Geostatistics Glossary E-M#
Michael J. Pyrcz, Professor, The University of Texas at Austin
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Chapter of e-book “Applied Geostatistics in Python: a Hands-on Guide with GeostatsPy”.
Cite this e-Book as:
Pyrcz, M.J., 2024, Applied Geostatistics in Python: a Hands-on Guide with GeostatsPy [e-book]. Zenodo. doi:10.5281/zenodo.15169133 
The workflows in this book and more are available here:
Cite the GeostatsPyDemos GitHub Repository as:
Pyrcz, M.J., 2024, GeostatsPyDemos: GeostatsPy Python Package for Spatial Data Analytics and Geostatistics Demonstration Workflows Repository (0.0.1) [Software]. Zenodo. doi:10.5281/zenodo.12667036. GitHub Repository: GeostatsGuy/GeostatsPyDemos 
By Michael J. Pyrcz
© Copyright 2024.
Ensemble Kalman Filter#
The Ensemble Kalman Filter (EnKF) is a sequential Bayesian updating method that updates an ensemble of model realizationss by conditioning simulated predictions on observed data.
The EnKF represents uncertainty using an ensemble of possible model states:
where:
\(\mathbf{m}^{(\ell)}\) is the model state for realization \(\ell\)
\(L\) is the number of realizations
Each realization is passed through a forward model:
where:
\(\mathbf{d}_{sim}^{(\ell)}\) is the simulated observation for realization \(\ell\)
\(G\) is the forward model or transfer function
When observations become available, each realization is updated:
where:
\(\mathbf{m}_{prior}^{(\ell)}\) is the prior model realization
\(\mathbf{m}_{post}^{(\ell)}\) is the updated posterior realization
\(\mathbf{d}_{obs}^{(\ell)}\) is the observed data, often perturbed to represent measurement uncertainty
\(K\) is the Kalman gain
The Kalman gain determines how strongly observations update the model:
where:
\(C_{md}\) is the model-data covariance
\(C_{dd}\) is the simulated data covariance
\(C_d\) is the observation error covariance
The EnKF is commonly used for dynamic history matching because it can update many uncertain model variables while accounting for correlations between model parameters and observations.
Advantages:
maintains an ensemble representation of uncertainty
updates models sequentially as new data become available
avoids requiring explicit calculation of large covariance matrices
Limitations:
assumes approximately Gaussian uncertainty
updates may create unrealistic model properties for strongly nonlinear systems
ensemble size may limit representation of uncertainty
Applications include:
reservoir history matching
groundwater model calibration
weather forecasting
ocean modeling
The Ensemble Kalman Filter is an application of inversion and Bayesian updating.
Used in:
Also see:
Ergodic Fluctuations#
Statistical fluctuations observed when calculating statistics from finite simulated realizations of an ergodic random function. The statistics calculated from an individual realization are expected to vary around the input model statistics. For example,
the histogram of an individual realization may not exactly reproduce the input histogram
the variogram of an individual realization may not exactly reproduce the input variogram
the correlation coefficient between primary and secondary paired realizations may not exactly reproduce the input correlation coefficient
Some general observations about ergodic fluctuations,
part of the uncertainty model - fluctuations in statistical reproduction, along with scenarios, are an important part of the uncertainty model because they represent natural variability among possible realizations
magnitude - controlled by the ratio of spatial continuity range to the size of the model domain
minimized - when the model domain is large relative to the spatial continuity range, providing many effective independent spatial samples
maximized - when the model domain is small relative to the spatial continuity range, providing fewer effective independent spatial samples
When checking simulated realizations, some fluctuation in the histogram, variogram, and correlation coefficients should be expected.
best practice is to evaluate the expectation of these statistics over many realizations and compare the ensemble statistics with the input model statistics
Used in:
Error#
Error is the difference between a true value and an estimated or predicted value.
The error is:
where:
\(e\) is the error
\(\hat{Z}\) is the estimated value
\(Z\) is the true value
In practice, the true value is generally unknown. Therefore, models use observed residuals:
where:
\(r_i\) is the residual
\(y_i\) is the observed value
\(\hat{y}_i\) is the predicted value
Residuals are measurable differences between observations and predictions, while errors include the unknown difference from the true value.
Error may include:
random error due to natural variability or measurement uncertainty
systematic error due to bias in a model or measurement process
Error is fundamental to:
Also see:
Estimation#
The paradigm and process of obtaining a single best value to represent a feature or variable at an unsampled location or time. The “best” estimate is determined by an objective criterion, such as minimizing estimation error or uncertainty.
Some additional estimation concepts,
local accuracy - estimation methods prioritize honoring local data and minimizing local uncertainty, often at the expense of reproducing the full range of global spatial variability
deterministic model - the same inputs always produce the same outputs
smoothness - estimation methods commonly produce values that are smoother than the true variability because local averaging reduces variance
nonlinear response - smooth estimates may not be appropriate when applying transforms or decision criteria that are sensitive to heterogeneity, such as flow response, connectivity, recovery, or economic metrics
examples - inverse distance weighting and kriging
many predictive machine learning models focus on estimation, including k-nearest neighbours, decision trees, and random forests
Used in:
Contrast with the simulation paradigm:
Estimation Variance#
The uncertainty of a spatial estimate quantified as the variance of the difference between the estimated value and the unknown true value,
where \(Z^*(\mathbf{u})\) is the estimated value at location \(\mathbf{u}\) and \(Z(\mathbf{u})\) is the unknown true value.
For simple kriging, the estimator is a weighted sum of nearby data,
Substituting the kriging estimator into the estimation variance,
Expanding the variance using covariance terms,
where \(C(\mathbf{u}_\alpha,\mathbf{u}_\beta)\) represents covariance between data locations and \(C(\mathbf{u}_\alpha,\mathbf{u})\) represents covariance between data locations and the estimation location.
Using the kriging system equations, the estimation variance simplifies to,
or,
where \(\sigma^2\) is the variance of the feature and \(C(\mathbf{h}_\alpha)\) is the covariance between each data location and the estimation location separated by lag vector \(\mathbf{h}_\alpha\).
The estimation variance has important properties,
minimized by the kriging weights, making kriging a minimum variance unbiased estimator.
depends only on the spatial configuration of the data, the covariance model, and the estimation location; it does not depend on the measured data values.
generally increases where data are sparse or spatial continuity is weak, and decreases where data are dense and spatial continuity is strong.
Used in:
See also:
Evidence#
In Bayes’ Theorem, the evidence term represents the overall probability of observing the data. It provides the normalization required to ensure probability closure of the updated posterior probability.
where:
\(P(A)\) is the prior probability representing uncertainty before observing new information,
\(P(B|A)\) is the likelihood function describing the compatibility of observations \(B\) with state or parameter \(A\),
\(P(B)\) is the evidence term representing the total probability of observing data \(B\) and normalizing the posterior probability,
\(P(A|B)\) is the posterior probability representing updated uncertainty after incorporating observations.
Expectation#
The expected value is the probability-weighted average outcome of a random variable. It is a measure of central tendency that represents the average value accounting for the likelihood of all possible outcomes.
for a discrete random variable, the expectation is the sum of all possible outcomes weighted by their probabilities,
for a continuous random variable, the expectation is the integral of all possible values weighted by the probability density function,
Expectation is also the mathematical foundation for the average when all realizations are considered equiprobable.
Expectation is extremely useful for doing mathematics with random variables,
expectation of a constant,
expectation of a random variable plus a constant,
expectation of a constant multiplied by a random variable,
expectation of the addition of two random variables,
Expectation is widely used for optimum decision making in the presence of uncertainty, i.e., selecting the choice that maximizes expected profit.
Used in:
Also see:
Experimental Variogram#
An empirical variogram calculated from sample data and visualized as discrete points, with lag distance on the x-axis and variogram value on the y-axis.
Each point represents one half the average squared difference between data pairs separated by a specified lag distance and direction,
where \(N(\bf{h})\) is the number of data pairs separated by lag vector \(\bf{h}\). Some salient points about experimental variograms,
experimental variogram results can be sensitive to variogram search parameters, including lag distance tolerance and azimuth tolerance, which control the data pairs included in each estimate.
experimental variograms are represented by discrete points because they are calculated only at selected lag distances and directions. This differentiates them from a variogram model, which is represented by a continuous mathematical function.
experimental variograms are often more difficult to calculate and interpret in horizontal directions due to sparse sampling and limited data pairs. Vertical experimental variograms are often easier to characterize because wells commonly provide dense sampling along vertical and sub-vertical directions.
Used in:
Contrast with:
F-distribution#
The F-distribution is a continuous statistical distribution describing the ratio of two independent variance estimates.
