Solving the equation

Equations (6) and (7) usually contain an enormous number of variables and, as will be shown later, are extremely ill-conditioned. However, good results may be obtained in a small number of iterations if a conjugate gradient method is used. To solve the equation

$$\bold L \bold u = \bold d$$ (8)

we can minimize the function

$$\bold f = {\Vert \bold L \bold u - \bold d \Vert }^2 ,$$ (9)

or for stochastic inversion,

$$\bold f = {{\Vert \bold L \bold u - \bold d \Vert }^2 \over \sigma_d^2 } + {{\Vert \bold u \Vert}^2 \over \sigma_u^2},$$ (10)

where $\sigma_d^2$ and $\sigma_u^2$ are the variances of the data and the model spaces respectively.

When a velocity analysis panel is obtained by the solution of equation (6) or equation (7), an inversion is easily done by application of the operator $\bold P$ or $\bold H^{\bold T}$ on the velocity analysis panel $\bold u$.

We can also consider solving the equation  

$$\bold H \bold d = \bold u,$$ (11)

where $\bold u$ was obtained by the application of $\bold H$ on (t,x)-space, and  

$$\bold P^{\bold T} \bold d = \bold u,$$ (12)

where $\bold u$ was obtained by the application of $\bold P^{\bold T}$ on (t,x)-space.