Next, we compareresults from the regular JMI without any regularization, JMI with conventional TV, and JMI with directional TV. The frequency bandwidth during the second step of JMI is Hz Hz. We use the same and for both the conventional TV and directional TV. is also increasing with iteration and . For directional TV, and . For the conventional TV, is an identity matrix. After 50 iterations for each method, the inverted results are shown in Figure 12 and Figure 13. Because of the inversion process included in JMI, all the images in Figure 13 are quite accurate compared to the true reflectivity structures. Furthermore, all the estimated velocity models in Figure 12 are also surprisingly stable and show some details.

In Figure 12a, the regular JMI without any regularization is slightly trapped in a local minimum, e.g., in the lower right area pointed by the red arrow. With the help of TV regularization, JMI with conventional TV in Figure 12b achieves a better result by smoothing the model via enhancing the sparsity of the spatial gradient of the velocity difference, which allows us to steer away from the local minimum. Instead of using the conventional TV, a much better inverted velocity with clearer edges of structures is obtained in Figure 12c using JMI with directional TV. This is because we consider the structural directions of the spatial gradient and their weights according to the local dip from the associated image. Please note some obvious improvements pointed out by the white arrows. In addition, compared to L1 directional TV, L2 directional Laplacian smoothing results in a smoother velocity model (Figure 12d); however, it intensifies the local minima issue and tends to produce models with blurred discontinuities. That is because the directional Laplacian smoothing may over-smooth the velocity and cannot preserve edges very well; it is also more sensitive to the accuracy of the estimated dip field, compared to L1 directional TV. As a result of the improvement of the inverted velocity, the inverted reflectivity also becomes more accurate (Figure 13): The inverted reflectivities highlighted with white arrows in Figure 13c have better focussing and less distortions than the other alternatives.

Note that the velocity field estimated from JMI has less details compared to that from FWI, as it only needs to describe propagation, not reflection. Similar as in the FWI example, we show in Figure 14 the modeled data generated from each of the final inverted velocities and the corresponding differences with the observed data. From this figure, we can see that regularizations on velocity do not make much difference in the data residual, because the velocity in JMI only explains propagation effects, and the reflectivities explain the scattering effects, which makes JMI less sensitive to the details in the velocity model compared to FWI.

JMI_fig1
JMI example: (a) Initial reflectivity model. (b) Initial velocity model.
Figure 11. |
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JMI_fig3
JMI example: The inverted velocity using (a) regular JMI without any regularization, (b) JMI with conventional TV, (c) JMI with directional TV, and (d) JMI with L2 directional laplacian smoothing.
Figure 12. |
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JMI_fig2
JMI example: The inverted reflectivity using (a) regular JMI without any regularization, (b) JMI with conventional TV, (c) JMI with directional TV, and (d) JMI with L2 directional laplacian smoothing.
Figure 13. |
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JMI_shots-01
JMI example: the modeled data generated at m and the corresponding difference with the observed data based on the inverted velocity and reflectivity using (a, b) regular JMI without any regularization, (c, d) regular JMI with conventional TV , and (e, f) regular JMI with directional TV.
Figure 14. |
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2020-12-07