The F-statistic is calculated as:
where:
\(s_1^2\) is the first sample variance
\(s_2^2\) is the second sample variance
The F-distribution is defined by two degrees of freedom:
\(\nu_1\) for the numerator variance
\(\nu_2\) for the denominator variance
The F-distribution is commonly used for:
comparing variances between populations
analysis of variance (ANOVA)
testing the significance of regression models
The F-distribution is asymmetric and bounded below by zero:
Used in:
Also see:
Facies#
A method of grouping rock into discrete categories, creating a new categorical feature. Facies are used to represent geological variability in a manner that improves,
characterization through statistics, e.g., distributions and variograms
prediction of subsurface features, e.g., porosity and permeability away from wells
For oil and gas, the term facies is commonly used, while mining commonly uses terms such as rock types or zones. In subsurface modeling, multiple types of facies may be considered,
lithofacies - based on rock-related characteristics, including lithology, sedimentary structures, and small-scale geological features that influence porosity and permeability, for example, shale, sandstone, dolomite, limestone, laminated sandstone, hummocky cross-stratification, etc.
depofacies - integrate multiple lithofacies with depositional geometry and reservoir-scale architecture that impacts flow behavior and well connectivity, for example, channel axis, channel margin, outer sheet, etc.
seismic facies - large-scale classifications based on acoustic and elastic properties and seismic geomorphological expressions that define the reservoir framework, for example, parallel continuous high amplitude, chaotic amplitudes, mounded discontinuous low amplitudes, truncation, onlap, offlap, etc.
Here are some important considerations for determining facies,
facies or rock type is an important decision for subsurface modeling. Facies determination should remain a collaborative decision integrating expertise from the entire project team (Geologists, Reservoir Modelers, Reservoir Engineers, Petro- and Geophysicists).
facies or rock types must improve subsurface prediction away from the data or they do not add value.
the number of facies is a balancing act between geological realism, statistical inference, and modeling effort.
reservoir modeling is often hierarchical, for example, geological elements contain multiple depofacies, depofacies contain multiple lithofacies, and lithofacies have specific porosity and permeability distributions.
often 80-90% of reservoir-scale heterogeneity may be captured by the facies model.
Here is a summary of criteria for facies, rock types, or any discrete grouping used in a subsurface model,
Separation of rock properties - facies must be separable based on features that impact subsurface environmental and economic performance, for example, grade, porosity, permeability, etc.
Identifiable in data - facies must be identifiable from the most commonly available data. For example, facies identifiable only from cores are not useful if most wells only have well logs.
Map-able away from data - facies must be easier to predict away from data than the rock properties of interest directly; otherwise, facies do not improve prediction.
Sufficient sampling - there must be enough data to infer reliable statistics within each facies, i.e., by-facies statistics.
Used in:
Facies Simulation#
Methods for generating multiple plausible spatial facies realizations of categorical features, commonly facies in oil and gas or rock types in mining.
Facies simulation represents geological uncertainty by generating alternative models that honor available data and spatial relationships.
There are four common approaches for simulating facies,
sequential indicator simulation - based on indicator variograms for each facies category and indicator kriging to calculate local conditional probabilities within the sequential simulation framework. These conditional probabilities define the local cumulative distribution function used to sample facies realizations.
multiple point simulation - a pattern reproduction method based on categorical training images. Spatial relationships and geological patterns are learned from the training image and used to calculate local conditional probabilities within a sequential simulation framework.
object-based simulation - a marked point process based on Monte Carlo simulation of geometric parameters and stochastic placement of geological objects within the volume of interest.
generative machine learning - trained deep learning generators used to calculate categorical realizations by learning complex geological patterns from training datasets.
Facies simulation methods differ primarily in how they represent and reproduce geological information,
variogram-based methods prioritize two-point spatial continuity.
multiple point methods prioritize complex spatial patterns and geological geometries.
object-based methods prioritize explicit geological objects and architecture.
generative machine learning methods prioritize learned patterns from large training datasets.
Used in:
Feature#
A property measured, observed, or calculated (i.e., engineered feature) for analysis in a study. Features represent the information used to characterize, model, or predict a system. Examples include,
porosity, permeability, mineral concentrations, saturations, contaminant concentration, etc.
derived properties such as seismic attributes, ratios, transformations, or other calculated quantities.
Different fields use different terminology,
in data mining and machine learning this is commonly called a feature
in statistics this is commonly called a variable
in geoscience this is often called a property or attribute
Feature values may require significant measurement, processing, interpretation, and analysis before they are suitable for modeling.
When features are modified, combined, or transformed to improve model performance, this is called feature engineering.
Also known as:
Variable
Used in:
Feature Engineering#
The process of creating, modifying, combining, transforming, or selecting features to improve model performance, interpretation, or statistical inference. Feature engineering may incorporate domain knowledge, physical understanding, and data analysis. Examples include,
adjusting total porosity to effective porosity
combining porosity and permeability into a single rock quality index measure
transforming data to account for volume support differences using a volume-variance relations model
weighting data samples for improved spatial representativity using declustering
transforming a data distribution to standard normal in sequential Gaussian simulation
Feature engineering is commonly applied before modeling to create inputs that better represent the controlling processes and improve prediction or estimation
Used in:
Forward Model#
A forward model is a mathematical or computational model that predicts observations or responses from a set of model parameters, features, or system states.
The forward relationship is:
where:
\(\mathbf{d}\) is the predicted observation or response
\(\mathbf{m}\) is the model state, parameters, or features
\(G\) is the forward model or transfer function
Forward models represent the process by which a system generates observable data.
Examples include:
reservoir model \(\rightarrow\) flow simulator \(\rightarrow\) production data
ore body model \(\rightarrow\) mining and processing model \(\rightarrow\) plant feed grade
subsurface properties \(\rightarrow\) seismic simulator \(\rightarrow\) seismic response
The inverse problem attempts to determine the model that produced observations:
However, inverse problems are often uncertain and nonunique because multiple models may produce similar observations.
Forward models are fundamental components of:
Also see:
Frequentist Probability#
A measure of the probability that an event occurs based on the long-run relative frequency observed from repeated experiments or repeated sampling. For random experiments and well-defined settings (such as coin tosses),
where:
\(n(A)\) = number of times event \(A\) occurred
\(n\) = number of trials
The frequentist interpretation assumes that probability represents an objective property of a repeatable process. Examples include,
probability of drilling a dry hole for the next well
probability of encountering sandstone at a location (\(\bf{u}_{\alpha}\))
probability of exceeding a rock porosity of \(15\%\) at a location (\(\bf{u}_{\alpha}\))
In geoscience, many processes cannot be repeated exactly; therefore, frequentist probabilities are often estimated from available samples under assumptions of representativity and stationarity.
Used in:
Contrast with:
Gammabar#
A volume-integrated variogram, \(\overline{\gamma}(v,V)\), that accounts for the spatial-continuity between two volumes of support. The tail locations are integrated over volume \(v\) and the head locations are integrated over volume \(V\),
where \(\mathbf{u}\) and \(\mathbf{u}^{\prime}\) represent all possible tail and head locations within the two volumes.
Since a continuous variogram function for all possible locations is not generally available, gammabar values are practically calculated by discretizing both volumes and averaging the variogram values over all possible pairs of discretized locations over the two volumes,
where \(n_v\) and \(n_V\) are the number of discretization points within volumes \(v\) and \(V\).
Gammabar values are used to calculate spatial continuity while accounting for the volume support of data samples and model cells. For kriging systems, the volume-integrated covariance, known as “c bar”, \(\overline{C}(v,V)\), is calculated as,
where \(\sigma^2\) is the feature variance.
Dispersion variance, which quantifies variance changes due to scale or volume support, can be calculated using gammabar values,
where \(\overline{\gamma}_{V,V}\) and \(\overline{\gamma}_{v,v}\) are variogram models integrated over volumes \(V\) and \(v\), respectively.
Used in:
Gaussian Distribution#
The Gaussian (or normal) distribution is a continuous statistical distribution defined by its mean, \(\mu\), and variance, \(\sigma^2\), which control the location and spread of the distribution.
The probability density function is:
The Gaussian distribution is symmetric:
The standard normal distribution is the specific case with:
mean = 0
standard deviation, \(\sigma = 1.0\) (variance, \(\sigma^2 = 1.0\))
The standard normal distribution is often represented as:
Gaussian error assumptions are commonly associated with homoscedasticity, where uncertainty is approximately constant.
