One-way wave-equation

weqrc.3d can be used to describe two-way propagation of acoustic waves in a semi-orthogonal Riemannian space. For one-way wavefield extrapolation, we need to modify the acoustic wave weqrc.3d by selecting a single direction of propagation.

In order to simplify the computations, I introduce the following notation:

&=& 1 Å^2 ,
&=& GJ^2 ,
&=& EJ^2 ,
&=& FJ^2 ,
&=& 1ÅJ JÅ ,
&=& 1ÅJ GÅJ - FÅJ ,
&=& 1ÅJ EÅJ - FÅJ .   All quantities in coefs.3d are only function of the chosen coordinate system, and do not depend on the extrapolated wavefield. They can be computed by finite-differences for any choice of Riemannian coordinates which fulfill the orthogonality condition indicated earlier. In particular, we can use ray coordinates to compute those coefficients. With these notations, the acoustic wave-equation can be written as:  

$$\czz \dtwo{\UU}{\qz} + \cxx \dtwo{\UU}{\qx} + \cyy \dtwo{\U... ...{\qy} + \cxy \mtwo{\UU}{\qx}{\qy} = - \frac{\ww^2}{v^2} \UU \;.$$ (7)

For the particular case of Cartesian coordinates ($\cx=\cy=\cz=0, \cxx=\cyy=\czz=1, \cxy=0$), the Helmholtz weqrc.3d.coefs reduces to its familiar form

$$\dtwo{\UU}{\qz} + \dtwo{\UU}{\qx} + \dtwo{\UU}{\qy} = -\frac{\ww^2}{v^2} \UU \;.$$ (8)

From weqrc.3d.coefs, we can directly deduce the modified form of the dispersion relation for the wave-equation in a semi-orthogonal 3D Riemannian space:  

$$- \czz \kqz^2 - \cxx \kqx^2 - \cyy \kqy^2 +i\cz \kqz +i\cx \kqx +i\cy \kqy - \cxy \kqx\kqy = - \ww^2 s^2 \;.$$ (9)

For one-way wavefield extrapolation, we need to solve the quadratic disp.3d for the wavenumber of the extrapolation direction $\kqz$,and select the solution with the appropriate sign to extrapolate waves in the desired direction:  

$$\kqz = i \frac{\cz}{2\czz} \pm \sqrt{ \frac{\lp\ww s\rp^2}{\... ...-i \frac{\cy}{ \czz}\kqy \rb - \frac{\cxy}{\czz} \kqx\kqy }\;.$$ (10)

The solution with the positive sign in oneway.3d corresponds to propagation in the positive direction of the extrapolation axis $\qz$.

For the particular case of Cartesian coordinates ($\cx=\cy=\cz=0, \cxx=\cyy=\czz=1, \cxy=0$), the one-way wavefield extrapolation equation takes the familiar form

$$\kqz = \pm\sqrt{\lp\ww s\rp^2 -\kqx^2 - \kqy^2}\;.$$ (11)

oneway.3d specialized for the case of 2D coordinate systems obtained by ray tracing is further discussed in Appendix A.

We can use the wavenumber $\kqz$ for recursive wavefield extrapolation of the data recorded on the acquisition surface using the relation

$$\UU_{\qz+\DEL\qz} = \UU_{\qz} e^{i \kqz \DEL\qz} \;,$$ (12)

where $\DEL\qz$ is the discrete extrapolation step.

We can simplify the Riemannian wavefield extrapolation method by dropping the first-order terms in oneway.3d. According to the theory of characteristics for second-order hyperbolic equations (27), these terms affect only the amplitude of the propagating waves. To preserve the kinematics, it is sufficient to keep only the second order terms of oneway.3d:  

$$\kqz = \pm \sqrt{ \frac{\lp\ww s \rp^2}{\czz} - \frac{\cxx}{... ...x^2 - \frac{\cyy}{\czz}\kqy^2 - \frac{\cxy}{\czz}\kqx\kqy }\;.$$ (13)

The coordinate system coefficients for Riemannian wavefield extrapolation given by coefs.3d have singularities at caustics, e.g., when the geometrical spreading term J, defining a cross-sectional area of a ray tube, goes to zero. In the following examples, I use simple numerical regularization to avoid this problem, by adding a small non-zero quantity to the denominators to avoid division by zero.