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Appendix A: Overview of tensor calculus

Transofrmation between general coordinates is conveniently handled by tensor calculus. We consider a genearl curvilinear, possibly nonorthogonal, coordinate system $ \lbrace\xi_1,\xi_2,\xi_3\rbrace$ and Cartesian coordinates $ \lbrace x_1,x_2,x_3\rbrace$ . At each point $ \mathbf{r}(\xi_1,\xi_2,\xi_3)$ in space, two sets of basis vectors exist: covariant vectors $ \mathbf{e}_i = \partial \mathbf{r} / \partial \xi_i$ and contravariant vectors $ \mathbf{e}^i = \nabla \xi_i$ . Basis vectors $ \mathbf{e}_i$ and $ \mathbf{e}^i$ are, in general, not of unit length. A metric tensor is a second-order symmetric tensor from which the unit arc length, unit area and unit volume can be computed easily. Each element of the covariant metric tensor $ g_{ij}$ is the inner product of a pair of covariant vectors, $ g_{ij} = \mathbf{e}_i \cdot \mathbf{e}_j$ . The contravariant metric tensor is defined similarly, $ g^{ij} = \mathbf{e}^i \cdot \mathbf{e}^j$ . The two metric tensors form a pair of inverse matrices $ g_{ij} = [g^{ij}]^{-1}$ . If the curvilinear coordinate system $ \xi_i$ is orthogonal, for example spherical coordinates, the two basis vectors coincide and metric tensors become diagonal matrices.

Transformations between coordinate systems are characterized by Jacobian matrix $ \mathbf{J}$ , defined as $ J_{ij} = \partial x^\prime_i / \partial x_j$ for the transformation from coordinate system $ x_i$ to $ x^\prime_i$ . In the case that $ x_i$ is Cartesian coordinates, we noticed that each row of $ \mathbf{J}$ is one contravariant basis vector $ \mathbf{e}^i$ , thus the metric tensors can be computed from Jacobian matrix, $ [g^{ij}] = \mathbf{J} \mathbf{J}^T$ and $ [g_{ij}] = (\mathbf{J} \mathbf{J}^T)^{-1}$ .

Once the basis vectors and metric tensors are known, differentiations in the curvilinear coordinates $ \xi_i$ is straightforward. The gradient of scalar $ \phi$ is (Riley et al., 2006)

$\displaystyle \nabla \phi = \frac{\partial \phi}{\partial \xi_j} g^{ij} \mathbf{e}_i ,$ (29)

and the divergence of $ \mathbf{f} = f_i\mathbf{e}_i$ is

$\displaystyle \nabla \cdot \mathbf{f} = \frac{1}{\sqrt{g}} \frac{\partial}{\partial \xi_i}\left(\sqrt{g} f_i\right)$ (30)

where $ g = \det(g_{ij})$ . Combining these two expression give the Laplacian of scalar $ \phi$

$\displaystyle \nabla^2 \phi = \frac{1}{\sqrt{g}} \frac{\partial}{\partial \xi_i}\left(\sqrt{g} g^{ij} \frac{\partial \phi}{\partial \xi_j}\right) .$ (31)


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Next: Appendix B: domain traveltime Up: Wavefield extrapolation in pseudodepth Previous: Acknowledgments

2013-04-02