Wave measurements from ship mounted sensors in the Arctic marginal ice zone

Trygve. K. Løkena,111E-mail address and phone number for corresponding:
<EMAIL_ADDRESS>(+47) 94885767 (T.K. Løken)
E-mail addresses<EMAIL_ADDRESS>(J. Rabault<EMAIL_ADDRESS>(A.
Jensen),
<EMAIL_ADDRESS>(G. Sutherland<EMAIL_ADDRESS>(K.H.
Christensen<EMAIL_ADDRESS>(M. Müller)., Jean Rabaultb,a, Atle Jensena, Graig
Sutherlandc, Kai H. Christensenb and Malte Müllerb

a Department of Mathematics, University of Oslo, Oslo, Norway
b Norwegian Meteorological Institute, Oslo, Norway
c Environment and Climate Change Canada, Québec, Canada

## Abstract

This study presents wave measurements in the marginal ice zone (MIZ) obtained
from ship mounted sensors. The system combines altimeter readings from the
ship bow with ship motion correction data to provide estimated single point
ocean surface elevation. Significant wave height and mean wave period, as well
as one-dimensional wave spectra are derived from the combined measurements.
The results are compared with integrated parameters from two spectral wave
models over a period of eight days in the open ocean, and with spectra and
integrated parameters derived from motion detecting instruments placed on ice
floes inside the MIZ. Mean absolute errors of the integrated parameters are in
the range $13.4$-$29.9$% when comparing with the spectral wave models and
$1.0$-$9.6$% when comparing with valid motion detecting instruments. The
spatial wave damping coefficient is estimated by looking at the change in
spectral wave amplitude found at discrete frequency values as the ship was
moving along the longitudinal direction of the MIZ within time intervals where
the wave field is found to be approximately constant in time. As expected from
theory, high frequency waves are effectively dampened by the presence of sea
ice. The observed wave attenuation rates compare favourably with a two-layer
dissipation model. Our methodology can be regarded as a simple and reliable
way to collect more waves-in-ice data as it can be easily added to any ship
participating to ice expeditions, at little extra cost.

Keywords: Sea ice dynamics, wave measurements, marginal ice zone, wave
attenuation

##  1 Introduction

Sea ice is a major feature in the polar environments, it has importance for
Arctic ecosystems as well as for global ocean and atmospheric circulation. A
decline in the Arctic ice cover has been observed over the past decades
(Feltham, 2015). Interactions between sea ice and surface gravity waves play
an important role in breakup and reduction of ice cover. The decline in ice
cover leads to a larger fetch where waves build up more energy and enhance the
breakup and melting process in a positive feedback mechanism (Thomson and
Rogers, 2014). The mixture of icebergs, floes and grease ice found in the
interface between solid ice, such as land fast ice or pack ice, and the open
ocean, is called the marginal ice zone (MIZ). Previous studies have found that
high frequency wind waves are effectively dampened in the MIZ (Weber, 1987;
Wadhams et al., 1988), which reduces the ice cover break up rate.

Recent changing conditions in the polar regions have allowed for increased
human activities, which raises the importance of better forecast models and
improved physical understanding of the environment to ensure safe operations
(Fritzner et al., 2019). In situ wave measurements can increase our
understanding of global climate systems and provide data for calibration of
numerical models. Also, mathematical models to describe wave attenuation in
ice, e.g. Weber (1987), Newyear and Martin (1997) and Sutherland et al.
(2019), need experimental data for validation and improvement. However,
experimental data are relatively sparse due to the inaccessibility of the
regions where sea ice is present, combined with the harsh and dangerous
environment for both researchers and instruments (Squire, 2007).

We present here results from shipborne wave measurements in the MIZ. This
system has not up until now been used deep into the MIZ, therefore, the
novelty of our observations. The methodology, first described in Christensen
et al. (2013), combines a bow mounted altimeter and a motion correction
device. We provide wave spectra and integrated parameters from spectra, which
are important quantities when considering wave-ice interactions. Significant
wave height and mean periods are compared with a spectral wave forecast model
over a period of eight days in the open ocean. We also compare measurements in
the MIZ with data from wave measuring instruments consisting of inertial
motion units (IMUs) placed on ice floes (Rabault et al., 2020). From the
spectra, the spatial damping coefficient is found as a function of wave
frequency, which can be compared to attenuation models.

In this paper, the data acquisition and processing methods are described in
Section 2. The results are presented in Section 3, which is divided in two
parts. A comparison with spectral models and in situ instruments is outlined
in Section 3.1. In Section 3.2, we present results on wave attenuation and the
spatial damping coefficient is compared to the theoretical model of
(Sutherland et al., 2019). Finally, a discussion follows in Section 4 and the
concluding remarks are given in Section 5.

##  2 Data and Methods

The data were obtained during a research campaign in the Barents Sea with R/V
Kronprins Haakon in September 2018 as part of the Nansen Legacy project
(Reigstad et al., 2017). The vessel is $100$ m long and with a beam of $21$ m.
In total, the cruise lasted two weeks, during which continuous measurements
were made. The results from the MIZ presented here were recorded on September
19 when the ship ventured approximately $28$ km into the MIZ. Four stops were
made to deploy in situ waves-in-ice (WII) instruments on ice floes (Rabault et
al., 2020), which were used to validate our ship mounted system. Upon the
return to the open ocean, a total of seven stops were made with more or less
equal spacing on a close to straight south-southwest heading where
measurements for wave damping estimates were carried out. Twenty minutes
samples were recorded as the ship was freely drifting on each station. The
location, starting time and wave travel distance (WTD) for each measurement in
the MIZ are summarized in Table 1. Prefix $1$ denotes the four stops into the
MIZ while prefix $2$ denotes the seven stops out of the MIZ.

Stop | Time | Position (N/E) | WTD [km]
---|---|---|---
1.1 | 04:22 | 82.126/20.736 |
1.2 | 06:28 | 82.246/20.245 |
1.3 | 08:59 | 82.355/19.803 |
1.4 | 12:20 | 82.436/19.674 |
2.1 | 13:11 | 82.421/19.579 | 92.7
2.2 | 14:32 | 82.359/19.544 | 69.2
2.3 | 15:40 | 82.294/19.389 | 58.3
2.4 | 16:54 | 82.228/19.275 | 20.8
2.5 | 18:00 | 82.163/19.183 | 6.7
2.6 | 19:08 | 82.099/19.046 | 0
2.7 | 20:09 | 81.994/18.982 | 0
Table 1: Time and location where measurements were carried out inside the MIZ.
Prefix $1$ for stop number indicate deployment of WII instruments while prefix
$2$ indicate measurement of damping coefficient. WTD through the MIZ are
listed for the stops with prefix $2$.

(a) Satellite image 14:55 UTC 19-09-2018.

(b) Wave attenuation measurements.

