Amplitude preserving one-way wave equation

The standard one-way wave equation is  

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

where k_x is the wave number in the x direction, and v is the velocity. From the dispersion relation, we have

$$k_z=\frac{\omega}{v}\sqrt{1-\frac{v^2k_x^2}{\omega^2}},$$ (2)

where k_z is the wavenumber in the z direction. Equation (1) mimics the phase behavior of the acoustic wave equation:  

$$\frac{1}{v^2}\frac{\partial^2 }{\partial t^2}P- \left( \fra... ... }{\partial x^2 } + \frac{\partial^2}{\partial z^2} \right)P=0,$$ (3)

although the amplitude is not correct. Zhang (1993) suggests that to maintain the dynamics information of equation (3), an extra amplitude correction term needs to be included into the standard one-way wave equation. The new one-way wave equation is  

$$\frac{\partial P}{\partial z}=\left [ ik_z-\frac{1}{2v}\fr... ...left( 1+\frac{(vk_x)^2}{\omega^2-(vk_x)^2} \right) \right]P.$$ (4)

We call it the amplitude preserving one-way wave equation. By equation (4), the wavefield can be extrapolated as the following:

$$\begin{eqnarray} P(z+\Delta z)&=&e^{ik_z\Delta z-\frac{1}{2v}\frac{\partial v}{\... ...\omega^2-(vk_x)^2} \right)\Delta z} \cdot e^{ik_z\Delta z}P(z) .\end{eqnarray}$$ (5) (6)

In equation (6), first term defines the amplitude information of the wavefield, and second term $e^{ik_z\Delta z}$ defines the phase information of the wavefield.