Stereonet
| Calculation | Published source |
|---|---|
| Equal-area (Schmidt) projection, R = √2 sin((90°−p)/2) | Lambert (1772); brought to petrofabrics by Schmidt (1925). Vector form as implemented: Allmendinger et al. (2012, ch. 2) |
| Equal-angle (Wulff) projection, R = tan((90°−p)/2) | Wulff (1902); Phillips (1971); Ragan (2009, ch. 2–4) |
| Choosing between them | Lisle and Leyshon (2004); Priest (1985) |
| Direction cosines from trend and plunge | Allmendinger et al. (2012, ch. 2, eq. 2.1–2.3); Mardia and Jupp (2000, ch. 9) |
| Right-hand-rule strike and dip, and its pole | Allmendinger et al. (2012, ch. 2); Marshak and Mitra (1988) |
| Quadrant notation (N45E) | Compton (1985, ch. 2) |
| Rake to trend and plunge | Aki and Richards (2002, sec. 4.4); Allmendinger et al. (2012, ch. 6) |
| Great-circle trace | Allmendinger et al. (2012, ch. 2); Lisle and Leyshon (2004, ch. 3) |
| Small circles and cones | Lisle and Leyshon (2004, ch. 4); Ragan (2009, ch. 5); Priest (1985, ch. 3) |
| Rodrigues’ rotation formula (rotation, unfolding) | Rodrigues (1840); Goldstein et al. (2001, sec. 4.8); Allmendinger et al. (2012, ch. 5, eq. 5.13) |
| Bedding restoration and partial tilt correction | Ramsay (1967); Allmendinger et al. (2012, ch. 5) |
| Plane intersections, dihedral angles, bisectors | Allmendinger et al. (2012, ch. 2) |
| Plane from two lineations or two apparent dips | Ragan (2009, ch. 7); Allmendinger et al. (2012, ch. 2) |
| Apparent dip, tan α = tan δ · sin β | Ragan (2009, ch. 1); Rowland et al. (2007, ch. 2); Marshak and Mitra (1988, ch. 7) |
| Calculation | Published source |
|---|---|
| Fisher mean direction, κ and α95 | Fisher (1953); Fisher et al. (1987, ch. 4–5); Mardia and Jupp (2000, ch. 10) |
| θ63, the half-angle containing 63% of the data | Fisher et al. (1987, sec. 5.3); Butler (1992, ch. 6) |
| α95 in paleomagnetic practice | Butler (1992, ch. 6); Tauxe (2010) |
| Folding axial data before Fisher statistics | Fisher et al. (1987, sec. 3.2); Mardia and Jupp (2000, sec. 9.2) |
| Orientation tensor and its eigenvalues | Watson (1965, 1966); Mark (1973) |
| Jacobi eigendecomposition | Jacobi (1846); Press et al. (2007, sec. 11.1) |
| Best-fit plane; π-axis from poles to bedding | Ramsay (1967, ch. 2); Fisher et al. (1987, ch. 3–4) |
| Woodcock shape K and strength C | Woodcock (1977) |
| Testing eigenvalue ratios against randomness | Woodcock and Naylor (1983) |
| Vollmer P–G–R fabric classification | Vollmer (1989, 1990) |
| Bingham distribution | Bingham (1974) |
| Bingham confidence ellipses | Onstott (1980); Fisher et al. (1987, ch. 10); Allmendinger et al. (2012, eq. 5.25) |
| Calculation | Published source |
|---|---|
| Kamb counting circle, density in standard deviations | Kamb (1959) |
| Linear-Kamb and exponential-Kamb weighting | Vollmer (1995) |
| Counting circles versus continuous weighting | Robin and Jowett (1986) |
| Exponential weighting as a Fisher-kernel density estimate | Diggle and Fisher (1985); Fisher et al. (1987, sec. 5.3) |
| Schmidt 1%-area point counting | Schmidt (1925); Turner and Weiss (1963, ch. 4) |
| Extension to fabric and finite-strain data | Vollmer (2018) |
| Marching-squares contour extraction | Lorensen and Cline (1987); Maple (2003) |
| Resolving ambiguous saddle cells | Nielson and Hamann (1991) |
| Terzaghi correction for scanline sampling bias | Terzaghi (1965); Priest (1993); Park and West (2002) |
| Calculation | Published source |
|---|---|
| Cylindrical best fit (π-axis) | Ramsay (1967, ch. 2); Allmendinger et al. (2012, ch. 6) |
| Conical (small-circle) fold-axis fit | Ramsay (1967, ch. 8); Gray et al. (1980) |
| Cylindrical, circular-conical or elliptical-conical folds | Charlesworth et al. (1976); Kelker and Langenberg (1976) |
| P, B and T axes from a fault–slip pair | Marrett and Allmendinger (1990, eq. 4–6); Allmendinger et al. (2012, ch. 6–7) |
| Tangent-lineation diagram (arrow convention) | Marrett and Allmendinger (1990); Twiss and Unruh (1998); originally Hoeppener (1955) |
| Fault-and-striae plots | Angelier (1979) |
| Auxiliary nodal plane and focal mechanism | Aki and Richards (2002, sec. 4.4); Allmendinger et al. (2012, eq. 7.34); Stein and Wysession (2003, ch. 4) |
| Moment tensors, for background | Jost and Herrmann (1989) |
