Measuring (Subglacial) Bedform Orientation, Length, and Longitudinal Asymmetry – Method Assessment

Geospatial analysis software provides a range of tools that can be used to measure landform morphometry. Often, a metric can be computed with different techniques that may give different results. This study is an assessment of 5 different methods for measuring longitudinal, or streamlined, subglacial bedform morphometry: orientation, length and longitudinal asymmetry, all of which require defining a longitudinal axis. The methods use the standard deviational ellipse (not previously applied in this context), the longest straight line fitting inside the bedform footprint (2 approaches), the minimum-size footprint-bounding rectangle, and Euler’s approximation. We assess how well these methods replicate morphometric data derived from a manually mapped (visually interpreted) longitudinal axis, which, though subjective, is the most typically used reference. A dataset of 100 subglacial bedforms covering the size and shape range of those in the Puget Lowland, Washington, USA is used. For bedforms with elongation > 5, deviations from the reference values are negligible for all methods but Euler’s approximation (length). For bedforms with elongation < 5, most methods had small mean absolute error (MAE) and median absolute deviation (MAD) for all morphometrics and thus can be confidently used to characterize the central tendencies of their distributions. However, some methods are better than others. The least precise methods are the ones based on the longest straight line and Euler’s approximation; using these for statistical dispersion analysis is discouraged. Because the standard deviational ellipse method is relatively shape invariant and closely replicates the reference values, it is the recommended method. Speculatively, this study may also apply to negative-relief, and fluvial and aeolian bedforms

Dependence of the LSL and RLA methods on footprint shape.

<p>Examples of Puget Lowland LSB footprints for which the orientation and length of the reference LA is better matched by the RLA than the LSL (A-F), and better matched by the LSL than the RLA (G-L). Grey lines are the minimum bounding rectangle mid-axes; dashed lines represent the LSL and its perpendicular bisector; solid black lines are the reference LA. Bars at the bottom are 100 m long for B and 200 m long for A and C-L.</p

Idealized LSB shapes.

<p>A) elliptical; B) half-lemniscate (oval); C) stadium; D) parabolic with symmetric crescentic lee; E) hyperbolic with symmetric convex lee; F) hyperbolic with asymmetric convex lee. A and C have 2 axes of symmetry; B, D and E have 1 axis of symmetry. Inside the shapes: solid lines represent the reference longitudinal axis and its perpendicular bisector; dashed lines in C, D and F are the longest straight line and its perpendicular bisector. Angles in C, D and F represent the difference in orientation between the reference LA and the longest straight line.</p

Derivation of automated longitudinal axes.

<p>A) LSL—longest straight line enclosed by the footprint; LSL-IP—longest straight line crossing the footprint’s innermost point (IP); RLA—rectangle longitudinal axis. B) Longitudinal axis of the standard deviational ellipse (SDE) [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174312#pone.0174312.ref018" target="_blank">18</a>] computed using the footprint’s structural vertices and their 180°-rotated version (one of three tested SDEs). In the illustrated case, the ellipse’s axis is longer than the drumlin and thus was cropped to the minimum bounding rectangle’s extent.</p

Derivation of the morphometric database for method evaluation.

<p>LSB = longitudinal subglacial bedform; DTM = digital terrain model; SDE = standard deviational ellipse; DEM = digital elevation model; <i>AS</i>_{<i>pl_A</i>}: the ratio between the footprints’ upflow area and total area [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174312#pone.0174312.ref011" target="_blank">11</a>]. With the exception of the longest straight line (derived in Geospatial Modelling Environment^® [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174312#pone.0174312.ref021" target="_blank">21</a>]), all steps were conducted in ArcMap^® 10.2.</p

Computation of the standard deviational ellipse.

<p>Two orthogonal coordinate systems centered on the mean center of the dataset are represented. The small circles are the vertices of a longitudinal subglacial bedform footprint. The ellipse’s long axis is a segment of the axis for which the standard deviation (SD in diagram) of the distances from the vertices to the axis (dashed lines) is smallest relative to any other axis orientation. The ellipse’s minor axis corresponds to a segment of the axis that yields maximum standard deviation. The two axes are always orthogonal to each other.</p

Footprints with a relatively large difference in orientation between the SDE2 and SDE3 methods.

<p>Short-dashed, long-dashed and solid lines represent the orientation of the SDE2, SDE3 and reference LAs, respectively. Black and white dots are the un-rotated and rotated footprints’ structural vertices. From left to right, angular divergence between SDE2 and SDE3 lines is 1.7°, 1.1°, 2.2°, 1.3°, 1.7° and 0.6°. Bars at the bottom are 100 m long.</p

Dependence of elliptical length (Euler’s approximation) error on footprint shape and perimeter complexity.

<p>A) Elongation = 3, and two axes (1, 2) or one axis (3, 4) of symmetry; B) Elongation = 3, and zero axes (1–3) or one axis (4) of symmetry; C) Elongation = 6, and zero axes (1–3) or one axis (4) of symmetry. D) Elliptical length error (%).</p

Error central tendency (MAE) and dispersion (MAD) for LSBs with elongation < 5.

<p>Error central tendency (MAE) and dispersion (MAD) for LSBs with elongation < 5.</p

Data from: Jorge M G, Brennand T A. Measuring (subglacial) bedform orientation, one-dimensional size, and directional shape - method accuracy. Submitted to PLOS ONE, July 2015

<p>Georeferenced (EPSG 32610) vector file (shapefile format) containing the longitudinal subglacial bedform footprint dataset used in the study.<br>  </p