Multipoint statistics, or MPS, is a stochastic simulation method that builds spatial models by reproducing patterns found in a training image. Unlike variogram-based methods, which describe relationships between pairs of locations, MPS can learn arrangements involving several locations at once.

This ability makes MPS particularly useful for modeling connected and geometrically complex features such as channels, reefs, dikes, veins, and lithological bodies. It can also honor observed data and reproduce relationships between categorical and continuous variables.

This article explains how MPS works, why training images matter, how the Direct Sampling algorithm generates realizations, and how to configure an MPS workflow in Isatis.neo.
 

Summary: What You Need to Know About MPS

• MPS reproduces spatial patterns from a training image rather than relying only on a variogram or covariance model.
• It is well suited to geological settings where the shape, arrangement, and connectivity of features influence the result.
• The Direct Sampling algorithm fills a simulation grid by searching the training image for patterns that resemble the known neighborhood around each unsimulated location.
• MPS can simulate categorical variables, continuous variables, or several related variables together.
• Conditioning data, trends, proportions, rotation, and scaling can help adapt a training image to the target area.
• A clear and representative training image is usually more useful than an unnecessarily detailed one.
• The main modeling challenge is balancing pattern fidelity, variability, and computation time.

What is multipoint statistics?

Multipoint statistics is a family of geostatistical simulation methods that generates spatial realizations from patterns observed in a reference dataset, commonly called a training image.

A training image is not necessarily a photograph. It is usually a two-dimensional or three-dimensional grid that represents the expected spatial organization of the features being modeled. It may show, for example, the geometry of channels, the continuity of mineralized veins, or the relationship between geological facies.

The algorithm uses the training image as a source of spatial patterns. It then creates one or more alternative realizations that reproduce those patterns while respecting the available conditioning data.

Because each realization is stochastic, MPS does not produce only one definitive interpretation. It produces several plausible models that can be analyzed to understand spatial uncertainty and its effect on technical decisions.

– Two-point statistics versus multiple-point patterns

A variogram describes how similarity changes with distance and direction between pairs of locations. This information is valuable, but it does not completely describe the geometry of a geological feature.

MPS considers patterns involving multiple neighboring locations. It can therefore represent spatial arrangements that are difficult to express through pairwise relationships alone.

For example, two models may reproduce the same histogram and variogram while displaying very different levels of connectivity. Those differences can lead to different flow, permeability, or transport responses, even when conventional statistical validation suggests that the models are similar.
 

Why Variogram-Based Methods May Not Capture Complex Geology

Classical simulation methods remain effective for many spatial modeling problems. However, they may struggle when the result depends on the morphology or connectivity of geological bodies.

Consider a river-delta environment. A multi-Gaussian simulation and an MPS realization may reproduce similar global proportions and two-point statistics. Nevertheless, the multi-Gaussian realization may break channels into disconnected segments, while the MPS realization preserves a more continuous channel network.

That structural difference becomes important when the models are used for flow simulation. A model with fragmented channels can produce a very different outflow response from a model with connected channels.

MPS addresses this limitation by placing the geological concept directly in the training image. The modeler can therefore represent the expected geometry instead of trying to describe it entirely through a covariance function.

– When should MPS be considered?

MPS is particularly relevant when:

• connectivity is a major modeling objective;
• geological bodies have recognizable shapes or arrangements;
• several facies have complex spatial relationships;
• the available conceptual knowledge cannot be represented adequately by a variogram;
• the model must reproduce structures at more than one spatial scale;
• categorical and continuous variables need to be simulated jointly.

MPS does not automatically replace classical geostatistical methods. The choice should depend on the geological problem, the available data, and the decisions the model must support.
 

How the Direct Sampling Algorithm Works

Direct Sampling is an MPS algorithm that populates the simulation grid by looking directly for comparable patterns in the training image. It does not need to build and store an exhaustive catalog of every possible training-image pattern before the simulation begins.

