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).

Theoretical direction of Horizontal Stress and Deviation From the Measured Stress

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():

deviation <- deviation_shmax(model, 90)
print(deviation)
#>   dev.gc dev.sc dev.ld.cw dev.ld.ccw
#> 1    -90      0       -45         45

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).

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).

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)
#> [1] 0.2790849

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:

data("nuvel1")
head(nuvel1)

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

data("cpm_models")
head(cpm_models)

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

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

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.