** Next:** Differential image perturbation
** Up:** Sava and Biondi: WEMVA
** Previous:** Introduction

In migration by downward continuation, the wavefield at depth
is obtained by phase-shift from the wavefield at
depth *z*:
| |
(1) |

Using a Taylor series expansion,
the depth wavenumber () depends linearly
on its value in the reference medium () and the
laterally varying slowness in the depth interval under consideration
| |
(2) |

*s*_{o} represents the constant slowness associated with the
depth slab between the two depth intervals.
represents the derivative of the depth wavenumber with
respect to the reference slowness and can be implemented
in many different ways, e.g by the Fourier-domain method
of Huang et al. (1999).
The wavefield downward continued through the *background* slowness
is

| |
(3) |

and the full wavefield depends on the
background wavefield by
| |
(4) |

where represents the difference between the
true and background slownesses .
Defining the *wavefield perturbation* as
the difference between the wavefield propagated through the
medium with correct velocity, , and the wavefield propagated
through the background medium, , we can write

| |
(5) |

| (6) |

Equation (5) represents the
foundation of the wave-equation migration velocity analysis
method. One major problem
with Equation (5) is that the wavefield and
slowness perturbations are not linearly related.
For inversion purposes, we need to find a linearization
of this equation around the reference slowness, *s*_{o}.
Biondi and Sava (1999) linearize
Equation (5) using the Born
approximation ().
With this choice, the WEMVA Equation (5) becomes
| |
(7) |

The wavefield perturbation in Equation (7)
turns into an image perturbation after we apply an
imaging condition, e.g. summation over frequencies.
The WEMVA objective function is
| |
(8) |

where is a linear operator defined recursively from
Equation (7) at every depth level and frequency.
We estimate the slowness update by minimizing this objective function
through an iterative conjugate gradient optimization technique.

** Next:** Differential image perturbation
** Up:** Sava and Biondi: WEMVA
** Previous:** Introduction
Stanford Exploration Project

7/8/2003