knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

This vignette teaches you how to handle large stress datasets and how to retrieve relative plate motions parameters from a set of plate motions.

library(tectonicr) library(ggplot2) # load ggplot library

**tectonicr** also handles larger data sets. A subset of the World Stress Map
data compilation (Heidbach et al. 2016)
is included as an example data set and can be imported through:

data("san_andreas") head(san_andreas)

Modeling the stress directions (wrt. to the geographic North pole) using the Pole of Oration (PoR) of the motion of North America relative to the Pacific Plate. We test the dataset against a right-laterally tangential displacement type.

data("nuvel1") por <- subset(nuvel1, nuvel1$plate.rot == "na") san_andreas.prd <- PoR_shmax(san_andreas, por, type = "right")

Combine the model results with the coordinates of the observed data

san_andreas.res <- data.frame( sf::st_drop_geometry(san_andreas), san_andreas.prd )

`ggplot2::ggplot()`

can be used to visualize the results. The
orientation of the axis can be displayed with the function
`geom_spoke()`

. The position argument `position = "center_spoke"`

aligns
the marker symbol at the center of the point. The deviation can be color
coded. `deviation_norm()`

yields the normalized value of the deviation,
i.e. absolute values between 0 and 90$^{\circ}$.

Also included are the plate boundary geometries after Bird (2003):

data("plates") # load plate boundary data set

Alternatively, there is also the NUVEL1 plate boundary model by DeMets et al.
(1990) stored under `data("nuvel1_plates")`

.

First we create the predicted trajectories of $\sigma_{Hmax}$ (more details in Article 3.):

trajectories <- eulerpole_loxodromes(por, 40, cw = FALSE)

Then we initialize the plot `map`

...

map <- ggplot() + geom_sf( data = plates, color = "red", lwd = 2, alpha = .5 ) + scale_color_continuous( type = "viridis", limits = c(0, 90), name = "|Deviation| in (\u00B0)", breaks = seq(0, 90, 22.5) ) + scale_alpha_discrete(name = "Quality rank", range = c(1, 0.4))

...and add the $\sigma_{Hmax}$ trajectories and data points:

map + geom_sf( data = trajectories, lty = 2 ) + geom_spoke( data = san_andreas.res, aes( x = lon, y = lat, angle = deg2rad(90 - azi), color = deviation_norm(dev), alpha = quality ), radius = 1, position = "center_spoke", na.rm = TRUE ) + coord_sf( xlim = range(san_andreas$lon), ylim = range(san_andreas$lat) )

The map shows generally low deviation of the observed $\sigma_{Hmax}$ directions from the modeled stress direction using counter-clockwise 45$^{\circ}$ loxodromes.

The *normalized* $\chi^2$ test quantifies the fit between the
modeled $\sigma_{Hmax}$ direction the observed stress direction
considering the reported uncertainties of the measurement.

norm_chisq( obs = san_andreas.res$azi.PoR, prd = 135, unc = san_andreas.res$unc )

The value is $\leq$ 0.15, indicating a significantly good fit of the model. Thus, the traction of the transform plate boundary explain the stress direction of the area.

The direction of the maximum horizontal stress correlates with plate motion direction at the plate boundary zone. Towards the plate interior, plate boundary forces become weaker and other stress sources will probably dominate.

To visualize the variation of the $\sigma_{Hmax}$ wrt. to the distance to the
plate boundary, we need to transfer the direction of $\sigma_{Hmax}$ from the
geographic reference system (i.e. azimuth is the deviation of a direction from
geographic North pole) to the **Pole of Rotation (PoR)** reference system
(i.e. azimuth is the deviation from the PoR).

The

PoR coordinate reference systemis the oblique transformation of the geographical coordinate system with the PoR coordinates being the the translation factors.

The azimuth in the *PoR reference system* $\alpha_{PoR}$ is the angular
difference between the azimuth in geographic reference system $\alpha_{geo}$
and the (initial) bearing of the great circle
that passes through the data point and the PoR $\theta$.

To calculate the distance to the plate boundary, both the plate boundary
geometries and the data points (in geographical coordinates) will be
transformed in to the *PoR* reference system.
In the *PoR* system, the distance is the latitudinal or longitudinal difference
between the data points and the inward/outward or tangential moving plate
boundaries, respectively.

This is done with the function `distance_from_pb()`

, which returns the angular
distances.

plate_boundary <- subset(plates, plates$pair == "na-pa") san_andreas.res$distance <- distance_from_pb( x = san_andreas, PoR = por, pb = plate_boundary, tangential = TRUE )

Finally, we visualize the $\sigma_{Hmax}$ direction wrt. to the distance to the plate boundary:

azi_plot <- ggplot(san_andreas.res, aes(x = distance, y = azi.PoR)) + coord_cartesian(ylim = c(0, 180)) + labs(x = "Distance from plate boundary (\u00B0)", y = "Azimuth in PoR (\u00B0)") + geom_hline(yintercept = c(0, 45, 90, 135, 180), lty = 3) + geom_pointrange( aes( ymin = azi.PoR - unc, ymax = azi.PoR + unc, color = san_andreas$regime, alpha = san_andreas$quality ), size = .25 ) + scale_y_continuous( breaks = seq(-180, 360, 45), sec.axis = sec_axis( ~., name = NULL, breaks = c(0, 45, 90, 135, 180), labels = c("Outward", "Tan (L)", "Inward", "Tan (R)", "Outward") ) ) + scale_alpha_discrete(name = "Quality rank", range = c(1, 0.1)) + scale_color_manual(name = "Tectonic regime", values = stress_colors(), breaks = names(stress_colors())) print(azi_plot)

