Geostatistics Glossary N-S#
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.
Normal Score Transformation#
A normal score transform is a distribution transformation that converts a feature to follow a standard Gaussian distribution while preserving the rank order of the data.
The transformation is performed by mapping cumulative probabilities to the inverse Gaussian cumulative distribution function:
where:
\(X\) is the original feature
\(F_X\) is the cumulative distribution function of \(X\)
\(\Phi^{-1}\) is the inverse cumulative distribution function of the standard Gaussian distribution
\(Y\) is the normal score transformed feature
Normal score transforms are commonly applied before Gaussian-based methods such as kriging and sequential Gaussian simulation.
Also see:
Normalized Histogram#
A bar chart of the univariate statistical distribution with probability over an exhaustive set of bins over the range of possible values. These are the steps to build a normalized histogram,
Divide the continuous feature range of possible values into \(K\) equal size bins, \(\delta x\):
or use available categories for categorical features.
Count the number of samples (frequency) in each bin, \(n_k\), \(\forall k=1,\ldots,K\)
Divide each by the total number of data, \(n\), to calculate the probability of each bin,
Plot the probability vs. the bin label (use bin centroid if continuous)
Additional comments:
step 3 converts a standard histogram to a normalized histogram with a y-axis of probability instead of frequency
for categorical features, a normalized histogram represents an empirical probability mass function
for continuous features, the normalized histogram represents the empirical probability of each bin interval and is an approximation to the underlying continuous distribution
Used in:
Also see:
Nugget Effect#
The discontinuity in the variogram at distances smaller than the minimum sampling spacing, representing variability that cannot be resolved by the available sampling.
The nugget effect may result from,
measurement error
spatial variability occurring at scales smaller than the sampling interval
It is often communicated as the ratio of the nugget effect to the sill, known as the relative nugget effect. For example,
a copper grade variogram model with a 20% relative nugget effect
The nugget effect is both,
a structure interpreted from the experimental variogram
a positive-definite variogram structure that may be included as one of the nested structures in a variogram model
A large relative nugget effect indicates weak short- spatial continuity range, while a small relative nugget effect indicates strong local continuity.
Used in:
Null Hypothesis#
The null hypothesis is the default assumption applied in hypothesis testing that there is no effect, relationship, or statistically significant difference. It is denoted by,
Examples include:
or
where the population means for sample groups 1 and 2 are assumed to be equal.
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
Sample data are used to determine whether there is sufficient evidence to reject the null hypothesis.
Failure to reject the null hypothesis does not prove that it is true.
It indicates only that there is insufficient evidence to conclude otherwise, and the null hypothesis lives on to be tested another day.
Rejecting the null hypothesis does not prove that it is false.
It indicates that the observed data would be unlikely if the null hypothesis were true, providing evidence in support of the alternative hypothesis.
Used in:
Also see:
Object-based Simulation#
A stochastic simulation approach that generates realizations by sequentially placing parameterized geological objects within the simulation domain. The objects are designed to represent geological shapes observed in nature while allowing variability through distributions of geometric parameters.
The general object-based simulation workflow is:
Specify the object templates along with their occurrence probabilities and geometric parameter distributions.
Initialize the model with a background facies.
Monte Carlo simulate an object realization, including the object type and its geometric parameters from the specified distributions.
Randomly place the geological object within the simulation grid. All grid cells falling inside the object’s geometry are assigned the object’s facies.
Repeat the placement process (return to step 3) until a target facies proportion, object count, or other stopping criterion is reached.
By repeatedly placing stochastic geological objects, realizations reproduce expected geological geometries while representing uncertainty in object occurrence, size, shape, orientation, and location.
conditioning to local data and representing spatial non-stationarity in object occurrence and geometry are often the most challenging aspects of object-based simulation
Used in:
Also see:
Order Relations Correction#
A correction applied to the local distributions estimated by indicator kriging to ensure they satisfy the axioms of probability.
Order relations correction is required for either,
continuous features - correction of the local cumulative distribution function
categorical features - correction of the local probability mass function
Because each threshold or category is estimated independently, the resulting local probability model may not be valid.
For continuous features,
the estimated cumulative distribution function may not be monotonic increasing, resulting in negative probabilities over portions of the distribution
For categorical features,
the estimated category probabilities may not sum to one, violating probability closure
For continuous features, the order relations correction is,
a two-pass algorithm that constructs two monotonic cumulative distribution functions and averages them to obtain the corrected cumulative distribution function
For categorical features, the order relations correction is equivalent to the L1 normalization commonly used in machine learning feature engineering,
where the estimated probability for each category is divided by the sum of all estimated category probabilities so that the corrected probabilities sum to one.
Used in:
p-value#
A p-value is the probability of observing a test statistic at least as extreme as the calculated value, assuming the null hypothesis is true.
The p-value is calculated from the appropriate sampling distribution:
where:
\(T\) is the test statistic
\(t_{observed}\) is the observed test statistic
\(H_0\) is the null hypothesis
A small p-value indicates that the observed result would be unlikely if the null hypothesis were true.
A hypothesis test typically compares the p-value to the selected significance level:
if \(p \leq \alpha\), reject the null hypothesis
if \(p > \alpha\), fail to reject the null hypothesis
The p-value does not represent:
the probability that the null hypothesis is true
the probability that the observed result occurred by chance
Used in:
Also see:
Parameter#
A numerical quantity that describes a population or probability model.
Examples include,
population variance or standard deviation
distribution parameters, such as the mean and variance of a Gaussian distribution
variogram parameters, such as the nugget effect, range, and sill
Population parameters are generally unknown because the entire population is rarely observed. Instead, they are inferred from available sample statistics.
Used in:
TBD
Contrast with:
Percentile#
A percentile is a value that divides a statistical distribution such that a specified proportion of observations are less than or equal to that value.
The \(p\) percentile is calculated from the inverse cumulative distribution function:
where:
\(X_p\) is the value at percentile \(p\)
\(F_X^{-1}\) is the inverse cumulative distribution function
\(p\) is the cumulative probability, where:
For example:
\(X_{0.50}\) is the 50th percentile, also called the median
\(X_{0.90}\) is the 90th percentile, where 90% of values are less than or equal to this value
Percentiles are commonly used to describe distributions without assuming a particular statistical distribution.
Common percentile summaries include:
\(P10\) - 10th percentile
\(P50\) - 50th percentile or median
\(P90\) - 90th percentile
In uncertainty analysis and post-processing realizations, percentiles summarize the local distribution of simulated values:
where the percentile is calculated over an ensemble of realizationss at location \(\mathbf{u}_{\alpha}\).
Percentiles are useful because they:
quantify uncertainty without assuming Gaussian behavior
describe skewed and non-symmetric distributions
identify extreme or threshold values for decision making
Used in:
Also see:
Permeability#
Permeability is the ability of a porous medium to transmit fluids through interconnected pore space.
Permeability is commonly estimated from Darcy’s Law:
where:
\(q\) is volumetric flow rate
\(k\) is permeability
\(A\) is cross-sectional flow area
\(\mu\) is fluid viscosity
\(\frac{dP}{dL}\) is the pressure gradient
Permeability is commonly measured in darcies (D) or millidarcies (mD).
Unlike porosity, permeability depends strongly on pore connectivity, pore throat geometry, and fractures.
