Completion of the Boundary Element Method in a Self-Gravitating Elastic Half-Space, with Application to Gravity Gradient Observations

Abstract:

The boundary element method (BEM) is important for the studies, of seismology, fault zone dynamics, and magma chamber evolution. Self-gravitation is especially important for gravity gradient imaging of underground deformations, as it is the dominant signal in the gravity gradient compared with the surface deformation. In contrast, surface deformation dominates the gravity signal.

Two notoriously critical issues must be resolved before the BEM convolution can be performed. The first is to establish the boundary integral equation (BIE) by dealing with the singularities on the boundary; and the second concerns the extraction of the Cauchy principle value (CPV) surrounding the singular point within the boundary element that contains it. Engineers have striven to overcome those difficulties with non-gravitating Green functions throughout the age of the BEM.

We have succeeded previously in establishing the BIE by partitioning the complicated self-gravitating Green function into an analytical part with the singularity and a numerical part without singularity. Here we present the method and result of extracting the CPV. In view of the non-spherical and unbounded topology of the singularities associated with the self-gravitating Green functions, we developed a direct and systematic procedure to extract the CPV and present the resultant expressions in the form of non-singular integrals that are readily implemented numerically. The CPV integral on a smooth singular boundary element is decomposed into the basic building blocks — the CPV integrals on the horizontal source plane and on the orthogonal vertical planes. The existence of distinctive CPV integrals on the horizontal and vertical singular elements is a consequence of cylindrical symmetry, and thus, not found in engineering problems with spherical symmetry. The CPV on a non-smooth polygonal singular element may undergo up to three steps of decomposition: from triangularization, to projection, to the vertical and horizontal decomposition mentioned above. Our new method can easily be extended to general problems of cylindrical symmetry with or without self-gravitation. Numerical implementation of the new BEM in surface deformation and especially the gravity gradient induced by pressure variations of reservoirs and magma chambers will be presented.