Geostatistics Glossary T-Z#

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 DOI

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 DOI

By Michael J. Pyrcz
© Copyright 2024.

t-critical#

A t-critical value is a quantile from the Student’s t distribution used to construct confidence intervals when the population standard deviation is unknown and estimated from the sample.

The t-critical value is calculated from the inverse cumulative distribution function (inverse CDF):

\[ t_{\alpha/2,\nu}=T_{\nu}^{-1}\left(1-\frac{\alpha}{2}\right) \]

where:

  • \(\alpha\) is the significance level

  • \(\nu\) is the degrees of freedom

  • \(T_{\nu}^{-1}\) is the inverse CDF of the Student’s t distribution

For a confidence interval:

\[ \bar{x}\pm t_{\alpha/2,\nu}SE \]

where \(SE\) is the standard error.

The t-critical value depends on:

  • the desired confidence level

  • the degrees of freedom, typically related to sample size

For increasing sample size, the t-distribution approaches the Gaussian distribution, and:

\[ t_{critical}\rightarrow z_{critical} \]

A t-critical value is a property of the selected confidence level and degrees of freedom, not the observed sample.

Used in:

Also see:

t-score#

The t-score measures how many estimated standard errors an observation or statistic lies above or below the estimated mean. Unlike the z-score, the t-score accounts for uncertainty in estimating the population standard deviation from a sample.

The t-score is calculated as:

\[ t=\frac{x-\mu}{SE} \]

or, for testing a sample mean,

\[ t=\frac{\bar{x}-\mu}{s/\sqrt{n}} \]

where:

  • \(\bar{x}\) is the sample mean

  • \(\mu\) is the hypothesized population mean

  • \(s\) is the sample standard deviation

  • \(n\) is the sample size

  • \(SE\) is the standard error

The t-score follows Student’s t-distribution with:

\[ \nu=n-1 \]

degrees of freedom.

where:

  • \(p\) is the cumulative probability

  • \(\nu\) is the degrees of freedom

  • \(T_{\nu}^{-1}\) is the inverse CDF of Student’s t-distribution with \(\nu\) degrees of freedom.

As sample size increases, the t-distribution approaches the Gaussian distribution, and t-scores become nearly equivalent to z-scores.

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Tabular Data#

A data representation where observations are organized into a table with,

Tabular data is the most common data format for machine learning, statistics, and data analytics. Examples include,

  • spatial data with one row for each sampled location

  • temporal data with one row for each time observation

  • multivariate laboratory measurements with one row for each sample

Pandas’ ‘DataFrame’ is the standard Python class for working with tabular data due to,

  • convenient storage, access, and manipulation of tabular data

  • built-in methods to load data from a variety of file formats, databases, and spreadsheets

  • built-in methods for summary statistics, visualization, grouping, filtering, sorting, and joining tables

  • built-in methods for cleaning, reshaping, and transforming data

  • built-in attributes describing the data structure, for example, dimensions, column names, data types, and missing values

Compare with:

Also see:

Test Statistic#

A test statistic is a statistic used in hypothesis testing to quantify the difference between an observed sample result and the value expected under the null hypothesis.

A test statistic is denoted as:

\[ T \]

or for a specific test:

\[ T_{statistic} \]

A test statistic is generally calculated as:

\[ \text{Test Statistic}= \frac{\text{Observed Statistic}-\text{Expected Statistic}} {\text{Standard Error}} \]

The test statistic expresses the observed difference in units of uncertainty, allowing comparison with a sampling distribution.

Common test statistics include:

  • z-score – based on the Gaussian distribution

  • t-score – based on the Student’s t distribution

  • \(\chi^2\) statistic – based on the chi-square distribution

  • F statistic – based on the F distribution

A hypothesis test uses the test statistic to determine the probability of observing the sample result if the null hypothesis were true.

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Training Image#

For multiple point simulation, reliable greater than 2-point spatial statistics are generally impractical to calculate directly from sparse subsurface data because the number of possible spatial configurations grows combinatorially while conditioning data remain limited. Instead, these higher-order spatial patterns are borrowed from a training image.