| Stress inversion as an alternative reading of the same data | Angelier (1984); Delvaux and Sperner (2003) |
| Cautions on fault-slip inversion | Twiss and Unruh (1998); Sperner and Zweigel (2010) |
| Markland test for planar and wedge sliding | Markland (1972); Hoek and Bray (1981); Wyllie and Mah (2004, ch. 2, 7) |
| Daylight envelope | Lisle (2004) |
| Flexural toppling and the slip limit | Goodman and Bray (1976); Goodman (1989) |
| Hemispherical projection in rock mechanics | Priest (1985); Hudson and Harrison (1997) |
| Delineating joint sets from clusters of poles | Shanley and Mahtab (1976); Priest (1993) |
Rose diagram
| Calculation | Published source |
|---|---|
| Bin width, axial versus vectorial, area scaling | Nemec (1988); Wells (2000); Sanderson and Peacock (2020) |
| Smoothed and moving-average roses | Munro and Blenkinsop (2012); Baas (2000) |
| Rose diagrams in paleocurrent work | Curray (1956); Potter and Pettijohn (1977) |
| Doubling axial angles, halving the mean | Krumbein (1939); Curray (1956); Fisher (1993, sec. 2.3) |
| Halving the circular standard deviation for axial data | Batschelet (1981, ch. 1) |
| Circular mean and resultant R̄ | Fisher (1993, sec. 2.2); Mardia and Jupp (2000, ch. 2–3) |
| Circular standard deviation σ = √(−2 ln R̄) | Mardia and Jupp (2000, ch. 2) |
| von Mises κ, piecewise maximum-likelihood estimate | Best and Fisher (1981); Fisher (1993, sec. 4.5.6); Mardia and Jupp (2000, sec. 5.3) |
| α95 confidence cone on the mean | Fisher (1953); Fisher (1993, sec. 4.4.3) |
| Rayleigh test for preferred orientation | Rayleigh (1880); Wilkie (1983); Mardia and Jupp (2000, sec. 6.3) |
| Alternatives for multimodal data (Kuiper, Watson U², Rao) | Fisher (1993, sec. 4.3); Mardia and Jupp (2000, ch. 6) |
| Detecting modal sets by circular kernel density | Fisher (1993, ch. 5); Taylor (2008); Oliveira et al. (2012) |
| Reflecting a bounded domain at its edges | Silverman (1986, sec. 2.10); Schuster (1985) |
Histogram
| Calculation | Published source |
|---|---|
| Bin-width rules against which fixed widths are judged | Sturges (1926); Scott (1979); Freedman and Diaconis (1981); Scott (1992, ch. 3) |
| Bin width for circular variables | Sanderson and Peacock (2020) |
| Quartiles by interpolation of order statistics | Hyndman and Fan (1996), their type 7 (the default in R and NumPy) |
| Skewness (Fisher–Pearson standardized third moment) | Joanes and Gill (1998) |
| Circular mean, R̄ and circular standard deviation | Fisher (1993, sec. 2.3); Mardia and Jupp (2000, ch. 2) |
| Circular median and quartiles | Mardia (1972, ch. 2); Fisher (1993, sec. 2.3.2) |
| Rayleigh test | Rayleigh (1880); Mardia and Jupp (2000, sec. 6.3) |
| Gaussian kernel density, h = 1.06 σ n−1/5 | Silverman (1986, eq. 3.28), the normal reference rule |
| von Mises kernel for circular variables | Fisher (1993, ch. 5); Mardia and Jupp (2000, sec. 12.2); Taylor (2008) |
| Modified Bessel function I₀ | Abramowitz and Stegun (1965, sec. 9.8); Press et al. (2007, sec. 6.6) |
| Empirical cumulative distribution | van der Vaart (1998, ch. 19); Davis (2002, ch. 2) |
Scatter plot
| Calculation | Published source |
|---|---|
| Pearson product-moment correlation | Pearson (1896) |
| Ordinary least squares; slope, intercept, r² | Legendre (1805); Draper and Smith (1998, ch. 1); Davis (2002, ch. 4) |
| Why least squares is not symmetric in x and y | Davis (2002, sec. 4.6); Warton et al. (2006) |
| Correlation for angular data, done properly | Mardia (1976); Johnson and Wehrly (1977); Fisher and Lee (1981, 1983) |
| Regression on an angular variable | Gould (1969); Fisher and Lee (1992) |
| 95% covariance ellipse, χ²(2 df) = 5.991 | Johnson and Wichern (2007, ch. 4–5); Friendly et al. (2013) |
| Convex hull (monotone chain) | Andrew (1979); cf. Graham (1972); de Berg et al. (2008, ch. 1) |
| Two-dimensional kernel density, h ∝ σ n−1/6 | Silverman (1986, sec. 4.3); Scott (1992, sec. 6.3); Botev et al. (2010) |
| Contour extraction and saddle-cell handling | Lorensen and Cline (1987); Nielson and Hamann (1991) |
| Smoothing the traced contour | Catmull and Rom (1974) |
| Axis tick stepping (the 1/2/5 rule) | Heckbert (1990); Wilkinson (2005, ch. 5); Talbot et al. (2010) |
| Marginal histograms and overplotting | Carr et al. (1987); Cleveland (1993, ch. 3); Tufte (2001) |
Map view
| Calculation | Published source |
