Uncertainty LHS Tab

The Latin Hypercube Sampling (LHS) tab is part of FracSTAT's Uncertainty Analyzer, a self-contained probabilistic workspace for evaluating fault and fracture stability under uncertain stress, pore-pressure, rock-strength, and orientation conditions. Unlike a deterministic calculation that returns one value for each input condition, the LHS workflow propagates uncertainty through the Mohr–Coulomb fault-stability calculation and produces probability distributions for Slip Tendency (Ts), Dilation Tendency (Td), Fracture Susceptibility (Sf), and Coulomb Failure Index (Fsi).

1. Purpose and Scope

The LHS module provides a stratified, variance-reduced probabilistic simulation environment for evaluating mechanical stability of faults and fractures. It propagates uncertainty in the three principal stresses, pore pressure, cohesion, friction, SHmax azimuth, strike and dip through the same Mohr–Coulomb mechanics used by the deterministic stability calculator.

The Uncertainty Analyzer is independent of the Static and Dynamic Analyzer workspaces. Its input tables, sampling settings, cached results and session state are private to the uncertainty workspace and are not written back into the Static/Dynamic shared project cache.

1.1 What the LHS module provides

• Stratified Latin Hypercube sampling with reproducible random seeds.
• Uncertain stress, strength, pressure and orientation inputs.
• Mohr–Coulomb fault-stability calculations for every realization.
• Probability distributions for Ts, Td, Sf and Fsi.
• Descriptive statistics, convergence diagnostics and analytical/reference validation.
• SRC and PRCC sensitivity analysis.
• Spatial probability mapping when coordinate data are supplied.
• Publication-oriented visualization, image export and tabular result export.

2. Interface Overview

The LHS tab occupies one page of the CTkTabview created by UncertaintyAnalyzerApp._build_workspace_tabs. It contains the same workspace structure as the Monte Carlo tab, but uses sampling_method="lhs".

2.1 Workspace header

The shared header provides Load Excel/CSV, mode switching between Dynamic and Static workspaces, and a live status bar. The status area reports loading results, simulation outcomes, convergence state, cancellation and export locations. SVG loading and scale calibration are disabled in this workspace because the uncertainty module works with tabular fault data rather than digitized fracture traces.

2.2 Nested analysis modules

Each sampling tab contains six nested modules in a fixed order: Geomechanics, Statistics, Sensitivity, Visualization, Spatial Probability Map, and Validation & Export. Only the currently visible module is redrawn after a run or fault switch, which keeps large ensembles responsive.

2.3 Shared control bar

3. Data Input

3.1 Loading fault/fracture orientation data

Use Load Excel/CSV in the shared header. Workbook loading is performed on a background thread so the Qt interface remains responsive. Multi-sheet workbooks open a worksheet selection dialog; CSV files and single-sheet workbooks proceed directly.

Column recognition is alias-driven. FracSTAT normalizes header names to lower-case, alphanumeric-only text and maps recognized aliases to canonical fields.

3.2 Input validation

3.3 Spatial coordinate files

The Spatial Probability Map has a separate coordinate-loading path. Two coordinate layouts are recognized automatically.

Units can be detected from bracketed or parenthesized headers such as X-Coordinate [m] or X [km]. Input and display units can be independently selected from km, m, cm and mm.

4. Settings and Controls

4.1 Geomechanics — uncertainty inputs

4.1.1 Per-row controls

4.1.2 Stress regime selector

The stress-regime selector provides Normal Faulting (Sv = σ1), Strike-Slip (Sv = σ2), and Reverse/Thrust (Sv = σ3). The selected regime controls which sampled principal stress is mapped to the vertical slot used by the stability calculation. The displayed σ1 > σ2 > σ3 ordering remains unchanged.

4.1.3 Effective-stress switch

Use effective principal stresses controls whether pore pressure is subtracted from the stress terms used by Ts, Td and Fsi. Changing the switch recalculates the active and cached metrics without re-running the sampler. Sf is not affected by it: a sustainable pore-pressure increase only exists under the effective-stress principle, so Sf — and the failure flag derived from it — is always computed on σ′_n = σ_n − Pp (Section 5.5).

