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Determination of Iceberg Draft, Mass and Cross-Sectional Areas
Barker, Anne; Sayed, Mohamed; Carrieres, T.
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Proceedings of The Fourteenth (2004) International Offshore and Polar
Engineering Conference, 1, ISOPE-2004-MS-02, 2004-05-23
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Determination of Iceberg Draft, Mass and Cross-Sectional Areas
A. Barker1, M. Sayed1 and T. Carrieres2
1Canadian Hydraulics Centre, National Research Council of Canada
Ottawa, ON, CANADA
2 Canadian Ice Service, Environment Canada
Ottawa, ON, CANADA
ABSTRACT
A new operational iceberg forecasting model is under development at
the Canadian Ice Service (CIS). The model deals with iceberg drift,
deterioration, and calving. One of the main features of the new model is
the utilization of detailed environmental conditions. In particular, the
vertical distribution of water current is used to calculate water drag
forces. An accurate description of keel geometry is, therefore, needed
in order to take advantage of the detailed water current information.
This paper describes the analyses done to determine the geometry of
iceberg keels and sails. The objective was to provide refined input for
the iceberg drift section of the forecasting model. Available iceberg
data were used to create empirical equations which describe keel cross-
sectional areas at different depth intervals from a given waterline
length. The equations also determine sail area, draft, and mass as
functions of waterline length. This is the first investigation that
determines geometry in detail.
KEY WORDS: iceberg forecasting; drift; iceberg geometry; draft;
length; mass; area.
INTRODUCTION
The development of the Grand Banks of Canada for oil and gas
production requires reliable forecasting of iceberg and bergy bit drift
and deterioration, to ensure the safety of offshore structures and
shipping operations. A new operational model has been developed at
the Canadian Ice Service (CIS) in response to emerging forecasting
requirements. The model deals with the drift, deterioration, and calving
of icebergs. One of the main features of the model is employing
detailed environmental forcing information in order to improve
accuracy of the forecasts. In particular, detailed vertical profiles of
water current are used to calculate water drag forces, which lead to
significant improvements in predicted drift tracks. Previous prediction
models assumed a uniform current independent of depth. Naturally, an
estimate of keel area variation with depth is needed for appropriate
evaluation of drag forces. At present, no such information concerning
keel geometry is available in the literature.
The present investigation was primarily aimed at determining the
detailed geometry of the keel. The observations of Smith and
Donaldson (1987) provide the most comprehensive description of
iceberg geometry, along with measurements of drift and environmental
conditions. Those observations include the only available
measurements of keel cross-sectional area variation with depth. This
data set was used to develop a parameterization of iceberg geometry.
The main objective of the study was to study the available data in order
to determine the variation of keel width with depth. Additionally, sail
width, keel depth, and iceberg mass were examined. Supplemental data
concerning the draft and mass of the icebergs from other sources were
also examined in the analysis (e.g. Canatec et al., 1999; Brooks, 1985;
El-Tahan and Davis, 1985; Hotzel and Miller, 1985; Robe, 1976;
among others). In turn, the results of the analyses were used to develop
predictive formulas that describe the above aspects of iceberg
geometry.
THE DATA SET
The measurements of Smith and Donaldson (1987) were conducted
from 1983 to 1985 over locations covering the Strait of Belle Isle, the
southern Labrador shelf, and the Grand Banks. The measurements were
taken from a ship, which followed icebergs at relatively close distances
(1 to 2 km). The data covers 12 track segments of 9 icebergs.
Measurements recorded the track of the icebergs, vertical profiles of
water current, wind speeds, and temperatures. The processed
information of those variables was recorded at 10-minute intervals. To
determine the geometry of the icebergs, sonar profiles of the keel were
analyzed to produce cross-sectional areas at 10-m depth intervals. Sail
cross-sectional areas were also surveyed, using photographs and taking
vertical sextant angles above the horizon at known radar ranges. The
cross-sectional areas were calculated along two perpendicular
directions (length and width). Fourteen cross-sections were taken in
total. Five of these were repeats of icebergs that had been previously
observed.
