Estimation of Ocean Wave Spectra from Radar
Aya Akahori, Emma Thomas, Kaede Watanabe, Cameron Watt
Academic Mentor: Toshiaki Yachimura
Industry Mentors: Takahashi Ryuhei, Kameda Hiroshi
g-RIPS Sendai

August 7th, 2023

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
AcademicWave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
1 / 39Ryuh

Table of Contents

1

Motivation2

Physical Background

3

Goal

4

Current Progress

5Additional Optimization

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
2 / 39RyuhMotivation

Understanding sea surface conditions is necessary for marinetransportation.
Observations of sea surface conditions can allow for safer boat
navigation and warning of extreme weather events.
In addition, sea surface conditions can help with identifying oil spills
and debris.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustryAugust 7th,Mentors:
2023 Takahashi
3 / 39Ryuh

Radar Range-Direction Grids

Radar Range-Direction Grids

The width of the direction and range separations is determined by theradar’s azimuth and range resolutions, which is approximately 15 deg
and 1500m. The total azimuth is 120 deg and the total range isaround 60 km.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
4 / 39Ryuh

Sea Surface Motion
Ocean waves on the sea’s surface can be broken down in component
waves, each with their own frequency and direction. These canrepresented by wavevectors.

Decomposition of Random Sea Surface Motions [1]Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
5 / 39Ryuh

First Order Bragg Scattering
When the radio wavelength is half of an ocean wavelength (aka theradio wave number is twice the ocean wave number), the constructive
interference of the reflected radio waves is at its strongest and Bragg
scattering occurs.

First Order Bragg Scattering

σ(1) (ω) = 26 πk04

XS(−2mk0 )δ(ω − mωB )

(1)

m=±1

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
6 / 39Ryuh

Second Order Bragg Scattering

Second Order Bragg Scattering occurs due to electromagnetic anddue to hydrodynamic effects. In the former, the radio wave isreflected twice, off of two different wave components. In the latter,
the radio wave is reflected off the intersection of two ocean waves.
Both types of second order Bragg scattering require that the twoocean wavevectors involved sum to twice the negative of the radiowavevector.
k1 + k2 = −2k0
(2)
The Bragg scattered wave is receive by the radar as a Doppler
spectrum. A Fourier transform is applied to convert it to a frequency
domain equation (σ(2) (ω) on slide 9).

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
OceanAcademic
Wave Spectra
Mentor:fromToshiakiRadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
7 / 39Ryuh

Second Order Bragg Scattering

Hydrodynamic effects
Electromagnetic effects

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
OceanAcademic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
8 / 39
Ryuh

Goal

6

σ(2) (ω) = 2 πk04

X

X ZZ

m1 =±1 m2 =±1|Γ(m1 k1 , m2 k2 )|2 S(m1 k1 )S(m2 k2 )R2× δ(ω − m1pp
gk1 − m2 gk2 ) dp dq(3)Aim
Given the Doppler spectrum for second-order Bragg scattering σ(2) (ω),
how can we determine the wave spectrum S(k) using this theoreticalmodel of the relationship between σ(2) (ω) and S(k).

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
9 / 39Ryuh

Tikhonov Regularization
We let A(S) be the operator L(R2 , R) → L(R, R) that takes a wavespectrum S as an input and outputs the resulting Doppler spectrum :X X ZZ
64A(S) = 2 πk0|Γ(m1 k1 , m2 k2 )|2 S(m1 k1 )S(m2 k2 )m1 =±1 m2 =±1R2× δ(ω − m1p

gk1 − m2pgk2 ) dp dq(4)

Define cost function
L(S) = ∥A(S) − σ(2) (ω)∥2L2 (R) + λ∥S∥2L2 (R2 ) .