4.1.4 Plot appearance

Geomechanics appearance controls include Mohr uncertainty diagrams, total-stress Mohr circles, stress-scatter styling, backgrounds, grid, colour scales, failed-point highlighting, Coulomb envelope and Mohr-circle visibility, legends, axis scale and explicit plot width/height in pixels.

4.2 Statistics module

4.3 Sensitivity module

4.4 Validation & Export

4.5 Spatial Probability Map

4.6 Visualization module

Visualization provides response-metric and input selectors, fifteen single-plot types, multiple visualization layouts and a four-panel dashboard. The module also provides KDE and P10–mean–P90 overlays, full-screen viewing, image saving and extensive appearance controls.

5. Mathematical Background

5.1 Latin Hypercube Sampling design

For dimensionality d = 9 and sample size N, LHS partitions the [0,1] probability range of each dimension into N equal-probability strata. Exactly one sample is taken from every stratum in each dimension, while the stratum indices are independently permuted between dimensions (McKay et al., 1979). FracSTAT generates the design using scipy.stats.qmc.LatinHypercube(d, seed) (Virtanen et al., 2020).

This produces evenly stratified one-dimensional projections and can reduce sampling variance relative to pseudo-random Monte Carlo at the same sample size, particularly for means of responses dominated by additive input effects (McKay et al., 1979). The benefit depends on the response and statistic: improved precision for every quantile or tail probability is not guaranteed. See Helton and Davis (2003) for uncertainty-propagation applications. The design is a plain Latin hypercube: the strata are permuted independently and no space-filling criterion is optimised over candidate permutations (cf. Morris and Mitchell, 1995), and no rank-correlation structure is induced across dimensions (cf. Iman and Conover, 1982).

5.2 Probability distributions

Physical bounds are enforced: friction is restricted to [0,1], cohesion and pore pressure are non-negative, and principal stresses are constructed so that all realizations remain non-negative. When a bounded quantity is assigned Normal/Gaussian, the sampler silently uses the equivalent Truncated Normal law and records the effective distribution in reproducibility metadata.

The values entered are the moments of the law that is actually sampled. SciPy's truncnorm takes loc/scale as the parent normal's μ and σ, and the truncated distribution's own mean and standard deviation differ from those whenever a bound is within reach (Johnson, Kotz and Balakrishnan, 1994, Vol. 1, sec. 13.10):

α = (a − μ)/σ,   β = (b − μ)/σ,   Z = Φ(β) − Φ(α)

E[X]   = μ + σ · (φ(α) − φ(β)) / Z

Var[X] = σ² · [ 1 + (α·φ(α) − β·φ(β))/Z − ((φ(α) − φ(β))/Z)² ]

distributions._truncated_normal_parent therefore solves these two relations for the parent (μ, σ) whose truncation reproduces the requested mean and SD, as nested bracketed root finds: E[X] is monotone in μ, and the realised SD is then monotone in σ. Both roots are verified before use.

A truncated-normal law cannot carry every (mean, SD) pair: on [0, ∞), SD/mean has a supremum of 1 (the exponential limit); on a finite [a,b], SD cannot exceed (b − a)/√12, with a tighter limit when the mean is away from the midpoint. These are family-specific limits, not limits on every distribution with those bounds. An unattainable request raises an error reporting an approximate attainable SD for the truncated normal rather than quietly sampling different moments. For positive unbounded inputs, consider Lognormal or Gamma; for friction on [0,1], consider Beta or a feasible Uniform law. The alternative must still satisfy its own moment and support constraints.

5.3 Orientation sampling

Strike and SHmax are circular/azimuthal quantities, so a linear normal distribution is unsuitable near the 0°/360° wrap-around. FracSTAT uses the Von Mises distribution (Mardia and Jupp, 2000, ch. 3), where concentration k controls clustering: k = 0 is uniform and larger k produces tighter concentration around the mean.

