Wavefield extrapolation in pseudodepth domain

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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 and Cartesian coordinates . At each point in space, two sets of basis vectors exist: covariant vectors and contravariant vectors . Basis vectors and 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 is the inner product of a pair of covariant vectors, . The contravariant metric tensor is defined similarly, . The two metric tensors form a pair of inverse matrices . If the curvilinear coordinate system 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 , defined as for the transformation from coordinate system to . In the case that is Cartesian coordinates, we noticed that each row of is one contravariant basis vector , thus the metric tensors can be computed from Jacobian matrix, and .

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

 (29)

and the divergence of is

 (30)

where . Combining these two expression give the Laplacian of scalar

 (31)

 Wavefield extrapolation in pseudodepth domain

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

2013-04-02