Used in:
Also see:
Polygonal Declustering#
A declustering method to assign weights to spatial samples based on local sampling density, such that the weighted statistics are likely more representative of the population. Data weights are assigned so that,
samples in densely sampled areas receive less weight
samples in sparsely sampled areas receive more weight
Polygonal declustering proceeds as follows:
Split up the area of interest with Voronoi polygons. These are constructed by intersected perpendicular bisectors between adjacent data points. The polygons group the area of interest by nearest data point
Assign weight to each datum proportional to the area of the associated Voronoi polygon
where \(w(\bf{u}_j)\) is the weight for the \(j\) data. Note, the sum of the weights is \(n\); therefore, \(w(\bf{u}_j)\) is nominal weight of 1.0, sample density if the data were equally spaced over the area of interest.
Here are some highlights for polygonal declustering,
polygonal declustering is sensitive to the boundaries of the area of interest; therefore, the weights assigned to the data near the boundary of the area of interest may change radically as the area of interest is expanded or contracted
polygonal declustering is the same as the Theissen polygon method for calculation of precipitation averages developed by Afred H. Thiessen in 1911, [Thi11]
Used in:
Also see:
Poisson Distribution#
The Poisson distribution is a discrete statistical distribution representing the number of independent events occurring within a fixed interval of space or time.
The distribution is defined by the parameter \(\lambda\), where:
The Poisson distribution is commonly used for count data, such as the number of events, occurrences, or objects within a defined area or interval.
Used in: TBA
Also see:
Population#
The complete set of values for a feature over the 2D area of interest or 3D volume of interest, represented at sufficient resolution to support decision making.
For example,
the exhaustive set of porosity values at every location within a reservoir
the exhaustive set of gold grades throughout an ore body
In practice, the entire population is rarely observed. Instead, a limited sample is collected and used to infer population parameters.
Used in:
Contrast with:
Porosity#
Porosity is the fraction of the total rock volume occupied by pore space available to store fluids.
Porosity is calculated as:
where:
\(\phi\) is porosity (fraction)
\(V_p\) is pore volume
\(V_b\) is bulk rock volume
Porosity ranges from 0 to 1 and is commonly reported as a percentage.
Porosity is a fundamental petrophysical property that controls the storage capacity of a reservoir but does not necessarily indicate its ability to transmit fluids.
Used in:
Also see:
Post-Processing Realizations#
Post-processing realizations is the analysis of an ensemble of realizations and scenarios to summarize spatial uncertainty and support decision making.
A stochastic simulation produces multiple equally probable models of the subsurface:
where:
\(Z^{(\ell)}(\mathbf{u}_{\alpha})\) is the simulated value for realization \(\ell\) at location \(\mathbf{u}_{\alpha}\)
\(\ell=1,\ldots,L\) is the realization index
\(\mathbf{u}_{\alpha}\) is a location vector in the model domain
Post-processing summarizes the local uncertainty distribution over the ensemble of realizations.
Common post-processing methods include:
E-type estimate - the E-type estimate is the local expected value calculated as the average over all realizations:
The E-type estimate provides a single representative model, but may smooth local variability and does not represent a single possible subsurface outcome.
Conditional variance - the local conditional variance measures uncertainty at location \(\mathbf{u}_{\alpha}\):
Higher conditional variance indicates greater uncertainty among the realizations.
Local percentiles - percentiles summarize the local uncertainty distribution from the ensemble.
The \(p\) percentile is:
where:
\(p\) is the cumulative probability
\(F^{-1}_{Z(\mathbf{u}_{\alpha})}\) is the inverse cumulative distribution function derived from the ensemble values
Common examples include:
\(P10\) - optimistic high case
\(P50\) - median case
\(P90\) - conservative low case
Probability of exceedance - the probability that a variable exceeds a threshold \(z_c\) is:
where \(I\) is an indicator function:
Post-processing is essential because individual realizations represent possible subsurface states, while summary models quantify uncertainty and support decisions.
Used in:
Also see:
Posterior#
In Bayes’ Theorem, the posterior probability represents updated knowledge or uncertainty about possible model assumptions, states, or parameters after incorporating new data. The posterior is calculated by combining the prior probability with the likelihood function describing the compatibility of observed data with possible models,
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.
Power Law Average#
A flexible family of averaging methods used to scale a feature from a smaller volume support, \(v\), to a larger support, \(V\), by calculating an effective value representative of the larger volume.
The power law average is,
where \(\omega\) is the averaging power.
Special cases include,
\(\omega = 1\) — arithmetic average
\(\omega = -1\) — harmonic average
\(\omega \rightarrow 0\) — geometric average (obtained in the limit)
The choice of averaging power depends on the physical process being modeled. For example, for permeability,
arithmetic averaging is appropriate for flow parallel to bedding
harmonic averaging is appropriate for flow perpendicular to bedding
near-geometric averaging is often appropriate for oblique flow directions
Used in:
PP Plot#
A probability-probability (PP) plot is a graphical method for comparing two statistical distributions by plotting corresponding cumulative probabilities against each other.
For two distributions \(X\) and \(Y\), the PP plot displays:
where:
\(F_X(x)\) is the cumulative distribution function of distribution \(X\) evaluated at value \(x\)
\(F_Y(x)\) is the cumulative distribution function of distribution \(Y\) evaluated at the same value \(x\)
The PP plot is created by plotting paired cumulative probabilities over the range of observed values:
If two distributions are identical:
and the PP plot follows the 45-degree line:
PP-plots are commonly used to:
compare an empirical cumulative distribution function with a modeled distribution
evaluate agreement between two probability models
identify differences in probability behavior
Common PP-plot interpretations:
deviations indicate differences in cumulative probability
systematic curvature indicates differences in distribution shape
deviations near the upper or lower ends indicate differences in tails or extreme probabilities
Compared with QQ plots:
PP plots compare cumulative probabilities using the forward CDF:
QQ plots compare corresponding values using the inverse CDF:
Used in:
Also see:
Prediction#
Estimate unknown or future sample values given assumptions about, or a model of, the population. For example,
given a model of the reservoir, predict the porosity, permeability, or production rate at a proposed well location before drilling
given historical production data, predict next month’s production rate
statistical or data-driven prediction uses inferred relationships from data, while physics-based prediction uses governing equations and physical models
Prediction is concerned with estimating unknown values, rather than inferring the parameterss of the underlying population.
Used in:
Compare with:
Prediction Interval#
An uncertainty range for a future prediction represented by lower and upper bounds based on a specified probability level, known as the confidence level.
For example, a 95% prediction interval may be communicated as,
given predictor feature values, \(X_1=x_1,\ldots,X_m=x_m\), there is a 95% probability that a future reservoir NTG observation will fall between 13% and 17%.
A prediction interval represents uncertainty in the next observation and integrates,
uncertainty in the estimated model prediction, \(\hat{Y}|X=x\)
irreducible variability or error in the response around the model prediction, \(Y-\hat{Y}\)
Therefore, prediction intervals are wider than confidence intervals for the estimated mean response because they include both model uncertainty and observation uncertainty.
The prediction interval depends on,
uncertainty in model parameters
uncertainty in the estimated conditional mean response
variability of observations around the conditional mean response
Used in:
Contrast with:
Predictor Feature#
A feature used as an input to predict a response feature in a predictive model.
A predictive machine learning model may be represented as,
where \(y\) is the response feature, \(x_1,\ldots,x_m\) are the predictor features, and \(\epsilon\) represents model error.
Additional comments,
predictor features are also commonly called input features or explanatory features
traditional statistical modeling often uses the term independent variable, although predictor feature is preferred because predictor features are not necessarily statistically independent
Used in:
Contrast with:
Primary Data#
Data samples of the feature being modeled.
the target feature for a geostatistical model
the response feature for a predictive machine learning model
Primary data are the observations that the model directly estimates or simulates.
