Wave Gradiometry and its Link with Helmholtz Tomography Applied to USArray Data

Abstract:

Wave gradiometry is a new technique utilizing the shape of seismic wave fields captured by USArray transportable stations to determine fundamental wave propagation characteristics. It is designed to take advantage of the wavefield variations and resolve structure that cannot be derived strictly from averaging methods. The spatial gradients, ground displacements and ground velocities are linearly linked by two unknown coefficients, which can be used to infer wave phase velocity, back azimuth, radiation pattern and geometrical spreading [Langston, 2007]. Spatial gradients are estimated from shifted displacement fields using bi-cubic spline interpretation [Beavan and Haines, 2001]. With singular value decomposition, the coefficients are solved iteratively at each seismic station, which are then inverted for wave parameters described above. However, the phase velocity, depends not only on the medium but also on the local geometry of the wavefield, which is referred as dynamical phase velocity [Wielandt, 1992]. In order to find the structural phase velocity determined only by the properties of the medium, an amplitude correction, involving the first and second derivatives of logarithmic amplitude, is required [Wielandt, 1992]. We show that the two coefficients that we are solving for actually correspond to the gradient of logarithmic amplitude and the gradient of phase, respectively. Treating this vector field in the same way that we treat displacement fields, the Laplacian of logarithmic amplitude and Laplacian of phase can be determined. This enables us to estimate pseudo-structural phase velocities through solutions to the Helmholtz equation, assuming that the structure is smoothly inhomogeneous compared to the heterogeneity of the wave field [Wielandt, 1992]. Furthermore, the density can be estimated by integrating in the direction of wave propagation, using also the first and second derivatives of the phase. Numerical experiments with synthetic data sets provided by Princeton University’s Neal Real Time Global Seismicity Portal are being conducted to test the algorithm stability and evaluate errors. Our modification of the wave gradiometry method is also being employed across the USArray to obtain structural parameters from mid-west to the eastern coast of the U.S.