|---|---|
| Mercator projection, forward and inverse | Snyder (1987, ch. 7) |
| Web Mercator (EPSG:3857) and what it distorts | Battersby et al. (2014) |
| Tile pyramid (z/x/y) scheme | Open Geospatial Consortium (2019b) |
| UTM: meridional arc and footpoint latitude | Snyder (1987, sec. 8, eq. 3-21 and 3-26), the equations as implemented |
| Zone definition, scale factor, false easting and northing | Snyder (1987, sec. 8); National Geospatial-Intelligence Agency (1989) |
| Norway and Svalbard grid-zone exceptions | National Geospatial-Intelligence Agency (1990) |
| Accuracy limits of the truncated series | Karney (2011) |
| WGS84 ellipsoid parameters | National Geospatial-Intelligence Agency (2000) |
| Coordinate reference system definitions | International Association of Oil and Gas Producers (2024) |
| Magnetic declination model | Chulliat et al. (2020); Alken et al. (2021) |
| Why declination moves azimuths but not rakes or dips | Compton (1985, ch. 2); Lisle and Leyshon (2004, ch. 1) |
| Strike-and-dip and lineation symbol set | Federal Geographic Data Committee (2006); U.S. Geological Survey (2020) |
| Field conventions the symbols encode | Compton (1985); Barnes and Lisle (2004) |
| Hillshade computation | Horn (1981) |
| Elevation data behind topographic and hillshade tiles | Farr et al. (2007) |
| Raster overlay import (GeoTIFF) | Ritter and Ruth (1997); Open Geospatial Consortium (2019a) |
| Calculation | Published source |
|---|---|
| Spatial index for viewport culling | Bentley and Friedman (1979); Samet (2006, ch. 1–2) |
| Thinning point features as scale decreases | Töpfer and Pillewizer (1966); McMaster and Shea (1992) |
| Screen-cell binning as a declutter strategy | Carr et al. (1987) |
| Keeping labels clear of the symbols they name | Imhof (1975); Christensen et al. (1995) |
| Cartographic design generally | Imhof (1982); Slocum et al. (2009) |
| Inverse-distance weighting for the strain grid | Shepard (1968) |
| Orientation tensor of P or T axes at each node | Watson (1966); Woodcock (1977); Marrett and Allmendinger (1990) |
| Grid-based distance-weighted strain from scattered data | Cardozo and Allmendinger (2009) |
| Summing many small deformations into a regional strain | Kostrov (1974); Molnar (1983) |
| Smoothed grids of stress and strain orientation | Heidbach et al. (2010, 2018) |
| Interpolation methods compared | Li and Heap (2014) |
| Point-in-polygon test for lasso selection | Shimrat (1962); Hormann and Agathos (2001) |
References cited
Abramowitz, M., and Stegun, I.A., eds., 1965, Handbook of mathematical functions with formulas, graphs, and mathematical tables: New York, Dover Publications, 1046 p.
Aki, K., and Richards, P.G., 2002, Quantitative seismology (2nd ed.): Sausalito, California, University Science Books, 700 p.
Alken, P., Thébault, E., Beggan, C.D., and others, 2021, International Geomagnetic Reference Field: The thirteenth generation: Earth, Planets and Space, v. 73, no. 49, https://doi.org/10.1186/s40623-020-01288-x.
Allmendinger, R.W., Cardozo, N., and Fisher, D.M., 2012, Structural geology algorithms: Vectors and tensors: Cambridge, UK, Cambridge University Press, 289 p.
Andrew, A.M., 1979, Another efficient algorithm for convex hulls in two dimensions: Information Processing Letters, v. 9, p. 216–219, https://doi.org/10.1016/0020-0190(79)90072-3.
Angelier, J., 1979, Determination of the mean principal directions of stresses for a given fault population: Tectonophysics, v. 56, p. T17–T26, https://doi.org/10.1016/0040-1951(79)90081-7.
Angelier, J., 1984, Tectonic analysis of fault slip data sets: Journal of Geophysical Research, v. 89, p. 5835–5848, https://doi.org/10.1029/JB089iB07p05835.
Baas, J.H., 2000, EZ-ROSE: A computer program for equal-area circular histograms and statistical analysis of two-dimensional vectorial data: Computers & Geosciences, v. 26, p. 153–166, https://doi.org/10.1016/S0098-3004(99)00072-2.
Barnes, J.W., and Lisle, R.J., 2004, Basic geological mapping (4th ed.): Chichester, UK, John Wiley & Sons, 184 p.
Batschelet, E., 1981, Circular statistics in biology: London, Academic Press, 371 p.
Battersby, S.E., Finn, M.P., Usery, E.L., and Yamamoto, K.H., 2014, Implications of Web Mercator and its use in online mapping: Cartographica, v. 49, p. 85–101, https://doi.org/10.3138/carto.49.2.2313.
