Example

In this example the conjugate algorithm is tested for a simple layered model. Figure  shows the velocity model and the source function at time equals zero.

 

model Figure 2 Input source wave field at time equals zero and velocity function used for the model.

The result of the forward time-extrapolation is shown in Figure , on the top of the snap shots. It contains the direct arrival, primary reflection, ghost and multiple reflection events. In order to avoid the inclusion of the boundary reflections, time extrapolation was performed before the arrival time of the boundary reflections (see the snap shots in Figure ). Figure  shows the result of the reverse-time migration using the algorithm presented in this paper. It can be seen that the conjugate operator does a good job. The backprojected source wave field recovered closely matching the original source function except for some artifacts. These artifacts are mainly caused by the wave fields propagated downward, and can be suppressed using a least-squares optimization technique (Ji, 1992) by solving an objective function as follows:

$$\min_{\bf m}\vert\vert{\bf d - Wm}\vert\vert^2$$ (8)

where ${\bf W}$ represents the forward time-extrapolation operator in equation (4), and ${\bf d}$ and ${\bf m}$ are the wave fields at depth equals zero and at time equals zero, respectively. The result of the least-squares inversion is obtained after 10 iterations using the conjugate gradient algorithm and is shown in Figure  along with the original wave field and the wave field obtained by the conjugate operator. All the artifacts are well reduced in the least-squares inversion result.

 

frd
Figure 3 Forward time extrapolation experiment: The time trace along with snap shots.

 

rt
Figure 4 Reverse time migration using the algorithm shown in equation (2): The depth section along with snap shots.

 

rtinv Figure 5 Comparison of wave fields: Top is the original wave field. Middle is the result of conjugate operator. Bottom is the result of the least-squares inversion (after 10 iteration).