– Step 1: Initialize the simulation grid

The process begins with the simulation grid and any available conditioning data. Conditioning data may include observations from drill holes, wells, samples, or other measured locations.

These known values are transferred to or associated with the simulation grid before the remaining cells are simulated.

– Step 2: Select a location to simulate

The algorithm follows a randomized path through the grid. At each stage, it selects a location whose value has not yet been simulated.

Using a random path helps generate different realizations when the simulation is repeated with different random seeds.

Step 3: Build a neighborhood pattern

The algorithm identifies nearby informed locations. These may include original conditioning data and cells that have already been simulated.

Their positions and values form a local pattern around the target location. In MPS terminology, this local configuration is often called a data event.

– Step 4: Search the training image

The algorithm selects locations in the training image and compares their surrounding patterns with the data event in the simulation grid.

A distance or mismatch measure determines how closely the two patterns correspond. The search continues until:

• a sufficiently close match is found; or
• the configured maximum proportion of the training image has been scanned.

– Step 5: Assign the simulated value

When an acceptable pattern is identified, the value at the center of that training-image pattern is transferred to the target location in the simulation grid.

The procedure is repeated until the simulation domain has been populated. Running the process several times produces alternative realizations of the same geological concept.

– Why Direct Sampling is flexible

Direct Sampling can be used with:

• categorical variables such as facies or rock types;
• continuous variables such as grade, porosity, permeability, or geophysical measurements;
• several variables simulated independently;
• several variables simulated together as a vector;
• conditioning data and auxiliary variables.

When variables are simulated together, the workflow can preserve relationships between categorical and numerical properties instead of modeling each variable in isolation.
 

Where Is Multipoint Statistics Used?

MPS is not limited to one industry or one type of spatial variable. Any application with repeatable spatial patterns may potentially benefit from the method.

– Hydrogeology and reservoir modeling

In hydrogeology and reservoir studies, MPS can reproduce connected aquifer units, channels, barriers, and other structures that control fluid movement.

The resulting realizations can be used to evaluate how uncertainty in geological architecture affects groundwater flow, contaminant transport, or reservoir performance.

– Mining and mineral resource modeling

In mining, MPS can help represent complex lithological domains, narrow mineralized veins, geological contacts, and relationships between geology and grade.

It is especially relevant when the continuity and morphology of geological bodies are difficult to reproduce with conventional techniques.

– Geophysics and spatial gap filling

MPS can be used to reconstruct missing information in spatially structured datasets, including geophysical surveys. The training dataset provides examples of the spatial patterns the algorithm should reproduce in unsampled or missing areas.

– Meteorology and environmental modeling

The same principle can be applied to rainfall fields and other environmental variables with recognizable spatial organization.

The applications highlighted in the original webinar include aquifer modeling, gold-deposit modeling, electromagnetic-survey gap filling, and rainfall simulation.
 

What is a Training Image?

A training image is a conceptual representation of the spatial structures expected in the target area. It teaches the MPS algorithm which patterns are geologically plausible.

A training image should not be treated as a complete geological model. Its role is to represent the main structures, relationships, orientations, and scales that the simulation should reproduce.

– What should a good training image contain?

An effective training image should:

• represent the geological concept relevant to the study;
• include the principal feature shapes and relationships;
• operate at a scale compatible with the simulation grid;
• contain enough repetitions of important patterns;
• distinguish the variables or facies being simulated;
• remain simple enough to understand and validate.

A visually detailed training image is not automatically a statistically useful one. If an important structure appears only once, the algorithm has few opportunities to learn alternative versions of that pattern.

– How can a training image be created?

Depending on the application and available knowledge, a training image can be:

• drawn manually and digitized;
• developed from an analog outcrop or a well-understood site;
• derived from an interpreted geological model;
• generated with mathematical equations;
• produced using object-based simulation;
• constructed by combining automated generation with manual editing.

The objective is not to reproduce the target deposit exactly. The objective is to provide representative examples of the spatial structures the model should consider plausible.

