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SURVEY AND STATISTICAL PARAMETERIZATION OF BED
ROUGHNESS OF A CORAL REEF
A THESIS SUBMITTED TO THE GRADUATE DIVISION OF THEUNIVERSITY OF HAWAI'I IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
OCEAN & RESOURCES ENGINEERING
AUGUST 2005
By
Vasco B. Nunes
Thesis Committee:
Geno Pawlak, ChairpersonKwok Fai CheungMark Merrifield
© Copyright (2005)
by
Vasco B. Nunes
III
IV
ACKNOWLEDGMENTS
I would like to thank my committee for all their assistance and interaction, without
their guidance I would have never reached my goal. A special thanks to my advisor,
Dr. Geno Pawlak, by all means this is the end result of our work and interaction. I
would like to thank Marion Bandet and Kimball Milikan for their much useful sug
gestions and comments as well as their help in the field work. All the encouragement
words were much welcomed. I would like to thank Monte Hansen, Pedro Garcia and
Rick Carter for their key help with the field instruments' setup, their input made
all the difference. I would like to thank Danny Merritt, Sonja DuPlessis and Andy
Hebert for their diving assistance. I would like to thank Chip Fletcher and Chris
Conger for their generosity in lending me the Force motorboat. I would like to thank
Kumar Rajagopalan, Volker Roeber, Yong Wei and Liujuan Tang for their help in
the data analysis process. I also would like to thank Edith Katada and her assistants
for their patience with my always late paperwork. Thanks to Amal Phadke as he
kindly supplied me with this I¥IEX template. Last but not least, a big thank you to
my family and girlfriend, Maria Byrne, for their constant and unconditional support.
This research was supported in part by a grant from the Office of Naval Research and
Sea Grant.
This personal achievement is entirely dedicated to my father.
v
ABSTRACT
Sea bed roughness was measured and quantified in an area of the south shore of O'ahu
using a boat-mounted acoustic altimetry system. The study site is characterized
by a highly inhomogeneous rough sea bed. Roughness was resolved in the 10 to
200 em range over an area extending from the wave breaking zone to a depth of
20 m. Theory suggests that both roughness length scales and distribution play an
important role as a wave energy dissipation mechanism. This paper investigates
methods for quantifying inhomogeneous bed roughness within this range of scales.
Various roughness parameterizations are examined including spectral and statistically
derived variables. These methods are validated qualitatively with side-scan imagery
and quantitatively by in-situ roughness surveys. Results show that wave number
spectra can differentiate smooth from rough areas. Results also suggest that the bed
roughness in the study site exhibits a high spatial variability, within the examined
range, with no dominant length scale but with a characteristic spectral slope. In
addition, roughness also exhibits a sharp contrast between the shallow and deep ends
of the study site.
TABLE OF CONTENTS
VI
Abstract ... V
List of Tables viii
List of Figures IX
1 Introduction . 1
1.1 Bed roughness and the wave friction factor 1
1.2 Research framework. 5
2 Study Area 9
3 Methodology 11
3.1 Data collection 11
3.2 Data quality control 14
3.3 Data analysis .... 16
3.3.1 Data gridding 16
3.3.2 Spectral analysis 16
3.3.3 Data validation 18
4 Results .. ...... 21
4.1 Spectral results 21
4.2 Validation of spectral results. 25
4.2.1 Side-scan survey ... 25
4.2.2 'Roughness' bar survey 26
4.3 Root mean square validation . 30
VB
References . . . . . . . . . . .
A Confidence intervals . . .
C Boat-bar spectral merges
COULWAVE simulations
32
35
1
2
6
7
Discussion . . . . . . .
Conclusions summary.
4.4
4.5
B
Vlll
LIST OF TABLES
B.1 Depth (m), significant wave height (m), peak period (s) and wave direc-tion (0) for each site. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3
B.2 Depth (m) and modelled significant wave height (m), peak period (s)and wave direction (0) for sites 1,2 and 3. . . . . . . . . . . . . . . . .. 4
B.3 Depth (m) and modelled significant wave height (m), peak period (s)and wave direction (0) for sites 4 and 5. . . . . . . . . . . . . . . . . .. 4
B.4 Depth (m) and modelled significant wave height (m), peak period (s)and wave direction (0) for sites 8 and 10. . . . . . . . . . . . . . . . .. 5
ix
LIST OF FIGURES
Figure Page
1.1 Typical bed roughness offshore of Honolulu Harbor. The roughness scalescan be roughly gauged by the diver in the background. . . . . . . . .. 6
2.1 Location of study area on the south shore of Oahu, courtesy of the U.S.Geological Survey and the EROS Data Center. . . . . . . . . . . . . .. 9
3.1 Detail of part of the boat mount setup (starboard side) with the ADCPlooking downwards. On the portside an identical setup was used for thealtimeter. 11
3.2 Survey site in the adopted projection along with the survey boat tracks.The 5 m bathymetry contours are also plotted. . . . . . . . . . . . . .. 13
3.3 An example of the altimeter beam (solid line) and boat tracks (dottedline) with a stable boat ride (left) and less stable boat leg interruptedby rejected points (segment between diamonds; x=56m, y=-26m). . .. 15
3.4 Measurements using the 'roughness' bar. A schematic of the apparatus(left) and typical measurement procedure (right). 18
3.5 Visual scale for the qualitative analysis of the side-scan images. Thesepictures, left (smooth=l) to right (rough=5), illustrate the adopted clas-sification. 20
4.1 Spectral curves for boxes 4, 28, 32 and 40. 22
4.2 Roughness rms values for four wave number bands. Lighter tones correspond to smoother beds while darker tones corresponding to rougherbeds. Bathymetry contours are also plotted. Boxes with insufficient dataare marked with targets. . . . . . . . . . . . . . . . . . . . . . . . . .. 23
4.3 Normalized rms values as a function of depth. The horizontal line to the12 m tick. The vertical line represents the mean normalized rms. . . .. 24
4.4 Qualitative roughness map based on a side-scan survey with density ofobservations show in the left and interpolated scores in the right. Boxeswith insufficient data are marked with targets. . . . . . . . . . . . . .. 26
x
4.5 Relationship between the side-scan and the roughness rms maps is exam-ined. Black dots are normalized rms values for the 63 to 200 cm interval. 27
4.6 Wavenumber spectra from the 'roughness' bar (dashed line) and the boatmeasurements (solid line) with 95% confidence intervals. 28
4.7 Spectral energy of the study area for the same wave number bands(37-47 cm, 47-63 cm, 63-100 cm and 100-200 cm) based on the 'roughness' bar data. Bathymetry contours are also plotted. Boxes with insuf-ficient data are marked with targets. 29
4.8 Spatial rms values of the study area for four wave number bands (37-47 cm,47-63 cm, 63-100 cm and 100-200 cm). Bathymetry contours are alsoplotted. Boxes with insufficient data are marked with targets. ..... 30
4.9 Relationship between rms values computed via spatial and wavenumberdomain:: circle = 30-60 cm, triangles = 60-90 cm, squares - 90-120 cm,diamonds = 120-220 em. . . . . . . . . . . . . . . . . . . . . . . . . .. 31
B.1 Location of wave gauges (white dots) with site numbering. 2
C.1 Wavenumber spectra from the 'roughness' bar (dashed line) and the boatmeasurements (solid line) with 95% confidence intervals. 6
1
CHAPTER 1
INTRODUCTION
Ocean waves lose energy at boundaries, via surface and bed stresses, and by breaking.