Used in:
Also see:
Geometric Anisotropy#
A variogram interpreted structure in spatial data where the range varies by direction. Some observations,
commonly, the vertical range of correlation is much less than the horizontal range due to the formation of “layering” due to sedimentary processes. Walter’s Law in stratigraphy states that the vertical sequence occurs horizontally at difference scales
the ratio of the horizontal to vertical range, \(a_{hori}:a_{vert}\) is commonly known as the horizontal to vertical anisotropy ratio, for example, a fluvial setting may have a 10:1 horizontal to vertical anisotropy ratio.
geometric anisotropy is common for the horizontal directions also, the ratio of horizontal major direction : horizontal minor direction range, \(a_{maj}:a_{min}\), is commonly known as the horizontal major to minor anisotropy ratio
Used in:
Also see:
Geometric Anisotropy Model#
We assume geometric anisotropy to model 2D and 3D variogram over all directions from experimental variograms calculated only in primary directions.
this model provides a valid interpolation of the variogram between the primary directions
the geometric anisotropy model is based on this lag distance,
where \(a_{maj}, a_{maj}, a_{vert}\) are the ranges in the major, minor and vertical directions and \(\bf{h}_{maj}, \bf{h}_{maj}, \bf{h}_{vert}\) are the lag distance components in the major, minor and vertical directions.
Used in:
Also see:
Geometric Mean#
The geometric mean is a univariate statistic that represents the central tendency of positive values using multiplication rather than addition.
For \(n\) positive observations, \(x_1, x_2, ..., x_n\), the geometric mean is:
or equivalently,
The geometric mean is appropriate for multiplicative processes and positively skewed variables, such as growth factors and some features with approximately lognormal distributions.
The geometric mean is less influenced by large values than the arithmetic mean.
Used in:
Also see:
Geostatistics#
A branch of applied statistics that integrates,
spatial (geological) context
spatial relationships and continuity
volume support and scale
Geostatistics provides methods to characterize, model, predict, and simulate spatial phenomena by incorporating the spatial structure of the data and the uncertainty in the subsurface or other spatial systems to support optimum decision making.
The boundary between geostatistics and spatial statistics is debated. In this course,
geostatistics includes many spatial statistics methods because, in practice, useful approaches for modeling spatial phenomena are adopted and integrated into the geostatistical toolkit.
Geostatistics is an expanding and evolving field of study that continues to incorporate new statistical, computational, and data-driven approaches.
Used in:
Global Accuracy#
Honoring (matching) global measures calculated over the entire volume of interest, for example,
Global accuracy is a primary objective for simulation, where the realizations should reproduce the input global statistics in expectation.
Used in:
Contrast with:
Global Measure#
A statistical or spatial summary calculated over the entire volume of interest. Examples include,
Global measures characterize the overall behavior of a feature or model and are commonly used for model checking and validation.
Used in:
Contrast with:
Grade#
Grade is the concentration or abundance of a valuable component within a rock, ore, or mineral deposit. Grade is the primary property used to quantify the economic value of a mineral resource.
Grade is calculated as:
where:
\(G\) is grade
\(M_v\) is the mass (or amount) of the valuable component
\(M_t\) is the total mass (or amount) of the sample
Grade may be reported in several units depending on the commodity, including:
weight percent (wt.%)
grams per tonne (g/t)
parts per million (ppm)
parts per billion (ppb)
ounces per ton (oz/ton)
Grade is commonly estimated and modeled using geostatistical methods to support mineral resource estimation, mine planning, and uncertainty assessment.
Examples include:
copper grade (% Cu)
gold grade (g/t Au)
iron grade (% Fe)
lithium grade (% Li₂O)
potash grade (% K₂O)
Used in:
Also see:
Gridded Data#
Data represented at regularly spaced locations over a 2D area of interest or 3D volume of interest, commonly used to represent maps and spatial models.
Gridded data are characterized by,
regularly spaced locations defined by grid dimensions and cell spacing
exhaustive coverage over the modeled domain, where every grid location contains a value or assigned missing value
implicit spatial relationships between neighboring grid locations
Gridded data may be stored as,
a .csv comma-delimited file, where data may be represented as a matrix with \(n_y\) rows and \(n_x\) columns
binary formats for more compact storage and faster processing, although these files are not human readable
Gridded data are commonly visualized directly, for example,
image representations using functions such as matplotlib’s imshow
contour maps and other spatial visualization methods
Examples of gridded data include,
geological interpretation-based maps
acoustic impedance models inverted from seismic reflection surveys
For spatial modeling problems,
model outputs or response features are often estimated over a regular grid to create continuous maps or 3D models
Used in: -Gridded Data Loading and Visualizing Demonstration
Also see:
Hard Data#
Data that is treated as certain due to having a high degree of certainty relative to other available information sources. Hard data usually comes from direct measurement or observation of the feature of interest, for example,
Core-based porosity, permeability, mineralogy, and facies observations
direct measurements of grade from drill core samples
laboratory measurements of rock or fluid properties
Hard data is considered sufficiently reliable that uncertainty in the measurement is commonly not explicitly modeled or integrated into subsequent workflows. Note, hard data is not necessarily error-free;
it represents information that is treated as certain for the purpose of the analysis.
Hard data generally has high resolution (small scale, volume support), but poor spatial coverage because only an extremely small proportion of the population is directly sampled. For example,
Core coverage deepwater oil and gas - well core may sample only one five hundred millionth to one five billionth of a deepwater reservoir, assuming 3 inch diameter cores with 10% core coverage in vertical wells with 500 m to 1,500 m spacing
Core coverage mining grade control - diamond drill hole cores may sample one eight thousandth to one thirty thousandth of an ore body, assuming HQ 63.5 mm diameter cores with 100% core coverage in vertical drill holes with 5 m to 10 m spacing
Hard data provides the most direct calibration of subsurface models but must be integrated with other information sources to overcome limited spatial coverage.
Used in: TBD
Contrast with:
Harmonic Mean#
The harmonic mean is a univariate statistic that emphasizes smaller values and is commonly used for averaging rates and ratios.
For \(n\) positive observations, \(x_1, x_2, ..., x_n\), the harmonic mean is:
The harmonic mean is useful when small values strongly influence the overall response, for example averaging velocities, rates, or flow properties in series systems.
For positive values:
where \(H\) is the harmonic mean, \(G\) is the geometric mean, and \(\bar{x}\) is the arithmetic mean.
Used in:
Also see:
Heterogeneity#
Heterogeneity is the degree of spatial or statistical variability of a geological property within a volume of interest.
Heterogeneity may occur due to variations in geological processes, depositional environments, diagenesis, structural deformation, or fluid flow behavior.
Examples of heterogeneity include:
permeability variations controlling preferential flow paths
facies changes controlling reservoir connectivity
grade variability affecting mineral resource estimation
Heterogeneity may be quantified using statistical measures such as:
Heterogeneity is a key consideration in estimation, simulation, and subsurface decision-making because spatial variability affects uncertainty and predicted system behavior.
Used in:
Also see:
Heteroscedasticity#
Heteroscedasticity is the property that the variance of a feature or the residuals from an estimation model changes over the range of the predictor features or estimated values.
Heteroscedasticity commonly occurs in subsurface data where uncertainty increases with the magnitude of the response, for example, permeability, production rate, or ore grade.
Accounting for heteroscedasticity is important for accurate uncertainty quantification and prediction interval estimation.
Used in:
Also see:
Histogram#
A bar chart representation of a univariate statistical distribution showing the frequency of samples over an exhaustive set of bins spanning the range of possible values.
These are the steps to build a histogram,
Divide the continuous feature range of possible values into \(K\) equal size bins, \(\Delta x\):
or use available category labels for categorical features.
Count the number of samples (frequency) in each bin, \(n_k\), \(\forall k=1,\ldots,K\).
Plot frequency versus the bin label (use bin centroid for continuous features).
The histogram y-axis represents frequency. When normalized by the total number of samples,
the result is a normalized histogram, where the y-axis represents probability.
for discrete features, the normalized histogram is an empirical probability mass function
for continuous features, the normalized histogram represents probability mass over intervals; dividing by bin width provides an estimate of the probability density function
Additional comments about histograms,
typically plotted as bar charts
2D histograms use a orthonormal view columns with 2 axes for features and along with the frequency axis, but not often used to attempt to visualize bivariate relationships
Used in:
Also see:
History Matching#
A workflow to update a subsurface model by adjusting uncertain model features and parameters such that the output of a production transfer function matches observed historical data.
History matching is:
a specific case of inversion, often known as historical production matching
an inversion problem that is challenging because the system is ill-posed and the solution is generally nonunique
For example:
update porosity and permeability features throughout a reservoir model such that flow simulation results match historical production data
update gold grade distributions throughout an ore body model such that the simulated mining schedule, mixing, dilution, and machine selectivity models match historical plant feed grade
A subsurface model produces predicted observations through a forward simulator:
where:
\(\mathbf{d}_{sim}\) is the simulated response
\(\mathbf{m}\) is the subsurface model
\(G\) is the forward simulator or transfer function
History matching seeks model parameters that minimize the difference between simulated and observed data:
where:
\(\mathbf{d}_{obs}\) is the observed data
\(W\) is a weighting matrix representing measurement uncertainty
History matching commonly updates uncertain reservoir properties including:
permeability distribution
porosity distribution
fault properties
relative permeability parameters
aquifer support
Because the inversion problem is nonunique, many subsurface models may reproduce historical observations. Therefore, history matching often generates multiple conditioned realizationss rather than a single best model.