Dip is bounded to 0–90° and is sampled from a bounded Truncated Normal distribution whose standard deviation is derived from the concentration parameter. Alternatively, Fisher 3D couples Strike and Dip through a single spherical orientation distribution using Strike's concentration parameter (Fisher, 1953).

SHmax always uses its own independent Von Mises concentration because the maximum horizontal stress direction is an independent geomechanical measurement rather than a structural attribute.

5.4 Principal-stress ordering and regime remapping

The sampler guarantees σ1 > σ2 > σ3 by drawing σ3 directly and constructing σ2 and σ1 from σ3 plus two non-negative stress gaps. The configured stress distributions are retained while the gap construction prevents principal-stress crossing. The three regimes follow the Andersonian classification, in which one principal stress is vertical because the free surface carries no shear traction (Anderson, 1951); see also Zoback (2007, ch. 1) for the modern stress-state formulation.

5.5 Fault-stability equations

For every realization, the downward-directed unit normal (pole) of the fault plane is constructed from strike and dip in an (East, North, Up) frame under the right-hand rule — dip direction = strike + 90°, so the pole trends strike − 90° and plunges 90° − dip — and then projected onto the horizontal principal axes defined by the SHmax azimuth:

n         = ( −sin(dip)·cos(strike),  sin(dip)·sin(strike),  −cos(dip) )

rotated_x = nx·sin(SHmax) + ny·cos(SHmax)        (component along SHmax)
rotated_y = nx·cos(SHmax) − ny·sin(SHmax)        (component along the other horizontal axis)

σn        = σ1·rotated_x² + σ2·rotated_y² + σ3·nz²
|T|²      = σ1²·rotated_x² + σ2²·rotated_y² + σ3²·nz²
τ         = √ max(|T|² − σn², 0)

Effective normal stress is σ′_n = σ_n − Pp, the Terzaghi effective-stress form (Terzaghi, 1943) as applied to faulting by Hubbert and Rubey (1959). The Effective-Stress switch determines whether total or effective stress is used for Ts, Td and Fsi.

The failure flag is Sf ≤ 0, equivalent to the Fsi ≥ 1 criterion, but it is stored as a three-valued boolean: True, False, or missing where Sf could not be evaluated:

result["failure"] = pd.array(
    np.where(np.isfinite(sf_values), sf_values <= 0, None), dtype="boolean"
)

5.6 Sensitivity coefficients: SRC and PRCC

SRC is obtained from ordinary least-squares regression of standardized target output against standardized inputs. PRCC measures rank correlation after the effect of all other inputs has been removed in rank space (Iman and Helton, 1988; Marino et al., 2008). Both range from −1 to +1; magnitude indicates relative influence and sign indicates direction.

Two preparation steps are applied to the design matrix before either coefficient is computed. Strike is folded modulo 180°, because it is axial and the artificial discontinuity at 180°/360° would otherwise enter the regression as if it were real variation in orientation. Multi-fault population runs are collapsed to one row per realization by aggregate_paired_faults, so the ranking represents the population rather than one fault's individual response.

Squared SRCs partition the output variance — that is, they satisfy ΣSRC² = R² and may be read as variance shares — only when the inputs are mutually uncorrelated and the response is linear in them (Iman and Helton, 1988; Saltelli et al., 2008, ch. 1; Marino et al., 2008, sec. 2). Neither condition is guaranteed here: the ordered-gap construction of Section 5.4 can draw σ1, σ2 and σ3 with strong mutual correlation. The module therefore reports three diagnostics with every run — model R², ΣSRC², and the largest |r| between inputs — and adds a warning when |r| > 0.1 and |ΣSRC² − R²| > 0.05 (read the SRCs as conditional regression weights and rank with PRCC instead), or when R² < 0.6 (the response is not well described by a linear model).

