The goals of the proposed research are (1) to measure surface displacements and use the measured surface displacements as data for the inverse problem of locating faults and portraying their geometry, and (2) to develop criteria for deciding whether a fault system is in its nucleation phase, which would suggest that an earthquake is imminent. First, the mathematical framework for a stability analysis of displacement fields near equilibrium will be developed, and the Green's tensor for the half-space fault eigenvalue problem will be derived. Next, using rigorous mathematical analysis, questions of convergence will be addressed as the depth of the medium increases, and convergence rates will be verified by high-order numerical schemes for hypersingular boundary integral equations. Finally, closed form recovery formulas hinging on the dominant term of the depth asymptotics will be derived and combined with a fast minimization method. The PI will infer robust algorithms for solving the fault inverse problem. Reconstruction methods will be tested first on synthetic data and then used on data coming from field measurements of surface dislocations.
This research project is expected to impact the advancement of scientific risk assessment for areas prone to devastating earthquakes. Earth science is seeking to understand the physics involved in seismic activity. It has established that most earthquakes occur near faults. If we knew the approximate geometry and location of faults, we could perform computer simulations of seismic activity, thus assessing risk in given areas. Unfortunately, the approximate location of faults is not always known. Due to recent developments in GPS technology, minute surface displacements can now be measured with remarkable accuracy. The proposed research will use these measurements to build methods for locating faults more precisely and for portraying their geometry. The PI will also investigate the exciting prospect of making progress in addressing a long-standing problem: is there a way to confidently assert whether an earthquake is imminent? For this question too, our work will be based on adequate data processing of surface displacements.
