- Let us study Sequential Gaussian Simulation. We are going to talk about Steps of Sequential Gaussian Simulation. Sequential Gaussian Simulation is a procedure that uses the kriging mean and variance to generate a Gaussian field. Normal Gaussian distribution is defined here. It uses input data and simulated data when computing a value at an unsimulated grid cell. It generates many equally probable realizations which can be post-processed to quantify and assess uncertainty. It generates multiple equiprobable realizations of a property, rather than simply estimating the mean, for example when using the kriging algorithm. Essentially, we are adding back in some variability to undo the smoothing effect of kriging. This result possibly gives a better representation of the natural variability of the property and provides a means for quantifying uncertainty. It is most commonly used for geostatistical simulation for reservoir modeling for continuous variables like porosity. The family of sequential simulation methods makes use of the same basic algorithm. There are following basic steps in the SGS process. 1. Transform the original well data to normal-score data with zero mean and unit variance. 2. Assign transformed data into simulation grid. 3. Generate a random path through the grid nodes. 4. Visit a node in the random path and use kriging to estimate a mean and standard deviation at that node based on surrounding data and variogram, it generates a local conditional probability distribution. Select at random a value from lcpd and set the node value to that number. 6. Include the newly simulated value as part of the conditioning data. 7. Repeat steps 4 to 6 until all grid nodes have values. 8. Back-transform the realization into the original space. For SGS it is important that the data actually follow a Gaussian distribution, so when SGS is used, the data are first transformed using the normal score transformation with zero mean and unit variance. For details, please refer to The normal Score Transformation. Here is a 2D porosity data, 7 data, x, y, original data, and transformed score data. Assign transformed data into simulation grid which will be used as conditioning data. Five conditioning data are assigned to their nearest gird cell centers. When multiple data belong to one cell, one option is to make an average for that. For example, 0.8 is the average of 0.78, 0.8 and 0.82. At the end, one cell has one datum. A random path is used to avoid the artifacts induced by visiting grid nodes in a regular fashion. The random path must go through each cell once and in random order. This is an example of a random path for nx=8, ny=7 grid. The order is from first to second, etc, until 56th. Input Data. Conditional data. Variogram parameters. Variogram are computed from normal score data. Search parameters. Seed, it is a number used to initialize a pseudorandom number generator. Number of realizations. Minimum and maximum. They are used to clip the result. The first node (marked as 1) is not yet simulated. Let's work on that. The underlying algorithm of Sequential Gaussian Simulation is kriging. Kriging gives us an estimate of both the mean of the property value and its standard deviation using conditional data. Variogram parameters, and Search parameters. Here we get µ = 0.1, σ = 0.7. Computed mean and standard deviation of input conditional data are assigned to the cell when there are no conditional data obtained in the search area. Rather than choosing the weighted mean as the estimate at each node, SGS uses µ = 0.1 and σ = 0.7 construct a normal distribution that is local conditional probability distribution, see the gray curve. Run uniform random number generator which represents the probability level, we get probability 0.71, follow the arrowed green lines, we get property value 0.85. Set the simulated value 0.85 to the node. Include the simulated value (x = 35, y = 45, score = 0.85) in the set of conditioning data. This ensures that closely spaced values have the correct short-scale correlation. The second node (marked as 2) is not yet simulated. The similar procedures as the first node will be conducted, one difference is the simulated 0.85, shown in a purple circle, can be used as the conditional data if it is in the search region in order to preserve the proper covariance structure, spatial continuity, between simulated values. Repeat steps 4 to 6 until all grid nodes have simulated values. Back transform the normal-score realization with zero mean and unit variance into the original data space. For details, please refer to The normal Score Transformation. The final results will be clipped using input Minimum and maximum values. Let's review the steps of SGS. 1. Transform the original well data to normal-score data with zero mean and unit variance. 2. Assign transformed data into simulation grid 3. Generate a random path though the grid nodes. 4. Visit a node in the random path and use kriging to estimate a mean and standard deviation. 5. Select at random a value from the lcpd and set the node value to that number. 6. Include the newly simulated value as part of the conditioning data 7. Repeat steps 4 to 6 until all grid nodes have results. 8. Back-transform the realization into the original space.