# 2. Handling large datasets

#### Tobias Stephan

#### 2025-05-22

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

## Larger Data Sets

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)
    #> Simple feature collection with 6 features and 9 fields
    #> Geometry type: POINT
    #> Dimension:     XY
    #> Bounding box:  xmin: -118.84 ymin: 35.97 xmax: -112.58 ymax: 39.08
    #> Geodetic CRS:  WGS 84
    #>         id   lat     lon azi unc type depth quality regime
    #> 1 wsm00892 38.14 -118.84  50  25  FMS     7       C      S
    #> 2 wsm00893 35.97 -114.71  54  25  FMS     5       C      S
    #> 3 wsm00894 37.93 -118.17  24  25  FMS     5       C      S
    #> 4 wsm00896 38.63 -118.21  41  25  FMS    17       C      N
    #> 5 wsm00897 39.08 -115.62  30  25  FMS     5       C      N
    #> 6 wsm00903 38.58 -112.58  27  25  FMS     7       C      N
    #>                geometry
    #> 1 POINT (-118.84 38.14)
    #> 2 POINT (-114.71 35.97)
    #> 3 POINT (-118.17 37.93)
    #> 4 POINT (-118.21 38.63)
    #> 5 POINT (-115.62 39.08)
    #> 6 POINT (-112.58 38.58)

> The full or an individually filtered world stress map dataset can be downloaded by `download_WSM()`

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
    )

## Stress map

`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
    )
    #> [1] 0.0529064

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.

## Variation of the Direction of the Maximum Horizontal Stress wrt. to the Distance to the Plate Boundary

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 system is 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_x_continuous(breaks = seq(-10, 10, 2)) +
      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)

Binned statistics (e.g. weighted mean and 95% confidence interval) of the transformed azimuth can be achieved through `distance_binned_stats()`. Here, this gives a summary statistic for every 2\(^{\circ}\).

    san_andreas_binned <- distance_binned_stats(
      azi = san_andreas.res$azi.PoR,
      distance = san_andreas.res$distance,
      unc = san_andreas$unc,
      prd = 135,
      width.breaks = 2
    )

    azi_plot +
      geom_step(
        data = san_andreas_binned,
        aes(distance_median, mean - CI),
        lty = 2
      ) +
      geom_step(
        data = san_andreas_binned,
        aes(distance_median, mean + CI),
        lty = 2
      ) +
      geom_step(
        data = san_andreas_binned,
        aes(distance_median, 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 \(\chi^2\):

    # 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 - .15) / 2 + .15, (.7 - .33) / 2 + .33, .7 + 0.15),
        labels = c("Good fit", "Acceptable fit", "Random", "Systematic\nmisfit")
      )) +
      scale_x_continuous(breaks = seq(-10, 10, 2)) +
      scale_color_manual(name = "Tectonic regime", values = stress_colors(), breaks = names(stress_colors())) +
      geom_step(
        data = san_andreas_binned,
        aes(distance_median, nchisq)
      )

We can see that the data in fact starts to scatter notably beyond a distance of 2\(^{\circ}\) and becomes random at 6.5\(^{\circ}\) away from the plate boundary. Thus, the North American-Pacific plate boundary zone at the San Andreas Fault is approx. 2–6.5\(^{\circ}\) (ca. 200–800 km) wide.

> The normalized \(\chi^2\) vs. distance plot allows to specify the width of the plate boundary zone.

## R base plots for quick analysis

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)
    #>      azi.PoR prd       dev     nchisq      cdist  lat.PoR   lon.PoR  distance
    #> 1 173.240166 135  38.24017 0.18053214 0.38311043 59.05548 -85.46364 -2.446542
    #> 2   1.058195 135 -46.05820 2.21486507 0.51846479 60.50445 -78.16683 -2.751949
    #> 3 147.609558 135  12.60956 0.01962975 0.04765752 59.38025 -84.55317 -2.705483
    #> 4 163.607388 135  28.60739 0.10103489 0.22925429 59.73632 -85.74431 -3.162091
    #> 5 152.233017 135  17.23302 0.03666381 0.08776909 61.68018 -84.31053 -4.988116
    #> 6 150.611386 135  15.61139 0.03008832 0.07242085 63.39714 -80.60932 -6.358818

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

    results$stats
    #>                      mean
    #> mean         138.90248502
    #> sd            18.57998667
    #> var            0.18967312
    #> median       138.94509179
    #> quasi_median 136.85808363
    #> skewness      -0.33779208
    #> kurtosis       2.78636137
    #> conf95         5.13534451
    #> dispersion     0.09858997
    #> norm_chisq     0.05290640

Statistical test results are shown by

    results$test
    #> $C
    #> [1] 0.8028201
    #>
    #> $statistic
    #> [1] 38.09802
    #>
    #> $p.value
    #> [1] 4.600542e-314

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

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

## References

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.