Estimation of Q from surface-seismic reflection data in data space and image space |

where is the slowness at the reference frequency .

Having the new velocity/slowness, I obtain the new single square root as

(12) |

This new SSR can be approximated into a simplified form by using Taylor expansion around reference slowness and reference quality factor :

where

(14) |

(15) |

and is the reference slowness at the reference frequency. The first two terms of equation 13 describe the split-step migration. The third term is the high-order correction, which allows for pseudo-screen migration.

The single square root for FFD migration is shown as follows:

where and for 45-degree migration.

In addition, I can rewrite Q migration in a matrix form to conveniently compare with the conventional migration. The conventional migration can be written in the following matrix form:

where is the data, is the model, is the migration operator, and the superscript indicates the matrix transpose.

The downward continuation migration with Q can be written as

where is the attenuation operator, which consists of real numbers less than .

Equation 18 indicates that the migrated model will be further attenuated, with the attenuation operator being applied to the attenuated modeled data. Therefore, Q migration will compensate for the phase change, but will not compensate for the amplitude loss due to attenuation.

In this section, I apply Q migration to the modeled data in Figures 1(a) and 1(b). Figure 3(a) shows the conventional migration of the non-attenuated data in Figure 1(b), which images the reflector at 1500 m depth. Figures 3(b) and 3(c) show the conventional migration and Q migration of the attenuated data in Figure 1(a). The wavelets in Figure 3(c) are stretched in comparison to the ones in Figure 3(a). This result confirms that Q migration further attenuates the data, instead of compensating for its amplitude loss.

Estimation of Q from surface-seismic reflection data in data space and image space |

2012-05-10