Geostatistics Glossary A-D#
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.
This chapter is a summary of essential Geostatistics Terminology. (UPDATED July 8, 2026)
Motivation for this Glossary#
Firstly, why create this glossary?
I received a request for a course glossary from students in my Data Analytics and Geostatistics undergraduate course. While I usually dedicate a definition slide in my lecture slide decks to important terms, several students requested a consolidated glossary of terminology as part of their course review materials. The e-book provides an excellent vehicle and motivation for creating this resource.
Let me begin with a confession. There is already an outstanding resource: the Geostatistical Glossary and Multilingual Dictionary written by my good friend Dr. Ricardo A. Olea, an excellent geologist and statistician from the USGS. For those seeking a comprehensive and in-depth reference of geostatistical terminology, this book remains an excellent resource.
So why create another glossary?
By writing my own glossary, I can limit the scope and descriptions to the concepts covered in this course. I believe many students would be overwhelmed by the size, breadth, and mathematical notation of a comprehensive geostatistics glossary.
By integrating the glossary directly into the e-book, I can link terminology to detailed chapter discussions, demonstrations, and examples. The goal is to eventually populate all chapters with hyperlinks to glossary entries, enabling students to move easily between concepts and applications.
Finally, like the rest of this e-book, I want the glossary to be an evergreen living document. It will continue to evolve with new concepts, improved explanations, and feedback from students and readers.
Addition Rule#
Method to calculate the probability of any event (the union of outcomes, represented by “or” grammar). For example, the probability of \(A\) or \(B\) is calculated with the probability addition rule,
given mutually exclusive events we can generalize the addition rule as,
Used in:
Affine Correction#
A distribution rescaling method that applies a shift and linear scaling (stretching or squeezing) to a univariate distribution, for example, histogram.
For the case of affine correction of feature \(X\) to feature \(Y\),
we correct \(X\) to match the mean and variance of \(Y\):
where \(\overline{x}\) and \(\sigma_x\) are the mean and standard deviation of the original distribution, and \(\overline{y}\) and \(\sigma_y\) are the target mean and standard deviation.
The procedure,
centers the data by subtracting the original mean
rescales the deviations by the ratio of standard deviations
shifts the result to the target mean
Affine correction does not alter the distribution shape; it preserves the relative ordering of values and applies only a linear transformation. For transformations that modify distribution shape, see distribution transformation.
Used in:
Alpha Level#
The alpha level is the numerical value of the significance level used in hypothesis testing.
The alpha level is denoted by:
and represents the probability of incorrectly rejecting a true null hypothesis:
Common alpha levels include:
\(\alpha=0.10\)
\(\alpha=0.05\)
\(\alpha=0.01\)
The alpha level controls the trade-off between sensitivity and false positive risk:
smaller \(\alpha\) values require stronger evidence before rejecting the null hypothesis
larger \(\alpha\) values increase sensitivity but increase the risk of type I error
The alpha level is selected before performing a hypothesis test.
For confidence intervals, the corresponding confidence level is:
Used in:
Also see:
Alternative Hypothesis#
The alternative hypothesis is the assumption applied in hypothesis testing that an effect, relationship, or statistically significant difference exists. It is denoted by
or
Examples include:
or
where the population mean differs from, is greater than, or is less than the hypothesized value.
The alternative hypothesis may be:
two-sided, testing for any statistically significant difference
one-sided, testing whether the population parameter is greater than or less than a specified value
Support for the alternative hypothesis is obtained by rejecting the null hypothesis.
Failure to reject the null hypothesis does not disprove the alternative hypothesis.
It indicates only that there is insufficient evidence from the sample to support the alternative hypothesis at the selected significance level.
Rejecting the null hypothesis provides evidence in support of the alternative hypothesis.
It does not prove that the alternative hypothesis is true, as there remains a possibility of a Type I error.
Used in:
Also see:
Anisotropic Variogram#
A variogram that exhibits directional dependence in spatial continuity, where spatial correlation varies with azimuth.
Anisotropy can be observed in both experimental variograms and variogram models:
For an anisotropic experimental variogram, direction is explicitly considered by restricting azimuth and bandwidth (e.g., using azimuth tolerance < 90°). This produces variograms that capture spatial continuity in specific directions.
For an anisotropic variogram model, directional dependence is represented by assigning different ranges in different directions (e.g., a major range greater than a minor range). This models stronger continuity along one direction than another.
An anisotropic variogram is therefore directionally dependent, in contrast to an isotropic variogram, which assumes the same spatial continuity in all directions.
anisotropy often reflects geologic structure such as stratification, channeling, or depositional trends.
Also known as a directional variogram.
Used in:
Contrast with:
Area of Interest#
The 2D spatial domain that is being characterized, modeled, and evaluated to support subsurface decision making. In general, the area of interest,
is the subsurface reservoir for oil and gas, the ore body for mining, or the aquifer for hydrogeological applications
may include volume away from the reservoir or ore body to support data integration and extraction modeling
may be further subdivided into local regions or facies and modeled separately
is represented by a grid with features populated from data, estimation, or simulation
in 3D modeling is commonly called the volume of interest
the extent and grid cell size are selected based on a trade-off between model accuracy and computational complexity
Average#
The average is a measure of central tendency. There are several useful interpretations of the average,
representative value - a single value to represent an entire distribution
estimate - given a distribution of values, the average minimizes the L2 norm (sum of squared error)
scale-up - under linear averaging of a spatial feature, the average is the correct upscaled value
expectation - if all outcomes are equiprobable, the average is equal to the expectation
For a sample, the average is,
Note, the average is quite sensitive to outliers.
Used in:
Also see:
Azimuth Tolerance#
Azimuth tolerance defines the angular search window used to select data pairs when calculating a directional experimental variogram. It controls the range of azimuths (directions) around a target direction that are included in the variogram calculation.
for a target azimuth (e.g., 90° in map view, aligned with the positive x-direction), an azimuth tolerance of ±20° includes all pairs with azimuths between 70° and 110°.
General guidelines for setting azimuth tolerance:
a common choice for directional experimental variograms is ±22.5°, which results in a 45° total angular window. This provides a balance between directional specificity and sufficient data pairs for stable, interpretable variograms.
larger tolerances may be used to increase the number of data pairs and smooth the experimental variogram, improving interpretability at the cost of directional resolution.
for isotropic (omnidirectional) variograms, an azimuth tolerance of 90° is typically used, effectively including all directions and removing directional filtering.
azimuth tolerance does not introduce anisotropy; it only controls how directional the data selection is for the experimental variogram.
Used in:
Bandwidth#
The maximum orthogonal deviation from the lag vector when identifying data pairs to calculate an experimental variogram.
