1 Radar Data Assimilation & Forecasts of Evolving Nonlinear Wave Fields Sina H. Aragh Ph.D. Research Prospectus Department of Naval Architecture and Marine

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Radar Data Assimilation &

Forecasts of Evolving Nonlinear Wave Fields

Sina H. AraghPh.D. Research Prospectus

Department of Naval Architecture and Marine Engineering

University of Michigan

Friday, May 4, 2007

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Overview

• Introduction Research Problem Motivation

• Literature Review Marine Radars Wave Model Data Assimilation

• Methodology Forward Wave Model Assimilation Scheme Adjoint Wave Model Minimization Scheme

• Summary

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Research Problem

Numerical Wave ModelRadar Image

Data Assimilation

Improved Estimate of the Wave Field

Short term Forecast

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Research Objectives

• Develop an efficient numerical wave model that:

1) Captures basic nonlinear interactions

2) Predict nonlinear evolution of

multi-directional sea states

• Determine an optimal strategy to assimilate real-time radar observations into short-term forecast wave model to:

1) Provide improved estimates of the wave field

2) Forecast the short-term evolution of sea state

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Motivation-Aircraft Landing-Optimal Path of Automated Vessels-Launch/Recovery

Eisenhower Carrier Aircraft Landing

Laun

ch/R

ecov

ery

http://www.naval-technology.com/projects/littoral/index.html

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Motivation(Rogue Waves)

t(sec)x(m)

Time history @ mid-length

Rock n’ Role

Rogue Waves

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Literature Review(Marine Radars)

Marine radars can be used as a remote sensing tool to survey ocean wave fields

The wave measurement due to interaction of electromagnetic waves with sea surface ripples

This interaction produces a backscatter phenomenon showing a wave pattern in the radar screen (“sea clutter”)

“Sea clutter” is the required signal to get appropriate sea state characterization

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Radar Imaging Mechanisms

Literature Review

(Marine Radars)

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Literature Review(Marine Radars)

• Ijima et al.(1965) & F. Wright(1965)Among the first to use marine radars for imaging ocean waves

• Oudshoorn(1960), Willis & Beaumont(1971), Evmenov et al.(1973)Mean wave direction, Wave length, and Period

• Hoogeboom & Rosenthal(1982), Ziemer et al.(1983)Digitized radar images / 2D F.T. / Compared with Buoy Data

Major Problem with 2D wave number spectra: 180 degree directional ambiguity

• Atanassov et al.(1985)Removed the ambiguity in direction

• Young and Rosenthal1985)Used 3D F.T. / 3D energy density spectrum (wave number-frequency space)

Found the Mean current through least squares

High signal-to-noise ratio

• Estimates of significant wave height/instantaneous wave fields, requires knowledge of “Transfer Function”

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Literature Review (Marine Radars)

• Alpers & Hasselmann(1978)Included shadowing/tilting, used signal processing in frequency domain to improve SNR, simpler hydrodynamic interaction / action balance • Ziemer & Rosenthal(1987)MTF in Fourier space, based on tilting/shadowing/hydrodynamicNonlinear influence mainly due to shadowing• Nieto Borge(1999)Directional spectrum/Significant wave height (based on its relation to SNR)Found correlation factor = 0.89 btw buoy and “WaMoS”• Nieto et al.(2004)Empirical scheme for inversion based on shadowing onlyApproach: 1) 3D Fourier decomposition amplitude & phase spectra 2) high-pass filtering remove long range effects 3) band-pass filtering extract wave-related components 4) I.F.T and re-scaling Wave amplitude

• Rosenthal & Dankert(2004)Based on determination of surface tilt angle in antenna look direction at each pixel of radar image

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Literature Review (Wave Models)

• Stationary Waves -Asymptotic expansion of velocity potential in terms of a small

parameter (steepness/shallowness) Stokes

-Fourier approximation methods (Chappelear, 1961 & Dean, 1965)

• Nonlinearly evolving -Boundary integral methods Longuet-Higgins & Cokelet(1976)

-Time-dependent Fourier series Fenton & Rienecker(1982) Solved 3D Laplace eq. numerically (initially inspected pseudo-spectral

method)

-Attempts to find formal solution of Laplace eq. analytically Dynamic & Kinematic FSBC can be reduced to a pair of evolution

equations (in two horizontal coordinates)

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Literature Review(Wave Models)

Two evolution equations

Need to close the system

Three surface variables

West et al.(1987) Asymptotic expansions (successive inversion of asymptotic sequences)

Matsuno(1992) 2nd order approximation

Craig & Sulem(1993) Expanding Dirichlet-Neumann operator / for constant depth, later extended by (Guyenne & Nickols, 2005) to variable depth

Choi(1995) 3rd order approximation (Similar to West et al.)

