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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