A successful history match should:
reproduce historical observations within uncertainty
maintain geological plausibility
quantify remaining uncertainty for forecasting
History matching is a form of inversion and is closely related to Bayesian updating.
Used in:
Also see:
Homoscedasticity#
Homoscedasticity is the property that the variance of a feature or the residuals from an estimation model remains approximately constant over the range of the predictor features or estimated values.
Homoscedasticity is a common assumption in many statistical and machine learning methods. Violations of this assumption may result in inefficient estimates and unreliable uncertainty quantification.
Used in:
Also see:
Hybrid Model#
A model that combines both deterministic model and stochastic model components.
Hybrid models separate predictable structure from uncertain variability by combining,
deterministic components - representing known relationships, physical processes, expert interpretation, or data-driven trends
stochastic components - representing spatial variability, uncertainty, and unresolved processes that cannot be deterministically modeled
Most geostatistical models are hybrid models. For example, an additive deterministic trend model and stochastic residual model:
where \(m(\mathbf{u})\) represents the deterministic trend and \(R(\mathbf{u})\) represents the stochastic residual.
Other examples include,
deterministic geological frameworks combined with stochastic property simulation
physics-based models calibrated or conditioned with stochastic uncertainty models
data-driven predictive models combined with probabilistic uncertainty models
Used in:
Hypothesis#
A hypothesis is a testable statement or assumption about a population parameter, the relationship between features, or the performance of a model. Statistical hypothesis testing uses sample data to determine whether there is sufficient evidence to reject the assumed hypothesis.
A hypothesis test consists of:
a null hypothesis – the assumption that there is no effect, relationship, or statistically significant difference
an alternative hypothesis – the assumption that an effect, relationship, or statistically significant difference exists
a test statistic – a standardized measure of the observed effect relative to its standard error
a standard error – quantifies the uncertainty in the observed effect by accounting for sample size and sampling variability
a decision based on a significance level or p-value
Hypothesis testing is widely used to compare populations, evaluate models, and determine whether observed differences are likely due to random sampling variation. It helps answer the practical question:
“Is the observed effect likely to be real, or could it reasonably be explained by random sampling variation?”
Hypothesis testing also protects us from the law of small numbers fallacy—the mistaken belief that small samples are as representative as large samples.
Used in:
Also see:
Hypothesis Test Result#
A hypothesis test has two possible outcomes:
reject the null hypothesis
fail to reject the null hypothesis
The decision is based on the calculated test statistic, the corresponding p-value, or comparison with a critical value.
There are four possible combinations of the statistical decision and the true state of nature:
True State |
Decision |
Outcome |
|---|---|---|
Null hypothesis is true |
Fail to reject |
Correct decision |
Null hypothesis is true |
Reject |
Type I error (false positive) |
Null hypothesis is false |
Reject |
Correct decision |
Null hypothesis is false |
Fail to reject |
Type II error (false negative) |
Hypothesis testing provides evidence for making decisions but cannot prove that a hypothesis is true or false.
Used in:
Also see:
Independence#
Two random events, \(A\) and \(B\), are independent if knowledge of one event provides no information about the likelihood of the other event. Mathematically, events are independent if and only if the following equivalent relationships are true,
Conditional probability:
If any of these relationships are violated, then the events are dependent, indicating that some form of relationship exists between them.
Note that dependence does not necessarily imply causation.
In spatial modeling, independence indicates that knowing one feature or event provides no additional information about another feature or event. For example,
spatially independent samples have no correlation over the specified lag distance.
Used in:
Indicator Kriging#
The application of simple kriging to a set of indicator transforms, one for each threshold of a continuous feature or one for each category of a categorical feature, to directly estimate the local uncertainty model. For continuous features, the result is a local cumulative distribution function (CDF); for categorical features, it is the local probability of each category.
The indicator kriging estimator is,
where \(\lambda_\alpha(k)\) is the indicator kriging weight for data \(\alpha\) and threshold or category \(k\), \(i(\mathbf{u}_\alpha; k)\) is the indicator transform at location \(\mathbf{u}_\alpha\), and \(p(k)\) is the global probability (or a local probability if a trend model is provided).
by estimating \(p^*_{IK}(\mathbf{u}; k)\) for every threshold or category, indicator kriging directly estimates the local uncertainty model without assuming any specific probability distribution (i.e., no Gaussian assumption).
The workflow for indicator kriging is,
Define thresholds or categories.
for categorical features, the categories are given.
for continuous features, select thresholds that adequately span the feature distribution so that the local CDF can be resolved.
thresholds may also correspond to meaningful engineering, environmental, or economic decision limits.
Apply the indicator transform to the data.
Calculate an indicator variogram for each threshold or category.
Perform indicator kriging independently for each threshold or category to estimate the local cumulative probability (continuous features) or category probability (categorical features).
Apply an order relations correction to ensure the final uncertainty model is mathematically valid.
for continuous features, independently estimated cumulative probabilities may violate monotonicity.
for categorical features, independently estimated probabilities may not sum to one.
Some general observations,
a separate variogram model is required for every threshold or category, making inference substantially more demanding than conventional kriging.
however, this additional flexibility allows spatial continuity to vary with feature value. For example, high-grade mineralization or high-porosity regions may exhibit different spatial continuity than the remainder of the distribution.
indicator kriging naturally accommodates multiple information types through soft-data encoding. For example, a probability distribution (random variable) may be assigned at a data location instead of a single hard value.
Used in:
See also:
Indicator Transform#
Indicator coding converts a random variable into a probability relative to a category or a threshold.
For a categorical feature, the indicator answers the question,
what is the probability that the data value or realization belongs to a specific category?
The indicator transform is,
For example,
given category \(z_2 = 2\), and data at \(\mathbf{u}_1\) with \(z(\mathbf{u}_1)=2\), then \(i(\mathbf{u}_1;z_2)=1\)
given category \(z_1 = 1\), and a random variable away from data at \(\mathbf{u}_2\), the probability that the realization belongs to category \(z_1\) is \(P(Z(\mathbf{u}_2)=z_1)=0.23\), therefore \(i(\mathbf{u}_2;z_1)=0.23\)
For a continuous feature, the indicator answers the question,
what is the probability that the realization is less than or equal to a threshold?
The indicator transform is,
For example,
given threshold \(z_1 = 6\%\), and data at \(\mathbf{u}_1\) with \(z(\mathbf{u}_1)=8\%\), then \(i(\mathbf{u}_1;z_1)=0\)
given threshold \(z_4 = 18\%\), and a random variable away from data, \(Z(\mathbf{u}_2)\sim N(\mu=16\%,\sigma=3\%)\), then
The indicator transform may be applied to an entire random function by transforming the random variable at every location. Indicator transforms provide the foundation for indicator kriging, indicator variograms, and indicator simulation.
Note, the indicator transform,
may be applied to encode data softness, in this case the data values are not strickly 0 or 1 with respect to each threshold or category at the a data location.
may also be applied to encode a constriant relationship, e.g., data value cannot be category 1, but there is no information about category 2 nor 3, or data value is between 0.1 and 0.4, but there is no information within this interval.
is analogous to the one-hot-encoding approach for feature engineering commonly used to deal with categorical features in machine learning.
Used in:
Indicator Variogram#
Variogram calculated and modeled from the indicator transform of spatial data and used for indicator kriging. The indicator variogram is,
where \(i(\mathbf{u}_\alpha; z_k)\) and \(i(\mathbf{u}_\alpha + \mathbf{h}; z_k)\) are the indicator transforms for threshold or category \(z_k\) at the tail location \(\mathbf{u}_\alpha\) and head location \(\mathbf{u}_\alpha + \mathbf{h}\), respectively.
For hard data the indicator transform, \(i(\mathbf{u}; z_k)\), is either 0 or 1. Therefore, \(\left[i(\mathbf{u}_\alpha; z_k)-i(\mathbf{u}_\alpha+\mathbf{h}; z_k)\right]^2\) is equal to 0 when the head and tail have the same indicator value (both \(\le z_k\) or both \(> z_k\) for continuous features, or both equal to or not equal to category \(z_k\) for categorical features), and equal to 1 when they differ.
the indicator variogram is one-half of the proportion of pairs that change across the specified lag. This makes the indicator variogram particularly intuitive, as it is directly related to the probability of change with increasing separation distance.
The sill of an indicator variogram is the indicator variance,
where \(p\) is the proportion of ones (equivalently zeros, since the variance is symmetric).
