Fig 1.
Derivation of the morphometric database for method evaluation.
LSB = longitudinal subglacial bedform; DTM = digital terrain model; SDE = standard deviational ellipse; DEM = digital elevation model; ASpl_A: the ratio between the footprints’ upflow area and total area [11]. With the exception of the longest straight line (derived in Geospatial Modelling Environment® [21]), all steps were conducted in ArcMap® 10.2.
Fig 2.
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.
Fig 3.
Manually and automatedly derived longitudinal axes for 5 drumlins.
For ease of visual comparison all lines start at the same location. The grey polygon is the footprint’s 180°-rotated version. LSL—longest straight line fitting inside the footprint; LSL-IP—longest straight line crossing the footprint’s innermost point; RLA—minimum bounding rectangle longitudinal axis; SDE1 –longitudinal axis derived based on the standard deviational ellipse computed using the footprint’s vertices.
Fig 4.
Derivation of automated longitudinal axes.
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) [18] 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.
Fig 5.
Computation of the standard deviational ellipse.
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.
Fig 6.
Relationship between footprint elongation and differences between morphometric measurement methods.
A) Difference in orientation relative to the LSL method. B) Difference in length relative to elliptical length (Euler’s approximation). C) Difference between longitudinal asymmetry (ASPL_A [11]) computed using the LSL and RLA methods. Footprint elongation was calculated based on the minimum bounding rectangle.
Fig 7.
Error distributions for LSBs with elongation < 5.
The box corresponds to the interquartile range; the horizontal line and the small square within the box plot the median and the mean, respectively; the whiskers represent the 5th and 95th percentiles; the crosses plot the minimum and maximum errors. Where maximum or minimum error exceeds chosen vertical scale (same across graphs for comparability), its value is provided within brackets. Errors correspond to the difference between the values from the automated method and from the reference LA. For length and longitudinal asymmetry (ASpl_A [11]), errors correspond to the % difference relative to value derived from the reference LA.
Table 1.
Error central tendency (MAE) and dispersion (MAD) for LSBs with elongation < 5.
Table 2.
Central tendency error—Difference between the means of the automated method and reference data for LSBs with elongation < 5.
Fig 8.
Dependence of the LSL and RLA methods on footprint shape.
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.
Table 3.
General adequacy of the LSL and RLA methods depending on footprint shape.
Fig 9.
Footprints with a relatively large difference in orientation between the SDE2 and SDE3 methods.
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.
Fig 10.
Dependence of elliptical length (Euler’s approximation) error on footprint shape and perimeter complexity.
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 (%).
Fig 11.
Dependence of elliptical length (Euler’s approximation) error on elongation of a rectangle and an ellipse.
Elliptical length % error = [(elliptical length–reference LA length) / reference LA length] x 100.