Inversion Procedure

The near sea-bottom properties can be estimated using inversion. Since the problem is highly non-linear, we applied an iterative damped least-squares approach Levenberg (1944); Lines and Treitel (1984); Marquardt (1963) which minimizes the misfit between the amplitude spectrum of the theoretical reflection (equation 2) or AVO coefficient (equation 4) and the calculated one (equation 1 and 3). The misfit function is defined as:

$$F \: = {1\over2}\sum{\left( {\sqrt{R_{theo} R_{theo}^{*}}} - {\sqrt{R_{calc} R_{calc}^{*}}}\right)}^2$$ (5)

The star (*) denotes the complex conjugate. R is either the reflection coefficient or the AVO coefficient, depending on the data being used. Because of the nonlinearity of the problem, it is important to make a reasonable initial guess of the parameters. Once the near seafloor parameters have been estimated, they can be used to calculate a predicted reflection or AVO coefficient for all angles of incidence.