Unlike the regular variogram,
an indicator variogram is required for every threshold (continuous features) or every category (categorical features). This provides additional modeling flexibility because spatial continuity may vary with threshold or category, but it also increases the inference effort.
Used in:
Inference#
The process of using a sample drawn from a population to infer properties of the entire population. For example,
given sparsely sampled well data (the sample) with porosity well log measurements, infer the porosity histogram of the entire reservoir (the population)
given sparsely sampled drill hole data (the sample) with gold grade measurements, infer the gold grade distribution and spatial variability throughout the ore body (the population)
in geostatistics, inference includes estimating statistical properties (e.g., histograms, variograms, and trends) as well as spatial models away from the sampled locations
the statistical discipline devoted to inference is known as statistical inference
Inference is a broad topic encompassing many statistical methods. For this e-book, we adopt this simplified, practical definition focused on subsurface characterization and modeling.
Used in:
Compare with:
Interpolation#
Interpolation is the estimation of a value at an unsampled location using nearby observed sampless.
Interpolation methods assume that nearby observations provide information about the unknown value. Different interpolation methods vary in how they define the influence of neighboring samples.
Common interpolation methods include:
Inverse Distance – estimates values using distance-based weights
Kriging – estimates values using weights derived from modeled spatial continuity
Interpolation produces a single estimated value at each location and is commonly applied to generate continuous maps from discrete observations.
Also see:
Intersection of Events#
The event in which two or more events occur together. For two events, \(A\) and \(B\), the intersection is denoted by,
The probability of the intersection is,
where we may state, “probability of A and B”. If \(A\) and \(B\) are independent, then the joint probability simplifies to,
Without the assumption of independence, we apply the multiplication rule,
or equivalently,
Used in:
See also:
Inverse Distance#
An inverse distance interpolator is a deterministic interpolation method that estimates an unknown value using nearby sampless weighted by their distance from the estimation location.
The estimated value at location \(u\) is calculated as:
where the inverse distance weight is:
where:
\(\hat{Z}(u)\) is the estimated value at location \(u\)
\(Z(u_i)\) is the observed value at sample location \(u_i\)
\(d_i\) is the distance between the estimation location and sample \(i\)
\(p\) is the inverse distance power parameter
A practical consideration in inverse distance interpolation is handling estimation locations that coincide with sample locations.
Since:
the weight becomes infinite when:
Common approaches include:
assign the sample value directly when the estimation location matches an observed sample location
add a small positive distance constant \(\epsilon\) to avoid division by zero:
The first approach preserves exact data honoring, while the second provides numerical stability in implementations.
The inverse distance power controls the influence of nearby samples:
larger \(p\) values increase the influence of nearby samples and create more local estimates, with possibility of overfit
smaller \(p\) values create smoother estimates with greater influence from distant samples, with possibility of underfit
Common choices include:
\(p=2\), a common inverse distance weighting (IDW) choice
\(p=1\), inverse distance weighting with linear distance decay
Inverse distance interpolation assumes that closer samples are more representative of an unsampled location, but it does not explicitly model spatial continuity.
Advantages:
simple and computationally efficient
honors sample data values when estimating at sample locations
requires no fitted variogram model
Limitations:
does not quantify estimation uncertainty
distance alone may not represent geological connectivity
does not account for anisotropy or spatial correlation structures, such as those represented by a variogram model
Inverse distance interpolation is commonly used as a benchmark for comparison with geostatistical methods such as kriging.
Used in:
Also see:
Inversion#
Inversion is the process of estimating unknown model parameters, states, or properties from observed data by finding the model that best explains the observations.
A forward model predicts observations from model parameters:
where:
\(\mathbf{d}\) is the observed data vector
\(\mathbf{m}\) is the model parameter vector
\(G\) is the forward model
In inversion, the objective is to estimate:
where \(\hat{\mathbf{m}}\) is the estimated model.
Because inversion problems are often uncertain, nonlinear, and under-constrained, multiple models may explain the observations. Therefore, inversion commonly includes:
prior knowledge about plausible models
a measure of data mismatch or likelihood
uncertainty quantification over possible solutions
A common optimization formulation is:
where:
\(L\) measures mismatch between observations and model predictions
\(R\) represents constraints or regularization on the model
In geoscience applications, inversion is used to estimate subsurface properties from indirect observations such as,
seismic data
production data
Used in:
Also see:
Isotropic Variogram#
A variogram calculated and modeled such that spatial continuity is independent of direction, where direction may include azimuth in 2D horizontal space or azimuth and dip in 3D space.
For an isotropic experimental variogram,
all directions are combined, resulting in a variogram that is insensitive to azimuth or dip direction.
in 2D, an omnidirectional experimental variogram can be calculated by setting the azimuth tolerance to 90 degrees.
For an isotropic variogram model,
in 2D, the major and minor ranges are equal, resulting in a model with the same spatial continuity in all directions.
an isotropic variogram is also called an omnidirectional variogram.
Used in:
Contrast with:
Contrast with: Anisotropic Variogram
Joint Probability#
Probability of an event in which two or more events occur together,
The probability of the intersection is,
where we may state, “probability of A and B”. If \(A\) and \(B\) are independent, then the joint probability simplifies to,
Without the assumption of independence, we apply the multiplication rule,
or equivalently,
Used in:
See also:
Kriging#
Spatial estimation approach that relies on linear weights that account for spatial continuity, data closeness and redundancy. The kriging estimate is,
the right term is the unbiasedness constraint, where one minus the sum of the weights is applied to the global mean.
In the case where the trend, \(t(\bf{u})\), is removed, we now have a residual, \(y(\bf{u})\),
the residual mean is zero so we can simplfy our kriging estimate as,
The simple kriging weights are calculated by solving a linear system of equations,
that may be represented with matrix notation as,
This system may be derived by substituting the equation for kriging estimates into the equation for estimation variance, and then setting the partial derivative with respect to the weights to zero.
we are optimizing the weights to minimize the estimation variance
this system integrates the,
spatial continuity - as quantified by the variogram (and covariance function to calculate the covariance, \(C\), values)
redundancy - the degree of spatial continuity between all of the available data with themselves, \(C(\bf{u}_i,\bf{u}_j)\)
closeness - the degree of spatial continuity between the available data and the estimation location, \(C(\bf{u}_i,\bf{u})\)
Kriging provides a measure of estimation accuracy known as kriging variance (a specific case of estimation variance).
Kriging estimates are best in that they minimize the above estimation variance.
Properties of kriging estimates include,
Exact interpolator - kriging estimates with the data values at the data locations
Kriging variance - a measure of uncertainty in a kriging estimate. Can be calculated before getting the sample information, as the kriging estimation variance is not dependent on the values of the data nor the kriging estimate, i.e. the kriging estimator is homoscedastic.
Spatial context - kriging takes integrates spatial continuity, closeness and redundancy; therefore, kriging accounts for the configuration of the data and structural continuity of the feature being estimated.
Scale - kriging by default assumes the estimate and data are at the same point support, i.e., mathematically represented as points in space with zero volume. Kriging may be generalized to account for the support volume of the data and estimate,
Multivariate - kriging may be generalized to account for multiple secondary data in the spatial estimate with the cokriging system. We will cover this later.
Smoothing effect - of kriging can be forecasted as the missing variance. The missing variance over local estimates is the kriging variance.
Used in:
Comparison of Simple and Ordinary Kriging Discussion and Demonstration
Comparison of Kriging and Simulation Discussion and Demonstration
Also see,
Kriging Simple vs. Ordinary#
The difference between simple kriging and ordinary kriging is related to the assumption of stationarity in the mean.
Simple Kriging - global stationary mean is an input provided by the user, i.e., the mean is assumed to be stationary.
for the kriging estimate, one minus the sum of the data weights is applied to the global stationary mean.
at data locations all weight is applied to the collocated datum, and beyond the variogram range from all data all weight is applied to the global mean.
the simple kriging weights, \(\lambda_1, \lambda_2, \dots, \lambda_n\), are calculated by solving this system of equations represented in matrix notation as,
Ordinary Kriging - local nonstationary mean calculated by the kriging system. The global mean is not an input, instead the local nonstationary mean is calculated by ordinary kriging. This relaxes the assumption of a stationary mean.
this is accomplished with the addition of a data kriging weights must sum to one constraint in the kriging system, if the weights must sum to one this removes the right hand unbiasedness constraint from the kriging estimate with the global mean,
at data locations all weight is applied to the collocated datum, and beyond the variogram range from all data, all weight is applied to the local mean calculated from local data
the ordinary kriging weights, \(\lambda_1, \lambda_2, \dots, \lambda_n\), are calculated by solving this system of equations, the simple kriging system with the added constraint that the weight sum to 1.0 represented in matrix notation as,
Used in:
Comparison of Simple and Ordinary Kriging Discussion and Demonstration
Comparison of Kriging and Simulation Discussion and Demonstration
Also see,
Kriging Variance#
A measure of accuracy and uncertainty for a kriging estimate, expressed as,
Kriging variance is a specific case of estimation variance,
Can be calculated before getting the sample information,
the kriging estimation variance is not dependent on the values of the data nor the kriging estimate, i.e. the kriging estimator is homoscedastic.