Figure 1: MIZ and location of measurements. a) Satellite image with ship
trajectory into (red) and out of (green) the MIZ. Cyan dots mark the location
where wave parameters from the spectral model WAM-4 are extracted to check for
constant sea state. White and orange squares indicate the horizontal
resolution of the spectral wave models ERA5 and WAM-4 respectively. Source:
MODIS Corrected Reflectance (250 m resolution, sharpened) b) Wave direction
from WAM-4 and WTD through the ice for each measurement (used for wave damping
coefficient). The ice edge found from the satellite image (blue) and the one
used in WAM-4 (pink) and ERA5 (orange) are indicated.

Figure 1(a) is a MODIS Corrected Reflectance satellite image, which shows the
ship trajectory into (red) and out of (green) the MIZ and the ice edge at
approximately $82.2$ °N below the thin cloud cover. The ice concentration for
the four stops $1.1$-$1.4$ was estimated to be roughly $10$%, $30$%, $90$% and
$100$% respectively in Rabault et al. (2020). From Fig. 1(a) the ice edge is
roughly located by visual inspection along the relevant longitudes and
recreated in Fig. 1(b) (blue) along with the location of the wave attenuation
measurements, i.e. stops $2.1$-$2.7$ (black). This figure also indicates the
ice edge used in the spectral models WAM-4 (pink) and ERA5 (orange), which are
further described in Section 2.5, and the mean wave direction on September 19
(gray) estimated by WAM-4. We use the ”going-to” convention for wave direction
and ship heading and the ”coming-from” convention for wind direction and
define directions as clockwise rotation from the geographic north. The gray
lines illustrate the WTD through the ice for each stop.

###  2.1 Data acquisition

The instrument setup consisted of an ultrasonic gauge (UG) that measured ocean
surface elevation relative to the ship bow. Figure 2(a) and 2(b) shows the
downward facing UG mounted on a rigid pole. Estimated absolute surface
elevation was obtained after correcting for ship motion by means of an IMU
placed on deck, also in the bow section of the ship. The horizontal and
vertical distances between the UG and the IMU were $x_{lev}=2.5$ m and
$y_{lev}=6$ m respectively.

(a) UG seen from the front.

(b) UG seen from above.

Figure 2: Installation of the downward facing UG on a rigid pole in the ship
bow.

We used a UG (Banner QT50ULB) with approximately $0.2$-$8$ m range. The
instrument emits $75$ kHz ultrasonic pulses at a $10.4$ Hz sampling rate.
Distance $L$ between the UG and the ocean surface is calculated internally
from the time delay of the echo, where the speed of sound in air is
temperature compensated with an integrated thermometer. Mean $L$ was
approximately $5$ m and the effective beam width, i.e. the diameter of the
circular area where the signal was reflected on the ocean surface, was
approximately $0.9$ m. This footprint is very low compared to the wavelength
$\lambda=156$ m of a typical $0.1$ Hz ocean wave, as the one shown in Fig. 4.
The wavelength is found from $\lambda=2\pi/k$, and the wavenumber $k$ from the
linear deep water dispersion relation:

$\omega^{2}=gk,$ (1)

where $\omega=2\pi f$ is the angular frequency, $f$ the wave frequency and $g$
the acceleration due to gravity. The depth dependency is neglected here since
the water depth was $\approx 3500$ m in the region. The use of (1) in the MIZ
can be justified by the fact that the dispersion relation in ice deviates
little from the deep water dispersion relation in the frequency band where
wave motion is present (Marchenko et al., 2017).

A feature of the UG called Auto-Window was enabled, meaning that a $1$ m
sensing window was centered around a taught length. The taught length
$\bar{L}$ was found automatically by time averaging $L$ over a couple of wave
periods before each measurement was initiated. We later display the output
signal of the instrument as $D=L-\bar{L}$ in Figures 4, 5(a) and 5(b). The
Auto-Window feature was chosen out of practical reasons and to increase
instrument accuracy at the expense of a reduced range ($|D|<0.5$ m), because
the intention was initially to only measure in the Barents Sea MIZ where low
wave heights are normally expected. However, it was decided to measure
continuously during the whole cruise, also in the open ocean where the waves
were generally larger than in the MIZ. Consequently, the instrument range was
exceeded (saturated) at times with large wave amplitudes. Note that saturation
only occured when $|D|>0.5$ m, not necessarely when the trough-to-crest
amplitude exceeded $1$ m, depending on ship responce. For future expeditions
we would recommend to pre-calibrate the UG to allow for a larger sensing
window if rougher waves are expected, for example in the Suthern Ocean MIZ, to
reduce the probability of saturation. With the right instrument calibration,
the measurement range could rather be limited by the actual instrument range
and the mean vertical distance between the instrument and the surface
($0.2$-$8$ and $5$ m respectively in our case). Christensen et al. (2013)
reports an equivalent setup of the system to be useful for significant wave
heights up to $4$ m.

We define a low saturation when $D<-0.5$ m and a high saturation when $D>+0.5$
m. In these cases, the output signal of $D$ is simply $\pm 0.5$ m, although
the distance is actually smaller or greater, and this can be seen as a ”cut”
graph in Fig. 5(b). We define the saturation proportion as the ratio of time
where saturation occurs over total sample time. Figure 3 shows the saturation
proportion of each sample for stops $2.1$-$2.7$, where ”total” saturation
proportion is the sum of ”low” and ”high” saturation proportion. We accept up
to $10$% total saturation and will therefore discard the sample recorded at
stop $2.7$, which had a total saturation of approximately $20$%, in our wave
attenuation analysis in Section 3.2.

Figure 3:  Saturation proportion (i.e. proportion of time when the UG range
was exceeded) for stops $2.1$-$2.7$, where ”total” saturation proportion is
the sum of ”low” and ”high” saturation proportion. Area of discarded data due
to total saturation exceeding $10$% threshold is shaded.

The system can measure trough-to-crest amplitudes up to $1$ m before the range
of the UG is exceeded. This constraint sets an upper limit for how large
significant wave height the instrument theoretically can record in a time
series without exceeding the measurement range. Significant wave height
($SWH$) from time series is defined as:

$SWH=4\sigma,$ (2)

where $\sigma$ is the standard deviation of the surface elevation $\eta$. In
the simplified case of a pure sine wave, the trough-to-crest amplitude is
equal to $2\sqrt{2}\sigma$. In this idealized example where ship motion is not
considered, the limiting upper value of $SWH$ is approximately $1.4$ m for the
UG.

As motion correction device, we used an IMU (VectorNav VN100) similar to the
ones in Rabault et al. (2020). It features 3-axis accelerometers and 3-axis
gyroscopes measuring at a rate of $800$ Hz. After an internal Kalman
filtering, the instrument gives an output frequency of $80$ Hz. The gyros
yield rotation angles about all three axis directly. Vertical acceleration is
integrated twice to obtain ship vertical displacement about the mean, where
positive values are displacement above the mean. Details on the integral
scheme, data filtering and other technical information on the instrument can
be found in Rabault et al. (2020).

