Synthetic Example

A synthetic data example is created to test the algorithm to make sure that it works properly. A simple layer-cake earth model is used, shown in Figure . RMS velocities are then created from this model as input data to the algorithm. If the inversion is run on this simple model, the result is almost perfect, as shown in Figure . The inversion is off by a maximum of 3%, which occurs at the bottom-most interface. This error could most likely be reduced further if we decrease the stopping criterion.

Now 1 and 5 percent Gaussian noise is added to the RMS velocities to simulate real data. The inversion of this noisy data with very little smoothness applied is shown in Figures and . The noise introduced to the model shows up as block features. As more noise is added the layers become harder to distinguish from each other.

 

model
Figure 1 A simple layer cake earth model

 

Modvint
Figure 2 Inversion result for simple model using the algorithm above. For the simple model shown in Figure , the result is almost perfect.

 

ModvintNoise
Figure 3 Inversion of the data with 1% noise. The layer boundaries are still visible, but the noise pollutes the results substantially.

 

ModvintNoise5
Figure 4 Inversion of noisy data with 5% noise. The noise is severe enough that the boundary layers are no longer discernible.

If we increase the smoothing parameters on the $\ell_1$ regularization, then much of the noise is smoothed out in the result (Figure and Figure ). If the regularization parameters, $\epsilon_{x,\tau}$ are increased further then the result will be even smoother (Figure and Figure ).

As seen in these examples, it is important to correctly choose the regularization parameter to get a good inversion result that is compromise between desired blockiness and introducing spurious elements into the model in the form high spatial frequency events.

 

ModvintNoiseSmooth
Figure 5 Smoothed version the data with 1% noise. Much of the noise has been smoothed out, but the sharp boundary contacts are still clear.

 

ModvintNoiseSmooth5
Figure 6 Smoothed version of the data with 5% noise. The smoothing parameter had to be increased to get rid of much of the noise. The boundaries are more evident here than in Figure , but the layer boundaries are smoothed out.

It can be seen that not all the noise is smoothed out in either Figure or Figure . This is because if boundaries are sharp then the $\ell_1$ regularization preserves them. The sharper the boundary, the higher the $\epsilon$ needs to be smooth them out. Much of the sharp contrast, however, is also smoothed away. From this test it became clear the smoothing along the midpoint is not currently working properly.

 

ModvintNoiseSmooth2
Figure 7 The smoothing parameters were increased further for the 1% noise. Most of the noise is no longer visible, but the layer boundaries are not as sharp.

 

ModvintNoiseSmooth2-5
Figure 8 Smoothing parameters were increased for the 5% noise to the point that almost no noise is visible. Doing this, however, has smoothed out the entire result.