Topography-Dependent Eikonal Traveltime Tomography for Upper Crustal Structure Beneath an Irregular Surface

Abstract

Seismic modeling of the crust with nonflat topography can be made by first-arrival traveltime tomography, which faces the challenge of an irregular free surface. A feasible way to deal with this problem consists of expanding the physical space by overlapping a low velocity layer above the irregular surface in order to have a flat topography, besides using the classical eikonal equation solver for traveltime computation. However, the undesirable consequences of this method include seismic ray deviations due to the transition from an irregular surface that is the free boundary to an inner discontinuity lying in the expanded computational space. An alternative solution, called irregular surface flattening, which involves the transformation between curvilinear and Cartesian coordinate systems, has been recently proposed through the formulation of the topography-dependent eikonal equation (TDEE) and a new solver for forward modeling of traveltimes. Based on the solution of this equation, we present topography-dependent eikonal traveltime tomography (hereafter TDETT) for seismic modeling of the upper crust. First-arrival traveltimes are calculated using the TDEE solver and the raypaths with the minimum traveltime that can be found by following the steepest traveltime gradient from the receiver to the source. By solving an algebraic equation system that connects the slowness perturbations with the already determined traveltimes, these variables can be obtained making use of the back-projection algorithm. This working scheme is evaluated through three numerical examples with different topographic complexities that are conducted from synthetic data and a fourth example with somewhat more complicated topography and real data acquired in northeastern Tibet. The comparison of the results obtained by both methods, i.e., physical space expansion above the irregular surface and irregular surface flattening, fully validates the tomography scheme that is proposed to construct seismic velocity models with nonflat topography.

Appendix: Irregular Surface Flattening and the Model Parameterization

Irregular surface flattening is achieved by a transformation between the curvilinear and the Cartesian coordinate system (Fig. 1b), which allow the irregular surface to be described using discrete grids that conform to the free surface in order to suppress artificial errors. Such a grid is termed a "boundary-conforming grid" (Thompson et al., 1985; Hvid, 1994), and has been used by a number of researchers (Fornberg 1988; Zhang and Chen, 2006; Appelo and Petersson, 2009; Lan and Zhang, 2011, 2013a, b). Under this transformation, the curvilinear coordinates q and r are mapped into Cartesian coordinates x and z within the gridded physical space, with both systems having a positive downward direction for the vertical coordinate. A boundary in the gridded physical space is represented by a constant value of one of the curvilinear coordinates, be it a curve in two dimensions or a surface in three dimensions. The transformation between the curvilinear and Cartesian coordinate systems has the following equations:

$$x = x(q, r),$$

(A1)

$$z = z(q, r),$$

(A2)

where the gridded physical space (x,z) with an irregular surface is converted to a computational space (q, r) with a flattened surface (Fig. 1b). Boundary-conforming grids may be of two fundamentally different types: structured and unstructured. A structured grid is characterized by a fixed number of elements along each coordinate direction, and the general element is a quadrilateral in 2D. Neighboring elements in the gridded physical space are also adjacent to one another in the computational space, which is one of the great advantages of this type of grid, and ensures its relative simplicity of application in a computer program. Structured grids are widely used in finite difference and finite volume schemes. Here, we focus on the use of structured boundary-conforming grids for the model parameterization. The irregular surface is still a free surface. A number of methods may be used to generate such grids, including partial differential equations (PDE), and algebraic, co-normal mapping, and variation methods (Thompson et al., 1985). Here, we utilize PDE methods (see Thompson et al., 1985 and Hvid, 1994 for details).

Using the chain rule to express the spatial derivatives in the Cartesian coordinate system (x,z) from the curvilinear coordinate system (q,r):

$$\partial_{x} = q_{x} \partial_{q} + r_{x} \partial_{r} ,$$

(A3)

$$\partial_{z} = q_{z} \partial_{q} + r_{z} \partial_{r} ,$$

(A4)

and:

$$\partial_{q} = x_{q} \partial_{x} + z_{q} \partial_{z} ,$$

(A5)

$$\partial_{r} = x_{r} \partial_{x} + z_{r} \partial_{z} ,$$

(A6)

where q _x denotes ∂q(x, z)/∂x (and its equivalents with other variables). These derivatives are known as metric derivatives. They can be written in the following forms:

$$q_{x} = \frac{{z_{r} }}{J},\,\,q_{z} = \frac{{ - x_{r} }}{J},\,\,r_{x} = \frac{{ - z_{q} }}{J},\,\,r_{z} = \frac{{x_{q} }}{J},$$

(A7)

where J is the Jacobian determinant of the transformation and can be written as J = x _q z _r −x _r z _q .

It is worthy to note that even if the mapping Eqs. (A1) and (A2) are expressed as an analytical function, the derivatives should still be calculated numerically to avoid spurious source terms that may be caused by the derivative coefficients when the conservative forms of the equations are used (Thompson et al., 1985). The mapping must be smooth and with a fixed number of grid cells (Zhou et al., 2012). In all the examples presented here, the metric derivatives are computed numerically using second-order finite-difference approximations.