Spectral factorization is the problem of finding a minimum-phase function with a given power spectrum. Minimum-phase functions have the property that they are causal with a causal (stable) inverse. In this thesis, I factor multidimensional systems into their minimum-phase components. Helical boundary conditions resolve any ambiguities over causality, allowing me to factor multi-dimensional systems with conventional one-dimensional spectral factorization algorithms.
In the first part, I factor passive seismic wavefields recorded in two-dimensional spatial arrays. The result provides an estimate of the acoustic impulse response of the medium that has higher bandwidth than autocorrelation-derived estimates. Also, the function's minimum-phase nature mimics the physics of the system better than the zero-phase autocorrelation model. I demonstrate this on helioseismic data recorded by the satellite-based Michelson Doppler Imager (MDI) instrument, and shallow seismic data recorded at Long Beach, California.
In the second part of this thesis, I take advantage of the stable-inverse property of minimum-phase functions to solve wave-equation partial differential equations. By factoring multi-dimensional finite-difference stencils into minimum-phase components, I can invert them efficiently, facilitating rapid implicit extrapolation without the azimuthal anisotropy that is observed with splitting approximations.
The final part of this thesis describes how to calculate diagonal weighting functions that approximate the combined operation of seismic modeling and migration. These weighting functions capture the effects of irregular subsurface illumination, which can be the result of either the surface-recording geometry, or focusing and defocusing of the seismic wavefield as it propagates through the earth. Since they are diagonal, they can be easily both factored and inverted to compensate for uneven subsurface illumination in migrated images. Experimental results show that applying these weighting functions after migration leads to significantly improved estimates of seismic reflectivity.