A training image is a dense conceptual representation of expected geological patterns, connectivity, and morphology in a 2D or 3D model. It provides the spatial configurations required to calculate multiple point statistics.

Important aspects of training images,

  • does not include local information or conditioning data

  • should be stationary with respect to the multiple point statistics and any lower order statistics, such as facies proportions and indicator variograms

  • must have the same cell size as the simulation model

  • does not need to have the same extent (number of model cells in each dimension), but should be large enough to provide sufficient replicates of geological patterns

  • larger training images provide more pattern examples but increase computational complexity

Training images are also used as conceptual pattern libraries for training generative AI models.

Used in:

Transfer Function#

A model, process, or calculation applied to spatial subsurface model realizations and scenarios to transform uncertain subsurface properties into a decision criteria.

The transfer function connects subsurface uncertainty models to decision making by calculating metrics that represent value, risk, health, environment, safety, or operational constraints.

Transfer functions may be physics-based, data-driven, or hybrid. Example transfer functions include,

  • transport and bioattenuation - numerical simulation to model soil contaminant concentrations over time during a pump-and-treat operation

  • volumetric calculation - estimate total oil-in-place from reservoir property models

  • heterogeneity metrics - calculate indicators related to recovery factor and estimate reserves from resources

  • flow simulation - generate pre-drill production forecasts for a planned well

  • Whittle pit optimization - calculate mineral resources and ultimate pit shell designs

Used in:

Trend#

An interpretation that a spatial feature is nonstationary over space, meaning that one or more statistics of the feature systematically change over the 2D area of interest or 3D volume of interest.

For example,

  • porosity decreases with depth

  • copper grade increases toward a highly faulted zone

Trend in spatial data may be identified by,

Trend is also used to describe a deterministic model of nonstationarity in a statistic or metric of interest, as in Trend Model.

Used in:

Also see:

Trend and Residual Workflow#

Most geostatistical modeling methods assume stationarity in the feature mean. Yet, nonstationarity, trend, in the mean is commonly observed in the subsurface.

  • to address this limitation, the common hybrid workflow is to deconvolve the spatial data into 2 components:

  1. known - deterministic trend model

  2. unknown - stochastic residual model

The known trend is calculated and then subtracted from the data, leaving a residual that is modelled stochastically with uncertainty (treated as unknown). The following steps are applied:

  1. model the nonstationary, spatial, deterministic trend for a feature of interest

  2. subtract the trend from the data to calculate the residual

  3. model the residual with geostatistical spatial estimation or simulation

  4. add the deterministic trend to the geostatistical (deterministic if kriging or stochastic if simulation) residual

  5. check the model

Used in:

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Trend Model#

A determistic model representing the spatial trend in a statistic that is applied as an input for a spatial simulation method, for example,

  • a linear function for reduction in average porosity with depth, based on local data and regional compaction trends

  • a moving window local average copper grade model to model the increase in copper grade toward the highly faulted zone

This provides a local value of the statistic at all model grid cells, so the simulation can apply the trend model to relax the assumption of statistionarity in the statistic.

Used in:

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Trend Structure#

Experimental variogram points rise approximately linearly above the sill,

  • indicates a trend in the data, e.g., fining upward, increased compaction with depth, etc.

  • could be interpreted as a fractal, i.e., model without a finite variance or sill fit with a power law function. Note, variogram models above the sill are not permissible for simulation methods like sequential Gaussian simulation, we must model to the sill

  • common workflow is to remove the trend, work with the residual, if the trend is removed the residual variogram will plateau at the sill

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Also see:

Type I Error#

A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true.

A Type I error is also known as:

  • false positive

  • false alarm

The probability of making a Type I error is controlled by the significance level:

\[ P(\text{Type I Error})=\alpha \]

Examples include:

  • concluding a geological relationship exists when the observed relationship is due to random sampling variation

  • identifying a significant difference between populations when no true difference exists

Reducing \(\alpha\) decreases the probability of Type I error but may increase the probability of Type II error.