For example,
porosity measurements from core are used to build a 3D geostatistical porosity model, supported by a 2D seismic acoustic impedance map. The core porosity measurements are the primary data because porosity is the feature being modeled.
Used in:
TBD
Contrast with:
Prior#
In Bayes’ Theorem, the prior probability represents knowledge or uncertainty about possible model assumptions, states, or parameters before considering new data. The prior is combined with the likelihood function describing new observations 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.
Probability Closure#
The normalization requirement of probability measures stating that the total probability over the entire sample space \(\Omega\) is equal to 1:
This property ensures that all possible outcomes collectively account for the entire probability mass.
Useful examples include:
Closure for complements:
Conditional complements:
Used in:
Probability Constraints#
The fundamental requirements for valid measures of probability include,
Boundedness, probabilities must be between zero and one,
Closure, the total probability over the entire sample space, \(\Omega\), is one,
Null set, the probability of the empty set is zero,
Additivity, the probability of mutually exclusive events is the sum of their individual probabilities,
These constraints are closely related to the Kolmogorov probability axioms.
Used in:
Probability Density Function#
A representation of a continuous statistical distribution with a density function, \(f(x)\), describing the relative density over the range of possible feature values, \(x\).
A univariate probability density function is denoted by,
and a bivariate probability density function is denoted by,
and extends to multivariate distributions.
For example, the Gaussian probability density function is specified as,
with probability density function,
parameterized by average, \(\mu\), and variance, \(\sigma^2\).
These are requirements for a valid probability density function,
non-negativity constraint, the density cannot be negative,
the density value may be greater than 1.0 because density is not probability
integrate density over a range of \(x\) to calculate probability,
probability closure, the total area under the PDF curve is equal to 1.0,
Nonparametric PDFs are commonly calculated with kernels (usually a small Gaussian distribution) that are summed over all data. Therefore, there is an implicit scale (smoothness) parameter when calculating a PDF.
too large of kernels will smooth out important information about the univariate distribution
too narrow a kernel will result in an overly noisy PDF that is difficult to interpret
This is analogous to the choice of bin size for a histogram or normalized histogram.
Parametric PDFs require model fitting to the data. The steps are,
Select a parametric distribution, e.g., Gaussian, lognormal, etc.
Calculate the parameters for the parametric distribution based on available data, using methods such as least squares or maximum likelihood.
It is very common to use the acronym PDF for probability density function.
Used in:
Contrast with:
Probability Mass Function#
A function that describes the probability distribution of a discrete feature. The probability mass function assigns a probability to each possible discrete outcome,
with the requirements,
and probability closure,
The normalized histogram of a discrete feature is an empirical probability mass function.
Contrast with:
Probability Operators#
A list of useful, common probability operators that are essential for working with probability and uncertainty problems.
Union of Events - the union of outcomes, the probability of \(A\) or \(B\) is calculated with the probability addition rule,
\(\quad\) where the intersection probability is subtracted to avoid double counting outcomes common to both events.
Intersection of Events - the intersection of outcomes, the probability of \(A\) and \(B\) is represented as,
\(\quad\) Under the assumption of independence of \(A\) and \(B\), the intersection probability can be calculated from the marginal probabilities,
\(\quad\) If there is dependence between \(A\) and \(B\), then conditional probability is required,
Complementary Events - the NOT operator for probability. If we define event \(A\), then the complement \(A^c\) represents all outcomes that are not \(A\).
\(\quad\) The resulting closure relationship is,
\(\quad\) Complementary events extend naturally to conditional probabilities, for example,
\(\quad\) Note, the conditioning event must remain the same.
Mutually Exclusive Events - events that do not intersect and have no common outcomes. Using set notation,
\(\quad\) and the joint probability is,
Used in:
Probability Perspectives#
The three primary perspectives for interpreting and calculating probability are:
Long-term frequencies - probability as the ratio of observed outcomes from repeated experiments. This perspective requires repeatable experiments and observations, and is the basis for frequentist probability.
Physical tendencies or propensities - probability based on knowledge of, or models for, the physical system. For example, the probability of a heads outcome from a coin toss can be known from the physical properties of the coin without performing repeated experiments.
Degrees of belief - probability representing our uncertainty about an outcome or proposition, allowing probabilities to be updated as new information becomes available. This perspective is the basis for Bayesian probability.
Used in:
Production Data#
Spatiotemporal subsurface engineering data including bottom hole pressure, fluid production rates, fluid composition, and temperatures.
Production data are important dynamic observations used to evaluate and calibrate subsurface models.
Some additional comments,
production from a single well may be commingled over multiple producing intervals unless production logging tool (PLT) data are available to allocate production by interval
production data provide important ground truth for matching reservoir model forecasts through the model calibration process known as history matching
As model outputs from a flow simulation transfer function, production data integration requires,
an inversion approach known as history matching, which is challenging because the system is ill-posed and the solution is generally nonunique
Used in:
TBD
Property#
Property is the physical, geological, engineering, or operational information measured or inferred from data.
Three basic aspects of spatial data include:
property or information
For example,
Core properties – Potential core properties include:
petrophysical properties (for example, mineralogy, porosity, permeability)
geomechanical properties (for example, compressive strength, P-wave and S-wave velocity)
stratigraphic and sedimentological properties (for example, facies, sedimentary structures, and depositional setting)
Well log properties – Potential well log properties include:
petrophysical properties (for example, gamma ray, resistivity, density, neutron porosity, mineralogy, and porosity)
stratigraphic and sedimentological properties (for example, facies, sedimentary structures, and depositional setting)
Seismic properties – Through seismic inversion, seismic data can provide inferred properties including:
petrophysical properties (for example, acoustic impedance, which is often negatively correlated with porosity)
stratigraphic and structural information (for example, depositional setting, major geological transitions, and faults)
Production data properties – Potential production data features include:
temporal trends
spatial regions and compartmentalization
reservoir connectivity and homogenization
drainage radius
flow heterogeneity
The value of a dataset depends not only on the number of measurements, but also on the property measured, the resolution, the coverage, and the uncertainty associated with the information.
Used in:
Also see:
Proportional Effect#
The proportional effect is the tendency for the uncertainty (variance or standard deviation) of a feature to increase as its mean or expected value increases. It is a common form of heteroscedasticity observed in many geological and engineering variables.
The proportional effect commonly results in positively skewed distributions because larger values have greater absolute variability. It is frequently observed for variables such as permeability, production rate, thickness, and ore grade.
For example, assuming a 10% relative uncertainty:
for a porosity estimate of 2%, the standard deviation of estimation error is 0.2%
for a porosity estimate of 20%, the standard deviation of estimation error is 2.0%
Recognition of the proportional effect is important when selecting estimation methods and uncertainty models, as prediction uncertainty should generally increase with the expected value.
Used in:
Also see:
Proportions#
The proportion of each possible category relative to the total number of observations. Proportions describe the categorical distribution of a feature and are equivalent to the probability of occurrence when observations are considered representative.
For a sample, the proportion of category \(k\) is,
where \(n_k\) is the number of observations in category \(k\) and \(n\) is the total number of observations.
The proportions satisfy probability closure,
where \(K\) is the total number of categories.
