# Bringing optical fluid motion analysis to the field: a methodology using an
open source ROV as camera system and rising bubbles as tracers

Trygve K. Løkena,111E-mail address and phone number for corresponding:
<EMAIL_ADDRESS>(+47) 94885767 (T.K. Løken)
E-mail addresses<EMAIL_ADDRESS>(T. J. Ellevold<EMAIL_ADDRESS>(R.
De La Torre),
<EMAIL_ADDRESS>(J. Rabault) and<EMAIL_ADDRESS>(A. Jensen), Thea J.
Ellevolda, Reyna G. Ramirez de la Torrea, Jean Rabaultb and Atle Jensena

###### Abstract

Detailed water kinematics are important for understanding atmosphere-ice-ocean
energy transfer processes in the Arctic. There are few in situ observations of
2D velocity fields in the marginal ice zone. Particle tracking velocimetry and
particle image velocimetry are well known laboratory techniques for measuring
2D velocity fields, but they usually rely on fragile equipment and pollutive
plastic tracers. Therefore, in order to bring these methods to the field, we
have developed a new system which combines a compact open-source remotely
operated vehicle as an imaging device, and air bubbles as tracing particles.
The data obtained can then be analyzed using image processing techniques tuned
for field measurements in the polar regions. The properties of the generated
bubbles, such as the relation between terminal velocity and diameter, have
been investigated under controlled conditions. The accuracy and the spread of
the velocity measurements have been quantified in a wave tank and compared
with theoretical solutions. Horizontal velocity components under periodic
waves were measured within the order of 10% accuracy. The deviation from
theoretical solutions is attributed to the bubble inertia due to the
accelerated flow. We include an example from an Arctic field expedition where
the system was deployed and successfully tested from an ice floe. This work is
an important milestone towards performing detailed 2D flow measurements under
the ice in the Arctic, which we anticipate will help perform much needed
direct observations of the dynamics happening under sea ice.

a Department of Mathematics, University of Oslo, Oslo, Norway
b Norwegian Meteorological Institute, Oslo, Norway

##  1 Introduction

The development of field studies of fluid flow in the ocean has increased
considerably in the last decades thanks to the technological advances within
measurement equipment. Nonetheless, working in the Arctic environment remains
yet difficult. Not only because of the technical aspects to execute a
successful controlled experiment in the field, but also because the
possibilities to use regular laboratory equipment are limited by the harsh
temperatures and conditions. On top of that, working on an ice layer brings
new challenges to the design of a field setup. Adaptation of well known
techniques to the field by accurately testing them in the laboratory first is
a reasonable approach. This paper introduces a new system for high resolution
velocity measurements of fluid flow specially designed for Arctic conditions,
where elements from already established methods are combined. The accuracy of
the instrumentation is determined to understand its capabilities in order to
adapt them to the desired object of study.

Although laboratory experiments can yield useful results due to the high level
of control over variables, scaling between wave tank conditions and the real
ocean is challenging as several nondimensional numbers are necessary to
describe the problem. In wave tank experiments, it is for example desirable to
moderate nonlinear effects and ensure deep-water waves, as this is the most
common situation in the Arctic. These conditions are achieved by respectively
reducing the wave steepness and the ratio of wavelength over depth. As pointed
out by Rabault et al. (2019), it is with such constraints difficult to
simultaneously obtain a realistic Reynolds number for turbulence under ocean
waves. It is therefore necessary to perform field measurements to complement
or confirm laboratory experiments, where scaling problems are usually
inevitable.

Ultrasonic velocimeters, such as Acoustic Doppler Velocimeters (ADVs) and
Acoustic Doppler Current Profilers (ADCPs) are well known instruments for
fluid velocity measurements in the ocean. However, these instruments provide
only single point velocity (ADVs), or in the best case an array of along-beam
velocities (ADCPs), which in many cases is insufficient to give a detailed
description of the dynamics of certain flow phenomena that occur in the
Arctic. An example of such a phenomenon is wave propagation underneath an ice
layer (Sutherland & Rabault, 2016; Rabault et al., 2017), which is interesting
for validation of wave attenuation models through the marginal ice zone, e.g.
Weber (1987); Newyear & Martin (1997); Sutherland et al. (2019). Another
example is the investigation of turbulent structures or jets induced by
colliding ice floes (Rabault et al., 2019), which may contribute to ice-ocean
energy transfer and increased eddy viscosity and viscous dissipation. In these
cases it is desirable to resolve a 2D velocity field for a better
understanding of the underlying physics.

Within fluid dynamics and wave studies in particular, researchers have used
two very established laboratory techniques to understand flow dynamics:
Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV). In
controlled environments, these techniques can accurately represent the
velocity fields in a wide range of flows. Conventional PIV and PTV systems
normally utilize a powerful light source, such as lasers, to illuminate a thin
sheet of tracer particles. In this way it can be ensured that the particles of
interest are situated in a plane with a fixed distance to the camera. In later
years it has been shown that these techniques can be applicable in field
measurements with certain reserves, PIV-laser systems have been developed and
deployed off docks and from ships (Bertuccioli et al., 1999; Smith et al.,
2002). However, it is difficult to arrange such a setup in the field,
especially in an Arctic environment, due to the large weight, high cost and
fragility of lasers.

Another complication in the use of traditional PIV and PTV in field
measurements is the use of tracers. As addressed in Bertuccioli et al. (1999),
some oceanic areas can be analyzed by the use of the natural suspended
particles. But, particularly in the Arctic Ocean, the water is mostly clear
during the winter. Therefore, the introduction of environmental-friendly
tracers is required. The adaptation of PIV using bubbles as tracers has become
popular throughout different applications (Delnoij et al., 1999; de la Torre
et al., 2015; Ryu et al., 2005; Rivillas-Ospina et al., 2012; Cheng et al.,
2005), and tested against more commonly seeding methods (Vargas et al., 2020).
The mentioned studies have brought to light different aspects that need to be
calibrated and considered to apply these techniques outside a controlled
environment. In this study, an array of small air bubbles has been introduced
as tracers, which provide a plane of particles and thus enables us to find a
two-dimensional velocity field.

A good way to assist underwater measurements is the use of ROVs (Remotely
Operated Vehicles). Since the early 1980s, the interest of utilizing ROVs for
field work has increased. Towards the end of that decade, the first major
field works using ROVs to retrieve information and samples were made (Ballard,
1993). Afterwards, ROVs have been used extensively for observations and sample
retrieving. By using an open source ROV, it is possible to control tools
underwater and retrieve images and information from different sensors, while
keeping costs down to a reasonable level. It is only in the last few years
that ROVs have been considered a tool in different fluid dynamics applications
and in situ measurements, where the reliability and limitations of the sensors
mounted remain the main issues (Pinkard et al., 2005; Keranen et al., 2012).

In this work, a methodology to perform reliable field measurements using PTV
and PIV techniques is described. Extensive laboratory tests have been included
to show the reliability and range of confidence of the bubble curtain as a
substitute for tracers and laser sheets. The use of an ROV as an imaging and
light source system in PTV/PIV field measurements has been introduced and
tested in extreme Arctic conditions. For simplicity, we introduce the
abbreviation PV (Particle Velocimetry) to address PTV/PIV and name the
introduced technique "ROV-PV system". The paper is organized in the following
manner. Section 2 describes the components and the assembly of the ROV-PV
system, the algorithms used for image processing and how fluid velocity is
interpreted from measured bubble velocity. Section 3 presents the results from
the laboratory experiments. An example of a field deployment in the Arctic is
included in Section 4. The methods and results are discussed in Section 5 and
conclusions are drawn in Section 6.

##  2 Data and methods

The ROV-PV system presented here is designed to capture dynamics generated
from air-water and ice-water interactions in the upper ocean. It is a portable
system build for Arctic field campaigns at remote locations and harsh
environments. The core concept is to perform optical recordings with the
camera of an ROV. A plane of rising air bubbles illuminated by the ROV’s
headlights are used as tracing particles. The bubbles rise inside a thin
aluminum frame with indicated reference coordinates. Images are calibrated to
obtain real world units. Image processing is applied to produce the 2D in-
plane velocity field of the bubbles. During post processing, the vertical
buoyancy driven velocity component of the bubbles is subtracted to obtain the
water velocity.