Used in:

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Type II Error#

A Type II error occurs when the null hypothesis is not rejected when it is actually false.

A Type II error is also known as:

  • false negative

  • missed detection

The probability of making a Type II error is denoted by:

\[ \beta \]

and represents:

\[ P(\text{Fail to Reject }H_0|H_0\text{ is false})=\beta \]

The statistical power of a hypothesis test is:

\[ Power=1-\beta \]

where power represents the probability of correctly detecting an effect when one exists.

Examples include:

  • failing to identify a real geological trend because the sample set is insufficient

  • failing to detect a difference between development strategies due to excessive uncertainty

Type II error probability is influenced by:

  • sample size

  • variability of the population

  • magnitude of the true effect

  • selected significance level

Used in:

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Uncertainty Modeling#

Characterization of the range of plausible values for a feature at a location, jointly over the entire subsurface model, or propagated through a transfer function to support decision making.

Uncertainty may be considered at different levels:

Common sources of uncertainty include:

  1. Data imprecision - measurement error, interpretation uncertainty, and imperfect observations

  2. Spatial offset from data - uncertainty from estimating unsampled locations away from available spatial data

  3. Model parameter inference - uncertainty in inferred parameters such as global mean, variance, variogram, and correlation structure

  4. Conceptual model uncertainty - uncertainty from choices about geological framework, modeling approach, and assumptions about the subsurface system

Uncertainty models are represented with ensembles of scenarios and realizations:

  • Scenarios - multiple spatial subsurface models calculated by changing input parameters or other modeling choices to represent uncertainty from model parameters and conceptual choices

  • Realizations - multiple spatial subsurface models calculated by holding input parameters and modeling choices constant and changing only the random number seed

How can we address each source of uncertainty?

  • data imprecision - model data uncertainty through data realizations, soft data integration, or indicator transforms

  • spatial offset from data - calculate multiple stochastic realizations by varying the simulation random number seed

  • model parameter inference - calculate scenarios by varying inferred model parameters

  • conceptual model uncertainty - develop and compare alternative geological interpretations or modeling workflows

Important considerations for uncertainty modeling,

  • uncertainty modeling is critical for quantifying limitations in sample precision and model predictions

  • uncertainty is itself a model; there is no objective uncertainty independent of assumptions, data, and modeling choices. Failure to recognize this leads to the circular pursuit of “uncertainty in the uncertainty”

  • uncertainty results from sparse sampling, measurement error, interpretation uncertainty, bias, and geological heterogeneity

  • uncertainty reflects our limited ability to observe subsurface features with sufficient accuracy, resolution, and coverage; it is not an intrinsic property of the geology itself

Used in:

Uniform Distribution#

The uniform distribution is a statistical distribution where all possible values have equal probability.

For a continuous uniform distribution between \(a\) and \(b\):

\[\begin{split} f(x)= \begin{cases} \frac{1}{b-a}, & a \leq x \leq b\\ 0, & \text{otherwise} \end{cases} \end{split}\]

The mean and variance are:

\[ \mu=\frac{a+b}{2} \]
\[ \sigma^2=\frac{(b-a)^2}{12} \]

For a categorical uniform distribution with \(k\) equally probable categories:

\[ P(X=x_i)=\frac{1}{k}, \quad i=1,\ldots,k \]

The categorical uniform distribution has:

\[ \mu = \frac{1}{k}\sum_{i=1}^{k}x_i \]

and probability mass distributed equally among all categories.

The uniform distribution is symmetric:

Uniform distributions are often used to represent limited knowledge where all values within a defined range or categories are considered equally likely:

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Union of Events#

The union of events represents all outcomes where event \(A\) occurs, event \(B\) occurs, or both events occur. The probability of the union is calculated with the probability addition rule,

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

The intersection probability is subtracted because outcomes where both \(A\) and \(B\) occur are included in both \(P(A)\) and \(P(B)\) and would otherwise be counted twice.

