`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.

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\(^{\circ}\) 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-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\(^{\circ}\) to the plate
motion trajectory. Geometrically, \(\sigma_{Hmax}\) direction follows along
45\(^{\circ}\) *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\(^{\circ}\).

Trajectories of theoretical directions can modeled by the following steps:

First, load the package:

```
library(tectonicr)
#> Registered S3 methods overwritten by 'spatstat.univar':
#> method from
#> mean.ecdf spatstat.geom
#> mean.ewcdf spatstat.geom
#> print.ewcdf spatstat.geom
#> quantile.density
#> quantile.ewcdf spatstat.geom
library(ggplot2) # load ggplot library
#> Warning: package 'ggplot2' was built under R version 4.3.3
```

Next, 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\(^{\circ}\)N/20\(^{\circ}\)E).

For example, the PoR has the coordinates: 90\(^{\circ}\)N/0\(^{\circ}\)E. Then \(\sigma_{Hmax}\) following great and small
circles and loxodromes geometries can be modeled with
`model_shmax()`

:

```
# Example:
point <- data.frame(lat = 45, lon = 20)
por <- data.frame(lat = 90, lon = 0)
model <- model_shmax(point, por)
print(model)
#> sc ld.ccw gc ld.cw
#> 1 90 135 0 45
```

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

:

The **normalized** \(\chi^2\) test quantitatively compares the
predicted (`model_shmax()`

) and observed \(\sigma_{Hmax}\) azimuth relative to the
reported \(\sigma\) standard deviation
(Wdowinski 1998).

The normalized \(\chi^2\) test yields a number in the range between 0-1 which indicates the quality of the fit. Low values of the normalized \(\chi^2\) test (\(\leq\) 0.15 indicate good agreement between predicted and observed directions. High values (\(>\) 0.7) indicate a systematic misfit between predicted and observed directions of about 90\(^{\circ}\). Random distribution of \(\sigma_{Hmax}\) directions results in Norm \(\chi^2 = 0.33\)

The test can be run using `norm_chisq(obs, prd, unc)`

.
`obs`

is a numeric vector with the observed \(\sigma_{Hmax}\); `prd`

is a
vector with the predicted \(\sigma_{Hmax}\) (vector must be of length
of `obs`

); and `unc`

is the uncertainty of
observed \(\sigma_{Hmax}\) (either
numeric vector of length of `obs`

or a number).

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

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

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

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](https://doi.org/10.1038/s41598-023-42433-2)

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