###  2.1 ROV-PTV system

An open source ROV has been chosen for image acquisition that will be used for
the PV analysis, as it is easily maneuvered below and around sea ice. The use
of open source instruments is an increasing trend in geophysics, see e.g.
Lockridge et al. (2016); Rabault et al. (2016, 2020). Open source instruments
provide flexibility in the sense that they offer the possibilities for
extensive modification to specific needs and easy interfacing with other open
source systems. Another advantage is the significantly lower cost than similar
closed source solutions. An ROV allows for preliminary inspections of the site
to find the most suitable location for measurements, as opposed to a
stationary camera. The high-performance BlueROV2 from BlueRobotics
(BlueRobotics, 2020) with open source software and electronics has been used.
It is installed with eight thrusters to increase the stability and obtain full
six degrees of freedom control. The ROV is physically connected to a field
computer by a tether, which carries and transfers video and data signal. An
open source application called QGroundControl (QGC) works as the user
interface and provides live video stream and various types of information for
the pilot.

The camera is a wide-angle low-light HD USB camera with 1920$\times$1080 pixel
resolution. It is mounted on a servo motor for user control of tilt angle. The
camera lens has a 2.97 mm focal length with low distortion (1%) and a
80°horizontal and 64°vertical Field Of View (FOV). The highest available frame
rate $\Delta f=$ 30 frames/s is used for maximum temporal resolution. Camera
settings such as exposure, brightness and gain are adjusted in QGC to optimize
the images in different environments and conditions, to ensure that the
bubbles appear as clear, circular particles and not as blurry streaks. The ROV
is equipped with four controllable headlights with a total capacity of 6000
lumens.

In the setup, which is illustrated in Fig. 1, bubbles are generated with a 5 m
long flexible silicon rubber hose. At the one end, the hose is fed with air of
approximately 0.5 bar from a 1.1 kW ABAC compressor. At the other end, the
last 0.75 m is perforated every 1 cm on the upward facing side with a 0.3 mm
needle and the tube is sealed with a plug to prevent air leakage. This
configuration provides an array of relatively small bubbles with a diameter of
0.6$-$1.4 mm. Bubbles have also been generated with a thin carbon fiber pipe
perforated with a 0.1 mm drill. This device was not used in the experiments
because the bubbles grew too large while sticking to the pipe before they
detached. The headlights of the ROV have proven sufficient to make the bubbles
visible during fieldwork. During laboratory experiments, the ROV has been
placed outside of the water tank due to space limitations, and the bubbles
have been illuminated with external LED lamps to avoid headlight reflections
in the glass wall.

Figure 1:  Schematic of the ROV-PV system in an Arctic environment. The 2D in-
plane water velocity is estimated from the bubble motion, which is recorded
with the ROV camera. The perforated hose can for example be suspended into the
water by means of a pulley system, as shown here, or directly through a hole
in the ice.

In order to obtain quantitative physical information from the experiments, it
is necessary to implement a coordinate system to be able to convert from
pixels to real-world units in the post processing. The coordinate system needs
to be in the framework of the bubble plane. A thin aluminum grid with a
combination of grid bars and woven string has been used as coordinate
indicators. The grid measures 55$\times$45 cm and the coordinate resolution is
8$-$12 cm in the horizontal and 9 cm in the vertical direction. In a
traditional particle tracking setup, the camera and the light-sheet are fixed
in one position throughout the whole experiment, making it easy to take a
picture of a coordinate system in advance of the measurements. In the current
system, the ROV will always be in some movement which leads to a varying
position and angle relative to the coordinate grid. Therefore, the perforated
hose is integrated at the bottom part of the grid so that each single image
can be converted into real-world units with a third order coordinate
transformation, e.g. Kolaas (2016). Upon deployment in the ocean, the
perforated hose and coordinate system are suspended from a constructed frame
or from the sea ice. Heavy bolts are attached to the bottom of the grid to
keep it vertical and in place when exposed to waves and ocean currents.

###  2.2 PTV and PIV methodology

In this work, both PTV and PIV were used in different cases. PTV differs from
PIV in a fundamental way. PIV relies on pattern matching in an essentially
Eulerian way, whereas PTV seeks to identify individual particles and follow
them in a Lagrangian sense. Therefore, it is important to choose the most
suitable technique based on the characteristics of the experiment, and on the
information which is desirable to obtain. A small review of the PTV and PIV
algorithms used for this work follows.

DigiFlow Software (Dalziel, 2006) has been utilized for PTV in this study.
After the image acquisition, the image is scanned for blobs that have an
intensity satisfying some threshold parameters. If a blob is found, its
characteristics are determined and compared with a set of requirements for the
blob to be considered a particle. If the blob satisfies these requirements, it
is recorded as a particle, if it does not, it is discarded. Once all the
particles in an image have been found at $t=t_{n}+1$, they need to be related
back to the previous image $t=t_{n}$. DigiFlow uses a modification of what is
known in operations research as the Transportation Algorithm, which was
developed by Dalziel (1992). The algorithm chooses a set of particles $P$ at
$t=t_{n}$, and the other the set of particles $Q$ at $t=t_{n}+1$. At
$t=t_{n}$, $p_{i}$ for $i=1,2,...,M$ represents the set of particles, while at
$t=t_{n}+1$, they are labeled $q_{j}$ for $j=1,2,...,N$. Each $p_{i}$ or
$q_{j}$ contains the location of the particle and other characteristics such
as size, shape, intensity, or any other desired information. A set of
association variables $\alpha_{ij}$ is defined. When $\alpha_{ij}=1$ then
$p_{i}$ at $t=t_{n}$ is the same particle as $q_{j}$ at $t=t_{n}+1$. If
$\alpha_{ij}=0$, then $p_{i}$ and $q_{j}$ represent different physical
particles. The number of particles in the images may be different at $t=t_{n}$
and $t=t_{n}+1$. To overcome this problem, $\alpha_{0j}$ and $\alpha_{i0}$ are
defined as dummy particles at times $t=t_{n}$ and $t=t_{n}+1$, respectively.
Unlike ordinary particles, more than one value of $j$ or $i$ may give a
nonzero value of $\alpha_{0j}$ and $\alpha_{i0}$, respectively. In this case,
a nonzero value of $\alpha_{0j}$ indicates that particle $p_{i}$ at $t=t_{n}$
has been lost from the image by $t=t_{n}+1$, either by moving out of the image
or for some other reason. Similarly, $\alpha_{0j}=1$ represents a particle
$q_{j}$ present at $t=t_{n}+1$ which was not there at $t=t_{n}$.

The in-house HydrolabPIV software developed at the University of Oslo has been
used for PIV in this work (Kolaas, 2016). This software combines different
techniques presented in previous works to optimize the PIV algorithm (Adrian,
1991; Westerweel & Scarano, 2005; Padfield, 2011; Raffel et al., 2018). Given
two consecutive images of a seeded flow, it is desirable to find an Eulerian
description of the velocity field. To achieve this, the images are divided
into a regular grid of subwindows, usually of size from 8$\times$8 to
64$\times$64 pixels. The use of a certain percentage of overlap is also
common, usually between 25% and 75%. For each subwindow in the first frame
$I_{1}$, a subwindow $I_{2}$ with a similar pattern is searched in the second
frame. A metric of the similarity of the patterns is required, a common choice
is the normalized cross-correlation

$R_{ncc}=\frac{\sum I_{1}I_{2}-\frac{\sum I_{1}I_{2}}{N}}{\sqrt{\left(\sum
I_{1}^{2}-\frac{(\sum I_{1})^{2}}{N}\right)\left(\sum I_{2}^{2}-\frac{(\sum
I_{2})^{2}}{N}\right)}}.$ (1)

To speed up the processing, fast Fourier transform is often used to calculate
the cross-correlation. An ensemble averaged velocity is found by dividing the
optimal match displacement of the pattern by $\Delta t$. After the velocity is
found, the result can be made more robust by checking the quality of the
image, and further validation of the vectors using outlier detection. This
detection uses a 3$\times$3 normalized local median filter. In addition, a
cubic B-spline is fitted to the velocity field using an iterative weighted
least squares fit. The local residuals are used with the biweight function in
the first iteration. The fit is then iterated a few times where the residuals
are used to update the weights in the least squares fit. The normalized
residuals from the last iteration together with the residuals from the local
median filter are used to mark the outliers. These outliers can be removed and
replaced by using the fitted B-spline.