In order to obtain time series of the surface elevation, UG and IMU data at
the same time instance were needed. We solved this by defining a common
sampling rate of $10$ Hz, which should be sufficient for resolving all
relevant ocean surface features. All data were then interpolated on the common
time base for the analysis.

###  2.2 Ship response

In general, if the wavelength $\lambda$ is large when compared with the
diameter $d$ of a floating body, the body will tend to follow the path of a
fluid particle at the free surface. In the opposite case when $\lambda/d$ is
small, the waves are losing their influence on the behavior of the body. The
response amplitude operator (RAO) of a vessel (or any object) is the ratio of
response amplitude over incoming wave amplitude. RAOs in all modes for R/V
Kronprins Haakon are reported in Ytterland (2016). RAOs depend on the wave
heading angle $\beta$, which is defined as the relative angle between the ship
heading and wave direction. Hence, $\beta=0\degree$ corresponds to following
seas, $\beta=90\degree$ corresponds to beam seas, and $\beta=180\degree$
corresponds to head seas.

###  2.3 Motion correction

Downwards facing altimeters mounted on fixed structures are common for
measuring ocean surface elevation $\eta$, which is a function of time (Reistad
et al., 2011; Magnusson and Donelan, 2013). Correction is required when the
altimeter is mounted on a floating structure like a ship. The parameter $\eta$
is a function of the position in the horizontal plane in addition to time when
the ship is free to move, but we assume the horizontal drift velocity to be
much smaller than the phase speed of the waves, so that the dependency of
position can be neglected. We define a coordinate system with the $(x,y,z)$
axis to be aligned horizontally in the direction from stern to bow, vertically
in upward direction and horizontally in the direction from port to starboard,
respectively. The origin coincides with the IMU. Along axis translation are
surge, heave and sway while rotation about the axis are roll, yaw and pitch,
respectively. Only heave $\xi$, roll angle $\phi_{R}$ and pitch angle
$\phi_{P}$ affect the distance $L$ measured by the UG and need to be addressed
when compensating.

In the (imaginary) case where the ship lies completely still, as if it was
fixed in position, surface elevation will be given by $\eta=\bar{L}-L$. When
heave response is present, we simply subtract the vertical displacement $\xi$,
which is positive when the ship is above its mean level and negative when the
ship is below. When rotation about the horizontal axis is included, two
aspects need to be considered. First, the UG measures the distance to the
surface with an angle about the vertical axis. The true vertical distance is
obtained by multiplying $L$ with the cosine of both $\phi_{R}$ and $\phi_{P}$
as shown in the second term on the R.H.S. of (3). Second, there is the lever
effect due to the horizontal and vertical distance between the $UG$ and the
$IMU$, $x_{lev}$ and $y_{lev}$ respectively. The UG is displaced in vertical
direction relative to the ocean surface when the UG rotates around the IMU
about the $x$-axis and the $z$-axis, which corresponds to ship roll and pitch
respectively. We name these displacements roll and pitch elevation effects,
and define them respectively as $y_{lev}(1-cos(\phi_{R}))$, which is always
positive, and $x_{lev}sin(\phi_{P})$, which is positive when $\phi_{P}$ is
positive and negative otherwise.

All corrections are combined and we obtain an expression for the ocean surface
elevation:

$\eta=\bar{L}-Lcos(\phi_{R})cos(\phi_{P})-\xi-
y_{lev}(1-cos(\phi_{R}))-x_{lev}sin(\phi_{P}),$ (3)

which is similar to Eq. 1 in Christensen et al. (2013) except from the roll
and pitch elevation effects in the fourth and fifth term on the R.H.S. of (3),
which are new considerations.

A typical example of a $150$ s time series inside the MIZ is shown in Fig. 4.
Trough-to-crest amplitudes up to approximately $0.4$ m and frequencies of
around $0.1$ Hz are found in this specific sample, as seen in the upper panel.
Ship heave response $\xi$ and UG reading $D=L-\bar{L}$ are also presented in
the upper panel and are in the same order of magnitude as the surface
elevation. The lower panel shows roll and pitch elevation effects (fourth and
fifth term on R.H.S. of (3) respectively) in addition to the UG angle
correction factor corresponding to the second term on R.H.S. of (3), here
presented as $1-cos(\phi_{R})cos(\phi_{P})$. These results indicate that the
roll elevation effect and the UG angle correction factor are
$\mathcal{O}(10^{-3})$ relative to the wave amplitude and thus negligible in
this case. The pitch elevation effect is on the other hand approximately
$0.02$ m and $\mathcal{O}(10^{-1})$ relative to the wave amplitude and is
therefore important to include in the processing.

Figure 4:  $150$ s time series of UG and IMU measurements from stop $2.4$. In
upper panel: Ship heave response $\xi$, UG-reading $D$ and surface elevation
$\eta$. In lower panel: UG angle correction factor (corresponds to the second
term on R.H.S. of (3), displayed with black dotted line), roll and pitch
elevation effect (fourth and fifth term on R.H.S. of (3) respectively).

A section of the time series from stop number $1.1$ is presented in Fig. 5(a),
where the mean value of the surface elevation $\eta$ jumps from zero up to
about $0.1$ m after roughly $35$ s and then jumps back to zero at roughly
$100$ s. This step is most likely caused by an ice floe drifting under the
sensing window of the UG for a short period, which for example also occurs at
stop number $2.4$ as illustrated in Fig. 2(b), and demonstrates the UG’s
ability to receive reflected signals off an ice cover. The crest-to-trough
amplitude is approximately $0.4$ m for the waves and $0.1$ m for the step
presented in Fig. 5(a). If the step is interpreted as a long wave, its
frequency is less than $0.008$ Hz, which is much lower than the lower cut-off
frequency $f_{min}$ explained in Section 2.4, and should therefore not affect
the spectral analysis presented in this study.

An example of a saturated sample can be seen in Fig. 5(b). This time series is
extracted from stop number $2.7$. The UG reading $D$, is clearly flattened out
at the peaks where the range of the instrument is exceeded. Total saturation
for this sample was approximately $20$%.

(a) Surface elevation over ice floe.

(b) Saturated UG reading.

Figure 5: Time series of UG-reading $D$ and surface elevation $\eta$. a) Stop
$1.1$ where an ice floe most likely drifts under the sensing window of the
instrument. b) Stop $2.7$ with approximately $20\%$ saturated UG signal.

###  2.4 Statistical parameters and spectrum

We obtain surface elevation from (3). Power spectrum density $PSD(f)$ of the
surface elevation is obtained with the Welch method (Earle, 1996) where the
samples are subdivided in $q$ consecutive segments and ensemble averaged.
Segment size is set to $200s$ with $50$ % overlap. With $20$ min sampling time
at $10$ Hz common sampling rate, this gives a segment size of $2000$ sampling
points and a total number of $12000$ sampling points for each measurement. A
Hanning window is applied to each segment to reduce spectral leakage.