In the intermediate and shallow water regions, before breaking occurs, bed friction
becomes the main energy dissipation mechanism (Ardhuin et al. 2003; Hay and Smith
2002). Quantifying this energy loss is important for determining wave heights and
estimating sediment transport. In the near shore environment, these quantities affect
coastal works, harbor maintenance, marine habitats (Friedlander and Parrish 1998;
McCormick 1994) and local wave refraction/shoaling patterns. In the last 50 years,
considerable research has been dedicated to this field in order to understanding these
processes.
1.1 Bed roughness and the wave friction factor
Quantifying the bed energy dissipation is done by means of an hydraulic roughness
parameter, of which the Manning or Chezy coefficients are common examples used
in river hydraulics. In the ocean environment, this energy loss depends on the wave
friction factor (fw), or the energy dissipation factor. The value of fw is a function
of the flow conditions above the bed, represented by the Reynolds number, and the
physical sea bed roughness. For a non-movable bed, the energy dissipation is the result
of the skin friction and the form drag. The former represents the friction resulting
2
from viscous forces along the bed while the latter represents the drag generated by
the pressure differences generated by the roughness elements and distribution. For a
movable bed, roughness further includes the contribution of the sediment in motion
over the bed (bed-load).
Historically, much work has been devoted towards parameterizing sea bed rough-
ness for homogeneous beds, e.g. sand ripples. Jonsson (1966) pioneered the research
on the relationships between roughness and wave energy dissipation and in his pa-
per he introduced two expressions for the energy dissipation. The energy dissipation
occurs within the boundary layer where the waves' orbital motion produces a shear
stress defined by:
where,
Ed - time averaged energy dissipation;
Tb - bed shear stress;
Ub - near bed maximum orbital velocity, defined by:
A·wUb = --,.---------,--
sinh(k . h)
where,
A - wave orbital amplitude immediately above the bed;
w - radian frequency (21r IT);
(1.2)
(1.1)
3
k - wave number (27f/L);
h - water depth.
Jonsson (1966) derived the expression (1.3) for energy dissipation assuming a
constant dissipation, or friction, factor:
where,
Ie - energy dissipation factor;
p - water density.
(1.3)
Using the relationship for the conservation of wave energy flux along with lin-
ear wave theory, an alternative expression for the energy dissipation was presented
(Jonsson 1966; Madsen,K., and Graber 1988):
1Ed = - . P . Iw • Ubr . u~
4
where,
Iw - wave friction factor;
Ubr = 8/3· 7f • Ub.
(1.4)
Jonsson (1966) also showed from dimensional analysis that the friction factor
depends on the Reynolds number and on the bed roughness. However, in real con-
ditions the flow is predominantly turbulent. In this regime the wave friction factor
4
solely depends on the bed roughness parameter (Kamphuis 1975), thus underlining
the importance of the bed roughness parametrization. For a rough turbulent regime a
commonly used expression for the wave friction factor was suggested by Swart (1974),
and subsequently adjusted by Nielsen (1992) as it over-predicted fw at the smaller
roughness scales:
(1.5)
where,
r - hydraulic roughness.
The term r / A is the relative roughness and r is the hydraulic roughness also
corresponding to the Nikuradse roughness (kN ; Nikuradse (1933)). This roughness
definition is widely used and represents the sea bed morphology in terms of single
length scale. In a rough turbulent flow, the bed roughness elements penetrate outside
the laminar sublayer and therefore the vertical velocity profile reduces to zero (no-slip
condition) somewhere between the lowest and the highest roughness feature. Defining
this point, or equivalently hydraulically defining where the bed sits at, is not trivial
and the Nikuradse roughness addresses this particularly issue. Nikuradse observed
that for a bed closely packed with spheres of diameter d, the logarithmic velocity
profile is zero at a depth proportional to d. A general approach is to assume a rough
ness factor related to some measurement of grain size such as r = D50 . Madsen et al.
(1990) and Hay and Smith (2002), among others, extended these results to movable
5
beds under irregular waves, and presented alternative expressions for estimating both
the bed roughness and the wave friction factor.
1.2 Research framework
Despite all the contributions to this field of research, bed roughness parameterization
is still not an exact science. Most of the assumptions used in deriving roughness
parameterizations are based on small scale homogeneous beds with sand grain geom
etry. Moreover, these parameterizations apply for sea beds where the scale of the bed
features is at least one-fold smaller than the wave orbital amplitude. Validation is
lacking when the scales of these two quantities are comparable.
Roux (2003) observed that bed roughness can vary considerably within a given
site. Consequentially these physical abrupt changes are reflected in the boundary
layer, thus affecting directly the estimate of the bed friction. At the same time, a
particular roughness scale can dissipate energy differently depending on the wave
characteristics. Pawlak and MacCready (2002), for example, examined how oscil
latory flows react to different rough boundaries. Their results show that both the
length scales and distributions of roughness elements introduce mechanisms that can
enhance the energy dissipation near the boundary. This suggests that for accurately
determining the friction processes, bed measurements are needed and with it, effective
measurement schemes.
For a wide range of conditions, the accuracy of existing formulas decreases, par-
6
ticularly when dealing with rough and inhomogeneous beds such as coral reefs. In
these environments (Figure 1.1) roughness is inhomogeneous and individual elements
can often be of the same order of magnitude as the wave orbital amplitude. These
conditions challenge some of the assumptions used in the existing roughness parame-
terizations schemes.
