Summary

A question frequently asked about reverse time migration, which uses the two-way wave equation, is – “ Is it more sensitive to velocity errors than one-way wave equation PSDM”?. We have investigated this issue, both theoretically and empirically, and the results of our study suggest that both one-way and two-way wave equation PSDM methods are equally sensitive to velocity errors. However, the two-way wave equation has greater scope than the one-way wave equation, by imaging multiply reflected events and highly dipping reflectors, and, therefore, it is not possible to make fair comparisons for those situations not covered by one-way methods.

Velocity sensitivity analysis

The acoustic homogeneous wave equation is generally solved in the frequency domain using a depth extrapolation scheme or in the space-time domain as a boundary value problem where source or receiver observations, the seismic source signature or recorded seismic data, are input from the boundaries. The migration scheme based on the first set of solutions is known as oneway wave equation PSDM, and the latter is known as reverse time migration. The major difference between these two is that the first can only image up-going waves whereas the second can image waves propagating in all directions. However, the most important question to ask is “How sensitive are these solutions to velocity field errors”?. The usual way to measure migration sensitivity analysis is qualitatively. In other words, we migrate data with different velocities and measure dislocation of the image points with respect to the true image point. For simple velocity fields, such as flat reflector models, we can come up with quantitative methods to measure this uncertainty. However, more complex geological models require qualitative analysis. Here we will use a simple model and investigate depth migration uncertainties for one-way and two way PSDM methods. Later, we will use the SIGSBEE2A model to make comparisons.

Depth migration uncertainty for simple models

The most direct approach to identify or quantify errors in migrated images due to the use of incorrect migration velocities is by modeling followed by migration. In other words, the impact of the wrong velocity field can be modeled by migrating the input data with correct and incorrect velocity fields and analyzing the results. Cognot et al. (1995), Thore et al. (2002) and Pon and Lines (2005) quantified depth migration uncertainties in terms of velocity errors for flat reflectors. Zhu et al. (1998) studied “smiles” and “frowns” in depth migrations.

Examples

Our first example is a flat layer model, where the equations shown above apply directly. The interface is at a depth 490 meters. The two layers have velocities of 2000 m/sec and 2200 m/sec respectively. We modeled a split spread shot section, on the surface. This shot record is migrated with both two-way wave equation based reverse time and oneway explicit finite difference PSDM’s. Figure 1 shows two CIGs, the top one is from reverse time migration and the bottom one is from a one-way wave equation explicit finite difference PSDM.

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