Since we are later comparing our measurements with data from a spectral model,
we find the statistical parameters from spectra. The mean wave period and the
significant wave height are obtained from the spectral moments:

$m_{j}=\int_{f_{min}}^{f_{max}}f^{j}PSD(f)df,$ (4)

where the cutoff frequencies $f_{min}$ and $f_{max}$ are set to $0.04$ Hz and
$1.0$ Hz respectively, which should include the most energetic ocean waves.
The same cutoff frequencies are applied in the spectral model we have compared
our results with. In our comparison of integrated parameters from bow
instruments and WII instruments at stops $1.1$-$1.4$, we use $0.05$-$0.25$ Hz,
which corresponds to the cutoff frequencies applied by the on-board processing
unit of the WII instruments. The spectral moments can be used to estimate the
mean ($T_{m01}$) and zero up-crossing ($T_{m02}$) periods from

$T_{m01}=\frac{m_{0}}{m_{1}},$ (5)

and

$T_{m02}=\sqrt{\frac{m_{0}}{m_{2}}},$ (6)

respectively.

We use significant wave height $H_{s}$ from spectra for most of our analysis
here. $SWH$ from time series of surface elevation (Eq. 2) is used for
comparison in a redundancy check for the measurements, as the two methods
ideally should yield the same result. $H_{S}$ is defined as:

$H_{s}=4\sqrt{m_{0}}.$ (7)

Wave attenuation is dependent on frequency. It is therefore convenient to have
a measure for wave amplitude as function of frequency when finding the spatial
damping coefficient. We define a spectral amplitude $a$ at discrete forcing
frequency $f_{0}$ as:

$a(f_{0})=\sqrt{\int_{f_{0}-\Delta f}^{f_{0}+\Delta f}PSD(f)df},$ (8)

where $\Delta f$ is the bandwidth frequency set to $0.005$ Hz. Six discrete
forcing frequencies ($0.076$-$0.128$ Hz with $0.0104$ Hz increments) are
chosen based on the spectra presented in Section 3.2. Equation 8 is similar to
the definition of Meylan et al. (2014), used for attenuation analysis in the
Antarctic MIZ. The spectral amplitude $a(f_{0})$ is not the physical wave
amplitude in the classical sense, but an interpretation of the energy content
in a finite frequency proportion of the PSD with unit meter.

Confidence intervals for both spectra and the integrated parameters,
significant wave height and amplitude, are calculated from the Chi-squared
distribution, following the methodology presented in Young (1995). For
spectra, the total degree of freedom (TDF) is calculated as $TDF=2q$, and for
significant wave height, TDF is found with Eq. 9 below (ITTC, 2017).

$TDF=\frac{2q\left[\int_{f_{min}}^{f_{max}}PSD(f)df\right]^{2}}{\int_{f_{min}}^{f_{max}}[PSD(f)]^{2}df}.$
(9)

Equation 9 is also used when finding the confidence intervals for $a(f_{0})$,
with the small modification of multiplying the bandwidth frequency $\Delta f$
to the denominator.

We use the mean absolute percentage error (MAPE) to compare our system with
either the spectral model or the WII instruments. Systematic bias is described
with the mean percentage error (MPE). The error statistics are defined as:

$MAPE=\frac{100\%}{n}\sum_{t=1}^{n}\left|\frac{X_{t}-Y_{t}}{X_{t}}\right|,$
(10)

$MPE=\frac{100\%}{n}\sum_{t=1}^{n}\frac{X_{t}-Y_{t}}{X_{t}},$ (11)

where $Y_{t}$ are parameters obtained from bow measurements and $X_{t}$ are
reference parameters.

###  2.5 Wave models

WAM-4 is a third generation spectral wave model (Komen et al., 1996; Saetra
and Bidlot, 2004), run operationally by the Norwegian Meteorological
Institute. Its performance has been validated against in-situ and EnviSat
Radar Altimeter observations (Carrasco and Gusdal, 2014; Gusdal et al., 2011).
Spatial and temporal resolution of the model is $4$ km and $1$ hour,
respectively. It runs twice a day with a forecast period of 66 hours per run.
In this study, we have concatenated forecasts from consecutive runs to provide
a continuous time series. A hard ice boundary based on satellite images (ice
concentration over 3/10th) is defined in the model.

ERA5 reanalysis of global atmosphere and ocean waves is produced by the
European Centre for Medium-Range Weather Forecasts (ECMWF). ERA5 combines
historical observations into past forecasts using data assimilation systems to
model meteorologic and oceanographic parameters. The operational wave model at
ECMWF assimilates satellite synthetic aperture radar observations and
altimeter wave heights into the numerical scheme of the WAM spectral wave
model, which is coupled with the IFS atmospheric model to account for wind
forcing (Janssen and Bidlot, 2003). The sea ice concentration (SIC) is based
on coupled atmosphere ocean simulations of the IFS model combined with
satellite observations (Hirahara et al., 2016), and wave parameters are not
given for $SIC>50\%$. Reanalysis wave data from ERA5 have hourly output and a
horizontal resolution of $40$ km.

No wave simulations are performed where the assumed ice cover is present,
although the models still provide wind information everywhere within the
domain. We have used model data from WAM-4 and ERA5 as a comparison to our
wave observations outside the MIZ. The model ice edge was defined at $82.065$
°N and $82.175$ °N by WAM-4 and ERA5 respectively at the relevant longitude on
September 19. The horizontal resolution of the models are indicated with a
white and orange square in Fig. 1(a) for ERA5 and WAM-4 respectively. From the
models, we extract total mean wave direction $THQ$, $H_{S}$, $T_{m01}$,
$T_{m02}$ (from WAM-4 only) and wind parameters. $THQ$ is defined as the mean
of the two-dimensional wave spectrum over all frequencies and directions. The
wave parameters are integrated parameters from spectra, but the raw spectra
were not available.

###  2.6 Waves-in-ice instruments

Four in situ WII instruments (Rabault et al., 2020), placed on ice floes close
to the ship at the stops $1.1$-$1.4$, have been used to cross-validate our
system inside the MIZ. The instruments measure waves with an integrated IMU,
and they have been tested and used in a series of previous works (Rabault et
al., 2016, 2017; Sutherland and Rabault, 2016; Marchenko et al., 2017). The
WII instruments are autonomous and designed to be deployed for long durations.
Compressed power spectra from $20$ min time series of surface elevation were
sent via Iridium satellite and used as comparison with the spectra obtained
from the bow measurements. We also compare the integrated parameters $H_{S}$
and $T_{m02}$.