5.7 Descriptive and confidence-interval statistics

The Statistics module reports mean, sample standard deviation, median, mode, P10, P50 and P90. The 95% confidence interval of the mean is approximated as:

CI95% = x̄ ± 1.96 × (s / √n)

Strike (period 180°, axial) and the SHmax azimuth (period 360°, directional) wrap, so neither has a meaningful arithmetic mean or standard deviation. When one of them is the displayed variable, the location and spread are computed on the circle instead — the mean resultant vector of the angles mapped onto the full circle, with mean = atan2(S, C) and s.d. = √(−2·ln R), scaled back by period/360° (Mardia and Jupp, 2000, sec. 2.2.1, 2.3.2 and 2.3.4; Fisher, 1993, sec. 2.3.1). The P10/mean/P90 overlay on those two inputs is likewise taken on the angular residual about the circular mean, and a band that straddles the wrap point is drawn as the two arcs it actually occupies. Measured on a von Mises sample centred on a true strike of 0.04°, the linear arithmetic mean read 89.89° — the perpendicular orientation; an SHmax azimuth concentrated on 359.98° read 180.28°. Dip is a plunge confined to 0–90° and does not wrap, so it keeps the linear summary.

6. Algorithms

7. Internal Workflow

7.1 End-to-end run sequence

1. Read and validate SAMPLES and SEED.
2. Assemble means, standard deviations, distributions, bounds, orientation concentrations, effective-stress setting and stress regime.
3. Snapshot the selected fault or the complete fault table for an All-Faults run.
4. Disable run/input controls and display the progress overlay.
5. Run sampling_engine.generate_samples and validation in a daemon worker thread.
6. Return progress updates to the Qt main thread through app.after(0, ...).
7. On success, cache results, refresh the visible module, mark the workspace dirty, autosave and report effective-distribution substitutions.
8. On cancellation, restore the idle interface while preserving previous cached results.
9. On other exceptions, restore the interface and show the error message.

7.2 Threading and responsiveness

File loading and simulation are performed on background threads. Qt UI updates are routed through app.after(0, ...), keeping window dragging, tab switching and cancellation responsive during long simulations.

7.3 Progress reporting

Progress is reported in coarse stages rather than once per realization. The progress workload includes distribution preparation, sampling and stability calculation for each fault, followed by convergence, analytical benchmark, reference comparison and metadata stages.

7.4 Cancellation

A threading.Event is checked at fault-level checkpoints inside generate_samples. Cancellation is therefore cooperative and normally becomes effective within roughly one fault's worth of computation.

7.5 Fault-selection caching and effective-stress toggle

Completed results are cached using the active fault-selection label. An existing All-Faults result can be filtered for an individual fault without redundant simulation. The effective-stress switch recalculates Ts, Td, Sf, Fsi and the failure flag for active and cached results without sampling again.

7.6 Autosave

Loading data, switching modes, changing the active module and completing a simulation trigger silent autosave through core.project_state.save_state. Uncertainty session state remains independent from Static and Dynamic project state.

8. Statistical Methods and Validation

8.1 Descriptive statistics

Statistics are recalculated whenever the selected metric, input variable or plot type changes. They always use the currently selected fault's cached or freshly computed results.

8.2 Sensitivity analysis

SRC and PRCC populate the numeric sensitivity table and tornado chart. The tornado can be sorted by ascending or descending absolute magnitude or left in input-declaration order. The sidebar note beneath the table always carries the three diagnostics of Section 5.6 — model R², ΣSRC², and the largest |r| between inputs — together with the number of paired samples the fit used.

8.3 Fault-level quantile envelopes

per_fault_quantiles groups realizations by fault ID and calculates P10, P50 and P90. These values define the uncertainty band and central-tendency line used by the Visualization module.

8.4 Validation diagnostics

Why an increment is not an error bound. The nested levels form a geometric ladder, so a single small step between two neighbouring sizes says only that the estimate happened to move little there. On heavy-tailed synthetic ensembles (Pareto Ts, normal Sf, 40 runs per size) the old final-increment-only rule "passed" 10 of 40 runs at 512 realizations, and 8 of those 10 were still more than 2% away from a 400,000-realization reference; the true remaining error reached 1.9× the increment that had been inspected. The precision condition supplies the missing piece — a Monte Carlo standard error is the actual measure of sampling precision (Helton and Davis, 2003; Saltelli et al., 2008, ch. 2) — while the two-increment stability requirement is the standard successive-refinement check (Roache, 1998, ch. 5).