Bentley, J.L., and Friedman, J.H., 1979, Data structures for range searching: ACM Computing Surveys, v. 11, p. 397–409, https://doi.org/10.1145/356789.356797.
Best, D.J., and Fisher, N.I., 1981, The bias of the maximum likelihood estimators of the von Mises–Fisher concentration parameters: Communications in Statistics - Simulation and Computation, v. 10, p. 493–502.
Bingham, C., 1974, An antipodally symmetric distribution on the sphere: Annals of Statistics, v. 2, p. 1201–1225, https://doi.org/10.1214/aos/1176342874.
Botev, Z.I., Grotowski, J.F., and Kroese, D.P., 2010, Kernel density estimation via diffusion: Annals of Statistics, v. 38, p. 2916–2957, https://doi.org/10.1214/10-AOS799.
Butler, R.F., 1992, Paleomagnetism: Magnetic domains to geologic terranes: Boston, Blackwell Scientific, 319 p.
Cardozo, N., and Allmendinger, R.W., 2009, SSPX: A program to compute strain from displacement/velocity data: Computers & Geosciences, v. 35, p. 1343–1357, https://doi.org/10.1016/j.cageo.2008.05.008.
Cardozo, N., and Allmendinger, R.W., 2013, Spherical projections with OSXStereonet: Computers & Geosciences, v. 51, p. 193–205, https://doi.org/10.1016/j.cageo.2012.07.021.
Carr, D.B., Littlefield, R.J., Nicholson, W.L., and Littlefield, J.S., 1987, Scatterplot matrix techniques for large N: Journal of the American Statistical Association, v. 82, p. 424–436, https://doi.org/10.1080/01621459.1987.10478445.
Catmull, E., and Rom, R., 1974, A class of local interpolating splines, in Barnhill, R.E., and Riesenfeld, R.F., eds., Computer aided geometric design: New York, Academic Press, p. 317–326.
Charlesworth, H.A.K., Langenberg, C.W., and Ramsden, J., 1976, Determining axes, axial planes, and sections of macroscopic folds using computer-based methods: Canadian Journal of Earth Sciences, v. 13, p. 54–65, https://doi.org/10.1139/e76-006.
Christensen, J., Marks, J., and Shieber, S., 1995, An empirical study of algorithms for point-feature label placement: ACM Transactions on Graphics, v. 14, p. 203–232, https://doi.org/10.1145/212332.212334.
Chulliat, A., Brown, W., Alken, P., Beggan, C., Nair, M., Cox, G., Woods, A., Macmillan, S., Meyer, B., and Paniccia, M., 2020, The US/UK World Magnetic Model for 2020–2025: Boulder, Colorado, NOAA National Centers for Environmental Information Technical Report, 112 p., https://doi.org/10.25923/ytk1-yx35.
Cleveland, W.S., 1993, Visualizing data: Summit, New Jersey, Hobart Press, 360 p.
Compton, R.R., 1985, Geology in the field: New York, John Wiley & Sons, 398 p.
Curray, J.R., 1956, The analysis of two-dimensional orientation data: The Journal of Geology, v. 64, p. 117–131, https://doi.org/10.1086/626329.
Davis, J.C., 2002, Statistics and data analysis in geology (3rd ed.): New York, John Wiley & Sons, 638 p.
de Berg, M., Cheong, O., van Kreveld, M., and Overmars, M., 2008, Computational geometry: Algorithms and applications (3rd ed.): Berlin, Springer, 386 p.
Delvaux, D., and Sperner, B., 2003, New aspects of tectonic stress inversion with reference to the TENSOR program, in Nieuwland, D.A., ed., New insights into structural interpretation and modelling: Geological Society, London, Special Publication 212, p. 75–100, https://doi.org/10.1144/GSL.SP.2003.212.01.06.
Diggle, P.J., and Fisher, N.I., 1985, SPHERE: A contouring program for spherical data: Computers & Geosciences, v. 11, p. 725–766, https://doi.org/10.1016/0098-3004(85)90015-9.
Draper, N.R., and Smith, H., 1998, Applied regression analysis (3rd ed.): New York, John Wiley & Sons, 706 p.
Farr, T.G., Rosen, P.A., Caro, E., and others, 2007, The Shuttle Radar Topography Mission: Reviews of Geophysics, v. 45, RG2004, https://doi.org/10.1029/2005RG000183.
Federal Geographic Data Committee, 2006, FGDC digital cartographic standard for geologic map symbolization: Reston, Virginia, FGDC-STD-013-2006, U.S. Geological Survey Techniques and Methods 11-A2, 290 p.
Fisher, N.I., 1993, Statistical analysis of circular data: Cambridge, UK, Cambridge University Press, 277 p.
Fisher, N.I., and Lee, A.J., 1981, Nonparametric measures of angular-linear association: Biometrika, v. 68, p. 629–636, https://doi.org/10.1093/biomet/68.3.629.
Fisher, N.I., and Lee, A.J., 1983, A correlation coefficient for circular data: Biometrika, v. 70, p. 327–332, https://doi.org/10.1093/biomet/70.2.327.
Fisher, N.I., and Lee, A.J., 1992, Regression models for an angular response: Biometrics, v. 48, p. 665–677, https://doi.org/10.2307/2532334.
