FWI Example

In order to demonstrate the effectiveness of FWI with directional TV, we consider the velocity model shown in Figure 2a, which is scaled from the top half of the Marmousi model. To avoid cycle-skipping, a Ricker wavelet with a dominant frequency of $14$ Hz is used as the source wavelet. Using a constant-density acoustic finite-difference modeling, we generated 23 shots with 334 receivers for each shot. The shot spacing is $180$ m and the receiver spacing is $12$ m. The horizontal and vertical grid size are $12$ m. In addition, some random noise with $SNR = 10$ is added to the modeled data. The middle shot gather is shown in Figure 3. The initial velocity model is shown in Figure 2b, which is a smoothed version of the model in Figure 2a.

First, with the initial model, we apply 6 iterations of Full Wavefield Migration (Davydenko and Verschuur, 2017; Berkhout, 2014b) at a maximum frequency of $25$ Hz to the dataset and then denoise the inverted image via a simple soft-thresholding in the curvelet domain (Figure 4a, Donoho (1995)). Note that full wavefield migration can be considered as a JMI process in which the velocity is assumed known and fixed. It will honour all multiples and transmission effects properly. Now, with this inverted reflectivity, we can estimate the dip field using plane-wave destruction algorithm proposed by Fomel (2002), shown in Figure 4b. This estimated dip field is used to build the directional TV operator for each gridpoint.

Next, we compare three methods: regular FWI without any regularization, FWI with conventional TV, and FWI with directional TV. We use the same $\mu$ and $\lambda$ for TV and directional TV. We choose a relaxation strategy to set $\mu$, which is increasing exponentially. $\lambda$ is chosen as $0.005$, which depends on the scale of data. For directional TV, $\alpha_1 : \alpha_2 = 3 : 1$ and $\alpha_1 + \alpha_2 = 2$. For TV, $\Lambda$ is an identity matrix. After 100 iterations, the inverted results are shown in Figure 8. Note that the regular FWI without any regularization is trapped into local minima very quickly, despite the accurate starting model (Figure 8a). With the help of TV regularization, FWI with TV 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 local minima. However, we can observe that the structures still remain vague in Figure 8b, especially in the deeper area, since traditional TV only uses horizontal- and vertical- and ignores the local structure. Compared to the regular TV, much weaker artifacts can be observed in the result of FWI with the directional TV, shown in Figure 8c, because we consider the structural directions of the spatial gradient and their weights according to the local dip. The convergence diagrams of the misfit function with iteration number corresponding to the three methods are shown in Figure 8d, in which it is visible that FWI with TV works well to mitigate the ill-posedness and non-uniqueness of FWI, and FWI with directional TV behaves clearly better than FWI with the conventional TV. Figure 6 shows a comparison between the different velocities at three different locations. The locations are annotated in Figure 8a-c. We can see the obvious improvement using directional TV.

To further illustrate the contribution of regularization in the inversion, we show the gradients from the residual at the first iteration based on the different methods in Figure 7. Compared to Figure 7a, Figure 7b has sharper structures, especially in the deeper part, by preserving the edges via the TV constraint. The gradient in Figure 7c shows even more blocky structures that correspond to the geologic information. Note that, in Figure 7c, there are imprints introduced by the imperfect raw reflectivity model and dip field (Figure 4). However, these imprints have been compensated and suppressed during inversion, and the proposed method ends up with a decent result shown in Figure 8c, which shows the proposed method is insensitive to the locally-incorrect dip field. Figure 8 and 9 demonstrate the corresponding depth migration images and common image gathers calculated using full wavefield migration. We can see that the reflectivity based on the velocity from FWI with directional TV has the best focusing resolution and less imprints, and their corresponding common image gathers are flatter than the alternative methods. Please note some obvious improvements pointed out by the red arrows. In the end, we show in Figure 10 the modeled data generated from each of the final inverted velocities and the corresponding differences with the observed data. From this figure, we note that FWI with directional TV approach can restore the observed data much better than the alternatives.

FWI_fig1-01
Figure 2. FWI example: (a) Real velocity model, at specific gridpoint $\left (i,j\right )$. The black dash arrows illustrate $\nabla _x \mathbf {p}\left (i,j\right )$ and $\nabla _z \mathbf {p}\left (i,j\right )$, the red solid arrows illustrate $\nabla _1 \mathbf {p}\left (i,j\right )$ and $\nabla _2 \mathbf {p}\left (i,j\right )$, based on the structural dip at $\left (i,j\right )$. (b) Initial velocity model.

shots_orig
Figure 3. FWI example: Recorded middle shot gather at $X = 2000$ m

FWI_fig2
Figure 4. FWI example: (a) The inverted reflectivity model after denoising using thresholding in the curvelet domain. (b) The estimated dip field (in degrees).

vinv,vinv_tv,vinv_dtv,objs
Figure 5. FWI example: The inverted velocity using (a) regular FWI without any regularization, (b) FWI with conventional TV, and (c) FWI with directional TV. (d) The convergence diagrams of the data misfit as a function of iteration. The blue line denotes the inverted velocity using regular FWI without any regularization. The red line denotes the inverted velocity using FWI with conventional TV. The yellow line denotes the inverted velocity using FWI with directional TV.

vel_profile-01
Figure 6. FWI example: Comparison of different velocities at three locations. Velocity comparison at (a) $X = 1000$ m, (b) $X = 2000$ m, (c) $X = 3000$ m. The purple line denotes the true velocity. The green line denotes the initial velocity. The blue line denotes the inverted velocity using regular FWI without any regularization. The red line denotes the inverted velocity using FWI with conventional TV. The yellow line denotes the inverted velocity using FWI with directional TV. The three locations are highlighted by the black dash lines in Figure a-c

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FWI_gradient
Figure 7. FWI example: The velocity gradient from the residual at the first iteration using (a) regular FWI without any regularization, (b) FWI with conventional TV, and (c) FWI with directional TV.

FWI_fig4
Figure 8. FWI example: The corresponding reflectivity based on the inverted velocity using (a) regular FWI without any regularization, (b) FWI with conventional TV, and (c) FWI with directional TV.

FWI_CIGs
Figure 9. FWI example: The corresponding angle gathers generated at $X = 1000$ m, $X = 2000$ m, and $X = 3000$, based on the inverted velocity using (a, b, c) regular FWI without any regularization, (d, e, f) regular FWI with conventional TV, and (g, h, i) regular FWI with directional TV.

FWI_shots-01
Figure 10. FWI example: The modeled data generated at $X = 2000$ m and the corresponding difference with the observed data based on the inverted velocity using (a, b) regular FWI without any regularization, (c, d) regular FWI with conventional TV, and (e, f) regular FWI with directional TV