Used in:
Kriging-based Declustering#
A declustering method to assign weights to spatial samples based on local sampling density using the spatial continuity model from a variogram model. The objective is to assign weights such that the weighted statistics are intended to be more representative of the population.
samples in densely sampled areas receive less weight
samples in sparsely sampled areas receive more weight
Kriging-based declustering proceeds as follows:
calculate and model the experimental variogram
apply kriging to calculate estimates over a high-resolution grid covering the volume of interest
accumulate the kriging weights assigned to each data sample over the entire grid
assign declustering weights proportional to the accumulated kriging weights
The kriging-based declustering weight for data sample \(j\) is calculated as,
where \(n\) is the number of data samples, \(n_x\) and \(n_y\) are the number of grid cells in the declustering grid, and \(\lambda_{j,ix,iy}\) is the kriging weight assigned to data sample \(j\) when estimating grid cell \(ix,iy\).
The resulting weights sum to the number of samples,
which allows the weights to be directly applied in weighted statistics.
Important considerations for kriging-based declustering,
like polygonal declustering, kriging-based declustering is sensitive to the boundaries of the area of interest; therefore, samples near the boundary may receive substantially different weights as the area of interest is expanded or contracted.
kriging-based declustering integrates the spatial continuity model from the variogram. Therefore, the variogram model selection can significantly impact the resulting weights.
if there is a 100% relative nugget effect, there is no spatial continuity and all samples receive equal weight. In this case, the kriging weights contain no spatial information and the declustering calculation must be handled separately to avoid division by zero.
geometric anisotropy may significantly impact the weights because data aligned along preferred directions may be considered spatially closer or farther based on the covariance function model.
Used in:
Also see:
Kolmogorov Probability Axioms#
The three axioms proposed by Andrey Kolmogorov that establish the rigorous mathematical foundation for probability theory.
Probability of an event is a non-negative number,
Probability of the entire sample space, all possible outcomes \(\Omega\), is one (unity), also known as probability closure,
Additivity of mutually exclusive events for unions,
For example, the probability of two mutually exclusive events \(A_1\) and \(A_2\) is,
Used in:
Kurtosis#
Kurtosis is a univariate parameter or univariate statistic that measures the tendency of a probability distribution to produce extreme values through tail behavior.
A common measure of kurtosis is:
For comparison with the normal distribution:
mesokurtic distributions have kurtosis approximately equal to 3
leptokurtic distributions have kurtosis greater than 3, indicating heavier tails and more extreme values
platykurtic distributions have kurtosis less than 3, indicating lighter tails and fewer extreme values
Kurtosis is important in geostatistical modeling because extreme values of geological properties, such as permeability, thickness, grade, or production rate, may strongly influence resource estimates and uncertainty.
Used in:
Also see:
Lag#
The separation between paired spatial data described by a vector, \(\mathbf{h}\), applied to specify spatial offset for calculation, modeling and plotting variograms.
the experimental variogram characterizes spatial continuity over a range of lags, including both lag magnitude and orientation
the variogram model is applied to calculate spatial continuity for any possible lag vector, including all separation distances and orientations
during variogram calculation, a unit lag distance is determined and a sequence of lags based on integer multiples is considered, i.e., the first lag is the unit lag, the second lag is two times the unit lag, etc.
Used in:
Also see:
Lag Distance#
The magnitude of the lag vector, \(\mathbf{h}\), describing the separation distance between paired data used for variogram calculation, modeling and plotting.
the experimental variogram characterizes spatial continuity over a discrete set of lag distances, while the variogram model is applied to calculate spatial continuity over any possible lag distance
lag distance is represented on the x-axis of a variogram plot
Used in:
Also see:
Lag Distance Tolerance#
For calculating an experimental variogram, the tolerance \(\pm \Delta\) around a target lag distance used to pool data pairs with similar separation distances for calculating a variogram,
for example, given a lag distance of 300 m, a lag tolerance of 50 m would include all data pairs separated by 250 m to 350 m to calculate the experimental variogram value at this lag distance
it is common practice to use half the unit lag distance as the lag tolerance, ensuring that adjacent lag bins overlap and there are no gaps in the variogram search template
lag tolerance may be increased beyond half the unit lag distance to include more data pairs and smooth the experimental variogram for improved interpretation
Used in:
Also see:
Likelihood Function#
In Bayes’ Theorem, the likelihood function describes the compatibility of new data with possible states or inferred model parameters. It quantifies how likely the observed data are for different model assumptions and is combined with the prior probability to calculate the posterior probability,
where:
\(P(A)\) is the prior probability representing uncertainty before observing new information,
\(P(B|A)\) is the likelihood function describing the compatibility of observations \(B\) with state or parameter \(A\),
\(P(B)\) is the evidence term used to normalize the posterior probability,
\(P(A|B)\) is the posterior probability representing updated uncertainty after incorporating observations.
Linear Regression#
Simple, linear parametric prediction model based on a linear data weighted combination of the predictor features with a constant term,
where the trainable model parameters are \(b_{\alpha}\), the feature weights, and \(b_0\), the constant intercept term.
The analytical solution for the model parameters, \(b_1,\ldots,b_m,b_0\), is available by minimizing the L2 norm loss function. This is known as least squares because the residual errors are squared and summed over all training data,
where \(y_i\) is the observed response feature value and \(\sum_{\alpha=1}^{m}b_{\alpha}x_{\alpha,i}+b_0\) is the model prediction for training sample \(i\).
The assumptions of linear regression include,
Error-free predictors - predictor features are treated as known values, not random variables
Linearity - the expected response is a linear combination of predictor features
Constant variance - the variance of the response error is constant over predictor feature values (homoscedasticity)
Iindependence of error - errors in the response are uncorrelated with each other
No multicollinearity - predictor features are not linearly redundant with each other
Used in:
Also see: TBA
Location Map#
A spatial data plot where the two axes represent spatial coordinates, showing the locations and values of a sampled feature.
location maps are always a good first step for spatial data analytics because they provide a direct view of data coverage, clustering, trends, anomalies, and potential sampling bias before applying any spatial model
Some additional comments,
location maps are used to visualize the spatial distribution of sampled feature values, often using color to represent feature magnitude
location maps allow interpretation of the data without a spatial model that may bias our understanding of the sampled feature distribution
location maps may be extended to 3D using three spatial coordinate axes; however, visualizing large spatial datasets in 3D is often challenging
Used in:
Lorenz Coefficient#
The Lorenz coefficient is a measure of inequality or concentration within a distribution. It quantifies how unevenly a cumulative quantity is distributed among observations.
The Lorenz curve compares the cumulative proportion of observations with the cumulative proportion of the total property.
The Lorenz coefficient is commonly calculated as:
where \(A\) is the area under the Lorenz curve.
The Lorenz coefficient ranges from:
0 for a completely uniform distribution
1 for maximum concentration where a small number of observations contain nearly all of the total quantity
The Lorenz coefficient is used to quantify heterogeneity in geological variables, including permeability, thickness, and resource distributions.
Used in:
Also see:
Loss Function#
A loss function quantifies the cost of an undesirable outcome, error, or mismatch between a model result and a desired outcome.
Loss functions are used in two common contexts:
model parameter estimation and training
decision making under uncertainty
Model Training Loss - in model training, a loss function measures the difference between observed values and model predictions.
For a set of observed values \(y_i\) and model predictions \(\hat{y}_i\):
The model parameters are selected to minimize the total loss:
where:
\(\theta\) represents model parameters
\(y\) represents observed values
\(\hat{y}\) represents model predictions
Common loss functions include, L1 loss (absolute error),
L1 loss emphasizes the magnitude of errors and is less sensitive to large outliers. L2 loss (squared error),
L2 loss penalizes large errors more strongly and is commonly used for linear regression.
The L1 and L2 losses are related to vector norms:
where \(e\) is the vector of prediction errors.
Decision Loss Under Uncertainty - in decision making, a loss function represents the consequence of making a decision when the true state of the system is uncertain.
The expected loss is:
where:
\(a\) is the decision action
\(Z\) is the uncertain true state
\(L(a,Z)\) is the loss from choosing action \(a\) when the true state is \(Z\)
Decision making seeks the action that minimizes expected loss:
Examples include:
overestimation loss - committing resources assuming a higher value than actually exists
underestimation loss - failing to develop or exploit a valuable opportunity
asymmetric loss - when overestimation and underestimation have different consequences
In geoscience applications, loss functions are used to incorporate uncertainty into decisions such as:
reservoir development planning
resource classification
production forecasting
operational decisions
A loss function defines what “best” means by connecting model accuracy or uncertainty to the consequences of decisions.