For mutually exclusive events, the intersection probability is zero,

\[ P(A \cap B)=0 \]

and the probability addition rule simplifies to,

\[ P(A \cup B)=P(A)+P(B) \]

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Unit Lag Distance#

The spacing between successive lag distance bins applied when calculating an experimental variogram.

For example,

  • if the unit lag distance is 50 m and 5 lags are calculated, the experimental variogram will have points centered approximately at lag distances of 50 m, 100 m, 150 m, 200 m, and 250 m

  • typically, the unit lag distance is selected as the nominal minimum data spacing in the specific direction of analysis

  • the absolute minimum data spacing should not be used because there may be only one or very few data pairs available, resulting in an unreliable first experimental variogram point

  • nominal minimum data spacing represents the smallest lag distance with sufficient pairs to calculate a reliable experimental variogram estimate

The unit lag distance, together with lag distance tolerance, controls the grouping of data pairs used to calculate experimental variogram points.

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Univariate#

Involving a single feature or event only.

Examples include:

Used in:

Compare with:

Univariate Parameter#

A univariate population summary measure describing a single feature.

Examples include:

In practice, the complete population is rarely available, so univariate parameters are inferred from available univariate statistics calculated from samples.

Used in:

  • TBD

Contrast with:

Univariate Statistic#

A summary measure calculated from samples of a single feature.

Examples include:

Univariate statistics describe the available sample and are used to infer the corresponding ppopulation parameter.

Used in:

Contrast with:

Variable#

Any property measured or observed in a study, for example,

  • porosity, permeability, mineral concentrations, saturations, contaminant concentration

  • in data mining / machine learning this is known as a feature

  • often requires significant analysis, interpretation, etc.

Used in:

Same as:

Variance#

A measure of distribution dispersion, the spread or variability of a feature about its average. Larger variance indicates greater variability.

For a sample, the variance is,

\[ s^2 = \frac{1}{n-1}\sum_{\alpha=1}^{n}\left(x_{\alpha}-\overline{x}\right)^2 \]

The equivalent population parameter is the population variance,

\[ \sigma^2 = \frac{1}{n}\sum_{\alpha=1}^{n}\left(x_{\alpha}-\overline{x}\right)^2 \]

Some comments about variance,

  • units - the units of variance are squares units of the feature, for more intuitive units consider using the standard deviation

  • additivity - variance are additive, enabling a lot of workflows like analysis of variance and trend + residual workflows, for example given \(X_{residual} + X_{trend} = X_{total}\), the variance is calculated as,

\[ \sigma^2_{X_{total}} = \sigma^2_{X_{trend}} + \sigma^2_{X_{residual}} + 2\,\mathrm{Cov}(X_{trend},X_{residual}) \]

  • momments - variance is the \(2^{nd}\) centered momment

  • outliers - variance is very sensitive to outliers

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Variance Reduction Factor#

A convenient factor used to correct variance when changing from the data volume support to a larger model volume support.

The variance reduction factor, \(f\), is defined as the ratio of variance at the larger volume support, \(v\), to the variance at the original data volume support, \(\cdot\):

\[ f = \frac{\sigma^2(v)}{\sigma^2(\cdot)} \]

Using volume-variance relations, this can be calculated as,

\[ f = 1 - \frac{\overline{\gamma}(v,v)}{\sigma^2} \]

where \(\overline{\gamma}(v,v)\) is the average variogram value within the volume support \(v\) and \(\sigma^2\) is the variance at the original data support.

Equivalently, using dispersion variance,

\[ f = \frac{D^2(v,V)}{D^2(\cdot,V)} = \frac{D^2(v,V)}{\sigma^2} \]

The variance reduction factor is applied to adjust the data histogram to represent the reduced variability expected at a larger model scale.

Without volume support correction, the original data-scale distribution will have excessive variance when applied directly to a larger model volume.

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Variogram#

A scatterplot with axes of difference or variance over distance.

  • Experimental variogram is calculated over integer multiples of the unit lag distance and then plotted as points, then permissible variogram models are fit to the experimental variogram while integrating other domain and local knowledge.