Fisher, N.I., Lewis, T., and Embleton, B.J.J., 1987, Statistical analysis of spherical data: Cambridge, UK, Cambridge University Press, 329 p.
Fisher, R.A., 1953, Dispersion on a sphere: Proceedings of the Royal Society of London, Series A, v. 217, p. 295–305, https://doi.org/10.1098/rspa.1953.0064.
Freedman, D., and Diaconis, P., 1981, On the histogram as a density estimator: L₂ theory: Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, v. 57, p. 453–476, https://doi.org/10.1007/BF01025868.
Friendly, M., Monette, G., and Fox, J., 2013, Elliptical insights: Understanding statistical methods through elliptical geometry: Statistical Science, v. 28, p. 1–39, https://doi.org/10.1214/12-STS402.
Goldstein, H., Poole, C., and Safko, J., 2001, Classical mechanics (3rd ed.): San Francisco, Addison Wesley, 638 p.
Goodman, R.E., 1989, Introduction to rock mechanics (2nd ed.): New York, John Wiley & Sons, 562 p.
Goodman, R.E., and Bray, J.W., 1976, Toppling of rock slopes, in Proceedings of the Specialty Conference on Rock Engineering for Foundations and Slopes: Boulder, Colorado, American Society of Civil Engineers, v. 2, p. 201–234.
Gould, A.L., 1969, A regression technique for angular variates: Biometrics, v. 25, p. 683–700, https://doi.org/10.2307/2528567.
Graham, R.L., 1972, An efficient algorithm for determining the convex hull of a finite planar set: Information Processing Letters, v. 1, p. 132–133, https://doi.org/10.1016/0020-0190(72)90045-2.
Gray, N.H., Geiser, P.A., and Geiser, J.R., 1980, On the least-squares fit of small and great circles to spherically projected orientation data: Journal of the International Association for Mathematical Geology, v. 12, p. 173–184, https://doi.org/10.1007/BF01091203.
Heckbert, P.S., 1990, Nice numbers for graph labels, in Glassner, A.S., ed., Graphics gems: Boston, Academic Press, p. 61–63.
Heidbach, O., Tingay, M., Barth, A., Reinecker, J., Kurfeß, D., and Müller, B., 2010, Global crustal stress pattern based on the World Stress Map database release 2008: Tectonophysics, v. 482, p. 3–15, https://doi.org/10.1016/j.tecto.2009.07.023.
Heidbach, O., Rajabi, M., Cui, X., and others, 2018, The World Stress Map database release 2016: Crustal stress pattern across scales: Tectonophysics, v. 744, p. 484–498, https://doi.org/10.1016/j.tecto.2018.07.007.
Hoek, E., and Bray, J.W., 1981, Rock slope engineering (3rd ed.): London, Institution of Mining and Metallurgy, 358 p.
Hoeppener, R., 1955, Tektonik im Schiefergebirge: Geologische Rundschau, v. 44, p. 26–58, https://doi.org/10.1007/BF01802930.
Hormann, K., and Agathos, A., 2001, The point in polygon problem for arbitrary polygons: Computational Geometry, v. 20, p. 131–144, https://doi.org/10.1016/S0925-7721(01)00012-8.
Horn, B.K.P., 1981, Hill shading and the reflectance map: Proceedings of the IEEE, v. 69, p. 14–47, https://doi.org/10.1109/PROC.1981.11918.
Hudson, J.A., and Harrison, J.P., 1997, Engineering rock mechanics: An introduction to the principles: Oxford, UK, Pergamon, 444 p.
Hyndman, R.J., and Fan, Y., 1996, Sample quantiles in statistical packages: The American Statistician, v. 50, p. 361–365, https://doi.org/10.1080/00031305.1996.10473566.
Imhof, E., 1975, Positioning names on maps: The American Cartographer, v. 2, p. 128–144, https://doi.org/10.1559/152304075784313304.
Imhof, E., 1982, Cartographic relief presentation: Berlin, De Gruyter, 389 p.
International Association of Oil and Gas Producers, 2024, EPSG geodetic parameter dataset: London, IOGP Geomatics Committee, https://epsg.org.
Jacobi, C.G.J., 1846, Über ein leichtes Verfahren, die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen: Journal für die reine und angewandte Mathematik, v. 30, p. 51–94.
Joanes, D.N., and Gill, C.A., 1998, Comparing measures of sample skewness and kurtosis: Journal of the Royal Statistical Society, Series D, v. 47, p. 183–189, https://doi.org/10.1111/1467-9884.00122.
Johnson, R.A., and Wehrly, T.E., 1977, Measures and models for angular correlation and angular-linear correlation: Journal of the Royal Statistical Society, Series B, v. 39, p. 222–229.
Johnson, R.A., and Wichern, D.W., 2007, Applied multivariate statistical analysis (6th ed.): Upper Saddle River, New Jersey, Pearson Prentice Hall, 773 p.
Jost, M.L., and Herrmann, R.B., 1989, A student’s guide to and review of moment tensors: Seismological Research Letters, v. 60, p. 37–57, https://doi.org/10.1785/gssrl.60.2.37.