What is the motivation for using bandwidth?
when working with a cross section (axes, y is vertical and x is horizontal) at large lag distances, \(\bf{h}\), azimuth (dip) tolerance may result in including data from adjacent (mixing between) stratigraphic units.
bandwidth is typically applied to the vertical direction to reduce the potential for mixing between adjacent units
When not to use bandwidth?
bandwidth should not be used to calculate isotropic (also called omnidirectional) experimental variograms
bandwidth is rarely used to calculate horizontal experimental variograms
bandwidth is removed by setting it very large relative to the model extent
Used in:
Bayes Theorem#
A theorem that relates conditional probabilities and provides the mathematical basis for Bayesian updating of uncertainty models given new information.
where:
\(P(A)\) is the prior probability representing uncertainty before considering new information,
\(P(B|A)\) is the likelihood function describing the compatibility of observations \(B\) with possible states or parameters \(A\),
\(P(B)\) is the evidence term used to normalize the posterior probability,
\(P(A|B)\) is the posterior probability representing updated uncertainty after incorporating observations.
Commonly applied in:
Bayesian Inference
Compare with:
Used in:
Bayesian Probability#
Probability framework that represents uncertainty using prior knowledge and new information. Prior information may be based on,
expert judgment
experience
historical data
physical understanding, or
previous observations.
Bayesian probability provides a formal approach to updating uncertainty as new information becomes available. The general approach,
start with a prior probability distribution representing uncertainty before considering new information
formulate a likelihood function describing the compatibility of new observations with possible states or parameters
combine the prior and likelihood using Bayes’ theorem to calculate the posterior probability distribution
continue updating uncertainty as additional information becomes available
Bayesian probability is applied to solve probability problems where prior knowledge, limited data, or sequential information updates are important,
Bayesian probability contrasts with the frequentist probability approach, which interprets probability primarily through long-run frequencies of repeated experiments
Used in:
Contrast with:
Bayesian Updating#
The process of revising a prior probability distribution using new observations or evidence to obtain a posterior probability distribution according to Bayes’ theorem.
The Bayesian updating workflow is:
Specify a prior distribution representing existing knowledge.
Observe new data.
Evaluate the likelihood of the observations.
Apply Bayes’ Theorem to obtain the posterior distribution.
Use the posterior as the prior when additional observations become available.
Used in:
Bayesian Inference
Sequential Learning
Data Assimilation
Used in:
Also see:
Beta Level#
The beta level is the probability of failing to reject the null hypothesis when it is actually false.
The beta level is denoted by:
and represents the probability of a Type II Error:
The beta level is influenced by:
sample size
variability of the population
magnitude of the true effect
selected alpha level
The statistical power of a hypothesis test is:
A smaller beta level indicates a lower probability of missing a real effect.
Reducing beta generally requires:
increasing the sample size
reducing measurement uncertainty
increasing the magnitude of the effect relative to uncertainty
The beta level represents the acceptable false negative risk in a hypothesis test.
Used in:
Also see:
Bias#
Bias is a systematic difference between an estimated value and the true value.
The bias of an estimator is:
where:
\(\hat{\theta}\) is an estimator
\(\theta\) is the true parameter
\(E[\hat{\theta}]\) is the expected value of the estimator
An unbiased estimator has:
Bias represents a systematic tendency to overestimate or underestimate.
Examples include:
preferential sampling causing biased estimates of a population mean
an interpolation method consistently smoothing high values
a model that systematically underpredicts production
Bias differs from random error:
bias is systematic and repeatable
random error varies unpredictably
Bias and variance together describe estimator performance:
where MSE is the mean squared error.
Used in: TBA
Also see:
Biased Spatial Sampling#
Spatial sampling such that the sample statistics are not representative of the population parameters. For example,
the sample mean is not the same as the population mean
the sample variance is not the same as the population variance
Of course, the population parameters are not accessible, so we cannot directly calculate sampling bias, i.e., the difference between the sample statistics and the population parameters. Methods we can use to check for biased sampling,
evaluate the samples for preferential sampling, clustering, filtering, or survivorship bias.
apply declustering as a diagnostic to check for biased sampling
Used in:
Big Data#
Identification of big data is based on a combination of these criteria:
Data Volume - many data samples and features, difficult to store, transmit and visualize
Data Velocity - high-rate collection, continuous data collection relative to decision making cycles, challenges keeping up with the new data while updating the models
Data Variety - data form various sources, with various types of data, types of information, and scales
Data Variability - data acquisition changes during the project, even for a single feature there may be multiple vintages of data with different scales, distributions, and veracity
Data Veracity - data has various levels of accuracy, the data is not certain
For common subsurface applications most, if not all, of these criteria are met. Subsurface engineering and geoscience are often working with big data!
Used in:
Big Data Analytics#
The process of examining big data using statistical, computational, and machine learning methods to discover patterns, extract insights, and support decision making.
Used in:
Bin Size#
Bin size is a modeling and visualization parameter that controls the scale over which data are grouped for analysis. Appropriate bin size selection balances resolution and statistical stability.
Examples of bin size include:
histogram and normalized histogram bin size – width of the histogram bars used to summarize the frequency distribution of a feature or variable.
experimental variogram lag distance or azimuth tolerance – size of the search interval used to identify data pairs separated by a lag vector, \(h\).
The impact of bin size choice includes:
too large bins / too few bins – smooth the data distribution, reduce resolution, and potentially mask important features such as multiple populations or local variability.
too small bins / too many bins – increase noise due to insufficient samples within each bin, resulting in unstable estimates of frequency, probability, or spatial continuity.
The general guidance is to select the highest resolution with the lowest possible noise.
At extreme bin sizes:
very large bins combine many observations into a small number of groups, potentially collapsing all observations into a single bin.
very small bins result in many empty or sparsely populated bins, reducing the statistical reliability of the estimate.
Bin size influences the apparent distribution and continuity of data and should be selected based on the available sample size, variability, and the purpose of the analysis.
Used in:
Also see:
Binomial Distribution#
The binomial distribution is a discrete statistical distribution representing the probability of observing a number of successes in a fixed number of independent trials.
The probability mass function is:
where,
\(n\) is the number of independent trials
\(k\) is the number of observed successes, \(k=0,1,\ldots,n\)
\(p\) is the probability of success for each trial
\((1-p)\) is the probability of failure for each trial
\(\binom{n}{k}\) is the number of possible combinations of \(k\) successes among \(n\) trials
The binomial distribution is used for binary outcomes with a fixed number of trials, such as:
number of successful predictions from a classification model
number of occurrences within a fixed sampling effort
number of wells encountering a target geological facies
The binomial distribution becomes approximately Gaussian for sufficiently large \(n\) and moderate \(p\).
Used in: TBA
Also see:
Bivariate#
Involving two features (variables) simultaneously, often to study their relationship, dependence, or correlation. For examples see bivariate statistic.
Used in:
Compare with:
Bivariate Statistic#
A summary measure calculated from two features (variables) measured over a collection of samples. Bivariate statistics describe the relationship, dependence, or correlation between two variables.