Figure 1.1: Typical bed roughness offshore of Honolulu Harbor. The roughness scalescan be roughly gauged by the diver in the background.
Coral reefs and rocky sea beds have not been studied as extensively as sandy beds
in the context of wave friction. Gerritsen (1980) examined pressure sensors' data to
derive wave friction factors for a site adjacent, and very similar in bed morphology,
to our study area. He obtained values of fw ranging from 0.1 to 0.5 in agreement
with the predictions using Jonsson (1966) formula. More recently, Falter (2002)
examined wave attenuation data over a coral reef on the east coast of O'ahu. The
fairly flat reef platform allowed wave refraction, shoaling and diffraction effects to
be neglected relative to wave attenuation induced by wave friction. Using a range of
parameterizations he obtained an average fw value of 0.22±0.03 and a hydrodynamic
7
roughness scale of 0.2 m. Nelson (1996) also investigated wave friction and bed
roughness of a coral reef on the north-east coast of Australia with a wave environment
similar to that examined by Falter (2002). A hydrodynamic roughness of 0.06 m was
determined, based on measured wave heights for two data sets with fw values of 0.13
and 0.18. This value agreed with the results given by Jonsson (1966)'s and Swart
(1974)'s formulas. This bed roughness value is an order of magnitude smaller than
that inferred by Falter (2002) and suggests an attenuation magnitude similar to that
observed in sandy beds. It is suggested that such differences could be related to
individual reef differences since detailed physical measurements were not performed
in both studies.
This discussion illustrates the need for robust parameterization schemes of bed
roughness in order to estimate wave friction factors. Current state-of-the-art models
like SWAN, developed at Delft University of Technology (Booij et al. 1999), or
COULWAVE, developed at Cornell University (Lynett et al. 2000), do not recommend
the use of friction factors above 0.1 m since there is little or no validation in this
roughness range. For a typical Hawaiian coral reef, parameterizations within this
range, or higher, may be suggested based just on visual observations (Figure 1.1).
Bed shapes can often be 0.5 to 1.0 m in height and width, thus presenting unanswered
questions on how to calculate wave and current friction factors.
All said, there is no scientific agreement on how to parameterize inhomogeneous
roughness larger than the common sand ripple, much less for validating its use in
a wave model. This thesis attempts to improve our understanding in this issue.
8
Following recent studies using spectral analysis to investigate sea bed morphology
(Ardhuin et al. 2003; Chae et al. 2004), we examined the bed roughness of a coral
reef using in-situ measurements. Potential parameterization schemes are suggested
and discussed.
9
CHAPTER 2
STUDY AREA
The study area was located offshore of Honolulu Harbor on the south shore of O'ahu,
covering approximately 1 km2 and ranging from 5 to 20 m deep. Figure 2.1 shows the
location of the study site along with a close up of the region of survey. The survey
'box' was rotated relative to north to be roughly aligned with the incoming wave
direction. Also shown are the 5 m contours with the surveyed area bounded by the
box-grid. Each box dimensions are 75 x 75 m. The sea bed in the area is comprised
of coral reef and sand, exhibiting a wide range of roughness scales.
Figure 2.1: Location of study area on the south shore of Oahu, courtesy of the U.S.Geological Survey and the EROS Data Center.
The wave environment is seasonal although energy is present year round. During
winter time, wave heights are small (less than 1 m) associated with weak south swells
and short period energy resulting from the wrap wind swell generated by north-
10
easterly trades. During summer, wave heights are higher but rarely larger than 2
m, associated with very long period swells generated by distant southern hemisphere
storms. Wave breaking only occurs within the shallowest portion of the site. The
offshore slope in the study area is relatively steep, ranging from 0 to 60 m deep within
about a kilometer. This suggests that local bathymetry accounts for most of the wave
transformation process, in contrast with broad shelf continental regions.
11
CHAPTER 3METHODOLOGY
3.1 Data collection
Data were collected using a boat-mounted altimetry system consisting of an acoustic
doppler current profiler (ADCP), a narrow beam altimeter (with a 2.5 0 nominal beam
width) and a GPS, all linked to a laptop. The ADCP and the altimeter were mounted
on an aluminum frame secured to the boat's hull. This frame was assembled using
scaffold equipment and attached to the boat using a pressure bolt system (Figure
3.1). Power to the ADCP and altimeter was provided by a 24 volt battery set, while
the GPS and laptop used their own power supply systems.
Figure 3.1: Detail of part of the boat mount setup (starboard side) with the ADCPlooking downwards. On the portside an identical setup was used for the altimeter.
12
A WAAS enabled GPS was used to provide a reference time stamp and an absolute
position within a 4 m accuracy. This accuracy is adequate for navigational purposes,
however, a 4 m accuracy was too coarse to resolve submeter roughness features. GPS
position data, therefore, was only used in the data gridding process, discussed in
section 3.3.1.
For accurate relative positioning, ADCP ouput was used. The ADCP yields high
accuracy vessel velocities via its bottom tracking capability (Fong and Monismith
2004). Relative vessel position can then be obtained by integrating the vessel velocity
with respect to time, which in its discrete form is represented by:
Sx = VN . COS(()i) . !:1Ti
Sy = VE . sin(()i) . !:1Ti
where,
Sx - along-track distance in the X direction;
Sy - along-track distance in the Y direction;
VN - north velocity;
VE - east velocity;
()i - ith angle of velocity magnitude with respect to the adopted projection;
!:1Ti - ith time increment.
(3.1)
(3.2)
The result is a time series of boat velocity converted into a time series of along
track distance series, Sxy(T), for the adopted projection. This projection has its zero
13
south of the site with positive X direction alongshore to the right, and positive Y
direction cross shore towards the coast. Figure 3.2 illustrates this projection along
with the boat survey tracks.
$6WIfboat_
JcXi .- ·200 ·100 0 100 200 300 <00 soo 800lIst-.lm)
Figure 3.2: Survey site in the adopted projection along with the survey boat tracks.The 5 m bathymetry contours are also plotted.
The time series of vessel velocitie was sampled at about 2 Hz (ADCP sampling
rate), with pitch and roll angles also provided by the ADCP sensors. Contrary to
the GPS output, the Sxy(T) error is cumulative but assumed to be negligible within
a survey track (Fong and Monismith 2004), which were always shorter than 330 m.
The boat provided a stable survey platform and the survey were performed in very
calm sea conditions when waves and wind were at their minimum.
