Package 'tectonicr'

Description

Calculates the absolute angular velocity of plate motion

Usage

abs_vel(w, alpha, r = earth_radius())

Arguments

Value

numeric (unit of velocity: km/Myr)

See Also

earth_radius()

Examples

abs_vel(0.21, 0)
abs_vel(0.21, 45)
abs_vel(0.21, 90)

Description

Calculates the angle between two vectors

Usage

angle_vectors(x, y)

Arguments

Value

numeric. angle in degrees

Examples

u <- c(1, -2, 3)
v <- c(-2, 1, 1)
angle_vectors(u, v) # 96.26395

Description

Helper functions to transform between angles in degrees and radians.

Usage

rad2deg(rad)

deg2rad(deg)

Arguments

Value

numeric. angle in degrees or radians.

Examples

deg2rad(seq(-90, 90, 15))
rad2deg(seq(-pi / 2, pi / 2, length = 13))

Description

Performs Angular Randomisation Test for homogeneity on two samples of circular data after Ruxton et al. (2023). P-values are estimated using permutation.

Usage

ar_test(x, y, n_perm = 1000L, axial = FALSE, alpha = NULL)

Arguments

Details

Null Hypothesis ($H_0$): The two circular samples share an identical underlying probability distribution.

Alternative Hypothesis ($H_{1}$): The two samples come from different distributions.

Interpretation

• Small p-value ($p < \alpha$, e.g., <0.05): Reject the null hypothesis. This indicates strong evidence that the two samples come from different circular distributions (differing in central tendency/mean direction or shape).

• Large p-value ($p \ge \alpha$): Fail to reject the null hypothesis; there is insufficient evidence to claim the two circular samples differ.

Value

list containing the test statistic, the p-value, the significance value alpha and a logical decision whether to reject the null hypothesis or not.

Note

Concentration Differences: The test can suffer from markedly lower statistical power if the underlying unimodal distributions differ by concentration (dispersion/spread) rather than location—especially with small, uneven sample sizes where the smaller sample comes from the more concentrated distribution.

Axial/Multimodal Data: ART performs poorly and loses power when applied to axially symmetric or symmetrically multimodal distributions.

References

Ruxton, G.D., Malkemper, E.P. & Landler, L. Evaluating the power of a recent method for comparing two circular distributions: an alternative to the Watson U2 test. Sci Rep 13, 10007 (2023). https://doi.org/10.1038/s41598-023-36960-1

See Also

Other Tests: kuiper_test(), norm_chisq(), rayleigh-test, watson_test(), watson_two_sample, watson_wheeler_test_perm(), weighted-rayleigh-test

Examples

set.seed(20250411)
x1 <- c(35, 45, 50, 55, 60, 70, 85, 95, 105, 120)
x2 <- c(75, 80, 90, 100, 110, 130, 135, 140, 150, 160, 165)
ar_test(x1, x2)

# San Andreas Fault Data:
data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
ar_test(sa.por$azi.PoR, rvm(100, 135, 10), axial = TRUE, alpha = 0.05)

Description

Show direction axes in a map

Usage

axes(
  x,
  y,
  angle,
  radius = 0.5,
  arrow.code = 1,
  arrow.length = 0,
  add = FALSE,
  ...
)

Arguments

Value

No return value, called for side effects

Examples

data("san_andreas")
axes(san_andreas$lon, san_andreas$lat, san_andreas$azi, add = FALSE)

Description

The circular mean difference is based on the sample circular distance

Usage

circular_mean_difference(x, w = NULL, axial = TRUE, na.rm = TRUE)

circular_mean_difference_alt(x, w = NULL, axial = TRUE, na.rm = TRUE)

Arguments

Value

numeric

References

Mardia, K.V., and Jupp, P.E (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, USA. doi:10.1002/9780470316979

See Also

sample_circular_distance()

Examples

data("san_andreas")
circular_mean_difference(san_andreas$azi)
circular_mean_difference(san_andreas$azi, weighting(san_andreas$unc))

circular_mean_difference_alt(san_andreas$azi)
circular_mean_difference_alt(san_andreas$azi, weighting(san_andreas$unc))

Description

Calculate the (weighted median) and standard deviation of orientation data.

Usage

circular_mean(x, w = NULL, axial = TRUE, na.rm = TRUE)

circular_var(x, w = NULL, axial = TRUE, na.rm = TRUE)

circular_sd(x, w = NULL, axial = TRUE, na.rm = TRUE)

circular_median(x, w = NULL, axial = TRUE, na.rm = TRUE)

circular_quantiles(x, w = NULL, axial = TRUE, na.rm = TRUE)

circular_IQR(x, w = NULL, axial = TRUE, na.rm = TRUE)

Arguments

Value

numeric vector

Note

Weighting may be the reciprocal of the data uncertainties.

Weightings have no effect on quasi-quantiles if length(x) %% 4 == 0.

References

Mardia, K.V. (1972). Statistics of Directional Data: Probability and Mathematical Statistics. London: Academic Press.

Mardia, K.V., and Jupp, P.E (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, USA. doi:10.1002/9780470316979

N.I. Fisher (1993) Statistical Analysis of Circular Data, Cambridge University Press.

Ziegler, M. O.; Heidbach O. (2019). Manual of the Matlab Script Stress2Grid v1.1. WSM Technical Report 19-02, GFZ German Research Centre for Geosciences. doi:10.2312/wsm.2019.002

Heidbach, O., Tingay, M., Barth, A., Reinecker, J., Kurfess, D., & Mueller, B. (2010). Global crustal stress pattern based on the World Stress Map database release 2008. Tectonophysics 482, 3<U+2013>15, doi:10.1016/j.tecto.2009.07.023

Examples

set.seed(1)
x <- rvm(10, 0, 100) %% 180
unc <- stats::runif(length(x), 0, 10)
w <- weighting(unc)
circular_mean(x, w)
circular_var(x, w)
circular_sd(x, w)
circular_median(x, w)
circular_quantiles(x, w)
circular_IQR(x, w)

data("san_andreas")
w2 <- weighting(san_andreas$unc)
circular_mean(san_andreas$azi)
circular_mean(san_andreas$azi, w2)
circular_median(san_andreas$azi)
circular_median(san_andreas$azi, w2)
circular_quantiles(san_andreas$azi)
circular_quantiles(san_andreas$azi, w2)
circular_var(san_andreas$azi)
circular_var(san_andreas$azi, w2)

data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
circular_mean(sa.por$azi.PoR, w2)
circular_median(sa.por$azi.PoR, w2)
circular_var(sa.por$azi.PoR, w2)
circular_quantiles(sa.por$azi.PoR, w2)

Description

Kernel density estimates for circular data from a given kernel (von Mises, wrapped Cauchy, and wrapped Normal distribution and bandwidth

Usage

circular_density(
  x,
  z = NULL,
  bw = NULL,
  weights = NULL,
  na.rm = TRUE,
  from = 0,
  to = 360,
  n = 512L,
  axial = TRUE,
  kappa = NULL,
  rho = NULL,
  sd = NULL,
  c = NULL,
  kernel = c("vonmises", "wrappedcauchy", "wrappednormal", "wrappedlevy"),
  adjust = 1,
  subdensity = FALSE
)

Arguments

Value

Object of class "density"

See Also

stats::density(), dvm(), dwcauchy(), dwnorm(), and plot_density()

Examples

w <-  weighting(san_andreas$unc)

# von Mises kernel density
circular_density(san_andreas$azi, kappa = 100)
circular_density(san_andreas$azi, weights = w, kappa = 100)

# wrapped Cauchy kernel density
circular_density(san_andreas$azi, rho = 0.9, kernel = "wrappedcauchy")

# wrapped Normal kernel density
circular_density(san_andreas$azi, sd = 5, kernel = "wrappednormal")

Description

Calculates bootstrapped estimates of the circular dispersion, its standard error and its confidence interval.

