In order to approximate the inverse of seismic wave propagation, we can use the migration operator in a least-squares inversion problem (, ). The seismic wave propagation is a linear operator (29#29) which gives us the seismic data (30#30)from the ideal subsurface model (31#31) as seen by:

32#32 | (15) |

Once again, a migration operator such as downward-continuation migration is the adjoint of the wave propagation (33#33). A direct least-squares solution for the model from this equation is

34#34 | (16) |

However, the size of the seismic imaging problem makes it unreasonable to invert 35#35 directly. Fortunately, we can approximate the inverse by using 33#33 and 29#29 in an iterative conjugate-gradient scheme. This amounts to minimizing the objective function:

36#36 | (17) |

I choose to write this equation in a more intuitive form known as a fitting goal:

37#37 | (18) |

10/31/2005