Proportions are,
the y-axis of a categorical feature probability density function
the marginal probability’s for a categorical feature
commonly used to summarize lithology, facies, rock type, mineral class, land use, or any categorical variable
Facies proportions are central to geostatistical modeling because facies often define stationary domains with distinct statistical and spatial characteristics. For example,
sand — high porosity and permeability occurring in relatively large connected bodies
shale — low porosity and permeability occurring in drapes and thin, laterally continuous beds
The determination of facies proportions is one of the most important modeling decisions because they control,
expected volumetrics of each facies
connectivity and geological architecture
flow simulation and production forecasts
uncertainty in downstream decision making
Facies proportions may be estimated from,
well or drill-hole observations
interpreted seismic data
geological analogs
conceptual geological models
Because available data are sparse, the proportions themselves are uncertain. This uncertainty is commonly represented with multiple scenarios by varying the global proportions within plausible limits and evaluating the impact on the resulting subsurface models.
Used in:
Also see:
QQ Plot#
A quantile-quantile (QQ) plot is a graphical method for comparing two statistical distributions by plotting corresponding quantiles against each other.
For two distributions \(X\) and \(Y\), the QQ plot displays:
where:
\(F_X^{-1}(p)\) is the inverse cumulative distribution function of distribution \(X\) evaluated at cumulative probability \(p\)
\(F_Y^{-1}(p)\) is the inverse cumulative distribution function of distribution \(Y\) evaluated at cumulative probability \(p\)
The QQ plot is created by plotting paired quantiles over a range of cumulative probability values:
If two distributions are similar, the QQ plot approximately follows a straight line:
where:
\(a\) represents a location shift
\(b\) represents a scale difference
For identical distributions:
and the QQ plot follows the 45-degree line:
QQ-plots are commonly used to:
compare a sample distribution with a theoretical distribution, such as a Gaussian distribution
identify departures from distributional assumptions
evaluate normality
Common QQ-plot interpretations:
shifts indicate differences in location or mean
curvature indicates differences in distribution shape
deviations in the tails indicate differences in extreme values
different slopes indicate differences in scale or variance
Used in:
Also see:
Contrast with:
Qualitative Feature#
Feature described by labels rather than numerical quantities. Qualitative features represent information that requires interpretation or classification and the values do not have inherent numerical meaning.
typically qualitative features cannot be directly measured from rock, but instead require interpretation steps
Examples of qualitative features include,
rock type = sandstone
facies = channel sand, levee, floodplain
zonation = bornite-chalcopyrite-gold higher grade copper zone
Qualitative features may be encoded numerically for analysis, but the numerical codes represent categories and do not imply magnitude or order. For example,
sandstone = 1 and shale = 2 are category labels, not measurements where shale is greater than sandstone.
In geostatistics and machine learning, qualitative features are commonly transformed using approaches such as indicator transform or one-hot encoding before modeling.
Used in: TBD
Contrast with:
Quantitative Feature#
A feature that can be measured and represented by numerical values with meaningful magnitude.
Examples of quantitative features include,
age = 10 Ma (millions of years)
porosity = 0.134 (fraction of volume is void space)
saturation = 80.5% (volume percentage)
Similar to qualitative feature, quantitative features often require interpretation. For example,
total porosity may be directly measured, but effective porosity may require geological interpretation or a petrophysical model.
Quantitative features may be continuous or discrete depending on whether the possible values are measured along a continuum or occur as countable values.
Used in: TBD
Contrast with:
R Squared#
Stated as coeficient of determination, “R-squared” or in mathatical notations as \(R^2\), a statistical measure of the proportion of the variability in a response feature that is explained by a regression model.
For ordinary least-squares linear regression with an intercept, the response feature variance can be partitioned into,
where,
\(\sigma^2_{tot}\) is the variance of the response feature, \(y_i\)
\(\sigma^2_{reg}\) is the variance explained by the regression model
\(\sigma^2_{res}\) is the residual variance, variance of prediction error
The coefficient of determination is therefore,
where,
\(R^2=0\) indicates the regression explains none of the response variability
\(R^2=1\) indicates the regression explains all of the response variability
For ordinary least-squares linear regression with a single predictor feature,
where \(\rho_{x,y}\) is the correlation coefficient between the predictor and response.
Some additional comments,
for many nonlinear regression models an \(R^2\) statistic can still be calculated, but it should be interpreted as a relative reduction in squared prediction error rather than a strict variance decomposition
\(R^2\) does not indicate whether predictions are unbiased or whether the model generalizes well to unseen data; prediction error metrics such as mean square error, root mean square error, or mean absolute error are required to evaluate predictive performance.
Used in:
Also see:
Random Function#
A set of random variables correlated over space or time. In geostatistics, a random function provides the mathematical framework for representing spatial uncertainty and variability.
The key concept introduced by Matheron is that a geological phenomenon is viewed as a realization of an underlying random function. The observed spatial data represent one possible outcome from this random process.
Important points on random function nomenclature,
random variables are denoted with upper-case, e.g., \(X\)
random functions are denoted with upper-case with location vectors, e.g.,
joint outcomes called realizations, or data samples are represented with lower case, e.g.,
realizations with the \(\ell\) notation, e.g.,
for \(\ell = 1,\ldots,L\) realizations.
Used in:
Random Sampling#
A representative spatial sampling method from a:
population – sample for statistical inference
data set – bootstrap sample generated by sampling with replacement
model – stochastic realization
Random sampling requires that:
every possible sample of the specified size is equally likely to be selected
each selection is made randomly and is not influenced by previous selections or outcomes
This minimizes selection bias and supports valid statistical inference.
Used in:
Also see:
Random Variable#
A mathematical representation of uncertainty where the value of a feature is unknown and can take a range of possible outcomes described by a statistical distribution, probability density function, or cumulative distribution function.
A random variable is denoted with upper-case notation, e.g., \(X\), while possible outcomes or observed values are represented with lower-case notation, e.g., \(x_{\alpha}\) or realization \(x^{\ell}\).
For spatial phenomena, a location vector, \(\mathbf{u}\), is added to represent the random variable at a specific location,
spatial random variable:
spatial data measure:
spatial realization:
The collection of correlated spatial random variables over many locations forms a random function.
Used in:
Range#
There are two common uses for the term range:
statistical range as a measure of dispersion for a distribution
variogram range as a parameter describing spatial continuity
These two uses describe different concepts and should be interpreted from context.
The statistical range is a measure of dispersion (spread) of a univariate distribution. It is calculated as:
or using the cumulative distribution function:
where \(F_X^{-1}\) is the inverse cumulative distribution function (quantile function) for feature \(X\).
The variogram range is the lag distance where the experimental variogram approximately reaches the sill, indicating the distance beyond which there is no additional modeled spatial continuity.
Important aspects of variogram range include:
for lag distances less than the range, nearby samples provide information about each other through spatial continuity.
for lag distances at and beyond the range, the variogram model indicates no additional spatial continuity.
range is a variogram model parameter used to define permissible, positive definite variogram models.
for nested variogram models, each spatial structure may have its own range.
when the range varies by direction, this is called geometric anisotropy.
Used in:
Realization#
An outcome from a random variable or a joint outcome from a random function.
an outcome from a random variable, \(X\), or a joint set of outcomes from a random function
represented with lower case notation, e.g., \(x\)
for spatial settings it is common to include a location vector, \(\mathbf{u}\), to describe the location, e.g., \(x(\mathbf{u})\), corresponding to the random variable \(X(\mathbf{u})\)
generated by simulation methods, e.g., Monte Carlo simulation, sequential Gaussian simulation, or any other method that samples jointly from a random function
in general, stochastic simulation assumes realizations are equiprobable, meaning each realization is considered an equally likely outcome of the modeled uncertainty
Used in:
Realizations#
A realization ensemble of spatial models generated by stochastic simulation by holding input parameters and model choices constant while changing only the random number seed.
realizations explore the spatial offset from data component of uncertainty modeling
A realizations represents spatial uncertainty by sampling multiple possible outcomes from the same random function.
