WKBJ asymptotic solutions to the acoustic wave-equation in 3-D Riemannian spaces

Neglecting wave propagation in the directions orthogonal to $\zeta$, one can reduce the wave equation (6) to the form of the ordinary differential equation  

$$\frac{1}{\AA\,J}\,\frac{d}{d\,\zeta}\, \left(\frac{J}{\AA}... ...}}{d\,\zeta^2} + \frac{\omega^2}{v^2}\,\mathcal{U}\approx 0\;.$$ (25)

The high-frequency (WKBJ) asymptotics for the solution of equation (25) can be obtained by using a trial solution $\mathcal{U}= A\,e^{i\,\omega\,\phi}$, substituting it in equation (25) and evaluating terms with the same order of $\omega$. The highest asymptotic order yields an equation for the phase function $\phi$: 

$$-\frac{1}{\AA^2}\,\left(\frac{d\,\phi}{d\,\zeta}\right)^2 + \frac{\omega^2}{v^2} = 0\;.$$ (26)

The next asymptotic order produces an equation for the amplitude function A:  

$$\frac{1}{\AA J}\,\frac{d\,(J/\AA)}{d\,\zeta}\, \frac{d\,\p... ...ta} + \frac{1}{\AA^2}\,\frac{d^2\,\phi}{d\,\zeta^2}\,A = 0\;.$$ (27)

Rearranging equation (26) to the form  

$$\frac{d\,\phi}{d\,\zeta} = \pm \frac{\AA}{v}$$ (28)

and equation (27) to the form  

$$\frac{d\,(\log{A})}{d\,\zeta} = - \frac{1}{2}\,\left[ \f... ...ta} + \frac{d\,\left(\log{(\AA/v)}\right)}{d\,\zeta}\right]\;,$$ (29)

we can solve them explicitly to obtain the WKBJ approximation for the wave traveling preferentially in the $\zeta$ direction:  

$$\mathcal{U}_1 \approx \mathcal{U}_0\,\left(\frac{v_1\,J_0}{... ... \int\limits_{\zeta_0}^{\zeta_1} \frac{\AA}{v}\,d\zeta \right]}$$ (30)

In the case of the ray coordinate system, equation (30) corresponds to the Green's function approximation commonly employed in Kirchhoff imaging.

Accounting for the wave propagation in the directions different from $\zeta$and constructing the solution numerically by finite differences allows us to account for the finite-bandwidth wave propagation effects.

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