Usage

circular_dispersion_boot(
  x,
  y = NULL,
  w = NULL,
  w.y = NULL,
  R = 1000,
  conf.level = 0.95,
  ...
)

Arguments

Value

list containing:

MLE

the maximum likelihood estimate of the circular dispersion

sde

standard error of MLE

CI

lower and upper limit of the confidence interval of MLE

See Also

circular_dispersion()

Examples

data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
circular_dispersion(sa.por$azi.PoR, y = 135, w = weighting(san_andreas$unc))
circular_dispersion_boot(sa.por$azi.PoR, y = 135, w = weighting(san_andreas$unc), R = 1000)

Description

MLE angle (maximum density) using a circular distribution kernel with specified concentration

Usage

circular_mode(x, ...)

Arguments

Value

numeric

References

N.I. Fisher (1993) Statistical Analysis of Circular Data, Cambridge University Press.

Examples

set.seed(20250411)
x <- rvm(10, 0, 100)

# Mode of von Mises kernel density (the default)
circular_mode(x)

# Mode of wrapped Cauchy kernel density
circular_mode(x, kernel = "wrappedcauchy")

Description

Polar diagram where angles increase clockwise.

Usage

circular_plot(
  main = NULL,
  sub = NULL,
  labels = TRUE,
  at = c(grid.lines, grid.lines + 180),
  cborder = TRUE,
  radius = 1,
  scale = 1.1,
  grid = FALSE,
  grid.lines = seq(0, 135, 45),
  grid.circles = seq(0.2, radius, 0.2),
  origin.text = "N",
  ...
)

Arguments

Value

none

Examples

circular_plot(grid = TRUE)

Description

Uniformly distributed orientations should yield a straight line through the origin. Systematic departures from linearity will indicate preferred orientation.

Usage

circular_qqplot(
  x,
  axial = TRUE,
  xlab = paste("i/(n+1)"),
  ylab = NULL,
  main = "Circular Quantile-Quantile Plot",
  add_line = TRUE,
  col = "#B63679FF",
  ...
)

Arguments

Value

plot

References

Borradaile, G. J. (2003). Statistics of earth science data: their distribution in time, space, and orientation (Vol. 351, p. 329). Berlin: Springer.

See Also

cunif()

Other circ-qqplot: vm_qqplot()

Examples

set.seed(20250411)

# von Mises distribution
x_vm <- rvm(100, mean = 0, kappa = 2)
circular_qqplot(x_vm, pch = 20)

x_wcauchy <- rwcauchy(100, mean = 0, rho = 0.5)
circular_qqplot(x_wcauchy, pch = 20)

# circular uniform data
x_cunif <- rcunif(100)
circular_qqplot(x_cunif, pch = 20)

Description

Length of the smallest arc which contains all the observations. The circular range is based on the sample circular distance.

Usage

circular_range(x, axial = TRUE, na.rm = TRUE)

Arguments

Value

numeric. angle in degrees

References

Mardia, K.V., and Jupp, P.E (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, USA. doi:10.1002/9780470316979

See Also

sample_circular_distance()

Examples

roulette <- c(43, 45, 52, 61, 75, 88, 88, 279, 357)
circular_range(roulette, axial = FALSE)

data("san_andreas")
circular_range(san_andreas$azi)

Description

Measure of the chance variation expected from sample to sample in estimates of the mean direction (after Mardia 1972). It is a parametric estimate of the the circular standard error of the mean direction by the particular form of the standard error for the von Mises distribution. The approximated standard error of the mean direction is computed by the mean resultant length and the MLE concentration parameter $\kappa$.

Usage

circular_sd_error(x, w = NULL, axial = TRUE, na.rm = TRUE)

Arguments

Value

numeric

References

• Batschelet, E. (1971). Recent statistical methods for orientation data. "Animal Orientation, Symposium 1970 on Wallops Island". Amer. Inst. Biol. Sciences, Washington.

• Mardia, K.V. (1972). Statistics of Directional Data: Probability and Mathematical Statistics. London: Academic Press.

• N.I. Fisher (1993) Statistical Analysis of Circular Data, Cambridge University Press.

• Davis (1986) Statistics and data analysis in geology. 2nd ed., John Wiley & Sons.

See Also

mean_resultant_length(), circular_mean()

Examples

# Example data from Davis (1986), pp. 316
finland_stria <- c(
  23, 27, 53, 58, 64, 83, 85, 88, 93, 99, 100, 105, 113,
  113, 114, 117, 121, 123, 125, 126, 126, 126, 127, 127, 128, 128, 129, 132,
  132, 132, 134, 135, 137, 144, 145, 145, 146, 153, 155, 155, 155, 157, 163,
  165, 171, 172, 179, 181, 186, 190, 212
)
circular_sd_error(finland_stria, axial = FALSE)

data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
circular_sd_error(sa.por$azi.PoR, w = weighting(san_andreas$unc))

Description

Circular mean, standard deviation, variance, quasi-quantiles, mode, 95% confidence angle, standardized skewness and kurtosis

Usage

circular_summary(
  x,
  w = NULL,
  axial = TRUE,
  mode = FALSE,
  kappa = NULL,
  fisher.CI = FALSE,
  conf.level = 0.95,
  na.rm = FALSE
)

Arguments

Value

named vector

See Also

circular_mean(), circular_sd(), circular_var(), circular_quantiles(), confidence_angle(), second_central_moment(), circular_mode()

Examples

data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
circular_summary(sa.por$azi.PoR)
circular_summary(sa.por$azi.PoR, w = weighting(san_andreas$unc))

Description

Filter smoothed stress field containing a range of search radii or kernel half widths to find shortest wavelength (R) with the least circular sd. or dispersion (or any statistic) for each coordinate, respectively.

Usage

compact_grid(x, type = c("stress", "dispersion"))

compact_grid2(x, ..., FUN = min)

Arguments

Value

sf object

See Also

stress2grid(), PoR_stress2grid(), kernel_dispersion(), stress2grid_stats(), dplyr::dplyr_tidy_select()

Examples

data("san_andreas")
res <- stress2grid(san_andreas)
compact_grid(res) |> head()

## Not run: 
res2 <- stress2grid_stats(san_andreas)
compact_grid2(res2, var, FUN = min)

## End(Not run)

Description

Probabilistic limit on the location of the true or population mean direction, assuming that the estimation errors are normally distributed.