The altimeter produced a time series of bottom range, Rxy(T), at a sampling rate
of about 10 Hz. The altimeter beam track was reproduced using pitch and roll values.
The altimeter unit sampled at about 10 Hz and therefore Sxy(T) was interpolated
to match the higher sampling rate of Rxy(T). These two time series resulted in a
14
spatial series of bottom range, i.e. along-track distance versus bottom range Rxy(S),
in the general direction of the local wave path, referenced to the long period south
swells (as opposed to short period trade wind swell). This time series, Rxy(S), was
the fundamental input for the data analysis.
3.2 Data quality control
The bottom range spatial series, Rxy(S), was filtered to remove questionable data
points. These points included locations where it was too shallow to navigate, the
altimeter exceeded its maximum range (> 20 m) or when the boat pitch and/or roll
angles were excessive. High values of these two angles were rare, caused primarily
by nearby boat traffic. As a result, all bottom range points from the altimeter track
were rejected if the altimeter track fell backwards in reference to the boat track, or
if the track moved laterally in reference to the boat track beyond a threshold value
(80% of the along-track distance between pings) or were obvious false readings, e.g.
Om.
It is shown in Figure 3.3 two examples of the altimeter (solid line) and the boat
tracks (dotted line). The altimeter track was usually offset from the boat track by
a small distance due to the instrument mount setup. This offset did not affect the
data acquisition since it was constant throughout the survey period. Ideally, smooth
boat legs yielded approximately parallel tracks between the boat and the altimeter
beam. Whenever a disturbance occurred, e.g. a ship wake, the boat would roll and
15
pitch making the tracks less parallel. The higher the disturbance the more likely the
existence of rejected data points. Figure 3.3 (right) shows some rejected points where
the altimeter (solid line) pinged backwards relative to the boat track (dotted line).
!t -17.5
.,~, 37.5 "
lMhllflCe(mJ
Figure 3.3: An example of the altimeter beam (solid line) and boat tracks (dottedline) with a stable boat ride (left) and less stable boat leg interrupted by rejectedpoints (segment between diamonds; x=56m, y=-26m).
The above criteria segmented the Rxy(S) spatial series, such that the rejected bot-
tom range points defined the start and end points of 'good' segments of data, hereafter
termed 'boat legs'. As a result, the bottom range spatial series was segmented into
several boat legs varying from a minimum of 12.5 to a maximum of 330 m long. All
of these boat legs were aligned with the incoming wave direction of about 2000 (long
period swell from SSE). The minimum length of 12.5 m of continuously 'good' data
was adopted with the spectral analysis in mind since our goal was analysis of the
characteristic spectral roughness energy at the order of magnitude of a typical wave-
length at depths shallower than 20 m (typically within the 50-300 m range). At the
same time, shorter segments maximized data use due to constraints on the acoustic
footprint and sampling rate, as will be discussed in Section 3.3.2.
16
3.3 Data analysis
3.3.1 Data gridding
The next step in the analysis consisted in spatially gridding, or spatially 'boxing', each
bottom range point based on its CPS reading. Data were organized into 75x75 m
boxes (Figure 2.1), a size determined by compromise between the order of magnitude
of a typical wavelength and the survey data density (Figure 3.3). The grid box was
rotated, relative to north, to be roughly aligned with the incoming wave direction.
3.3.2 Spectral analysis
Spectral analysis was used to investigate the different roughness scales. The fast
Fourier transform (FFT) was applied to the various segments of bottom range. The
input signal consisted of a 12.5 m running window at an average sampling rate of 10
Hz. The FFT was centered at every point of each boat leg, starting 6.25 m ahead
of the first point, continuing at every single point onwards and ending 6.25 m before
the last bottom range point of the segment in order to maintain the 12.5 m input
for the FFT. As an example, a 100 m boat leg entailing 1,000 points produces 875
spectra. It is noted that by using this approach not all spectra are independent. This
approach maximized data use and coverage and it was considered in determining the
confidence intervals for each box (see Appendix A).
The FFT dealt with two key variables: the sampling rate and the acoustic foot-
17
print. The altimeter sampling rate was about 10 Hz but not constant, such that
bottom range data were unevenly spaced. Thus, the sampling rate input for the FFT
corresponded to the largest spacing between points in each 12.5 m window and the
data input corresponded to the original data interpolated at this largest spacing dis
tance. Each 12.5 m window was multiplied by a Hanning window to minimize data
leakage. Data trend and mean were removed prior to the Hanning filtering.
The acoustic footprint (or equivalently the altimeter horizontal resolution) is dic
tated by the altimeter beam angle and the depth of measurement. For a constant
beam angle (set at 2.5°), the resolution of measurements is reduced in deeper regions.
For example, the altimeter can resolve 13 em at 3 m but only 87 em at 20 m, while
the altimeter sampling rate is predominantly lower than 20 em. For these surveys, the
footprint was predominantly larger than the altimeter sampling rate. Nonetheless,
the footprint was accounted for so that spectral estimates were limited to wavelengths
larger than the greater of the footprint or the altimeter sampling rate. Thus, we were
able to compare equal resolutions independently of the depth of measurement, with
the drawback that shallower regions tend to have more data compared to deeper re
gions. The term sampling rate term will be used onwards and will refer to the actual
sampling rate used in the EFT analysis, i.e. the maximum (in the space domain)
between the acoustic footprint and the altimeter sampling rate at each box.
Large and single occurring features, on the order of a meter in height and width, or
higher, are seldom but do exist in the study area. These features' size are considerably
lower than a typical wavelength and hence do not affect the wave transformation
18
processes since they are several orders of magnitude smaller that a typical wavelength
and at the same time their distribution is scarce. However, they can influence the
spectral analysis considerably. As a result, each box was repeatedly surveyed and the
obtained results were averaged together for a given box.
3.3.3 Data validation
Data validation was performed to assess the consistency of our methods. At the same
time, the validation provided a helpful insight in relating spectral output to a more
meaningful quantity. These validation techniques included the use of a 'roughness'
bar, visual observations and side-scan imagery.
Figure 3.4: Measurements using the 'roughness' bar. A schematic of the apparatus(left) and typical measurement procedure (right).
The 'roughness bar' (McCormick 1994) is a diver based measurement device con-
sisting in a 2.8 m long perforated aluminum bar, which measures elevation every 5 ern
using aluminum rods that slide vertically through the bar (Figure 3.4). The bar is
placed over a section of the bed, letting the 68 ern long rods slide down until they
19
hit the bed. As the rods fall down they take the natural shape of that bed section.