Kamb, W.B., 1959, Ice petrofabric observations from Blue Glacier, Washington, in relation to theory and experiment: Journal of Geophysical Research, v. 64, p. 1891–1909, https://doi.org/10.1029/JZ064i011p01891.
Karney, C.F.F., 2011, Transverse Mercator with an accuracy of a few nanometers: Journal of Geodesy, v. 85, p. 475–485, https://doi.org/10.1007/s00190-011-0445-3.
Kelker, D., and Langenberg, C.W., 1976, A mathematical model for orientation data from macroscopic conical folds: Journal of the International Association for Mathematical Geology, v. 8, p. 549–559, https://doi.org/10.1007/BF01042995.
Kostrov, V.V., 1974, Seismic moment and energy of earthquakes, and seismic flow of rock: Izvestiya, Academy of Sciences of the USSR, Physics of the Solid Earth, v. 1, p. 23–44.
Krumbein, W.C., 1939, Preferred orientation of pebbles in sedimentary deposits: The Journal of Geology, v. 47, p. 673–706, https://doi.org/10.1086/624827.
Lambert, J.H., 1772, Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten: Beyträge zum Gebrauche der Mathematik und deren Anwendung, pt. III, sec. 6.
Legendre, A.M., 1805, Nouvelles méthodes pour la détermination des orbites des comètes: Paris, Firmin Didot, 80 p.
Li, J., and Heap, A.D., 2014, Spatial interpolation methods applied in the environmental sciences: A review: Environmental Modelling & Software, v. 53, p. 173–189, https://doi.org/10.1016/j.envsoft.2013.12.008.
Lisle, R.J., 2004, Calculation of the daylight envelope for plane failures of rock slopes: Géotechnique, v. 54, p. 279–280, https://doi.org/10.1680/geot.2004.54.4.279.
Lisle, R.J., and Leyshon, P.R., 2004, Stereographic projection techniques for geologists and civil engineers (2nd ed.): Cambridge, UK, Cambridge University Press, 112 p.
Lorensen, W.E., and Cline, H.E., 1987, Marching cubes: A high resolution 3D surface construction algorithm: ACM SIGGRAPH Computer Graphics, v. 21, p. 163–169, https://doi.org/10.1145/37402.37422.
Maple, C., 2003, Geometric design and space planning using the marching squares and marching cube algorithms, in Proceedings of the 2003 International Conference on Geometric Modeling and Graphics: London, IEEE, p. 90–95.
Mardia, K.V., 1972, Statistics of directional data: London, Academic Press, 357 p.
Mardia, K.V., 1976, Linear-circular correlation coefficients and rhythmometry: Biometrika, v. 63, p. 403–405, https://doi.org/10.1093/biomet/63.2.403.
Mardia, K.V., and Jupp, P.E., 2000, Directional statistics: Chichester, UK, John Wiley & Sons, 429 p.
Mark, D.M., 1973, Analysis of axial orientation data, including till fabrics: Geological Society of America Bulletin, v. 84, p. 1369–1374.
Markland, J.T., 1972, A useful technique for estimating the stability of rock slopes when the rigid wedge sliding type of failure is expected: London, Imperial College Rock Mechanics Research Report 19, 10 p.
Marrett, R., and Allmendinger, R.W., 1990, Kinematic analysis of fault-slip data: Journal of Structural Geology, v. 12, p. 973–986, https://doi.org/10.1016/0191-8141(90)90093-E.
Marshak, S., and Mitra, G., 1988, Basic methods of structural geology: Englewood Cliffs, New Jersey, Prentice Hall, 446 p.
McMaster, R.B., and Shea, K.S., 1992, Generalization in digital cartography: Washington, D.C., Association of American Geographers, 134 p.
Molnar, P., 1983, Average regional strain due to slip on numerous faults of different orientations: Journal of Geophysical Research, v. 88, p. 6430–6432, https://doi.org/10.1029/JB088iB08p06430.
Munro, M.A., and Blenkinsop, T.G., 2012, MARD: A moving average rose diagram application for the geosciences: Computers & Geosciences, v. 49, p. 112–120, https://doi.org/10.1016/j.cageo.2012.07.012.
National Geospatial-Intelligence Agency, 1989, The universal grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS): Defense Mapping Agency Technical Manual 8358.2, 92 p.
National Geospatial-Intelligence Agency, 1990, Datums, ellipsoids, grids, and grid reference systems: Defense Mapping Agency Technical Manual 8358.1, 74 p.
National Geospatial-Intelligence Agency, 2000, Department of Defense World Geodetic System 1984 (3rd ed., amendment 1): NIMA Technical Report 8350.2, 175 p.
Nemec, W., 1988, The shape of the rose: Sedimentary Geology, v. 59, p. 149–152, https://doi.org/10.1016/0037-0738(88)90105-4.
Nielson, G.M., and Hamann, B., 1991, The asymptotic decider: Resolving the ambiguity in marching cubes, in Proceedings of IEEE Visualization ’91: Los Alamitos, California, IEEE Computer Society Press, p. 83–91.
Oliveira, M., Crujeiras, R.M., and Rodríguez-Casal, A., 2012, A plug-in rule for bandwidth selection in circular density estimation: Computational Statistics & Data Analysis, v. 56, p. 3898–3908, https://doi.org/10.1016/j.csda.2012.05.021.