Usage

confidence_angle(x, conf.level = 0.95, w = NULL, axial = TRUE, na.rm = TRUE)

confidence_interval(x, conf.level = 0.95, w = NULL, axial = TRUE, na.rm = TRUE)

Arguments

Details

The confidence angle gives the interval, i.e. plus and minus the confidence angle, around the mean direction of a particular sample, that contains the true mean direction under a given level of confidence.

Value

Angle in degrees

References

• Batschelet, E. (1971). Recent statistical methods for orientation data. "Animal Orientation, Symposium 1970 on Wallops Island". Amer. Inst. Biol. Sciences, Washington.

• Mardia, K.V. (1972). Statistics of Directional Data: Probability and Mathematical Statistics. London: Academic Press. (p. 146)

• Davis (1986) Statistics and data analysis in geology. 2nd ed., John Wiley & Sons.

• Jammalamadaka, S. Rao and Sengupta, A. (2001). Topics in Circular Statistics, Sections 3.3.3 and 3.4.1, World Scientific Press, Singapore.

See Also

mean_resultant_length(), circular_sd_error()

Examples

# Example data from Davis (1986), pp. 316
finland_stria <- c(
  23, 27, 53, 58, 64, 83, 85, 88, 93, 99, 100, 105, 113,
  113, 114, 117, 121, 123, 125, 126, 126, 126, 127, 127, 128, 128, 129, 132,
  132, 132, 134, 135, 137, 144, 145, 145, 146, 153, 155, 155, 155, 157, 163,
  165, 171, 172, 179, 181, 186, 190, 212
)
confidence_angle(finland_stria, axial = FALSE)
confidence_interval(finland_stria, axial = FALSE)

data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
confidence_angle(sa.por$azi.PoR, w = weighting(san_andreas$unc))
confidence_interval(sa.por$azi.PoR, w = weighting(san_andreas$unc))

Description

For large samples (n >=25) i performs are parametric estimate based on sample_circular_dispersion(). For smaller size samples, it returns a bootstrap estimate.

Usage

confidence_interval_fisher(
  x,
  conf.level = 0.95,
  w = NULL,
  axial = TRUE,
  na.rm = TRUE,
  boot = FALSE,
  R = 1000L,
  quiet = FALSE
)

Arguments

Value

list

References

N.I. Fisher (1993) Statistical Analysis of Circular Data, Cambridge University Press.

Examples

# Example data from Davis (1986), pp. 316
finland_stria <- c(
  23, 27, 53, 58, 64, 83, 85, 88, 93, 99, 100, 105, 113,
  113, 114, 117, 121, 123, 125, 126, 126, 126, 127, 127, 128, 128, 129, 132,
  132, 132, 134, 135, 137, 144, 145, 145, 146, 153, 155, 155, 155, 157, 163,
  165, 171, 172, 179, 181, 186, 190, 212
)
confidence_interval_fisher(finland_stria, axial = FALSE)
confidence_interval_fisher(finland_stria, axial = FALSE, boot = TRUE)

data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
confidence_interval_fisher(sa.por$azi.PoR, w = weighting(san_andreas$unc))
confidence_interval_fisher(sa.por$azi.PoR, w = weighting(san_andreas$unc), boot = TRUE)

Description

Corrects the longitudes or latitudes to value between -180&deg and 180&deg or -90&deg and 90°

Usage

longitude_modulo(x)

latitude_modulo(x)

Arguments

Value

numeric

Examples

longitude_modulo(-361 + 5 * 360) # -1
latitude_modulo(-91 + 5 * 180) # 89

Description

Converts vector between Cartesian and geographical coordinate systems

Usage

cartesian_to_geographical(n)

geographical_to_cartesian(p)

geographical_to_spherical(p)

Arguments

Value

Functions return a (2- or 3-dimensional) vector representing a point in the requested coordinate system.

See Also

cartesian_to_spherical() and spherical_to_cartesian() for conversions to spherical coordinates

Examples

n <- c(1, -2, 3)
cartesian_to_geographical(n)
p <- c(50, 10)
geographical_to_cartesian(p)

Description

Converts vector between Cartesian and spherical coordinate systems

Usage

cartesian_to_spherical(n)

spherical_to_cartesian(p)

spherical_to_geographical(p)

Arguments

Value

Functions return a (2- or 3-dimensional) vector representing a point in the requested coordinate system.

See Also

cartesian_to_geographical() and geographical_to_cartesian() for conversions to geographical coordinates

Examples

n <- c(1, -2, 3)
cartesian_to_spherical(n) # 36.699, -63.435
p <- c(50, 10)
spherical_to_cartesian(p) # 0.75, 0.13, 0.64

Description

Compilation of global models for current plate motions, including NUVEL1 (DeMets et al. 1990), NNR-NUVEL1A (DeMets et al., 1990), NNR-MORVEL56 (Argus et al., 2011), REVEL (Sella et al., 2002), GSRM2.1 (Kreemer et al., 2014), HS2-NUVEL1 (Gripp and Gordon, 1990), HS3-NUVEL1A (Gripp and Gordon, 2002), P073 (Chase 1978), AM1-2 (Minster and Jordan, 1978), ITRF2020-PPM (Altamimi et al. 2023), and PB2002 (Bird, 2003)

Usage

data('cpm_models')

Format

list containing objects of class data.frame

plate.name

The rotating plate

plate.rot

The abbreviation of the plate's name

lat,lon

Coordinates of the Pole of Rotation

angle

The amount of rotation (angle in 1 Myr)

plate.fix

The anchored plate, i.e. plate.rot moves relative to plate.fix

model

Model for current global plate motion

References

Altamimi, Z., Métivier, L., Rebischung, P., Collilieux, X., Chanard, K., Barnéoud, J., 2023. ITRF2020 Plate Motion Model. Geophys. Res. Lett. 50, 1–7. doi:10.1029/2023GL106373

Argus, D.F., Gordon, R.G., 1991. No-net-rotation model of current plate velocities incorporating plate motion model NUVEL-1. Geophys. Res. Lett. 18, 2039–2042. doi: 10.1029/91GL01532

Argus, D. F., Gordon, R. G., & DeMets, C. (2011). Geologically current motion of 56 plates relative to the no-net-rotation reference frame. Geochemistry, Geophysics, Geosystems, 12(11). 10.1029/2011GC003751.

Chase, C.G. (1978). Plate kinematics: The Americas, East Africa, and the rest of the world. Earth Planet. Sci. Lett. 37, 355–368. doi: doi:10.1016/0012-821X(78)90051-1

Bird, P. (2003), An updated digital model of plate boundaries, Geochem. Geophys. Geosyst., 4, 1027, doi: 10.1029/2001GC000252, 3.

DeMets, C., Gordon, R. G., Argus, D. F., & Stein, S. (1990). Current plate motions. Geophysical Journal International, 101(2), 425-478. doi:10.1111/j.1365-246X.1990.tb06579.x.

Gripp, A. E., & Gordon, R. G. (2002). Young tracks of hotspots and current plate velocities. Geophysical Journal International, 150(2), 321<U+2013>361. doi:10.1046/j.1365-246X.2002.01627.x.

Kreemer, C., Blewitt, G., & Klein, E. C. (2014). A geodetic plate motion and Global Strain Rate Model. Geochemistry, Geophysics, Geosystems, 15(10), 3849<U+2013>3889. doi: 10.1002/2014GC005407.