###  2.3 Air bubbles as tracers

Bubbles as tracing particles are challenging because they do not passively
follow the fluid motion as opposed to conventional tracers. The buoyancy
driven motion of the bubbles must be subtracted in order to achieve the
absolute water velocity. In the following, the $\mathbf{x}=(x,y)$,
$\mathbf{V}=(u,v)$ and $\mathbf{F}=(F_{x},F_{y})$ conventions will be used for
2D position, velocity and force, respectively. The objective of this study is
to introduce a technique where the velocity of bubbles $\mathbf{V_{b}}$ is
measured in order to determine the velocity of the water $\mathbf{V_{w}}$. An
important quantity in this method is the slip velocity $\mathbf{V_{slip}}$
which is the relative velocity between the water and the bubbles, defined as

$\mathbf{V_{slip}}=\mathbf{V_{b}}-\mathbf{V_{w}}.$ (2)

Small bubbles are beneficial because they have rectilinear vertical
trajectories, whereas larger bubbles tend to oscillate in the horizontal
direction (Haberman & Morton, 1953), which raises the need for further
velocity corrections. It is additionally advantageous to generate uniform
bubbles along the array, which will allow for subtraction of the mean buoyant
velocity across the entire velocity field. As pointed out by Oguz &
Prosperetti (1993), it is difficult to produce small bubbles of uniform size
and rate. In the present study, the variability in bubble diameter was large
over the span of the perforated hose. From visual inspection, the bubble
diameter seemed to be randomly distributed in size. It is likely that the
difference in bubble size was caused by the non-uniform size of the outlet
holes, since the silicon rubber material may have contracted differently from
hole to hole after the needle was subtracted.

For steady uniform water flow, bubbles will approximately follow the water
motion in the horizontal direction after a relaxation time. An additional
vertical component is present due to the buoyancy force
$\mathbf{B}=\frac{4}{3}(\rho_{w}-\rho_{b})g\pi R^{3}$, where $\rho_{w}$
denotes the water density, $\rho_{b}$ the bubble density, $g$ the acceleration
due to gravity and $R$ the bubble radius. In this case,
$\mathbf{V_{slip}}=v_{slip}$ only has a vertical component driven by
$\mathbf{B}=B_{y}$. On the other hand, $\mathbf{V_{slip}}$ also depends on the
bubble inertia when the water flow $\mathbf{V_{w}}(\mathbf{x},t)$, where $t$
denotes time, is accelerated. Since $\frac{\rho_{b}}{\rho_{w}}\ll 1$, the
effect of added mass becomes important to determine the inertia. Drag force on
a spherical bubble is expressed as $\mathbf{D}=\frac{1}{2}\rho_{w}\pi
R^{2}C_{D}\mathbf{V_{slip}}|\mathbf{V_{slip}}|$. The drag coefficient $C_{D}$
depends on the Reynolds number $Re=\frac{2R|\mathbf{V_{slip}}|}{\nu}$, where
$\nu$ denotes the water kinematic viscosity. Following Grue et al. (1999), an
estimate of the slip velocity can be obtained from momentum balance of a
bubble

$\mathbf{D(V_{slip})}-\mathbf{B}=M_{b}\left[\frac{\partial\mathbf{V_{w}}}{\partial
t}\right]_{max},$ (3)

where $M_{b}=\rho_{w}\frac{2}{3}\pi R^{3}$ is the added mass of a spherical
bubble, and the mass of the bubble is neglected. The expression for
$\mathbf{V_{slip}}$ depends on the relation between $C_{D}$ and $Re$ and on
the maximum water acceleration $[\partial\mathbf{V_{w}}/\partial t]_{max}$. In
Section 3, the horizontal velocity component under the crests of water waves,
where there exist analytical approximations for the acceleration, is
investigated. An analytical relation for $C_{D}(Re)$ is used to solve Eq. (3)
for $\mathbf{V_{slip}}$ and it is shown that the observations are in agreement
with the error analysis.

Analytical solutions exist for $C_{D}(Re)$ for spherical bubbles moving in
infinite mediums at low Reynolds numbers. Stokes’ solution, i.e.
$C_{D}=\frac{24}{Re}$ is a good approximation for $Re<1$ (Haberman & Morton,
1953). For moderate Reynolds numbers, boundary-layer theory approximations
have resulted in analytical solutions. Batchelor (2000) assumed an
irrotational flow outside a thin boundary layer around spherical bubbles and
use potential theory to obtain $C_{D}=\frac{48}{Re}$, which is a fair estimate
for 20 $<Re<$ 200$-$500, depending on the fluid. Bubbles tend to stay
spherical up to and including moderate Reynolds numbers due to sufficiently
strong surface tension forces. In the spherical regime, the bubble trajectory
is usually rectilinear and the vertical velocity increases with the radius
squared. Above $Re\approx$ 450, bubbles enter the elliptical regime where they
start to deform and flatten as viscous and hydrodynamic forces become more
prominent. Consequently, the drag increases, and approximations of $C_{D}$
primarily rely on numerical simulations and experimental data. In the
ellipsoidal regime, the rising motion is typically helical or oscillatory and
the terminal velocity has been observed to increase more slowly or even
decrease with radius (Clift et al., 2005). Haberman & Morton (1953) reported a
smooth transition in vertical velocity between the spherical and the
elliptical regime in fresh water. For $Re>$ 5000, bubbles normally take form
as spherical caps, but this region is not relevant for the present study.

The terminal velocity $v_{T}$ of a bubble can be found theoretically by
balancing the drag and the buoyancy force in the vertical direction. For a
comparison with the observed bubble terminal velocity presented next, the
analytical relation for $C_{D}(Re)$ of Batchelor (2000) is used, i.e.
$C_{D}=\frac{48}{Re}$. This gives a drag force $D_{y}=12\mu_{w}\pi Rv_{T}$.
The force balance is rearranged and solved for the terminal velocity

$v_{T}=\frac{(\rho_{w}-\rho_{b})gR^{2}}{9\mu_{w}}.$ (4)

The relation between the size and the rising velocity of bubbles produced by
the perforated hose has been investigated in the Hydrodynamical Laboratory at
the University of Oslo. The experiments were performed in a small glass tank
measuring 1.5 m long and 0.4 m wide with water depth $h=$ 27 cm. The
perforated hose was placed on the bottom, parallel to the long wall. In this
experiment, the motion of the rising bubbles was of interest, and not the ROV-
PV system as a whole. Therefore, the ROV camera was substituted with a Photron
FASTCAM high speed camera for increased accuracy. Fresh water was used in the
first case and salt water with salinity equal to typical Arctic conditions was
used in the second case. A LED lamp placed on the back side of the tank
illuminated the bubbles. An integrated laboratory Kaeser compressor produced a
stable airflow to the perforated hose. A schematic of the experimental setup
is shown in Fig. 2. The water temperature was 18∘C. Other fluid parameters are
summarized in Table 1.

Parameter | Fresh water ($f$) | Salt water ($s$)
---|---|---
Density $[kg/m^{3}]$ | $\rho_{w}=998$ | $\rho_{w}=1026$
Dynamic viscosity $[Pa$ $s]$ | $\mu_{w}=1.027\times 10^{-3}$ | $\mu_{w}=1.103\times 10^{-3}$
Table 1: Parameters used in the experiments. These values are used for
calculating the theoretical terminal velocity in both fresh and salt water.
Figure 2:  Schematic of the experimental setup to determine the terminal
velocity of rising bubbles in stagnant fluids with $(x,y)$ axis aligned
horizontally and vertically, respectively. The bubble motion was recorded with
a high speed camera and the bubble velocity and size were found from image
processing techniques.

The image resolution was 1024$\times$1024 pixels. The frame rate and shutter
speed were set to 500 frames/s and 1/9000 s, respectively. The camera was
located outside the tank, 15 cm from the tank wall. The FOV for the validation
was 40.1$\times$40.1 mm in fresh water and 48.8$\times$48.8 mm in salt water.
To be certain that the terminal velocity was reached, the center of FOV was
placed 18 cm above the tank bottom. The vertical displacement of a bubble
between two frames varied 5$-$20 pixels, depending on the bubble speed.

Throughout the experiments, bubbles interacted with each other and in some
situations, the tracking algorithm mistook two bubbles for one. These bubbles
were filtered out. Additionally, bubbles which entered an elliptical regime
were removed in order to compare the results with theoretical solutions for
spherical bubbles. Elliptical bubbles and misinterpretations were filtered out
by applying a ratio parameter $\beta=\frac{\text{vertical
radius}}{\text{horizontal radius}}$. $\beta$ contains information on the
bubble shape, e.g. $\beta\approx$ 1 implies a spherical shape and $\beta<$ 1
implies a horizontal elliptic shape. Table 2 lists the threshold values of
$\beta$ used in the validation. The horizontal and vertical bubble diameters
were determined with 4 pixel accuracy, which corresponds to an error up to
approximately 14%.