Used in:
Also see:
Local Accuracy#
Accuracy of a spatial estimate at individual locations, typically assessed by minimizing the local estimation uncertainty or estimation error.
For kriging, local accuracy is achieved by minimizing the estimation variance,
where \(Z^*(\mathbf{u})\) is the estimated value and \(Z(\mathbf{u})\) is the unknown true value.
Some general observations about local accuracy,
estimation methods prioritize local accuracy by honoring nearby data and minimizing estimation variance.
locally accurate estimates are often smooth because they represent the best estimate at each location, but they generally do not reproduce the spatial variability of the modeled feature.
estimates optimized for local accuracy may not reproduce global measures, such as the histogram, variogram, or correlation coefficient.
Used in:
Comparison of Estimation and Simulation Discussion and Demonstration
Model Checks for Global Accuracy Description and Demonstration
Contrast with:
Local Measure#
A statistical measure that evaluates model performance at individual locations or over a local neighborhood.
Local measures are used to assess local accuracy.
local measures focus on location-specific agreement between estimates and observations or local uncertainty.
local measures do not assess reproduction of global measures, such as the entire histogram, variogram, or correlation coefficient.
Used in:
Comparison of Estimation and Simulation Discussion and Demonstration
Model Checks for Global Accuracy Description and Demonstration
Contrast with:
Major Direction#
The direction with the largest variogram range, representing the strongest spatial continuity, used for calculating and modeling a variogram.
the major and minor direction describe the horizontal spatial continuity for 2D phenomena
for 3D phenomena, the major and minor directions are augmented with a vertical direction, orthogonal to the horizontal directions, to describe spatial continuity in three dimensions
in stratigraphically controlled systems, the major and minor directions may be defined relative to stratigraphic coordinates rather than geographic coordinates
Used in:
Contrast with:
Margin of Error#
The margin of error is the uncertainty range added and subtracted from an estimate when constructing a confidence interval.
The margin of error is calculated as:
For a population mean:
or:
The confidence interval is then:
The margin of error increases with:
higher confidence levels
greater variability in the population
smaller sample sizes
The margin of error decreases with:
larger sample sizes
improved measurement precision
Used in:
Also see:
Marginal Probability#
Probability that considers only a single event occurring. For example, the probability of event \(A\),
Marginal probabilities may be calculated from joint probabilities through the process of marginalization,
where we integrate over all cases of the other event, \(B\), to remove its influence. Given discrete, categorical or binned continuous cases of event \(B\) we can simply sum the probabilities over all possible cases of \(B\),
Used in:
Contrast with:
Markov Chain Monte Carlo#
Known widely by the acronym MCMC, a family of algorithms used to estimate complicated probability distributions by generating samples from those distributions.
MCMC methods are based on,
Markov chain - a sequence of samples where each sample depends only on the previous sample
Markov property - the assumption that the current sample contains all required information from previous samples, such that future samples are conditionally independent of earlier samples given the current sample, this is also known as Markov screening
Monte Carlo - a simulation process that uses random sampling to approximate probability distributions and expectations
The general workflow is,
Initialize - start at an initial point in the distribution.
Propose - generate a candidate new sample based on a proposal distribution.
Evaluate - calculate the probability of accepting the proposed sample relative to the current sample.
Accept or Reject - accept the new sample according to an acceptance probability. Higher probability samples are favored, while lower probability samples may still be accepted to explore the full distribution.
Repeat - generate many samples to form a Markov chain that approximates the target probability distribution.
There are a variety of MCMC methods, including,
Metropolis-Hastings - proposes random steps and uses a stochastic acceptance rule to sample from the target distribution
Gibbs Sampler - samples each feature or variable sequentially from its conditional probability distribution
Also see:
Mean#
The mean (or average) is a measure of central tendency. There are several useful interpretations of the mean,
representative value - a single value to represent an entire distribution
estimate - given a distribution of values, the mean minimizes the L2 norm (sum of squared error)
scale-up - under linear averaging of a spatial feature, the mean is the correct upscaled value
expectation - if all outcomes are equiprobable, the mean is equal to the expectation
For a sample, the mean is,
Note, the mean is quite sensitive to outliers.
Used in:
Also see:
Median#
The median is a univariate statistic that represents the central value of an ordered set of observations. The median divides the data such that approximately half of the observations are smaller and half are larger.
For \(n\) ordered observations:
the median is:
The median is less sensitive to extreme values than the arithmetic mean, making it useful for describing the central tendency of skewed geological and engineering variables.
For a symmetric distribution:
For positively skewed distributions:
Used in:
Also see:
Minor Direction#
The direction with the smallest variogram range, representing the strongest spatial continuity, used for calculating and modeling a variogram.
the major direction and minor direction describe the horizontal spatial continuity for 2D phenomena
for 3D phenomena, the major and minor directions are augmented with a vertical direction, orthogonal to the horizontal directions, to describe spatial continuity in three dimensions. Minor direction is selected over horizontal and not vertical directions.
in stratigraphically controlled systems, the major and minor directions may be defined relative to stratigraphic coordinates rather than geographic coordinates
Used in:
Contrast with:
Mode#
The mode is a univariate statistic representing the most frequently occurring value or range of values in a distribution.
For continuous variables, the mode is commonly estimated as the location of the maximum probability density:
where \(f_X(x)\) is the probability density function of feature \(X\).
The mode is useful for identifying dominant populations, such as geological facies, mineral grade populations, or peaks in multimodal distributions.
Unlike the mean and median, the mode does not necessarily represent the center of the distribution.
A distribution may have:
one mode (unimodal)
multiple modes (multimodal), indicating multiple populations or processes
Used in:
Also see:
Model Checking#
A set of critical steps in any spatial modeling workflow to ensure the models are ready to support decision making. Model checking evaluates whether the model honors the available information, accurately predicts known data, and provides a reliable representation of uncertainty.
Examples of model checks include,
Model Inputs - data and statistics integration
check that the model honors the input data and statistical assumptions, generally evaluated over all realizations
for example, compare output histograms, variograms, and correlation coefficients from the realizations against the input statistics
Accurate Spatial Estimates - ability of the model to predict away from available sample data
evaluate predictive performance using cross validation, where some data are withheld and then predicted by the model
predictive accuracy is generally summarized with a truth versus predicted cross plot and measures such as mean square error,
Accurate and Precise Uncertainty Models - the uncertainty model is consistent with the amount of information available and the sources of uncertainty
evaluate uncertainty using cross validation by withholding data and checking whether the observed values occur within the predicted probability intervals at the expected frequency
summarize uncertainty goodness with observed proportion within interval versus the predicted probability interval
points on the 45 degree line indicate a good uncertainty model
points above the 45 degree line indicate an overly conservative uncertainty model, where uncertainty intervals are too wide
points below the 45 degree line indicate an under-estimated uncertainty model, where uncertainty intervals are too narrow or the model is biased
Used in:
Model Grid#
A model grid discretizes the volume of interest into cells over which geostatistical spatial estimation, simulation, and engineering calculations are performed.
A model grid has the following aspects:
regular grid with approximately constant cell volume support. Regular grids satisfy the assumptions of most geostatistical estimation and simulation algorithms. Direct simulation on unstructured grids remains an active area of research due to a variety of unresolved technical challenges.
grid cell volume support (model resolution), determined by balancing the modeling purpose, available data resolution, computational and storage resources.
grid extents that include the volume of interest together with important geological and engineering boundaries that may impact volumetrics, fluid flow, or mining.
The Geo-DAS model grid format used by GSLIB and GeostatsPy is a cell-centered regular grid defined by the following parameters:
\(n_x\), \(n_y\), and \(n_z\) – number of cells in the \(x\), \(y\), and \(z\) coordinates.
\(min_x\), \(min_y\), and \(min_z\) – coordinates of the center of the lower-left-bottom grid cell.
\(siz_x\), \(siz_y\), and \(siz_z\) – cell dimensions in the \(x\), \(y\), and \(z\) coordinates.
Following coordinate transformation, the \(X\), \(Y\), and \(Z\) coordinates may actually represent transformed coordinates such as \(I\), \(J\), and \(K\). For consistency, the Geo-DAS format retains the parameter names \(X\), \(Y\), and \(Z\).
Alternative model grid formats include:
\(Z\) increasing downward, with the model origin located in the uppermost layer. This convention is common in many reservoir simulators.
\(Y\) increasing downward, the default orientation for two-dimensional NumPy ndarrays.
local grid refinement (LGR), where selected grid cells are subdivided to better resolve steep gradients, for example, near injection and production wells.
corner-point grids, which conform to structural surfaces and faults while maintaining logical grid connectivity.
tetrahedral and other unstructured grids, which better represent complex geometries and reduce artifacts associated with strictly orthogonal grids.