Proceedings of The Fourteenth (2004) International Offshore and Polar Engineering Conference
Toulon, France, May 23
−
28, 2004
Copyright © 2004 by The International Society of Offshore and Polar Engineers
ISBN 1-880653-62-1 (Set); ISSN 1098-6189 (Set)
899
ICEBERG DRAFT AND MASS
A summary of the Smith and Donaldson (1987) measured iceberg
dimensions, mass and cross-sectional areas is given in Table 1. The
accuracy of the data was ±5% above water and ±10% below water.
Table 1. Waterline dimensions, draft, mass, and sail cross-sectional
area (CSA), from Smith and Donaldson (1987).
Measured
Height
Measured
Length
Measured
Width
Measured
Draft
Estimated
Mass
Mean
Sail
CSA
Iceberg Type m m m m kilotonnes m²
83-1 pinnacle 19 66 37 54 85 445
83-2A drydock 32 146 86 96 800 2646
83-2B drydock 33 137 86 83 860 2510
83-3A domed 25 129 71 84 620 1871
83-3C domed 27 99 67 89 530 1548
83-5 drydock 20 77 56 67 147 624
84-5A drydock 43 198 181 120 2100 4039
84-5B drydock 44 204 136 110 1700 4254
84-6A domed 19 90 70 70 320 1033
84-6B domed 20 86 73 75 270 1191
84-7 domed 32 178 137 110 1700 4055
85-1A blocky 23 118 92 110 570 1820
85-1B blocky 23 118 92 110 570 1820
85-4 drydock 16 61 41 40 33 387
The focus was on using the waterline length (the largest horizontal
distance across the iceberg at the waterline) of the icebergs to
determine the variation of keel width with depth, sail width, keel depth,
and iceberg mass. This waterline length was chosen because most
observations of icebergs are determined from aircraft. Often iceberg
length and shape are the only values estimated. Additionally, as
icebergs are generally observed at weekly intervals, the length and
geometry evolve between observations. This effect is included in the
deterioration portion of the drift model, and it is essential to feed this
information back into the model.
Initially, the data were plotted to examine relationships between draft
and each of the iceberg length, width, height, mass and volume values.
As there are no present means of determining these values theoretically,
one must use curve-fitting techniques with what data is available.
Fitting a power curve to the draft versus length data led to a reasonable
definition:
D=2.91L 0.71 (1)
where D is the draft of the iceberg in metres, and L is the waterline
length in metres. This is similar to other empirical relationships
between draft and length found in the literature (see El-Tahran, 1982;
Hotzel and Miller, 1983; Buckley et al., 1985; Canatec, 1999; for
example). Equation (1), however, involves dimensional parameters.
Subsequently, a satisfactory linear relationship, involving a
dimensionless parameter, between draft and length was found from
regression analysis to be:
D=0.7L (2)
The advantage of Equation 2 is that it is dimensionless, which helps to
minimize the effects of erroneous data, and makes better use of a
limited data set.
A comparison of the results from Eq.1 and Eq.2 is shown in Figure 1.
In comparing the two relationships for relating draft to waterline length,
it can be seen in Figure 1 that the linear equation (Eq.2) underestimates
the draft for smaller waterline lengths, and overestimates the larger
lengths compared to Eq.1. However, the determination that iceberg
draft is also 70% of the waterline length is in keeping with results from
Hotzel and Miller (1985).
A similar analysis was carried out for the length-mass relationship,
resulting in the equation:
M=0.43L 2.9 (3)
where M is the mass in tonnes and L is the length in metres. The
equation was simplified to the following dimensionally correct form
M=0.5
ρ
iceL 3 (4)
where
ρ
ice is ice density, taken here to be 910 kg/m³. Figure 2 shows a
comparison of the two empirical mass equations developed here, Eq.3
and Eq.4. It can be seen that Eq.4, the dimensionless equation,
overestimates at waterline lengths greater than 100 m. For example,
with a waterline length of 220 m, the calculated mass ranges from
approximately 2 million tonnes to 4 million tonnes, depending upon
whether Eq. 3 or Eq.4 is used.
