# Introduction

#### Tobias Stephan

#### 2025-12-11

`tectonicr` is a free and open-source R package for modeling and analyzing the direction of the maximum horizontal stress based on the empirical link between the direction of intraplate stress and the direction of the relative motion of neighboring plates.

## Prerequisites

Before you start you need to have R and and R-compatible IDE installed on your computer:

1. You will need R installed on your computer. You can download R from the CRAN website: https://cran.r-project.org/

2. Once you downloaded and successfully installed R, you can use R in your preferred IDE. I recommend installing and using RStudio. However, Positron and VisualStudio are also great alternatives. You are also allowed to use R’s native GUI or use R in your terminal only.

3. Open your IDE and now you can install the {tectonicr} package by typing into the commando console the following code. This will install the package and every required library as well.

4. Next you must “load” the library to use the functions of {tectonicr}:

    library(tectonicr)

## Theoretical background

The theory of intraplate tectonics (Wdowinski 1998) allows for calculating the first-order intraplate deformation induced by horizontal displacement of deformable plate boundaries (Stephan et al., 2023). It is based on empirical link between the directions of relative plate motion and the displacement and deformation fields within a plate interior adjacent to three types of deformable plate boundaries: inward-, outward-, and tangential-displaced boundaries. The model predicts the direction of intraplate displacement, displacement rate, strain, and stress fields in terms of small circles, great circles, and 45° loxodromes around the pole of rotation of two adjacent plates. According to the theory, the principal axis of the maximum horizontal stress follows small circles for inward-displaced boundaries, great circles for outward-displaced boundaries, and loxodromes for tangential-displaced boundaries.

The theory assumes that the first-order intraplate deformation is predominantly induced by horizontal forces acting on plate boundaries and by buoyancy forces that arise from lateral density variations between mid-ocean ridges and plate interiors (ridge push).

### Inward, Outward and Tangential Displaced Boundaries

Inward-moving plate boundaries induce compressional horizontal tractions from the plate boundary towards the plate’s interior along the direction of relative plate motion. These compressional tractions are produced by forces related to subduction, collision, or ridge-push. Thus, stresses across convergent plate boundaries are characterized by the dominance of thrusting or strike-slip faulting (\(\sigma_1 \approx \sigma_{Hmax}\)) with \(\sigma_{Hmax}\) (maximum horizontal stress) trending parallel to the plate convergence, i.e. parallel to small circles around the pole of the relative plate motion (pole of rotation, PoR).

Outward moving plate boundaries produce tensional tractions and displacements directed away from the plate interior. Along spreading ridges and intracontinental rifting stresses are dominated by normal faulting (\(\sigma_1 \approx \sigma_{vertical}\), \(\sigma_2 \approx \sigma_{Hmax}\)) with \(\sigma_{Hmax}\) trending perpendicular to the plate motion trajectories (i.e. along great circles). In the case of intracontinental setting, stresses and displacements may be associated to slab-retreat, back-arc extension, or the release of the excess of gravitational potential energy stored in thickened crust through, e.g., gravitational collapse.

Along transform boundaries (tangential displaced boundaries), the two neighboring plates exert shear tractions tangential to the orientation of the boundary (Forsyth and Uyeda, 1975). Faulting and displacement adjacent to these plate boundaries are characterized by strike-slip parallel to the plate motion, and thus, the principal axes of maximum and minimum stress are orientated at an angle of c. 45° to the plate motion trajectory. Geometrically, \(\sigma_{Hmax}\) direction follows along 45° loxodromes (lines of constant bearing) which diverge —depending on the sense of the transform boundary— clockwise or counterclockwise from the relative PoR and intersect both small and great circles at an angle of 45°.

## Theoretical direction of Horizontal Stress and Deviation From the Measured Stress

Trajectories of theoretical directions can modeled by the following steps:

First, we need to specify coordinates of the Pole of Rotation (PoR) to get the directions of the great circles, small circles, and loxodromes around the PoR at the given point (e.g. at 45°N/20°E).