Minster, J. and Jorda, T. (1978). Present-day plate motions. Journal of Geophysical Research, 83, doi:10.1029/jb083ib11p05331.

Sella, G. F., Dixon, T. H., & Mao, A. (2002). REVEL: A model for Recent plate velocities from space geodesy. Journal of Geophysical Research: Solid Earth, 107(B4). doi: 10.1029/2000jb000033.

Examples

data("cpm_models")
head(cpm_models[[1]])

Description

Density, probability distribution function, quantiles, and random generation for the circular uniform distribution.

Usage

rcunif(n, axial = FALSE)

dcunif(theta, axial = FALSE, log = FALSE)

pcunif(theta, axial = FALSE, lower.tail = TRUE, log.p = FALSE)

qcunif(p, axial = FALSE, lower.tail = TRUE, log.p = FALSE)

Arguments

Value

dcunif gives the density, pcunif gives the probability of circular uniform distribution function, rcunif generates random deviates (in degrees), and qcunif provides quantiles (in degrees).

See Also

wnorm, wcauchy, and vonmises

Examples

set.seed(1)
x <- rcunif(5)

dcunif(x)
dcunif(x, axial = TRUE)

pcunif(x)
qcunif(c(.25, .5, .75))

Description

Convenience function to add PoR coordinates and PoR azimuths to data

Usage

data2PoR(x, PoR)

Arguments

Value

sf object in PoR CRS with additional columns lon.PoR, lat.PoR, and azi.PoR

Examples

por <- subset(nuvel1, nuvel1$plate.rot == "na")
data2PoR(san_andreas, por)

Description

Normalizes the angle between two directions to the acute angle in between, i.e. angles between 0 and 90°

Usage

deviation_norm(x, y = NULL)

Arguments

Value

numeric vector, acute angles between two directions, i.e. values between 0 and 90°

Author(s)

Tobias Stephan

Examples

deviation_norm(175, 5)
deviation_norm(c(175, 95, 0), c(5, 85, NA))
deviation_norm(c(-5, 85, 95, 175, 185, 265, 275, 355, 365))

Description

Calculate the angular difference between the observed and modeled direction of maximum horizontal stresses ($\sigma_\text{Hmax}$) along great circles, small circles, and loxodromes of the relative plate motion's Euler pole

Usage

deviation_shmax(prd, obs)

Arguments

Details

Deviation is positive for counterclockwise deviation of observed azimuth wrt. predicted azimuth.

Value

An object of class data.frame

dev.gc

Deviation of observed stress from modeled $\sigma_\text{Hmax}$ following great circles

dev.sc

Small circles

dev.ld.cw

Clockwise loxodromes

dev.ld.ccw

Counter-clockwise loxodromes

Author(s)

Tobias Stephan

References

Stephan, T., Enkelmann, E., and Kroner, U. "Analyzing the horizontal orientation of the crustal stress adjacent to plate boundaries". Sci Rep 13. 15590 (2023). doi:10.1038/s41598-023-42433-2.

See Also

model_shmax() to calculate the theoretical direction of $\sigma_\text{Hmax}$.

Examples

data("nuvel1")
# North America relative to Pacific plate:
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")

# the point where we want to model the SHmax direction:
point <- data.frame(lat = 45, lon = 20)

prd <- model_shmax(point, PoR)
deviation_shmax(prd, obs = 90)

Description

Circular distance between two angles and circular dispersion of angles about a specified angle.

Usage

circular_distance(x, y, axial = TRUE, na.rm = TRUE)

circular_dispersion(
  x,
  y = NULL,
  w = NULL,
  w.y = NULL,
  axial = TRUE,
  na.rm = TRUE
)

circular_sd2(x, y, w = NULL, axial = TRUE, na.rm = TRUE)

Arguments

Details

Circular dispersion is a measure for the spread of data like the variance. Dispersion measures the spread about a given angles, whereas the variance measures the spread about the mean (Mardia and Jupp, 1999). When y = NULL the dispersion is identical to the variance.

Circular standard deviation in circular_sd2() is the transformed dispersion instead of the variance as for circular_sd().

Value

circular_distance returns a numeric vector of positive numbers, circular_dispersion and circular_sd2() return a positive number.

Note

If y is NULL, than the circular variance is returned.

References

Mardia, K.V. (1972). Statistics of Directional Data: Probability and Mathematical Statistics. London: Academic Press.

Mardia, K.V., and Jupp, P.E (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, USA. doi:10.1002/9780470316979

See Also

circular_mean(), circular_var().

Examples

a <- c(0, 2, 359, 6, 354)
circular_distance(a, 10) # distance to single value

b <- a + 90
circular_distance(a, b) # distance to multiple values

data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
circular_dispersion(sa.por$azi.PoR, y = 135)
circular_dispersion(sa.por$azi.PoR, y = 135, w = weighting(san_andreas$unc))
circular_sd2(sa.por$azi.PoR, y = 135, w = weighting(san_andreas$unc))

Description

Returns the great circle distance between a location and all grid point in km

Usage

dist_greatcircle(
  lat1,
  lon1,
  lat2,
  lon2,
  r = earth_radius(),
  method = c("haversine", "orthodrome", "vincenty", "euclidean")
)

Arguments

Value

numeric vector with length equal to length(lat1) or length(lat2)

See Also

orthodrome(), haversine(), vincenty()

Examples

# Haversine: (4149.157, 2296.583) km
dist_greatcircle(lat1 = 20, lon1 = 12, lat2 = c(50, 30), lon2 = c(40, 32))

# Orthodrome: (4149.157, 2296.583) km
dist_greatcircle(
  lat1 = 20, lon1 = 12, lat2 = c(50, 30), lon2 = c(40, 32),
  method = "orthodrome"
)

# Vincenty: (4149.157, 2296.583) km
dist_greatcircle(
  lat1 = 20, lon1 = 12, lat2 = c(50, 30), lon2 = c(40, 32),
  method = "vincenty"
)

# Euclidean (4076.220, 2284.169) km
dist_greatcircle(
  lat1 = 20, lon1 = 12, lat2 = c(50, 30), lon2 = c(40, 32),
  method = "euclidean"
)

Description

Circular summary statistics over intervals of distances.

Usage

distance_binned_stats(
  azi,
  distance,
  n.breaks = 10,
  width.breaks = NULL,
  unc = NULL,
  prd = NULL,
  prd.error = NULL,
  kappa = 2,
  R = 1000,
  conf.level = 0.95,
  ...
)

Arguments

Value

tibble containing the n values for aziin each bin, min/median/max distance of the bin, and the summary statistics for azi. If prd is specified, the normal Chi-squared statistic, dispersion and its standard error are returned as well.

See Also

circular_summary(), circular_dispersion(), and circular_dispersion_boot()

Examples

data("plates")
plate_boundary <- subset(plates, plates$pair == "na-pa")
data("san_andreas")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
san_andreas$distance <- distance_from_pb(
  x = san_andreas,
  PoR = PoR,
  pb = plate_boundary,
  tangential = TRUE
)
dat <- san_andreas |> cbind(PoR_shmax(san_andreas, PoR, "right"))

distance_binned_stats(dat$azi.PoR,
  distance = dat$distance, width.breaks = 1,
  unc = dat$unc, prd = 135
) |> head()

Description

Absolute distance of data points from the nearest plate boundary

Usage

distance_from_pb(x, PoR, pb, tangential = FALSE, km = FALSE, ...)

