(13) |

mul_sktch4
Imaging of water-bottom multiple
for a flat water-bottom. Notice that and that the apparent
reflector at the image point is flat.
Figure 4 |

From Figure it is clear that due to the symmetry of the
problem, *t*_{s1}=*t*_{s2}=*t*_{r1}=*t*_{r2}=*t*_{m}/4 and ,
which in turn means . Furthermore, from
Equations 5 and 6 it immediately follows that
and which says that
the traveltimes of the refracted rays are equal to the corresponding
traveltimes of the multiple. Equation 2 thus simplifies to

(14) |

From Equation 3, the depth of the image point can be easily computed as

(15) |

(16) |

Equations 14-16 give the image space coordinates in terms of the data space coordinates. An important issue is the functional relationship between the subsurface offset and the image depth, since it determines the moveout of the multiples in the subsurface-offset-domain common-image-gathers (SODCIGs). Replacing and in Equation 15 we get

(17) |

odcig1
Subsurface offset domain common
image gather of a water-bottom multiple from a flat water-bottom. Water
velocity is 1500 m/s, water depth 500 m, sediment velocity 2500 m/s and
surface offsets from 0 to 2000 m. Overlaid is the residual moveout curve
computed with Equation 17.
Figure 5 |

Figure shows an SODCIG for a non-diffracted water-bottom multiple from a flat water-bottom 500 m deep. The data was migrated with a two-layer velocity model: the water layer of 1500 m/s and a sediment layer of velocity 2500 m/s. Larger subsurface offsets (which according to Equation 14 correspond to larger surface offsets) map to shallower depths (for the normal situation of ), as we should expect since the rays are refracted to increasingly larger angles until the critical reflection angle is reached. Also notice that the hyperbola is shifted down by a factor with respect to the image point when migrated with water velocity.

In angle-domain common-image-gathers (ADCIGs), the half-aperture angle reduces to , which in terms of the data space coordinates is given by

(18) |

(19) |

adcig1
Angle domain common
image gather corresponding to the SODCIG shown in Figure .
Overlaid is the residual moveout curve computed with equation 20.
Figure 6 |

As we did with the SODCIG, it is important to find the functional
relationship between and since it dictates the
residual moveout of the multiple in the ADCIG. Plugging
Equations 14 and 15 into equation 12,
using Equations 13, and 18 to eliminate *h*_{D} and
simplifying we get

(20) |

11/1/2005