Response of an Elastic Half Space to an Arbitrary 3-D Vector Body Force Smith and Sandwell, JGR 2003 Develop the three differential equations relating 3-D vector displacement to a 3-D vector body force. Take the 3-D Fourier transform to reduce the partial differential equation to a set of linear algebraic equations. Solve the linear system using the symbolic capabilities in Matlab. Perform the inverse fourier transform in the z-direction (depth) by repeated application of the Cauchy Residue Theorem. Check the analytic solution using the symbolic capabilities in Matlab. Solve the Boussinesq Problem to correct the non-zero normal traction on the half-space. Construct screw dislocation and test with analytic line-source solution. Integrate the point-source Green's function to simulate a vertical fault and check with the analytic fault-plane solution. Develop an equivalent body force for a general fault model. Modify the solution to account for surface topography. Modify the solution to have a layered half-space?? Modify the solution to have a visco-elastic rheology?? Objective: calculate the displacement vector u(x, y, z) on the surface of the Earth due to a vector body force at depth

Full Displacement Solution: (Source)(Image)(Boussinesq) Components:

Force couple Magnitude ~ slip rate Magnitude ~ slip rate Direction || to plate motion Direction || to plate motion Sketch of 3-D fault in an elastic half-space Analytic form of the force couple is the derivative of a Gaussian function with half-width equal to cell spacing Analytic form of the force couple is the derivative of a Gaussian function with half-width equal to cell spacing Cosine transform in x-direction is used for constant velocity difference across the plate boundary Cosine transform in x-direction is used for constant velocity difference across the plate boundary Uniform far-field velocity is simulated by arranging the fault trace to be cyclic in the y-direction Uniform far-field velocity is simulated by arranging the fault trace to be cyclic in the y-direction User defines: d 1, d 2, z obs, x 1, x 2, y 1, y 2, and F

Assign slip rates from literature Paralleling segments: sum to 40 mm/yr Locking depths? depths? San Andreas Fault Segments Use 1099 horizontal GPS velocity measurements to solve for locking depth

Locking Depth Results rms model misfit: 2.43 mm/yr

Predicted Vertical Uplift Geologic estimates San Gabriel Mts 3-10mm/yr 3-10mm/yr [Brown, 1991] San Bernadino Mts 2 mm/yr [Yule and Seih, 1997] Geodetic estimates

Coulomb Stress