For example,
hold the porosity average, variogram model, conditioning data, and simulation parameters constant
generate multiple porosity models by changing only the random number seed
differences between the realizations represent spatial uncertainty in porosity away from conditioning data
Used in:
Sequential Gaussian Simulation Description and Demonstration
Sequential Indicator Simulation Description and Demonstration
Contrast with:
Regular Sampling#
A representative spatial sampling method from a:
population – observations collected at fixed intervals
data set – every kth observation
model – values extracted on a regular grid of predictor features
Regular sampling:
collects samples at fixed intervals in space, time, or sequence
provides uniform coverage of the population or domain
may introduce bias if the sampling interval aligns with periodic patterns in the population, data set, or model
Used in:
Also see:
Representative Spatial Sampling#
The sample and resulting sample statistics are representative of the population, by sampling theory we have 2 options:
Random sampling - each potential sample from the population is equally likely to be sampled as samples are collected. This includes,
selecting a specific location has no impact on the selection of subsequent locations.
assumption that the population size that is much larger than the sample size; therefore, significant correlation between samples is not imposed due to without replacement sampling (the constraint that you can only sample a location once). Note, generally this is not an issue for the subsurface due to the sparsely sampled massive populations
Regular sampling - sampling at equal space or time intervals. While random sampling is prefered, regular sampling is robust as long as,
the regular sampling intervals do not align with natural periodicity in the data, e.g., the crests are systemally sampling resulting in biased high sample statistics
Used in:
Contrast with:
Residual#
A residual is the difference between an observed value and a model prediction or estimate.
The residual is calculated as:
where:
\(r_i\) is the residual for observation \(i\)
\(y_i\) is the observed value
\(\hat{y}_i\) is the predicted or estimated value
Residuals represent the portion of the observed data not explained by the model.
For a well-performing model:
residuals should have approximately zero mean
residuals should not show systematic patterns
residual variability should be consistent with the assumed uncertainty model
Residual analysis is used to evaluate:
model assumptions
unexplained variability
spatial continuity of remaining variation
In spatial modeling, residuals may be analyzed using:
residual histograms
residual variograms
residual maps
The relationship between error and residual is:
error is the difference from the unknown true value:
residual is the measurable difference between observed and predicted values:
Residuals are commonly used in:
model validation
Also see:
Resolution#
Resolution is the smallest spatial scale at which a measurement or model can distinguish meaningful variation.
achievable resolution is limited by the physics of the measurement system, while the selected model resolution is often a practical choice based on modeling objectives, data availability, computational resources, and storage constraints.
volume support describes the physical volume represented by an individual measurement or model value. Resolution and volume support are related, but resolution may be limited by measurement physics even when smaller volume supports are selected.
Three basic aspects of spatial data include:
Examples of data resolution include,
core – scanned core provides geological information, including sedimentary structures, textures, fractures, and mineralogy, at camera resolutions of approximately 100–250 μm/pixel.
well log – conventional wireline gamma ray measurements are vertically averaged over approximately 30–60 cm (1–2 ft) along the borehole.
seismic – deepwater Gulf of Mexico inverted seismic acoustic impedance volumes typically have a dominant frequency of 20–50 Hz and P-wave velocities of 2,500–4,000 m/s, giving a quarter-wavelength vertical resolution of approximately 10–25 m. Model-based inversion constrained by well logs estimates acoustic impedance with an effective vertical resolution of approximately 5–15 m.
An example of model resolution includes,
Based on the modeling objectives and available computational and storage resources, a subsurface model is discretized into 50,000,000 cells (1,000 × 1,000 × 50). To represent the reservoir, each model cell has dimensions of 100 × 100 × 2 m, which defines the model resolution.
Resolution spans many orders of magnitude in subsurface characterization. For example, core images may resolve features on the order of 100 μm, well logs average over 30–60 cm, while inverted seismic resolves features on the order of 5–15 m. Successful subsurface modeling integrates information across these multiple scales.
There is often a resolution–coverage trade-off with spatial data:
High-resolution data sources tend to have limited coverage – core data provides measurements at micrometer to millimeter scales, but only samples a very small fraction of the reservoir volume. A core may represent only approximately one ten-billionth to one hundred-billionth of the reservoir volume.
High-coverage data sources tend to have lower resolution – remote sensing and seismic data may cover the entire reservoir or field area, but typically represent geological properties at resolutions of tens to hundreds of meters.
Used in:
Also see:
Response Feature#
The output or target feature for a predictive machine learning model. A predictive machine learning model can be generalized as,
where the response feature is \(y\), the predictor features are \(x_1,\ldots,x_m\), and \(\epsilon\) represents model error or unexplained variability.
The response feature is the quantity that the predictive model attempts to estimate or predict.
traditional statistical modeling uses the term “dependent variable” instead of response feature
Used in:
Contrast with:
Rock Quality Index#
The Rock Quality Index (RQI) is a petrophysical measure of rock quality that combines permeability and porosity to quantify the effectiveness of pore networks for fluid flow.
The Rock Quality Index is calculated as:
where:
\(k\) is permeability (mD)
\(\phi\) is porosity (fraction)
0.0314 is a unit conversion factor when permeability is in millidarcies
Higher Rock Quality Index values indicate greater pore connectivity and improved fluid flow capacity.
The Rock Quality Index is commonly used to:
compare rock quality between reservoirs or geological facies
identify hydraulic flow units
calculate the Flow Zone Indicator (FZI)
The Rock Quality Index normalizes permeability by porosity, reducing the influence of pore volume alone and emphasizing pore connectivity.
Used in: TBA
Also see:
Sample#
A subset of values and locations measured from a population and used to infer parameter(s) of the population.
Examples include,
sparse spatial samples - 1,000 porosity measures from well log data in a reservoir with high measurement precision, but very small volume support and limited spatial coverage.
dense spatial samples - 1,000,000 acoustic impedance measurements over a 1,000 x 1,000 2D grid for a reservoir unit of interest with lower measurement precision and larger volume support.
In spatial modeling, the information content of a sample depends not only on the number of measurements, but also on the spatial distribution, measurement precision, and volume support.
also the amount of information in the sample set may be related to data locations and spatial continuity, i.e., the degree of redundancy between the samples.
Used in:
Contrast with:
Sampling Distribution#
A sampling distribution is the probability distribution of a statistic calculated from repeated samples drawn from the same population.
For example, the sampling distribution of the sample mean describes the expected variation in:
over many possible samples of size \(n\) from a population.
The sampling distribution depends on:
the population distribution
the sample size
the statistic being calculated
Sampling distributions are fundamental to statistical inference because they describe the expected variability due to random sampling.
The standard error is the standard deviation of a sampling distribution, for example,
where \(\sigma_{\bar{X}}\) is the standard deviation of the sampling distribution of the sample mean.
The Central Limit Theorem states that the sampling distribution of the sample mean approaches a Gaussian distribution as sample size increases, even when the original population distribution is not Gaussian.
Used in:
Also see:
Saturation#
Saturation is the fraction of pore volume occupied by a specific fluid phase.
