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Deterministic and stochastic modeling of the
end-to-end interdisciplinary system,and its
errors and uncertainties P.F.J. Lermusiaux April
9, 2003

• END-TO-END SYSTEMS AND COUPLED MODELS
• UNCERTAINTIES IN END-TO-END COMPONENTS
• SOURCES, FORWARD TRANSFERS, BACKWARD TRANSFERS
• RESEARCH SUBJECTS FOR END-2-END UNCERTAINTY
  MODELING
• TOPICS AND DIRECTIONS
• ILLUSTRATIVE QUESTIONS AND CHALLENGES

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AD Acoustical DataMD Meteorological DataPD
Physical DataGD Geological DataND Noise
DataSD Sonar Data
PMD Physical Model DataBMD Bottom Model
DataNMD Noise Model DataAPMD Acous. Prop.
Model DataSMD Sonar Model DataTMD Target
Model Data
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Coupled (Dynamical) Models and Outputs

• BOTTOM MODELS
• Hamilton model, Sediment flux models (GG), etc
• Statistical/stochastic models fit-to-data
• OUTPUTS
• Wave-speed, density and attenuation coefficients
• NOISE MODELS
• Wenz diagram, empirical models/rule of thumbs
• OUTPUTS
• f-dependent ambient noise (f,x,y,z,t) due to
  sea-surface, shipping, biologics
• SONAR SYS. MODELS AND SIGNAL PROCES.
• Sonar equations (f,t)
• Detection, localization, classification and
  tracking models and their inversions
• OUTPUTS
• SNR, SIR, SE, FOM

• PHYSICAL MODELS
• Non-hydrostatic models (PDE, x,y,z,t)
• Primitive-Eqn. models (PDE, x,y,z,t)
• Quasi-geostrophic models, shallow-water
• Objective maps, balance eqn. (thermal-wind)
• Feature models
• OUTPUTS
• T, S , velocity fields and parameters, C field
• Dynamical balances
• ACOUS. PROP. MODELS
• Parabolic-Eqn. models (x,y,z,t/f)
• (Coupled)-Normal-Mode parabolic-eqn. (x,z,f)
• Wave number eqn. models (x,z,f OASIS)
• Ray-tracing models (CASS)
• OUTPUTS
• Full-field TL (pressure p, phase ?)

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DEFINITION AND REPRESENTATION OF UNCERTAINTY

• x estimate of some quantity (measured,
  predicted, calculated)
• x t actual value (unknown true nature)
• e x - x t (unknown error)
• Uncertainty in x is a representation of the error
  estimate e
• e.g. probability distribution function of e
• Variability in x vs. Uncertainty in x
• Uncertainties in general have structures, in time
  and in space

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MAIN SOURCES OF UNCERTAINTIES INEND-TO-END
COMPONENTS

• Physical model uncertainties
• Bathymetry
• Initial conditions
• BCs surface atmospheric, coastal-estuary and
  open-boundary fluxes
• Parameterized processes sub-grid-scales,
  turbulence closures, un-resolved processes
• e.g. tides and internal tides, internal waves and
  solitons, microstructure and turbulence
• Numerical errors steep topographies/pressure
  gradient, non-convergence
• Bottom/geoacoustic model uncertainties
• Model structures themselves parameterizations,
  variability vs. uncertainty
• Measured or empirically-fit model parameters
• BCs (bathymetry, bottom roughness) and initial
  conditions (for flux models)
• 3-D effects, non-linearities
• Numerical errors e.g. geological layer
  discretizations, interpolations

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Uncertainties in bathymetry (from data
differences and statistical model)
NOAA soundings combined with Smith and Sandwell
(overlaid with GOM bathymetry)
Smith and Sandwell
(predicted topography based on gravity anomaly
not well compensated for regions with thick
sediments)
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Uncertainties in atmospheric forcings (from
buoy-data/3d-model differences)
Baugmarter and Anderson, JGR (1996)
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Three-Hourly Atmospheric Forcings Adjusted
Eta-29 model, 21 July 1996, 2pm EST
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Uncertainties in un-resolved processes
Stochastic forcing model of sub-grid-scale
internal tides
Hovmoller diagram
Sample effects of sub-grid-scale internal tides
difference between non-forced and forced model
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MAIN SOURCES OF UNCERTAINTIES INEND-TO-END
COMPONENTS (Continued)

• Acoustical model uncertainties
• Sound-speed field (c)
• Bathymetry, bottom geoacoustic attributes
• BCs Bottom roughness, sea-surface state
• Scattering (volume, bottom, surface)
• 3-D effects, non-linear wave effects
  (non-Helmholz)
• Numerical errors e.g. c-interpolation,
  normal-mode at short range
• Computation of broadband TL

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MAIN SOURCES OF UNCERTAINTIES INEND-TO-END
COMPONENTS (Continued)

• Sonar system model and signal processing
  uncertainties
• Terms in equation SL, TL, N, AG, DT
• Sonar equations themselves 3D effects,
  non-independences, multiplicative noise
• Beamformer posterior uncertainties, Beamformer
  equations themselves
• Noise model uncertainties
• Ambiant noise frequencies, directions,
  amplitudes, types (manmade, natural)
• Measured or empirically-fit model parameters
  (Wenz, 1962)
• Target model uncertainties
• Source level, target strength (measured or
  empirically-fit model parameters)
• Reverberation model uncertainties (active)
• Scattering models themselves parameterizations
  (bottom scattering, bubbles, etc)
• Measured or empirically-fit model parameters

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METHOLOGIES FOR UNCERTAINTY MODELING

