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Next: Anisotropic extrapolation Up: pseudodepth domain wave equation Previous: Vertical time coordinate frame

Isotropic extrapolations

Alkhalifah et al. (2001) obtained $ \tau $ -domain wave equation by applying a coordinate transformation to the conventional eikonal equation and then develop the dynamic part by inverse Fourier transform in space and time to formulate a wave equation in $ \tau $ domain. Because of the high frequency assumption of eikonal equation, the resulting wave equation is not accurate in amplitudes. Here, we derive the coordinate transformation to the differentiations in wave equation directly.

From Equations in 5, we can obtain the Jacobian matrix associated with coordinate transformation from Cartesian domain $ x_i$ to $ \tau $ -domain $ \xi_i$

$\displaystyle \mathbf{J} = \left[\frac{\partial \xi_i}{\partial x_j}\right] = \...
...{bmatrix}1 & 0 & 0  0 & 1 & 0  \sigma_1 & \sigma_2 & 1 / v_m\end{bmatrix} ,$ (7)

where we have denoted horizontal variation of vertical time by $ \sigma_i = \partial \tau / \partial x_i \; (i=1,2)$ . The nonzero off-diagonal elements in the Jacobian matrix $ \mathbf{J}$ indicate that $ \xi_i$ coordinates are nonorthogonal. From the Jacobian matrix we can compute its metric tensor

$\displaystyle [g^{ij}] = \mathbf{J} \mathbf{J}^T = \begin{bmatrix}1 & & \sigma_...
...\sigma_1 & \sigma_2 & \sigma_1^2 + \sigma_2^2 + \frac{1}{v_m^2} \end{bmatrix} .$ (8)

and its determinant $ g = 1 / \det(g^{ij}) = v_m^2$ . Using the Jacobian matrix defined in 7, we obtain the following derivative transformations:

$\displaystyle \frac{\partial}{\partial x_1}$ $\displaystyle = \frac{\partial}{\partial \xi_1} + \sigma_1\frac{\partial}{\partial \xi_3}$    
$\displaystyle \frac{\partial}{\partial x_2}$ $\displaystyle = \frac{\partial}{\partial \xi_2} + \sigma_2\frac{\partial}{\partial \xi_3}$ (9)
$\displaystyle \frac{\partial}{\partial x_3}$ $\displaystyle = \frac{1}{v_m} \frac{\partial}{\partial \xi_3} .$    

A brief overview of the relevant tensor calculus theory is enclosed in Appendix A.

The two-way wave equation may be written in the following first order system

$\displaystyle \frac{\partial p}{\partial t} = v^2 \nabla\cdot\mathbf{q} \quad \mathrm{and} \quad \frac{\partial \mathbf{q}}{\partial t} = \nabla p ,$ (10)

where $ p$ is stress and $ -\mathbf{q}$ is particle momentum. The gradient of a scalar $ \phi$ in a general curvilinear coordinate frame $ \xi_i$ is

$\displaystyle \nabla \phi = \frac{\partial \phi}{\partial \xi_j} g^{ij} \mathbf{e}_i ,$ (11)

which upon substitution of $ \tau $ domain metric tensor 8 gives the following form of the $ \tau $ domain gradient operator

$\displaystyle \nabla \phi$ $\displaystyle = \left(\frac{\partial \phi}{\partial \xi_1} + \sigma_1 \frac{\partial \phi}{\partial \xi_3} \right) \mathbf{e}_1$    
  $\displaystyle + \left(\frac{\partial \phi}{\partial \xi_2} + \sigma_2 \frac{\partial \phi}{\partial \xi_3} \right)\mathbf{e}_2$ (12)
  $\displaystyle + \left[\sigma_1 \frac{\partial \phi}{\partial \xi_1} + \sigma_2 ...
...frac{1}{v_m^2}\right)\frac{\partial \phi}{\partial \xi_3} \right]\mathbf{e}_3 .$    

The divergence of vector $ f = f_i\mathbf{e}_i$ in a general curvilinear coordinate frame $ \xi_i$ is

$\displaystyle \nabla \cdot \mathbf{f} = \frac{1}{\sqrt{g}} \frac{\partial}{\partial \xi_i}\left(\sqrt{g} f_i\right) .$ (13)