Fluid saturation is calculated as:
where:
\(S_i\) is the saturation of fluid phase \(i\)
\(V_i\) is the volume of fluid phase \(i\)
\(V_p\) is pore volume
Common fluid saturations include:
water saturation, \(S_w\)
oil saturation, \(S_o\)
gas saturation, \(S_g\)
The saturations sum to one:
Saturation is a fundamental reservoir property used to estimate hydrocarbon volumes, fluid flow, and production performance.
Used in:
Also see:
Scatter Plot#
A graph that displays paired observations of two features as points to visualize their relationship, dependence, trends, clusters, and outliers.
For paired samples of two features, each observation is represented as a point,
where one feature is plotted on the x-axis and the other feature is plotted on the y-axis.
Scatter plots are used to visually assess,
relationship and dependence between features
trends and nonlinear patterns
clusters and populations within the data
outliers and anomalous observations
Note that association observed in a scatter plot does not necessarily imply causation.
Used in:
Scenarios#
Multiple subsurface models calculated by changing input parameters, assumptions, or modeling choices to represent uncertainty due to incomplete knowledge of the system.
Examples of scenario uncertainty include,
changing the input feature distributions, for example, modeling low, mid, and high porosity mean scenarios and generating subsurface models from each distribution
changing geological interpretations, such as alternative facies proportions, structural interpretations, or depositional models
changing model parameters, such as variogram parameters, trend models, or spatial continuity assumptions
Each scenario may include an ensemble of realizations generated by varying the random number seed in stochastic simulation to represent spatial uncertainty within that scenario.
Used in:
Contrast with:
Score#
A score is a standardized measure that quantifies how far an observation or estimate lies from a reference value relative to a measure of uncertainty or variability.
Practical rules of thumb to determine a small sample size:
for estimating a population mean, a sample size of approximately \(n < 30\) is often considered a small sample, and Student’s t-score is typically preferred over the z-score when the population standard deviation is unknown.
for estimating a population proportion, the normal approximation is generally appropriate only when both,
and
where \(p\) is the population proportion. Otherwise, exact or small-sample methods are preferred.
Scores are generally calculated as:
where the measure of variability depends on the application.
Common examples include:
Scores are commonly used to:
compare observations from different populations or distributions
calculate probabilities and quantiles
construct confidence intervals
perform statistical hypothesis tests
Also see:
Secondary Data#
Data samples of a feature other than the feature being modeled, used to improve estimation or simulation of the primary feature.
Secondary data are integrated through a model of the relationship between the secondary and primary data.
For example,
acoustic impedance measurements from seismic data (secondary data) are used to support calculation of a 3D porosity model, where porosity is the feature of interest
porosity measurements (secondary data) are used to support calculation of a permeability model, where permeability is the feature of interest
Secondary data may provide additional spatial information, trends, or constraints, but are not the direct observations of the feature being modeled.
Used in:
TBD
Contrast with:
Seismic#
A geophysical measurement technique that uses controlled acoustic sources and receivers to measure subsurface reflections and infer geological structure and rock properties.
Reflection seismic data provide high spatial coverage but generally lower resolution compared with direct measurements such as well log and core data.
Some important details include,
seismic reflection amplitudes are processed and inverted to estimate rock properties, such as acoustic impedance, calibrated and positionally aligned with well sonic logs
seismic provides a geological framework by identifying bounding surfaces, structural features, and reservoir extents
seismic provides soft information for reservoir properties, such as porosity and facies, through relationships established between seismic attributes and available primary data
In geostatistical modeling, seismic is commonly used as secondary data to improve spatial prediction and uncertainty models.
Used in:
TBD
Sequential Gaussian Simulation#
A stochastic simulation method used to calculate equiprobable spatial model realizations by sequentially sampling from local conditional uncertainty distributions.
The name describes the three fundamental principles,
Sequential - previously simulated values are sequentially added to the conditioning data set so that the simulated realization honors the modeled spatial covariance.
Gaussian - data are transformed to Gaussian space so that local uncertainty distributions can be calculated using the kriging mean and kriging variance under the Gaussian random function assumption. After simulation, values are back-transformed to reproduce the original feature distribution.
Simulation - local conditional distributions are sampled using Monte Carlo simulation to add the missing spatial variability and calculate multiple equiprobable realizations. The random seed controls both the local random draws and the sequential simulation path.
The complete sequential Gaussian simulation workflow is,
Establish the simulation grid and coordinate system, including geological framework transformations such as flattening folds and restoring faults.
Assign available data to the simulation grid, accounting for scale changes between data support and grid cell support.
Transform the data to Gaussian space using Gaussian anamorphosis.
Calculate and model the variogram of the Gaussian transformed data.
Establish a random simulation path through all grid nodes. At each node,
identify nearby conditioning data and previously simulated grid nodes
calculate the local conditional distribution using kriging, where the mean is the kriging estimate and the variance is the kriging variance
Monte Carlo simulate a realization from the local conditional distribution
add the simulated value to the conditioning data set for subsequent simulation steps
Check the realization in Gaussian space. The realization should honor,
conditioning data at sampled locations
the standard normal distribution, \(N[0,1]\), with mean zero and variance one
the modeled variogram
Back-transform simulated values from Gaussian space to the original feature distribution.
Restore the original geological framework, including folds and faults.
Check that the realization honors,
geological concepts
geophysical data
historical production data
Repeat steps 5 through 9 to calculate multiple realizations.
The critical algorithmic steps of sequential Gaussian simulation are,
Transform data to Gaussian space with mean 0.0 and variance 1.0 (standard normal).
Apply a random simulation path through the grid. At each grid node,
use kriging to calculate the local conditional distribution
Monte Carlo simulate a local realization
add the simulated value to the conditioning data
Back-transform the simulated values to the original feature distribution.
Used in:
Also see:
Sequential Indicator Simulation#
A stochastic simulation method to calculate continuous feature or categorical feature realizations for spatial models based on the following principles,
Sequential - previously simulated values are sequentially added to the conditioning data set so that the simulated realization honors the modeled spatial covariance.
Indicator - indicator transform is applied to the data over a set of thresholds for continuous features or categories for categorical features. This allows flexible probability encoding of hard and soft data and direct estimation of local cumulative distribution functions or categorical probability distributions.
Simulation - Monte Carlo simulation is applied to the local distributions to add the missing variance and construct multiple, equiprobable realizations. The random seed determines the individual Monte Carlo simulations and the random path for the sequential simulation.
The sequential indicator simulation workflow is,
Establish the simulation grid and coordinate system, including geological framework transformations such as flattening folds and restoring faults.
Assign available data to the simulation grid, accounting for scale changes between data support and grid cell support.
Apply the indicator transform to all data for all thresholds or categories, \(k = 1,\ldots,K\).
Calculate and model the indicator variogram for each threshold or category, \(k = 1,\ldots,K\).
Establish a random simulation path through all grid nodes. At each node,
identify nearby conditioning data and previously simulated grid nodes
for each threshold or category, calculate the local conditional probability distribution using indicator kriging
apply order relations correction to ensure the local cumulative distribution function or probability distribution is valid
Monte Carlo simulate a realization from the local conditional distribution
add the simulated value to the conditioning data set and apply the indicator transform for subsequent simulation steps
Check the realization. Does it honor,
conditioning data at sampled locations?
the global histogram or categorical feature proportions?
the modeled indicator variograms?
Restore the original geological framework, including adding back folds and faults.