Examples include:
Scatter Plot – to visualize the relationship between two variables.
Joint Probability – to quantify the probability of outcomes from two variables occurring together.
Joint Probability Density Function – a complete probabilistic model of the relationship between two variables.
Used in:
Compare with:
Bootstrap#
A statistical resampling procedure used to quantify uncertainty in a calculated statistic by repeatedly resampling from the available sample data. Some general comments,
sampling with replacement - \(n\) (number of data samples) Monte Carlo simulations from the empirical distribution of the dataset produces a realization of the resampled data.
simulates the data collection process - the fundamental idea is to approximate repeated sampling from the population by repeatedly resampling from the available data instead of collecting new samples.
bootstrap any statistic - the bootstrap is flexible because uncertainty can be characterized for almost any calculated statistic.
computationally efficient - repeated resampling generates realizations of the statistic that can be used to build an uncertainty distribution. A large number of realizations, \(L\), improves characterization of the uncertainty model.
calculates the entire uncertainty distribution - for any statistic, summary statistics of the uncertainty distribution can be calculated, such as the mean, P10, and P90 uncertainty in an estimated mean.
bagging for machine learning - bagging applies bootstrap resampling to create multiple training datasets, train multiple predictive models, and aggregate predictions from the model ensemble to reduce prediction variance.
What are the limitations of bootstrap?
biased sample data will likely result in a biased bootstrap uncertainty model; samples should first be corrected for known bias, e.g., declustering
bootstrap requires a sufficient sample size to reliably characterize uncertainty
classical bootstrap quantifies uncertainty due to limited sampling but does not explicitly account for spatial context, including sample locations, volume of interest, or spatial continuity
a variant called spatial bootstrap accounts for spatial relationships during resampling
Used in:
Categorical Feature#
A feature or variable that can take one of a limited and usually fixed number of possible categories. Categories often represent qualitative classes and generally do not have inherent numerical meaning or ordering, unless categorical ordinal feature,
categories may have qualitative names, but are often represented by integer labels or binary indicator variables for computational analysis and modeling.
Used in:
Categorical Nominal Feature#
A categorical feature without a natural ordering relationship between categories. Examples include,
facies = {boundstone, wackestone, packstone, breccia}
minerals = {quartz, feldspar, calcite}
Categories may be assigned labels or integer codes for analysis, but the labels do not represent magnitude or ranking.
Used in:
Contrast with:
Opposite:
Categorical Ordinal Feature#
A categorical feature with a natural ordering relationship between categories. The ordering provides relative ranking information, but category differences may not represent equal numerical intervals. Examples include,
geologic age = {Miocene, Pliocene, Pleistocene} - ordered from older to younger rock
Mohs hardness = \(\{1, 2, \ldots, 10\}\) - ordered from softer to harder minerals
Used in:
Contrast with:
Cell-based Declustering#
A declustering method that assigns weights to spatial samples based on local sampling density to reduce sampling bias and produce statistics that are more representative of the population (i.e., improve geostatistical sampling representativity]) in the presence of clustered spatial sampling. Data weights are assigned such that,
samples in densely sampled areas receive less weight
samples in sparsely sampled areas receive more weight
The goal of declustering is to reduce the influence of uneven sample locations on statistical estimates. For example, infill drilling or blast hole samples should not significantly change statistics for the area of interest simply because some locations have been sampled more densely.
Cell-based declustering proceeds as follows:
a cell mesh is placed over the spatial data and initial weights are assigned proportional to the inverse of the number of samples in each cell
the cell mesh size is varied, and a cell size is selected based on the resulting declustered statistics. Typically, the cell size that minimizes the declustered mean is selected when the sample mean is biased high, and the cell size that maximizes the declustered mean is selected when the sample mean is biased low
to reduce sensitivity to cell mesh position, the cell mesh is randomly shifted multiple times and the resulting declustering weights are averaged for each datum
The weights are calculated as:
where \(n_l\) is the number of data in the current cell, \(L_o\) is the number of cells containing data, and \(n\) is the total number of data.
Some highlights for cell-based declustering,
expert judgement to assign cell size based on nominal sample spacing (e.g., data spacing before infill drilling) may improve performance compared with automated cell size selection based only on minimizing or maximizing the declustered mean
cell-based declustering does not account for boundaries of the area of interest; therefore, samples near the boundary may appear more sparsely sampled and receive larger weights
cell-based declustering was introduced by Professor André Journel in 1983, [Jou83], and remains a foundational geostatistical declustering method.
Used in:
Also see:
Central Limit Theorem#
The Central Limit Theorem (CLT) describes the behavior of the sampling distribution of a sample statistic as the number of samples increases.
For sufficiently large sample sizes, the sampling distribution of the sample mean (or average) approaches a Gaussian distribution, regardless of the shape of the original population distribution.
The sampling distribution of the sample mean is:
where:
\(\bar{X}\) is the sample mean
\(\mu\) is the population mean
\(\sigma^2\) is the population variance
\(n\) is the sample size
The Central Limit Theorem explains why Gaussian-based methods are widely used for statistical inference, including:
estimation of population parameters
The Central Limit Theorem does not state that the original data are Gaussian. Instead, it states that the distribution of sample means becomes increasingly Gaussian as sample size increases.
Important considerations:
larger sample sizes generally provide more reliable Gaussian approximations
strongly skewed or heavy-tailed populations may require larger sample sizes
dependent or spatially correlated samples may violate the assumptions of independent sampling
Used in: TBA
Also see:
Chi-square Distribution#
The chi-square distribution is a continuous statistical distribution commonly used for inference involving variance, frequency, and categorical data.
A chi-square statistic is calculated as:
where:
\(O_i\) is the observed frequency or value
\(E_i\) is the expected frequency or value
\(k\) is the number of categories or groups
The chi-square distribution is defined by degrees of freedom, \(\nu\).
The chi-square distribution is commonly used for:
testing whether observed frequencies match expected frequencies
comparing histograms or categorical distributions
evaluating variance estimates for normally distributed data
Important characteristics:
bounded below by zero
becomes more symmetric as degrees of freedom increase
Used in: TBA
Also see:
Cloud Transform#
A cosimulation approach based on the bivariate relationship, or scatter plot, between primary and secondary variables (or features) to simulate realizations.
the primary variable, \(z\), is simulated by conditioning on a previously simulated realization of the secondary variable, \(y\)
For all locations in the model, \(\alpha = 1, \ldots, nx \cdot ny\), \(\bf{u}_{\alpha}\),
find the collocated secondary value, \(y(\bf{u}_{\alpha})\)
calculate the conditional distribution,
from the scatter plot-based joint distribution function, \(f_{y,z}\), and the collocated secondary value, \(y(\bf{u}_{\alpha})\)
draw a simulated value from \(f_{z|y=y(\bf{u}_{\alpha})}(z)\) using a field of correlated p-values (known as a p-field)
Some comments about cloud transform,
prioritizes reproduction of the cloud, the bivariate relationship (scatter plot), between the primary and secondary variables
may not reproduce the spatial continuity or histogram of the primary variable well because the primary variable spatial model is not explicitly simulated
cloud transform is commonly used in practice to simulate permeability conditioned to secondary porosity realizations. Permeability data are often sparsely sampled, resulting in poorly defined permeability distributions and variograms, while porosity data often have better defined distributions and spatial continuity models. The porosity-permeability relationship may therefore provide more reliable conditioning information.
the general approach of calculating simulated realizations by applying p-fields to local conditional distributions is called p-field simulation. It simplifies information integration by separating local conditioning from the imposition of spatial correlation and can provide computational advantages for large models.
