# Reflector curvature

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Define the ratio CE of the reflection amplitude at normal incidence from a curved boundary to the reflection amplitude from a flat boundary at the same depth [1] as

 ${\displaystyle CE={\frac {1}{\sqrt {1+A^{-1}z}}},}$ (13a)

where z is the depth to the reflector, and A is the reflector curvature — negative for synclines and positive for anticlines. Note that for a flat reflector, whatever its depth, CE = 1. But for a curved reflector, the reflection amplitude at normal incidence to a synclinal interface is greater than the case of a flat reflector, and the reflection amplitude at normal incidence to an anticlinal interface is smaller than the case of a flat reflector. The physical basis of equation (13a) is that a synclinal interface focuses the energy associated with the reflecting wave, whereas an anticlinal interface defocuses it.

The effect of reflector curvature on amplitude variation with offset is quantified as [2]

 ${\displaystyle CE(\theta )={\frac {1}{\sqrt {1+A^{-1}z/\cos ^{2}\theta }}},}$ (13b)

where θ is the angle of incidence. Note that equation (13b) reduces to equation (13a) for the case of normal incidence (θ = 0). Both equations are for the case of a 2-D reflecting interface. Equation (13b) was extended by Bernitsas [3] to the case of a 3-D reflecting interface with curvature in both the inline and crossline directions.

To understand the effect of reflector curvature on amplitude variation with offset, it is convenient to study the ratio CE(θ)/CE(θ = 0) as a function of angle of incidence [2] [4]. Figure 11.2-7 shows the behavior of this ratio for the cases of a syncline and an anticlne with varying degrees of curvature. Note that, for an anticlinal interface, the curvature effect defined by the ratio CE(θ)/CE(θ = 0) decreases with angle of incidence or offset (Figure 11.2-7c). In practice, this means that the reflection amplitudes from an anticlinal interface at far offsets are lower than those from a flat interface. Also, note that the larger the curvature defined by the ratio z/x denoted in Figure 11.2-7, the more prominent the curvature effect. For a synclinal interface with a mild curvature (0 > z/x > −1), the curvature effect on amplitudes increases with angle of incidence, and it decreases with angle of incidence for a tight syncline (z/x < −1). Again, note that the larger the curvature, the more prominent the curvature effect.

Figure 11.2-7  Effect of reflector curvature on angular dependence of reflected wave amplitudes. The incident wave is of compressional type. (a) A synclinal reflector (x < 0), (b) an anticlinal reflector (x > 0), (c) effect of curvature on the angle-dependent reflection amplitudes associated with the reflector as in (a), and (d) the same as in (c) for the reflector as in (b). Adapted from [2].

It is clear from the model studies summarized above that reflection amplitudes are influenced by the reflector curvature. This is true for both normal incidence (equation 13a) and non-normal incidence (equation 13b). Reflection amplitudes must be corrected for the reflector curvature by prestack time migration before prestack amplitude inversion to derive the AVO attributes which are discussed in this section. Similarly, reflection amplitudes must be corrected for the reflector curvature by poststack time migration before poststack amplitude inversion to derive the acoustic impedance attribute which is discussed in the next section.

## References

1. Hilterman, 1975, Hilterman, F. J., 1975, Amplitudes of seismic waves — A quick look: Geophysics, 40, 745–762.
2. Shuey et al., 1984, Shuey, R. T., Banik, N. C., and Lerche, I., 1984, Amplitude from curved reflectors: 54th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 664–665.
3. Bernitsas (1990), Bernitsas, N., 1990, Curvature effect of a reflecting surface for arbitrary offset and azimuth: 60th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1053–1055.
4. Castagna, 1993, Castagna, J. P., 1993, AVO analysis — tutorial and review: in Offset-dependent reflectivity — theory and practice, Soc. Expl. Geophys.