Precision analysis of first-break times in grid models

Rectangular grid velocity models and their derivatives are widely used in geophysical inversion techniques. Specifically, seismic tomographic reconstruction techniques, whether they be based on raypath methods (Bregman et al., 1989; Moser, 1991; Schneider et al., 1992; Cao and Greenhalgh, 1993; Zhou, 1993) or full wave equation methods (Vidale, 1990; Qin and Schuster, 1993; Cao adn Greenhalgh, 1994) for calculating synthetic arrival times, involve propagation through a grid model. Likewise, migration of seismic reflection data, using asymptotic ray theory or finite difference/pseudospectral methods (Stolt and Benson, 1986; Zhe and Greenhalgh, 1997) involve assigning traveltimes to upward and downward propagating waves at every grid point in the model. The traveltimes in both cases depend on the grid specification. However, the precision level of such numerical models and their dependence on the model parameters is often unknown. In this paper, we describe a two-dimensional velocity model and derive an error bound for first-break times calculated with such a model. The analysis provides clear guidelines for grid specifications. The technique developed in this paper for first-arrival time calculations is equally applicable to diffractions and reflections. It is simply a case of adding times from source to grid point (diffractor) to that from grid point to receiver to recover the later arrivals. Therefore, the conclusions from this study concerning accuracy of first-break times are equally applicable to reflection tomography and seismic migration as they are to transmission tomography and reflection. It should be stressed that the errors due to grid discretization are more severe than is generally appreciated. The reason is that many traveltime calculation schemes work just with fixed primary cells (albeit with adjustable spacing) rather than using the more flexible nodal subdivision scheme adopted here and described below. Our model discretizes a velocity field into a two-dimensional rectangular grid of constant velocity elements, called cells (see Figure 1). Cells are square and of equal size. We place primary nodes at the corners of all cells, and secondary nodes along the boundaries of all cells such that all nodes are equally spaced along boundaries. Sources and receivers are positioned at primary nodes. We shall call two nodes neighbors if they have at least one cell in common. Note tht nodes along the sme boundary always have two cells in common.; A 2D velocity model is described and an error bound is derived for first break times calculated with such model. The model discretizes a velocity field into a 2D grid of equally sized square constant velocity elements called cells. Primary nodes are placed at the corners of all cells and secondary nodes along the boundaries. Sources and receivers are positioned at primary nodes. The upper error bounds are calculated from a given number of secondary nodes and are determined primarily by the maximum angular between exact and approximated travelpaths, unless additional information about the wavefront is stored with each node. The analysis provides clear guidelines for grid specifications.