• Representations
• Random numbers
• Statistical moments
• Bayesian, Bayesian hierarchical, Maximum entropy
  methods (Erickson and Smith,1988)
• Error subspace (EOFs, Polynomial Chaos, ESSE,
  etc)
• Fuzzy uncertainties (Klir and Wierman, 1999)
• Belief functions (Dempster, 1990)
• Evolutions/propagations/forward transfers
• Deterministic/Stochastic calculus (e.g.
  Jazwinski, 1970)
• Statistics (pdf convolutions, etc)
• Information theory (Cover, 1991)
• Deterministic differentials (outputs wrt inputs)
• Inversion methods/backward transfers
• Adjoint methods, Generalized inverse, Smoothing
  methods (KS, ESSE)

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RESEARCH SUBJECTS FOR END-2-END UNCERTAINTY
MODELING

• Current and anticipated research organized in
  major subjects
• Modeling Approaches and Methodologies
• End-to-End Scales and Nonlinearities
• Error Estimation, Error Models and Error
  Reductions
• Sensitivities, Prioritizations and Idealized
  Uncertainty Modeling
• Uncertainty Complex Systems and Fleet Operations

• Table provided for each subject
• List of research topics and directions (left
  column)
• Series of illustrative research questions and
  challenges (right column, not intended to be
  comprehensive)

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Modeling Approaches and Methodologies
Research Topics and Directions End-2-end models Full deterministic coupling of advanced end-2-end models to be done Separate components ok, advanced coupling starting, uncertainty modeling limited Since different components in various situations - for bottom models, uncertainty close to variability - for parts of sonar models, dynamical models are pdfs amplitudes, shapes of pdfs are then uncertainties careful transfer is essential! Illustrative Questions and Challenges     What are essential review references on models of the end-2-end components? Is the limit in uncertainty modeling x x(r,t) ?x(r,t) ??(r,t) ? Etc      
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Modeling Approaches and Methodologies (continued)
Research Topics and Directions   Uncertainty representation/transfer methods Deterministic, statistic and stochastic models Representations for efficient computations sub-optimal reduction of uncertainty space to be optimal (error subspace) Evaluations and benchmarks for both idealized and realistic situations   Lessons from other fields Information theory, fuzzy statistics Atmospheric/weather forecasting Illustrative Questions and Challenges     What methods of representing and transferring uncertainties are in use today? What methods are most promising? Different methods for different purposes? How should methods be evaluated? What about methods that utilize the structure of end-2-end PDEs? Etc   What are useful uncertainty representations? What can be learned methods, systems? Etc
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End-to-End Scales and Nonlinearities
Research Topics and Directions   Multiple scales and multivariate  Environmental vs. acoustical time and space scales, in 3D/2D models Measurement models linking multi-resolution data to relevant coupled models Research, real-time, operational, crisis-response Nonlinear effects Multiplicative noise and stochastic calculus Impacts of nonlinearities on forward and backward/inverse uncertainty transfers and data assimilation Predictive capabilities and ultimate predictability limits for e-2-e systems Illustrative Questions and Challenges     How to best combine relocatable 2D acoustic models with 3D ocean models? How efficiently utilize internal wave data, bottom data, in 3D? Etc       How nonlinear is the wave equation wrt its parameters? Should this affect uncertainty modeling? What are and how to estimate the predictability limits of sonar systems dynamics? Etc
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Error Estimation, Error Models and Error
Reductions
Research Topics and Directions Error models   Stochastic, deterministic, adaptive (for both dynamics/data) Structural errors and parameter errors Error models for unresolved processes, forcing and boundary condition errors, environmental noise Measurement models and data uncertainties for end-2-end (physical, geological, acoustical and sonar) data bases Efficient error reductions Data assimilation methods Control, estimation, inverse and optimization theories, and stochastic/hybrid methods Model state, model parameters and model structures estimations End-2-end adaptive sampling and model improvements Illustrative Questions and Challenges   How to quantitatively prioritize uncertainties? How to differentiate between structural and parameter errors in such complex systems? How to estimate accurate stochastic forcings? How to account for and model interdisciplinary measurement errors? Etc   Why should uncertainty representations and uncertainty reduction criterion be compatible?
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Sensitivities, Prioritizations and Idealized
Uncertainty Modeling
Research Topics and Directions   Sensitivity studies Impact of same uncertainty onto different system components (e.g. bathymetry) Impact of different or variable uncertainties (amplitude, pdf shape, types) on same components? On end-2-end system? Idealized end-2-end models and systems   Applied math and theoretical research for representing, characterizing, capturing and reducing (end-to-end) uncertainty for scientific and Naval purposes Truncation issues and divergence Illustrative Questions and Challenges     How different are the impacts of environmental uncertainties on target detection, localization, classification and tracking? Is the broadband TL more sensitive to volume than bottom uncertainties? Etc What are the effects of simplifying assumptions? What is a parsimonious parameterization in a range dependent environment Etc
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Uncertainty Complex Systems and Fleet Operations
Research Topics and Directions  Computations, technologies and systems   Coupling of end-2-end systems components Generic versus regional systems Visualization of uncertainties (and uncertainties in visualization) Information technology, scientific distributed computing Fleet applications/operational systems   Automated systems for uncertainty predictions, skill evaluations Efficient research-to-operation and operation-to-research transitions/feedbacks Research, real-time, operational, crisis-response Typical scenarios and rules of thumb Illustrative Questions and Challenges     How to couple end-2-end components for efficient computing? How to benefit from Fleet experiences? How to downscale scientific descriptions to useful operational uncertainties? Can uncertainty models lead to improved and more efficient TDA, tactical advantage? How to usefully estimate accuracies/errors of an operational system? Should operator overload uncertainties be modeled?