THE PARABOLIC EQUATION

Here we derive the most basic migration equation via the dispersion relation, equation (). Recall this equation basically says $\cos\theta=\sqrt{1-\sin^2\theta}$. 

$$k_z \eq { \omega \over v} \ \sqrt{ 1 \ \ -\ \ {v^2k_x^2 \over \omega^2 } \ }$$ (1)

The dispersion relation above is the foundation for downward continuing wavefields by Fourier methods in chapter . Recall that nature extrapolates forward in time from t=0 whereas a geophysicist extrapolates information in depth from z=0. We get ideas for our task, and then we hope to show that our ideas are consistent with nature. Suppose we substitute $ik_z=\partial/\partial z$into equation (1), multiply by P, and interpret velocity as depth variable.  

$${\partial P \over \partial z } \eq { i \, \omega \over v(z)} \ \sqrt{ 1 \ \ -\ \ {v(z)^2\, k_x^2 \over \omega^2 } \ } \ P$$ (2)

Since the above steps are unorthodox, we need to enquire about their validity. Suppose that equation (2) were valid. Then we could restrict it to constant velocity and take a trial solution $P=P_0\exp(-ik_z z)$ and we would immediately have equation (1). Why do we believe the introduction of v(z) in equation (2) has any validity? We can think about the phase shift migration method in chapter . It handled v(z) by having the earth velocity being a staircase function of depth. Inside a layer we had the solution to equation (2). To cross a layer boundary, we simply asserted that the wavefield at the bottom of one layer would be the same as the wavefield at the top of the next which is also the solution to equation (2). (Let $\Delta z \rightarrow 0$be the transition from one layer to the next. Then $\Delta P=0$ since $\partial P/\partial z$ is finite.) Although equation (2) is consistent with chapter , it is an approximation of limited validity. It assumes there is no reflection at a layer boundary. Reflection would change part of a downgoing wave to an upcoming wave and the wave that continued downward would have reduced amplitude because of lost energy. Thus, by our strong desire to downward continue wavefields (extrapolate in z) whereas nature extrapolates in t, we have chosen to ignore reflection and transmission coefficients. Perhaps we can recover them, but now we have bigger fish to fry. We want to be able to handle v(x,z),

lateral velocity variation. This requires us to get rid of the square root in equation (2). Make a power series for it and drop higher terms.  

$${\partial P \over \partial z } \eq { i \, \omega \over v(z)... ...{v(z)^2\,k_x^2 \over 2\, \omega^2 } \ \right) \ P \ + \ \cdots$$ (3)

The first dropped term is $\sin^4\theta$where $\theta$ is the dip angle of a wavefront. The dropped terms increase slowly with angle, but they do increase, and dropping them will limit the range of angles that we can handle with this equation. This is a bitter price to pay for the benefit of handling v(x,z), and we really will return to patch it up (unlike the transmission coefficient problem). There are many minus signs cropping up, so I give you another equation to straighten them out.  

$${\partial P \over \partial z } \eq \left( { i \, \omega \ov... ... -\ \ {v(z)\,k_x^2 \over - \, i \, \omega \, 2 } \ \right) \ P$$ (4)

Now we are prepared to leap to our final result, an equation for downward continuing waves in the presence of depth and lateral velocity variation

v(x,z). Substitute $\partial_{xx}=-k_x^2$into equation (4) and revise interpretation of P from $P(\omega,k_x,z)$to $P(\omega, x,z)$.

 

$${\partial P \over \partial z } \eq { i \, \omega \over v(x,... ...er - \, i \, \omega \, 2 } \ {\partial^2 P \over \partial x^2}$$ (5)

As with v(z), there is a loss of lateral transmission and reflection coefficients. We plan to forget this minor problem. It is the price of being a data handler instead of a modeler. Equation (5) is the basis for our first program and examples.