Clamond & Grue(2001) Formulated higher order effect by an integral representation

Similar approaches for long waves by Wu(1998, 2001), Madsen, Bingham, and Liu(2002), and Madsen, Bingham, Schaffer(2003)

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Literature Review(Data Assimilation)

“Data Assimilation” is defined as finding the optimal initial/boundary conditions that minimize the difference between the measurements and model predictions over a time interval.

Data Assimilation

SequentialVariational

• Sequential Schemes:

Kalman Filter(Gelb, 1974)

+Optimal for linear problems

Direct Insertion(Daley, 1991)

+Spurious Oscillations

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Sequential schemes

Nudging (Kistler, 1974, Hoke & Anthes, 1976) choose nudging time scale is important

Successive Corrections (Bergthorsson & Doos, 1955) weighting functions/radius of influence

Variational Methods

Cost function is defined as the difference between model predictions and observations

Goal minimize the cost function over assimilation period

Gradient of the cost function with respect to initial conditions is needed

Good for large-scale

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Le Dimet & Talagrand(1986) Gradient of the cost function with respect to all control variables can be obtained by one single integration of the “adjoint” of the evolution equations backward in time. The value of the adjoint variable at t = 0 will give use the desired gradient ( Proof given in appendix (A) of the prospectus report)

Direct method need to perturb components of the initial condition one at a time (very expensive)

Minimization Methods for large-scale problems:

-Conjugate Gradients( Liu & Nocedal, 1989) good for solving very large problems with relatively inexpensive objective function, can be suitable for multi-processor computers (Nocedal, 1990)

Search direction an estimate of the relative change in each component of the control vector to produce max. reduction in the cost function

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Path to Methodology

Challenges:

• Large set of observational data (High resolution)• Complexity of modulation mechanisms • Nonlinearity of the sea state• Large sets of control parameters to optimize• Cost function does not directly depend on the

initial/boundary conditions• Need fast wave model / Efficient optimization scheme

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Methodology(Forward Wave Model)

Zakharov(1968)(Hamiltonian Formulation)

P.E.K.E.

Canonical Pair

Free surface evolution equations

=

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Forward Wave ModelExpanding about z = 0

Where :

Closure:

Solving the linear boundary value problem

(Matsuno, 1992)

Fourier Space

Physical Space

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Forward Wave Model

Inversion

First order evolution eq.

Third order evolution eq.

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Numerical Solution of Forward Wave ModelPseudo-spectral method used to solve the evolution equations

Linear operators Fourier Space

NL operators Physical space

RK-4 used to integrate in time

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Validating wave model with experimental data(NRC)

• Two wave makers at both sides / totally 168 segments

• 25m long boom with 20 probes

• 5 fixed probes

• Generating various multi-directional sea states

• Nonlinear interactions between crossing waves

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Validating numerical wave model with experimental data(NRC)

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Assimilation Scheme

•Determine optimal set of initial conditions, boundary conditions and model parameters that minimize differences between model predictions and radar observations over assimilation interval

Cost Function:

Objective:

•Use optimal set of initial conditions, boundary conditions and model parameters to forecast wave field evolution

•After preliminary tests of different schemes, chose adjoint-based method

•Conjugate gradients used for minimization

•Gradient of the cost function w.r.t. initial condition obtained from adjoint technique

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Adjoint Data Assimilation

InitializeModel

Prediction - Observation



Forward Model

Adjoint Model




InitializewithError

Grad ofCost

Function

Conjg.Gradient

Min.

