PROPOSED APPROACH

I propose to use 3-D inversion of the seismic data to estimate a 3-D velocity model that matches the given well-log information and follows the trend of the seismic data. To do so, I plan to integrate a set of irregularly-spaced 1-D well-log velocity curves and a 3-D seismic stacking velocity cube to produce a 3-D velocity model. This approach combines the processing and geostatistical analysis of seismic and well-log data.

First, I plan to analyze and preprocess the well-log data from a particular seismic survey. Ideally, this step would require the following well-log curves: sonic log, gamma ray log, and checkshot. The gamma ray curve is very useful for interpreting the geology and defining where the major boundaries separating the geological macro-layers are. If the sonic log is not available, it can be inferred from the gamma ray curve at the same well location. The sonic curve, along with a checkshot, will allow me to estimate the seismic (stacking) velocities at the well.

I will then process the 3-D seismic data in order to determine an initial 3-D velocity model. This processing comprises noise filtering, sorting of the traces from shot gathers into common depth-point gathers (CMP), and velocity analysis. I also plan to apply normal moveout to the CMP gathers before stacking them and to perform a depth migration in order to get an initial depth image of the geology. This stacked and/or depth-migrated image will help me visualize boundaries of the geological macro-layers and the presence of structural features such as faults, folds, and diapirs. The macro-layers observed in the section will help me make visual correlations with the processed well-log data.

Following this processing sequence for both data sets, I will define an inversion scheme based on the seismic and the well-log data (the data space) in order to derive a corresponding 3-D velocity model (the model space). The initial scheme I propose is a least-square inversion method that minimizes the function

$$\Phi \, (m) \; = \; E \, (m) \: + \: \epsilon^2 \, L \, (m)$$ (1)

where $E \, (m)$ measures the prediction error between the estimated 3-D velocity model m and the well-log-derived velocity that is linearly interpolated on a regular grid. The presence of $L \, (m)$, which measures the length of the model, ensures that the trend of the estimated model does not move far away from the trend of the seismic velocity:

$$\begin{array} {lcl} E \, (m) & = & (d_w \, - \, L m)^T \, C_... ...& [A \, (m \, - \, m_0)]^T \, [A \, (m \, - \, m_0)]\end{array}$$ (2)

where d_w is the velocity data derived from the well, L is the linear interpolation operator, and C_w^-1 is the inverse covariance matrix of the well velocity. A is a roughening operator, and m₀ is the initial model, which is the seismic velocity data mapped onto the model space by linear interpolation ($m_0 \, = \, L^T d_s$).

The factor $\epsilon$ determines the relative importance given to the prediction error and the model length, which is also the relative importance given to the well data over the seismic data in this scheme. I will use a prediction error filter (PEF) determined from the seismic data for the roughening operator A. The role of this PEF is to ensure that the global trend of the estimated model follows roughly that of the initial seismic velocity.

Another approach to this problem is that provided by geostatistics. Ultimately, I want to derive a measure of the global and local uncertainties about the estimated 3-D velocity model using sequential Gaussian simulation algorithm. The same algorithm can be applied to obtain an estimated 3-D velocity model by means of the well-log-derived velocity and the initial seismic velocity model. First I will calculate the histogram of the cumulative distribution function (cdf) of the well velocity V_w and the initial seismic velocity V_s. I will apply a normal score transform $\varphi$ to transform these velocity values V_w and V_s into $\Omega_w$ and $\Omega_s$, following a standard normal distribution ${\cal N}(0,1)$ Deutsch and Journel (1992). Then I will perform a sequential Gaussian simulation on the $\Omega$ values, as follows:

1.

I will define a random path that visits all the grid locations once. At each location i, I will define a neighborhood around the location $\vec{x_i}$ to retain only a portion of the data points.

2.

At the location $\vec{x_i}$, I will infer a covariance model from the spatial variability of the data.

3.

I will then estimate $\omega_K^{\ast}(\vec{x_i})$, the best least-square estimated value of $\Omega$ at the location $\vec{x_i}$, and $\sigma_K^{2}(\vec{x_i})$, the variance of the conditional cumulative distribution function (ccdf) of the random function $\Omega$. This estimation is realized by a kriging with a trend (KT) algorithm, which requires the covariance model infered at step 2. I plan to perform KT on the well velocities using the trend of the seismic velocities to constrain the estimation.

4.

I will go to the next location and repeat steps 1 to 3 until all the $\omega(\vec{x_i})$ are simulated.

Finally, I plan to back transform all the simulated normal values $\omega$into simulated values for the velocity V.

Performing this conditional simulation will allow me to determine a measure of local and global uncertainties about the estimated velocity model. To obtain L equally probable simulated images of the velocity model, I will use the best least-square estimated value $\omega_K^{\ast}(\vec{x_i})$ in step 3 as the mean of the ccdf at location $\vec{x_i}$, draw L simulated values $\omega^{(l)}(\vec{x_i}) \;\; l \, = \, 1,L$ based on the ccdf ${\cal N}(\omega_K^{\ast}(\vec{x_i}),\sigma_K^{2}(\vec{x_i}))$, and back transform them into the velocity space. At each location $\vec{x_i}$, the L simulated values provide a histogram of possible outcomes for that particular location, which is a model of local uncertainty about the velocity. In addition, for each of the L simulated images, I can average the simulated velocities over the entire space and organize the result in a histogram that provides a measure of global uncertainty about the average estimated velocity.