  • the variogram is calculated as one half the average squared difference over lag distance, 𝐡, over all possible pairs of data,

\[ \gamma_z(\bf{h}) = \frac{1}{2 \cdot N(\bf{h})} \sum_{\alpha = 1}^{N(\bf{h})} \left( z(\bf{u}_{\alpha}) - z(\bf{u}_{\alpha} + \bf{h}) \right)^2 \]

  • the precise term is semivariogram (or variogram if you remove the \frac{1}{2} in the equation above), but in practice, the semivariogram is only used and the term variogram is always used for the semivariogram

  • the \(\frac{1}{2}\) term is added to the semivariogram so that the covariance function, \(C_z(\bf{u})\), and variogram, \(\gamma_z(\bf{h})\), may be related as:

\[ C_z(\bf{h}) = \sigma_z^2 - \gamma_z(\bf{h}) \]

Note the correlogram, \(\rho_z(\bf{u})\), is related to the covariance function, \(C_z(\bf{u})\), as:

\[ \rho_z(\bf{u}) = \frac{C_z(\bf{h})}{\sigma_z^2} \]

Here are some general observations about the variogram,

  1. Often increasing - as the lag Distance, \(\bf{h}\), increases, variability over the lag distance increase (in general).

  2. Not a local measure - the variogram is calculated with over all possible pairs separated by lag vector, \(\bf{h}\).

  3. Interpret relative to the sill - we need to plot the sill on with the experimental variogram to know the degree of correlation.

  • the sill is the variance, \(\sigma^2\), given stationarity of the variance and variogram, \gamma_z(\bf{h})):

\[ C_z(\bf{h}) = \sigma_z^2 - \gamma_z(\bf{h}) \]

\(\quad\) and given a standardized feature, \(\sigma_z^2 = 1.0\),

\[ \rho_z(\bf{u}) = \sigma_z^2 - \gamma_z(\bf{h}) \]

\(\quad\) the distance from the sill to the experimental variogram is the correlation coefficient over the specific lag distance.

  1. Range - the lag distance at which the variogram reaches the sill is know as the range.

  • at the range, knowing the data value at the tail provides no information about a value at the head.

  1. Nugget effect - sometimes there is a discontinuity in the variogram at distances less than the minimum data spacing. This is known as nugget effect.

  • the ratio of nugget divided by sill, is known as relative nugget effect, reported in percentage, e.g., 10% relative nugget effect

  • we model the nugget effect as a no correlation structure over all lags greater than an infinitesimal distance, \(\bf{h} \gt \epsilon\)

  • measurement error, causes an apparent nugget effect, if this is suspected do not add nugget effect to the variogram model

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Variogram Interpretation#

The process of analyzing characteristics of the experimental variogram to understand spatial continuity and develop an appropriate variogram model for geostatistical estimation or simulation.

Variogram interpretation considers,

  • nugget effect - discontinuity at short lag distances due to measurement error, microscale variability, or unresolved spatial variation

  • sill - variance represented by the spatial model and reference level for evaluating spatial continuity

  • range - lag distance beyond which there is no modeled spatial correlation

  • nested structures - multiple spatial continuity scales represented by combining permissible variogram structures

  • directional behavior - changes in spatial continuity with direction, indicating anisotropy

  • geological consistency - interpretation of spatial continuity based on depositional processes, geology, and the intended modeling application

The experimental variogram is a statistical estimate from available samples; variogram interpretation is the process of converting this estimate into a spatial continuity model suitable for prediction and uncertainty modeling.

  • also known as variography.

Used in:

Variogram Map#

A diagnostic method for calculating the experimental variogram over multiple lag distances and directions simultaneously.

A variogram map is calculated by applying a two-dimensional lag template, where,

  • template cell size controls the spatial resolution of the variogram calculation, analogous to the unit lag distance (assuming the lag tolerance is approximately \(\frac{1}{2}\) the lag distance)

  • number of template cells controls the maximum lag distance or extent of the variogram map calculation

Variogram maps are useful to,

Variogram maps generally require,

  • more data than conventional isotropic or directional experimental variograms because many lag-distance and direction combinations must be populated with sufficient data pairs.