The WII instruments measured with five hours intervals to conserve battery
power. Data from the first measurement were corrupted due to disturbances
during carrying and placement on the ice and were discarded. The results
presented here are from the second measurement of each instrument, i.e. five
hours after the respective bow measurement. The temporal change of sea state
over these five hour periods are addressed in Section 3.1. The WII instruments
drifted $2.4$ km, $1.7$ km, $3.5$ km and $2.2$ km during the five hours
between first sample taken at stop $1.1$-$1.4$ respectively, and second
sample. These are relatively small displacements compared to the distance
between the WII instruments, which were an order of magnitude larger. Hence,
the measurements with the two different systems were made at approximately the
same place with five-hour time delay.

###  2.7 Attenuation modeling

Previous field measurements indicate that waves decay exponentially in ice
(Squire and Moore, 1980; Wadhams et al., 1988). For each frequency bin
corresponding to the six forcing frequencies $f_{0}$, the spectral amplitudes
$a$ are fitted to decreasing exponentials on the shape $a=Ce^{-\alpha x}$,
where $C$ and $\alpha$ are estimated parameters and $x$ is wave traveling
distance through the MIZ, by means of non-linear least squares. From the
fitted curves, the spatial damping coefficients $\alpha$, which describe wave
attenuation are determined from:

$\frac{\partial a}{\partial x}=-\alpha a,$ (12)

for each frequency bin. Non-linear least squares are applied to fit a power
function $\alpha=Af^{p}$ to the measured values of $\alpha$ as function of
frequency $f$, where $A$ and $p$ are the best-fit parameters. The standard
error in the exponent is obtained from the square root of the variance of $p$.

Spatial damping coefficients found in the field measurements are compared to
the model of Sutherland et al. (2019), which presents a parameterization for
wave dissipation that allows for a two-layer structure within the ice. The
lower layer with thickness $\epsilon h_{i}$, where $h_{i}$ is the total ice
thickness, is defined as a highly viscous layer where wave motion exists. The
upper layer is defined as impermeable. With a no-slip boundary condition
imposed on the bottom of the ice, Equation (16) of Sutherland et al. (2019)
can be written as:

$\alpha_{mod}=\frac{1}{2}\epsilon h_{i}k^{2},$ (13)

where $k$ is the wavenumber, here calculated with (1). Total ice thickness is
set to $h_{i}=1.1$ m based on visual observations from the field campaign. We
use $\epsilon=0.02$ due to the large ice thickness, meaning that wave motion
is assumed to exist in a small fraction of the total ice thickness. Note that
(13) is proportional to frequency to the fourth power.

##  3 Results

###  3.1 Data comparison

Independent in-situ wave measurements were not available for validation of the
observations in the open ocean on this cruise. Therefore, we compare our
results with WAM-4 and ERA5 spectral model data. It is worth mentioning that
Christensen et al. (2013) performed and documented validation of an equivalent
setup of the system in open water against a moored Waverider buoy, model data
from the ERA Interim Reanalysis and satellite observations from AVISO.

Bow measurements outside the MIZ are compared with spectral model data at the
ship location in Fig. 6. The period with no available model data due to the
ice cover is highlighted with red background color and is not included in the
comparison between observations and the spectral models. Valid measurements,
i.e. time periods where total UG saturation is below $10$% and ship speed over
ground (SOG) is below $0.5$ m/s, are highlighted with green background color.
The whole comparison spans over eight days, and periods of valid measurements
are found $23$% of the time. Error statistics are summarized in Table 2 where
MAPE describes the mean absolute error and MPE describes the mean error. In
general, there is a good agreement between observations and models, also for
the conditions not defined as valid. The MPEs of WAM-4 do not exceed $\pm 10$%
for $H_{S}$ and $T_{m01}$ in the valid periods, which indicate low systematic
bias. ERA5 systematically overestimates $H_{S}$ compared to the observations,
but matches the observed $T_{m01}$ better than WAM-4. A surprising result is
that significant wave height has a higher MAPE in the valid periods than
outside, although the difference is not large ($18.9$% vs $13.3$% for WAM-4
and $29.2$% vs $21.3$% for ERA5). MAPEs for $T_{m01}$ are about the same
inside the valid periods as outside. Note that significant wave height does
not exceed the theoretical limit (for sine waves) of $1.4$ m as described in
Section 2.1 within the valid data periods. MAPE between measured $H_{S}$ and
$SWH$ was $11.7$% during valid measurements (not shown in Table 2).

Figure 6:  Long time comparison between observations and the models WAM-4 and ERA5. Significant wave height from time series and spectra (upper), and periods from spectra (lower). Time periods where total saturation is below $10$% and ship speed over ground (SOG) is below $0.5$ m/s are highlighted with green background color and the period with an ice cover (hence no model data) is highlighted with red background color. Note that ERA5 provides wave data further into the MIZ than WAM-4. Error | WAM-4 | ERA5
---|---|---
$H_{S}$ | $T_{m01}$ | $T_{m02}$ | $H_{S}$ | $T_{m01}$
Valid | MAPE [%] | 18.9 | 15.0 | 17.2 | 29.2 | 13.4
MPE [%] | 9.2 | -8.9 | -14.4 | 28.5 | 3.4
Not val. | MAPE [%] | 13.3 | 15.7 | 14.7 | 21.3 | 16.4
MPE [%] | 3.2 | -5.5 | -4.4 | 19.5 | 3.1
Table 2: Long time error statistics for significant wave height $H_{S}$ and
periods $T_{m01}$ and $T_{m02}$ when comparing bow measurements with the WAM-4
and the ERA5 model. Valid periods are defined as when total UG saturation is
less than $10$% and ship speed over ground is less than $0.5$ m/s ($23$% of
the time). All other times are considered not valid.

A stationary sea state over the measurement period inside the MIZ is an
advantage in our analysis. The sea state is investigated in Fig. 7 where time
series of $H_{S}$, $T_{m01}$ and $THQ$ from WAM-4 are presented. The model
data are extracted from the six points arranged in a grid configuration
outside the MIZ, shown in cyan in Fig. 1(a). Time series from each grid point
are plotted in gray and the mean value in a different color. All parameters
were close to constant between 11:00 and 17:00 but varying with time outside
of this period. This indicates a variability in sea state in the comparison of
bow and WII instruments, due to the five-hour time delay between the
measurements. However, the change was not dramatic. From 04:22 to 09:22 at
stop $1.1$, the standard deviation in the mean samples (containing five
values) were $0.06$ m, $0.10$ s and $1.00\degree$ for $H_{S}$, $T_{m01}$ and
$THQ$ respectively. From 12:20 to 17:20 at stop $1.4$, the standard deviation
in the mean samples are an order lower for all parameters. The results in Fig.
7 also indicate a variability in sea state over the period from 13:11 to 19:08
where wave damping was measured. In this period, the standard deviations in
the mean samples (containing seven values) were $0.03$ m, $0.10$ s and
$1.64\degree$ for $H_{S}$, $T_{m01}$ and $THQ$ respectively.

