Overview of NRL Uncertainty Program Steven Finette Acoustics

Overview of NRL Uncertainty Program Steven Finette Acoustics Division Naval Research Laboratory Washington DC 20375 Quantifying, Predicting and Exploiting Uncertainty DRI San Francisco Dec 8, 2006 Work supported by ONR

Overview of NRL Uncertainty Program Kevin Le. Page, Roger Oba, Yu Yu Khine, Rich Keiffer, Colin Shen Naval Research Laboratory Washington DC 20375 Quantifying, Predicting and Exploiting Uncertainty DRI San Francisco Dec 8, 2006 Work supported by ONR

Long Term Research Objective / Approach OBJECTIVE • Develop a physically consistent, predictive modeling capability for acoustic propagation in littoral regions to estimate ASW system performance. General Approach: • Integration of hydrodynamics / acoustic modeling -- sub-mesoscale hydrodynamic model to compute space/time sound speed distribution in 2 -D, 3 -D. -- 2 -D, 3 -D acoustic propagation through environmental snapshots. • Quantify environmental uncertainty, integrate it with computations of the ocean dynamics and acoustic propagation.

Definition of Uncertainty (after Oberkampf and Helton) • mathematical model • data / environment Irreducible: inherent variation linked to the physical system or environment (aka stochastic, aleatory, variability) Reducible: potential deficiency in any phase of modeling process due to lack of knowledge (aka epistemic) Error: recognizable deficiency in any phase of modeling process not due to lack of knowledge MAIN ISSUE: quantify reducible introduction amplification propagation • interaction

Definition of Uncertainty (after Oberkampf and Helton) • mathematical model • data / environment Irreducible: inherent variation linked to the physical system or environment (aka stochastic, aleatory, variability) Reducible: potential deficiency in any phase of modeling process due to lack of knowledge (aka epistemic) Error: recognizable deficiency in any phase of modeling process not due to lack of knowledge ocean model acoustic model se

Strategies for Modeling Uncertainty Statistical methods: Bayesian prediction, maximum entropy Stochastic basis expansions: polynomial chaos Sensitivity methods: e. g. adjoint Fuzzy Sets Possibility Theory Evidence Theory Interval Analysis Alternatives to probability theory

Defining Models in a Probabilistic Framework Set of possible models for ocean acoustic propagation governing eqn(s). for physical model initial conditions boundary conditions parameters / fields parabolic eqn. acoustic propagation starting field at r = 0 for all z model pressure release, soft bottom sound speed distribution (e. g. EOF) probability distribution on

Probabilistic Interpretation of Uncertainty Complete knowledge of m: P(m) Incomplete knowledge of m uncertainty m Probabilistic description of environmental uncertainty measured as a “spread” or moments of distribution Random variable: Ex: water depth, source frequency Random process: Ex: sound speed field, bathymetry

Coupling Uncertainty and Dynamics space time events / realizations Uncertainty (additional “dimension”) Vector space of space-time functions Probability space of 2 nd order random variables Hilbert Space Stochastic Dynamics

Stochastic Dynamics and Uncertainty Set of possible models for ocean acoustic propagation Space of system inputs Deterministic Dynamics Space of system outputs Deterministic variables and fields random variables and processes Stochastic Dynamics Polynomial chaos representation

Two Examples of Hilbert Spaces Deterministic Functions Weierstrass Approximation Theorem (1885) Random Processes Cameron-Martin Theorem (1947) polynomial chaos basis functionals

Stochastic Basis Expansion arbitrary 2 nd order random process can be expanded in a complete set of orthogonal random polynomials [Cameron and Martin 1947] Generalized polynomial chaos of order n Complete orthogonal basis in Hilbert space of random variables that 2 nd order random variables represent uncertainty: volume surface bottom deterministic expansion coefficients Pressure field: random multi-variate polynomial complete probabilistic description of system uncertainty linked to propagation

Autovariance at (r, z): 21 modes

Sensitivity : Analysis of Polynomial Chaos input process [sound speed]: output process [pressure field]: Jacobian “sensitivity” matrix

True or False? Conventional wisdom… adding more physics and / or increasing grid resolution “Better” predictions “Decreasing” uncertainty Not necessarily true… • Additional parameters / fields may be incompletely known • Increased resolution may not be consistent with physics at that grid resolution • “Simpler” model incorporating uncertainty might improve predictive capabilities�

Latitude (deg) Example of Acoustic Based Metric μ: Horizontal Array Bearing Wander and Bias Prediction 155º Source S 22. 2 100 m 22. 0 Beamforming angle from endfire R S R 0º 21. 8 200 m Mode 3 21. 6 117. 4 117. 2 Longitude (deg. ) 117. 6 Mean bearing bias (~2°N) 0 0. 25 0. 75 phase speed gradient (m/s)/m ASIAEx 2001(Orr & Pasewark) 125. 875 26 TIME (Julian Day) TEMPERATURE (deg C) 28 Source (300 Hz) Receiver 24 2. 5 h 47 m 22 20 IW @S 126. 000 IW @ R 77 m 126. 125 18 18 20 22 00 02 JD 126 TIME (hours) 04 06 Using archival sound speed only: predicts slope induced bearing bias but not wander 145 150 155 160 165 BEARING (deg) 75 85 95 105 BEAM POWER (d. B re 1 m. Pa) 170 Sarchive = Reference score