Eq.1: D = 2.91L0. 71
R2 = 0.76
Eq.2: D = 0.7L
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250
Measured Length (m)
Measured Draft (m)
Figure 1: Comparison of Equations 1 and 2, relating iceberg draft to
waterline length. The measured draft values used to calculate Eq.1 are
indicated in the plot.
Eq.3: M = 0.43L2.9
R2 = 0.92
Eq.4: M = 0.5ρiceL³
0.0E+00
5.0E+05
1.0E+06
1.5E+06
2.0E+06
2.5E+06
3.0E+06
3.5E+06
4.0E+06
0 50 100 150 200 250
Measured Length (m)
Measured Mass (tonnes)
Figure 2: Comparison of equations relating iceberg mass to waterline
length. The measured mass values used to calculate Eq.3 are indicated
in the plot.
Previous empirical equations for relationships between draft and
waterline length and mass and waterline length have yielded a variety
of results, as shown in Table 2. Given the diverse data sets, scatter in
900
the results is to be expected. This is especially evident in the mass
calculations. Even so, the results had reasonable agreement and
generally fell within ±10% of each other for icebergs less than 200 m.
Large icebergs (with a waterline length greater than 200 m)
encountered in the regions of interest to the present work are usually
tabular. Non-tabular large icebergs would ground in the relatively
shallow water depths. Obviously, geometry of tabular icebergs is not
the focus of this study. Given this criteria, it was determined that the
present study gave results that were in general agreement with the
previous studies, within that range, for use as input into a numerical
model for predicting iceberg drift.
Table 2: A sample of other empirical equations for determining iceberg
mass and draft from waterline length, and their source reports
Equation and Source Report
MassHibernia, Canatec (1999): M=1.03L^2.67
Labrador, Canatec (1999): M=2.25L^2.58
Singh et al. (1998): M=0.97L^2.78
Fuglem et al. (1995): M=0.81L^2.77
Hotzel and Miller (1983): M=2.009L^2.68
DraftHibernia, Canatec (1999): D=1.95L^0.79
Labrador, Canatec (1999): D=3.9L^0.63
Hotzel and Miller (1983): D=3.781L^0.63
SAIL AND KEEL AREAS
The Smith and Donaldson (1987) data set included vertical cross-
sectional areas of the keel of each iceberg at 10 m depth intervals, as
well as sail areas, taken from two views. These area data were then
averaged. An example of the reported average data for two of the
icebergs is shown in Table 3. The focus of the work for the iceberg
drift model was to establish a formulation for determining keel cross-
sectional area(s) based upon waterline length. To establish a
correlation, plots were made of waterline length versus cross-sectional
area, for each of the vertical sections contained in the data set.
Table 3: Example vertical cross-sectional areas from data set. Areas
are averages of two cross-sections taken at different angles.
Iceberg
83-1
Iceberg
83-3C
Depth Area (m²) Depth Area (m²)
Sail 445 1548
Layer 1 0-10m 546 0-10m 976
Layer 2 10-20m 500 10-20m 1071
Layer 3 20-30m 427 20-30m 1091
Layer 4 30-40m 403 30-40m 1115
Layer 5 40-50m 348 40-50m 1062
Layer 6 50-54m 51 50-60m 970
Layer 7
60-70m 795
Layer 8
70-80m 531
Layer 9
80-89m 221
A plot of sail areas versus waterline length is shown in Figure 3. The
relationship representing the best fit of the data is expressed as
Asail = a0L+b0 (5)
where Asail is the cross-sectional area of the sail (m2), and a0 (m) and b0
(m²) are parameters determined by curve-fitting the data as shown in
Figure 3.