Onstott, T.C., 1980, Application of the Bingham distribution function in paleomagnetic studies: Journal of Geophysical Research, v. 85, p. 1500–1510, https://doi.org/10.1029/JB085iB03p01500.
Open Geospatial Consortium, 2019a, OGC GeoTIFF standard, version 1.1: OGC Document 19-008r4.
Open Geospatial Consortium, 2019b, OGC two dimensional tile matrix set: OGC Document 17-083r2.
Park, H.J., and West, T.R., 2002, Sampling bias of discontinuity orientation caused by linear sampling technique: Engineering Geology, v. 66, p. 99–110, https://doi.org/10.1016/S0013-7952(02)00034-0.
Pearson, K., 1896, Mathematical contributions to the theory of evolution. III. Regression, heredity, and panmixia: Philosophical Transactions of the Royal Society of London, Series A, v. 187, p. 253–318, https://doi.org/10.1098/rsta.1896.0007.
Phillips, F.C., 1971, The use of stereographic projection in structural geology (3rd ed.): London, Edward Arnold, 90 p.
Potter, P.E., and Pettijohn, F.J., 1977, Paleocurrents and basin analysis (2nd ed.): Berlin, Springer-Verlag, 425 p.
Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P., 2007, Numerical recipes: The art of scientific computing (3rd ed.): Cambridge, UK, Cambridge University Press, 1235 p.
Priest, S.D., 1985, Hemispherical projection methods in rock mechanics: London, George Allen & Unwin, 124 p.
Priest, S.D., 1993, Discontinuity analysis for rock engineering: London, Chapman & Hall, 473 p.
Ragan, D.M., 2009, Structural geology: An introduction to geometrical techniques (4th ed.): Cambridge, UK, Cambridge University Press, 602 p.
Ramsay, J.G., 1967, Folding and fracturing of rocks: New York, McGraw-Hill, 568 p.
Rayleigh, Lord (Strutt, J.W.), 1880, On the resultant of a large number of vibrations of the same pitch and of arbitrary phase: Philosophical Magazine, ser. 5, v. 10, p. 73–78, https://doi.org/10.1080/14786448008626893.
Ritter, N., and Ruth, M., 1997, The GeoTIFF data interchange standard for raster geographic images: International Journal of Remote Sensing, v. 18, p. 1637–1647, https://doi.org/10.1080/014311697218340.
Robin, P.-Y.F., and Jowett, E.C., 1986, Computerized density contouring and statistical evaluation of orientation data using counting circles and continuous weighting functions: Tectonophysics, v. 121, p. 207–223, https://doi.org/10.1016/0040-1951(86)90044-2.
Rodrigues, O., 1840, Des lois géométriques qui régissent les déplacements d’un système solide dans l’espace: Journal de Mathématiques Pures et Appliquées, v. 5, p. 380–440.
Rowland, S.M., Duebendorfer, E.M., and Schiefelbein, I.M., 2007, Structural analysis and synthesis (3rd ed.): Malden, Massachusetts, Blackwell, 301 p.
Samet, H., 2006, Foundations of multidimensional and metric data structures: San Francisco, Morgan Kaufmann, 993 p.
Sanderson, D.J., and Peacock, D.C.P., 2020, Making rose diagrams fit-for-purpose: Earth-Science Reviews, v. 201, 103055, https://doi.org/10.1016/j.earscirev.2019.103055.
Schmidt, W., 1925, Gefügestatistik: Tschermaks Mineralogische und Petrographische Mitteilungen, v. 38, p. 392–423, https://doi.org/10.1007/BF02993943.
Schuster, E.F., 1985, Incorporating support constraints into nonparametric estimators of densities: Communications in Statistics - Theory and Methods, v. 14, p. 1123–1136, https://doi.org/10.1080/03610928508828965.
Scott, D.W., 1979, On optimal and data-based histograms: Biometrika, v. 66, p. 605–610, https://doi.org/10.1093/biomet/66.3.605.
Scott, D.W., 1992, Multivariate density estimation: Theory, practice, and visualization: New York, John Wiley & Sons, 317 p.
Shanley, R.J., and Mahtab, M.A., 1976, Delineation and analysis of clusters in orientation data: Journal of the International Association for Mathematical Geology, v. 8, p. 9–23, https://doi.org/10.1007/BF01039681.
Shepard, D., 1968, A two-dimensional interpolation function for irregularly-spaced data, in Proceedings of the 23rd ACM National Conference: New York, Association for Computing Machinery, p. 517–524, https://doi.org/10.1145/800186.810616.
Shimrat, M., 1962, Algorithm 112: Position of point relative to polygon: Communications of the ACM, v. 5, p. 434, https://doi.org/10.1145/368637.368653.
Silverman, B.W., 1986, Density estimation for statistics and data analysis: London, Chapman & Hall, 175 p.
Slocum, T.A., McMaster, R.B., Kessler, F.C., and Howard, H.H., 2009, Thematic cartography and geovisualization (3rd ed.): Upper Saddle River, New Jersey, Pearson Prentice Hall, 561 p.
Snyder, J.P., 1987, Map projections: A working manual: U.S. Geological Survey Professional Paper 1395, 383 p., https://doi.org/10.3133/pp1395.