The standard error resamples realizations, not rows. In a multi-fault run every fault of a realization shares one sampled stress state, so the rows are clustered and an ordinary row bootstrap would treat n_faults correlated values as independent evidence and understate the SEM. _cluster_bootstrap_standard_errors resamples whole realizations with replacement instead (Efron and Tibshirani, 1993, ch. 8; Field and Welsh, 2007), using 200 resamples, reduced to 60 above 20,000 rows for runtime; a checkpoint with fewer than two realizations returns NaN. In a single-fault run it reduces to the ordinary bootstrap.

The reference is deliberately built a different way. The vectorized path uses squared direction cosines, and that shortcut is exactly what concealed the pole-convention error of Section 5.5; a reference sharing the same algebra could not have detected it. The tensor path also computes Sf unconditionally on effective normal stress, matching the calculator. When the reference instead followed the effective-stress switch, reference_comparison(use_effective_ts=False) reported a maximum absolute error of 12.0 MPa and "passed": False on a correctly working calculator. The analytical benchmarks (σ1 = 60, σ2 = 40, σ3 = 25 MPa, Pp = 10, μ = 0.6, C0 = 0) expect Sf = 15 MPa for a horizontal plane and Sf = 50 MPa for the vertical case; the previously expected 25 and 60 MPa were the values the buggy calculator produced with Pp dropped, so the benchmark had been confirming the defect.

9. Visualization

9.1 Geomechanics plots

9.2 Statistics plots

9.3 Sensitivity tornado

A horizontal bar chart ranks the nine sampled inputs by SRC or PRCC magnitude for the selected target output. A zero-reference line and optional numeric labels can be enabled.

9.4 Spatial Probability Map

Each fault is rendered as a coloured point or line segment and mapped using P(Fsi ≥ 1), mean Fsi, or P10/P50/P90. The companion Fracture Uncertainty Map ranks faults by P90−P10 width, making the least certain outcomes visually prominent.

9.5 Visualization single-plot catalog

9.6 Multi-panel layouts

10. Advanced Features

10.1 Full-screen visualization

The Visualization module can detach the active figure into a borderless always-on-top full-screen window with a compact duplicate toolbar. Closing the window restores the same figure inside the embedded canvas.

10.2 Layout editing mode

Move Layout Elements allows plot axes and legends to be click-dragged into custom positions for publication-ready composite figures. Twinned axes belonging to one panel move together.

10.3 Effective vs total stress

This switch changes the physical basis of every Ts, Td and Fsi result across the workspace, while allowing the same sampled realization set to be reused. Sf and the failure flag stay on effective normal stress in either position (Section 5.5), so a comparison run isolates the effect of the basis on the ratio metrics without changing the pore-pressure margin.

10.4 Stress-regime selection

Normal, strike-slip and reverse/thrust Andersonian environments are supported through stress-column remapping without duplicating the underlying stability calculator.

10.5 Fisher 3D orientation sampling

Fisher 3D provides a coupled strike-dip perturbation model when the observed fracture/fault population is better represented by a spherical orientation cluster than two independent one-dimensional distributions.

10.6 Custom plot pixel sizing

Plotting modules expose explicit width and height controls in pixels, allowing figures to be prepared at a specified export resolution or aspect ratio.

11. Performance Optimization

12. Practical Examples

12.1 Population-level reactivation screening

1. Load the fault orientation workbook with Load Excel/CSV.
2. Set the Geomechanics means, SDs and distributions for σ1, σ2, σ3, friction, cohesion, pore pressure and SHmax.
3. Leave Strike/Dip means blank when each fault's loaded orientation should be used.
4. Select All Faults, choose a sample size such as 5,000, and run LHS.
5. Check the Validation & Export convergence card. Increase SAMPLES and repeat if convergence fails.
6. Use Visualization → Fault-Level Uncertainty Envelopes to inspect Fsi bands and the Fsi = 1 threshold.
7. Load coordinates and use Spatial Probability Map to view P(Fsi ≥ 1) spatially.