Check that the realization honors,
geological concepts
geophysical data
historical production data
Repeat steps 5 through 8 to calculate multiple realizations.
Used in:
Also see:
Significance Level#
The significance level is the probability threshold used in hypothesis testing to determine whether to reject the null hypothesis.
The significance level is also called the alpha level:
The significance level also determines the corresponding confidence level for confidence intervals.
Common choices include:
\(\alpha=0.10\) for 90% confidence
\(\alpha=0.05\) for 95% confidence
\(\alpha=0.01\) for 99% confidence
The significance level defines the probability of incorrectly rejecting a true null hypothesis:
This error is known as a type I error.
The significance level is selected before performing the hypothesis test and represents the acceptable false positive risk.
Selecting a significance level requires balancing sensitivity and false positive risk:
too small — the test becomes conservative, requiring very strong evidence before identifying an effect; real effects may be missed
too large — the test becomes permissive, identifying more outcomes as unusual but increasing false positive risk
The fundamental question is:
How rare must an event be before we consider it evidence against our assumption?
How often should our process generate a red alert when the process is actually operating normally?
Used in:
Also see:
Sill#
For a stationary random function, the theoretical variogram sill is equal to the variance of the feature of interest.
The sill provides the reference for interpreting spatial continuity,
at lag distances where the experimental variogram reaches the sill, spatial correlation approaches zero. This lag distance is called the range.
experimental variogram values above the sill may indicate negative covariance function, but are also commonly caused by sampling variability, insufficient data, or non-stationarity. In practice, variogram models are generally constrained to approach the sill and assume no spatial correlation beyond the range.
an experimental variogram that continues to increase approximately linearly beyond the sill may indicate a spatial trend or other violation of stationarity assumptions.
For a feature with variance, \(\sigma^2\), the variogram sill is,
as the lag distance increases beyond the range.
Used in:
Simulation#
A stochastic process of obtaining one or more possible values of a feature at unsampled locations that are consistent with available data and a spatial uncertainty model.
Simulation models are designed to reproduce global characteristics and spatial variability, known as Global accuracy, where the model reproduces specified global measures, including,
feature univariate distributions, including proportions, mean, variance, and the complete cumulative distribution function
feature spatial models, including the variogram, training image, or geological object geometries
Unlike estimation methods, simulation produces multiple equiprobable realizations that represent uncertainty rather than a single optimal prediction.
Examples of simulation models include,
subsurface heterogeneity models, including sequential Gaussian simulation, sequential indicator simulation, multiple point simulation, and object-based simulation
uncertainty propagation through a transfer function, including Monte Carlo simulation
Use simulation when,
reproducing feature distributions is important, especially when extreme values influence decisions
realistic spatial models are required for applications such as flow simulation
uncertainty in the decision criteria must be quantified through multiple possible models
Used in:
Sequential Gaussian Simulation Description and Demonstration
Sequential Indicator Simulation Description and Demonstration
Contrast with:
Simulation Post-processing#
A variety of operations to calculate statistical summaries over multiple realizations and scenarios. At each location in the model, \(\bf{u}_{\alpha}\), we pool the local realizations, \(\ell = 1, \ldots, L\),
and then calculate the nonparametric local cumulative distribution function as,
Here are common summaries and how they are used,
e-type - is the local expectation (since each realization is assumed to be equiprobable, this is the same as the local average) over the realizations,
often calculated to visualize the trends after integration of all information sources
Conditional standard deviation - is the local standard deviation over the realizations,
often calculated to visualize the level of local uncertainty
Local percentile - is the local percentile over the realizations,
often calculated to show maps of local upper and lower bounds, for example, local P10 and P90 models
Local probability of exceedance - the probability of exceeding a specified threshold, \(z_k\), calculated as one minus the cumulative probability of the threshold at each location over the realizations,
often calculated to communicate risk, for example, the probability of locally exceeding an environmental threshold for a contaminant
A sufficient number of realizations, \(L\), is required to calculate reliable summaries,
the expected value is the easiest to calculate and is generally reliable with 20 or more realizations
percentiles on the tails, e.g., P10 and P90, require 100 or more realizations for reliable results
More realizations generally result in more reliable simulation post-processing and improved decision support.
Used in:
Skewness#
Skewness is a univariate parameter or univariate statistic that measures the asymmetry of the probability distribution of a feature or variable.
A common measure of sample skewness is:
Interpretation:
positive skewness indicates a longer upper tail with occasional large values
negative skewness indicates a longer lower tail with occasional small values
zero skewness indicates a symmetric distribution
Skewness is important for subsurface variables because many geological and engineering properties are not symmetric and may contain extreme values.
Also see:
Soft Data#
Data with significant uncertainty such that the information is represented probabilistically and uncertainty must be integrated into the spatial model.
For example,
a local porosity probability density function calibrated from acoustic impedance measurements
a probability of facies’s occurrence interpreted from seismic data
Soft data integration requires workflows that incorporate uncertainty in the conditioning information, such as,
p-field simulation
workflows that randomize or transform soft information into data realizations compatible with simulation methods that traditionally assume hard data, such as sequential Gaussian simulation
Soft data integration is an advanced topic and an active area of research; however, many standard subsurface modeling workflows and commercial software packages include approaches for incorporating soft information.
Used in:
TBD
Contrast with:
Spatial Estimation#
The process of obtaining a single best value to represent a feature at an unsampled location or time, \(\bf{u}\).
Given spatial data, \(z(\bf{u}_1), \dots, z(\bf{u}_n)\), we estimate the unknown feature value at location \(\bf{u}\) with a linear combination of the available data,
An unbiasedness constraint may be added by assigning the remainder of the weight (one minus the sum of weights) to the global average. Therefore, if no informative data are available, the estimate approaches the global average of the feature,
Some additional concepts,
local accuracy takes precedence over global accuracy, meaning that spatial estimation methods prioritize matching nearby observations over reproducing global statistics such as the histogram and variogram
spatial estimation maps and models generally have reduced variance and increased spatial continuity, resulting in smoother models than the true heterogeneous feature distribution
estimation models are not appropriate for transfer functions that are sensitive to heterogeneity and feature distributions, such as flow simulation or economic optimization
spatial estimation produces a single deterministic model and therefore does not provide multiple realizations required to sample uncertainty in the decision criteria; simulation methods are required for comprehensive uncertainty modeling and decision support
Examples of spatial estimation methods include,
inverse distance
There are also general non-spatial estimation methods; for example, many predictive machine learning models perform estimation by focusing on local predictive accuracy rather than global distribution reproduction,
k-nearest neighbors
decision tree
random forest
Used in:
Contrast with:
Spatial Continuity#
Also known as spatial correlation, is the correlation of a feature over distance.
Spatial continuity can be calculated from a variogram. The correlogram, \(\rho(\bf{h})\), is related to the variogram, \(\gamma(\bf{h})\),
where \(\rho(\bf{h})\) is the spatial correlation at lag vector \(\bf{h}\), and \(\sigma^2\) is the feature variance, the variogram sill.
the greater the difference between the variogram value and the sill, the greater the spatial continuity
at the sill, the variogram indicates no correlation over that separation distance and the correlogram approaches zero
Spatial continuity may be considered,
between sample data during calculation of an experimental variogram
while inferring a variogram model, including selecting nested structures, contributions, and ranges
between simulated values during construction of a spatial simulation model
How do we interpret spatial continuity?
no spatial continuity - values are uncorrelated over distance, meaning knowing a value at one location provides no information about the value at another location
homogeneous phenomena - values are identical or nearly identical over space and therefore exhibit perfect conceptual continuity over all distances; note that a perfectly homogeneous feature has zero variance, so the correlogram formulation is not defined
strong spatial continuity - nearby values are highly correlated, and the variogram increases slowly with lag distance
weak spatial continuity - nearby values have limited correlation, and the variogram approaches the sill over short distances
Used in:
Standard Deviation#
The square root of the variance. Standard deviation measures the spread of a feature about its average in the same units as the original feature.
