---
title: "1. Introduction to tectonicr"
author: "Tobias Stephan"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{1. Introduction to tectonicr}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  markdown: 
    wrap: 72
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

`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.

## 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$^{\circ}$ 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, 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$^{\circ}$ 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$^{\circ}$.

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

<!-- **tectonicr** basically calculates the bearing from a point of  -->
<!-- interest to a given Euler pole of the relative plate motion by the formula: -->

<!-- ```{=tex} -->
<!-- \begin{equation}  -->
<!--   \theta = \arctan2 (\sin \Delta\lambda * \cos\psi_2, \cos\psi_1 \sin\psi_1-\sin\psi_1 \cos\psi_2 \cos\Delta\lambda) -->
<!-- \end{equation} -->
<!-- ``` -->

<!-- where $\psi_1$, $\lambda_1$ are the longitude and latitude of the point,  -->
<!-- $\psi_2$, $\lambda_2$ the longitude and latitude of the Euler pole  -->
<!-- ($\Delta\lambda$ is the difference in longitude).  -->

<!-- > The bearing $\theta$ is the orientation of a great circle that passes through  -->
<!-- the point and the Euler pole. Thus, it reflects the theoretical direction of  -->
<!-- $\sigma_{Hmax}$ that follows great circles trajectories.  -->
<!-- Small circle directions then are $90^{\circ}$ and loxodrome directions  -->
<!-- are $\pm45^{\circ}$ to $\theta$. 

<!-- --- -->

Trajectories of theoretical directions can modeled by the following 
steps:

First, load the package:

```{r setup, echo=TRUE}
library(tectonicr)
library(ggplot2) # load ggplot library
```

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()`:

```{r direction_of_plate_motion, echo=TRUE}
# Example:
point <- data.frame(lat = 45, lon = 20)
por <- data.frame(lat = 90, lon = 0)
model <- model_shmax(point, por)
print(model)
```

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()`:

```{r deviation_of_plate_motion, echo=TRUE}
deviation <- deviation_shmax(model, 90)
print(deviation)
```

## Quantitative Comparison Between Predicted and Observed Maximum Horizontal Stress

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).

<!-- ```{=tex} -->
<!-- \begin{equation}  -->
<!--   \textrm{Norm} \chi^2_i =  -->
<!--  \frac{\sum_{i=1}^{n} \left(\frac{\alpha_i - \alpha_{predict}}{\sigma_i}\right)^2} -->
<!--  {\sum_{i=1}^{n} \left(\frac{90}{\sigma_i}\right)^2} \end{equation} -->
<!-- ``` -->
<!-- with $\alpha_i$ and $\alpha_{predicted}$ are the observed and predicted -->
<!-- directions of $\sigma_{Hmax}$ with respect to the pole of rotation,  -->
<!-- respectively. -->
<!-- The $\sigma_i$ parameter is the reported uncertainty of the observed -->
<!-- azimuth and $M$ is the number of observations that are used in each -->
<!-- test. -->

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).

```{r shmax_test, echo=TRUE}
data("nuvel1") # import example data set for Euler rotations
por <- subset(
  nuvel1, nuvel1$plate.rot == "na"
) # North America relative to Pacific plate
point <- data.frame(lat = 45, lon = 20)

prd <- model_shmax(point, por)
norm_chisq(obs = 90, prd$sc, unc = 10)
```

# Models of current plate motion
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:

```{r nuvel1, eval=FALSE, include=TRUE}
data("nuvel1")
head(nuvel1)
```

Other current plate motion models, in particulars NNR-MORVEL-56, GSRM2.1, REVEL, PB2002, and HS3-NUVEL1A, are available through
```{r cpm_models, eval=FALSE, include=TRUE}
data("cpm_models")
head(cpm_models)
```

Any desired relative plate motion can be extracted via the following:
```{r equivalent_rotation, eval=FALSE, include=TRUE}
gsrm <- subset(cpm_models, model == "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](https://doi.org/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](https://doi.org/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.
<!---[10.1029/97jb03390](https://doi.org/10.1029/97JB03390).-->
