I plan to use GSLIB Deutsch and Journel (1992) to perform both the kriging of the well velocity with the seismic velocity as a trend and the sequential Gaussian simulation (sGs) to calculate a measure of the local and global uncertainties about the estimated model.
The kriging estimated value of the velocity based on the well data and using the seismic velocity model as a linear trend is
where coefficients are estimated by solving the kriging with a trend (KT) system (9), and are the velocities of the N neighboring points located at position used to determine . The KT system is made of (N+3) equations:
where is the covariance of the velocity at location and location , and ai are the coefficients of the linear trend:
A covariance model is therefore needed and will be inferred from the semi-variogram model of the well velocity. By definition, is equal to C(0)-C(h). The sGs will also allow me to estimate a range of equiprobable velocity models that will require some interpretation in order to produce a measure of the uncertainty about the estimated velocity model.
The sGs procedure is based on the normal score transform of the velocity random variable , assuming that the transformed variable is multi-normal. When the multi-normality hypothesis cannot be retained, I will use an indicator simulation instead of the Gaussian approach.