Figure 7:  Time series of $H_{S}$ (upper), $T_{m01}$ (middle) and $THQ$
(lower) from WAM-4 spectral model. Values are extracted from locations
indicated by the six cyan colored dots in Fig. 1(a) and plotted in gray. Mean
values are plotted in different colors.

A comparison of ship measurements and available WAM-4 model data from stops
$2.1$-$2.6$ is presented in Fig. 8. The upper panel shows ship heading and
speed over ground (SOG), extracted from the ship navigation system. Forecasted
wind direction and speed presented in the lower panel are compared to $10$ min
average values (corrected for ship motion) acquired by the ship anemometer.
The six stops when measurements are performed are clearly visible as when the
SOG graph flats out at close to zero value. These periods are highlighted with
green background color.

As the ship cruised between stops, the heading direction was about
$190\degree$. During the measurements, the heading direction was $240\degree$
at stop $2.1$, roughly $140\degree$ at stop $2.2$-$2.5$ and about $315\degree$
at stop $2.6$. The wind direction was almost constantly $45\degree$ over the
six-hour period. It is clear that the ship was oriented approximately
perpendicular to the wind direction when it drifted freely during the stops,
except from stop $2.1$ where the high ice concentration most likely prevented
the ship from rotating. For stop $2.2$-$2.5$, the wind came in on port side of
the vessel, while it came in from starboard side at stop $2.6$.

There is generally a good agreement between forecast and measured wind
direction ($WD$) and speed ($WS$), although the model overestimated $WS$ by
$50$-$75$% in the period from roughly 15:30 to 18:30. $THQ$ from model and
$H_{S}$ from model and bow measurements are also presented in the lower panel
of Fig. 8. The overestimated wind speed could possibly explain why forecast
$H_{S}$ is $95$% higher than measured value at 20:09 when wave model data were
available, as WAM-4 is forced with winds at $10m$ height from the atmospheric
model UM4. Also, occasional ice floes were observed at this point and WAM-4
does not take attenuation caused by the presence of ice into considerations.
However, the measured wave data were saturated at this point and cannot be
completely trusted. THQ was $282.1\degree$ at 20:09. Due to the low variation
in THQ (in time and space) over the period when attenuation measurements were
sampled, as seen in Fig. 7, this value ($282.1\degree$) is used to find the
WTD through the MIZ, presented in Table 1 and Fig. 1(b). The wave heading
angle $\beta$ was $40\degree$ at stop $2.1$, $140\degree$ at the four
intermediate stops $2.2$-$2.5$ and $35\degree$ at stop $2.6$. Wave heading
angle will be further assessed in Section 4.

Figure 8:  Comparison of observations and WAM-4 model. Ship heading and SOG
(upper), and wind direction (WD) and wind speed (WS) from ship measurements
and model, total mean wave direction $THQ$ from model and significant wave
height $H_{S}$ (multiplied with a factor of 10) from bow measurements and
model (lower). Directions increase clockwise from geographic north (zero
degrees). Times with valid data are shaded.

Power spectra from bow measurements (black) and WII instruments (blue) with
their respective $95$% confidence intervals are compared in Fig. 9, going
successively from stop number $1.1$ (upper panel) to $1.4$ (lower panel). The
power spectra from the two methods are consistent for most frequencies in the
two first samples closer to the open ocean. The third sample is consistent for
some frequencies up to $0.09$ Hz. For higher frequencies, the WII spectrum
flattens out while the bow spectrum peaks at $0.1$ Hz. Investigation of bow
spectra from samples taken respectively twenty and forty minutes after the PSD
presented here while the ship was still stationary around the WII instruments,
reveals similar spectral peaks at the same frequency. This consistency
substantiates the validity of the bow measurements. A discussion on this
discrepancy follows in Section 4. The fourth sample furthest into the ice
displays a fair agreement between the two methods.

Figure 9:  Validation of bow measurements in the MIZ with WII instruments
placed on ice floes. PSD are presented from bow measurements (black) and from
WII instruments (blue) with their respective $95$% confidence intervals as
shaded regions, going further into the ice zone from upper (stop $1.1$) to
lower panel (stop $1.4$).

Integrated parameters from the spectra in Fig. 9 are summarized in Table 3.
There is a good match for both significant wave height and mean period where
MPEs do not exceed $\pm 10$% for all stops except $1.3$ where the error in
$H_{S}$ is $-87.8$%. We use MPE as defined in (11), although it is not
strictly a mean error in this context where only single point parameters are
compared.

Stop | Significant wave height | Mean wave period
---|---|---
|  $H_{S\\_bow}$ [m] |  $H_{S\\_WII}$ [m] | MPE [%] |  $T_{m02\\_bow}$ [s] |  $T_{m02\\_WII}$ [s] | MPE [%]
1.1 | 0.36 | 0.39 | 8.4 | 8.85 | 8.65 | -2.4
1.2 | 0.29 | 0.28 | -4.3 | 9.03 | 9.12 | 1.0
1.3 | 0.16 | 0.09 | -87.8 | 10.14 | 10.89 | 6.9
1.4 | 0.04 | 0.04 | -5.1 | 11.36 | 12.57 | 9.6
Table 3: Significant wave height and zero up-crossing period from bow
measurements inside the MIZ are compared with WII measurements. Errors are
included.

###  3.2 Wave attenuation

One dimensional power spectra of the surface elevation with $95$% confidence
intervals from stops $2.1$-$2.6$ are presented in Fig. 10. Most of the energy
is found at either low (swell) or high (wind wave) frequencies. Averaged peak
frequency is found to be $0.076$ Hz for swell and $0.128$ Hz for wind wave.
Spectral amplitudes $a(f_{0})$ for a set of six finite frequency bins are
found from (8) and investigated at stops $2.1$-$2.6$. The highest and lowest
$f_{0}$ are set to $0.076$ Hz and $0.128$ Hz respectively. Four intermediate
forcing frequencies are evenly distributed between the swell and wind wave
frequency.

Figure 10:  Power spectra at different latitudes with $95$% confidence
intervals, and ship resonance frequencies in vertical modes from Ytterland
(2016).

It is evident that wave energy content increases as the ship approaches the
open ocean. Figure 11(a) shows an exponential decrease in $H_{S}$ (with $95$%
confidence intervals) as function of WTD through the MIZ for stops
$2.1$-$2.6$. Ideally, the measurements should have been performed
simultaneously to ensure stationary wind forcing throughout the attenuation
run. Fortunately, wind conditions did not change dramatically over the period
from 13:11 to 19:08 as can be seen in Fig. 8, and we therefore assume the wave
generation to be approximately stationary.

Spectral amplitude for swell, the third intermediate wave, and wind wave are
plotted versus wave travel distance through the MIZ and presented in Fig.
11(b) along with their $95$% confidence intervals. A clear exponential
attenuation for wind wave can be seen. The exponential shape is less obvious
for the intermediate frequency and swell, but the trend is visible. Also,
swell amplitude seems to decrease from stop $2.5$ to $2.6$, which is not
physical, and this problem is addressed in Section 4. Exponential fits lie
within the confidence intervals of $a(f_{0})$ for almost all frequencies.