The bar is then photographed digitally for post analysis, where one bar reading is
measured using eight photos to limit the image distortion. Scaling is determined by
the black tape stripes and heights by the rod offset. Laboratory tests using artificial
roughness elements resulted in an accuracy better than 5%. However, the bar spatial
coverage is limited and at the same time data collection was also limited by dive
time. The 'roughness' bar was used in locations with contrasting surfaces in both
shallow and deep spots. Sandy areas were not surveyed by the 'roughness' bar since
its accuracy is reduced at these sites. Surveys were performed within eight boxes
resulting in 93 bar measurements. Although the spots were previously selected, the
actual surveyed sections were randomly chosen by dropping a marker from the surface
prior to the measurement.
Over the course of the project, numerous dives gave a qualitative idea of the exist
ing roughness length scales. These visual observations provided a helpful background
for qualitative analysis of a side-scan survey data. The side-scan data were produced
by a REMUS autonomous underwater vehicle, that surveyed part of the study site.
The result of this survey is a high resolution acoustic picture of the bed. Based on
the images from this survey a classification with scores ranging from 1 to 5 was used:
smooth bed textures or mostly sand (1), some textures or mostly sand with small reef
coverage (2), medium bed textures or equal coverage of reef and sand (3), significant
bed textures or mostly reef with small sand pockets (4) and strong bed texture or
mostly rough reef (5). The roughness distribution was accounted by the density of
20
the side-scan classifications, which were approximately one per every 20 x 20 m area.
Figure 3.5 illustrates the aforementioned scaling.
Figure 3.5: Visual scale for the qualitative analysis of the side-scan images. Thesepictures, left (smooth=l) to right (rough=5), illustrate the adopted classification.
21
CHAPTER 4
RESULTS
4.1 Spectral results
Spectral results were spatially 'boxed' and averaged so that for each box there is one
representative spectral curve. No dominant roughness scales were identified but an
average characteristic spectral slope was observed. The average slope was of -3.0 ±
0.7 in the logarithmic domain. The observed spectral shape is typically 'red' as lower
wavenumbers are associated with higher spectral energy and higher wavenumbers are
associated with lower spectral energy. From all the existing boxes, boxes 4, 28, 32
and 40 are used to exemplify the spectral curves. Their representative spectral curves
are shown in Figure 4.1. These boxes range from deep (box 32) to shallow (box 28) as
well as from smooth sandy beds (box 28) to rough coral reefs (box 4). Roughness was
quantified by integrating the areas bounded by wavenumber bands of equal width.
The adopted intervals were: 37-47 cm, 47-63 cm, 63-100 cm and 100-200 cm.
The result of the integration corresponds to a spectral energy value and its square
root corresponds to the roughness root mean square (rms) value. The rms values
obtained are shown in Figure 4.2. Boxes with insufficient data are marked by targets,
which were either too shallow to be surveyed, deeper than the altimeter range (20 m)
or the acoustic footprint was too coarse for the wavenumber band in question. Due to
this last issue, deeper boxes tend to have more data than the ones located shallower.
BOX 4 BOX 28
22
lcfL.----~--- ........10" 10"
BOXS2
1cfL-------~---'10" 10"
BOX 40
~.
:: _: ~...'. ~ .. ' .
10'
10'\lo"L.--------........
10'" 10"
Wa~ !lli'nber (;1ianj
Figure 4.1: Spectral curves for boxes 4, 28, 32 and 40.
Bed roughness was observed to a transition from smooth to rough, going shallower.
Figure 4.3 illustrates this conclusion where the normalized roughness rms values are
plotted versus depth. The horizontal dashed line represent the 12 m contour and the
vertical solid line represents the mean rms value. The black dots correspond to the
rms values for the wavenumber 63 to 200 cm band since it has values in both deep and
shallow areas. Black circles correspond to the side-scan values which are presented in
the next section. The side-scan values were used to validate this observation.
• Figure 4.2 also shows that spectral energy is able to differentiate bed roughness
scales. Between 5 and 10 m deep, inside boxes 28,33,34 and 35 (see also Figure 2.1),
there was a large sand patch. Although this sand patch did not cover these boxes
entirely it was large enough to mark its signature. The rms value for these boxes is
23
RMS 47 to 63cm (boat)
o40Q200o
900
800
600
500
400,
300
200-200
RMS 37 to 47cm (boat)900
800
~ 700 ..E';; 600·uc:.!! 500I/)
is 400
300
200 L...L::.----'-__~_ __'__
-200 '0 200 400
2
o400
RMS 100 to 200cm (boat)
200 '---"'---'-----'-.........,,--"'-----200
900,
1.5
1
0.5
o
RMS 6~to 100cm (boat)900
800
,.....0; 700 •E';600uc:,~ 500
is 400
300
200 '----'.........-'-------'-...........,~-"'----200 0 200 400
Distance (m)
Figure 4.2: Roughness rms values for four wave number bands. Lighter tones correspond to smoother beds while darker tones corresponding to rougher beds. Bathymetry contours are also plotted. Boxes with insufficient data are marked with targets.
low and comparable to the rms values observed in the deeper and more smooth boxes,
hence contrasting with the nearby shallow rough boxes. This result is particularly
evident in the wavenumber interval of 63 to 100 em, suggesting that sand and reef are
more distinguishable at these scales. This contrast tends to fade at scales higher than
one meter suggesting that some lower order geological conditions may playa role, for
example a gentle substrate variations on which the present sand and reef substarte
lies upon. Other boxes exhibit more energetic roughness (e.g. box 4 and 40). There
24
• • b
·~4~----C.3!:---.2!:---.7-1--+-0--":-1---0-2 _----!-_---lNormaHz~rmsend s1Qeo.s~_\l8lues
Figure 4.3: Normalized rms values as a function of depth. The horizontal line to the12 m tick. The vertical line represents the mean normalized rms.
is also a small trend for smoothing at the shallower areas. It could be suggested that
this area is too close to the wave breaking zone, where too much stress ex.ists for coral
to develop. It is also interesting to note the sudden increase of rms values above the
meter scales.
All the length scales examined, below and above 1 m, can potential be parame-
terized as roughness parameters since they are all comparable to the wave orbital
amplitude at the bed. This discussion relates to the link between physical and hy-
drodynamic roughness and, despite falling beyond the scope of this analysis, it is still
worthwhile to note and certainly is one of the main motivations for this work. Some
comments on this issue are presented in Section 4.3.
