670
The stochastic assessment used repeated random sub-division of the drumlin populations into 671
LGM and YD sub-populations of appropriate sizes. 100,000 iterations were used to generate the 672
probability density functions for
!
"ln(Vr) (Fig. 10). Three such distributions were generated, one 673
for the original data and DEM (Smith et al., 2009), and one for each of the sets of 10 synthetic 674
DEMs related to Methods 1 and 2. For the latter, one of the 10 DEMs was selected randomly 675
each iteration. Implicitly, such indiscriminate assignment of drumlins to sub-populations 676
employed the null hypothesis, H0: no difference exists. Using 100,000 iterations critical values 677
(e.g.
!
"ln(V )crit) at the 95% significance level are 5,000th in lists of generated differences (e.g.
678
!
"ln(Vr)) ordered in descending magnitude: These are the levels that, if exceeded, occur rarely by 679
chance and imply a statistically significant difference.
!
"ln(Vr) of Smith et al. (2009) is 1.38 680
times that of the 95% critical value of its distribution, so instances where
!
"ln(Vr) was more that 681
1.38 ×
!
"ln(V )crit for the synthetic DEMs were taken as more extreme than those reported by 682
Smith et al. (2009). The similarity in the shape of the distributions that makes this test valid is 683
evidenced visually (Fig. 10), and by similar numbers of simulations containing more extreme 684
differences than the threshold: 715 for the data of Smith et al. (2009), and 628 and 750 for 685
Methods 1 and 2 respectively. Methods 1 and 2 use 178 and 173 drumlins respectively, 686
bracketing the n = 175 of Smith et al. (2009). Agreement between the methods, therefore, 687
indicates insensitivity to these variations. Consequently, no correction was made for the 688
difference in n.
689 690
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692
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875
Figures 876
877
Figure 1: Idealised profiles to illustrate the process used to create synthetic DEMs. There are 878
three ‘components’ (Hillier and Smith, 2008): Drumlins (dark grey shade) sit upon a regional 879
trend (dotted line). Both are overlain by ‘clutter’ or ‘noise’ (light grey shade). a) Upper and 880
lower surfaces of drumlin are estimated to define it, and it is removed (height subtracted). b) 881
Two Gaussian shaped drumlins are inserted (height added). Noise as Fig. 4a.
882 883
Figure 2: Location maps. a) Study area is located at (4°28’ W, 56°02’ N), white circle:
884
England (E), Scotland (S), Ireland (I). Coastlines of seas and major inland water bodies are 885
shown. b) DEM of a sub-region of the study area, with height displayed as greyscale, located in 886
c): Woodland (W), Tree (T). c) Study area, with main geomorphic features of interest 887
highlighted; drumlins (black outlines), rivers (grey), water (grey shade). A division between 888
Younger Dryas (YD) and Last Glacial Maximum (LGM), white and stippled areas respectively, 889
is also shown (Smith et al., 2006). Arrows indicate approximate ice flow trends in the LGM 890
(Sissons, 1967) and YD (Rose, 1987). Map coordinates for b) and c) are of the British National 891
Grid. 5 m grid.
892 893
Figure 3: Example of an inspection used to assess the ability of widths of median filter to 894
separate noise, drumlins, and hills. Also, a drumlin illustrating the estimation of parameters 895
(x,y,L,W,θ), and selection of primary axes, from its digitized outline. a) Centre of drumlin (bold 896
line) is at x,y (white circle), determined by Method 1. Digitized outlines of other drumlins (thin 897
lines) overlay the measured DEM. b) Computed axes (lines) overlie ‘decluttered’ topography to 898
which a 60 m wide median filter has been applied. Arrows indicate directions of increasing 899
distance along profiles in Figure 4. c) ‘Clutter’ removed i.e. difference between a) and b). d) 900
Hhills estimated using a 500 m median filter. Map coordinates are of the British National Grid.
901 902
Figure 4: Height profiles across drumlin in Fig. 3. Profiles start from the directions θL and θS 903
such that, assuming a conventional stoss-lee form, traditional interpretations of ice flow are as 904
indicted. Lines are: measured height (thin lines), estimates of the decluttered surface using 60 m 905
wide median filter (thick black lines), linear interpolation between heights at the digitized edges 906
of the drumlin (dotted lines), 500 m wide median filter (grey lines). Grey boxes delimit the 907
drumlins. a) and b) are heights along the long and short axes, and c) and d) are normalized 908
profiles along the same axes.
909 910
Figure 5: Stacked (i.e. average) de-trended and normalized height profiles (black lines). Dashed 911
lines are ±1σ of height and dotted lines are ±2 standard errors of the individual profiles when 912
binned in 0.01 intervals of distance. Thick grey lines are Gaussian curves centred upon x = 0.5.
913
Grey box is the bounding box for all drumlins. a) and b) are respectively for long and short axes 914
found using Method 1, whilst c) and d) are for Method 2. All profiles start from the directions θL 915
Idealised drumlin consisting of interpolation (black line) between digitized points (black dots), 919
with the centre Cxy (open circle). At each point on the outline, θ and l are calculated. b) θ and l 920
for all points on drumlin #15 in Figure 2. θL (open circle) is θ for the largest l i.e. L, and θs = θL + 921
90° (cross): Compare to θ for the smallest L (filled black circle). ‘Ice flow’ arrow indicates a 922
traditional interpretation of a conventional stoss-lee form.
923 924
Figure 7: Illustrative synthetic drumlin. a) Profile across the centre of drumlin. Drumlin interior 925
(grey shade) and topography (solid line), are the top 90% of a Gaussian curve (dashed line). b) 926
Plan view, grey shaded relief of drumlin, where black is zero. White line locates profile in a), 927
and dark lines trending N-S and E-W are given for reference.
