# Gravity Field (2 GB) __TOP__

It is important to note that given GRAIL's high precision and resolution, the topography needs to be expressed in the same reference frame as the GRAIL gravity solutions, namely the Principal Axes frame (MOON_PA) as opposed to the Mean Earth (MOON_ME) frame used for most cartographic uses. The LOLA team archived a spherical harmonics expansion of the lunar topograhy in the MOON_PA frame specifically for this reason (SHA, LBL).

## Gravity Field (2 GB)

Subtracting that correction from the free-air gravity yields the Bouguer gravity field. As shown below, subsurface features and crustal thickness variations then appear clearly. The spherical harmonics coefficients of the Bouguer gravity can be used to perform geophysical spectral analysis, such as crustal thickness modeling. These coefficients are archived in the SHA format, with a supporting description (LBL).

As explained above, Bouguer gravity removes from the free-air gravity the signal expected purely from surface relief, assuming a (constant) crustal density. Over the whole spectrum, the Bouguer map is dominated by the crustal thickness dichotomy between nearside and farside, and the South Pole-Aitken basin is also prominent. It also helps to identify burried, interior mass anomalies, such as those created by large impact craters. Filtering out the longest wavelengths from the Bouguer map highlights the lunar mascons, prominent on the nearside but also present in the lunar highlands of the farside.

In Horndeski theories containing a scalar coupling with the Gauss-Bonnet (GB) curvature invariant RGB2, we study the existence and linear stability of neutron star (NS) solutions on a static and spherically symmetric background. For a scalar-GB coupling of the form Î±Î¾(Ï•)RGB2, where Î¾ is a function of the scalar field Ï•, the existence of linearly stable stars with a nontrivial scalar profile without instabilities puts an upper bound on the strength of the dimensionless coupling constant Î±. To realize maximum masses of NSs for a linear (or dilatonic) GB coupling Î±GBÏ•RGB2 with typical nuclear equations of state, we obtain the theoretical upper limit Î±GB

We study a general field theory of a scalar field coupled to gravity through a quadratic Gauss-Bonnet term Î¾(Ï†)RGB2. The coupling function has the form Î¾(Ï†)=Ï†n, where n is a positive integer. In the absence of the Gauss-Bonnet term, the cosmological solutions for an empty universe and a universe dominated by the energy-momentum tensor of a scalar field are always characterized by the occurrence of a true cosmological singularity. By employing analytical and numerical methods, we show that, in the presence of the quadratic Gauss-Bonnet term, for the dual case of even n, the set of solutions of the classical equations of motion in a curved FRW background includes singularity-free cosmological solutions. The singular solutions are shown to be confined in a part of the phase space of the theory allowing the non-singular solutions to fill the rest of the space. We conjecture that the same theory with a general coupling function that satisfies certain criteria may lead to non-singular cosmological solutions.

Two types of high-resolution data sets, the terrestrial and the airborne gravity measurements, are combined in this study. However, the GRAV-D (Gravity for the Redefinition of the American Vertical Datum) airborne gravity data require additional editing or low-pass filtering before being used (see e.g., GRAV-D Science Team 2018). Various low-pass filtering methods exist and have been applied to the airborne gravity data, such as the spatial Gaussian filter, the fast Fourier transform (FFT, Childers et al. 1999), and the Butterworth filter (Forsberg et al. 2001). Lieb et al. (2015) proposed a low-pass filtering in the spectral domain by SRBFs. Li (2018) demonstrated that the SRBFs show certain de-noising or smoothing properties of the high-frequency noise in the airborne data. In this study, we apply the low-pass filter to the airborne gravity data by using the cubic polynomial (CuP) function, and the smoothing features in this type of SRBFs are used for filtering the high-frequency noise in the airborne data. An advantage of using the CuP function for low-pass filtering is that the filtering process is automatically done when establishing the observation equations, i.e., no extra computation efforts are required. For the terrestrial gravity data, the non-smoothing Shannon function is preferred to avoid the loss of spectral information (Bucha et al. 2016). Schreiner (1999) showed that it is possible to use different types of SRBFs for different types of observations, since the coefficients are independent on the choice of the SRBFs, as long as they cover the same frequency range. Klees et al. (2018) achieved an improvement by applying a truncated SRBF for the terrestrial data but a tapered SRBF for the satellite data, based on simulations. However, to the best of our knowledge, the idea of combining different types of SRBFs for different types of observations has not been applied to real data sets yet. Thus, our results based on real data also indicate the validity of this idea.