Arguments

Details

The distance to the plate boundary is the longitudinal or latitudinal difference between the data point and the plate boundary (along the closest latitude or longitude) for inward/outward or tangential plate boundaries, respectively.

Value

Numeric vector of the great circle distances in units defined by km.

Note

Stresses emanate from the plate boundary along great circles, small circles or loxodromes associated with the pole of rotation. Hence the emanation distance is not necessarily the shortest distance to the plate boundary, which is measured along a great circle unrelated to the pole of rotation. The differences are particularly notable when the plate boundary is kinked or for convergent and divergent plate boundaries.

References

Wdowinski, S. (1998). A theory of intraplate tectonics. Journal of Geophysical Research: Solid Earth, 103(3), 5037<U+2013>5059. http://dx.doi.org/10.1029/97JB03390

Examples

data("nuvel1")
na_pa <- subset(nuvel1, nuvel1$plate.rot == "na")

data("plates")
plate_boundary <- subset(plates, plates$pair == "na-pa")

data("san_andreas")
res <- distance_from_pb(
  x = san_andreas, PoR = na_pa, pb = plate_boundary, tangential = TRUE
)
head(res)

res.km <- distance_from_pb(
  x = san_andreas, PoR = na_pa, pb = plate_boundary, tangential = TRUE, km = TRUE
)
range(res.km)

Description

IERS mean radius of Earth in km (based on WGS 84)

Usage

earth_radius()

Value

numeric value

Description

Transforms a sequence of rotations into a new reference system

Usage

equivalent_rotation(x, fixed, rot)

Arguments

Value

sequence of plate rotations in new reference system. Same object class as x

See Also

relative_rotation()

Examples

data(nuvel1) # load the NUVEL1 rotation parameters

# all nuvel1 rotation equivalent to fixed Africa:
equivalent_rotation(nuvel1, fixed = "af")
# relative plate motion between Eurasia and India:
equivalent_rotation(nuvel1, "eu", "in") # lat = 24.58, lon = 18.07, angle = 0.528

Description

Estimates the concentration parameter of a von Mises distribution, given a set of angular measurements.

Usage

est.kappa.MLE(x, w = NULL, bias = FALSE)

est.kappa(x, w = NULL, p = 2)

Arguments

Details

est.kappa.MLE() is the maximum likelihood estimate for MLE for $\kappa$.

est.kappa() uses an approximation based on the empirical equation:

$\kappa = \frac{\bar{R}(p-\bar{R}^2)}{1-\bar{R}^2}$

where $\bar{R}$ is the mean resultant length and $p$ is the dimensionality of the data (2 for circular data).

Value

numeric. Concentration of a von Mises distribution

See Also

vonmises()

Examples

set.seed(123)
x <- rvm(100, 90, 10)
w <- weighting(runif(100, 0, 10))

est.kappa(x, w)

est.kappa.MLE(x, w)

Description

Creates an object of the orientation of the Euler pole axis

Usage

euler_pole(x, y, z = NA, geo = TRUE, angle = NA)

Arguments

Value

An object of class "euler.pole" containing the Euler pole axis in both geographical and Cartesian coordinates and the angle of rotation in radians.

Examples

euler_pole(90, 0, angle = 45)
euler_pole(0, 0, 1, geo = FALSE)

Description

geom_azimuth() visualizes axial-directional vector fields using a geom to produce a new graphical layer, which allows aesthetic options. This layer can be overlaid on a map to improve visualisation of mapped data. The geom draws line segments (spokes) centered at (x, y) with a given orientation (angle in degrees) and length (radius). By default the spoke is centered using PositionCenterSpoke, so that the given coordinates mark the middle of the line. The azimuths are given as angles in degrees increasing clockwise from North.

Usage

geom_azimuth(
  mapping = NULL,
  data = NULL,
  stat = "azimuth",
  center = TRUE,
  radius = NULL,
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  ...
)

Arguments

Value

A ggplot2 layer that adds axis-like spokes.

Aesthetics

geom_azimuth() understands the following aesthetics (required aesthetics in bold):

• x

• y

• angle (in degrees, transformed internally)

• radius

• colour

• alpha

• linewidth

• linetype

See Also

ggplot2::geom_spoke(), geom_azimuthpoint()

Examples

set.seed(20250411)
df <- data.frame(
  x = runif(5), y = runif(5),
  angle_deg = rvm(5, mean = 90, kappa = 10),
  radius = runif(5, 0.1, 2)
)

if (require("ggplot2")) {
  ggplot(df, aes(x, y)) +
    geom_azimuth(aes(angle = angle_deg), radius = .1, linewidth = 1.2, colour = "blue")
  if (require("grid")) {
    ggplot(df, aes(x, y, radius = radius)) +
      geom_azimuth(aes(angle = angle_deg), center = FALSE, colour = "red", arrow = grid::arrow())
  }
}

Description

geom_azimuthpoint() draws line segments (spokes) like geom_azimuth(), but also places a point (marker) at the spoke's center ⁠(x, y)⁠.

Aesthetic rules:

• linewidth, linetype affect the spoke only

• shape affects the point only

• colour, alpha affect both spoke and point

• size sets the size of the point only

Usage

geom_azimuthpoint(
  mapping = NULL,
  data = NULL,
  stat = "identity",
  center = TRUE,
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  size = 2,
  ...
)

Arguments

Value

A list of ggplot2 layers (spokes + points).

Aesthetics

geom_azimuthpoint() understands the following aesthetics (required aesthetics in bold):

• x

• y

• angle (in degrees, transformed internally; spoke only)

• radius (spoke only)

• colour (shared)

• alpha (shared)

• linewidth (spoke only)

• linetype (spoke only)

• shape (point only)

• size (point only, or via argument)

• fill (point only, for shapes that accept fill)

See Also

geom_azimuth(), ggplot2::geom_spoke(), ggplot2::geom_point()

Examples

set.seed(20250411)
df <- data.frame(
  x = runif(5), y = runif(5),
  angle_deg = rvm(5, mean = 90, kappa = 10),
  radius = runif(5, 0.1, 1),
  group = rep(1:2, length.out = 5)
)

if (require("ggplot2")) {
  ggplot(data = df, aes(x, y, angle = angle_deg, radius = radius)) +
    geom_azimuthpoint(aes(colour = factor(group), shape = factor(group)),
      linewidth = 1.1, linetype = "dashed",
      size = 3, alpha = 0.8
    )
}

Description

Calculate initial bearing (or forward azimuth/direction) to go from point a to point b following great circle arc on a sphere.

Usage

get_azimuth(lat_a, lon_a, lat_b, lon_b)

Arguments

Details

get_azimuth() is based on the spherical law of tangents. This formula is for the initial bearing (sometimes referred to as forward azimuth) which if followed in a straight line along a great circle arc will lead from the start point a to the end point b.

