Least-squares Inversion of Spatial Seismic Refraction Data by Ernest R. Kanasewich and Stephen

A method is demonstrated for recording and inverting in-line and broadside seismic refraction and reflection data to obtain three-dimensional structure and velocity information. This technique of spatial seismic refraction recording produces a superabundance of common ray intersections at the target horizon. The full power of the linear least-squares inversion method may be used in the overconstrained case. A large number of recording instruments are required to perform the experiment efficiently but this has the advantage of reducing the number of sources and difficult source corrections. A damped least-squares inversion method has been found to give rapid convergence even in the case of complex faulted models using noisy data. The spatial seismic refraction recording method has been applied in a suboptimal experiment over the Williston Basin in southern Saskatchewan. The crustal structure consists of several normally faulted blocks defined by aeromagnetic anomalies over an area notable for its thick crust, local seismicity and a linear conductive anomaly. INTRODUCTION Over the past decade, considerable effort has been directed toward methods of geophysical inversion. The purpose of seismic inversion is to extract information about the earth's structure and physical parameters, including an estimate of the resolution and variance, from a set of seismic observations. When the recorded data is, in the words of Jackson (1972), "inaccurate, insufficient and inconsistent," the mathematical robustness of the least-squares approach is one of the standard tools for the geophysical inversion. The linear least-squares inversion method (Backus and Gilbert 1967, 1968, 1970) has been widely applied to diverse geophysical problems. Some of the studies include the inversion of global earth data by using free oscillation periods (Wiggins, 1972; Wiggins et al., 1976), the inversion of teleseismic travel-time data (Aki and Lee, 1976; Hawley et al., 1981), the inversion and migration of three-dimensional seismic data (Gjoystdal and Ursin, 1981), and the determination of an impedance profile (Cooke and Schneider, 1983). Recently, Lines and Treitel (1984) gave a detailed review of linear least-squares inversion and its applications to geophysical problems. The purpose of this paper is to describe a procedure for obtaining a threedimensional structural and velocity model from a combination of in-line and broadside seismic refraction and reflection data. It involves a new iterative processing method based on the least-squares inversion technique. A two-dimensional surface spatial array of sources and receivers is used to generate the seismic observations. The spatial superabundance of common intersections at the target horizon may be interpreted with efficient and flexible ray-generating algorithms. The computation of seismic travel times from a forward model is based on the three-dimensional ray tracing method (Chander, 1977), and the linear inversion is formulated as an iteratively damped least-squares technique (Levenberg, 1944; Marquardt, 1963). The damping factor, which adds to the diagonal parameters of the matrix, is computed automatically for each iteration (Hoerl and Kennard, 1970; Hoerl et al., 1975). 865 866 ERNEST R. KANASEWICH AND STEPHEN K. L. CHIU The formulation of Chander's algorithm provides a flexible method of incorporating desirable features in a seismic problem. These include: (1) any number of layer interfaces with arbitrary dips; (2) low-velocity layers interbedded with higher velocity layers; (3) critical refraction or multiple reflections along the ray path; (4) the incorporation of geological constraints; and (5) the lateral variations of velocity along the ray path. Although, the restrictions of this approach require a constant velocity within a layer and plane interfaces, it is easy to modify the solution to allow for continuous velocity variation with depth and curvilinear interfaces if there is sufficient observational data. SPATIAL SEISMIC REFRACTION RECORDING The ideal seismic refraction or reflection experiment is one that yields accurate, high resolution, three-dimensional structure with a minimum number of sources. An effective method of obtaining three-dimensional structure of the solid crust is ~ S o u r c e s ( 7 ) I n t o Al l R e c e i v e r s + R e c e i v e r s ( 4 8 0 ) ~',II',I',',II~ S c a l e : lO a n d 100 K~a. F1G. 1. A spatial seismic refraction recording array to obtain three-dimensional structure on the Mohorovi~i5 discontinuity. The triangle of receivers has dimensions of 350 km and a receiver spacing of 2.5 km. The lines are surface projections of rays traversing below the Moho. Each source is recorded by all the receivers. to place elastic wave sources at the vertices and mid-points of a equilateral triangle of receivers. Each source is recorded sequentially by all the receivers around and interior to the triangle (Figure 1). In an actual field experiment, the deployment of receivers and sources would be constrained by the availability of access routes. However, the method is very robust, and major variations in the geometry do not complicate the inversion process. The reversed in-line refraction data gives the control on the velocities while the broadside refraction and wide angle reflection data yield the detailed structure over the area of interest. In continental crustal refraction experiments, the deepest horizon of interest (Moho) may vary in depth from 30 to 60 km. Therefore, source to receiver distances may need to extend to a maximum of 300 to 400 km. However, intermediate layers at depths of 10 to 25 km are of great interest to tie near vertical incidence reflection surveys and geological outcrops. This objective may be met with sources at the midpoints of the triangle of receivers. The projection of the head wave portion of the ray path for one third of the area is shown in Figure 2. In marine refraction surveys, the role of source and receiver is often reversed. A limited number of ocean bottom LEAST-SQUARES INVERSION OF SEISMIC REFRACTION DATA 867 seismometers is used together with a mobile (air gun) source moving along the triangular path. A large number of identical independent event recording instruments are desirable to reduce the environmental and monetary cost of chemical sources. There is also a scientific advantage to having only one or two explosions at each source location since this reduces the complication of making many difficult source corrections. For good phase correlation, the spacing of receivers for crustal studies should be between 0.5 to 3 km. For a 350-km equilateral triangle and 1 km spacing, about 1100 recorders are required. This is about five times the capability of any national seismic resource base at the present time. Present plans in Canada call for the acquisition of 240 identical digital recording seismic refraction instruments for PROJECT LITHOPROBE. Scientists in the United States are recommending that 1000 such instruments be made available (Report to the National Academy of Sciences, the Panel on Seismological Studies of the Continental Lithosphere, Committee on Seismology, George Thompson, Chairman). In Figures 1 and 2, an example of the deployment of 480 (2 x 240) such instruments is shown with a receiver spacing of