This study is conducted between \(-\,110^\circ \) and \(-\,102^\circ \) longitude and between 35\(^\circ \) and 40\(^\circ \) latitude (Fig. 1a), majorly located in Colorado, USA. It is a mountainous area, with an average elevation of 2017 m. The highest location reaches 4386 m, the lowest 932 m. The eastern part of the study area is more flat than the western and the central part, while it is still higher than 1000 m. The larger the topographic heights are, the worse the accuracy of the geoid becomes (Foroughi et al. 2019). Thus, this is a challenging study area, due to the rugged terrain, high elevation, and varying gravity field.

a Terrain map of the study area; b given terrestrial (blue points) and airborne (green flight tracks) gravity data, GSVS17 benchmarks (223 points of the red line) as well as the model grid area (black rectangle)

We compute two sets of output gravity functionals, and the results will be presented and discussed. The first one is at the Geoid Slope Validation Survey 2017 (GSVS17) benchmarks (red line in Fig. 1b), and the second one is the quasi-geoid and geoid model for the target area from \(-\,109^\circ \) to \(-\,103^\circ \) and 36\(^\circ \) to 39\(^\circ \) (black box in Fig. 1b) with a spatial resolution of \(1'\times 1'\).

Transfer the observations in terms of absolute gravity g to gravity disturbance \(\delta g\) by subtracting the normal gravity \(\gamma \) at the ellipsoidal height h of the observations

The general expression (Eq. 5) needs to be adapted for describing different gravitational functionals (Lieb et al. 2016). In this study, the observations are given in terms of gravity disturbances \(\delta g\), which can be expressed as the gradient of the disturbing potential T. In spherical approximation, the magnitude of the gravity disturbance can be written as (Heiskanen and Moritz 1967).

where \(\gamma \) is the normal gravity at the normal height of point \(P_\mathrmc\). Following SÃ¡nchez et al. (2018), we use the ellipsoid GRS80 (Moritz 2000) for the computation of U and \(\gamma \). According to the error propagation law, the standard deviation of the quasi-geoid vector \(\sigma _\zeta \) can be calculated by

In this study, the long-wavelength component is computed from the global gravity model XGM2016 (Pail et al. 2018) up to maximum degree 719 for both the terrestrial and the airborne data. The topographic model dV_ELL_Earth2014 (Rexer et al. 2016) from degree 720 to degree 2159 and a residual terrain model ERTM2160 (Hirt et al. 2014) from degree 2160 to degree \(\sim \) 80,000 (equivalent to a spatial resolution of 250 m) are removed from the terrestrial data; the dV_ELL_Earth2014 from degree 720 to degree 5480 is removed from the airborne data. We use two different topographic models above degree 2160 for the terrestrial and airborne data; this is justifiable due to the fact that the two models (dV_ELL_Earth2014 and ERTM2160) are calculated using the same original data and contain the same signal (Hirt et al. 2014; Rexer et al. 2016). For airborne data, the effect of dV_ELL_Earth2104 from degree 2160 to degree 5480 is equal to the ERTM2160, but the ERTM2160 is only available as a grid on the Earth surface, i.e., not as spherical harmonic coefficients with which the gravity values can be computed at any height.

Figure 2 visualizes the remove step. Comparing the last two rows, it is clear that after the GGM reduction, the gravity field is dominated by the topographic effect, which is very large in this study. This implies the importance of including the topographic effect in the RCR, especially in mountainous areas. After subtracting this topographic effect, the gravity field becomes much smoother, especially in regions with varying elevation (mid part of the study area). As shown by the statistics listed in Table 1, the terrestrial observations are smoothed by 42% in terms of the standard deviation (SD) by subtracting the GGM, and by 82% after including the topographic model. The airborne observations are smoothed by 72% by subtracting the GGM and by 89% after including the topographic model. Such significant smoothing effects enable a better least-squares fit. Including the topographic model gives larger smoothing effects for the terrestrial observations than for the airborne observations. This could be explained by the fact that the high-frequency signal of the gravity field decreases with height, and the airborne gravity measurement is less sensitive to the high-frequency part than the terrestrial gravity measurement. Thus, subtracting the topographic effect affects the terrestrial gravity data more than the airborne gravity data.

In regional gravity modeling, the extension of the target area \(\partial \varOmega _\mathrmT\), the observation area \(\partial \varOmega _\mathrmO\), and the computation area \(\partial \varOmega _\mathrmC\) needs to be defined carefully. In the border of the observation area \(\partial \varOmega _\mathrmO\), the unknown coefficients \(d_k\) cannot be appropriately estimated, due to the lack of fully surrounding observations. This fact provokes edge effects. The observation area \(\partial \varOmega _\mathrmO\), where the observations are given, should be larger than the target area \(\partial \varOmega _\mathrmT\), in which the final output gravitational functionals are computed. Furthermore, the computation area \(\partial \varOmega _\mathrmC\), where the SRBFs are located, should be larger than the observation area \(\partial \varOmega _\mathrmO\). The reason for this extension is due to the oscillation of the SRBFs, especially at the boundaries of the computation area \(\partial \varOmega _\mathrmC\), where the oscillations cannot overlap and balance with each other (see Naeimi et al. 2015; Lieb et al. 2016 for more details). Thus, \(\partial \varOmega _\mathrmT\subset \partial \varOmega _\mathrmO\subset \partial \varOmega _\mathrmC\). 041b061a72