The instantaneous horizontal ($u_{i}$) and vertical ($v_{i}$) bubble
velocities were calculated from the position ($x_{i},y_{i}$) for each frame
$i$ a bubble was tracked. Here, $i=1,2,...,N-1$, where $N$ is the number of
frames, typically 50$-$200 depending on the bubble speed. The minimum, mean
and maximum horizontal and vertical velocity were found for each bubble. The
standard deviation of vertical velocity ($\sigma_{v}$) and horizontal radius
($\sigma_{r}$) for each bubble were used as indicators for consistency, and
bubbles which did not satisfy the quality parameters listed in Table 2 were
not included in the analysis. The bubbles included in the results were very
consistent with $\sigma_{v}$ and $\sigma_{r}$ both in the order of $10^{-3}$
in fresh and salt water. This indicates that the vertical velocity was quite
constant and that the terminal velocity was reached.

Parameter | Fresh water | Salt water
---|---|---
$Ratio$ | $0.6\leq\beta\leq 1.0$ | $0.6\leq\beta\leq 1.0$
$\sigma_{r}$ $[mm]$ | $\sigma_{r}\leq 0.012$ | $\sigma_{r}\leq 0.015$
$\sigma_{v}$ $[ms^{-1}]$ | $\sigma_{v}\leq 0.007$ | $\sigma_{v}\leq 0.01$
Table 2: Threshold parameters applied when processing the data from the bubble
terminal velocity experiments.

Figure 3 presents the tracked terminal velocity of the rising bubbles (black
dots) in fresh water (left panel) and salt water (right panel). A logarithmic
increase in terminal velocity with respect to the bubble diameter can be
observed. A logarithmic curve fit has been applied to the terminal velocity in
both stagnant fluids by means of linear least squares (red line). Tracked
maximum and minimum absolute horizontal velocities are displayed with purple
and blue dots, respectively. There is a slight increase in maximum absolute
horizontal velocity with respect to bubble diameter in the case of fresh
water. This increase is more prominent in salt water. The theoretical terminal
velocities calculated from Eq. (4) with values listed in Table 1 and
$\rho_{b}=$ 1.2 kg/m3 are displayed as blue lines.

Figure 3: Measured bubble terminal velocity vs diameter (black dots) from
experiments in fresh water (left panel) and salt water (right panel) at 18∘C.
Only bubbles satisfying the quality criteria in Table 2 are included. The
theoretical model of Batchelor (2000) (blue line) is a fair estimate up to
$Re\approx$ 300 for fresh water and $Re\approx$ 500 for salt water. Reynolds
numbers of the tracked bubbles are indicated. The experimental results of
Aybers & Tapucu (1969) (Fig. 15, curve I) are included for comparison with the
fresh water experiment (black circles) and agree well with the current
observations. A logarithmic curve fit has been applied to the tracked terminal
velocity (red line), and is used in Section 3 as a relation between bubble
terminal velocity and diameter. Blue dots show minimum and purple dots show
maximum tracked absolute horizontal velocity.

Aybers & Tapucu (1969) conducted a similar study on the terminal velocity of
rising bubbles in stagnant water. Their results (Fig. 15, curve I), included
in Fig. 3 (left panel) as black circles for comparison, are in agreement with
the present findings. Aybers & Tapucu (1969) found that the bubbles entered
the ellipsoidal regime for $Re\approx$ 530, where the terminal velocity
started to decrease. Whether the present bubbles enter the ellipsoidal regime
in fresh water is not clear from visual inspections, the change in regime may
occur at a higher Reynolds number than investigated here. The salt water
bubbles possibly enter the ellipsoidal regime above $Re\approx$ 540, where we
observe the terminal velocity to flatten out. The theoretical terminal
velocity from Eq. (4) is a fair estimate up to $Re\approx$ 300 for fresh water
and $Re\approx$ 500 for salt water, although the curve shape is quite
different, most likely because the bubbles investigated are in the transition
between the spherical and ellipsoidal regime.

##  3 Validation in a wave tank

The ROV-PV system was evaluated under propagating periodic surface waves. A
comparison to theoretical solutions to determine the accuracy of the
measurements follows. The experiments were carried out in a wave tank in the
Hydrodynamical Laboratory at the University of Oslo.

Plane progressive waves are characterized by the acceleration of gravity $g$,
the angular wave frequency $\omega$, the amplitude $a$, the wave number $k$
and the corresponding wavelength $\lambda$, where $\omega=2\pi f$ and
$\lambda=2\pi/k$. In the case of $h>\frac{1}{2}\lambda$, the effects of the
bottom are negligible. Periodic waves with steepness $ak\ll$ 1 can be well
approximated by third order Stokes theory, where the velocity potential $\phi$
is given by

$\phi=\frac{ag}{\omega}e^{ky}sin(kx-\omega t)+O(a^{4}),$ (5)

where $x$ denotes the horizontal axis pointing in the direction of wave
propagation and $y$ the vertical axis pointing upwards with $y=$ 0 as the
undisturbed free surface (Newman, 2018). Equation (5) is accurate up to and
including terms of $O(a^{3})$ as long as the second order dispersion relation

$\omega^{2}=gk(1+a^{2}k^{2})+O(a^{3}k^{3}),$ (6)

which describes the relation between $\omega$ and $k$, is applied (Newman,
2018). From Eq. (5), the following terms for horizontal and vertical velocity
components, respectively, are obtained up to and including third order

$u_{w}=\frac{\partial\phi}{\partial x}=\frac{gak}{\omega}e^{ky}cos(kx-\omega
t),$ (7)

$v_{w}=\frac{\partial\phi}{\partial y}=\frac{gak}{\omega}e^{ky}sin(kx-\omega
t).$ (8)

Following Jensen et al. (2001), the velocities are nondimensionalized with
$gak/\omega$ for the comparison of waves with different amplitudes. The
velocities in vertical columns directly underneath the wave crests are
investigated in the further analysis. Here, the phase functions are $kx-\omega
t=2\pi n$, where $n$ is any integer. The nondimensional form of Eqs. (7)$-$(8)
evaluated under the wave crest reduce to $u_{w}/(gak/\omega)=e^{ky}$ and
$v_{w}/(gak/\omega)=0$, respectively. In the following, the non-zero
horizontal velocity component is investigated.

In Section 2.3, it was shown from the momentum equation that the slip velocity
is related to the bubble inertia due to added mass. Here, it is attempted to
estimate $u_{slip}$ by evaluating the momentum equation in the horizontal
direction. As a wave undergoes an entire period, the maximum horizontal water
acceleration up to and including third order terms is

$\left[\frac{\partial u_{w}}{\partial t}\right]_{max}=gake^{ky}.$ (9)

It is assumed that viscosity dominates the effect of inertia of the fluid in
the horizontal direction, due to the small bubble size and horizontal slip
velocity. Therefore, the drag force $D_{x}$ acting on the bubbles is
approximated with the Stokes drag, i.e. $D_{x}=6\rho_{w}\nu\pi Ru_{slip}$. By
applying the Stokes drag approximation, a linear relation between the drag
force and the slip velocity is obtained. It should therefore be reasonable to
perform a decomposition of the drag force into the vertical and horizontal
direction. Equation (9) and the drag force approximation are inserted into Eq.
(3), which is rearranged to estimate the slip velocity

$u_{slip}\approx\frac{R^{2}}{9\nu}\left[\frac{\partial u_{w}}{\partial
t}\right]_{max}=\frac{R^{2}}{9\nu}gake^{ky},$ (10)

where the buoyancy term is neglected, as we are only looking at the horizontal
component. Equation (7) and (10) are inserted into Eq. (2) to obtain an
estimate for the horizontal bubble velocity

$u_{b}\approx\frac{gak}{\omega}e^{ky}\left(1-\frac{\omega
R^{2}}{9\nu}\right).$ (11)

Figure 4:  Schematic drawing of the experimental setup in the wave tank. A
coordinate system is defined with the $(x,y)$ axes aligned horizontally in the
direction of wave propagation and vertically in upward direction,
respectively, with $y=$ 0 as the undisturbed free surface. The velocity under
the waves is estimated from the motion of the bubbles, which is recorded with
the ROV camera.

