Controlling incidence angle

In the previous sections the slanted wavefront was synthesized by using the ray parameter p. If the reflector in the subsurface is generally gentle and the lateral velocity change is small, the slanted wavefronts with constant ray parameters are enough to retrieve the angle-dependent reflectivity. However, if the reflector in the subsurface has a complex structure or the lateral velocity change is not small, a slanted wavefront with a constant ray parameter is not enough to retrieve the angle-dependent reflectivity because one need to calculate the incidence angle using the velocity of the subsurface location, which corresponds to the ray parameter of the subsurface location. We then need a plane-wave with a constant-incidence angle to reflectors instead.

In order to synthesize an areal shot record for a slanted wavefront with a constant-incidence angle at a depth level, the impulse in each trace of the predefined wavefield, ${\bf \bar s} (z_n)$, must have time lag, $\Delta \tau_j$,with respect to every other trace. If the trace interval is $\Delta x$ 

$$\Delta \tau_j = \pm {\Delta x \over v_h },$$ (11)

where v_h is the horizontal phase velocity of the propagating wavefront at depth level z_n. Snell's law states that for a stratified earth  

$$p = {\sin(\theta(z)) \over v(z)} = {1 \over v_h} = {\rm constant},$$ (12)

where p is the ray parameter, and v(z) is the depth-dependent velocity of the medium. Combining equations  () and  (), we can obtain the time lag, $\Delta \tau_j$, between the traces to simulate a plane-wave which has a constant incidence angle to the depth level as shown below:

$$\Delta \tau_j = {\Delta x \sin(\theta) \over v(z_n,x_j) }.$$ (13)

Now the synthesized wavefront becomes

$$\bar s_j(z_n) = \exp [i\omega \tau_j]$$

with

$$\tau_j = \tau_{j-1} - {\Delta x \sin(\theta) \over v(z_n,x_j) }.$$

 

marm-syn-pos-2
Figure 14 Synthesis operator that synthesizes a plane-wave with 10 degrees angle at 1200 m depth.

 

marm-stk-pos-2
Figure 15 Wave-stacked section of Marmousi data set with synthesis operator shown in Figure .

The sign convention of the angle, $\theta$, for the slanted wavefront used in this thesis is shown in Figure . Figure  is the stacked section obtained by synthesizing a plane wave with 10 degrees of incidence angle at 1200 m depth (Figure ). It looks very similar to the synthesized stack with a constant ray parameter (Figure ) at the same depth level, but the difference between them becomes clear in the images shown in Figures  and  . The image obtained by synthesizing a constant incidence angle is readily interpretable in terms of an angle-dependent reflectivity of the reflectors where the plane-wave is synthesized, since we can easily calculate the incidence angle to reflector from the angle of synthesized plane-wave and the dip of reflector.

 

anglesign
Figure 16 The sign convention used in this thesis for the slanted plane wavefront.