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Modelling Waves in the Ocean

Modelling Waves in the Ocean

Jack McGinnSupervisor: Jake Grainger

September 2020

Modelling Waves in the Ocean

Time Series

Wind generated waves treated as stochastic processSampling displacement over time of some point in the oceanis denoted by stochastic variable X∆ = [Xt∆]t∈Z formingdiscrete time series

Modelling Waves in the Ocean

Spectral Density

Continuous Spectral Density

f (ω) =1

2π

∫ +∞

−∞c(τ)exp(−iωλ)dτ

Discrete Spectral Density

f∆(ω) =∆

2π

∞∑τ=−∞

c(τ∆)exp(−iωτ∆)

Aliasing

f∆(ω) =∞∑

k=−∞f (ω +

2πk

∆)

Modelling Waves in the Ocean

Aliasing

Modelling Waves in the Ocean

Periodogram

Periodogram estimate for spectral density from wave displacementtime series

I (ω) =∆

2πN

∣∣∣∣∣N−1∑t=0

Xt∆exp(−it∆ω)

∣∣∣∣∣2

Modelling Waves in the Ocean

JONSWAP

JONSWAP model for spectral density of wind generated waves inthe ocean

SG (ω|θ) = αω−rexp

(−rs

(ω

ωp

)−s)γδ(ω|θ)

Modelling Waves in the Ocean

JONSWAP

Modelling Waves in the Ocean

Estimating parameters - Whittle Approximation

Whittle Likelihood function

`W (θ|X∆,N) = −∑ω∈Ω

log(f (ω|θ)) +I (ω)

f (ω|θ)

De-biased Whittle Likelihood function, replace f (ω|θ) infraction with,

E [I (ω)] =1

2πRe

(2∆

N−1∑τ=0

(1− τ

N

)c(τ |θ)exp(−iωτ∆)−∆c(0|θ)

)

Modelling Waves in the Ocean

Simulated Data

Modelling Waves in the Ocean

Real Data

Modelling Waves in the Ocean

Real Data continued

Modelling Waves in the Ocean

Swell

Modelling Waves in the Ocean

Robustness of Removing swell

Modelling Waves in the Ocean

Real Parameters

Modelling Waves in the Ocean

Further Work

Improve method of testing robustness

Improve method for removing swell