Figure 4 shows a schematic drawing of the experimental setup in the wave tank.
It measures 24.6 m long and 0.5 m wide and was filled with fresh water to a
depth $h=$ 0.6 m. Waves were generated in one end of the tank with a computer
controlled hydraulic piston wave maker. The perforated hose was centered on
the bottom of the tank in the span-wise direction 11 m from the wave maker.
Two powerful LED lamps were suspended from the tank walls and angled downwards
to illuminate the bubbles from the sides. The ROV was placed outside the tank
with the camera axis perpendicular to the bubble plane. The camera lens was
located 30 cm from the tank wall, meaning that the distance between camera and
bubbles was approximately 55 cm. The FOV in the tank center was approximately
80$\times$45 cm. Before the experiments were initiated, a Perspex plate marked
with reference coordinates was placed vertically on the same location as the
bubble plane and recorded with the camera. The reference coordinates, 88 in
total, had a horizontal resolution of 5 cm and a vertical resolution of 2 cm
close to the surface and 5 cm further down in the depth. Pixel coordinates
were converted to real world coordinates with a cubic transform (Kolaas,
2016). Fluid motion was analyzed within a section of 45 cm in the span-wise
direction, located approximately in the center of the camera FOV, and 30 cm in
the vertical direction from the undisturbed surface and downwards. This
section was well within the domain of the pixel-to-world transformation. The
distortion effect of the camera, which is most prominent in the outer edges,
was also reduced in the center of the FOV.

A series of about 70 monochromatic waves was generated in each run. After a
short transient build-up, the wave train became periodic and this is the part
included in the analysis. An absorbing beach damped the waves at the far end
of the tank and the analysis was terminated before any reflected waves reached
back to the FOV. Figure 5 shows the surface elevation $\eta$ in the position
of the ROV with respect to time for 1.4 Hz waves and three different
amplitudes. The 10 investigated periods are highlighted. Surface elevation at
the test section of the tank was measured with an array of three ULS Advanced
Ultrasonic wave Gauges (UGs) from Ultralab. The UG situated directly above the
ROV was mainly used, and the two others were applied as redundancy in case the
first one failed, which happened in a few runs. The UGs sampled at 250 Hz and
they have a technical resolution of 0.18 mm.

Figure 5:  Time series of surface elevation $\eta$ obtained with ultrasonic
gauges for 1.4 Hz monochromatic waves. The amplitude was approximately 7, 15
and 23 mm for top, middle and bottom panel, respectively. The shaded area
marks the 10 periods investigated, which were periodic, almost constant in
amplitude and unaffected by any reflected waves from the beach.

Wave amplitude $a$ from each run was determined as the mean value of
$\eta_{max}$ over the 10 periods investigated. Three different wave
frequencies $f=$ 1.4, 1.6 and 1.8 Hz, and three different amplitudes of
approximately 7, 15 and 23 mm were investigated. Each combination was repeated
three times, which means 27 runs in total. The wavenumber $k$ of each run was
determined from Eq. 6, which was used to find the wavelength $\lambda$. For
all frequencies, $\lambda$ (0.51$-$0.82 m) was comparable to $h$. Hence, the
waves were considered to be deep-water waves. In most runs, the wave steepness
$ak<$ 0.2. Only the two runs where the highest amplitude and the two highest
frequencies were combined yielded 0.2 $<ak<$ 0.24. Grue et al. (2003) showed
that the velocity profile beneath a wave crest is very close to Stokes third
order theory for $ak=$ 0.23, and still a good approximation for $ak<$ 0.30.
Therefore, it is reasonable to use Eq. (7) as an approximation of theoretical
$u_{w}$.

Only bubbles located in vertical columns (0.04$\lambda$ wide) directly below
the wave crests were considered. In the following results, $u_{b}$ of the
smallest tracked bubbles is presented in order to minimize the horizontal
oscillating motion (Haberman & Morton, 1953). The bubble radius was estimated
from the observed vertical velocity, based on the findings presented in
Section 2.3. The vertical velocities accepted were 0.15 $<v_{b}<$ 0.25 m/s,
which should correspond to bubble radius 0.33 $<R<$ 0.47 mm according to the
logarithmic curve fit applied in the left panel of Fig. 3. This range of
bubble radius is marginally smaller than the bubbles investigated in Section
2.3, most likely due to a slightly different input pressure on the perforated
hose. For the 1.4 Hz waves, each run contained on average 683 analyzed bubbles
which satisfied the quality criteria (small bubbles located directly below the
crest) during the 10 periods investigated. In the case of 1.4 Hz waves, the 10
investigated periods consisted of approximately 214 image frames, meaning that
3.2 relevant bubbles were analyzed on average per image frame.

Figure 6:  Horizontal velocity profiles divided into 10 equally spaced
vertical bins below wave crests. Wave frequency: 1.4, 1.6 and 1.8 Hz from
upper to lower row, respectively. Wave amplitude: approximately 7, 14 and 21
mm from left to right column, respectively. Blue dots: observed horizontal
bubble velocity $u_{b}$. Green error bars: mean and $2\sigma$ uncertainty of
the observed $u_{b}$ within each bin (at least 10 observations must be present
within the bin for the error bar to be displayed). Red line: theoretical
horizontal water velocity $u_{w}$ given by Eq. (7). Orange shaded region:
expected $u_{b}$ from estimated bubble size and Eq. (11). The observed mean
$u_{b}$ lies within the expected region for most of the vertical bins.

Figure 6 shows the horizontal velocity profiles below wave crests with
increasing frequencies from upper to lower panels and increasing amplitudes
from left to right panels. All three repetitions of the same frequency-
amplitude combination are included in each panel. The blue dots indicate
observed $u_{b}$. Each profile is divided into 10 equally spaced vertical
bins. A global filter was applied to remove outliers which were mostly
situated around $y=$ 0, probably caused by local surface tension effects or
from bubbles accumulating on the surface. Horizontal velocities which deviated
more than a certain threshold off the mean of the same bin were discarded. The
threshold was set to 70% of the total mean $u_{b}$ of all observed particles
in the whole vertical profile. For the 1.4 Hz waves, this filter removed on
average 2.9% of the observations. After filtering, the mean and two standard
deviations ($2\sigma$) of $u_{b}$ within each bin were calculated (green error
bars). Each bin must contain at least 10 tracked velocities for the error bar
to be shown. The theoretical fluid velocity given in Eq. (7) is displayed with
a red line. The expected bubble velocity was calculated from Eq. (11), with
the range of estimated bubble radius and the kinematic viscosity $\nu=$ 10-6
m2/s inserted, and displayed as an orange highlighted region.

From Fig. 6, it can be seen that the horizontal bubble velocity in general has
a smaller magnitude than the fluid velocity, but the exponential profile is
clearly visible. The simple error estimate given in Section 2.3 and Eq. (11)
seems to be a good approximation for the slip velocity, as the observed mean
lies within the expected bubble velocity for most of the vertical bins. The
theoretical bubble velocity deviates 10.6$-$21.6, 12.2$-$24.7 and 13.7$-$27.7%
from the theoretical fluid velocity for the 1.4, 1.6 and 1.8 Hz waves
respectively, depending on the bubble radius. Statistics from the second
vertical bin from the top of all the 1.4 Hz waves are displayed in Table 3.
This bin was chosen because the highest bin in terms of y-position and
velocity does not contain enough tracked bubbles in the case of the highest
wave amplitude. Here, the mean observed $u_{b}$ deviates with 1.3$-$6.5% from
the expected $u_{b}$ of the average estimated bubble radius. The spread in the
observed data, exemplified through $\sigma$ in Table 3, slightly increases
with increasing fluid velocity (i.e. higher wave amplitude). However, the
relative standard deviation of the observations, that is $\sigma$ divided by
mean observed $u_{b}$ in the respective bin, decreases from 22.4% to 10.9%
from the lowest to the highest amplitude for the 1.4 Hz waves. Deeming from
Fig. 6, the relative standard deviation of the observed velocities decrease in
general with increasing fluid velocity (i.e. higher wave amplitude and/or
frequency). The mean observed $u_{b}$ is 11.7$-$16.0% smaller than theoretical
$u_{w}$.

$a$ [mm] | Mean obs. $u_{b}$ [m/s] |  $\sigma$ obs. $u_{b}$ [m/s] | PE $u_{w}$ [%] | PE $u_{b}$ [%]
---|---|---|---|---
7 | 0.038 | 0.008 | -13.9 | 3.8
15 | 0.079 | 0.010 | -11.7 | 6.5
23 | 0.115 | 0.013 | -16.0 | 1.3
Table 3: Statistics of observed $u_{b}$ for 1.4 Hz waves in the second
vertical bin from the top. The fourth column from the left is the percentage
error between observed $u_{b}$ and theoretical $u_{w}$ (from Eq. 7) and the
fifth column is the percentage error between observed and expected $u_{b}$
(from Eq. 11).