Mentioned but not demonstrated (to be added later) in:
Clustered Spatial Sampling#
Spatial samples with locations preferentially selected or concentrated in certain areas, resulting in potentially biased statistics.
spatial samples are often clustered in locations associated with higher or more desirable values, for example, high porosity and permeability, good quality shale for unconventional reservoirs, or low acoustic impedance indicating higher porosity
Because the true population parameters are generally unknown, sampling bias cannot be directly calculated as the difference between sample statistics and population parameters. Methods to diagnose and address biased sampling include,
evaluate samples for preferential sampling and spatial clustering
apply declustering as a diagnostic method to evaluate the impact of clustered sampling on statistics
Used in:
Coefficient of Variation#
The coefficient of variation (CV) is a dimensionless univariate statistic that measures relative variability by comparing the standard deviation to the mean.
The coefficient of variation is calculated as:
where:
\(\sigma\) is the standard deviation
\(\mu\) is the mean
The coefficient of variation allows comparison of variability between distributions with different units or magnitudes by standardizing the variability relative to the mean.
A larger coefficient of variation indicates greater relative variability and may indicate increased heterogeneity.
Common applications include:
comparing variability of geological properties with different magnitudes, such as permeability, porosity, thickness, or grade
quantifying reservoir heterogeneity, such as coefficient of variation of permeability
Some observations about the coefficient of variation:
for positively skewed geological variables, such as permeability, the coefficient of variation is often large due to high-value observations in the distribution tail.
for distributions where variability increases proportionally with the mean, the coefficient of variation may provide a more comparable measure of variability across different populations.
Used in:
Also see:
Cognitive Biases#
Cognitive biases are automatic mental shortcuts, or heuristics, that influence human reasoning and decision making. These shortcuts help humans efficiently process information under uncertainty, but they can also systematically distort interpretation of data, scientific evidence, and engineering decisions.
Common cognitive biases include:
Anchoring Bias - excessive influence of initial information or assumptions, even when later information suggests alternatives.
Availability Heuristic - overestimating the importance of information that is easily recalled or available, such as anecdotes.
Bandwagon Effect** - increasing confidence in a belief because many others hold the same belief.
Blind-spot Effect - failing to recognize one’s own cognitive biases.
Choice-supportive Bias - favoring information that supports previous decisions or commitments.
Clustering Illusion - perceiving patterns in random data.
Confirmation Bias - preferentially considering information that supports existing beliefs or models.
Conservatism Bias - favoring established information over new evidence.
Recency Bias - giving excessive weight to recently acquired information.
Survivorship Bias - focusing only on successful or visible examples while ignoring missing cases.
Mitigating cognitive biases requires deliberate uncertainty analysis, quantitative evaluation of evidence, diverse perspectives, and critical review of assumptions.
Used in:
Cokriging#
A generally bivariate and possibly multivariate extension of kriging that simultaneously accounts for primary and secondary variable(s) to build multivariate spatial models.
like regular kriging, a spatial estimation approach that relies on linear weights that account for spatial continuity, data closeness, and redundancy
the weights are determined to provide an unbiased estimator with minimum estimation variance
The simple cokriging weights are calculated by solving a linear system of equations that may be represented with matrix notation as,
where we assume that there are \(1, 2, \ldots, n_z\) primary data \(z\), and \(1, 2, \ldots, n_y\) secondary data \(y\).
note, I could be more rigorous with the notation and indicate the location \(\bf{u}_{1}\) for \(z\) and \(y\) may not be the same location, but the notation was already getting quite dense.
This system integrates the,
spatial continuity - as quantified by the variogram model (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, \(C(\bf{u}_i,\bf{u})\)
relationship - between the primary and secondary features over lag distance from the cross covariance terms, \(C_{z,y}\) and \(C_{y,z}\).
Once the weights are calculated from the above linear system of equations, the cokriging estimator is expressed as,
assuming that each feature is detrended such that the mean is 0.0.
Some general comments about cokriging,
secondary and primary data may be collocated, not collocated, or a mixture of collocated and non-collocated data
direct variograms, \(\gamma_z\) and \(\gamma_y\), and cross variograms, \(\gamma_{z,y}\), must be calculated and modeled. To ensure all direct and cross variograms are jointly positive definite, constraints from the linear model of coregionalization must be applied.
the model can be expanded to consider any number of secondary variables
Cokriging provides a rigorous framework for multivariate analysis spatial estimation; however, simplified approaches such as collocated cokriging and cloud transform are more commonly applied in practice.
Mentioned but not demonstrated (to be added later) in:
Compare with:
Collocated Cokriging#
A simplified variant of cokriging that honors primary variable hard data and spatial continuity while incorporating the bivariate relationship between primary and secondary variables.
Collocated cokriging introduces two assumptions that greatly simplify the cokriging method:
Markov screening - only one (the collocated) secondary variable datum is considered. The secondary variable value at the estimation location is assumed to screen all secondary data at other locations.
as a result, the secondary variogram is not required and the cokriging system is greatly reduced
Bayesian updating - this relationship can be viewed as a Bayesian updating analogy, the correlation coefficient between primary and secondary variables provides prior information about their relationship, while the primary covariance function model provides the spatial continuity framework used to construct the updated cross covariance model,
where \(\rho_{z,y}\) represents the prior relationship between the primary and secondary variables, \(C_z(\bf{h})\) represents the spatial covariance model, and \(C_{z,y}(\bf{h})\) represents the resulting cross covariance model.
as a result, we do not need to calculate the cross variogram
The collocated cokriging system of equations in matrix notation are,
where \(C_{z,y}(0)\) is the cross covariance at lag distance \(\bf{h} = 0\), for standardized features (variance of 1.0) this is the correlation coefficient, \(C_{z,y}(0) = \rho_{z,y}\).
Used in:
Compare with:
Complementary Events#
The logical NOT relationship in probability. For a simple example, if we define event \(A\), then the complementary event, \(A^c\), represents NOT \(A\), and the resulting probability closure relationship is,
Complementary events can also be considered for multivariate and conditional probabilities. For example, for a bivariate conditional relationship,
Note that the conditioning event must remain the same for complementary probability closure.
