global residuals are also provided in terms of the gravitational gradients as they provide better spatial localization than gravity, and there is strong observational support from ESA’s satellite gradiometry mission GOCE down to the spatial resolution of 80–90 km.

Appendices

Appendix 1: Choice of Volume Elements

Triangular parameterizations such as that used in LITHO1.0 are not often used for calculating the Earth’s gravitational signal from a global density distribution; see, however, the exception (Haagmans 2000). The situation is complicated by the fact that each node might be equipped with a unique value of the density and individual triangles may differ in area. This causes an ambiguity in a way the volume element can be set up.

To reduce the effect of triangulation in LITHO1.0, we define the volume element of each node according to Fig. 7 as space between two spherical surfaces ($R_1$ and $R_2$) for a specific solid angle:

$$\begin{aligned} {\hbox{d}}V' = \frac{1}{3} S_a(R_2^3 - R_1^3), \end{aligned}$$

(8)

where $R_1<=R_2$ and $S_a$ is the average solid angle from the all spherical triangles containing the particular node. The gray area on the left of Fig. 7 depicts $S_a$, white radii bound the volume element vertically and the green area shows the discretized layer passing from node to node. Because the number of nodes is half of the number of triangles (40,962 nodes vs. 81,920 triangles), the local average $S_a$ must be multiplied by two (or exactly by 81,920/40,962) to cover $4\pi$. Nonetheless, a global sum of such average $S_a$ will deviate from $4\pi$. In our case, the error is $\delta =4.4\cdot 10^{-5}$ steradian that corresponds to more than 5 significant decimal digits of $4\pi$. To compare, a sum of all spherical triangles provided with LITHO1.0 gives $4\pi$ with the error $\delta =-3.5\cdot 10^{-10}$ steradian. Hence, the volume numerical integration with models like LITHO1.0 is a sort of a trade-off between a way the local average is made (to assign the density to a volume) and preservation of the whole Earth surface/mass.

The variation of the average surface elements in percent is shown on the right of Fig. 7. The patterns agree well with the process of tesselation used in LITHO1.0; nearly each tesselation level can be recognized (from large to small patterns). The area variation is computed with respect to the largest value (red) and it reaches up to 22%. For example, omitting this area variation and using a constant value for each node (i.e., $S_a=4\pi /40962$) affects all significant digits of the resulting gravitational signal.

Fig. 7

On the left, the average solid angle of each node for setting up volume elements according to Eq. (8). On the right, the global variation of $S_a$ in percent (relatively to the largest value) obtained from LITHO1.0

Appendix 2: Comparison of 1D and 3D Integration

One dimensional approximations such Eq. (7) are often being used to calculate the gravity signal from a density column $\rho _i$ (Turcotte and Schubert 2002; Hees 2000). Although not by definition, the shell formula may also provide a quasi-lateral variation if each density column is evaluated separately. Hence, by using all the non-homogeneous layers in LITHO1.0 a 3D approach based on Eq. (4) can be confronted with Eq. (7).

Figure 8 shows such a difference in terms of $g_U$. It is seen the large discrepancies are located over the thick parts of the lithosphere and over the oceanic ridges where the shell formula simply does not account for a lateral density variation. The magnitudes reach $\pm 1000\,\hbox {mGal}$ that is more than twice as much as the difference between GOCO05s and LITHO1.0 (including ak135) evaluated with Eq. (2); see Fig. 5 and Table 1. The 1D approximation would introduce a large methodological error in our context. On the other hand, Eq. (7) is not singular (compare with Eq. (2) when $L\rightarrow 0$) and it is much faster to evaluate than Eq. 2 (seconds vs. hours on a single PC).

Fig. 8

A difference in $g_U$ (in mGal) between Eqs. (2) and  (7) at 6621 km by using all the non-homogeneous layers in LITHO1.0