The velocity profiles were nondimensionalized and the different amplitudes
with the same frequency are plotted together with increasing frequency from
left to right panel in Fig. 7. Outliers were filtered in the same manner as in
Fig. 6. Here, the mean and error bar is presented if a bin contains at least
30 tracked bubbles. A clear collapse of the different amplitudes can be seen
for each frequency. The mean of the observed $u_{b}$ is located within the
expected region for most vertical bins. The wavenumbers from the different
amplitudes are averaged to show theoretical $u_{w}$ and expected $u_{b}$ for
each frequency. Visually, the spreading of tracked horizontal velocities seems
to decrease with increasing frequency. This observation is consistent with the
decreasing relative standard deviation for higher velocities, which was the
case for the dimensional graphs in Fig. 6.

Figure 7: Horizontal non-dimensional velocity profiles divided into 10 equally
spaced vertical bins below wave crests. Wave frequency: 1.4, 1.6 and 1.8 Hz
from left to right panel respectively. Blue dots: observed horizontal bubble
velocity $u_{b}$. Green error bars: mean and $2\sigma$ uncertainty of the
observed $u_{b}$ within each bin (at least 30 observations must be present
within the bin for the error bar to be displayed). Red line: $e^{ky}$. Orange
shaded region: expected $u_{b}$ from estimated bubble size and Eq. (11). The
observed mean $u_{b}$ lies within the expected region for most of the vertical
bins.

##  4 Field deployment

The ROV-PV system was tested close to an ice floe in the North-West Barents
Sea on April 26, 2019. The objectives of the field experiment were to test the
setup under Arctic conditions and to investigate the flow around floating ice
subjected to wave motion. The site was near Hopen Island, which is a part of
the Svalbard Archipelago shown in Fig. 8, and the geographical coordinates
were 76.18∘N, 25.77∘E (indicated by the red dot). The setup was lowered down
from a beam by a pulley system into the water next to the floe, which had a
diameter of approximately 20 m. Figure 9 shows the ROV positioned in front of
the grid (left panel) and the bubble plane seen from the ROV (right panel).
The bubbles were illuminated by both ambient light and the ROV headlights.

Figure 8:  A map of the area where the field work was carried out. The red dot
indicates the location of the ice floe. Source: TopoSvalbard (2020).

Figure 9:  The ROV located in front of the bubble plane and the coordinate
grid, which were lowered into the water by the pulley system, seen from above
(left panel) and from the ROV perspective (right panel). A spirit level was
used to align the coordinate grid with the horizontal.

Due to the dense bubble plane and the relatively long distance from the ROV to
the bubbles, it proved difficult to track single bubbles between frames
obtained next to the ice floe. Therefore, PIV was applied to obtain a 2D
velocity field. The PIV processing was performed with subwindows of
48$\times$48 pixels and 75$\%$ overlap. A search range of 1/3 of the subwindow
was used. This configuration yielded approximately 20$\times$16 2D velocity
vectors inside the bubble plane. Each velocity vector can be decomposed into
$u_{b}$ and $v_{b}$, i.e. a horizontal and a vertical component, respectively.

In the attempt of finding $v_{w}$, two different approaches have been tested
to compensate for the vertical buoyancy driven velocity component of the
bubbles; by subtracting either the most frequently observed terminal velocity
in the laboratory experiment with salt water, or the mean vertical velocity
observed in the field. In the first case, the velocities observed in the right
panel of Fig. 3 are distributed into bins with 0.02 m/s resolution. The mode
of the vertical velocity, $\overline{v_{b,lab}}=$ 0.37 m/s, is subtracted from
$v_{b}$, i.e. $v_{w}=v_{b}-\overline{v_{b,lab}}$. In the second case, a
reference image couple from the field experiment where the horizontal velocity
component is very small, is chosen. From the PIV analysis of this image
couple, the mean velocity components $\overline{u_{b,field}}=$ 0.0058 m/s and
$\overline{v_{b,field}}=$ 0.15 m/s are found by spatially averaging the
horizontal and vertical velocity components, respectively. Then,
$\overline{v_{b,field}}$ from the reference image couple is subtracted from
all $v_{b}$ in the image couple that is analyzed, i.e.
$v_{w}=v_{b}-\overline{v_{b,field}}$. In the latter case, it is assumed that
$v_{w}$ is small at the time of the reference image couple, since the observed
$u_{w}$ is small.

An example from an instantaneous velocity field obtained next to the ice floe
is presented in Figure 10. Measured bubble velocities are distributed into
bins with 0.02 m/s resolution and displayed (blue) in the left ($u_{b}$) and
center ($v_{b}$) panels. The probability distribution for the vertical water
velocity obtained from the two different approaches, i.e.
$v_{b}-\overline{v_{b,field}}$ (orange) and $v_{b}-\overline{v_{b,lab}}$
(yellow), are shown in the center panel. The same procedure is followed in the
horizontal direction in the left panel, although this is only for illustrative
purpose, as we assume that $u_{b}\approx u_{w}$. The two different approaches
yield quite different results in the vertical direction. The terminal velocity
observed in the laboratory was in general more than twice as high as in the
field. A possible explanation for this deviation is the temperature
difference, which may have changed the flexibility of the perforated hose. For
example, the diameter of the holes where the bubbles escape may decrease in
lower temperature, or they could be partly clogged by ice. The most realistic
results are obtained with $v_{b}-\overline{v_{b,field}}$ because the
conditions in the field differ so much from the laboratory. Additionally, this
approach gives similar water velocity magnitudes in the horizontal and
vertical direction.

Figure 10:  Example of an instantaneous velocity field obtained next to the
ice floe. Probability distributions of the tracked bubble velocity (blue), and
estimates for the water velocity from laboratory (yellow) and reference field
(orange) experiments in the horizontal (left panel) and vertical (center
panel) direction. Quiver plot estimated from field reference experiment (right
panel).

A quiver plot is presented in the right panel of Fig. 10. It is produced with
the latter approach, i.e. subtraction of the mean reference velocity from the
field, in both horizontal and vertical direction. The undisturbed ocean
surface is set to $y=$ 0 and the ice floe is located at a negative x-value.
The direction of the water flow is towards the ice floe. In fact, it was
observed from the ROV images that the horizontal velocity oscillated towards
and away from the ice floe with periods in the order of 10 s. This motion is
either direct wave motion or flow generated from the heaving ice floe. There
may also exist a circulating structure covering about 30% of the right side of
the quiver plot, which could be a turbulent eddy generated by the ice floe.

##  5 Discussion

In order to determine the accuracy of the introduced ROV-PV system, potential
sources of error must be identified and quantified. Fluid velocity is
estimated from the motion of bubble tracers by identifying the bubble position
with an image processing technique. Two central questions that need to be
addressed are how accurately the bubble position can be determined and how
well a particle follows the fluid motion. According to Dalziel (1992), the
position of particles spanning over one pixel is usually determined with pixel
accuracy, while the uncertainty decreases for particles spanning over several
pixels. In the wave tank experiments, the bubbles typically covered at least a
couple of pixels. Therefore, one pixel is used as a conservative accuracy
estimate of the bubble position, which corresponds to approximately 0.4 mm in
real world coordinates. The bubble velocity is determined from a central
differencing scheme, i.e. over three image frames. With a typical vertical
velocity of 0.25 m/s, a bubble travels 0.025 m during three frames. This gives
1.6% relative error in velocity.

When it comes to the passivity of the particles or their capability of
following the motion of the surrounding fluid, it has been shown that the
horizontal bubble velocity lags behind the fluid velocity under periodic
waves. This effect is attributed to the fluid acceleration and the bubble
inertia due to the effect of added mass. For the 1.4 Hz waves, the
discrepancies between observed bubble velocity and theoretical water velocity
were 11.7$-$16%. The relative error in velocity caused by the bubble inertia
increases with the bubble radius and is theoretically found to be 10.6$-$21.6%
for the bubbles investigated under the 1.4 Hz waves. Hence, the error due to
the bubble inertia is an order of magnitude larger than the error due to the
image processing technique. The estimated horizontal slip velocity agrees with
the observations. Note that the observed horizontal slip velocity yields
Reynolds numbers of unity order of magnitude, which suggests that Stokes drag
is a reasonable approximation.

