Frequency-selection LSRTM

Due to data redundancy, the frequency domain data can be represented by a coarser sampling than the Nyquist rate. Figure 3.5 illustrates how to estimate the necessary sampling. According to Mulder and Plessix (2004b), the maximum frequency sampling

$$\Delta f_{max} = \frac{1}{T_{max}-T_{min}}.$$ (313)

In this example, as explained in Figure 3.5, $T_{min}$ is about 1.49 sec and $T_{max}$ is about 2.91 sec calcuated with average velocity 2.5 km/s, so that the $\Delta f_{max}$ is about 0.70 Hz. I choose 0.625 Hz as the frequency sampling for convenience, and thus the range of 0-50 Hz can be represented by 80 frequency samples. The frequency-selection encoding scheme is as follows. Each shot is assigned one out of 400 frequencies, and then blended to form a supergather. For the next iteration, the frequency assigned to each shot is increased by 0.625 Hz. When the frequency of a shot exceeds 50 Hz, it will be wrapped around to the low frequencies. Therefore, 80 such LSRTM iterations can sample the frquency domain of interest.

A reflectivity model is derived from the true velocity by high-pass filtering and shown in Figure 3.6 as the benchmark. A pseudo-spectral method is used to compute the true data of 400 CSGs and Figure 3.7 depicts a shot gather that is fired at $x=3~km$ after muting the direct waves. Those 400 shot gathers are then transformed into the frequency domain and 80 supergathers are formed with the above frequency-selection encoding strategy.

First, the iterative stacking method is applied to the 80 supergathers, where those 80 supergathers are migrated with the migration velocity in Figure 3.1(b) and stacked together. The image is shown in Figure 3.8(b) and it is almost identical to the conventional shot domain RTM image in Figure 3.8(a).

To reduce the migration artifacts and improve the image resolution, the same 80 supergathers are migrated with the LSRTM algorithm with one supergather for each iteration. Figure 3.9(a) plots the image for the first iteration, which contains strong ringing artifacts. As iterations proceed, the LSRTM image quality gradually improves (see Figure 3.9(b) for 20-iteration result). Figure 3.9(c) shows the image after 80 iterations which is of higher resolution than the conventional RTM image in Figure 3.8(a). The zoom views of the shallow part (Figure 3.11) and the deep part (Figure ) show similar resolution enhancement. The drawback is that there is high frequency noise present in the LSRTM image.