Short-term forecast

Assimilation Interval

(0 T)

t = Tt = 0

Adjoint Data Assimilation

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Deriving Adjoint Model(for linear forward model)

Adjoint Model

1st order Forward Model

Perturb

Tangent Linear Model

Multiplyby Adjoint Variable andIntegrate by Parts

+

+

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Third-Order Equations Forward Model

Tangent Linear Model

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Third-Order Equations

Adjoint Model

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Evaluation of Gradient Computations

• Domain length = 3066m; Grid spacing = 6m, Time Step = 0.25s

• Added 10% uncorrelated noise to 16 model predictions at intervals of 2.5s

• Selected JONSWAP sea state with Hs = 2m, Tp = 10s

• Compared adjoint prediction of gradient of cost function with direct sensitivity method

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Proposed Approach(Adjoint vs. Direct method – 1st order)

400 500 600 700 800 900 1000-1.5

-1

-0.5

0

0.5

1

1.5

spatial domain

Gra

die

nt

of

the

co

st

fun

cti

on

w.r

.t. i

nit

ial c

on

dit

ion

Adjoint method

Direct method

Numerical tests with JONSWAP sea state

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Proposed Approach(Adjoint vs. Direct method – 3rd order)

Numerical tests with JONSWAP sea state

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Ability to Reproduce Initial Condition with Noisy Model Prediction used as “Observation”

2400 2450 2500 2550 2600 2650-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

spatial domain(x)

Su

rfa

ce

ele

va

tio

n @

t =

0

noisy initial cond.

true state

assimilated

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

1

2

3

4

5

k

S(k

) @

t =

0

noisy initial condition

true state

assimilated

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Time history of the surface elevation at x = L/4, forecasted from the produced optimal initial condition.

Noisy model predictions were used as “observation”

0 10 20 30 40 50 60 70 80-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time(s)

Su

rfa

ce

ele

va

tio

n(m

)time history @ x = 0.25L

with noisy I.C.

true stateassimilated

350 10 20 30 40 50 60 70 80

-1.5

-1

-0.5

0

0.5

1

1.5

time(s)

Su

rfa

ce

ele

va

tio

n(m

)time history @ x = 0.5L

with noisy I.C.

true state

assimilated

history of the surface elevation at x = L/2, forecasted from the produced optimal initial condition. Noisy model

predictions were used as “observation”.

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Application to Synthetic Radar Data

600 m824 m

39 m

Modeled section

Line source array

• True sea surface that was used in EM simulations of radar backscatter• Retrieved wave profiles from radar backscatter• Filtered profiles based on the linear wave dispersion relation

Provided Information by OSU

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Synthetic Radar Data (OSU)

From: Johnson et al. (2006)

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0 100 200 300 400 500 600-4

-3

-2

-1

0

1

2

3

4

5

x-domain

surf

ace

elev

atio

n(m

)

Surface elevation @ t = 0

retrieved from radartrue stateassimilated

Ability to Reproduce Initial Condition with True Sea State used as “Observation”

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Ability to Reproduce Initial Condition with True Sea State used as “Observation”

0 20 40 60 80 100 120 140 160 18010

0

101

102

103

iteration

Co

st F

un

ctio

n


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0 5 10 15 20 25 30 35-3

-2

-1

0

1

2

3

4

time(sec)

surf

ace

elev

atio

n

time history at mid-length

true state filteredretrievedassimilated

Ability to Forecast with Retrieved Radar Data used as “Observation” (x=L/2)

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Application to Synthetic Radar DataAbility to Forecast with Filtered Radar Data used as “Observation”

(x=L/2)

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Field Data (Alaska, Sept. 2006)

•Evaluate the performance and limitations of both Doppler (coherent) and non-coherent X and S-band radars for the real-time measurements of ocean wave fields

•Validate of nonlinear wave evolution model

•Validate data assimilation and forecast scheme

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Future Plans

Test different optimization schemes with synthetic data

Extend assimilation scheme to boundary conditions

Improve efficiency of the assimilation scheme by defining

control variables in Fourier space instead of physical space

Develop 2-D model and validate model with lab data

Develop 2-D assimilation scheme and compare with field data

Documents

1 Radar Data Assimilation & Forecasts of Evolving Nonlinear Wave Fields Sina H. Aragh Ph.D. Research Prospectus Department of Naval Architecture and Marine