For each layer of the keel, the area is expressed, in a similar manner, as
A(k) = a(k)L+b(k) (6)
where A(k) is the cross-sectional area of layer k, which extends from (k-
1) x 10 m depth to k x 10 m depth. Figure 4 shows, as an example, a
plot of the areas versus waterline length for Layer 1, and the associated
best-fit line.
A = 28.194L - 1420.2
R
2 = 0.97
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 50 100 150 200 250
Measured Length ( m)
Sail Cross-Sectional Area (m²)
Figure 3: Plot of vertical cross-sectional area versus length for iceberg
sail values, where R² is the correlation coefficient.
A = 9.5181L - 26.11
R
2 = 0.93
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 50 100 150 200 250
Measur ed Len gt h ( m)
Layer 1 Cross-Sectional Area (m²)
Figure 4: Plot of iceberg Layer 1 (0-10 m keel depth) cross-sectional
area versus length, where R² is the correlation coefficient.
Curve fits were done for each layer to determine the parameters a(k)
and b(k). The results showed that Eq. (6) represents the data with
sufficient accuracy. Hence, a set of empirical equations was developed
for each “layer” of an iceberg, based solely on waterline length. This
set is shown in Table 4.
The largest draft in the data-set of Smith and Donaldson (1987) was
120 m. Drafts up to approximately 230 m, however, are reported in the
Grand Banks area of Canada (e.g. Miller and Hotzel, 1985). In order to
expand the equations for deeper drafts (but still less than 200 m, as
previously discussed), plots of the slope and intercept values from the
vertical cross-sectional area equations versus maximum draft depth for
each layer were created (Figures 5 and 6). In these figures, an obvious
change in slope of each graph occurs at the draft depth of 90 m. It was
not evident why this change occurred. The data below these depths
901
should be used with caution, as it is possible that the change is due to
the small number of data points at these deeper depths. Fitting a power
equation to each section of the plot above and below this 90 m depth
gave equations for calculating the slope and intercept for a given layer
depth. In this case, equations were developed for drafts up to 160 m,
which are also shown in Table 4, for Layers 12-16.
Using the equations found in Table 4, combined with Eq.2 (the
calculation for estimating iceberg draft based upon iceberg length),
composite icebergs can be created. Two such icebergs, with drafts of
70 m and 105 m, are shown in Figure 7. The pictures show only the
keel of the icebergs. These composites, and the equations they were
derived from, were used in the calibration of the CIS iceberg drift
numerical model, discussed in the following section.
Table 4: Parameters for calculating vertical cross-sectional areas
Height/Depth (m) a(k) b(k)
Sail 0+ 28.194 -1420.2
Layer 1 0-10 9.5181 -26.11
Layer 2 10-20 11.17 -107.42
Layer 3 20-30 12.482 -232.44
Layer 4 30-40 14.004 -407.02
Layer 5 40-50 14.327 -456.91
Layer 6 50-60 14.8 -599.7
Layer 7 60-70 14.68 -720.56
Layer 8 70-80 16.098 -1168.1
Layer 9 80-90 17.136 -1662.9
Layer 10 90-100 13.223 -1199
Layer 11 100-110 6.4432 -503.5
Layer 12 110-120 4.50 -319.1
Layer 13 120-130 3.05 -198.9
Layer 14 130-140 2.13 -128.4
Layer 15 140-150 1.53 -85.47
Layer 16 150-160 1.12 -58.39
0
2
4
6
8
10
12
14
16
18
0 102030405060708090100110120
Maxi mum depth of layer ( m)
Slo
p
e
Figure 5: Plot of vertical cross-sectional area slope parameter versus
maximum depth of layer
-1800
-1600
-1400
-1200
-1000
-800
-600
-400
-200
0
0 102030405060708090100110120
Maxi mum depth of layer ( m)
Interce
p
t
Figure 6: Plot of vertical cross-sectional area intercept parameter
versus maximum depth of layer
Iceberg Length = 100m
Keel Depth = 70m
Iceberg Length = 150m
Keel Depth = 105m
Figure 7: Composite icebergs, created using equations from Table 3
USING THESE RESULTS TO PREDICT ICEBERG DRIFT
The new Canadian Ice Service iceberg forecasting model deals with
iceberg drift, deterioration and calving, as well as the drift and
deterioration of calved bergy bits. The formulation of the model was
discussed by Savage (2001). Carrieres et al. (2001) also gave an
overview of the model implementation and testing. The analysis of
iceberg geometry presented in this paper was used in the section for
modelling iceberg drift, which is carried out by considering the various
forces that act on each iceberg, and solving the linear momentum
equations. This was incorporated into the model in 2000. The model
includes forces resulting from water drag, air drag, wave radiation
pressures, and water pressure gradient. The linear momentum
902
equations of each iceberg include the sum of those forces. Added mass
and Coriolis force are considered in the equations. An implicit solution
is used in the present version of the program to obtain the velocities
and update the positions.