12.2 Sensitivity-driven data acquisition

1. Select a specific fault and run the LHS sampler.
2. Open Sensitivity and choose Ts as the target.
3. Use SRC or PRCC to rank the nine inputs.
4. Identify the largest-magnitude sensitivity bar as the strongest contributor to Ts variability.
5. Prioritize tighter field/laboratory measurement or a reduced uncertainty range for that input in a follow-up run.

12.3 Comparing total and effective stress

1. Run with the Effective-Stress switch enabled and record the Fsi distribution.
2. Disable the switch. Cached results are recalculated without re-sampling.
3. Compare the two Fsi histograms to quantify the effect of pore-pressure assumptions on predicted failure probability.

13. Geological Interpretation

The four response metrics are Mohr–Coulomb fault-mechanics indicators used in reservoir geomechanics, induced-seismicity assessment and structural-diagenesis studies.

13.1 Slip Tendency (Ts)

Ts is the ratio of resolved shear stress to normal stress (Morris et al., 1996). A plane reaches frictional sliding when Ts approaches the static friction coefficient μ. Values around or above approximately 0.6 are commonly treated as near-critical in cohesionless applications, following Byerlee's experimental μ ≈ 0.6–0.85 for most rock types (Byerlee, 1978), but the actual interpretation should follow the friction and cohesion assumptions used in the simulation.

13.2 Dilation Tendency (Td)

Td identifies planes favourably oriented to open under the applied stress state (Ferrill and Morris, 2003). High Td can indicate fracture orientations that are more favourable for dilation and fluid-flow pathways.

13.3 Fracture Susceptibility (Sf)

Sf is expressed in MPa and is read as the pore-pressure increase the fault can sustain before the Coulomb criterion is met at the resolved shear stress (Mildren et al., 2002; Streit and Hillis, 2004): an Sf of 4 MPa means roughly 4 MPa of additional pressure would be expected to reactivate it. Positive Sf indicates a stable margin; Sf ≤ 0 indicates predicted shear failure at current conditions. Because it is a pressure margin it is always computed on effective normal stress, whatever the Effective-Stress switch is set to, and it is reported as missing rather than as a number where it cannot be evaluated (σ′_n ≤ 0, or μ ≤ 0).

13.4 Coulomb Failure Index (Fsi)

Fsi is the ratio of resolved shear stress to Mohr–Coulomb frictional strength. Fsi = 1 lies on the failure envelope and Fsi ≥ 1 predicts shear failure. It is the principal metric used by the module's population-level fault-behaviour classification and spatial risk mapping.

13.5 Stress regime and orientation uncertainty

The stress-regime selector follows Andersonian normal, strike-slip and reverse faulting environments. Orientation concentration k provides a way to represent tight or diffuse structural clustering: higher k means a tighter orientation distribution, while lower k represents greater uncertainty. Fisher 3D provides a coupled strike-dip representation where appropriate.

13.6 Why uncertainty quantification matters

Deterministic stability calculations use single best-estimate values and can conceal measurement and interpretive uncertainty. LHS propagates those uncertainties through the same Mohr–Coulomb physics and produces probability distributions, allowing statements such as the probability that a particular fault is critically stressed under the assumed uncertainty model.