A considerable spread in the observed $u_{b}$ under the wave crests can be
seen from Figs. 6-7, where $\sigma\approx$ 0.01 m/s. This spread is partly
related to the varying bubble size and therefore varying slip velocity, as
indicated by the orange shaded regions in the figures, but also due to
secondary motion, typically oscillation of rising bubbles. The perforated hose
was designed to produce as small bubbles as possible to obtain rectilinear
trajectories. Although this was achieved to a certain extent, Fig. 3 shows
maximum absolute horizontal velocity values of approximately 0.03 m/s for
terminal velocity around 0.25 m/s ($Re\approx$ 300). This observation is in
agreement with Clift et al. (2005), who described onset of secondary motion
from $Re=$ 200$-$1000, where the typical horizontal component is 5% of the
vertical component. Oscillations are most likely caused by vortex shedding.
The magnitude of the horizontal velocity component of a rising bubble can be
considered analogous to the oscillating part of surface waves, see e.g. Eq.
(7). Its magnitude will depend on y-position and time. For bubbles of the same
size, we can expect a uniform distribution in the horizontal velocity
component due to oscillations. Since the bubble size varies, the distribution
should be denser around the mean, which appears to be the case by visual
inspections of Fig. 7.

There is a large uncertainty associated with measurements of the vertical
water velocity component since the bubbles are naturally buoyant. Bubbles
rising in calm water and vertical columns directly below wave crests have been
investigated in laboratory experiments. In both cases, the vertical water
velocity is known to be zero, so that the vertical bubble velocity equals the
vertical slip velocity. However, such controlled environment is seldom
obtained in the field. For accurate measurements of the vertical velocity
component, it is necessary to generate uniform bubbles in size. In this way,
the terminal velocity due to the buoyancy can be found in advance and assumed
within reasonable accuracy to be constant for all bubbles. The vertical fluid
velocity can then be found from Eq. (2) in the case of steady flow, or from a
combination of Eq. (2)-(3) in the case of accelerated flow.

The current ROV-PV system with bubbles as tracing particles is suitable for
measuring flows where the horizontal velocity component is of interest.
Examples of such situations are waves propagating under an ice cover and
horizontal shear flow in the boundary layer beneath an ice cover. As discussed
in the previous paragraph, a uniform bubble size is necessary to accurately
measure the vertical velocity component. Further effort must be spent in order
to improve the perforated hose to overcome this challenge in future studies.
At the same time, focus should be directed towards keeping the construction
simple in use and robust for field conditions. Hydrogen bubbles generated from
hydrolysis could be an option as these tend to be small (diameter around 0.1
mm), although Weier & Landgraf (2013) reported a distribution in bubble size
due to coalescence phenomena. On the other hand, bubbles cannot be identified
by the particle tracking software if they are too small. PIV might be suitable
in this case.

##  6 Conclusions

In this paper, a new method for measuring 2D velocity fields in the upper
ocean by utilizing image processing technology and air bubbles as tracing
particles has been presented. The novelty of this method is the combination of
bubbles and an ROV, which gives a simple and lightweight setup which is
suitable for field measurements in the polar regions. The ROV-PV system has
been demonstrated to measure the flow in the vicinity of an ice floe during an
Arctic field campaign. There is a need for detailed ocean kinematics in the
marginal ice zone, as a lot of important atmosphere-ice-ocean energy transfer
processes occur here and relatively few in situ observations of 2D velocity
fields exist. The introduced technique could be suitable for visualization and
quantification of many interesting flow phenomena in the Arctic, for example
wave propagation underneath various ice layers or turbulent structures induced
by colliding ice floes. Detailed observations within this field are important
for an improved understanding of the underlying physics and the design and
validation of numerical sea ice models.

Detailed laboratory experiments have been carried out to quantify the relation
between diameter and terminal velocity of the generated bubbles. Although many
approximations and models exist for this relation, the present results
demonstrate the importance of a thorough investigation of bubble properties
since substantial deviations to models may occur. The ROV-PV system has been
utilized to measure horizontal velocities under periodic water waves with an
accuracy in the order of 10%. The deviations are mainly attributed to the slip
velocity caused by the bubble inertia due to the added mass, and is important
to consider when measuring accelerated flow. In addition, there is a spread in
the data expressed through a relative standard deviation in the order of 10%,
due to the bubble size distribution and the oscillatory motion of rising
bubbles. Future studies should focus on generating smaller and more uniform
bubbles, which could decrease the spread in observed velocities and improve
the reliability in measurements of the vertical velocity component.

## Acknowledgement

The authors are grateful to Aleksey Marchenko for inviting us to the Hopen
cruise. We also thank the crew of Polarsyssel for their assistance. 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.

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# Adversarial Defense by Suppressing High-frequency Components

Zhendong Zhang  Cheolkon Jung&Xiaolong Liang
Xidian Media Lab, Xidian University, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recent works show that deep neural networks trained on image classification
dataset bias towards textures. Those models are easily fooled by applying
small high-frequency perturbations to clean images. In this paper, we learn
robust image classification models by removing high-frequency components.
Specifically, we develop a differentiable high-frequency suppression module
based on discrete Fourier transform (DFT). Combining with adversarial
training, we won the 5th place in the IJCAI-2019 Alibaba Adversarial AI
Challenge. Our code is available online.

##  1 Introduction

Deep neural networks (DNNs) have achieved state-of-the-art performances on
many tasks, such as image classifications. However, DNNs have been shown to be
vulnerable to adversarial attacks Szegedy et al. (2014) Goodfellow et al.
(2015). Adversarial attacks are carefully designed small perturbations to
clean data which significantly change the predictions of target models. The
lack of robustness w.r.t adversarial attacks of DNNs brings out security
concerns.

In this paper, we focus on defending DNNs from adversarial attacks for image
classifications. Many algorithms have been proposed to achieve this purpose.
Roughly, those algorithms fall into three categories:

  * •

data preprocessing, such as JPEG compression Das et al. (2017) and image
denoise Xu et al. (2018).

  * •

adding stochastic components into DNNs to hide gradient information Athalye et
al. (2018).

  * •

adversarial training Madry et al. (2018).

Data preprocessing or stochastic components are usually combined with
adversarial training since it is the most successful defense algorithm.

Recent works show that deep neural networks trained on image classification
dataset bias towards textures which are the high-frequency components of
images Geirhos et al. (2019). Meanwhile, researchers empirically find that the
perturbations generated by adversarial attacks are also high-frequency
signals. This means DNNs are mainly fooled by carefully designed textures.
Those facts suggest that suppressing high-frequency components of images is
helpful to reduce the effects of adversarial attacks and improve the
robustness of DNNs. On the other hand, the basic information on clean images
will be retained when suppression high-frequency components because it
converges on low frequencies. In this paper, we aim to develop a high-
frequency suppressing module which is expected to have the following
properties:

  1. 1.

separability: it should suppress high-frequency components while keep low-
frequency ones.

  2. 2.

efficiency: it should have low computational costs compared with the standard
DNNs.

  3. 3.

differentiability: it should be differentiable which allows to jointly
optimize with adversarial training.

  4. 4.

controllability: it should be easy to control the degree of high-frequency
suppression and the degree of how the original images are modified (e.g.
$L_{2}$ distance).

Discrete Fourier transform (DFT) which maps images into frequency domain is a
good tool to achieve those goals. Based on (inverse) DFT, we propose a high-
frequency suppressing module which has all those properties. We evaluate our
method in the IJCAI-2019 Alibaba Adversarial AI Challenge AAAC . Our code is
available on https://github.com/zzd1992/Adversarial-Defense-by-Suppressing-
High-Frequencies.

(a) AAAC

(b) CIFAR-10

Figure 1: Cumulative spectrum energy for $5,000$ images of AAAC in (a) and for
test images of CIFAR-10 in (b). Blue line for clean images and orange line for
adversarial perturbations.

##  2 Method

###  2.1 High-frequency suppression

As mentioned earlier, suppressing high-frequency components is helpful to
reduce the effects of adversarial attacks and improve the robustness of DNNs.
Given an input image, we transform it into frequency domain via DFT. Then we
reduce the high-frequency components in frequency domain. Finally, we
transform the modified frequency image back to time domain.

Formally, denote $\mathbf{x}\in\mathcal{R}^{M\times N}$ as the input image and
$\mathbf{\hat{x}}\in\mathcal{C}^{M\times N}$ as its frequency representation.