The equation parameters shown in Table 4 were used to refine
calculations of the air and water drag forces. Water drag forces are
calculated over 10 m-depth sections of the keel. The present formulas
are used to calculate the areas of those sections of the keel (using
waterline length). Water current at those levels is obtained from an
ocean model (see Carrieres, 2001), and is then used to calculate the
drag forces. The vector sum of those forces gives the resultant drag
force on the keel.
An example of a test of the model showing the predicted and observed
drift tracks for one case from Smith and Donaldson (1987) is shown in
Figure 8. In that test, measured water current and wind values were
used as input. In Smith and Donaldson (1987), the iceberg’s waterline
length is listed as 66 m and the duration of the drift was 14 hours. A
drag coefficient of 1.5 was used in calculations of both wind and water
drag forces. The best fit to the observed track occurred when the value
for L used in the model was chosen as an average value between the
measured waterline length and the average of the waterline length and
width values combined (in this case, 58 m, also shown in Figure 8).
We note that recent parametric studies by Kubat and Sayed (2003)
showed that using surface water current values leads to substantial
inaccuracy. Their conclusions indicate that calculations of keel
geometry are essential for acceptable forecasts.
-7
-5
-3
-1
1
3
5
7
02468101214
East (km)
North (km)
Model
L=58 m
Observed
Model
L=66 m
Figure 8: Predicted and measured iceberg trajectories, Smith
and Donaldson (1987) iceberg 83-1, with a waterline length of
66 m and 58 m. It can be seen that the two results are very
similar.
RECENT MODIFICATIONS TO THE EMPIRICAL MODEL
A number of refinements were made to the above empirical formulas
of the geometry at CIS, in the four years after the model’s initial
development in 2000. These changes examined the effects of small
sail areas, the bottom layer of keel areas and the extrapolation of larger
keel depths from the existing data set. The above formulas are
obviously limited to the range of waterline lengths of the Smith and
Donaldson (1987) data. Considerably smaller lengths may cause
inaccuracies. Based on the equations found in Table 3, the model can
only accept input for icebergs with a waterline length such that the
equations will not give “negative” heights or drafts. Rather than a
linear equation, a power law relationship was fit to the smallest three
icebergs in the data set. This resulted in the following equation for the
cross-sectional area, Asail, for icebergs with waterline length smaller
than 65 m:
Asail = 0.077L² (7)
For the bottom keel layer of each iceberg, the area (of the bottom
layer) was plotted versus length, regardless of depth. That led to a poor
correlation. Subsequently, the amount that the keel projects into the
bottom layer was included to created a new power law relationship:
Ab= 0.279(Ldk)0.989 (8)
where Ab is the bottom layer area and dk is the amount that the keel
projects into the bottom layer of the iceberg. Equation (8) was thus
used to handle cases where the bottom layer of the keel was relatively
small (depth smaller than 10 m).