928
929
Figure 8: Cumulative distributions (cdfs) of parameters (θL, V, H, L, W). a) Orientation: Solid 930
black line is input values for the synthetics measured from non-Gaussian outlines in the real 931
DEM using Method 1 (n = 178 of H > 0). This is overlain, indistinguishably, by the curve 932
combining the 10 synthetic DEMs generated from this input. Grey lines are for the individual 933
synthetic DEMs, and vertical bars are
!
y ± 2σ of the mean. Dashed line is Method 2 (n = 173 of 934
H > 0). b) and c) Volume and height: Solid black and black dashed lines as a), with black dotted 935
lines for the data of Smith et al. (2009). Dashed and dotted grey lines are values recovered by 936
the cookie cutter from synthetic DEMs created using Method 1 and 2 respectively, although they 937
mainly underlie the curves of the inputs in b). In b) solid grey line is an estimate of H, W and L 938
using Method 2, which is converted to V using an idealised Gaussian geometry. d) and e) are 939
distributions for length and width, with lines as in a). For consistency with the data of Smith et 940
al. (2009) only V > 0 are used throughout this figure. Smith et al. (2009) do not determine θL, L, 941
W for comparison.
942 943
Figure 9: Spatial distribution of drumlins. a) Observed distribution. b) Spatial density (i.e. % of 944
area) covered by drumlin, smoothed with a three kilometre wide boxcar filter, and displayed as a 945
proportion of the highest density within the area. c) Illustrative spatial distribution from one of 946
20 synthetic DEMs in study. d) Four synthetic DEMs neighbouring c) in generation process to 947
illustrate variability. Map coordinates are of the British National Grid.
948 949
Figure 10: Visualisation of method used to assess claims about differences in mean volume 950
between sub-populations. Probability density curves generated by stochastic analysis, n = 951
100,000, for the observations of Smith et al. (2009) (dotted line), Method 1 (solid line), and 952
Method 2 (dashed line). Vertical bars are 95% critical value and the difference observed by 953
Smith et al. (2009). For simplicity, curves are scaled so that 95% critical values align with that 954
generated by the data of Smith et al. (2009).
955 956
Figure 11: Effectiveness of the extraction of drumlin H and V. a) Histogram of individual 957
height recoveries, expressed as ratio of recovered height over input height, by Method 1. Arrow 958
is correct recovery. Circle is mean ratio, bar indicated ±1σ. Grey bars represent all drumlins, 959
Figure 12: Reliability of recovered population parameters
!
Method 1. Input mean height (light grey bar) is significantly less than recoveries from the 10 964
synthetic DEMs (grey dots), whose mean and range (±2σ) is displayed by the black dot and bar.
965
b) as a), except about
!
V and shows recoveries consistent with input values. c) and d) are as for a) 966
and b), but for Method 2.
967 968
Figure 13: Comparison of recovered V distributions. a) Distributions recovered as Smith et al.
969
(2009) (dotted line), and using idealised Gaussian drumlins from synthetic DEMs generated from 970
Methods 1 and 2 (solid and dashed lines respectively). Means of the curves,
!
Vr (solid vertical 971
line), are aligned to that of the extraction as Smith et al. (2009) by linear multipliers applied:
972
‘scale factor’. c) Corresponding input distributions. Grey lines are for extractions of DEMs 973
created by Method 1 with errors of 1.0, 2.0, 3.0, 4.0 and 5.0 times those actually recovered, 974
‘error multiplier’: higher factors are lighter. b) and d) mean absolute misfit (i.e.
975
form
!
| x1" x2 |
#
) between curves for varied ‘Scale Factor’ and ‘Error Multiplier’ and the 976recoveries as Smith et al. (2009). Best fit is white circle. Contour 25% higher than the best fit 977
defines error ellipse (black line). Search increments 0.02.
978 979
Figure 14: Schematic illustration of this action of a bi-cubic spline. Topography (solid lines) is 980
from Fig. 1, including observed noise from Fig. 4a. Arrows indicate gradients at the outline, and 981
dashed line is a minimum curvature (bi-cubic) interpolation (T = 0) to estimate the basal surface 982
(dotted line). Tensioning the spline reduces the amplitude of deviations.
983 984
Tables 984
985
Table 1: Percentage of ‘good’ estimates, with recovered values of H and V within ±25% of 986
actual values. Values in brackets use a variant of the cookie cutter recovery method using an un-987
Table 2: Input and recovered population parameters. The mean and standard deviation (σ) of 991
the estimates of
!
Vr and
!
Hr from 10 synthetic DEMs is reported in each case. Individual data 992
plotted on Fig. 12, grey dots. Note, σ is not a standard error. Values in brackets use a variant of 993
the cookie cutter recovery method using an un-tensioned spline.
994
Table 3: Spatial effects upon extraction methodology, EV, evaluated by comparing the difference 997
between input and recovered values in the LGM and YD populations. Subscripts denote volume 998
and height. Note that the standard deviation of E, σ, is the standard error of the differences 999
between LGM and YD for individual DEMs. Square brackets are for the un-tensioned spline 1000
variant.
1001
Method EV (m3) EH (m) 1 1,828 ± 5,146 (1σ)
[6,272 ± 5,756]
1.34 ± 0.22 (1σ) [1.28 ± 0.23]
2 -5,094 ± 2,862 (1σ) [1,033 ± 4,566]
1.15 ± 0.32 (1σ) [1.41 ± 0.39]
1002 1003
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
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Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14