Given the sample variance,
the sample standard deviation is,
The equivalent population parameters are, the population variance,
the population standard deviation is,
Used in:
TBD
Also see:
Standard Error#
The standard error (SE) is the estimated standard deviation of a sample statistic, most commonly the sample mean. It quantifies the uncertainty associated with estimating a population parameter from a sample.
For the sample mean, the standard error is:
where:
\(s\) is the sample standard deviation
\(n\) is the sample size
As sample size increases, the standard error decreases, indicating greater confidence in the estimated population parameter.
Standard error should not be confused with standard deviation, which measures the variability of the data rather than the uncertainty of a statistic estimated from the data.
Used in:
Also see:
Stationarity#
The decision that a subset of the subsurface is the same “stuff” and therefore can be pooled to calculate statistics and build models.
Replicates are required to calculate any statistic. In many applications, replicates are obtained by repeated measurements through time, for example,
air or water samples collected repeatedly from a monitoring station
For subsurface resource models,
repeated samples are generally not available at the same location; only one sample is available at each location
instead of pooling measurements through time, we must pool samples over space to calculate statistics
Why must we pool data? Ultimately, it is required to make inference about the population from a limited sample,
to calculate statistics
to build spatial models
The choice of stationary domain is an expert geological decision. Without a stationarity decision, we are restricted to the measured locations (well bores or drill holes) and cannot calculate statistics or make predictions between samples.
An example geological definition of stationarity could be:
The rock within the stationary domain is sourced, deposited, preserved, and post-depositionally altered in a similar manner. The domain is mappable and may be used for local prediction or as information for analogous locations within the subsurface; therefore, information may be pooled over this expert-defined volume of the subsurface.
This expert geological interpretation defines a domain over which statistical stationarity is assumed for modeling.
There are two aspects of any stationarity decision:
Import license - the choice of which samples are allowed to contribute to the calculation of a statistic
Export license - the choice of where the resulting statistic is applicable within the subsurface
To state a stationarity decision, we must specify:
the statistic assumed stationary, for example, the mean, variance, cumulative distribution function , or spatial continuity
the spatial domain over which the statistic is assumed stationary, for example, the entire model, a facies, a depositional environment, or a geological region
Examples of statistical definitions of stationarity include:
stationary mean
stationary cumulative distribution function
stationary semivariogram
The stationarity decision may be extended to any statistic of interest, including,
bivariate distributions
multiple point statistics
Additional considerations for stationarity include:
Stationarity is a decision, not a hypothesis - therefore, it is not directly tested. Instead, data may demonstrate that a chosen stationarity decision is inappropriate.
Stationarity depends on scale - the appropriate modeling scale should be selected based on the geological process, decision objective, and project requirements.
A stationarity decision cannot be avoided - without stationarity, spatial statistics cannot be calculated and modeling cannot progress beyond measured locations. Conversely, assuming broad stationarity over very large regions of the Earth is generally unrealistic.
Geomodeling stationarity is a domain decision - defining (1) where data may be pooled (import license) and (2) where resulting statistics may be applied (export license).
Nonstationary trends may be modeled explicitly - deterministic trends can be removed and the remaining stationary residual variation can be modeled stochastically. This is the hybrid modeling approach.
Used in:
Statistic#
A function of sample data that summarizes a property of the sample. Examples include,
sample mean - \(\overline{x}\)
sample standard deviation - \(s\)
sample variance - \(s^2\)
cumulative distribution function - \(F_x(x)\)
experimental variogram - \(\gamma(\mathbf{h})\)
Statistics are calculated from available samples because the complete population is generally unknown.
How do we use statistics?
Inference - calculate statistics from a sample and use them to estimate unknown population parameters
Prediction - use a model of population parameters and relationships to predict future observations or outcomes
Used in:
Compare with:
Statistical Distribution#
A description of the frequency or probability behavior of a feature over the range of possible values.
A univariate statistical distribution describes how feature values are distributed without considering their spatial or temporal arrangement. We represent the statistical distribution with,
What do we learn from a statistical distribution? For example,
what are the minimum and maximum values?
what is the most common range of values?
do we have many low values?
do we have many high values?
are there outliers or values that do not make geological or physical sense and require explanation?
what is the variability and uncertainty in the feature values?
Statistical distributions are fundamental for inference, simulation, and uncertainty modeling.
Used in:
Statistical Power#
Statistical power is the probability that a hypothesis test correctly rejects the null hypothesis when a real effect exists.
Statistical power is defined as:
where:
\(\beta\) is the probability of a Type II Error
Higher statistical power increases the probability of detecting meaningful effects.
Statistical power increases with:
larger sample size
lower variability
larger effect size
appropriate selection of significance level
Low statistical power may result in failing to detect real differences or relationships.
Used in:
Also see:
Statistics#
The theory and practice for collecting, organizing, and interpreting data, as well as drawing conclusions and making decisions.
Used in: Entire book
Same as:
Stochastic Model#
A model of a system or process that includes uncertainty and is represented by multiple possible outcomes, including realizations and scenarios, constrained by available data, statistics, and modeling assumptions.
Stochastic models represent uncertainty by describing a range of plausible outcomes rather than a single deterministic prediction.
Examples include,
data-driven models that integrate uncertainty, such as geostatistical simulation models
Monte Carlo models that propagate uncertainty through a transfer function
ensemble machine learning models that represent prediction uncertainty
Advantages:
computational speed compared with many physics-based models
explicit uncertainty assessment
ability to report confidence intervals, prediction intervals, and risk measures
ability to integrate many sources of data and information
flexible data-driven approaches
Disadvantages:
limited representation of underlying physics unless explicitly incorporated
dependence on statistical model assumptions and simplifications
uncertainty models may be incomplete if important processes or information are not represented
Used in:
Contrast with:
Student’s t Distribution#
Student’s t distribution is a continuous statistical distribution used for inference when estimating a population mean with unknown population standard deviation.
The t distribution is defined by its degrees of freedom, \(\nu\).
A t-score is calculated as:
where:
\(\bar{x}\) is the sample mean
\(\mu_0\) is the hypothesized population mean
\(s\) is the sample standard deviation
\(n\) is the number of samples
The t distribution accounts for additional uncertainty from estimating the population standard deviation using the sample standard deviation.
Important characteristics:
symmetric around zero
heavier tails than the Gaussian distribution
approaches the Gaussian distribution as degrees of freedom increase
The t distribution is commonly used for:
small sample inference
confidence intervals
hypothesis tests involving sample means
Used in:
Also see:
Subsurface Modeling Workflow#
A common geostatistical workflow for integrating subsurface data, modeling uncertainty, and supporting development decision making. The workflow proceeds from data to decisions through the following steps:
Integrate all available information to build multiple subsurface scenarios and realizations that sample the uncertainty space.
Apply all realizations through the transfer function to sample the uncertainty in the decision criteria.
Assemble the distribution of the decision criteria from the ensemble of realizations and scenarios.
Make the optimum reservoir development decisions while accounting for the modeled uncertainty.
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