– Should a training image be 2D or 3D?

Both two-dimensional and three-dimensional training images can be used. However, a 2D training image is often easier to create, inspect, and populate with repeated structures.

A reliable 3D training image may require a large grid to include enough examples of each geological feature. Larger training images also increase search and computation requirements.

For layered or stratigraphic geology, one alternative is to simulate successive 2D sections and condition each new section to previously simulated sections. This approach can build a coherent 3D model from representative 2D patterns.

The decision should be based on the dimensionality of the geology. A 2D solution should not be used when essential three-dimensional relationships cannot be represented adequately in sections.
 

How MPS Honors Data and Handles Non-Stationarity

A single training image rarely represents every local variation in a large study area. MPS workflows therefore provide controls for adapting the training-image patterns to the target domain.

– Conditioning to hard data

Conditioning requires the realizations to respect observed values such as drill-hole samples, well data, or known facies contacts.

During simulation, conditioning data contribute to the neighborhood pattern used to search the training image. A conditioning weight can increase the importance assigned to these observed values.

The objective is to reproduce training-image patterns without contradicting reliable field observations.

– Using trends and proportions

A trend variable can guide where particular patterns should occur. The trend must be defined in both the training image and the simulation domain so that the algorithm can match locations with comparable trend values.

Global or local proportion constraints can also guide the expected abundance of facies or classes of continuous values. These controls are useful when proportions vary across the study area.

– Applying rotation and scaling

Rotation changes the orientation of structures reproduced from the training image. Scaling changes their dimensions.

These transformations may be global across the simulation domain or vary locally through auxiliary variables. They allow one conceptual training image to represent structures whose orientation or size changes spatially.

– Adding connectivity constraints

For categorical variables, connectivity controls can be used to indicate that selected conditioning points belong to the same connected feature.

Such constraints can be valuable when geological interpretation shows that separate observations are part of one channel, vein, or other continuous body.
 

How to obtain reliable MPS results

The quality of an MPS model depends on more than selecting an algorithm and running the default parameters.

– Start with a clear geological concept

The training image should communicate a specific geological interpretation. Avoid combining incompatible concepts in one image unless their spatial relationship is explicitly understood.

– Validate patterns, not only global statistics

A histogram or facies proportion can indicate whether the overall distribution is reasonable, but it cannot confirm that the geometry is correct.

Validation should also examine:

• connectivity;
• feature width and length;
• orientation;
• transition relationships;
• spatial proportions;
• agreement with conditioning data;
• behavior in downstream flow or transport models.

– Compare several realizations

One realization shows only one possible spatial outcome. Review several realizations to determine whether the simulation produces both geological realism and meaningful uncertainty.

If every realization appears almost identical, the settings may be too restrictive. If the results vary excessively or lose the expected structures, the training image or search parameters may need adjustment.

– Test the main parameters systematically

Acceptance threshold, neighborhood size, maximum scan fraction, conditioning weight, and multiresolution settings should be included in a sensitivity analysis.

Record each tested configuration and the reason for accepting or rejecting it. This improves reproducibility and makes the final parameter choices easier to defend.

– Balance realism, variability, and runtime

Very strict matching criteria can increase computation time and reduce variability between realizations. Settings that are too permissive may generate patterns that do not reproduce the intended geological concept.

The goal is not to find the strictest possible configuration. It is to find a configuration that reproduces the relevant geology, respects observed data, and retains plausible uncertainty.
 

Key Takeaway

Multipoint statistics provides a practical way to include geological concepts directly in stochastic simulation.

A relatively simple training image can generate complex and connected realizations when it contains representative patterns and is combined with appropriate conditioning and non-stationarity controls. This makes MPS valuable in settings where conventional two-point methods do not adequately reproduce the geometry that controls system behavior.

Successful application still requires geological judgment. The training image, parameters, and resulting realizations should all be validated against observed data, expected structures, and the intended use of the model.