Green's function operator matrix of intraplate faults



This research presents a numerical tool to estimate the Green's function operator matrix of intraplate faults. Having this matrix and its inverse, spatial distribution of fault slippage could be investigated through the inverse analysis of geodetic data. This information could be employed to predict the location of future powerful earthquakes. To implement fault sliding in FE calculations, Soft Material Technique as a simple method is applied. In this technique, the fault is modeled by a flexible (very low elasticity modulus) thin element. This material not only prevents fault planes overlapping, but exhibits a good consistency with the physical behavior of fault. In other words, this material ignores the strength of neighboring rocks ready to trigger sliding. In this research, without involving the nonlinear contact problem, two sides of the fault are dislocated as one unit, and the ground surface deformation is measured.
Available geodetic data provides a proper opportunity to detect underground interactions. We can express observation equations with m observation data as:
Bi = Aij Xj + Ei (i = 1, …, m ) (1)
where Bi are the observed surface deformations, Xj the slippage components along the fault, Ei the random errors, and Aij Green’s function operators (i.e., the elastic response at a point i to a unit source at a point j on the model source region). This equation can be rewritten in a matrix form as:
B = AX + E (2)
where A is an m×n coefficient matrix.
To minimize the errors, the length of the vector E has to be minimized. It may be shown that, standard inversion equations based on the least-square method are obtained as:
X = (ATA)-1ATB (3)
where superscripts T and –1 indicate the transpose and inverse of a matrix, respectively. This relation offers a straightforward way for finding the source vector X. One of the new aspects of the present study is the calculation of the Green’s function operator matrix A by means of FEM. This issue enables us to overcome all limitations of traditional inverse methods.
How can the Green’s function operators be found by FEM? By applying unit source vectors in each degree of freedom, the relevant response of ground surface nodes is the corresponding component of the coefficient matrix A.
The proposed numerical model is first compared by available analytical approaches, and gains its proper validity, one of the Tehran faults is modeled by this method to calculate the corresponding Green's function operator matrix.