Sequential Gaussian simulation

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

$$V^{\ast}(\vec{x}) \; = \; \sum_{\alpha = 1}^N \lambda_\alpha V(\vec{x}_\alpha)$$ (8)

where coefficients $\lambda_\alpha$ are estimated by solving the kriging with a trend (KT) system (9), and $V(\vec{x}_\alpha)$are the velocities of the N neighboring points located at position $\vec{x}_\alpha$ used to determine $V^{\ast}(x)$. The KT system is made of (N+3) equations:

 

$$\left\{ \begin{array} {lclclclcl} \sum_{\beta=1}^N \lambda_\... ...}^N \lambda_\beta y_\beta & & & & & & & = & y\end{array}\right.$$ (9)

where $C(\vec{x}_\alpha - \vec{x}_\beta)$ is the covariance of the velocity at location $\vec{x}_\alpha$ and location $\vec{x}_\beta$, and a_i are the coefficients of the linear trend:

$$m(\vec{x_\alpha}) \; = \; a_0 \, + \, a_1 x_\alpha \, + \, a_2 y_\alpha$$ (10)

A covariance model is therefore needed and will be inferred from the semi-variogram $\gamma(h)$model of the well velocity. By definition, $\gamma(h)$ 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 $V(\vec{x})$, 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.