Wavefield extrapolation in pseudodepth domain |

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 |

2013-04-02