Used in:
Conditional Distribution#
A conditional distribution describes the probability distribution of a variable given available information or conditioning data.
The conditional distribution is written as:
where:
\(X\) is the uncertain variable
\(Y\) is the conditioning information
\(f_{X|Y}\) is the conditional distribution of \(X\) given \(Y\)
In geostatistics, a local conditional distribution describes uncertainty at a specific location given available spatial data:
where:
\(Z(\mathbf{u}_{\alpha})\) is the variable at location \(\mathbf{u}_{\alpha}\)
\(data\) represents available samples, trends, and model constraints
\(F\) is the local cumulative distribution function
The local conditional distribution provides:
expected value
conditional variance
local percentiles
probability of exceeding a threshold
Conditional distributions are the foundation of:
uncertainty quantification
Used in:
Also see:
Conditional Probability#
The probability of an event given that another event has occurred. For example,
We read \(P(A|B)\) as the probability of \(A\) “given” \(B\) has occurred. Conditional probability is calculated as the joint probability divided by the marginal probability of the conditioning event.
Conditional probabilities can be extended to multivariate cases by including additional conditioning events. For example,
Used in:
Confidence Interval#
A range of values that quantifies the uncertainty in a population parameter estimated from a sample. The range is constructed so that, over many repeated random samples, a specified proportion of the intervals contain the true population parameter. This proportion is called the confidence level.
Confidence intervals are commonly, but incorrectly, communicated as,
there is a 95% probability that the model slope, \(b_1\), is between 0.5 and 0.7.
Instead, confidence intervals should be interpreted as,
if we repeatedly drew random samples from the same population and calculated a 95% confidence interval for each sample, approximately 95% of those intervals would contain the true population slope.
The probability applies to the procedure used to construct the interval,
not to the unknown parameter itself. Once the interval has been calculated, the true parameter either lies within the interval or it does not.
To make the probabilistic statement that there is a 95% probability the parameter lies within an interval, Bayesian methods are required to calculate a credible interval.
Confidence intervals may be calculated with,
analytical methods, when available. For example, the confidence interval for the population mean is
where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, \(n\) is the sample size, and \(t_{1-\alpha/2,\,n-1}\) is the t-critical value from the Student’s \(t\)-distribution.
bootstrap methods, which estimate confidence intervals directly from repeated resampling of the observed data and can be applied when analytical solutions are unavailable or difficult to derive.
Used in:
Contrast with:
Confidence Level#
The confidence level is the probability associated with a confidence interval method describing how frequently intervals contain the true population parameter over repeated samples.
The confidence level is denoted by:
where:
\(\alpha\) is the significance level
Common confidence levels include:
90% confidence: \(\alpha=0.10\)
95% confidence: \(\alpha=0.05\)
99% confidence: \(\alpha=0.01\)
Increasing the confidence level increases the interval width because greater uncertainty must be included.
For example:
99% confidence intervals are wider than 95% confidence intervals
95% confidence intervals are wider than 90% confidence intervals
The confidence level represents the reliability of the estimation method, not the probability that a particular calculated interval contains the true parameter.
Used in:
Also see:
Continuous Feature#
A feature that can take any value within a continuous range of possible values. For example,
porosity = \(\{13.01\%, 5.23\%, 24.62\%\}\)
gold grade = \(\{4.56 \text{ g/t}, 8.72 \text{ g/t}, 12.45 \text{ g/t}\}\)
Used in:
Contrast with:
Continuous Interval Feature#
A continuous feature where differences between values are meaningful and equally spaced, but the zero point is arbitrary and does not represent the absence of the quantity.
For example,
Celsius temperature scale (the zero point is defined by convention)
calendar year (there is no objective zero year)
Continuous interval features can be compared using addition and subtraction operations, but multiplication and division comparisons are not meaningful.
Used in:
Contrast with:
Continuous Ratio Feature#
A continuous feature where differences between values are meaningful, the zero point represents absence of the measured quantity, and ratios are physically meaningful.
For example,
Kelvin temperature scale
porosity
permeability
saturation
Because ratio features have a true zero, multiplication and division operations are meaningful. For example, a permeability of 200 mD can be described as twice the permeability of 100 mD.
Used in:
Contrast with:
Coordinate Transformation#
Coordinate transformations are applied to spatial data and model grids to:
align the model with the true volume of interest
represent complex geological geometry in a regular modeling coordinate system
Without coordinate transformation, subsurface models would often:
waste computational and storage resources by including large volumes outside the reservoir or ore body
violate geostatistical assumptions, such as approximately constant volume support and stationary spatial relationships
distort distances, directions, and spatial (stratigraphic) correlation structures between wells, layers, and faults
Common coordinate transformations include:
2D and 3D rotation
proportional flattening
onlap flattening
truncation flattening
Coordinate transformations are applied in two directions:
forward transformation – from real-world coordinates (X, Y, Z) to model coordinates (I, J, K)
back transformation – from model coordinates (I, J, K) back to real-world coordinates (X, Y, Z)
Forward transformation is used for modeling, simulation, interpolation, and spatial analysis, while back transformation is used to visualize and communicate results in their original geological context.
Used in:
Also see:
Core#
The primary direct sampling method for characterizing subsurface resources.
In oil and gas exploration and development, core samples are obtained by replacing the drill bit with a specialized core barrel to recover a continuous rock sample. This process is expensive and time-consuming, so core data are typically sparsely and selectively acquired, often targeting specific geological intervals of interest.
In mining exploration and grade control, core drilling (commonly diamond drilling with a core barrel) is widely used because ore bodies may be accessed through surface drilling or underground workings such as drifts or stopes. As a result, core data are often more common than indirect measurements such as well logs in these settings.
In soft sediment environments, gravity, piston, and similar coring methods are used to sample unconsolidated sediments in lakes and oceans.
What do we learn from core data?
Petrological properties (e.g., sedimentary structures, mineralogy, and grade), petrophysical properties (e.g., porosity and permeability), and geomechanical properties (e.g., elastic moduli and Poisson’s ratio).
Stratigraphic relationships and geological geometry through direct observation and spatial interpolation between wells and drill holes.
Core data are critical for subsurface resource interpretation. They provide the most direct observations available, anchor geological and reservoir models, and provide calibration data for indirect measurements.
For example, core data are often used to calibrate well log responses, such as facies classification and porosity estimation.
Used in: TBD
Also see:
Correlation Coefficient#
A standardized measure of the strength and direction of the linear relationship between two features.