All said, these results suggest a good degree of consistency regarding wavenumber
spectra as a measuring scheme. This will be supported by the validation methods,
the 'roughness' bar and the side-scan analysis, discussed in the following section.
25
4.2 Validation of spectral results
4.2.1 Side-scansurvey
A visual roughness map was produced using the results provided by a side-scan survey
of a REMUS autonomous underwater vehicle. A side-scan survey consists of a sharp
acoustic picture with no direct information regarding bottom range. The analysis
of the side-san images consisted of assigning roughness values for areas of roughly
20 x 20 m based on apparent bed textures. Scores were interpolated onto the adopted
box-grid and a qualitative roughness map was produced (Figure 4.4). Although these
data are limited by what the human eye can distinguish when looking at an acoustic
picture, these appear to create a reliable picture of the site based on boat and diver
observations carried out by the approximate 2-year long research.
The roughness rms pattern is qualitatively similar to the one obtained by the boat
survey (Figure 4.2). This is particularly evident in the 63 to 100 em wavenumber
interval (74% correlation) suggesting that these are the scales better distinguished
by the human eye. The sand patch in the shallow region, the contrast between deep
and shallow areas and the rougher boxes on the right side, (e.g. box 40), are all
clearly distinguishable in the side-scan map. This suggests that wavenumber spectra
is consistent and effective in differentiating sea bed roughness. A scatter plot between
the side-scan and the rms maps is shown in Figure 4.5, showing the level of correlation
(~70%) between the the wavenumber band of 63 to 200 em in black.
26
Map of side-scan analysis900.----~--~--~__,
Scores from sidescan900
400o 200Distance (m)
200 ............'---~--'---" .........--'-----200
800
500
400 .
ado300 ~
200 ~ r-:--...-200 0 200 400
Di~ance(m)
Figure 4.4: Qualitative roughness map based on a side-scan survey with densityof observations show in the left and interpolated scores in the right. Boxes withinsufficient data are marked with targets.
4.2.2 'Roughness' bar survey
Several field surveys produced a collection of bar measurements, resulting in eight
boxes with 93 in-situ roughness measurements. From these eight boxes, three were
selected (boxes 4, 32 and 40; see Figure 4.2) to present the results for this in-situ
method (results for four others boxes are shown in Appendix C). These three boxes
range from medium to rough beds and from 9 to 13 m deep. It is noted that the
roughness bar method was not performed on sandy areas and for comparison purposes
the spectrum for box 28 is included, where the bed is predominantly covered with
sand and located in the shallow region of the study site.
Figure 4.6 shows the merge between the 'roughness' bar and the boat measure-
ments' spectra. The reasonably good agreement in terms of energy and slope indi-
cates the existence of a characteristic spectral slope for each site. An average slope
27
•
"~
1..E •«i .-t" 0lit •lJ ••,~
«i •6-1 • •z •• •• •
.2
•
• •• ••••......
•• •• ••
••• • ••
-~3~------:_2:-----C_1---00----:-1--72-----c1NonnaUled side-scan value
Figure 4.5: Relationship between the side-scan and the roughness rms maps is examined. Black dots are normalized rms values for the 63 to 200 em interval.
of -3.0±0.7 was obtained for the 50 to 120 em interval. At the same time, it further
suggests that wavenumber spectra is consistent in its measurements. The spectral
merge is observed to occur between 63 and 100 em wavenumber band. Since not all
boxes were surveyed using the bar technique, an extrapolation criteria into higher
wavenumbers was investigated based on the spectral agreement between boat and
bar measurements. However, the observed significant variability, based on data from
these eight boxes, yielded no relationship between higher and lower wavenumber
bands. Figure 4.7 shows the same rms maps as shown in Figure 4.2 with the same
equal width. The 47 to 200 em wavenumber intervals allows for a direct comparison
to the boat data. However, the 100 to 200 em band has broader confidence intervals
due to fewer points in the wavenumber domain. Results match satisfactorily for the
limited coverage of the 'roughness' bar, with a 63 to 84% correlation. This result,
BOX 4
BOX 32
BOX28
BOX40
28
toO '-- --.-.,..........."'-'"-- ........,.......................,...'10'3 10'~
Wave·number(11cm)
100 '-:--""--"......,-'---..........'""-"-:.,---------......................,...
10.3 10'2 10.1
Wave number{1/cm)
Figure 4.6: Wavenumber spectra from the 'roughness' bar (dashed line) and the boatmeasurements (solid line) with 95% confidence intervals.
along with the spectral agreement shown in Figure 4.6, suggests that wavenumber
spectra provides a realistic image of the bed roughness, varied in length scales entail-
ing both coral reef and sandy areas. However results also indicate that in order to
have an accurate estimate of the bed roughness a comprehensive survey is required
as bed roughness can vary greatly and randomly within short distances, a fact that
greatly limits the applicability of the 'roughness' bar compared to the boat mounted
altimetry system.
900
900
RMS 3710 47cm (bar)
RMS 6310 100cm (bar)
o 200 400Distance (m)
2
1.5
0.5
3.5
3
2.5
2
1.5
0.5
o
900
RMS 47 to 63cm (bar)
RMS 100 10 200cm(bal1
o 200 400Distance (m)
2
1.5
0.5
o
10
8
.8
4
2
o
29
Figure 4.7: Spectral energy of the study area for the same wave number bands(37-47 cm, 47-63 cm, 63-100 cm and 100-200 cm) based on the 'roughness' bar data.-Bathymetry contours are also plotted. Boxes with insufficient data are marked withtargets.
4.3 Root mean square validation
The root mean square values obtained via spectral analysis were also validated with
rms values obtained via analysis in the spatial domain. In order to perform a com-
parison within the different wavenumber bands, low-pass filtering was applied to the
spatial series of bottom range to remove wavenumbers outside the interval of com-
parison. The low-pass filter consisted of a Butterworth 10th order filter. Filtering
was then set at 37, 47, 63, 100 and 200 cm and rms values were computed with the
purpose of comparing them to the results in Figure 4.2. Figure 4.8 shows the results
using the same wavenumber bands used in Section 4.1.