The original data set contained only a few icebergs that had keel depths
greater than 100 m. To improve on the equations found in Table 3, the
relationship between keel cross-sectional areas at adjacent depths were
examined. That was done for layers below the depth of maximum
width, which produced the highest correlation (i.e. given the cross-
sectional area of the upper layer, the lower layer may be calculated
from a linear relationship - see Figure 9). Areas near the bottom of the
berg were excluded (to remove the effect of partial keel projection into
a layer depth). The resulting equation was:
AL=0.961AU+111.67 (9)
where AL is the cross-sectional area of the lower layer and AU is that of
the upper layer. Figure 10 shows a plot of the relationship between the
areas of each two adjacent layers. From Equation (9), simulated keel
areas for depths larger than 100 m were generated, and then used to
refine equations describing the areas of sections of the keel up to 200
m in depth. Table 5 shows the effects of these modifications on the
equations for vertical cross-sectional area with length. These changes
helped to better define iceberg drift in the CIS model.
Depth of
maximum width
Lower la yer
i
Upper layer
i
Figure 9: Variables used to study the relationship between cross-
sectional areas at adjacent depths, below the depth of maximum
iceberg width.
0
500
1000
1500
2000
2500
0 500 1000 1500 2000 2500
Upper Area (m²)
Lower Area (m²)
Figure 10: Relationship between cross-sectional areas at adjacent
layers, below the depth of maximum iceberg width
903
Table 5: Modified vertical cross-sectional area parameters
Depth (m) a(k) b(k)
Layer 1 0-10 9.5173 -25.94
Layer 2 10-20 11.1717 -107.50
Layer 3 20-30 12.4798 -232.01
Layer 4 30-40 13.6010 -344.60
Layer 5 40-50 14.3249 -456.57
Layer 6 50-60 13.7432 -433.33
Layer 7 60-70 13.4527 -519.56
Layer 8 70-80 15.7579 -1111.57
Layer 9 80-90 14.7259 -1125.00
Layer 10 90-100 11.8195 -852.90
Layer 11 100-110 11.3610 -931.48
Layer 12 110-120 10.9202 -1007.02
Layer 13 120-130 10.4966 -1079.62
Layer 14 130-140 10.0893 -1149.41
Layer 15 140-150 9.6979 -1216.49
Layer 16 150-160 9.3216 -1280.97
Layer 17 160-170 8.9600 -1342.95
Layer 18 170-180 8.6124 -1402.52
Layer 19 180-190 8.2783 -1459.78
Layer 20 190-200 7.9571 -1514.82
CONCLUSIONS
This paper documents an analysis of iceberg geometry. The objective
was to improve the accuracy of estimating water and air drag forces on
iceberg, which, in turn, would improve the forecasts of iceberg drift.
The present analysis is the first to establish a detailed description of
keel width variation with depth. It also provides reliable estimates of
sail cross-sectional areas, and iceberg mass. The preceding analysis
has primarily relied on the data of Smith and Donaldson (1987), which
was the most complete and detailed information on iceberg geometry,
drift, and environmental conditions available. These data were
supplemented with other relevant information from various sources.
The analysis showed that geometry information could be adequately
described using waterline length of the iceberg. That finding is
particularly useful since waterline length is relatively easier to observe
than other attributes of an iceberg. The mass and cross-sectional area
of the sail were related to waterline length. For the keel, areas of
sections, each 10 m in depth, were determined as linear functions of
the length. Correlation was relatively high for all the formulas obtained
in this analysis. Further refinement of the initial analysis correlated the
areas of adjacent layers of the keel. That correlation was the basis for
extending the data to describe keel geometry for larger depths
(>100m), where the data is relatively sparse.
The present results are used in an operational iceberg forecasting
model. Ongoing work aims at further validation of the results.
Additionally, further data is being acquired through data mining and
from profiling studies conducted in 2002 and 2003. This new data will
hopefully cover waterline length values beyond those of Smith and
Donaldson (1987) and would be of particular interest to add to this
study. This would increase the confidence in the present analysis.
ACKNOWLEDGEMENTS
The authors would like to thank S.B. Savage for valuable suggestions
throughout the study. The financial support of the Program on Energy
Research and Development (PERD) is gratefully acknowledged,
through the Operational Ice Modelling project and the Offshore
Environmental Factors POL.
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