(5)

We want to minimize this function over S.
λ > 0 is a regularization parameter needed to avoid overfitting.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
10 / 39RyuhOptimize Using Gradient Descent
Our strategy to minimize thecost function over S is to use
gradient descent in order toattempt to find a minimizer ofthe cost function that closely
approximates the correct wavespectrum.
For the algorithm, we start withan initial guess S0 .
We iterate by
Sn+1 = Sn − t∇L(Sn )(6)

for a chosen step size t.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
OceanAcademic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
11 / 39Ryuh

Necessary calculations for Gradient Descent
Since we have an infinite dimensional input space, we need to define
the gradient for our cost function.
In order to do this, we need to use the Fréchet derivative:lim∥h∥V →0∥f (x + h) − f (x) − Df (h)∥W= 0.
∥h∥V

(7)

The Riesz Representation Theorem says that for a linear operator
T : L2 (R2 ) → R, there exists a v ∈ L2 (R2 ) such that, for any h,T (h) = ⟨v , h⟩L2 (R2 ) .

(8)

As the derivative is a linear operator, we can apply this to the Frechet
derivative of L to get: ∃vDf such that for any h
Df (x)[h] = ⟨vDf (x), h⟩.

(9)Since the directional derivative gives Du f (x) = ⟨∇f (x), u⟩, we canconclude ∇f (x) = vDf (x).

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
12 / 39RyuhCalculation of Fréchet Derivative

The Fréchet derivative of L(S) isL′ (S)[h] = 2⟨A(S) − σ(2) , A′ (S)[h]⟩L2 (R) + 2λ⟨S, h⟩L2 (R2 )(10)

As we can see, this requires us to calculate the derivative of A(S),which we find to be:
X X ZZ′642A (S)[h] =2 πk0|Γ(m1 k1 , m2 k2 ) | S(m1 k1 )h(m2 k2 )m1 =±1 m2 =±1R2

pp+ S(m2 k2 )h(m1 k1 ) δ(ω − m1 gk1 − m2 gk2 ) dp dq.
(11)

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
13 / 39RyuhCalculation of Riesz Representation and Gradient
The Riesz Representation Theorem gives that the Riesz representationof L′ (S) is a function vS ∈ L2 (R2 ) such thatZZ′
L (S)[h] =vS (p, q)h(p, q)dpdq.
(12)R2

In order to calculate vs , we factor out h to get:Z ∞∇L(S) =
4(A(S) − σ(2) (ω)|×−∞

XX

m1 =±1 m2 =±22m1
Γ − (x + 2m2 k0 ), y ), (x, y )
×m2

s 
2p1
x + 2k0 + y 2 − m2 x 2 + y 2 δ ω − m 1 gm2

m1(13)
× S − (x + 2m2 k0 , y dω + 2λS.
m2

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
14 / 39RyuhApplying Gradient Descent
We start out with data consisting of discreet values from a Doppler
spectrum of second order Bragg scattering σ(2) (ω) whereω ∈ [−4.6116π, 4.6116π]. To find S(k), we proceed with thefollowing:
1We interpolate these values in order to obtain a function on R.

2

Chose an initial function S0 (n=0).

3Partition the pq-plane in order to numerically calculate (anapproximation of) the integral of A(Sn )4

Numerically calculate the Riesz representation.

5

Calculate the new value of S:
Sn+1 = Sn − αn vSn .

6(14)Repeat steps 3-5 until the cost function is sufficiently small.

Aya Akahori, Emma Thomas, Kaede Watanabe,Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
15 / 39Ryuh

Tuning the Gradient Descent Algorithm

In order to determine parameters for the gradient descent we will
construct Doppler spectra from known wave spectra and try toreconstruct the starting wave spectra
We want to use functions that could reasonably represent a physicalwave spectra.
We will consider a model that represents a dominant wavefront with acertain frequency and direction

Aya Akahori, Emma Thomas, Kaede Watanabe,Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
16 / 39
RyuhWhat functions should we train the gradient descent on

We use a model that separates the wave spectra into a frequency
spectra and directional distribution [2]S(f , θ) = Sf (f )G (θ|f )(15)

The frequency spectra is given by
Sf (f ) = 0.257Hs2 Ts (Ts f )−5 e −1.03(Ts f )−4(16)

where Hs is the significant wave height and Ts is the significant wave
periodAya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
17 / 39RyuhDirectional Distribution

G (θ|f ) = G0 cos2s̃(f )θ − θ02
(17)