14. Implementation Notes

14.1 Primary Python modules

15. Limitations

• The stability model assumes planar faults with one strike/dip and does not model curvature, roughness or heterogeneous strength along a single fault.
• Cohesion, friction and pore pressure are spatially uniform per fault or population; depth-dependent and spatially varying property models are not implemented.
• Convergence checks combine a bootstrap precision bound with a two-increment stability test on five named metrics (Section 8.4); they are a practical stopping rule for those statistics, not a formal confidence statement about every derived quantity.
• SRC/PRCC can under-represent strongly non-monotonic input-output relationships, and squared SRCs are a variance decomposition only when the inputs are uncorrelated and the response is linear — read the reported R², ΣSRC² and largest input |r| alongside the tornado chart.
• Fracture susceptibility is always computed on effective normal stress and is exempt from the Effective-Stress switch, because a sustainable pore-pressure increase has no total-stress form; it is undefined where σ′_n ≤ 0 or μ ≤ 0, and those realizations are excluded from the failure probability rather than counted as non-failures.
• Stress regimes are realised by permuting which sampled magnitude enters each slot of a fixed geometry (Section 5.4), not by re-deriving the geometry; an unrecognised regime string falls back to reverse/thrust.
• A truncated normal cannot reach every (mean, SD) pair: its SD ceiling is (b − a)/√12 on a finite range (lower away from the midpoint), and SD/mean < 1 on [0, ∞). Consider Lognormal or Gamma for positive unbounded inputs, or Beta or a feasible Uniform law for friction on [0,1]; changing the law does not remove its support constraints.
• The ±1.96·s/√n interval on the statistics cards assumes independent, identically distributed rows, which a stratified LHS design pooled across faults does not satisfy; treat it as indicative and use the convergence diagnostics for decisions.
• Circular treatment covers the mean, dispersion and overlay bands of strike and SHmax and the folding of strike before sensitivity fitting; formal circular-uniformity tests (Rayleigh, Watson) on the sampled orientations are not computed.
• Fisher 3D uses one concentration parameter and cannot represent anisotropic elliptical orientation dispersion.
• Spatial mapping requires exact fault-ID matching; fuzzy or partial ID matching is not performed automatically.
• Computation is single-machine and single-process; practical ensemble size is limited by available memory and Matplotlib/Qt rendering responsiveness.

16. References

Sources are listed alphabetically within each group. In-text citations throughout this page link to the corresponding entry below.

16.1 Rock Mechanics, Effective Stress and Fault Stability

• Anderson, E. M. (1951). The Dynamics of Faulting and Dyke Formation with Applications to Britain (2nd ed.). Oliver and Boyd, Edinburgh. (Classification of normal, strike-slip and reverse faulting regimes.)
• Byerlee, J. (1978). Friction of rocks. Pure and Applied Geophysics, 116(4–5), 615–626. https://doi.org/10.1007/BF00876528
• Ferrill, D. A., & Morris, A. P. (2003). Dilational normal faults. Journal of Structural Geology, 25(2), 183–196. https://doi.org/10.1016/S0191-8141(02)00029-9
• Hubbert, M. K., & Rubey, W. W. (1959). Role of fluid pressure in mechanics of overthrust faulting. Geological Society of America Bulletin, 70(2), 115–166. https://doi.org/10.1130/0016-7606(1959)70[115:ROFPIM]2.0.CO;2
• Jaeger, J. C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell Publishing, Oxford. ISBN 978-0-632-05759-7. (Mohr–Coulomb failure theory, effective stress.)
• Mildren, S. D., Hillis, R. R., & Kaldi, J. (2002). Calibrating predictions of fault seal reactivation in the Timor Sea. The APPEA Journal, 42(1), 187–202. https://doi.org/10.1071/AJ01011
• Morris, A., Ferrill, D. A., & Henderson, D. B. (1996). Slip-tendency analysis and fault reactivation. Geology, 24(3), 275–278. https://doi.org/10.1130/0091-7613(1996)024<0275:STAAFR>2.3.CO;2
• Streit, J. E., & Hillis, R. R. (2004). Estimating fault stability and sustainable fluid pressures for underground storage of CO₂ in porous rock. Energy, 29(9–10), 1445–1456. https://doi.org/10.1016/j.energy.2004.03.078
• Terzaghi, K. (1943). Theoretical Soil Mechanics. John Wiley & Sons, New York. https://doi.org/10.1002/9780470172766 (Effective-stress principle.)
• Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press. https://doi.org/10.1017/CBO9780511586477