$\theta = \arctan2 (\sin \Delta\lambda \cos\psi_2, \cos\psi_1 \sin\psi_1-\sin\psi_1 \cos\psi_2 \cos\Delta\lambda)$

where $\psi_1, \lambda_1$ is the start point, $\psi_2$, $\lambda_2$ the end point ($\Delta\lambda$ is the difference in longitude).

Value

numeric. Azimuth in degrees

References

http://www.movable-type.co.uk/scripts/latlong.html

Examples

berlin <- c(52.517, 13.4) # Berlin
tokyo <- c(35.7, 139.767) # Tokyo
get_azimuth(berlin[1], berlin[2], tokyo[1], tokyo[2]) # 41.57361

Description

Directions of 10 pidgeons, example data from Mardia and Jupp (1999), pp. 93

Format

numeric vector of length 10

References

Mardia and Jupp (1999). Directional Statistics. John Wiley and Sons.

Examples

data("homing")
homing

Description

Download WSM2025 or WSM2016 database from the GFZ sever and applies optional filters. If destdir is specified, the data can be reloaded in a later R session using load_WSM() using the same arguments.

Usage

download_WSM(
  destdir = tempdir(),
  load = TRUE,
  version = c("2025", "2016"),
  ...
)

load_WSM(
  file,
  quality = c("A", "B", "C", "D", "E", "X"),
  lat_range = c(-90, 90),
  lon_range = c(-180, 180),
  depth_range = c(-Inf, Inf),
  type = c("BO", "BOC", "BOT", "BS", "DIF", "FMA", "FMF", "FMS", "GFI", "GFM", "GFS",
    "GVA", "HF", "HFG", "HFM", "HFH", "HFP", "HFS", "OC", "PC", "SWB", "SWL", "SWS"),
  regime = c("N", "NS", "T", "TS", "S", NA)
)

download_WSM2016(destdir = tempdir(), load = TRUE, ...)

load_WSM2016(
  file,
  quality = c("A", "B", "C", "D", "E"),
  lat_range = c(-90, 90),
  lon_range = c(-180, 180),
  depth_range = c(-Inf, Inf),
  type = c("BO", "BOC", "BOT", "BS", "DIF", "FMA", "FMF", "FMS", "GFI", "GFM", "GFS",
    "GVA", "HF", "HFG", "HFM", "HFP", "OC", "PC", "SWB", "SWL", "SWS"),
  regime = c("N", "NS", "T", "TS", "S", NA)
)

Arguments

Value

sf object of and the parsed numeric uncertainty (unc) based on the reported standard deviation and the quality rank. If load=FALSE, the path to the downloaded file is returned.

Note

Because of R-compatibility and easy readability reasons, the downloaded dataset is a modified version of the original, WSM server version: All column names have been changed from uppercase (in the original dataset) to lowercase characters. Unknown azimuth values are represented by NA values instead of 999 in the original. Unknown regimes are represented by NA instead of "U" in the original.

Source

https://datapub.gfz.de/download/10.5880.WSM.2025.001-Scbwez/WSM_Database_2025.csv

https://datapub.gfz-potsdam.de/download/10.5880.WSM.2016.001/wsm2016.csv

References

Heidbach, O., M. Rajabi, X. Cui, K. Fuchs, B. M<U+00FC>ller, J. Reinecker, K. Reiter, M. Tingay, F. Wenzel, F. Xie, M. O. Ziegler, M.-L. Zoback, and M. D. Zoback (2018): The World Stress Map database release 2016: Crustal stress pattern across scales. Tectonophysics, 744, 484-498, doi:10.1016/j.tecto.2018.07.007.

Heidbach, Oliver; Rajabi, Mojtaba; Di Giacomo, Domenico; Harris, James; Lammers, Steffi; Morawietz, Sophia; Pierdominici, Simona; Reiter, Karsten; von Specht, Sebastian; Storchak, Dmitry; Ziegler, Moritz O. (2025): World Stress Map Database Release 2025. GFZ Data Services. doi:10.5880/WSM.2025.001

Examples

## Not run: 
download_WSM(
  quality = c("A", "B", "C"), lat_range = c(51, 72),
  lon_range = c(-180, -130), depth_range = c(0, 10), type = "FMS"
)

## End(Not run)

Description

Check if object is euler.pole

Usage

is.euler(x)

Arguments

Value

logical

Description

A statistical test used to determine whether a set of angular or circular data points (such as times of day, compass directions, or degrees) are spread out evenly around a circle or if they cluster in some way.

Usage

kuiper_test(x, alpha = 0, axial = TRUE, quiet = FALSE)

Arguments

Details

The Null Hypothesis ($H_0$): The data are distributed completely uniformly (randomly and evenly) around the circle.

The Alternative Hypothesis ($H_1$): The data are not uniform and show a preference, clustering, or pattern somewhere on the circle.

The Test Statistic (V or $D^{+} + D^{-}$): It measures the greatest positive and negative differences between your data's empirical cumulative distribution and a theoretical uniform distribution.

Interpreting the Results

• High Test Statistic / Low p-value ($p < \alpha$, typically 0.05): You reject the null hypothesis. This means your data are not uniform; they have a significant preferred direction, grouping, or non-random pattern on the circle.

• Low Test Statistic / High p-value ($p \ge 0.05$): You fail to reject the null hypothesis. There is no strong evidence to say the data are different from a flat, uniform distribution. The points appear random across the circle.

Value

list containing the test statistic statistic and the significance level p.value.

Note

Kuiper's test statistic is a rotation-invariant Kolmogorov-type test statistic. The critical values of a modified Kuiper's test statistic are used according to the tabulation given in Stephens (1970).

See Also

Other Tests: ar_test(), norm_chisq(), rayleigh-test, watson_test(), watson_two_sample, watson_wheeler_test_perm(), weighted-rayleigh-test

Examples

# Example data from Mardia and Jupp (1999), pp. 93
kuiper_test(homing, alpha = .05)

# San Andreas Fault Data:
data(san_andreas)
data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
kuiper_test(sa.por$azi.PoR, alpha = .05)

Description

Extract azimuths of line segments

Usage

line_azimuth(x, warn = TRUE)

lines_azimuths(x)

Arguments

Details

It is recommended to perform line_azimuth() on single line objects, i.e. type "LINESTRING", instead of "MULTILINESTRING". This is because the azimuth of the last point of a line will be calculated to the first point of the next line otherwise. This will cause a warning message (if warn = TRUE). For "MULTILINESTRING" objects, use lines_azimuths().

Value

sf object of type "POINT" with the columns and entries of the first row of x

Examples

data("plates")

# one line:
subset(plates, pair == "af-eu") |>
  smoothr::densify() |>
  line_azimuth() |>
  head()

# multiple lines:
lines_azimuths(plates[1:5, ]) |> head()

Description

Measure of spread around the circle. It should be noted that: If $R=0$, then the data is completely spread around the circle. If $R=1$, the data is completely concentrated on one point.

Usage

mean_resultant_length(x, w = NULL, na.rm = TRUE)

Arguments

Value

numeric.

References

Mardia, K.V. (1972). Statistics of Directional Data: Probability and Mathematical Statistics. London: Academic Press.

