# Least-squares migration

Least squares migration is a process by which the reflectivity model is error minimized with respect to some objective function. In the work of Luo and Hale [1] the objective function is

${\displaystyle E(u)={\frac {1}{2}}\parallel Su-d\parallel ^{2}}$

where d is the real data, u is the modeled wavefield, S is a receiver sampling operator. So to minimize the error, one simply finds where the derivative goes to zero. Under the Born approximation [2] this method can be represented by

${\displaystyle {\frac {\partial E}{\partial u}}={S}^{T}\left(Su-d\right)}$

where S transpose represents the migration operator. By iterating through conjugate gradient iterations or similar we can generate a resulting image which is superior to conventional depth migration.

The output of the least squares migration algorithm operating on the Marmousi velocity model is highly resolved in the case of an accurate starting velocity model.

Least squares migration is inherently sensitive to velocity errors. Using preconditioning filters to stabilize the reflectivity image solution, velocity sensitivity can be reduced. Hale's dynamic warping can also be used to decrease LSM's velocity error sensitivity.