To understand the correlation coefficient, start with variance, a measure of the dispersion of a single feature,
We can replace one squared deviation with the deviation of a second feature, \(y\), to obtain the covariance,
Covariance measures how features \(x\) and \(y\) vary together. However, covariance depends on the units and scale of both features. We standardize covariance by the product of the standard deviations of \(x\) and \(y\) to calculate the correlation coefficient,
or equivalently,
The correlation coefficient ranges from \(-1.0\) to \(1.0\),
\(\rho_{xy}=1.0\) indicates a perfect positive linear relationship
\(\rho_{xy}=-1.0\) indicates a perfect negative linear relationship
\(\rho_{xy}=0.0\) indicates no linear relationship
The correlation coefficient is useful because it is,
independent of the dispersion or standard deviation of both features
dimensionless, allowing comparison of relationships between features with different units and scales
related to the coefficient of determination, \(R^2\), for a simple linear regression model with an intercept, where \(R^2=\rho_{xy}^2\)
when we replace covariance with a covariance function, correlation becomes the correlogram, \(\rho(\mathbf{h})\), a measure of spatial correlation over separation distance
Some cautionary notes about the correlation coefficient,
correlation does not imply causation - causal analysis requires careful experiments with sufficient replicates and control of confounding features.
correlation is sensitive to outliers - a single extreme value can substantially change the magnitude and direction of the correlation coefficient.
Used in:
Also see:
Correlogram#
A measure of similarity between a feature and itself separated by a lag vector. The correlogram is the covariance between values separated by lag distance normalized by the feature variance,
For standardized features with a mean of 0.0 and variance of 1.0, the correlogram simplifies to,
The correlogram is the covariance normalized by the variance,
where \(C_x(\bf{h})\) is the covariance function and \(C_x(0)\) is the variance. Therefore, for standardized features with variance equal to 1.0,
The correlogram is also related to the variogram,
and for standardized features,
The correlogram is easy to interpret since it represents the correlation between samples separated by a specified lag distance.
Used in: TBD
Also see:
Cosimulation#
A set of simulation methods for simulating realizations of a primary feature conditional on a secondary feature realization. All cosimulation methods attempt to capture primary feature spatial continuity, honor local conditioning data, and reproduce the relationship with the secondary feature.
Each cosimulation method has a conditioning priority.
Collocated cokriging prioritizes the primary feature histogram and variogram while honoring the relationship between the primary and secondary features through their correlation coefficient.
The relationship between the primary and secondary features is limited to a correlation coefficient after Gaussian transformation of both features (i.e., in Gaussian space).
In the case of dense conditioning data, the relationship observed at the data locations will override the correlation coefficient.
Cloud transform prioritizes the specific form of the bivariate relationship (cloud) between the two features but may not reproduce the primary feature histogram or spatial continuity.
The precise scatter plot relationship between the primary and secondary features is prioritized.
These methods start with a completed realization of the secondary feature. For example,
first simulate a copper realization and then cosimulate the zinc (primary feature) realization given the copper (secondary feature) realization using collocated cokriging and the correlation coefficient between Gaussian transformed copper and zinc data.
first simulate a porosity realization and then cosimulate the permeability (primary feature) realization given the porosity (secondary feature) realization using cloud transform and the scatter plot relationship between Gaussian transformed porosity and permeability.
With cosimulation, there is an increasing likelihood that multiple information sources are contradictory. When this occurs, the lower-priority information source is preferentially sacrificed.
While the full cokriging approach for cosimulation is available, due to the inference burden of modeling all direct and cross variograms, it is typically not used in practice.
Used in:
Also see:
Covariance#
A measure of how two features vary together.
positive covariance indicates the features tend to increase together
negative covariance indicates that as one feature increases, the other tends to decrease
covariance near zero indicates little or no linear relationship
For a sample,
The covariance can be interpreted relative to the more familiar correlation coefficient.
the correlation coefficient is the covariance standardized by the product of the standard deviations of the two features
Some other observations about correlation,
unlike the correlation coefficient, covariance depends on the units of the two features and is therefore most useful for mathematical calculations rather than direct interpretation.
by replacing the second feature with the same feature offset in space, we get the covariance function, a useful measure of spatial similarity.
Used in:
Also see:
Covariance Function#
A spatial covariance as a measure of similarity between a feature and itself separated by a lag vector. The covariance function is calculated as the average product of deviations from the mean for values separated by the lag vector,
The covariance function, \(C_z(\bf{h})\), is the variogram, \(\gamma_z(\bf{h})\), flipped upside down relative to the sill, \(\sigma_z^2\),
The covariance function is also related to the correlogram,
where \(\rho_x(\bf{h})\) is the correlogram. For standardized features with a mean of 0.0 and variance of 1.0,
We model variograms, but inside kriging and simulation methods they are often converted to covariance values for numerical convenience,
Covariance matrices are typically diagonally dominant because the variance occurs on the diagonal, improving numerical stability when solving the linear systems used to calculate kriging weights.
Used in:
Also see:
Coverage#
Coverage is the amount of the volume of interest or population that is sampled, conveniently represented as a spatial, volumetric, or temporal fraction.
Due to the typically high cost of direct sampling, such as core and well log measurements, direct data coverage is often a very small proportion of the volume of interest.
Three basic aspects of spatial data include:
coverage
property or information
For example,
Core coverage – At the development stage of a reservoir, assuming 500 m well spacing, a 0.05 m core diameter, and 20% core recovery within wells, the volumetric coverage of core samples is approximately one billionth of the reservoir volume.
Well log coverage – Well logs provide continuous measurements along boreholes but sample only a small number of discrete locations throughout the reservoir.
Seismic coverage – Seismic surveys provide extensive spatial coverage of the reservoir and surrounding under- and overburden, but the measured response represents a lower-resolution, indirect estimate of subsurface properties.
Production data coverage – Production measurements provide temporally continuous information at producing wells, with spatial coverage controlled by well locations, drainage radius, and active reservoir response.
There is often a resolution–coverage trade-off with spatial data:
High-coverage data sources tend to have lower resolution – seismic and remote sensing data may cover the entire reservoir or field area, but typically represent geological properties at resolutions of tens to hundreds of meters.
High-resolution data sources tend to have limited coverage – core data provides measurements at micrometer to millimeter scales, but samples only a very small fraction of the reservoir volume.
The resolution–coverage trade-off motivates the integration of multiple data sources. High-resolution, low-coverage data provides detailed local information, while lower-resolution, high-coverage data provides spatial continuity.
Used in:
Also see:
Credible Interval#
A range of values that quantifies the uncertainty in an unknown parameter using its posterior probability distribution. The interval is constructed so that a specified probability of the posterior distribution lies within the interval.
Credible intervals are interpreted probabilistically, for the example of Bayesian Linear Regression,
there is a 95% probability that the model slope, \(b_1\), lies between 0.5 and 0.7, given the observed data and the prior information.
Unlike a frequentist confidence interval, the probability applies directly to the unknown parameter,
because Bayesian inference represents uncertainty about the parameter with a posterior probability distribution.
