THEORY

If an upward-propagating wave field is sampled uniformly at a given depth level, it can be accurately continued upwards to an arbitrary point on the earth's surface by a Kirchhoff summation operator. Let P(t,x_i,y_j,0) be the wave field recorded by an array of unevenly spaced receivers on the surface. We want to resample the wave field with an uniform sampling interval. We consider the wave field recorded on the earth's surface to be the upward continuation of a wave field P(t,x_k,y_l,z₀) that is recorded by an array of uniformly spaced receivers at the depth z₀. This relation is described mathematically by

$$P(t,x_i,y_j,0) = {\bf C} P(t,x_k,y_l,z_0),$$ (1)

where ${\bf C}$ is a Kirchhoff upward continuation operator. For a given P(t,x_i,y_j,0), we can estimate P(t,x_k,y_l,z₀) by minimizing the errors between the given input data and the data modeled by upward continuation, which is as follows:

 

$$E = \Vert P(t,x_i,y_j,0)-{\bf C} P(t,x_k,y_l,z_0)\Vert^2.$$ (2)

This is a linear least squares inverse problem. It can be efficiently solved by using the conjugate gradient method. Once P(t,x_k,y_l,z₀) is known, we can upward continue the wave field to the surface and resample it, as follows:

$$P(t,x_k,y_l,0) = \hat{\bf C} P(t,x_k,y_l,z_0)$$ (3)

where $\hat{\bf C}$ is another upward continuation operator.

The choice of z₀ is determined from two aspects. The aperture of the upward continuation operator for a given maximum time-dip of input data is inversely proportional to z₀. Therefore, to achieve high efficiency, we want to use small z₀. However, if the trace sampling intervals of the original data are large, using small z₀ causes the conjugate operator of the upward continuation to be aliased.

An assumption made by this algorithm is that the velocities above the depth level z₀ are known, which is much weaker than the assumption of the full knowledge of the velocity structure made by the least squares migration (Cole, 1992; Ji, 1992). The formulation presented above is well suited for resampling marine data because the depth level z₀ can be set within the water layer that has a constant velocity and does not generate any useful events. It should also work for land data recorded in an area of simple near-surface velocity structure. For land data with complicated near surface structures, a similar scheme can be formulated, in which the unevenly sampled traces are modeled as the reverse time propagation of the wave field sampled uniformly at a level above the earth's surface.