Introduction

Attenuation of multiples in the image space is attractive because prestack wave-equation migration accurately handles the complex wave propagation of primaries. In subsurface-offset-domain common-image-gathers (SODCIG) the primaries are imaged at zero subsurface offset at the depth of the reflector if migrated with the correct velocity. Correspondingly, in angle-domain common-image-gathers (ADCIG) the primaries are imaged with flat moveout. Attenuation of multiples in image space depends on the difference in residual moveout between the primaries and the multiples, either in SODCIGs or ADCIGs Alvarez et al. (2004); Hargreaves et al. (2003); Sava and Guitton (2003). Understanding how wave-equation migration maps the multiples into SODCIGs and ADCIGs is therefore of paramount importance in order to design a proper strategy to attenuate the multiples in the image space.

Non-diffracted water-bottom multiples from a flat or dipping water-bottom are imaged as primaries. Thus, if the migration velocity is that of the water, they are mapped to zero subsurface-offset in SODCIGs. Consequently, in ADCIGs, these multiples exhibit flat moveout just as primaries do Alvarez (2005). In the usual case of migration with velocities faster than water velocity, these multiples are mapped to subsurface offsets with the opposite sign with respect to the sign of the surface offsets. I will analytically show the moveout curve of these multiples in SODCIGs and ADCIGs.

Water-bottom diffracted multiples, on the other hand, even if from a flat water-bottom, do not migrate as primary reflections Alvarez (2005). That is, they do not focus to zero subsurface offset even if migrated with the water velocity. Obviously this happens because at the diffractor the reflection is not specular. I will show that these multiples migrate to both positive and negative subsurface offsets in SODCIGs depending on the relative position of the diffractor with respect to the receiver (for receiver-side diffracted multiples).

The next section presents a general formulation for computing the kinematics of diffracted and non-diffracted water-bottom multiples for both SODCIGs and ADCIGs. The following section then looks in detail at the special case of flat water-bottom where the equations simplify and some insight can be gained as to the analytical representation of the residual moveout of the multiples in both SODCIGs and ADCIGs. The next section presents a similar result for multiples from a dipping water-bottom. Although the equations are more involved and difficult to encapsulate in one single expression than those for the flat water-bottom, I show that we can still compute the image space coordinates of both the diffracted and non-diffracted multiples in terms of their data space coordinates. The last section discusses some of the implications of the results and the possibility that they can be used to attenuate the multiples in the image space. Detailed derivation of all the equations is included in the appendices.