Exploring substitution random functions composed of stationary multi-Gaussian processes.

Area of Science:

• Earth sciences
• Geostatistics
• Stochastic modeling

Background:

• Simulation of random fields is crucial for Earth science modeling and uncertainty quantification.
• Multi-Gaussian random fields are common but struggle to represent highly or poorly connected spatial structures.
• Existing models often isolate low or high value regions, impacting flow and transport simulations.

Purpose of the Study:

• To investigate the properties of Substitution Random Functions (SRFs) constructed from stationary multi-Gaussian fields.
• To develop a technique for controlling the connectivity of specific value ranges (low, intermediate, high) within SRFs.
• To enhance the flexibility of random field models for improved spatial feature representation.

Main Methods:

• Utilizing stationary multi-Gaussian random fields for both the directing (T) and coding (Y) processes.
• Combining these processes to form SRFs with bounded variograms.
• Introducing a control point in the coding process (Y) to guide realizations and using a Gibbs sampler for local value conditioning.

Main Results:

• The resulting SRFs (Z) are stationary but non-ergodic for mean and covariance due to the finite variance of the directing field (T).
• Demonstrated a method to control connectivity of low, intermediate, or high values within SRF realizations.
• Successfully conditioned SRFs to local values using a Gibbs sampler.

Conclusions:

• SRFs provide a more flexible alternative to multi-Gaussian fields for modeling spatial structures with controlled connectivity.
• The proposed control point technique enhances the practical applicability of SRFs in Earth sciences.
• This approach improves the representation of crucial spatial features like connectivity in flow and transport problems.