Credible intervals are calculated from the posterior distribution,
analytically, when a closed-form posterior distribution is available
numerically, using methods such as Markov chain Monte Carlo when analytical solutions are unavailable.
Used in: TBA
Contrast with:
Also see:
Cumulative Distribution Function#
Commonly known by its acronym CDF, it describes the accumulation of probability up to a specified value. The CDF is calculated as the cumulative sum of a discrete probability mass function or the integral of a continuous probability density function.
Important concepts about CDFs,
the CDF is stated as \(F_x(x)\), while the PDF is stated as \(f_x(x)\)
the CDF is the probability that a random sample, \(X\), is less than or equal to a specific value \(x\); therefore, the y-axis represents cumulative probability,
for discrete distributions, the CDF is calculated by summing probabilities up to and including the value \(x\)
for CDFs there is no bin assumption; therefore, bins are defined by the resolution of the available data
the CDF is a monotonically non-decreasing function because a negative slope would indicate decreasing cumulative probability over an interval
The requirements for a valid CDF include,
bounded probability:
non-decreasing probability:
probability closure at the limits:
Used in:
Cyclicity#
A variogram interpretation based on oscillating experimental variogram values that may indicate underlying geological periodicity, such as repeating depositional cycles or layered structures.
Important considerations,
oscillations in an experimental variogram may be caused by insufficient data, sampling artifacts, or random variability and should not automatically be interpreted as geological cyclicity.
the wavelength of the oscillations in the experimental variogram represents the characteristic spatial scale of the repeating geological structure.
cyclicity may be incorporated into a Variogram Model using a hole effect positive definite variogram structure.
Used in:
Data#
Data are observations collected to characterize a population or process. In spatial data analytics and geostatistics, three fundamental aspects determine the value of a dataset:
Data coverage - what proportion of the population has been sampled? In general, hard data have high resolution (small volume support), but poor spatial coverage. For example,
Core coverage in deepwater oil and gas may sample only one five hundred millionth to one five billionth of a reservoir, assuming 3-inch diameter cores with 10% core recovery in vertical wells spaced 500–1,500 m apart.
Core coverage for mining grade control may sample approximately one eight thousandth to one thirty thousandth of an ore body, assuming HQ (63.5 mm) cores with complete recovery in drill holes spaced 5–10 m apart.
\(\quad\) In contrast, soft data often provide excellent (sometimes complete) spatial coverage, but lower resolution, for example,
Seismic and other remote sensing measurements commonly cover the entire area of interest but have substantially lower spatial resolution, generally decreasing with depth.
Volume support (data scale) - what volume or scale is represented by each measurement? Examples include,
core tomography imaging at the pore scale (approximately 1–50 \(\mu\)m)
gamma ray well log sampled every 0.3 m with approximately 1 m radial investigation
ground-based gravity gradiometry with an effective resolution of approximately 20 m × 20 m × 100 m
Information content - what does the dataset tell us about the subsurface? Examples include,
grain size distributions used to calibrate permeability and saturation
fluid contacts used to identify oil-water contacts
structural dip and continuity used to infer reservoir connectivity
mineral grades used to delineate ore shells for mine planning
Used in: TBD
Data Analytics#
In this book, data analytics is used synonymously with:
Used in:
Data Weight#
A data weight is a numerical value assigned to a sample that represents its relative influence when calculating a weighted statistic, distribution, or model.
For example, a weighted mean is calculated as:
where:
\(X_i\) is the value of sample \(i\)
\(w_i\) is the data weight assigned to sample \(i\)
\(n\) is the number of samples
Data weights are commonly applied when samples do not equally represent the population of interest.
In geostatistics, data weights are commonly used for declustering:
densely sampled regions receive lower weights
sparsely sampled regions receive higher weights
the weighted distribution better represents the volume of interest
For example, in spatial sampling:
10 samples collected closely together may collectively represent a small volume
1 isolated sample may represent a much larger volume
Data weights may also be used for:
weighted histograms
weighted cumulative distribution functions
weighted estimation methods
accounting for unequal measurement reliability
Important considerations:
data weights correct for sampling influence, not measurement error
data weights should reflect the intended representation of the population or volume of interest
normalized weights are often used such that:
Used in:
Also see:
DataFrame#
A convenient Pandas data structure for working with tabular data. A DataFrame is a two-dimensional labeled data structure with rows representing samples or observations and columns representing features or variables.
DataFrames provide a convenient structure to,
store, access, and manipulate tabular data
load data from a variety of sources, including files, Python objects, databases, and Excel spreadsheets
calculate summary statistics and visualize data
perform data queries, sorting, filtering, and selection operations
complete data manipulation tasks, including cleaning, transformation, merging, and reformatting
store metadata and information about the dataset, such as dimensions, column names, data types, and missing values
Used in:
Compare with:
Debiasing with Secondary Data#
When the full range of a primary feature is not sampled, declustering alone cannot remove sampling bias because parts of the feature distribution are completely missing. Instead, we use,
a secondary data feature that provides coverage over the entire area of interest
the relationship between the Primary Data feature and the secondary feature
to infer the unsampled portion of the primary feature distribution.
The relationship between the primary and secondary features may be established using,
a statistical model that extrapolates the primary feature into the unsampled range
a physical model based on scientific or engineering understanding
expert knowledge of the underlying process
Unlike declustering, which corrects for clustered spatial sampling, debiasing with secondary data addresses situations where part of the primary feature distribution has not been sampled at all.
Mentioned but not demonstrated (to be added later) in: TBD
Compare with:
Decision Criteria#
An engineered feature or metric calculated from one or more subsurface models to support decision making. Decision criteria quantify the consequences of alternative decisions and may represent economic value, technical performance, environmental impact, health and safety, or combinations of these objectives. For example,
contaminant recovery rate to support the design of a pump-and-treat soil remediation project
oil in place to determine whether a reservoir should be developed
Lorenz coefficient as a heterogeneity measure to classify a reservoir and identify appropriate analogs
recovery factor or production rate to schedule production and optimize facilities
recovered mineral grade and tonnage to determine the economic ultimate pit shell
In quantitative decision workflows, the decision criterion is used to rank competing alternatives. Common approaches include,
maximizing a profit metric
minimizing a loss function
Best practice is to define decision criteria that directly represent project value. For example,
rather than stopping at hydrocarbon in place, continue through engineering and economics to estimate project profit (currency).
Used in:
Decision Making#
The ultimate objective of geostatistics and data analytics is to support better decisions. Estimation, prediction, uncertainty modeling, and machine learning are intermediate steps whose value is realized only when they improve decisions.
Decision making involves selecting the best,
estimate
choice
from a set of alternatives.
In quantitative decision workflows, a decision criteria is used to rank competing alternatives. Common approaches include,
maximizing a profit metric
minimizing a loss function
to identify the optimum estimate or decision.
when accounting for uncertainty, this optimization is performed over an ensemble of subsurface realizations and scenarios.
