Unified multi‐depth‐level field decomposition. (22nd July 2015) - Grobbe, Niels; van der Neut, Joost; Slob, Evert; Wapenaar, Kees; Almagro Vidal, Carlos; Drijkoningen, Guy
ABSTRACT: Wavefield decomposition forms an important ingredient of various geophysical methods. An example of wavefield decomposition is the decomposition into upgoing and downgoing wavefields and simultaneous decomposition into different wave/field types. The multi‐component field decomposition scheme makes use of the recordings of different field quantities (such as particle velocity and pressure). In practice, different recordings can be obscured by different sensor characteristics, requiring calibration with an unknown calibration factor. Not all field quantities required for multi‐component field decomposition might be available, or they can suffer from different noise levels. The multi‐depth‐level decomposition approach makes use of field quantities recorded at multiple depth levels, e.g., two horizontal boreholes closely separated from each other, a combination of a single receiver array combined with free‐surface boundary conditions, or acquisition geometries with a high‐density of vertical boreholes. We theoretically describe the multi‐depth‐level decomposition approach in a unified form, showing that it can be applied to different kinds of fields in dissipative, inhomogeneous, anisotropic media, e.g., acoustic, electromagnetic, elastodynamic, poroelastic, and seismoelectric fields. We express the one‐way fields at one depth level in terms of the observed fields at multiple depth levels, using extrapolation operators that are dependent on the medium parameters between the two depth levels. Lateral invariance at the depth level of decomposition allows us to carry out the multi‐depth‐level decomposition in the horizontal wavenumber–frequency domain. We illustrate the multi‐depth‐level decomposition scheme using two synthetic elastodynamic examples. The first example uses particle velocity recordings at two depth levels, whereas the second example combines recordings at one depth level with the Dirichlet free‐surface boundary condition of zero traction. Comparison with multi‐component decomposed fields shows a perfect match in both amplitude and phase for both cases. The multi‐depth‐level decomposition scheme is fully customizable to the desired acquisition geometry. The decomposition problem is in principle an inverse problem. Notches may occur at certain frequencies, causing the multi‐depth‐level composition matrix to become uninvertible, requiring additional notch filters. We can add multi‐depth‐level free‐surface boundary conditions as extra equations to the multi‐component composition matrix, thereby overdetermining this inverse problem. The combined multi‐component–multi‐depth‐level decomposition on a land data set clearly shows improvements in the decomposition results, compared with the performance of the multi‐component decomposition scheme.