θ0 is the significant wave directions̃ is the wave index given by 5
smax ff
 p −2.5s̃(f ) =s
fmax fp

if f ≤ fp(18)

if f > fp

Aya Akahori, Emma Thomas, Kaede Watanabe,Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
18 / 39RyuhExample Spectrum
Hs = 1, Ts = 3, smax = 10, fp = 0.317, θ0 = 0.5

Model wave spectrum infrequency-direction plane pθ = arctanqModel Wave spectrum in wave numberplane

f =1 √g (p 2 + q 2 )1/4 .
2π(19)Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
19 / 39
Ryuh

Constructed Second Order Doppler Spectrum

Constructed Second Order Doppler Spectrum

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
20 / 39
Ryuh

Numerical simulation for inverse problem
Tried to get the initial condition from constructed σ(2) with
Hs = 1, Ts = 3, smax = 10, fp = 0.317, θ0 = 0 of this initial condition.
These are the image of initial condition(max value is around 0.24),
σ(2) constructed from initial condition and reconstruction from σ(2) .

Doppler spectrum
Initial wave spectrum. constructed from initialwave spectrum.

reconstruction

Parameters for reconstruction are step size = 0.1, iteration number = 30,
regularization term = 10−4 and the time for calculations is about 54 sec.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
21 / 39RyuhNumerical simulation for inverse problem
Next we add perturbation to true data by sin curve and we get thesereconstruction. We express the amplitude by ϵ.
ϵ = 0.001 and after changing the value less than 0.002 to 0. We
calculated it by changing step number to 200 and the time forcalculation is 267 sec.

initial data with perturbation.

Reconstruction after setting
values less than 0.002 to 0.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
22 / 39RyuhNumerical simulation for inverse problem
ϵ = 0.003 and after changing the value less than 0.0 to 0. We
calculated it by same conditions of ϵ = 0.001 and the time forcalculation is 263 sec

initial data with perturbation.

Reconstruction after setting
values less than 0.0 to 0.

In the future, we need to construct stronger system to reconstruct
wave spectrum.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
23 / 39RyuhDynamic Step Size
In order to get the gradient descent method to converge faster rate ofconvergence for the gradient descent, we would like to use dynamic
step size.
We want large step sizes when we far from minima and small step
sizes when we are close to minima

Example of gradient descent.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
24 / 39Ryuh

Decreasing Cost Function
We want to ensure that the value of the cost function decreases witheach iteration
L(Sn+1 ) < L(Sn )(20)
However, this condition is not sufficient to ensure quick convergence

Gradient descent algorithm with insufficient decrease. Reprinted from [3].

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
25 / 39RyuhArmijo Condition
We introduce another term to demand further decrease
L(Sn+1 ) ≤ L(Sn ) − c1 αn ∥∇L(Sn )∥L2 (R2 )(21)

for c1 ∈ (0, 1) and αn is the step size.

Example of Armijo Condition. Reprinted from [3].

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
26 / 39Ryuh

Wolfe Condition
We also want the directional derivative to decrease in magnitude⟨∇L(Sn+1 ), ∇L(Sn )⟩L2 (R2 ) ≥ −c2 ∥∇L(Sn )∥2L2 (R2 )(22)for c2 ∈ (c1 , 1)

Example of Wolfe Condition. Reprinted from [3].

The strong Wolfe condition excludes slopes that are too positive aswell|⟨∇L(Sn+1 ), ∇L(Sn )⟩L2 (R2 ) | ≤ c2 ∥∇L(Sn )∥2L2 (R2 )(23)

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
OceanAcademic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
27 / 39Ryuh

Existence of Desired Step Size
We assume that the cost function L is bounded below
Definel(α) = L(Sn ) − c1 α∥∇L(Sn )∥L2 (R2 )ϕ(α) = L(Sn + α∇L(Sn ))(24)