16.2 Probability Distributions and Directional Statistics

• Fisher, R. A. (1953). Dispersion on a sphere. Proceedings of the Royal Society of London A, 217(1130), 295–305. https://doi.org/10.1098/rspa.1953.0064
• Fisher, N. I. (1993). Statistical Analysis of Circular Data. Cambridge University Press. https://doi.org/10.1017/CBO9780511564345 (Sec. 2.3.1: circular mean and circular standard deviation.)
• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Continuous Univariate Distributions (Vol. 1, 2nd ed.). John Wiley & Sons. ISBN 978-0-471-58495-7. (Sec. 13.10: moments of the truncated normal.)
• Mardia, K. V., & Jupp, P. E. (2000). Directional Statistics. John Wiley & Sons. https://doi.org/10.1002/9780470316979 (Sec. 2.2.1, 2.3.2, 2.3.4 and ch. 3: circular and axial summaries, von Mises law.)

16.3 Sampling, Sensitivity and Convergence

• Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall. https://doi.org/10.1007/978-1-4899-4541-9 (Ch. 8: bootstrapping clustered and non-iid data.)
• Field, C. A., & Welsh, A. H. (2007). Bootstrapping clustered data. Journal of the Royal Statistical Society: Series B, 69(3), 369–390. https://doi.org/10.1111/j.1467-9868.2007.00593.x
• Helton, J. C., & Davis, F. J. (2003). Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliability Engineering & System Safety, 81(1), 23–69. https://doi.org/10.1016/S0951-8320(03)00058-9
• Iman, R. L., & Conover, W. J. (1982). A distribution-free approach to inducing rank correlation among input variables. Communications in Statistics — Simulation and Computation, 11(3), 311–334. https://doi.org/10.1080/03610918208812265
• Iman, R. L., & Helton, J. C. (1988). An investigation of uncertainty and sensitivity analysis techniques for computer models. Risk Analysis, 8(1), 71–90. https://doi.org/10.1111/j.1539-6924.1988.tb01155.x
• Marino, S., Hogue, I. B., Ray, C. J., & Kirschner, D. E. (2008). A methodology for performing global uncertainty and sensitivity analysis in systems biology. Journal of Theoretical Biology, 254(1), 178–196. https://doi.org/10.1016/j.jtbi.2008.04.011
• McKay, M. D., Beckman, R. J., & Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(2), 239–245. https://doi.org/10.2307/1268522
• Morris, M. D., & Mitchell, T. J. (1995). Exploratory designs for computational experiments. Journal of Statistical Planning and Inference, 43(3), 381–402. https://doi.org/10.1016/0378-3758(94)00035-T
• Roache, P. J. (1998). Verification and Validation in Computational Science and Engineering. Hermosa Publishers, Albuquerque. ISBN 978-0-913478-08-0. (Ch. 5: successive-refinement stability as a convergence test.)
• Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., & Tarantola, S. (2008). Global Sensitivity Analysis: The Primer. John Wiley & Sons. https://doi.org/10.1002/9780470725184

16.4 Software

• Harris, C. R., Millman, K. J., van der Walt, S. J., Gommers, R., Virtanen, P., Cournapeau, D., … Oliphant, T. E. (2020). Array programming with NumPy. Nature, 585(7825), 357–362. https://doi.org/10.1038/s41586-020-2649-2
• Hunter, J. D. (2007). Matplotlib: A 2D graphics environment. Computing in Science & Engineering, 9(3), 90–95. https://doi.org/10.1109/MCSE.2007.55
• McKinney, W. (2010). Data structures for statistical computing in Python. In Proceedings of the 9th Python in Science Conference (pp. 56–61). https://doi.org/10.25080/Majora-92bf1922-00a
• Virtanen, P., Gommers, R., Oliphant, T. E., Haberland, M., Reddy, T., Cournapeau, D., … van Mulbregt, P. (2020). SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods, 17(3), 261–272. https://doi.org/10.1038/s41592-019-0686-2