(a) $H_{S}$ versus wave traveling distance into the MIZ.

(b) $a(f_{0})$ versus wave traveling distance into the MIZ.

Figure 11: Wave damping with $95$% confidence intervals. a) Significant wave
height $H_{S}$. b) Spectral amplitudes $a(f_{0})$ for swell, one of the
intermediate wave frequencies and wind wave. Exponential decay functions are
fitted to the data.

The spatial damping coefficients $\alpha$ are plotted versus their respective
$f_{0}$ with black asterisks in Fig. 12. We obtain values of $\alpha$ from
$0.67\times 10^{-5}\mathrm{m}^{-1}$ at $0.076$ Hz to $4.44\times
10^{-5}\mathrm{m}^{-1}$ at $0.128$ Hz. These values are within the same order
of magnitude as other attenuation observations within the same frequencies
reported in the literature where the measurements are done over approximately
the same wave traveling distance into the MIZ, e.g. Squire and Moore (1980)
($\alpha=2.7-8.6\times 10^{-5}\mathrm{m}^{-1}$) and the February 26, 1983
event reported in Wadhams et al. (1988) ($\alpha=1.6-4.8\times
10^{-5}\mathrm{m}^{-1}$).

Observed wave attenuation increase with frequency in the shape of a power
function. The fitted power curve is plotted with a dashed gray line in Fig.
12, and the $1\sigma$ uncertainty associated with the power law exponent is
marked with a gray shaded area. The fitted exponent is $3.51\pm 0.41$, where
the uncertainty is the standard error from the curve fit. Values of $\alpha$
obtained from (13) are plotted with a blue line in Fig. 12. The two-layer
model of Sutherland et al. (2019) describes the frequency dependence of
$\alpha$ quite well for the frequencies considered here. It can be noted that
the model slightly overestimates the frequency dependence for frequencies
above $0.12$ Hz.

Figure 12:  Spatial damping coefficient $\alpha$ as function of frequency from
measurements (black) and two-layer model (blue). Power coefficients for the
fitted curve and the model are displayed in the legend. $1\sigma$ errors
associated with the power coefficient estimate for the fitted curve (gray) are
shown as the gray shaded region.

##  4 Discussion

A culture of sharing the design of instruments within the scientific community
would enable a more cost effective and increased quantity of data collection
in the Arctic. We would like to stimulate researchers to use off-the-shelf
sensors and open-source code and therefore make all our material available
(See Appendix A). The components used to obtain wave data in this study are
relatively cheap compared to industry made black-box instruments. The system
can record autonomously, hence, expensive expedition time does not have to be
allocated as measurements can be done in parallel. Also, cruises in the Arctic
with other purposes than measuring waves in ice, for example studies within
biology or meteorology, could benefit from installing the system presented
here, which provides wave measurements without the need for human
intervention. Other devices such as floaters and buoys offer the advantage of
multiple simultaneous measurements at different locations, which is not
possible with bow mounted sensors. However, bow mounted sensors do not require
costly and time-consuming deployment and retrieval operations, and there is
low risk associated with losing instruments at sea.

The primary focus of this study is to present a new method for wave
measurements in the MIZ. Spectral wave models have been included only as a
supplement for comparison with the observations. Nevertheless, we provide a
short discussion on the difference between the models and how well they
correspond to the bow system. As mentioned in Section 2.5, ERA5 is a
reanalysis of previous observations and simulations, and is therefore
considered more reliable than a forecast model, such as WAM-4. However, WAM-4
matches the observations better than ERA5 in terms of significant wave height,
especially in the vicinity of the MIZ. No wave models are the true realization
of the ocean, but there are arguments that support the accuracy of WAM-4 over
ERA5 in our situation. ERA5 provides wave information for sea ice
concentration up to 50%, while the hard ice boundary in WAM-4 is defined at
30%. In other words, wave information is available further north from ERA5
than from WAM-4 as seen from Fig. 6, although none of the models consider the
effect of wave damping due to the presence of sea ice. Consequently, the
modeled significant wave height furthest into the MIZ deviates the most from
the real values. Additionally, ERA5 is a global model with a horizontal
resolution of 40 km, while WAM-4 is 10 times finer resolved, as seen in Fig
1(a). Considering the fact that the measurements were done relatively close to
the Svalbard and Franz Josef Land archipelagos, the horizontal resolution is
likely to be more important for the accuracy of the models than for example in
the middle of the Pacific.

Data recorded while a ship is in motion will contain a Doppler shifted
frequency according to the wave heading angle $\beta$ (Collins III et al.,
2017), which could be problematic in the calculation of periods. Peak and mean
periods were reported strongly biased by the Doppler shift in Christensen et
al. (2013), while the estimates of significant wave heights were reasonable
since they only depend on the sea surface variance. In the present study, we
consider the measurements invalid either when the ship is cruising (and a
Doppler shift could be introduced depending on $\beta$) or when the UG is
saturated (exceeded measure range), as we have not differentiated between
these two cases. We see little difference in MAPE inside versus outside the
time periods we consider valid, when comparing periods from bow measurements
with model data. MPEs are actually higher in the periods where data is
considered valid. There is also a slightly better match outside the valid
periods in the comparison of significant wave height. This is somewhat counter
intuitive, but there is of course uncertainty associated with the spectral
models, and $\beta$ is not included in the analysis. It is possible to correct
the Doppler shift, but this requires accurate observations of the wave
direction, or more precisely $\beta$ (Collins III et al., 2017). This would
allow for possibly improved mean period estimates during cruising.

Significant wave energy for $f>0.1$ Hz was detected by the bow mounted sensors
and not by the WII instruments at stop $1.3$ as seen in Figure 9. Yiew et al.
(2016) have performed experiments with thin floating disks and applied two
theoretical models to determine the hydrodynamic responses of ice floes with
diameter $d$ in regular waves with wavelength $\lambda$. They find RAO in
heave to be $0.5$ at approximately $\lambda/d=1.5$ and rapidly decreasing for
smaller $\lambda/d$. For the peak frequency measured by the bow sensor, i.e.
$f=0.1$ Hz, $\lambda$ is found to be $156$ m with Eq. 1. We have no exact
measure of the size of the ice floe at stop $1.3$, but deeming from Fig. 1(a)
and Fig. 9 in Rabault et al. (2020) its diameter seems to be at least in the
order of hundred meters. The flexural rigidity of the ice floe will of course
differ from the disks applied in Yiew et al. (2016), but the results in this
paper suggest a low heave response of such a large floe at this short
wavelength. As the WII instruments essentially calculate the power spectra
from the heave motion of the floe, a plausible explanation for the discrepancy
between the two instruments could be that waves in this frequency range were
effectively damped by the floe. The bow sensor on the other hand, measured at
points where the ice was broken up by the ship and was therefore able to
detect higher frequencies and give a more accurate observation of the sea
state.