With sparse spatial data, variogram maps may become noisy and unreliable for interpretation.

A variogram map is a diagnostic view of spatial continuity; the interpreted directions and structures must still be evaluated using geological understanding and modeled with permissible variogram models.

Used in:

  • TBD

Also see:

Variogram Model#

A mathematical model of spatial continuity for a random function, interpreted and parameterized from experimental variograms with geological knowledge.

The variogram model is required because the experimental variogram only provides estimates of spatial continuity over limited lag distances and directions. Variogram modeling provides a continuous, valid representation of spatial continuity for all possible lag vectors, \(\bf{h}\).

Reasons for variogram modeling include:

  • Interpolate all distances and directions - the variogram must be defined for all possible lag distances and directions, not only the limited lags calculated from experimental variograms.

  • Integrate geological knowledge - variogram modeling provides an opportunity to incorporate geological understanding, for example, depositional controls on geometric anisotropy.

  • Ensure a valid measure of spatial difference - the variogram model must be positive definite so that the variance of any linear combination of random variables remains non-negative. Additive nested variogram structures provide a practical approach to constructing valid models.

Variogram modeling with nested structures is applied to describe variance contributions and spatial continuity scales using the following workflow:

  1. Nugget effect - assign the isotropic short-scale variance contribution represented by the nugget effect.

  2. Number of structures - select the number of variogram structures required to represent the most complex direction of spatial continuity. The same number of structures is applied in all directions.

  3. Structure contributions - assign the variance contribution (partial sill) for each structure. Contributions should be consistent across directions, and the total modeled variance should equal the sill. Models should approach the sill but not exceed it.

  4. Apparent nugget effect - represent direction-dependent short-scale variability by applying a zero range structure in directions where apparent nugget behavior occurs.

  5. Geometric Anisotropy - represent direction-dependent spatial continuity by varying range parameters through anisotropy ratios.

  6. Zonal Anisotropy - represent directions with different sill contributions by assigning very large ranges where spatial correlation persists beyond the available experimental variogram extent.

Additional variogram modeling considerations:

  • coordinate transformations may be required before variogram calculation, for example, flattening folded beds or restoring faults. Incorrect geometry may underestimate spatial ranges.

  • interpret fundamental variogram behaviors including trends, cyclicity, geometric anisotropy, and zonal anisotropy. If trend is present, calculate a spatial trend model and model the residual spatial continuity.

  • short-scale structures are often most important because most predictions and simulated values are interpolated between available data locations.

  • measurement error may contribute to an observed nugget effect. If the goal is to model geological variability, measurement error should be separated from geological variance when possible.

  • vertical directions are often better informed due to dense sampling along wells. A common workflow is to model the vertical variogram first and then infer horizontal anisotropy ratios from limited horizontal experimental variogram information.

The variogram model is not simply a smooth curve through experimental points; it is an interpretable model of spatial continuity used for kriging and simulation.

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Also see:

Venn Diagram#

A visual tool for communicating probability relationships using set notation and the probability of events.

A Venn diagram contains:

  • a box labelled as \(\Omega\) representing the sample space, including all possible outcomes

  • enclosed labelled shapes representing events, which are subsets of the sample space

What do we learn from a Venn diagram?

  • the size of regions is proportional to the probability of occurrence

  • the entire sample space, \(\Omega\), represents all possible outcomes and therefore has probability:

\[ P(\Omega)=1.0 \]

  • individual regions represent marginal probabilities, for example:

\[ P(A) \]

  • overlapping regions represent joint probabilities, for example:

\[ P(A \cap B)=P(A,B) \]

  • overlapping regions relative to a conditioning event represent conditional probabilities, for example:

\[ P(A|B)=\frac{P(A\cap B)}{P(B)} \]

Venn diagrams are an excellent tool to visualize marginal probability, joint probability, and conditional probability relationships and are especially useful for understanding probability operators.

Used in:

Volume-Variance Relations#

The relationship between volume support and variance. In general, as the volume support increases, the variance of a feature decreases because larger volumes average over more spatial variability.