Examples

# Example data from Davis (1986), pp. 316
finland_stria <- c(
  23, 27, 53, 58, 64, 83, 85, 88, 93, 99, 100, 105, 113,
  113, 114, 117, 121, 123, 125, 126, 126, 126, 127, 127, 128, 128, 129, 132,
  132, 132, 134, 135, 137, 144, 145, 145, 146, 153, 155, 155, 155, 157, 163,
  165, 171, 172, 179, 181, 186, 190, 212
)
mean_resultant_length(finland_stria, w = NULL, na.rm = FALSE) # 0.800

Description

Models the direction of maximum horizontal stress $\sigma_\text{Hmax}$ along great circles, small circles, and loxodromes at a given point or points according to the relative plate motion in the geographical coordinate reference system.

Usage

model_shmax(df, euler)

Arguments

Details

$\sigma_\text{Hmax}$ following great circles is the (initial) bearing between the given point and the pole of relative plate motion. $\sigma_\text{Hmax}$ along small circles, clockwise, and counter-clockwise loxodromes is 90°, +45°, and +135° (or -45°) to this great circle bearing, respectively.

Value

data.frame

gc

Azimuth of modeled $\sigma_{Hmax}$ following great circles

sc

Small circles

ld.cw

Clockwise loxodromes

ld.ccw

Counter-clockwise loxodromes

Author(s)

Tobias Stephan

References

Stephan, T., Enkelmann, E., and Kroner, U. "Analyzing the horizontal orientation of the crustal stress adjacent to plate boundaries". Sci Rep 13. 15590 (2023). doi:10.1038/s41598-023-42433-2.

See Also

deviation_shmax() to compute the deviation of the modeled direction from the observed direction of $\sigma_\text{Hmax}$. PoR_shmax() to calculate the azimuth of $\sigma_\text{Hmax}$ in the pole of rotation reference system.

Examples

data("nuvel1")
# North America relative to Pacific plate:
euler <- subset(nuvel1, nuvel1$plate.rot == "na")

# the point where we mant to model the SHmax direction:
point <- data.frame(lat = 45, lon = 20)

model_shmax(point, euler)

Description

A quantitative comparison between the predicted and observed directions of $\sigma_\text{Hmax}$ is obtained by the calculation of the average azimuth and by a normalized $\chi^2$ test (Wdowsinki, 1998)

Usage

norm_chisq(obs, prd, unc)

Arguments

Details

The normalized $\chi^2$ test is

$\text{Norm} \chi^2_i = \frac{ \sum^M_{i = 1} \left( \frac{\alpha_i - \alpha_{{predict}}}{\sigma_i} \right) ^2} {\sum^M_{i = 1} \left( \frac{90}{\sigma_i} \right) ^2 }$

The value of the chi-squared test statistic is a number between 0 and 1 indicating the quality of the predicted $\sigma_\text{Hmax}$ directions. Low values ($\le 0.15$) indicate good agreement, high values ($> 0.7$) indicate a systematic misfit between predicted and observed $\sigma_\text{Hmax}$ directions.

Value

Numeric vector

References

Wdowinski, S., 1998, A theory of intraplate tectonics. Journal of Geophysical Research: Solid Earth, 103, 5037-5059, doi: 10.1029/97JB03390.

See Also

Other Tests: ar_test(), kuiper_test(), rayleigh-test, watson_test(), watson_two_sample, watson_wheeler_test_perm(), weighted-rayleigh-test

Examples

data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na") # North America relative to
# Pacific plate
data(san_andreas)
point <- data.frame(lat = 45, lon = 20)
prd <- model_shmax(point, PoR)
norm_chisq(obs = c(50, 40, 42), prd = prd$sc, unc = c(10, NA, 5))

data(san_andreas)
prd2 <- PoR_shmax(san_andreas, PoR, type = "right")
norm_chisq(obs = prd2$azi.PoR, 135, unc = san_andreas$unc)

Description

NNR-NUVEL-1 global model of current plate motions by DeMets et al. 1990

Usage

data('nuvel1')

Format

An object of class data.frame

plate.name

The rotating plate

plate.rot

The abbreviation of the plate's name

lat,lon

Coordinates of the Pole of Rotation

angle

The amount of rotation (angle in 1 Myr)

plate.fix

The anchored plate, i.e. plate.rot moves relative to plate.fix

source

Reference to underlying study

References

DeMets, C., Gordon, R. G., Argus, D. F., & Stein, S. (1990). Current plate motions. Geophysical Journal International, 101(2), 425-478. doi:10.1111/j.1365-246X.1990.tb06579.x.

Examples

data("nuvel1")
head("nuvel1")

Description

Global set of present plate boundaries on the Earth based on NUVEL-1 model by DeMets et al. 1990

Usage

data('nuvel1_plates')

Format

An object of class sf

References

DeMets, C., Gordon, R. G., Argus, D. F., & Stein, S. (1990). Current plate motions. Geophysical Journal International, 101(2), 425-478. doi:10.1111/j.1365-246X.1990.tb06579.x.

Examples

data("nuvel1_plates")
head("nuvel1_plates")

Description

Spectral decomposition of the 2D orientation tensor into two Eigenvectors and corresponding Eigenvalues provides provides a measure of location and a corresponding measure of dispersion, respectively.

Usage

ot_eigen2d(x, w = NULL, scale = FALSE)

principal_direction(x, w = NULL)

axial_strength(x, w = NULL)

axial_dispersion(x, w = NULL)

Arguments

Details

The Eigenvalues ($\lambda_1 > \lambda_2$) can be interpreted as the fractions of the variance explained by the orientation of the associated Eigenvectors. The two perpendicular Eigenvectors ($a_1, a_2$) are the "principal directions" with respect to the highest and the lowest concentration of orientation data.

The strength of the orientation is the largest eigenvalue $\lambda_1$ normalized by the sum of the eigenvalues (scale=TRUE). Then $\lambda_2 = 1-\lambda_1$ is a measure of dispersion of 2D orientation data with respect to $a_1$.

Value

ot_eigen2d returns a list of the Eigenvalues and the axial angles corresponding to the Eigenvectors. principal_direction(), axial_strength() and axial_dispersion() are convenience functions to return the orientation of the largest eigenvalue, the orientation strength, the axial dispersion respectively.

Note

Eigenvalues and Eigenvectors of the orientation tensor (inertia tensor) are also called "principle moments of inertia" and "principle axes of inertia", respectively.

See Also

ortensor2d()

Examples

test <- rvm(100, mean = 0, k = 10) / 2
ot_eigen2d(test)

data("nuvel1")
PoR <- subset(nuvel1, nuvel1$plate.rot == "na")
sa.por <- PoR_shmax(san_andreas, PoR, "right")
sa_eig <- ot_eigen2d(sa.por$azi.PoR, w = weighting(san_andreas$unc), scale = TRUE)
print(sa_eig)

rose(sa.por$azi.PoR, muci = FALSE)
rose_line(sa_eig$vectors,
  col = c("red", "green"),
  radius = sa_eig$values, lwd = 2
)
graphics::legend("topright",
  legend = round(sa_eig$values, 2),
  col = c("red", "green"), lty = 1
)

principal_direction(sa.por$azi.PoR)

axial_strength(sa.por$azi.PoR)

axial_dispersion(sa.por$azi.PoR)

Description

2D orientation tensor characterizes distribution of axial angles using the Eigenvalue method (Watson 1966, Scheidegger 1965).