Used in:
Declustered Statistics#
Once declustering weights are calculated for a spatial dataset, the unweighted (also called naive) statistics are replaced with weighted statistics that account for the declustering weights. These corrected statistics are then used as input for all subsequent analysis and modeling to mitigate sampling bias. For example,
the declustered mean is used as the stationary global mean for simple kriging.
the weighted cumulative distribution function is used in sequential Gaussian simulation so that the back-transformed realizations reproduce the declustered distribution.
Any sample statistic can be computed using declustering weights, including the entire cumulative distribution function (CDF). Examples include:
weighted sample mean,
where \(n\) is the number of data.
weighted variance,
where \(\overline{x}_{wt}\) is the declustered mean.
weighted Covariance,
where \(\overline{x}_{wt}\) and \(\overline{y}_{wt}\) are the declustered means for features \(X\) and \(Y\).
weighted cumulative distribution function (CDF),
This expression represents the empirical weighted CDF evaluated at the observed data values. Between observations, the CDF is obtained by interpolation.
No declustering method can guarantee improved estimates of the population parameters for every dataset.
however, when preferential sampling is present, declustering methods generally reduce sampling bias and provide improved statistical estimates in Expectation.
Used in:
Declustering#
A family of methods that assign weights to spatial samples based on local sampling density so that weighted statistics are more representative of the inaccessible population. Data weights are assigned so that,
samples in densely sampled areas receive less weight
samples in sparsely sampled areas receive more weight
There are various declustering methods:
It is important to note that no declustering method can prove that for every data set the resulting weighted statistics will improve the prediction of the population parameters, but in expectation these methods tend to reduce the bias.
Used in:
Deterministic Model#
A model that assumes a system or process is completely specified such that the same inputs always produce the same outputs. Deterministic models do not explicitly represent uncertainty in the system or process; therefore,
uncertainty is neglected and the system is treated as known or certain.
Deterministic models may be based on,
engineering and geoscience physics
expert interpretation and knowledge
data-driven estimation methods
Examples include,
numerical flow simulation for a specified set of reservoir properties
stratigraphic bounding surfaces interpreted from seismic data
kriging estimates
machine learning prediction models that return a single prediction
Advantages:
integrates physics, expert knowledge, and available data
integrates multiple information sources
often straightforward to interpret and apply
Disadvantages:
provides a single model or prediction without explicitly representing uncertainty
may underestimate decision risk when uncertainty is significant
often time consuming to construct, calibrate, and validate
Contrast with:
Used in:
Discrete Feature#
A feature that can only take one of a countable set of distinct values. Discrete features may be naturally discrete (categorical feature) or created by grouping (or binning) a continuous feature. For example,
binned continuous feature – porosity between 0% and 20% assigned to 10 bins:
\(\quad\) represented by the bin centroids,
categorical feature – facies:
ordinal feature – Mohs hardness:
Used in:
Contrast with:
Dispersion Variance#
A generalized measure of variability that accounts for the Volume Support of the data or model and the area of interest. Dispersion variance describes how variability changes when moving from a smaller support size (e.g., core plug, grid block) to a larger volume of interest.
For data support size represented by \(\cdot\),
and for model support size represented by \(v\),
both over the volume of interest, \(V\). For point-support data over the volume of interest, dispersion variance simplifies to the familiar variance,
therefore, the variance, \(\sigma^2\), is a special case of dispersion variance.
Under the assumptions of stationary mean, variance and variogram, dispersion variance is calculated as,
where \(\overline{\gamma}_{V,V}\) and \(\overline{\gamma}_{v,v}\) are variogram models integrated over volumes \(V\) and \(v\), respectively, known as Gammabar values.
Dispersion variance is fundamental for understanding the impact of Volume Support on variability, including the reduction of variance when moving from small-scale measurements to larger model blocks.
Used in:
Distribution Transformation#
A mapping from one probability distribution to another through corresponding percentile values, also called a quantile transformation. The transformation preserves the rank ordering of the data while changing the distributional shape, resulting in new,
Distribution transformations are commonly applied in geostatistical methods and workflows because,
inference - to transform a sample distribution toward an expected distribution when data are sparse, biased, or insufficient to characterize the full distribution
theory - to satisfy a distributional assumption required by a workflow step, for example, a Gaussian distribution with mean 0.0 and variance 1.0 is required for sequential Gaussian simulation
data preparation - to reduce the influence of extreme values by mapping them into the target distribution while preserving their rank relationship
How do we perform distribution transformations?
Values are transformed from the original cumulative distribution function (CDF), \(F_X\), to a target CDF, \(G_Y\), using percentile matching. This quantile transformation is applied to all sample values:
Forward transform:
Reverse transform:
This approach may be applied to any distribution, including parametric and nonparametric distributions, as long as percentile values can be mapped between the distributions.
The key property is:
rank preserving transform - the percentile position of a value is maintained, for example, P25 remains P25 after transformation
Contrast with affine correction, which only adjusts distribution location and scale (mean and variance),
distribution transformation modifies the complete distribution, including higher-order statistics and distribution shape
Used in:
Drill Cuttings#
Direct samples of subsurface material generated during drilling operations.
drill cuttings are fragments of rock produced by the drill bit and continuously transported to the surface, where they are collected, described, and logged during drilling.
Drill cuttings provide broader spatial coverage than core data because they are commonly recovered along much of the well or borehole trajectory during routine drilling operations. However, compared with core data, drill cuttings,
represent small, irregular, and mixed rock fragments rather than a continuous sample volume. Individual fragments may range approximately from 0.1 mm to 5 cm, although larger fragments (cavings) may occur due to mechanical failure along the borehole or wellbore.
lose orientation and large-scale structural information during recovery and transport because fragments are mixed and disrupted during pneumatic or hydraulic lifting from the borehole or well.
provide lithological and compositional information but generally cannot preserve continuous sedimentary structures, fracture orientations, or fine-scale spatial relationships.
Drill cuttings represent a trade-off between core and indirect measurements; they provide extensive direct sampling coverage but with reduced spatial resolution and geological context.
Used in: TBD
Also see:
Dykstra-Parsons Coefficient#
The Dykstra-Parsons coefficient is a measure of permeability heterogeneity commonly used in reservoir characterization.
The Dykstra-Parsons coefficient is calculated from permeability percentiles as:
where:
\(k_{50}\) is the median permeability
\(k_{84.1}\) is the permeability at the 84.1 percentile
The Dykstra-Parsons coefficient ranges from 0 to 1:
values near 0 indicate homogeneous permeability
larger values indicate greater permeability heterogeneity
The Dykstra-Parsons coefficient is related to the spread of a permeability distribution and is commonly applied for reservoir flow characterization.
Used in:
Also see:
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