For example, a PoR has the coordinates: 48.7°N/-78.2°E (relativ emotion of North America and Pacific plate). Then \(\sigma_{Hmax}\) following great and small circles and loxodromes geometries can be modeled with `model_shmax()`:

    # import example data set for Euler rotations
    data("nuvel1")

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

    # Example stress:
    point <- data.frame(lat = 45, lon = 20)
    prd <- model_shmax(point, por)
    print(prd)
    #>         sc   ld.ccw       gc    ld.cw
    #> 1 42.45436 87.45436 132.4544 177.4544

If there is an observed stress direction at the point, e.g. azimuth of \(\sigma_{Hmax}\) is 90°, the angle deviation from the modeled stress directions can be calculated through `deviation_shmax()`:

    deviation <- deviation_shmax(prd, 90)
    print(deviation)
    #>     dev.gc    dev.sc dev.ld.cw dev.ld.ccw
    #> 1 42.45436 -47.54564  87.45436  -2.545636

> More details on calculating the theoretical stress orientations are given in this tutorial.

## Quantitative Comparison Between Predicted and Observed Maximum Horizontal Stress

The circular dispersion \(D\) quantitatively compares the predicted (`model_shmax()`) and observed \(\sigma_{Hmax}\) azimuth relative to the reported \(\sigma\) standard deviation (Stephan and Enkelmann, 2025). The measure is (weighted) average of the circular distance \(d\) defined as \[d = 1 - \cos{\left[ k(\theta - \mu)\right]}\] where \(\theta\) are the observed angles (here \(\sigma_{Hmax}\)), \(\mu\) is the theoretical angles, and \(k=1\) for directional data and \(k=2\) for directional data. The weighted dispersion is

\[D = \frac{1}{Z} \sum_{i=1}^{n} w_i d_i\] where \(n\) s the number if observations, \(w_i\) are weights of each observation, and \(Z\) is the sum of all weights \(Z=\sum_{i=1}^{n} w_i\).

The dispersion parameter yields a number in the range between 0-1 which indicates the quality of the fit. Low dispersion values (\(D \le 0.15\)) indicate good agreement between predicted and observed directions (angle difference \(\le 22.5^\circ\)). High values (\(D > 0.5\)) indicate a systematic misfit between predicted and observed directions of about \(> 45^\circ\). A misfit of \(90^\circ\) and/or a random distribution of \(\sigma_{Hmax}\) directions results in \(D = 1\)

Assuming \(\sigma_{Hmax}\) has an azimuth of 90° at the given coordinate with a angle precision of 10°, we can compare all test all theoretical observations using `circular_dispersion()`:

    sapply(as.numeric(prd), function(p) {
      circular_dispersion(90, y = p, w = weighting(10))
    }) |>
      setNames(nm = names(prd))
    #>          sc      ld.ccw          gc       ld.cw
    #> 0.544371290 0.001972703 0.455628710 0.998027297

`ld.ccw` (counter-clockwise loxodrome) yields the smallest dispersion from the observation, indicating that counter-clockwise loxodrome geometry has a good fit with the observed stress orientation.

> More details on the statistical treatment, including variance estimation, statistical testing, and confidence intervals, are given in this tutorial.

# Models of current plate motion

The plate motions relative to the Pacific plate according to the NUVEL-1A model (DeMets et al. 1990) are implemented in the package and can be imported through:

    data("nuvel1")
    head(nuvel1)

Other current plate motion models, in particulars NNR-MORVEL-56, GSRM2.1, REVEL, PB2002, and HS3-NUVEL1A, are implemented and available through

    data("cpm_models")
    head(cpm_models)

Any desired relative plate motion can be extracted via the following:

    gsrm <- cpm_models[["GSRM2.1"]]
    equivalent_rotation(gsrm, rot = "na", fixed = "eu")

# References

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

Forsyth, D., and S. Uyeda. 1975. “On the Relative Importance of the Driving Forces of Plate Motion” Geophysical Journal International 43 (1): 163–200. doi: 10.1111/j.1365-246x.1975.tb00631.x

Stephan, T., Enkelmann, E., and Kroner, U. (2023). “Analyzing the horizontal orientation of the crustal stress adjacent to plate boundaries” Scientific Reports (13), 15590. doi:[10.1038/s41598-023-42433-2](doi:%5B10.1038/s41598-023-42433-2){.uri}

Stephan, T., & Enkelmann, E. (2025). All Aligned on the Western Front of North America? Analyzing the Stress Field in the Northern Cordillera. Tectonics, 44(9). https://doi.org/10.1029/2025TC009014

Wdowinski, Shimon. 1998. “A Theory of Intraplate Tectonics” Journal of Geophysical Research: Solid Earth 103 (B3): 5037–59. doi: 10.1029/97jb03390. https://doi.org/10.1029/97JB03390