The model grid does not increase the information content of the available data. Instead, it provides a computational framework for integrating data and representing spatial variation throughout the volume of interest.
Used in:
Also see:
Model Hyperparameter#
Model settings that are specified prior to training or estimation and control the structure, flexibility, or smoothness of the model.
Hyperparameters are not directly estimated from the data but are often tuned using validation data or expert knowledge.
Examples include:
regularization strength in regression models
tree depth in decision trees
nugget effect, range, and sill in variogram models (geostatistics)
Trend Model order or complexity
parameters controlling data conditioning or softness in spatial models
Used in:
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Model Parameter#
Quantities estimated from data that define a model and control its fit to observations.
Model parameters are typically obtained through optimization, analytical solutions, or statistical estimation methods such as least squares, maximum likelihood, or kriging.
Examples include regression coefficients, covariance values, and trend coefficients.
Used in:
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Modeling Purpose#
There is always a purpose to subsurface modeling. Before modeling begins, establishing the current and potential future modeling purpose(s) is critical, as the purpose determines the appropriate model extent, resolution, properties, complexity, uncertainty representation, calibration strategy, and computational effort.
Common modeling purposes include:
Common Earth Model – integrate geological, geophysical, petrophysical, engineering, and operational data into a consistent understanding of the subsurface, reconcile multiple interpretations, identify knowns and unknowns, and provide a common basis for communication.
Assess Resources – estimate the in situ resource or ore in place.
Quantify Resource Uncertainty – estimate the uncertainty in resource quantity, quality, and spatial distribution.
Investigate Geological Risk – evaluate the impact of geological uncertainty on exploration, appraisal, development, or mining decisions.
Forecast System Performance – predict future production, pressure, recovery, water breakthrough, geomechanical response, or other system behavior under alternative scenarios.
Evaluate Development Strategies – compare alternative well locations, completion designs, production strategies, mine plans, or infrastructure options.
Assess Reserves – estimate economically recoverable resources and classify reserves according to applicable reporting standards.
Transfer Geological Knowledge – capture geological understanding from mature assets to provide analogs and prior information for less mature projects.
Support Decision Making – integrate available information, quantify uncertainty, evaluate alternative scenarios, and improve technical, economic, and operational decisions.
Used in:
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Monte Carlo Simulation#
A method for generating random samples from one or more statistical distributions. A random sample from a distribution is defined as a random variable, \(X\). The steps for Monte Carlo simulation are:
Model the feature cumulative distribution function, \(F_x(x)\).
Draw a random value from a uniform \([0,1]\) distribution, representing a random cumulative probability value, \(p^{\ell}\).
Apply the inverse cumulative distribution function to calculate the associated sample value,
Repeat steps 2 and 3 to calculate enough realizations for subsequent analysis.
Monte Carlo simulation is a fundamental building block of stochastic simulation and uncertainty workflows. Examples include,
Monte Carlo simulation workflow - apply Monte Carlo simulation over all uncertain features, then apply a transfer function to calculate a realization of the decision criteria. Repeat this process to generate many realizations and propagate uncertainty through the transfer function.
Bootstrap - applies Monte Carlo simulation to generate realizations of the sample data, allowing estimation of uncertainty in sample statistics or ensembles of prediction models for ensemble-based machine learning.
Monte Carlo methods - use random sampling to approximate solutions to complex problems, with the solution generally converging as the number of random samples increases.
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Monte Carlo Simulation Workflow#
A general stochastic Monte Carlo simulation workflow for propagating uncertainty through a transfer function. The workflow includes the following steps,
Model the uncertainty distributions or cumulative distribution functions for all input features,
Monte Carlo simulate realizations for all input features,
Apply the transfer function to calculate a realization of the output, often the decision criteria,
Repeat steps 2 and 3 to calculate enough realizations to model the output uncertainty distribution,
The input feature realizations may be simulated independently or with relationships between features included through multivariate uncertainty models.
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Multiplication Rule#
The joint probablity of \(A\) and \(B\) as the product of the conditional probability of \(B\) given \(A\) with the marginal probability of \(A\),
The multiplication rule is axiomatic as it is derived as a simple manipulation of the definition of conditional probability, in this case,
and the definition of conditional probability is readily obseved from a simple Venn diagram.
Used in: Probability Multiplication Rule Defintion and Demonstration
Multiple Point Data Event#
For a Multiple Point Template, \(\tau\), the set of observed data and previously simulated nodes located at the multiple point template offsets,
The multiple point data event is the spatial pattern used to condition the simulation at the unknown location.
the first point in the template, \(\mathbf{u}_k\), is excluded from the multiple point data event because it is the unknown location where the conditional probability distribution is calculated
the conditional distribution for this multiple point data event is estimated by scanning the training image and identifying matching spatial patterns
Used in:
Multiple Point Simulation#
A sequential simulation approach similar to sequential Gaussian simulation that extends spatial modeling beyond two-point relationships (represented by the variogram) by using higher-order spatial patterns involving multiple points, commonly known by the acronym MPS.
in practice, calculating reliable statistics for spatial configurations with more than two points directly from sparse subsurface data is generally impractical because the number of possible spatial configurations grows combinatorially while conditioning data remain limited
MPS addresses this limitation by borrowing higher-order spatial statistics from a training image, which provides a dense conceptual representation of expected geological patterns, connectivity, and morphology
the training image does not include local conditioning information and should be stationary with respect to the multiple point statistics and lower-order statistics, such as facies proportions and indicator variograms
MPS integrates into the sequential simulation workflow by replacing the probability calculation step in sequential indicator simulation with a training image scan to estimate the conditional probability distribution
Used in:
Also see:
Multiple Point Template#
The spatial pattern used in multiple point simulation to define the relative locations of conditioning data and previously simulated nodes around an unknown location and to calculate the local conditional distributions by pooling matches with the multiple point data event in the training image.
The template is represented by a set of grid offsets,
where each \(\mathbf{h}_i\) is an offset in grid cells from the unknown location, defined by \(\Delta^{ix}_i\) and \(\Delta^{iy}_i\).
For example, a two-point template may include the unknown location,
and one neighboring location,
representing a node one cell upward and one cell left of the unknown location.
Used in:
Multivariate#
Involving more than two features (variables) considered together, often to study their relationships, dependence, or correlation.
For examples, see multivariate analysis.
Used in:
Compare with:
Multivariate Analysis#
The analysis of more than two features (variables) measured over a collection of samples to investigate their relationships, dependence, and correlation.
Examples include:
multivariate statistical models such as multilinear regression
scatter plot matrices showing relationships among multiple features
covariance and correlation coefficient matrices
principal components analysis
A common approach to multivariate analysis is to evaluate pairwise relationships among features using covariance, correlation, and scatter plots.
note, this is a simplified multivariate analysis because it considers only pairwise relationships among features. Higher-order relationships involving three or more features simultaneously are not explicitly modeled.
Used in:
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Mutually Exclusive Events#
Events that cannot occur together; they have no common outcomes. Using set notation, events \(A\) and \(B\) are mutually exclusive if,
Therefore, the probability of the intersection of mutually exclusive events is,
For mutually exclusive events, the probability of a union simplifies to the sum of the individual probabilities,
Used in:
Want to Work Together?#
I hope this content is helpful to those that want to learn more about subsurface modeling, data analytics and machine learning. Students and working professionals are welcome to participate.
Want to invite me to visit your company for training, mentoring, project review, workflow design and / or consulting? I’d be happy to drop by and work with you!
Interested in partnering, supporting my graduate student research or my Subsurface Data Analytics and Machine Learning consortium (co-PIs including Profs. Foster, Torres-Verdin and van Oort)? My research combines data analytics, stochastic modeling and machine learning theory with practice to develop novel methods and workflows to add value. We are solving challenging subsurface problems!
I can be reached at mpyrcz@austin.utexas.edu.
I’m always happy to discuss,
Michael
Michael Pyrcz, Ph.D., P.Eng. Professor, Cockrell School of Engineering and The Jackson School of Geosciences, The University of Texas at Austin
More Resources Available at: Twitter | GitHub | Website | GoogleScholar | Geostatistics Book | YouTube | Applied Geostats in Python e-book | Applied Machine Learning in Python e-book | LinkedIn
Comments#
I hope this glossary is helpful. I must admit that I enjoyed writing it and making the effort to communicate terms and related concepts in a clear and concise manner. Remember to,
follow the links to find the demonstration workflows, interactive dashboards and links to related lectures on YouTube.
Also, the author also acknowledges “Мотанка” (Ukrainian pronounciation moh-TAHN-kah) for countless conversations, persistent questions, and unwavering encouragement throughout the development of this glossary.
Sincerely,
Michael