Predicting volume-variance relations is central to integrating data collected at different scales and building subsurface models that represent the appropriate level of heterogeneity.

General observations and assumptions:

  • Under linear averaging and stationary conditions, the mean does not change with volume support; only the variance changes.

  • The distribution shape may change with volume support. This should be evaluated empirically. Common approaches include assuming no shape change with an affine correction or applying a distribution-specific correction such as an indirect lognormal correction.

  • Variance reduction is controlled by spatial continuity. Features with shorter correlation ranges experience faster variance reduction as volume support increases, while features with longer ranges retain variability over larger volumes.

Over common changes in subsurface modeling scale, the impact may be significant. Therefore, volume-variance relations should not be ignored.

  • Perfect scale-up accounting is rarely achieved because sufficient data are generally unavailable to fully characterize variability across all scales. This is commonly referred to as the missing scale problem.

  • A model is required to predict how variance changes with volume support.

Common methods to model and apply volume-variance relations include:

  1. Empirical - build a high-resolution model and numerically upscale to the larger volume support. For example,

  • calculate a fine-scale permeability model

  • apply flow simulation to estimate effective permeability over larger block volumes

  1. Power Law Average - a flexible averaging approach for changing support,

\[ z_V = \left[ \frac{1}{n} \sum z_v^{\omega} \right] ^{\frac{1}{\omega}} \]

\(\quad\) where \(\omega\) is the power of averaging:

  • \(\omega = 1\) is arithmetic averaging

  • \(\omega = -1\) is harmonic averaging

  • \(\omega = 0\) is geometric averaging, obtained as the limit as \(\omega \rightarrow 0\)

\(\quad\) The appropriate \(\omega\) may be determined from:

  • theoretical understanding, for example, harmonic averaging of permeability for flow perpendicular to beds

  • numerical upscaling with flow simulation followed by calibration of an effective averaging exponent

  1. Statistical Model - directly adjust statistical properties for the change in volume support. For linear averaging with a stationary variogram model, the Variance Reduction Factor is:

\[ f = 1 - \frac{\overline{\gamma}(v,v)}{\sigma^2} \]

\(\quad\) where \(f\) is the ratio of variance at larger volume support to variance at the original data support:

\[ f = \frac{D^2(v,V)}{D^2(\cdot,V)} = \frac{D^2(v,V)}{\sigma^2} \]

\(\quad\) The variance reduction factor is calculated from,

  • the variogram model representing spatial continuity

  • the original data support

  • the target model volume support

Used in:

Also see:

Volume of Interest#

The 3D spatial domain that is being characterized, modeled, and evaluated to support subsurface decision making. In general, the volume 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 2D modeling is commonly called the area of interest

  • the extent and grid cell size are selected based on a trade-off between model accuracy and computational complexity

Volume Support#

The spatial volume over which a variable is measured or averaged. Volume support defines the physical extent of a measurement and directly influences variability, smoothing, and scale dependence. In practice, it is often related to (but not identical with) the concept of scale.

Examples include:

Core volume support is:

\[ \pi r_{core}^2 \times l_{core} \]

where \(r_{core}\) is core radius and \(l_{core}\) is core length.

Well Log volume support is:

\[ \pi r_{log}^2 \times l_{log} \]

where \(r_{log}\) is the logging tool radius (or effective radius of investigation) and \(l_{log}\) is the vertical resolution or sampling interval.

Seismic volume support is:

\[ \delta x_{\text{inline}} \times \delta y_{\text{crossline}} \times \delta z_{\text{vertical}} \]

where \(\delta x_{\text{inline}}\) is inline resolution, \(\delta y_{\text{crossline}}\) is crossline resolution, and \(\delta z_{\text{vertical}}\) is vertical resolution.

It is critical to explicitly state volume support when describing data or models:

  • Volume support strongly influences measured statistics and spatial variability.

  • Consistent comparison between datasets requires accounting for differences in support through change-of-support or upscaling methods.

