Absorbing boundary conditions for LIWEQ

We develop absorbing boundary conditions for Zhiming Li's (1986) migration equation. In the space-time domain, the transforms are:   

$$\pmatrix{x^{'} \cr z^{'} \cr t^{'} \cr } = \pmatrix{ x \c... ... \sqrt{2}} (z-vt) \cr {1 \over \sqrt{2}} (t+{z\over v}) \cr },$$ (14)

and the inverse transforms are:   

$$\pmatrix{x \cr z \cr t \cr } = \pmatrix{ x^{'} \cr {1 \o... ...}+vt^{'}) \cr {1 \over \sqrt{2}} (t^{'}-{z^{'}\over v}) \cr }.$$ (15)

In the frequency-wavenumber domain, the transforms are     

$$\pmatrix{k_{x} \cr k_{z} \cr {\omega \over v} \cr } = \pm... ...\cr {1 \over \sqrt{2}} (k_{z}^{'}+{\omega^{'} \over v}) \cr }.$$ (16)

Then the original dispersion relation changes to

$$k_{x}^{'2}-{2 \over v}k_{z}^{'}\omega^{'}=0.$$ (17)

Because in coordinate transformation rotation, the k_x axis is fixed, so the transformation is actually a rotation with k_x as the symmetry axis. After coordinate rotation, the conic surface for k_x^'>0 is the same as the conic surface for k_x>0, similarly for k_x^'<0 versus k_x<0. So coordinate transformation does not change the propagation directions of the leftward and rightward traveling waves. We apply the leftgoing (or rightgoing) wave equation in the new coordinate system as a boundary condition, then we obtain the absorbing boundary conditions:

$$k_{x}^{'}=\pm \sqrt{ {2 \over v} k_{z}^{'} \omega^{'} }.$$ (18)

To get the difference equation in the space-time domain, we must approximate the square root in (18). The order of k_z^' in the approximate equation must be no more than one, for only upcoming waves are to be present in our migration wavefield. Because under coordinate transformation only our viewing angle is changed, the object (cone) itself is not changed. We can use the approximate formulas in the last section. After coordinate rotation, A1 and A2 change into:

$$B1: k_{x}^{'}= ak_{z}^{'}+ b{\omega^{'} \over v}$$ (19)

and

$$B2: k_{x}^{'}= { {{a \over v}k_{z}^{'}\omega^{'}} \over {ck_{z}^{'}+b{\omega^{'} \over v} } }.$$ (20)

In the space-time domain, B1 and B2 correspond to:

$$B1: P_{x'}+ {b \over v}P_{t'}- aP_{z'}=0$$ (21)

and

$$B2: cP_{x'z'}- {b \over v}P_{x't'}+ {b \over v}P_{z't'}= 0.$$ (22)

Because of coordinate transformation, the (x-z) plane changes to the (x^'-z^') plane, then the rightgoing wave must be considered in thethe new (x^'-z^') plane.

In the (x,z,t) domain, the rightgoing wave is

$$P_{I}= e^{i(kx\cos\theta-kz\sin\theta-\omega t)}$$ (23)

where $k={\omega \over v}$, and $0<\theta<90^{0}$. P_I could not be a downgoing wave, for in our migration wavefield there is only an upcoming wave component, no downgoing wave. After coordinate transformation,

$$P_{I}^{'}=e^{i[kx^{'}\cos\theta+{k \over \sqrt{2}}z^{'}(1-\sin\theta) -{\omega \over \sqrt{2}}t^{'}(1+\sin\theta)]}.$$ (24)

Let the right boundary reflection coefficient be R, then the wavefield near the boundary is:

$$P(x^{'},z^{'},t^{'})=P_{I}^{'}+ P_{R}^{'} =e^{i[kx^{'}\cos... ...'}(1-\sin\theta) -{\omega \over \sqrt{2}}t^{'}(1+\sin\theta)]}$$

$$+e^{i[-kx^{'}\cos\theta+{k \over \sqrt{2}}z^{'}(1-\sin\theta) -{\omega \over \sqrt{2}}t^{'}(1+\sin\theta)]}.$$ (25)

Substituting the above equation into boundary equations, we get:

$$B1:\vert R(\theta)\vert=\left\vert{{\sqrt{2}\cos\theta-b(1+\... ...sqrt{2}\cos\theta+b(1+\sin\theta)+a(1-\sin\theta)} }\right\vert$$ (26)

and

$$B2:\vert R(\theta)\vert=\left\vert{ {c(1-\sin\theta)+b(1+\si... ...ta)+b(1+\sin\theta)+{a\over \sqrt{2}}\cos\theta} }\right\vert.$$ (27)

For a reflection coefficient R₀=0.1,we find the coefficients a, b, and c, by trial and error, so that $\vert R(\theta)\vert \leq R_{0}$ in over as large a $\theta$ range as possible.

B1: a=1, b=0.35

B2: a=17, b=2, c=13.

In Figure 3, the reflection coefficients for B1, B2 are plotted. We apply the condition B2 to migrate three different frequency wavelets, and the results are displayed in Figure 4.

 

fig3
Figure 3 Graph of reflection coefficients for boundary conditions B1 and B2.

 

Figure 4: Migration of three different frequency wavelets,(left) zero-value boundary, (right) absorbing boundary condition B2.