$\mathbf{\hat{x}}_{u,v}=\sum_{a=0}^{M-1}\sum_{b=0}^{N-1}\mathbf{x}_{a,b}e^{-j2\pi\left(\frac{u}{M}a+\frac{v}{N}b\right)}$
(1)

To suppress the high-frequency components, we modify $\mathbf{\hat{x}}$ as
follows:

$\mathbf{\hat{x}}\leftarrow\mathcal{M}\odot\mathbf{\hat{x}}$ (2)

where $\mathcal{M}\in\mathcal{R}^{M\times N}$ and $\odot$ is element-wise
multiplication. $\mathcal{M}$ controls how different frequency is scaled.
Intuitively, $\mathcal{M}$ should close to $0$ for high-frequency components
and close to $1$ for low-frequency ones. In this paper, we set $\mathcal{M}$
to a box window with fixed radius $r$. That is

$\mathcal{M}_{u,v}=\left\\{\begin{array}[]{lc}1,&0<=|u|,|v|<=r\\\
0,&else\end{array}\right.$ (3)

To simplify the notation, we set
$\mathcal{M}_{-u,\cdot}=\mathcal{M}_{M-u,\cdot}$ and
$\mathcal{M}_{\cdot,-v}=\mathcal{M}_{\cdot,N-v}$. The overall function of our
high-frequency suppression module is

$\mathbf{x}\leftarrow\mathcal{F}^{-1}\left(\mathcal{M}\odot\mathcal{F}(\mathbf{x})\right)$
(4)

where $\mathcal{F}$ means DFT. An image is processed by this module and a
standard DNN in order.

Now we analyze the properties of our proposed module.

separability: because $\mathcal{M}$ is a box window, high-frequency components
are completely removed and low-frequency ones are perfectly reserved.

efficiency: the computational costs are dominated by DFT. For an $M\times N$
(we suppose $M>=N$) image, the time complexity of DFT is
$\mathcal{O}(MN\log_{2}M)$. In practice, DFT of a color image is faster than a
convolutional layer. Thus the costs of our proposed module are cheap compared
with DNNs.

differentiability: DFT can be expressed in matrix form:

$\mathcal{F}(\mathbf{x})=\mathbf{F_{M}xF_{N}}$ (5)

where $\mathbf{F}_{M}\in\mathcal{C}^{M\times M}$ is the so-called Fourier
transform matrix. Clearly, DFT is differentiable. Instead of an image pre-
processing method, this property makes it possible to integrate our module
into DNNs and optimize with adversarial training jointly.

controllability: denote $\mathbf{x}_{o}$ as the output of the proposed module.
Based on Parseval theory, we have

$\lVert\mathbf{x}-\mathbf{x}_{o}\rVert_{2}^{2}=\lVert\mathbf{\hat{x}}-\mathcal{M}\odot\mathbf{\hat{x}}\rVert_{2}^{2}$
(6)

Thus the degree of high-frequency suppression and the $L_{2}$ norm between the
original image and the modified image are easily controlled by varying $r$ of
the box window. For nature images, spectral energy is converged on low-
frequency regions. Thus $\lVert\mathbf{x}-\mathbf{x}_{o}\rVert_{2}$ is small
enough even when most of the frequency components are suppressed ($r$ is
small).

###  2.2 Adversarial training

The idea of adversarial training is optimizing DNNs w.r.t both clean samples
and adversarial samples.

$\min_{\mathbf{w}}\left\\{\mathcal{L}(f_{\mathbf{w}}(\mathbf{x}),y)+\beta\max_{\lVert\mathbf{\delta}\rVert<\epsilon}\mathcal{L}(f_{\mathbf{w}}(\mathbf{x+\delta}),y)\right\\}$
(7)

where $f$ maps an image into classification probability, $\mathbf{w}$ is the
parameters of $f$ and $\mathcal{L}$ is the cross-entropy loss.
$\mathbf{\delta}$ is obtained by (iteratively) projected gradient descent
(PGD). $\beta$ controls the tradeoff between clean samples and adversarial
samples.

Recently, Zhang et al. (2019) propose a novel adversarial training method
called TRADES. TRADES is formalized as follows:

$\min_{\mathbf{w}}\left\\{\mathcal{L}(f_{\mathbf{w}}(\mathbf{x}),y)+\beta\max_{\lVert\mathbf{\delta}\rVert<\epsilon}\mathcal{L}(f_{\mathbf{w}}(\mathbf{x}),f_{\mathbf{w}}(\mathbf{x+\delta}))\right\\}$
(8)

Instead of minimizing the difference between
$f_{\mathbf{w}}(\mathbf{x+\delta})$ and the true label, TRADES minimizes the
difference between $f_{\mathbf{w}}(\mathbf{x+\delta})$ and
$f_{\mathbf{w}}(\mathbf{x})$ which encourages the output to be smooth. In this
paper, we use TRADES as the adversarial training method because it has a
better tradeoff between robustness and accuracy. Refer Zhang et al. (2019) for
more information.

High-frequency suppression | Adversarial training | Model ensemble | Score
---|---|---|---
$\times$ | $\times$ | $\times$ | 2.0350
$\times$ | $\surd$ | $\times$ | 9.9880
$\surd$ | $\times$ | $\times$ | 14.9736
$\surd$ | $\surd$ | $\times$ | 19.0510
$\surd$ | $\surd$ | $\surd$ | 19.7531
Table 1: Ablation study for three strategies and their combinations.  Figure
2: We show the trade-off between robustness and accuracy with high-frequency
suppression modules whose $r$ are different. Robustness is measured by the
score of AAAC.

##  3 Experiments

We first analyze the statistics of clean images and adversarial images in
frequency domain. Then we evaluate the proposed method in the IJCAI-2019
Alibaba Adversarial AI Challenge (AAAC).

In AAAC, models are evaluated by image classification task for electric
business. There are about $11,000$ color images from $110$ classes for
training. There are $550$ images for test. Given an image, the score of a
defense model is calculated as follows:

$score=\left\\{\begin{array}[]{cl}0,&P_{y}\neq y\\\
mean(\mathbf{\lVert\delta}\rVert_{2}),&P_{y}=y\end{array}\right.$ (9)

where $P_{y}$ is the predicted label. The final score is averaged over all
images and all black-box attack models. Note that before computing the score,
images are resized to $299\times 299$.

We use ResNet-18 as the DNN architecture for all experiments. Our method is
implemented with PyTorch.

###  3.1 Statistics in frequency domain

We analyze the statistics of clean samples and adversarial samples in
frequency domain. Specifically, we study the distributions of cumulative
spectrum energy (CSE) w.r.t frequency. Given a 2D signal
$\mathbf{\hat{x}}\in\mathcal{C}^{M\times N}$ in frequecy domain, we define CSE
as follows:

$CSE(r)=\sum_{i=-r}^{r}\sum_{j=-r}^{r}\mathbf{\hat{x}}_{i,j}^{*}\mathbf{\hat{x}}_{i,j}$
(10)

where $r<=\frac{min(M,N)}{2}$. We randomly select $5,000$ images from AAAC. We
calculate the CSE score of each image and average all scores. We also
calculate the averaged CSE score for the corresponding adversarial
perturbations which are generated by iteratively PGD. The results are shown in
Fig. 1(a). As we can see, CSE for clean images converges on low-frequency
regions while CSE for adversarial perturbations is nearly uniform. Thus, when
we suppress the high-frequency components, the effects of adversarial attacks
will be significantly reduced while most of the information on clean images
will be retained. This is the main motivation of our work.

We calculate CSE score for CIFAR-10, as shown in Fig. 1(b). The distribution
is similar to AAAC’s.

###  3.2 AAAC results

As analyzed earlier, when we remove the high-frequency components, the model
will be more robust w.r.t adversarial attacks while the accuracy on clean
images will be decreased. We evaluate this phenomenon with different $r$
_without_ adversarial training. The accuracy is obtained on $5,000$ validation
clean images and the robustness is measured by the score of AAAC. We show the
results in Fig. 2. As $r$ decreased, the robustness w.r.t adversarial attacks
is substantially increased.

Then we do ablation study for three strategies and their combinations: 1) the
proposed high-frequency suppression module; 2) adversarial training via
TRADES; 3) ensembles of models with different $r$. As we can see in Tab. 1,
our proposed module is even better than adversarial training in this challenge
and those two methods are complementary to each other. The best score is
obtained by ensembling models with different $r$ each of which is trained
together with the proposed module and adversarial training. We secure the 5th
place in this challenge (the score for the 1st solution is $20.13$).

##  4 Conclusions and discussions

Motived by the difference of frequency spectrum distributions between clean
images and adversarial perturbations, we have proposed a high-frequency
suppression module to improve the robustness of DNNs. This module is
efficient, differentiable and easy to control. We have evaluated our method in
AAAC.

We list several directions or questions which are worth to be further
explored:

  * •

Is it helpful to change the radius of box window dynamically?

  * •

Is it helpful to suppress the high-frequency components of intermediate
convolutional features?

  * •

We evaluate our method for image classification. Does this method work for
other tasks or other kinds of data, such as speech recognition?

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