Note that l ′ (0) > ϕ′ (0), so there exists an α1 such thatl(α1 ) > ϕ(α1 ).
However, ϕ is bounded below, but l is not, so there exists an α2where l(α2 ) < ϕ(α2 ).
Therefore, there exists a α ∈ (α1 , α2 ) where l(α) = ϕ(α).
Let α′ be the smallest such α.
For α ∈ (0, α′ ), the Armijo condition is satisfied.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
28 / 39RyuhExistence of Desired Step Size Continued
For α ∈ (0, α′ ), the Armijo condition is satisfied.
By the mean value theorem, there exists an α′′ ∈ (0, α′ ) such thatϕ(α′ ) − ϕ(0) = α′ ϕ′ (α′′ )′′(25)l(α ) − L(Sn ) = −α ⟨∇L(Sn+1 ), ∇L(Sn )⟩(26)−c1 α ∥∇L(Sn )∥2L2 (R2 ) = −α′ ⟨∇L(Sn+1 ), ∇L(Sn )⟩.
c1 ∥∇L(Sn )∥2L2 (R2 ) = |⟨∇L(Sn+ ), ∇L(Sn )⟩|(27)′(28)c2 > c1 , soc2 ∥∇L(Sn )∥2L2 (R2 ) ≥ |⟨∇L(Sn+1 ), ∇L(Sn )⟩|.

(29)

Therefore, the Wolfe condition is satisfied for α′′ .

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
29 / 39
RyuhBacktracking Line Search
For each iteration,
αn = Mβ mn(30)where M is the maximum possible step length, β ∈ (0, 1), and mn isthe first natural number such that both the Armijo and Wolfe
conditions are true.

Example of Wolfe Condition. Reprinted from [3]Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWattOcean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
30 / 39
Ryuh

Environmental aspects of the study

The main goal for studying the environmental effects is to determine
a good initial guess for our gradient descent.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
31 / 39
Ryuh

Throughout site visit...

Send the wave spectra with four transmitting array antennas (left side) andreceive the reflected wave spectra with eight receiving array antennas (right side).
One radar is composed of these transmit and receive antennas. They are placed afew meters from the coast.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
32 / 39RyuhThroughout site visit...

Saw the border between darker and lighter blue area of the ocean
from airplane.
Information about fishes and animals from deep sea around Kochi
such as whales. Also the facility for pumping up deep sea water.
Experienced strong wind from sea.
Unique Terrain around the cost and the existence of the geopark.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
33 / 39Ryuh

Environmental aspects of the study

Topographic map of Shikoku region

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
34 / 39
Ryuh

Environmental aspects of the study

Waves move faster in deeper water but as they enter shallow water,
they slow and conservation of energy increases wave height.
The slowing waves causes refraction which can change the directionof the wave fronts.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
35 / 39
Ryuh

Environmental aspects of the study

Ocean current map. Reprinted from [4].

Waves tend to align with the currents. We will use this fact to determine
the wave direction of our initial guess.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
36 / 39RyuhSummary

We attempted to implement a gradient descent algorithm to solve theproblem
We hoped to optimize the gradient descent algorithm by adding
dynamic step sizes and by using environmental factors to get a goodinitial guess
Hopefully, these factors can be implemented in the future to obtain
more efficient algorithm
Then the knowledge of the wave spectra can be used to help withfishing and transportation

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
37 / 39
RyuhReferences[1]

J.M.J. Journée and W.W. Massie. Offshore Hydromechanics. Delft
University of Technology, 2001.

[2]

Tomoya Kataoka. “A Bayesian Inversion for a Directional Spectrumof Ocean Waves in Shallow Water Using HF Radar”. In: (2016).

[3]

Jorge Nocedal and Stephen J. Wright. Numerical Optimization.
Springer, 2006.

[4]

Japan Meteorological Agency. Daily 50m Current. 2023. url:
https://www.data.jma.go.jp/kaiyou/data/db/kaikyo/daily/
current_HQ.html?areano=8&stryy=2023&strmm=08&strdd=03.

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,Mentors:
2023 Takahashi
38 / 39RyuhThank you for listening

Aya Akahori, Emma Thomas, Kaede Watanabe,
Estimation
CameronofWatt
Ocean
Academic
Wave Spectra
Mentor:fromToshiaki
RadarYachimuraIndustry
August 7th,
Mentors:2023 Takahashi
39 / 39Ryuh