Simulation of underground gravity gradients from stochastic seismic fields

We present results obtained from a finite-element simulation of seismic displacement fields and of gravity gradients generated by those fields. The displacement field is constructed by a plane-wave model with a 3D isotropic stochastic field and a 2D fundamental Rayleigh field. The plane-wave model provides an accurate representation of stationary fields from distant sources. Underground gravity gradients are calculated as the acceleration of a free test mass inside a cavity. The results are discussed in the context of gravity-gradient noise subtraction in third generation gravitational-wave detectors. Error analysis with respect to the density of the simulated grid leads to a derivation of an improved seismometer placement inside a 3D array which would be used in practice to monitor the seismic field.

Figure 1

The two plots show the absolute values (solid lines) and phases (dash-dot lines) of the reflection coefficients. The range of horizontal slownesses is $p\in [0,1/c_S]$. The cross-mode (conversion) coefficient at slownesses greater than $p=1/c_P$ is solely relevant for the conversion of SV into evanescent $P$ modes.

Figure 2

The figure shows a cross section of the displacement field of a Rayleigh wave propagating from left to right. The rate of exponential amplitude decay is different for the pressure and shear part so that the direction of displacement depends on depth.

Figure 3

Density perturbations are characterized by a correlation length. If the correlation length is high as represented by the sphere on the right-hand side, then larger volumes of rock are perturbed in a similar way and fewer of those coherent volumes fill the spherical shell. Therefore, integrating density perturbations over shells around the test mass, one would expect that fields with smaller correlation lengths generate less GGN since perturbations add up incoherently. However, as explained in the text, this conclusion is wrong for most of the fields met in nature.

Figure 4

The figure displays the spatial coherence of the $P$-wave field as a function of distance between the two points of the rock volume forming the correlation pair. We choose those points to be at the same depth (to make the calculation easier) and to correlate the respective $z$ displacements. A similar result would be obtained for the horizontal displacements, although the height of the side maxima is different.

Figure 5

The plot shows the integrated gravity gradients. The curves represent a particular realization of the isotropic field. The test mass is located inside a cavity of radius 2 m, deep inside the rock (no GGN from the Earth’s surface). More than 90% of the GGN is contributed from shells within a radius of ${\lambda }_P$. Although the absolute value of the gravity gradient can change from one realization to the next, the curves always converge in the same characteristic way.

Figure 6

The contour plot shows the relative error if the limit of the GGN is calculated by averaging a finite stretch of the convergence curve instead of carrying out the integration over the entire rock volume. The error is below the 1% level for starting distances beyond one wavelength and averaging lengths of $0.6{\lambda }_P$ and longer.

Figure 7

Gravity gradients fall exponentially with increasing depth. Each point in this graph is an average over 50 realizations of the Rayleigh field, each composed of 20 plane Rayleigh waves. Assuming that the surface GGN is difficult to model from seismic data and that the surface seismicity is a factor of 10–30 stronger than the ambient body field, one needs to bring the test masses to underground levels where surface GGN is suppressed by a factor of 1000–3000 to reach the 1% level. However, by means of a surface array of seismometers, it should be possible to model part of the GGN from Rayleigh waves and to relax the requirements.

Figure 8

The figure shows the error of the gravity-gradient integral as a function of integration distance. In this case, the original uniform fine grid is compared with a uniform coarse grid. Integration errors of the coarse grid can accumulate to high values as in $\delta g_y$ for this particular realization of the seismic field.

Figure 9

When the displacement field represented by the uniform coarse grid is interpolated back to the original grid density, the error of the gravity-gradient integral decreases. In the majority of realizations, like this one, the improvement is not significant.

Figure 10

A linear selection of grid points of the fine grid leads to the uniform, coarse grid, which is investigated at the beginning of this section. Instead, one can pick more grid points close to the test mass and fewer grid points at greater distances. For example, in what we call the cubed selection, the distance between neighboring grid points is increased to about 500 m close to the boundary of the grid, whereas the distance of grid points close to the test mass is about 40 m. Grid points in the uniform coarse grid have constant distance of about 220 m from their neighbors.

Figure 11

The cubed selection flattens the error curves compared to the linear selection, but does not lead to a decreased error of the gravity-gradient integral. The reason is that grid density is too small starting at distances greater than about ${\lambda }_P/2$. Although the error is significantly better than in the uniform case at intermediate distances (near ${\lambda }_P$), convergence of the error curves towards increasing distance is very poor. The cubed selection does not produce satisfying results, at least not without applying more sophisticated models (yet to be developed) of the seismic field (instead of simple interpolation between grid points).

Figure 12

The squared selection produces very good results. The error curves are flat, and errors do not build up at greater distances despite the decreased grid density. These properties were observed in all realizations of the isotropic field.