Adding a rolling statistics (e.g. weighted mean and 95% confidence interval) of the transformed azimuth:

san_andreas.res_roll <- san_andreas.res[order(san_andreas.res$distance), ] san_andreas.res_roll$r_mean <- roll_circstats( san_andreas.res_roll$azi.PoR, w = 1 / san_andreas.res_roll$unc, FUN = circular_mean, width = 51 ) san_andreas.res_roll$r_conf95 <- roll_confidence( san_andreas.res_roll$azi.PoR, w = 1 / san_andreas.res_roll$unc, width = 51 ) azi_plot + geom_step( data = san_andreas.res_roll, aes(distance, r_mean - r_conf95), lty = 2 ) + geom_step( data = san_andreas.res_roll, aes(distance, r_mean + r_conf95), lty = 2 ) + geom_step( data = san_andreas.res_roll, aes(distance, r_mean) )

Close to the dextral plate boundary, the majority of the stress data have a strike-slip fault regime and are oriented around 135$^{\circ}$ wrt. to the PoR. Thus, the date are parallel to the predicted stress sourced by a right-lateral displaced plate boundary. Away from the plate boundary, the data becomes more noisy.

This azimuth (PoR) vs. distance plot also allows to identify whether a less known plate boundary represents a inward, outward, or tangential displaced boundary.

The relationship between the azimuth and the distance can be better visualized
by using the deviation (normalized by the data precision) from the the predicted
stress direction, i.e. the *normalized* $\chi^2$:

# Rolling norm chisq: san_andreas.res_roll$roll_nchisq <- roll_normchisq( san_andreas.res_roll$azi.PoR, san_andreas.res_roll$prd, san_andreas.res_roll$unc, width = 51 ) # plotting: ggplot(san_andreas.res, aes(x = distance, y = nchisq)) + coord_cartesian(ylim = c(0, 1)) + labs(x = "Distance from plate boundary (\u00B0)", y = expression(Norm ~ chi^2)) + geom_hline(yintercept = c(0.15, .33, .7), lty = 3) + geom_point(aes(color = san_andreas$regime)) + scale_y_continuous(sec.axis = sec_axis( ~., name = NULL, breaks = c(.15 / 2, .33, .7 + 0.15), labels = c("Good fit", "Random", "Systematic\nmisfit") )) + scale_color_manual(name = "Tectonic regime", values = stress_colors(), breaks = names(stress_colors())) + geom_step( data = san_andreas.res_roll, aes(distance, roll_nchisq) )

We can see that the data in fact starts to scatter notably beyond a distance of 3.8$^{\circ}$ and becomes random at 7$^{\circ}$ away from the plate boundary. Thus, the North American-Pacific plate boundary zone at the San Andreas Fault is approx. 4--7$^{\circ}$ (ca. 380--750 km) wide.

The

normalized$\chi^2$ vs. distance plot allows to specify the width of the plate boundary zone.

The data deviation map can also be build using base R's plotting engine:

# Setup the colors for the deviation cols <- tectonicr.colors( deviation_norm(san_andreas.res$dev), categorical = FALSE ) # Setup the legend col.legend <- data.frame(col = cols, val = names(cols)) |> dplyr::mutate(val2 = gsub("\\(", "", val), val2 = gsub("\\[", "", val2)) |> unique() |> dplyr::arrange(val2) # Initialize the plot plot( san_andreas$lon, san_andreas$lat, cex = 0, xlab = "PoR longitude", ylab = "PoR latitude", asp = 1 ) # Plot the axis and colors axes( san_andreas$lon, san_andreas$lat, san_andreas$azi, col = cols, add = TRUE ) # Plot the plate boundary plot(sf::st_geometry(plates), col = "red", lwd = 2, add = TRUE) # Plot the trajectories plot(sf::st_geometry(trajectories), add = TRUE, lty = 2) # Create the legend graphics::legend( "bottomleft", title = "|Deviation| in (\u00B0)", inset = .05, cex = .75, legend = col.legend$val, fill = col.legend$col )

A quick analysis the results can be obtained `stress_analysis()`

that returns a list. The transformed coordinates and azimuths as well as the deviations can be viewed by:

results <- stress_analysis(san_andreas, por, "right", plate_boundary, plot = FALSE) head(results$result)

Statistical parameters describing the distribution of the transformed azimuths can be displayed by

```
results$stats
```

Statistical test results are shown by

```
results$test
```

... and the associated plots can be displayed by setting `plot = TRUE`

:

stress_analysis(san_andreas, por, "right", plate_boundary, plot = TRUE)

Bird, Peter. 2003. “An Updated Digital Model of Plate Boundaries”
*Geochemistry, Geophysics, Geosystems* 4 (3).
doi: 10.1029/2001gc000252.

DeMets, C., R. G. Gordon, D. F. Argus, and S. Stein. 1990. “Current Plate Motions”
*Geophysical Journal International* 101 (2): 425–78.
doi: 10.1111/j.1365-246x.1990.tb06579.x.

Heidbach, Oliver, Mojtaba Rajabi, Karsten Reiter, Moritz Ziegler, and WSM Team. 2016. “World Stress Map Database Release 2016. V. 1.1.” GFZ Data Services. doi: 10.5880/WSM.2016.001.

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