30
2
1.5
RMS 47 to 63cm (spatial)900
0.5
0
RMS 100 to 200cm (spalla!)900 '10
8
6
4
2
00 200Distance (m)
2
0.5
1.5
9·5
o
3.5
3
2.5
2
1.5
o 200 400Distance (m)
RMS 37 to 47cm (spatial)
RMS 63 to 100cm (spatial)
900
900
Figure 4.8: Spatial rms values of the study area for four wave number bands(37-47 em, 47-63 em, 63-100 em and 100-200 em). Bathymetry contours are alsoplotted. Boxes with insufficient data are marked with targets.
Again, the matching results support the consistency of the wavenumber spectral
analysis. A good correlation is observed both in terms of value as well as distribution
pattern through out the survey site. The correlation between rms from the spatial
domain and rms from the wavenumber domain is particularly high at the 47 to 100 em
band, with a 62% correlation, and at the 100-200 em band with a 84% correlation.
A reduced correlation of 6% was observed on the 37-47 em wavenumber band due to
the fact that the number of boxes with data is less and at the same time there is a
lower boat coverage on some of the right side boxes (see Figure 3.3), which greatly
offset the correlation value. These relationships are illustrated in Figure 4.9.
31
4.-----,----,------r---"..-----,--r----,------...."
3
o
oo
o
o
o
o
o
o
-2 ~1 0 1 2NormalizedRMS(spatial)
o
-3
2
3
-2
Figure 4.9: Relationship between rms values computed via spatial and wavenumberdomain: circles = 37-47 em, triangles = 47-63 em, squares = 63-100 em, diamonds= 100-200 cm.
It is also observed that a handful of data points tend to be underestimated via
spatial domain (relative to the spectral analysis). These points, which fall above
the one-to-one line, correspond to boxes 4, 5 and 6 where data density from the boat
surveys is minimal (see Figure 3.3). Therefore, low confidence is given to these points.
Overall, however, a good correlation is apparent.
4.4 Discussion
One relevant and innovative result presented here is the development of bed rough-
ness maps for a sand/coral reef site in terms of rms bed variability. These maps
32
provide a numerical approach towards quantifying bed roughness scales and although
a clear single representative wavenumber was not observed, wavenumber spectra is a
potential candidate for a hydrodynamic parametrization scheme. However, it is still
valid to ask which rms scale relates to a wave friction factor/roughness parameter,
or in other words, how One obtains a wave friction factor/roughness parameter from
rms values. This question, which presented'the primary motivation for this research,
remains unanswered. All roughness scales examined in this study (37 to 200 em) are
of the same order of the magnitude of the orbital wave amplitude, thus any values
in this range can contribute towards a potential roughness parameter depending on
the wave height, period and depth. Orbital wave amplitudes can vary from about
0.10 to 5 m within the site and for a broad range of wave conditions. This variability
in orbital wave amplitudes in addition to the variability in the roughness rms within
the study site, illustrates that there is not a straightforward solution. The results
presented here attempt to improve our understanding on this issue and much more
can be investigated and improved.
The most obvious continuation of this research is establishing the link between the
rms values, or spectral energy, and the wave friction factor. Computer simulations
using the COULWAVE model were performed to examine how wave heights varied
with different wave friction values. These simulations were compared to wave data
collected in the study site using 10 wave pressure gauges. Results were inconsistent,
however, across all of the simulations. Using a 20 minute wave directional spectrum,
different model runs were performed using a range of bed friction values from 4x 10-4
33
to 0.7. Wave direction and particularly, wave heights, were on average underestimated
by the model which greatly hindered the analysis. No conclusion was obtained from
this part but a far more detailed modelling can be done. Other approaches, may in
clude laboratory tests of oscillating flows over a varied distribution of roughness scales
to investigate which wavenumber bands are prone to contribute more predominantly
to the energy dissipation process. This practical approach, along with computer sim
ulations, could result in a more valuable strategy. The major goal is still to derive a
parametrization scheme for the wave friction factor, based on field measurements of
bed roughness, and the results and methods presented discussed here constitute the
first steps towards such milestone.
Limitations regarding the methods used here and possible improvements are also
identified. By far, the more restricting factor to this study was the altimeter acoustic
footprint. The altimeter 2.50 nominal beam width associated with a user controlled
ping rate, allowed to resolve roughness up to 15 em in the shallower regions but the
loss of resolution in the deeper regions could not be avoided. Moreover, the deeper the
site the smaller the orbital wave amplitude, translating in a two-fold loss of accuracy.
Furthermore, the smaller roughness scales observed are very close to the altimeter
threshold which may add some unaccountable noise to the results. In the framework
of this research, an autonomous underwater vehicle equipped with a narrow beam
altimeter presents the most attractive tool to date. Such vehicle can navigate above
the bed at a constant height that can be as low as 3 m. For instance, this operational
altitude, for a 2.50 beam, results in a 13 em footprint at all depths and with a far
34
greater coverage in terms of area and depth range.
The spectral analysis methods also affected the results to some extent. The vari
able bottom range sampling rate, data interpolation and filtering as well as smoothing
and averaging techniques used all influenced the data to some extent. Nonetheless,
the methods used are commonly applied in similar studies with proven reliability.
In the particular field of spectral analysis, a multitude of techniques exist and this
study did not perform a thorough task in comparing which ones performed better
than others. Further fine tuning of these particular issues can potentially improve
the quality of our results. Limitations regarding the applicability of the Fourier trans
form to analyze a coral reef bed is also debatable since it loses all phase information
of the original signal. For homogeneous beds like sand ripples, phase information is
less important provided ripples occur periodically. Periodicity is not observed in in
homogeneous beds such as coral reef or rocky bottoms, however. In these cases, some
fundamental information may be lost, since random signals can yield equal spectra
(Armi and Flament 1985) consequentially overlooking a more accurate parameter
ization of the bed roughness that may depend on roughness element distributions
(Pawlak and MacCready 2002). Nonetheless, this an issue that requires further in
vestigation and the conclusions obtained by this research do not reject, in any way,
wavenumber spectra as a potential candidate for a parameterization scheme of sea
bed roughness.
35
4.5 Conclusions summary
Wavenumber spectra proved to be an effective method for distinguishing different bed
roughnesses. The roughness at the study site exhibits an apparent transition from
smooth to rough going shallower. In addition, a smoothing near the wave breaking
zone, in the shallower region of the survey site, was also observed. No dominant
roughness wavenumber was observed in the study area. The bed roughness of the
study site exhibits a characteristic spectral slope of -3.0 ± 0.7. The study area exhibits
a high spatial variability and no relationships were observed between different orders
of magnitude of wavenumbers. These methods were validated qualitatively with side
scan imagery and quantitatively by in-situ roughness surveys.
