EXAMPLES

For the sake of simplicity, we demonstrate the method by resampling a 2-D, post stack section. Figure  shows a synthetic reflectivity model. A zero reflectivity layer at the top of the model represents a water layer. To test our resampling algorithm, we generate an unevenly sampled dataset from this model. Figure  shows the surface positions of the receivers. The receiver spacing is randomly chosen from an uniform distribution. The average error in receiver position is 0.02 km, and the variance is 0.0195 km. Figure shows the zero-offset section generated with a constant velocity of 2 km/s. The influence of the errors in receiver position can be clearly identified. We solve the equation (2) using the conjugate gradient method. The implementation of the Kirchhoff upward-continuation operator is similar to the one described by Bevc (1992). After 10 iterations, we obtain the uniformly sampled wave field at the depth level of 0.1 km, which is shown in Figure . It is clear that the effect of irregular spacing has been removed by the inversion process. Figure  is a plot of the total energy of the residual as a function the iterations of the conjugate gradient method. Convergence is reached after 10 iterations, when the total energy of the residual drops close to zero. Figure  displays the synthetic data resampled at surface and the synthetic data modeled directly from the reflectivity model for comparison. The differences are imperceptible.

 

sigmoidm
Figure 1 A synthetic reflectivity model.

 

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Figure 2 Surface positions of receiver array with random spacing.

 

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Figure 3 The top panel shows the data sampled irregularly at the surface. The bottom panel shows the data after being downward continued to 0.1 km below the surface using the least squares inverse of the upward continuation operator.

 

convg Figure 4 The total energy of the residuals of the inversion process versus the iterations of the conjugate gradient method.

 

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Figure 5 The top panel shows the data resampled at the surface. The bottom panel shows the data directly model from the reflectivity model shown in Figure .