Synthetic example

The simple adjoint transform is not sufficient to provide as input to LA. Figure  shows results when supplying LA with model spaces produced by adjoint plane-wave decomposition verses 40 iterations of inversion with a single synthetic planewave used as data. LA, told to choose at most 10, returned two picks for both models despite only a single planewave existing in the data. LA will not return one pick if the data supplied is not perfectly single valued. In this example the data supplied to LA are coordinates with energy in the figure times the duration of the wavelet in/out of the plane. This leads to 3449 coordinate triples, some 500 being unique since the quantization of the amplitudes used 10 levels.

The adjoint model space has a diffuse character and diagonal streaks away from main blob of energy (due to limited rectangular surface acquisition). The model space was parameterized so that the energy would not be symmetrically located on the p_x,p_y-plane. To balance the distribution of energy, LA selected two points for the adjoint model space that an interpreter would recognize as inappropriate. After the inversion has clipped most of the acquisition tails and moved the edge of the distribution away from the boundary of the domain, the two picks returned are identical $(-0.0005006, 0.0001002) \& (-0.0004993, 9.997e-05)$.

 

adj.inv
Figure 2 Coordinates selected by Lloyd's Algorithm on the adjoint plane-wave decomposition model space and on 40 iterations of least-squares inversion.

Six planewaves were modeled in a regular 3D acquisition geometry as data. The data were inverted with the linear Radon transform. The inversion was stopped after 1, 20, and 80 iterations. These data were then supplied to the modified LA. Figure  shows inversion results after 20 iterations. The two panels show versions clipped for display at 99 and 100% of the maximum value. Three picks from LA are also plotted. The picks exactly overlay the maximum amplitude of the energy. LA was initialized to select 10 coordinate triples. Only six picks were returned. Note, however, that the lower right coordinates are actually two picks very close together. One planewave from the data space has not been picked. One of the planewaves had a ray parameter p_x,p_y=(0.0005,0.0001) s/m, while the range of p_x used for the inversion extended to only p_x,p_y=(0.00045,0.0003) s/m. This plane thus falls outside of the model space on the 2-axis, and the inversion is not able to focuse the energy. Unfortunately, LA does not recognize its significance either, and places an extra pick semi-randomly close to another well established pick.

 

rays
Figure 3 Ray parameter model space inverted from six synthetic planewaves. 20 iterations of least-squares inversion applied.

The approach seems robust for noisy data as well. Figure  shows a section of the model space where a uniform distribution of noise was added to the data space with the picks selected from it by LA overlain. The picks are identical to the previous runs when the variance of the added noise is less than 0.001. The threshold value used was 1% of the maximum value in the data. By increasing to higher levels (approximately 50%), the algorithm remains stable to variance values another order higher. When the level of noise is too high so that the thresholding of the data is not robust, the LA picks constantly distribute themselves roughly evenly along the one-axis and about centered across higher dimensions.

 

noise
Figure 4 Uniformly distributed noise with variance 0.001 added to model space shown in Figure . Threshold for LA was 1% of the maximum value in the input. Output picks are still reliable at this level.