Similarly we can find the $ \tau $ domain divergence using equation 8 as follows

$\displaystyle \nabla \cdot \mathbf{f} = \frac{1}{v_m} \frac{\partial}{\partial \xi_i} \left(v_m f_i\right)$ (14)

A $ \tau $ domain two-way wave equation is established by substituting the gradient and divergence operators in equations 12 and 14 into equation 10,

$\displaystyle \frac{\partial p}{\partial t}$ $\displaystyle = \frac{v^2}{v_m} \sum_{i=1}^3 \frac{\partial}{\partial \xi_i} \left(v_m q_i\right)$    
$\displaystyle \frac{\partial q_i}{\partial t}$ $\displaystyle = \frac{\partial p}{\partial \xi_i} + \sigma_i \frac{\partial p}{\partial \xi_3} \quad (i=1,2)$ (15)
$\displaystyle \frac{\partial q_3}{\partial t}$ $\displaystyle = \sigma_1\frac{\partial p}{\partial \xi_1} + \sigma_2\frac{\part...
...\sigma_1^2+\sigma_2^2+\frac{1}{v_m^2}\right)\frac{\partial p}{\partial \xi_3} .$    

This new wave equation is seemingly more complex then the normal two-way wave equation 10. It, however, does not raise the computational cost significantly because the number of differentiations on the right-hand side of the system is $ 6$ for both Equations 15 and 10. The only cost increase comes from the multiplication with the $ \sigma_i$ terms, which is less costly than differentiations. As will be shown in the next section, the additional cost due to $ \sigma_i$ terms is in practice offset by an efficiency gain due to reduced vertical sampling.

If the $ \tau $ coordinate system is orthogonal, in other words, $ v_m$ is laterally constant, and thus, $ \sigma_i = 0$ , then the Jacobian matrix becomes diagonal $ \mathbf{J} = \mathrm{diag}(1,1,1/v_m)$ and the metric tensor $ g^{ij} = \mathrm{diag}(1,1,1/v_m^2)$ . The two-way wave equation 15 simplifies to

$\displaystyle \frac{\partial p}{\partial t}$ $\displaystyle = v^2 \sum_{i=1}^2\frac{\partial q_i}{\partial \xi_i} + \frac{v^2}{v_m}\frac{\partial}{\partial \xi_3}\left(v_m q_3\right)$    
$\displaystyle \frac{\partial q_i}{\partial t}$ $\displaystyle = \frac{\partial p}{\partial \xi_i} \quad (i=1,2)$ (16)
$\displaystyle \frac{\partial q_3}{\partial t}$ $\displaystyle = \frac{1}{v_m^2} \frac{\partial p}{\partial \xi_3} .$    

The simplicity of this equation allows us to reorganize it into a second-order form

$\displaystyle \frac{\partial^2 p}{\partial t^2} = v^2 \left(\frac{\partial^2 p}...
...l}{\partial \tau} \left(\frac{1}{v_m} \frac{\partial p}{\partial \tau}\right) ,$ (17)

which upon expansion becomes

$\displaystyle \frac{\partial^2 p}{\partial t^2} = v^2 \left(\frac{\partial^2 p}...
...2}{v_m^3} \frac{\partial v_m}{\partial \tau} \frac{\partial p}{\partial \tau} .$ (18)

Since the first-order derivatives affect only the amplitude of the solution (Courant and Hilbert, 1989), the last term in Equation 18 can be dropped while retaining a kinematically correct solution, the resulting wave equation is

$\displaystyle \frac{\partial^2 p}{\partial t^2} = v^2 \left(\frac{\partial^2 p}...
...\partial y^2}\right) + \frac{v^2}{v_m^2} \frac{\partial^2 p}{\partial \tau^2} .$ (19)

When both $ v$ and $ v_m$ are constants, the wavefront described by Equation 19 is an ellipse. This suggests that elliptical anisotropy can be viewed as a linear change of the variable $ \tau = z / v_m$ to isotropic velocity.

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Next: Anisotropic extrapolation Up: pseudodepth domain wave equation Previous: Vertical time coordinate frame