Used in:

Also see:

Well Image Log#

A special case of well logs where the well logs are repeated at various azimuthal intervals within the well bore resulting in a 2D (unwrapped) image instead of a 1D line along the well bore. For example, Fullbore formation MicroImager (FMI) with:

  • with 80% bore hole coverage

  • 0.2 inch (0.5 cm) resolution vertical and horizontal

  • 30 inch (79 cm) depth of investigation

can be applied to observe lithology change, bed dips and sedimentary structures.

Used in: TBD

Also see:

Well Log#

A cheaper method to sample wells that does not interrupt drilling operations like cores, well logs are very common over the wells. Typically all wells have various well logs available. For example,

  • gamma ray on pilot vertical wells to assess the locations and quality of shales for targetting (landing) horizontal wells

  • neutron porosity to assess location high porosity reservoir sands

  • gamma ray in drill holes to map thorium mineralization

Well log data are critical to support subsurface resource interpretations. Once anchored by core data they provide the essential coverage and resolution to model the entire reservoir concept / framework for prediction, for example,

  • well log data calibrated by core data collocated with well log data are used to map the critical stratigraphic layers, including reservoir and seal units

  • well logs are applied to depth correct features inverted from seismic data that have location imprecision due to uncertainty in the rock velocity over the volume of interest

Used in: TBD

Also see:

z-critical#

A z-critical value is a quantile from the standard Gaussian distribution used to construct confidence intervals when the population standard deviation is known or when the Gaussian approximation is appropriate.

The z-critical value is calculated from the inverse cumulative distribution function (inverse CDF):

\[ z_{\alpha/2}=\G^{-1}\left(1-\frac{\alpha}{2}\right) \]

where:

  • \(\alpha\) is the significance level

  • \(\G^{-1}\) is the inverse CDF of the standard Gaussian distribution

For a confidence interval:

\[ \bar{x}\pm z_{\alpha/2}SE \]

where \(SE\) is the standard error.

Common two-sided confidence levels:

  • 90% confidence: \(z_{critical} \approx 1.645\)

  • 95% confidence: \(z_{critical} \approx 1.960\)

  • 99% confidence: \(z_{critical} \approx 2.576\)

A z-critical value is a property of the selected confidence level, not the observed sample.

Used in:

Also see:

z-score#

The z-score (standard score) measures how many population standard deviations an observation lies above or below the population mean.

The z-score is calculated as:

\[ z=\frac{x-\mu}{\sigma} \]

where:

A z-score of:

  • 0 indicates the observation equals the mean

  • +1 indicates one standard deviation above the mean

  • −2 indicates two standard deviations below the mean

Used in:

Also see:

Zonal Anisotropy#

A type of anisotropy where the experimental variogram does not reach the sill in all directions.

In zonal anisotropy, the experimental variogram in one or more directions levels off below the sill within the available lag distances.

  • this is often called an apparent sill.

Zonal anisotropy is commonly associated with geological layering and stratification, where spatial variability is partitioned differently along and across geological structures.

Common interpretations include:

  • the direction aligned with layers may show a lower apparent sill because variability within layers is smaller

  • the orthogonal direction may contain additional variance due to differences between layers

  • zonal anisotropy may occur together with cyclicity or trend structure in another direction, for example, zonal anisotropy in the major direction with trend in the minor direction

Zonal anisotropy provides a geological interpretation of variance partitioning over space:

  • the variance contribution up to the apparent sill represents variability within the geological layers

  • the variance contribution from the apparent sill to the total sill represents variability between geological layers

Used in:

Also see:

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

The Author:#

Michael Pyrcz, Professor, The University of Texas at Austin Novel Data Analytics, Geostatistics and Machine Learning Subsurface Solutions

With over 17 years of experience in subsurface consulting, research and development, Michael has returned to academia driven by his passion for teaching and enthusiasm for enhancing engineers’ and geoscientists’ impact in subsurface resource development.

For more about Michael check out these links:

Twitter | GitHub | Website | GoogleScholar | Geostatistics Book | YouTube | Applied Geostats in Python e-book | Applied Machine Learning in Python e-book | LinkedIn

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