Sperner, B., and Zweigel, P., 2010, A plea for more caution in fault-slip analysis: Tectonophysics, v. 482, p. 29–41, https://doi.org/10.1016/j.tecto.2009.07.019.
Stein, S., and Wysession, M., 2003, An introduction to seismology, earthquakes, and Earth structure: Malden, Massachusetts, Blackwell, 498 p.
Sturges, H.A., 1926, The choice of a class interval: Journal of the American Statistical Association, v. 21, p. 65–66, https://doi.org/10.1080/01621459.1926.10502161.
Talbot, J., Lin, S., and Hanrahan, P., 2010, An extension of Wilkinson’s algorithm for positioning tick labels on axes: IEEE Transactions on Visualization and Computer Graphics, v. 16, p. 1036–1043, https://doi.org/10.1109/TVCG.2010.130.
Tauxe, L., 2010, Essentials of paleomagnetism: Berkeley, University of California Press, 489 p.
Taylor, C.C., 2008, Automatic bandwidth selection for circular density estimation: Computational Statistics & Data Analysis, v. 52, p. 3493–3500, https://doi.org/10.1016/j.csda.2007.11.003.
Terzaghi, R.D., 1965, Sources of error in joint surveys: Géotechnique, v. 15, p. 287–304, https://doi.org/10.1680/geot.1965.15.3.287.
Töpfer, F., and Pillewizer, W., 1966, The principles of selection: The Cartographic Journal, v. 3, p. 10–16, https://doi.org/10.1179/caj.1966.3.1.10.
Tufte, E.R., 2001, The visual display of quantitative information (2nd ed.): Cheshire, Connecticut, Graphics Press, 197 p.
Turner, F.J., and Weiss, L.E., 1963, Structural analysis of metamorphic tectonites: New York, McGraw-Hill, 545 p.
Twiss, R.J., and Unruh, J.R., 1998, Analysis of fault slip inversions: Do they constrain stress or strain rate?: Journal of Geophysical Research, v. 103, p. 12,205–12,222, https://doi.org/10.1029/98JB00612.
U.S. Geological Survey, 2020, GeMS (Geologic Map Schema): A standard format for the digital publication of geologic maps and databases: U.S. Geological Survey Techniques and Methods 11-B10, 74 p., https://doi.org/10.3133/tm11B10.
van der Vaart, A.W., 1998, Asymptotic statistics: Cambridge, UK, Cambridge University Press, 443 p.
Vollmer, F.W., 1989, A triangular fabric plot with applications for structural analysis: Eos (Transactions, American Geophysical Union), v. 70, p. 463.
Vollmer, F.W., 1990, An application of eigenvalue methods to structural domain analysis: Geological Society of America Bulletin, v. 102, p. 786–791.
Vollmer, F.W., 1995, C program for automatic contouring of spherical orientation data using a modified Kamb method: Computers & Geosciences, v. 21, p. 31–49, https://doi.org/10.1016/0098-3004(94)00058-3.
Vollmer, F.W., 2018, Automatic contouring of geologic fabric and finite strain data on the unit hyperboloid: Computers & Geosciences, v. 115, p. 134–142, https://doi.org/10.1016/j.cageo.2018.03.006.
Warton, D.I., Wright, I.J., Falster, D.S., and Westoby, M., 2006, Bivariate line-fitting methods for allometry: Biological Reviews, v. 81, p. 259–291, https://doi.org/10.1017/S1464793106007007.
Watson, G.S., 1965, Equatorial distributions on a sphere: Biometrika, v. 52, p. 193–201, https://doi.org/10.1093/biomet/52.1-2.193.
Watson, G.S., 1966, The statistics of orientation data: The Journal of Geology, v. 74, p. 786–797, https://doi.org/10.1086/627211.
Wells, N.A., 2000, Are there better alternatives to standard rose diagrams?: Journal of Sedimentary Research, v. 70, p. 37–46, https://doi.org/10.1306/2DC408FC-0E47-11D7-8643000102C1865D.
Wilkie, D., 1983, Rayleigh test for randomness of circular data: Journal of the Royal Statistical Society, Series C, v. 32, p. 311–312, https://doi.org/10.2307/2347954.
Wilkinson, L., 2005, The grammar of graphics (2nd ed.): New York, Springer, 690 p.
Woodcock, N.H., 1977, Specification of fabric shapes using an eigenvalue method: Geological Society of America Bulletin, v. 88, p. 1231–1236.
Woodcock, N.H., and Naylor, M.A., 1983, Randomness testing in three-dimensional orientation data: Journal of Structural Geology, v. 5, p. 539–548, https://doi.org/10.1016/0191-8141(83)90058-5.
Wulff, G., 1902, Untersuchungen im Gebiete der optischen Eigenschaften isomorpher Kristalle: Zeitschrift für Kristallographie, v. 36, p. 1–28.
Wyllie, D.C., and Mah, C.W., 2004, Rock slope engineering: Civil and mining (4th ed.): London, Spon Press, 431 p.
Zar, J.H., 2010, Biostatistical analysis (5th ed.): Upper Saddle River, New Jersey, Prentice Hall, 944 p.
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