LEAST-SQUARES DATUMING

Datuming is a method of processing to extrapolate a known wavefield at a specified datum of arbitrary shape to another specified datum, also of arbitrary shape. Ji and Claerbout (1992) presented a datuming algorithm based on the phase-shift depth extrapolation elsewhere in this report. In the datuming problem, we use upward and downward extrapolation operators instead of the modeling and migration operators. For a given wavefield on the irregular surface, we can estimate new wavefield on a datum by minimizing the errors between the given input and the data modeled by the extrapolation, as follows:

$$\min_{\bf d_0} \vert\vert {\bf d - Ed_0} \vert\vert$$ (7)

where ${\bf d}$ is a given wavefield on an irregular topography, ${\bf d_0}$ is a wavefield on a new datum, and ${\bf E}$ represents an extrapolation operator either upward or downward. For example, if a new datum is located above the given datum, the extrapolation operator ${\bf E}$ will be the downward extrapolation operator.

For demonstration, the phase shift datuming operator, explained by Ji and Claerbout (1992), was used. Figure (a) is the wavefield on a datum. This wavefield was extrapolated to the nonflat surface shown in Figure (a), and the wavefield on the nonflat surface is shown in Figure (b). The purpose is to datum the wavefield to the original surface. By using the downward extrapolation operator, I obtained datumed wavefield; the result is shown in Figure (c). Compared to the original wavefield, the datumed wavefield is correctly positioned, but you can see that the amplitude of the datumed wavefield is different from that of the original wavefield. Figure (d) shows the datumed wavefield obtained by using the least-squares datuming method. It has a very nice amplitude recovery.

 

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Figure 9 Least-squares inverse datuming: (a) Wave field on the datum surface. (b) Wave field on the undulating surface shown in Figure (a), (c) Datumed wavefield from the undulating surface shown in (b), (d) wavefield obtained by the least-squares inverse datuming from the undulating surface shown in (b) (after 10 iterations).