5:15 PM - 6:45 PM
[SGD01-P04] Numerical calculation of ground gravity change based on Lagrangian form: a case study of a spherical pressure source
Keywords:gravity change, finite element method, spherical pressure source, crustal deformation, Lagrangian form
Geodetic observations such as crustal deformation and ground gravity observations are effective in understanding internal physical processes associated with seismic and volcanic activities. When modeling those solid-earth phenomena using observed geodetic data, analytical solutions for the fault slip and pressure change in a spherical source (Yamakawa, 1955; Hagiwara1977; Okada, 1992; Okubo, 1992) are often used as Green functions. However, these analytical solutions were derived by assuming a half-infinite homogeneous elastic medium, so the estimated models may be biased by the analytical solutions under circumstances where the effects of heterogeneity, topography or the Earth’s curvature cannot be ignored. In addition, the analytical solution of Hagiwara (1977) may not reproduce the gravity change due to the pressure change in a spherical source accurately, because it approximates the ground surface deformation by the increment/decrement of an infinite flat plate.
Finite element method (FEM) can solve the above problems, because it can calculate crustal deformation and gravity change by considering complexity such as topography and inhomogeneous underground structure. In recent years, FEM have often been used for modeling crustal deformation (Iinuma et al., 2016), and the limitations of the analytical solutions were assessed by utilizing FEM (Sakai et al., 2007). Some of the previous studies have also estimated synthetic gravity changes from the numerical results calculated by FEM (Currenti, 2007; Trasatti and Bonafede, 2008). However, these studies may not reproduce actual ground gravity changes precisely, because they simplified the displacement of observation points on the basis of the Eulerian form. In addition, they assumed that the free-air gravity gradient is constant, although it can change spatially due to topography and medium heterogeneity.
Therefore, we propose a new method to calculate ground gravity changes due to the medium deformation by employing FEM as follows. Firstly, we obtain the density change at each element of the FE model, based on the Lagrangian form. Secondly, we calculate the attraction force caused by the density fields before/after the deformation (g1 and g2, respectively), by fixing a gravity measurement point on the ground of the FE model. Then, we obtain the gravity change relating to the FE model as dgFEM = g2 – g1. We additionally calculate the free-air effect due to the existence of mass outside the FE model as dgres = dhFEM * βres, where dhFEM is the elevation change of the measurement point obtained from the FE analysis and βres is the free-air gradient due to the mass outside the FE model. Finally, we obtain the sum of gravity changes as dg = dgFEM + dgres, which can be compared with those observed on the ground.
We applied our method to a spherical pressure source embedded in a homogeneous elastic half-space, in order to evaluate the reproducibility on [1] the ground displacement, [2] the gravity change at a fixed point in the air, and [3] the gravity change at a fixed point on the ground. We found that [1]-[3] calculated in our method agree with those obtained from analytical solutions (Yamakawa, 1955; Hagiwara, 1977) within a few percent deviation, which corresponds to the calculation error of FEM. This result indicates that the infinite-plate approximation applied to the surface deformation by Hagiwara (1977) is precise enough to calculate ground gravity changes, as long as the homogeneous elastic half-space is assumed for the deforming medium. To assess the reproducibility of our method in other cases, we will next apply our method to an elastic medium having topography, and compare the calculated results with those obtained from analytical solutions (e.g., Nishiyama, 2022).
Finite element method (FEM) can solve the above problems, because it can calculate crustal deformation and gravity change by considering complexity such as topography and inhomogeneous underground structure. In recent years, FEM have often been used for modeling crustal deformation (Iinuma et al., 2016), and the limitations of the analytical solutions were assessed by utilizing FEM (Sakai et al., 2007). Some of the previous studies have also estimated synthetic gravity changes from the numerical results calculated by FEM (Currenti, 2007; Trasatti and Bonafede, 2008). However, these studies may not reproduce actual ground gravity changes precisely, because they simplified the displacement of observation points on the basis of the Eulerian form. In addition, they assumed that the free-air gravity gradient is constant, although it can change spatially due to topography and medium heterogeneity.
Therefore, we propose a new method to calculate ground gravity changes due to the medium deformation by employing FEM as follows. Firstly, we obtain the density change at each element of the FE model, based on the Lagrangian form. Secondly, we calculate the attraction force caused by the density fields before/after the deformation (g1 and g2, respectively), by fixing a gravity measurement point on the ground of the FE model. Then, we obtain the gravity change relating to the FE model as dgFEM = g2 – g1. We additionally calculate the free-air effect due to the existence of mass outside the FE model as dgres = dhFEM * βres, where dhFEM is the elevation change of the measurement point obtained from the FE analysis and βres is the free-air gradient due to the mass outside the FE model. Finally, we obtain the sum of gravity changes as dg = dgFEM + dgres, which can be compared with those observed on the ground.
We applied our method to a spherical pressure source embedded in a homogeneous elastic half-space, in order to evaluate the reproducibility on [1] the ground displacement, [2] the gravity change at a fixed point in the air, and [3] the gravity change at a fixed point on the ground. We found that [1]-[3] calculated in our method agree with those obtained from analytical solutions (Yamakawa, 1955; Hagiwara, 1977) within a few percent deviation, which corresponds to the calculation error of FEM. This result indicates that the infinite-plate approximation applied to the surface deformation by Hagiwara (1977) is precise enough to calculate ground gravity changes, as long as the homogeneous elastic half-space is assumed for the deforming medium. To assess the reproducibility of our method in other cases, we will next apply our method to an elastic medium having topography, and compare the calculated results with those obtained from analytical solutions (e.g., Nishiyama, 2022).