Initializing the inversion

As in the previous chapter, the value of $\sv n$ is initialized to $\st S\sv d$, which is just the prediction-error filter estimate of the noise. For poststack data, this is generally a good estimate, since it is just the common prediction-error estimate of the noise used in f-x or t-x prediction filtering. For prestack data, this estimate will be less accurate, since many more traces are likely to be missing in prestack data than in poststack data, but it is better than a starting estimate of zero for $\sv n$.Using $\st S\sv d$ for the initialization is necessary to improve the results and reduce the cost of the inversion by reducing the number of iterations needed by the solver. Although the code is complicated slightly by this initialization, the time taken by the inversion is reduced by an order of magnitude. Even with this increased speed, the result is superior to the result with the noise initialized to zero. Details of the initialization and its implementation with the conjugate-gradient solver used here is discussed in Abma 1995. More details about the conjugate-gradient method itself may be found in Claerbout 1995, Strang 1986, and Luenburger 1984.