What is Gaussian Simulation?

Gaussian simulation, specifically Sequential Gaussian Simulation (SGS), is a powerful geostatistical technique used to model the spatial distribution and variability of a natural phenomenon. It is a computer-based method designed for the generation of multiple plausible scenarios, known as realizations, of a spatially distributed property. These realizations, represented as z(x) values, are drawn from an underlying multiGaussian random function Z(x). This function is typically defined on a finite point set D, which often involves discretizing a one-, two-, or three-dimensional area of interest into N small volumetric units called voxels.

Understanding Sequential Gaussian Simulation (SGS)

At its core, SGS aims to quantify and visualize the uncertainty associated with estimating values at un-sampled locations across a spatial domain. Unlike simple interpolation methods that provide a single best estimate, SGS generates many equally probable maps or models, each honoring the original data points and the observed spatial correlation structure.

The "sequential" aspect is crucial:

• Conditional Simulation: Values are simulated one at a time at un-sampled locations.
• Dynamic Conditioning: Each new simulated value is conditioned not only on the original measured data points but also on all previously simulated values within its neighborhood. This ensures that the simulated realizations reproduce the spatial continuity (variogram) of the data.

The MultiGaussian Assumption

A fundamental principle behind SGS is the assumption that the spatial phenomenon can be represented by a multiGaussian random function. This implies that any linear combination of the values at different locations follows a Gaussian (normal) distribution. Often, the original data must first be transformed into a Gaussian distribution (e.g., using a normal score transformation) before the simulation can proceed.

How Sequential Gaussian Simulation Works

The general procedure for SGS involves several key steps:

1. Data Transformation: If the original data is not Gaussian, it is transformed into a normal (Gaussian) distribution. This ensures that the simulation can leverage the properties of Gaussian random fields.
2. Variogram Modeling: A variogram or correlogram is calculated and modeled from the transformed data. This model describes the spatial correlation structure—how similarity between data points changes with distance and direction.
3. Define a Random Path: A random path is established that visits every un-sampled location in the area of interest exactly once.
4. Local Conditional Distribution: For each location along the random path, a local conditional probability distribution is determined. This distribution is calculated using Kriging, considering the original data and all previously simulated values within a defined search neighborhood. This Kriging provides a conditional mean and a conditional variance for that location.
5. Random Draw: A value is then randomly drawn from this local conditional Gaussian distribution.
6. Add to Data Set: The simulated value is added to the data set and treated as a "hard data" point for subsequent simulations.
7. Iteration: Steps 4-6 are repeated for all un-sampled locations along the random path.
8. Back Transformation: If the data was transformed, the entire simulated realization is back-transformed to the original data units.
9. Multiple Realizations: The entire process (steps 3-8) is repeated multiple times, each with a different random path and different random draws, to generate numerous equally probable realizations.

Why is Gaussian Simulation Used? Applications and Benefits

Gaussian simulation is widely applied in various fields to understand and quantify spatial uncertainty, rather than just providing a single best estimate.

• Environmental Sciences:
  • Mapping contaminant plumes in soil or groundwater.
  • Assessing the spread of pollutants.
  • Modeling soil properties for agricultural or ecological studies.
• Earth Sciences (Geology, Mining, Petroleum Engineering):
  • Characterizing subsurface reservoirs (e.g., porosity, permeability distribution for oil and gas).
  • Estimating ore body grades and volumes in mining.
  • Modeling hydraulic conductivity in aquifers for hydrogeological studies.
  • Generating geological facies models.
• Remote Sensing and Image Processing:
  • Super-resolution mapping.
  • Gap filling in satellite imagery.

Benefits of using Gaussian simulation include:

• Uncertainty Quantification: It explicitly provides a measure of uncertainty by generating multiple possible outcomes, allowing for risk assessment.
• Realistic Spatial Patterns: Realizations reproduce the spatial variability and continuity observed in the original data, creating more realistic maps than simple smoothing interpolators.
• Data Honoring: All realizations honor the original measured data points at their exact locations.
• Constraint Incorporation: Can incorporate soft data and geological interpretations as constraints.
• Decision Making: Multiple realizations allow decision-makers to evaluate different scenarios and the associated risks, leading to more robust decisions in resource management, environmental remediation, and engineering design.

Key Concepts in Gaussian Simulation

While specific terminology can vary, understanding these concepts is vital:

• Random Function (RF): A set of random variables, one for each location in space, describing the spatial distribution of a property.
• MultiGaussian Random Function: An RF where any collection of its random variables follows a joint multivariate normal distribution. This is the underlying statistical model for SGS.
• Realization: A single, specific outcome or simulated map generated from the random function. Each realization represents one possible configuration of the spatial phenomenon.
• Voxel: A three-dimensional pixel, typically used to discretize a volume of interest in spatial modeling. The reference notes that the finite point set D is generally discretized into N voxels.
• Variogram/Correlogram: A geostatistical tool that quantifies the spatial correlation or dissimilarity between data points as a function of the distance and direction separating them.

Limitations of Gaussian Simulation

Despite its advantages, Gaussian simulation has some limitations:

• MultiGaussian Assumption: The assumption of multi-normality may not always hold true for all natural phenomena, especially those with highly skewed distributions or extreme values.
• Computational Cost: Generating many realizations for large datasets can be computationally intensive.
• Parameter Sensitivity: The quality of the simulation depends heavily on the variogram model parameters and the number of conditioning data points.
• Edge Effects: Simulations can sometimes exhibit artifacts or less reliable results near the boundaries of the simulated domain.

Gaussian simulation remains a cornerstone technique in geostatistics for its ability to provide a comprehensive understanding of spatial variability and the associated uncertainty, moving beyond single-best estimates to a rich representation of plausible realities.