Usage

ortensor2d(x, w = NULL, norm = FALSE)

Arguments

Details

The moment of inertia can be minimized by calculating the Cartesian coordinates of the orientation data, and calculating their covariance matrix. This yields

$I = x \cdot x^\intercal$

where $x$ is the Cartesian vector of the orientations. Orientation tensor $T$ and the inertia tensor $I$ are related by

$I = E - T$

where $E$ denotes the unit matrix, so that

$T = \frac{1}{n} \sum_{i=i}^{n} x_i \cdot x_i^\intercal$

Value

2x2 matrix

References

Watson, G. S. (1966). The Statistics of Orientation Data. The Journal of Geology, 74(5), 786–797.

Scheidegger, A. E. (1964). The tectonic stress and tectonic motion direction in Europe and Western Asia as calculated from earthquake fault plane solutions. Bulletin of the Seismological Society of America, 54(5A), 1519–1528. doi:10.1785/BSSA05405A1519

Bachmann, F., Hielscher, R., Jupp, P. E., Pantleon, W., Schaeben, H., & Wegert, E. (2010). Inferential statistics of electron backscatter diffraction data from within individual crystalline grains. Journal of Applied Crystallography, 43(6), 1338–1355. https://doi.org/10.1107/S002188981003027X

See Also

ot_eigen2d()

Examples

test <- rvm(100, mean = 0, k = 10)
ortensor2d(test)

Description

Assigns numeric values of the precision (sd.) of each measurement to the categorical quality ranking of the World Stress Map (A, B, C, D, E, X).

Usage

parse_wsm_quality(x)

quantise_wsm_quality(x)

Arguments

Value

"numeric". the standard deviation of stress azimuth

References

Heidbach, O., Barth, A., M<U+00FC>ller, B., Reinecker, J., Stephansson, O., Tingay, M., Zang, A. (2016). WSM quality ranking scheme, database description and analysis guidelines for stress indicator. World Stress Map Technical Report 16-01, GFZ German Research Centre for Geosciences. doi:10.2312/wsm.2016.001

Examples

parse_wsm_quality(c("A", "B", "C", "D", NA, "E", "X"))
data("san_andreas")
head(parse_wsm_quality(san_andreas$quality))

Description

PB2002 global model of current plate motions by Bird 2003

Usage

data('pb2002')

Format

An object of class data.frame

plate.name

The rotating plate

plate.rot

The abbreviation of the plate's name

lat,lon

Coordinates of the Pole of Rotation

angle

The amount of rotation (angle in 1 Myr)

plate.fix

The anchored plate, i.e. plate.rot moves relative to plate.fix

source

Reference to underlying study

References

Bird, P. (2003), An updated digital model of plate boundaries, Geochem. Geophys. Geosyst., 4, 1027, doi: 10.1029/2001GC000252, 3.

Examples

data("pb2002")
head("pb2002")

Description

Global set of present plate boundaries on the Earth based on PB2002 model by Bird (2003). Contains the plate boundary displacement types such as inward, outward, or tangentially displacement.

Usage

data('plates')

Format

An object of class sf

References

Bird, P. (2003), An updated digital model of plate boundaries, Geochem. Geophys. Geosyst., 4, 1027, doi: 10.1029/2001GC000252, 3.

Examples

data("plates")
head("plates")

Description

Plots multiples of a von Mises, wrapped Cauchy, and wrapped Normal density distribution in a circular plot

Usage

plot_density(
  x,
  bw = NULL,
  kernel = c("vonmises", "wrappedcauchy", "wrappednormal"),
  weights = NULL,
  axial = TRUE,
  n = 512L,
  norm.density = TRUE,
  kappa = NULL,
  rho = NULL,
  sd = NULL,
  c = NULL,
  fill = FALSE,
  scale = 0,
  shrink = 1,
  add = TRUE,
  main = NULL,
  labels = TRUE,
  at = seq(0, 360 - 45, 45),
  cborder = TRUE,
  grid = FALSE,
  ...
)

Arguments

Value

plot or calculated densities

See Also

circular_density()

Other circular-plot: plot_points(), rose(), rose_geom, rose_stats()

Examples

set.seed(20250411)
# Filled von Mises kernel density curve inside the plot
plot_density(san_andreas$azi,
  kappa = 100,
  fill = TRUE, col = "#51127C80", border = "#51127CFF",
  grid = TRUE,
  add = FALSE
)

# Superimpose a wrapped Cauchy kernel distribution curve
plot_density(san_andreas$azi,
  rho = 0.9, kernel = "wrappedcauchy",
  fill = FALSE, col = "#FB8861FF",
  add = TRUE
)

# Superimpose a wrapped Normal kernel distribution curve
plot_density(san_andreas$azi,
  sd = 2, kernel = "wrappednormal",
  fill = FALSE, col = "#E65164FF",
  add = TRUE
)

# Superimpose a von Mises kernel density curve on a rose diagram:
rose(san_andreas$azi, grid = TRUE)
plot_density(san_andreas$azi,
  bw = 100, col = "#51127CFF",
  add = TRUE, lwd = 3
)

# Corona plot (density curve outside of a rose diagram plot):
w <- weighting(san_andreas$unc)
rose(san_andreas$azi, weights = w, dots = TRUE, stack = TRUE, dot_cex = 0.5, dot_pch = 21)
plot_density(san_andreas$azi, weights = w,
  bw = 100,
  scale = 1.1, shrink = 3, xpd = NA,
  col = "#51127CFF"
)

Description

Add points to a plot of circular data points on the current graphics device.

Usage

plot_points(
  x,
  axial = TRUE,
  stack = FALSE,
  binwidth = 1,
  cex = 1,
  sep = 0.025,
  jitter_factor = 0,
  ...,
  scale = 1.1,
  add = TRUE,
  main = NULL,
  labels = TRUE,
  at = seq(0, 360 - 45, 45),
  cborder = TRUE
)

Arguments

Value

A list with information on the plot

See Also

Other circular-plot: plot_density(), rose(), rose_geom, rose_stats()

Examples

set.seed(20250511)
x <- rvm(100, mean = 90, k = 5)

# plot poinit without jitter
plot_points(x, add = FALSE)

# with some jitter
plot_points(x, jitter_factor = .2, add = FALSE)

# stacked dots:
plot_points(x, stack = TRUE, binwidth = 3, add = FALSE, xpd = TRUE)

Description

Transforms azimuths and models the direction of maximum horizontal stress $\sigma_\text{Hmax}$ in the Euler pole (Pole of Rotation) coordinate reference system. When type of plate boundary is given, it also gives the deviation from the theoretically predicted azimuth of $\sigma_\text{Hmax}$, the circular distance, and the normalized $\chi^2$ statistics.

Usage

PoR_azimuth(x, PoR, axial = TRUE)

PoR_shmax(x, PoR, type = c("none", "in", "out", "right", "left"), axial = TRUE)