Overview of NRL Uncertainty Program

1. 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

2. Overview of NRL Uncertainty Program Kevin LePage, 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

3. 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.

4. •mathematicalmodel • data / environment Uncertainty 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(afterOberkampf and Helton)

5. •mathematicalmodel • data / environment Uncertainty 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 sensor model Definition of Uncertainty(afterOberkampf and Helton)

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

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

8. P(m) m Probabilistic Interpretation of Uncertainty Complete knowledge of m: Incomplete knowledge of m  uncertainty 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

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

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

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

12. arbitrary 2nd 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 2nd order random variables random variables that represent uncertainty: volume  surface  bottom deterministic expansion coefficients Pressure field: random multi-variate polynomial complete probabilistic description of system uncertainty linked to propagation Stochastic Basis Expansion

13. Autovariance at (r,z): 21 modes

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

15. Conventional wisdom… adding more physics and / or increasing grid resolution “Better” predictions “Decreasing” uncertainty  True or False? 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

16. Example of Acoustic Based Metric μ: Horizontal Array Bearing Wander and Bias Prediction 125.875 22.2 Latitude (deg) TIME (Julian Day) 126.000 100m 22.0 126.125 21.8 155º Source 200m S S 21.6 Beamforming angle from endfire R R Longitude (deg.) 0º 117.4 117.2 117.6 0 0.25 0.5 0.75 Mode 3 Mean bearing bias (~2°N) 28 phase speed gradient (m/s)/m ASIAEx 2001(Orr & Pasewark) 26 Using archival sound speed only: predicts slope induced bearing bias but not wander Sarchive = Reference score IW @S Source (300 Hz) 24 Receiver TEMPERATURE (deg C) 47m 2.5 h 22 IW @ R 20 77m 18 18 20 22 00 02 04 06 145 150 155 160 165 170 JD 126 BEARING (deg) TIME (hours) 75 85 95 105 BEAM POWER (dB re 1 mPa)