Seismic data interpolation with shaping inversion to zero offset and least-squares flattening

Insufficient data often require regularized solutions to inverse problems in geophysics. In other words, we wish to find a model that best matches the data and obeys expected model behavior. Inadequate midpoint sampling is one insufficient data example where large trace intervals can lead to spatial aliasing in seismic data. Inversion to zero offset (IZO) is a solution to this problem that relates zero-offset traces with nonzero-offset traces to find a well sampled model that best matches the inadequately sampled data. IZO can significantly reduce spatial aliasing in stacked seismic data and is typically constrained by the widely known Tikhonov regularization. However, selecting parameters for traditional regularization techniques can be difficult, and the opposing roles of error minimization and model constraint can lead to slow convergence. Shaping regularization is an alternative method that explicitly maps the estimated model to the space of admissible models and, in some cases, can lead to faster convergence. In the first part of this thesis, I present a technique, called shaping IZO, that incorporates inversion to zero offset into the shaping regularization inversion scheme. I show that shaping IZO can reduce spatial aliasing and improve the resolution of stacked seismic data using synthetic and field examples. My results prove that shaping IZO is a viable alternative to traditional IZO.

Insufficient data can also create large areas of missing traces in seismic surveys. In extreme cases, only 2D lines might be available where 3D data are desired. Interpolating over such large areas can be beneficial for later seismic data processing and interpretation but doing so is a challenge. Previous methods, such as interpolation with projection onto convex sets algorithms, prediction error filters, and local slopes have struggled with the sparse and irregular inputs of 2D-to-3D interpolation. In the second part of this thesis, I present interpolation by least-squares flattening as a way to estimate a 3D seismic volume from 2D seismic lines. Interpolation by least-squares flattening first finds the shifts that best align corresponding reflectors of the input traces by incorporating dynamic time warping into a least-squares inversion scheme. After applying these shifts to align reflectors in the input traces, the method then interpolates the flattened reflectors and their associated shifts to a regular 3D grid. Finally, the interpolated shifts are used to unflatten the interpolated reflectors, yielding a 3D volume estimate. I apply interpolation by least-squares flattening to decimated versions of the Teapot Dome land dataset to validate the technique and compare it to other forms of interpolation. My results show that interpolation by least-squares flattening is an effective method to interpolate sparse, irregular traces to a 3D volume.