Tomographic inversion with matrix operators governing rays: Its application to Lake Superior refraction and reflection velocity reconstruction

This research presents a new algorithm for ray tracing tomography. Matrix operators are introduced in the ray tracing algorithm to make the method general, accurate, robust, and simple. Two matrix operators are derived for tracing rays: one for propagation rays and the other for reflection rays. Propagation matrix elements depend on the velocity gradient field and the ray direction; reflection matrix elements only depend on the reflector orientation. All matrix elements are derived from the basic ray equation. Any exit ray can be obtained by simply operating the matrix on its incident ray. Representation of the formulation in vector and matrix form avoids solving differential equations and makes it consistent and simple to implement the algorithm in both 2-D and 3-D regardless of the complexity of the velocity field in space. In the course of ray calculations, error controls are imposed on the algorithm. Any step of the ray tracing is guaranteed to be within a given error tolerance which can be set to any small value. Thus, ray tracing accuracy can meet any preset error tolerance at the expense of computer time.; Analytical formulae are presented for calculating interceptions of any ray with any plane surface in 3-D space. Finding those interceptions is almost indispensable in performing ray tracing in space. All the necessary ray calculation tools are presented for 3-D ray tracing. Any 3-D model parameterization method that defines its model with plane surfaces will be able to take advantage of the matrix operator ray tracing scheme.; A convenient block parameterization of the velocity structure is adopted. This parameterization method is very suitable for crustal models. It greatly reduces the number of parameters that are inverted. Synthetic tests show the suggested ray tracing algorithm coupled with this parameterization method is very accurate and effective.; This developed inversion method is applied to the velocity reconstruction under Lake Superior. The tomographic inversion has yielded a velocity structure that shows good agreement with previous forward modeling results except that it reveals that two important units of the Lake Superior basin have a somewhat shallower depth (by 3-4 km) and slower velocity (by {dollar}sim{dollar}0.5 km/s). This might indicate that the sedimentary interflow deposition process was even stronger than that suggested in past studies. Considering so many strong reflections that are present in the GLIMPCE multichannel reflection profile data that have not been fully explained in structure interpretations, this finding and the explanation are highly possible.; To support the velocity structure found by seismic tomography, gravity modeling has been done. The velocity values used in the seismic inversion are converted one-to-one into density values using the Nafe-Drake velocity-density relationship. The modeling results show that the structural model defined from the tomographic inversion produces a gravity anomaly that matches the observed anomaly profile on Lake Superior, whereas the model from past forward modeling research fails to generate one that matches the observations. The gravity anomaly match is best in the graben area and becomes poorer away from the graben. Thus, the velocity structure produced is considered reasonably accurate, especially in the graben area. The poor match on the two ends of the graben is probably caused by less seismic control because fewer rays sampled those regions.