1
APPENDIX A
CONFIDENCE INTERVALS
The ratio of the estimated wavenumber spectrum, Syy(k), and the expected values of
the wavenumber spectrum, Syy (k ):
(A.I)
is distributed as a chi.square variable with v degress of freedom. It then follows
that the true spectrum is expected to fall in the interval with 95% confidence:
v· Syy(k) S (k) v . Syy.(k)2 < yy < ---;2:O-=~
XO.025,v XO.975,v(A.2)
2
APPENDIX B
COULWAVE SIMULATIONS
In July 2004, 10 wave gauges were deployed within the study area as shown in Figure
B.lo Seven of these instruments recorded wave directional data. Using a wave model,
we were able to simulate the wave transformation over the are and create an output
of wave heights at each gauge site. This output was compared to the actual recorded
wave heights with the goal of isolating the contribution from the bed friction.
Figure B.l: Location of wave gauges (white dots) with site numbering.
Data was collected for a 2-week period. On Table B it is shown the observed data
for each site concerning a 20 minute interval (16/July/2004, 16HOO to 16H20). Sites
6,7 and 9 correspond to instrument with no directional output and were not included.
The Carnell University Long WAVE (COULWAVE) model was developed by
Patrick Lynett and Philip Liu ?? This model is able to simulate fully non-linear and
two-dimensional dispersive irregular waves over variable bathymetry. The model also
3
Site IDepth Hsig Pp()
1 17.6 1.29 18.5 32 9.0 1.62 18.2 3493 8.2 2.06 18.2 44 10.2 1.73 18.2 65 11.0 1.30 18.5 3496 8.17 9.58 5.4 1.61 18.5 19 7.310 3.3 1.38 18.5 18
Table B.l: Depth (m), significant wave height (m), peak period (s) and wave direction(0) for each site.
accounts for bottom friction. Model runs for a range of wave friction values from 4x
10-4 to 0.7 are shown in the following tables for each site excluding the sites with
non-directional wave data (sites 6, 7 and 9).
Results show that refraction and shoaling effects are not accurately simulated by
the model. Site 10 and 5 shown an underestimate in wave direction of about 10°
suggesting that refraction tends to be smaller in the model. At the same time, wave
heights are observed to be smaller than measured values suggesting an underestimate
of the shoaling transformation even for the runs with very low values of bed friction.
Despite the fact that this accuracy can be good or bad, depending on the purpose of
the analysis, the variability observed did not permitted more conclusive observations
within the scope of the bed friction attenuation.
4
fw ISite Hsig Pp () I Site Hsig Pp () ISite Hsig Pp ()
0.0004 1 1.22 18.9 0 2 1.41 18.9 2 3 1.29 18.9 20.0007 1 1.26 18.9 0 2 1.47 18.9 0 3 1.35 18.9 3580.001 1 1.29 18.9 0 2 1.49 18.9 0 3 1.38 18.9 3580.004 1 1.23 18.9 358 2 1.41 18.9 2 3 1.37 18.9 20.007 1 1.21 18.9 358 2 1.36 18.9 358 3 1.33 18.9 00.01 1 1.19 18.9 0 2 1.36 18.9 356 3 1.28 18.9 00.04 1 1.11 18.9 0 2 1.22 18.9 358 3 1.19 18.9 3580.07 1 1.08 18.9 0 2 1.14 18.9 358 3 1.12 18.9 3580.1 1 1.04 18.9 0 2 1.08 18.9 0 3 1.07 18.9 3580.4 1 0.79 18.9 0 2 0.76 18.9 0 3 0.77 18.9 3580.7 1 0.65 18.9 0 2 0.60 18.9 0 3 0.62 18.9 2
Table B.2: Depth (m) and modelled significant wave height (m), peak period (s) andwave direction (0) for sites 1,2 and 3.
fw I· Site Hsig Pp () ISite Hsig Pp ()
0.0004 4 1.06 18.9 2 5 1.26 18.9 20.0007 4 1.07 18.9 2 5 1.26 18.9 20.001 4 1.11 18.9 2 5 1.26 18.9 20.004 4 1.18 18.9 358 5 1.16 18.9 20.007 4 1.16 18.9 358 5 1.16 18.9 40.01 4 1.16 18.9 358 5 1.20 18.9 40.04 4 1.01 18.9 0 5 1.10 18.9 20.07 4 0.97 18.9 2 5 1.05 18.9 20.1 4 0.93 18.9 2 5 1.00 18.9 20.4 4 0.68 18.9 2 5 0.72 18.9 20.7 4 0.54 18.9 0 5 0.58 18.9 2
Table 8.3: Depth (m) and modelled significant wave height (m), peak period (s) andwave direction (0) for sites 4 and 5.
5
fw ISite Hsig Pp () ISite Hsig Pp ()
0.0004 8 1.60 18.9 4 10 1.72 18.9 20.0007 8 1.60 18.9 358 10 1.66 18.9 80.001 8 1.59 18.9 0 10 1.69 18.9 80.004 8 1.51 18.9 358 10 lAO 18.9 60.007 8 1.52 18.9 358 10 1.39 18.9 60.01 8 1048 18.9 358 10 1.35 18.9 80.04 8 1.33 18.9 0 10 1.31 18.9 80.07 8 1.21 18.9 0 10 1.20 18.9 80.1 8 1.12 18.9 0 10 1.12 18.9 8004 8 0.67 18.9 0 10 0.72 18.9 80.7 8 0.50 18.9 0 10 0.54 18.9 8
Table BA: Depth (m) and modelled significant wave height (m), peak period (s) andwave direction (0) for sites 8 and 10.
6
APPENDIX C
BOAT-BAR SPECTRAL MERGES
Four additional merges, between the spectral curves obtained by the boat measure-
ments and the 'roughness' bar measurements, are shown.
BOX31 BOX 39.
10' 10'..~-8.~ 10' 10'c:""~ 10' 10'"c:W
10Q 10'10" 10~ 10'\ 10" 10" 10"
BOX 41 BOX 43
10' 10'
%,.e~ 10' 10'Ii"~ 10' 10'"c:w
10' 10'10" 10" 10" '10" 10" 10"
VVaVllIll,lTlb~r(1/gn) vvaVllnuniber (1leni)
Figure C.l: Wavenumber spectra from the 'roughness' bar (dashed line) and the boatmeasurements (solid line) with 95% confidence intervals.
"
7
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