Another relevant problem is the response of the ship itself. Figure 10 shows
the resonance frequencies of the ship in the vertical modes with
$\beta=150\degree/30\degree$ (due to symmetry) (Ytterland, 2016), which almost
corresponds to $\beta$ at stops $2.1$-$2.5$ ($10\degree$ deviation) and stop
$2.6$ ($5\degree$ deviation). The RAO values are $1.8$ and $1.9$ for these
angles, meaning a possible maximum amplification of $80$% and $90$% for roll
and pitch motion respectively. Heave mode has only got a peak in RAO for
$\beta=90\degree$, which did not occur during stops $2.1$-$2.6$ and is
therefore not shown. None of the spectral peaks in Fig. 10 coincides with the
natural frequencies in roll. However, the high frequency peaks are in the same
range as the ship’s natural frequency in pitch. Pitch motion was likely
amplified up to $90$% when exposed to $0.12$ Hz waves. However, this effect
should be compensated for by the pitch motion correction performed on the
data.

Stop $2.6$ has been included in our wave attenuation analysis even though it
was located outside of the defined ice edge in Figure 1(b). This ice edge was
estimated from coarsely resolved satellite images and is therefore not very
accurate. Reflected wave energy from the ice edge could have been a problem at
stop $2.6$. However, the ice edge was not a strictly defined line, it was
rather observed as a gradually decreasing ice concentration towards the open
ocean and ice floes were still present around stop $2.6$. Swell energy was
observed to decrease from stop $2.5$ to $2.6$ where the opposite is expected.
Our assumption of stationary wave field may not have been valid in this case,
although wind measurements and WAM-4 model data suggest otherwise. There is
also uncertainty associated with the total mean wave direction $THQ$. There
could have been discrepancies between $THQ$ and mean swell direction
$THQ_{swell}$, although $90\degree<THQ_{swell}<270\degree$ (swell traveling
south) is unlikely. Another possible explanation is that ship yaw (i.e.
rotation in the horizontal plane) and/or a changing $\beta$ might have
influenced the low frequencies in the measurements. At stop $2.6$ the ship
heading increased continuously from approximately $200\degree$ until it
stabilized at $315\degree$ towards the end of the $20$ min sampling period,
whereas both ship heading and $\beta$ were almost constant for the other
stops, as seen in Figure 8.

We emphasize that we are not trying to show a universal power law in Fig. 12,
which would not make sense for frequencies spanning over less than a decade.
Our intention is to compare the observed wave attenuation with the two layer
waves-in-ice model of Sutherland et al. (2019), and not to verify the model.
The power function curve fit gives a frequency dependence of $\alpha\propto
f^{3.51\pm 0.41}$, which is similar to (13) where $\alpha\propto f^{4}$. Wind
energy input is not considered by the model. With a wind speed of
approximately $6$ m/s on average over the measurement period as seen in Fig.
8, it is reasonable to assume that wave generation has taken place, which will
reduce the slope of the observed attenuation at higher frequencies. This could
explain why the model predicts the observation quite well for the lower
frequency bins, while it slightly overestimates attenuation for the higher
frequency bin, based on the six data points.

##  5 Conclusions

We have presented shipborne wave measurements from the MIZ with a system
combining an altimeter (UG) and a motion correction device (IMU). The UG was
able to reflect signals off an ice-covered surface. The current system was
designed for the present cruise in the MIZ of the Barents Sea where waves are
normally quite small. Hence, the choice of the present UG and its auto
calibration with a $1$ m range. If some cruise would plan to go to rougher
seas, one could either use the same UG over its full range ($8$ m), or another
UG with larger range. The methodology is both cost effective as components are
off-the-shelf and it can be installed on a cruise with other objectives. The
system has proven to be robust as instruments have measured continuously over
the two-week period without suffering any damage. Our setup provides single
point time series of ocean surface elevation outside of and inside the MIZ,
which enables us to produce 1D power spectra and integrated parameters.

Measured data have been compared to the WAM-4 and the ERA5 spectral wave
models over a period of eight days. We have found good agreement in mean and
zero up-crossing periods and significant wave height. Mean absolute percentage
errors (MAPE) were $18.9$%, $15.0$% and $17.2$% for $H_{S}$, $Tm_{01}$ and
$Tm_{02}$ respectively when compared with WAM-4 and $29.2$% and $13.4$% for
$H_{S}$ and $Tm_{01}$ respectively when compared with ERA5 during valid
measurements ($23$% of the time). Errors were about the same during periods
where measurements were not considered valid, indicating that the system is
insensitive to exceeded instrument range and/or Doppler shifted wave
frequencies. Forecast significant wave height was $95$% higher than the value
measured inside the MIZ. This deviation might have been caused by an
overestimated model wind speed compared to wind speed measured with the ship
anemometer, or by the fact that occasional ice floes were present, which were
not considered by the forecast model. Power spectra and integrated parameters
from bow measurements have been compared with values obtained from waves-in-
ice instruments placed on ice floes. A good match was found when the sensors
were placed on smaller floes further out in the MIZ. The spectra agree fairly
well further into the MIZ. A major discrepancy was likely caused by a large
ice floe that filtered out higher frequencies where the waves-in-ice
instrument was placed. Errors in $H_{S}$, $Tm_{02}$ were smaller than $10$% in
all cases except for one of the measurements.

We observed an exponential wave attenuation when going through the MIZ. The
spatial damping coefficients obtained from measurements were within the same
order of magnitude as observations reported in previous studies with a similar
wave traveling distance through the ice (Squire and Moore, 1980; Wadhams et
al., 1988). We have found a strongly frequency dependent attenuation, where
$\alpha\propto f^{3.51\pm 0.41}$, which is similar to a two-layer attenuation
model where $\alpha\propto f^{4}$ (Sutherland et al., 2019).

## Acknowledgement

The authors are grateful to Øyvind Breivik for inviting us to the cruise. We
also thank the crew of RV Kronprins Haakon for their assistance and Yurii
Batrak at MET Norway for providing satellite images. The Nansen Legacy project
helped funding the cruise and loggers. Funding for the experiment was provided
by the Research Council of Norway under the PETROMAKS2 scheme (project DOFI,
Grant number $28062$). The data are available from the corresponding author
upon request.

## Appendix A: Open source code and designs

All the designs and files used for setting up the system, including the code
used to extract and process data, and general instructions for mounting the
instruments, are made available on the Github of the author under a MIT
license that allows full re-use and further development
(https://github.com/jerabaul29/Ultrasound_IMU_boat_waves_system [Note: The
material will be available upon publication in peer-reviewed literature]). All
software and designs are based entirely on open source tools, so that the
designs can be easily modified and built upon.

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