Self-adaptive Reference velocity choice with lateral velocity variations

The background velocity or reference velocity should be chosen to match the true velocity distribution as closely as possible. The more closely the reference velocity field mimics the true velocity, the more accurately the wave-field propagation can be calculated and the higher the modeling and imaging quality will be.

We redefine the velocity perturbation as  

$$\triangle S\left( x_{i},y_{j},z\right) = S\left( x_{i},y_{j},z\right)- S_{0}^{l}\left( x_{i},y_{j},z\right),$$ (17)

where l is the index of the selected reference velocity. $S_{0}^{l}\left( x_{i},y_{j},z\right)$ means that there is a reference velocity with the index l at a spatial point $\left( x_{i},y_{j}\right)$. In the extreme case, the maximum number of the reference velocities equals to the number of discrete spatial points in the extrapolation layer. Theoretically, every spatial point could be assigned a reference velocity, but that would entail a huge calculation cost. Generally, a set of reference velocities is chosen, with which the wavefield extrapolations are carried out. Then the extrapolated wavefields are merged together with some chosen methods Gazdag and Sguazzero (1984); Kessinger (1992). However, it is difficult to choose a set of reasonable reference velocities for modeling and imaging, if the velocity laterally varies severely and irregularly. Therefore, we propose the following self-adaptive strategy for medium with lateral velocity variations. The procedure is as follows:

(1) Assign a threshold value of the ratio of two adjoining reference velocities, according to numerical experiments and experience. The threshold reflects how severe the lateral velocity variation is. Generally, we define the ratio to be greater than 1.0. Of course the threshold could be less than 1.0, but this would require sorting the discrete velocity values in decreasing order in the third step.

(2) Filter the 2D or 3D velocity slice of the depth layer with a median filter to eliminate possible wild velocity values.

(3) Sort the discrete velocity values into an array in increasing order.

(4) Set the summation value equal to zero, then sum the sorted discrete velocity values from left to right and point-by-point, calculating a cumulative velocity average, if a velocity value is added, with the formula $v_{avg}^{l}=\frac{\sum\limits_{k=1}^{K^{l}}v_{k}^{\tilde{m}}\left( x_{i},y_{j}\right) }{K^{l}}$ . Here, $\tilde{m}=1,...,NX$ in the 2D case and $\tilde{m}=1,...,NX \times NY$ in the 3D case, where $\tilde{m}$ are the sequence numbers of the discrete velocity points that were disordered by sorting. K^l is the number of discrete velocity points in the $l^{\mbox{th}}$ region, which is known after a dividing point is determined; l is the number of the reference velocity. If the ratio between the velocity value at the next point and the velocity average at the current point is greater than the preset threshold, we can judge that there exists a velocity boundary at the current point. This point is a dividing point between two velocity regions.

(5) Starting from the dividing point, repeat Step 4 to finding each successive dividing point until the end of the sorted velocity array is reached.

(6) Repeat Step 4 and Step 5. This time, the ratio between the cumulative velocity average at the current point and the velocity average at the adjacent and next point (which is calculated in the last iteration) is used, and if it is greater than the preset threshold value, the dividing point between two velocity regions is determined.

(7) If the velocity dividing points are not changed and the velocity averages are not changed, stop the iterative procedure. Otherwise, repeat Step 6.

We can see that the number of velocity regions is the number of chosen reference velocities. The velocity average in a velocity region is a reference velocity value. With the set of reference velocities, wavefield extrapolation is carried out with SSF, FFD, or GSP operators. For computational efficiency, the number of reference velocities should be kept small, usually less than 5 or 6. It should be mentioned that Step 6 and 7 are optional.

In order to merge extrapolated wavefield, the corresponding sequence number between the input discrete velocity slice and the sorted discrete velocity value should be recorded. This is important for the approach.

The approach can be used in 2D or 3D, poststack or prestack, and time or depth migration or modeling, as long as the seismic-wave propagation is characterized with perturbation theory. This method can be used for the image edge detecting as well.

If necessary, the velocity variances may be used as the criterion for dividing the velocity regions. Usualy, it is sufficient to choose a set of reference velocities with velocity averages for migration.