CERTAIN THOUGHTS ON UNCERTAINTY ANALYSIS FOR DYNAMICAL SYSTEMS

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1 CERTAIN THOUGHTS ON UNCERTAINTY ANALYSIS FOR DYNAMICAL SYSTEMS Puneet Singla Assistant Professor Department of Mechanical & Aerospace Engineering University at Buffalo, Buffalo, NY-1426 Probabilistic Analysis of Volcanic Hazards: Current Methodologies and Vision for Future Efforts A Workshop at the University at Buffalo May 16 May 19, 211 Collaborators: M. Bursik, M. Jones, A. Patra, E. B. Pitman, P. Scott, T. Singh H. Bjornsson, S. Carn, K. Dean, M. Pavolonis, M. Ripepe, P. Webley

2 INTRODUCTION PROBLEM STATEMENT Stochastic System: ẋ = A(Θ)x + B(Θ)u + G(Θ)η, x(t ) = µ. Source of Uncertainties: system parameters, initial conditions, input to the system, modeling error. Robust modeling of the propagation of these uncertainties is important to accurately quantify the uncertainty in the solution at any future time. X 3 η i f(x 2,ηη 2,θ) X 1 f(x 1,η 1,θ) f(x,η,θ) X X2 p(x3,t 3,θ) p(x 1,t 1,θ) p(x 2,t 2,θ) ) p(x,t,θ) FIGURE: State and pdf transition. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 2 / 24

3 INTRODUCTION PROBLEM STATEMENT LINEAR SYSTEMS WITH KNOWN PARAMETER: ẋ = A(Θ )x + B(Θ )u + G(Θ )η State pdf is Gaussian assuming initial condition and input uncertainty to be Gaussian in nature. Drawback: appending unknown parameters as additional state variables lead to non-linear system with non-gaussian pdfs. LINEAR SYSTEMS WITH UNKNOWN PARAMETER: ẋ = A(Θ)x + B(Θ)u + G(Θ)η Generalized Polynomial Chaos (gpc): propagate time-invariant parametric uncertainty through an otherwise deterministic system of equations. Drawback: Using gpc series expansion for the time-varying stochastic forcing terms is computationally expensive. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 3 / 24

4 INTRODUCTION PROBLEM STATEMENT LINEAR SYSTEMS WITH KNOWN PARAMETER: ẋ = A(Θ )x + B(Θ )u + G(Θ )η State pdf is Gaussian assuming initial condition and input uncertainty to be Gaussian in nature. Drawback: appending unknown parameters as additional state variables lead to non-linear system with non-gaussian pdfs. LINEAR SYSTEMS WITH UNKNOWN PARAMETER: ẋ = A(Θ)x + B(Θ)u + G(Θ)η Generalized Polynomial Chaos (gpc): propagate time-invariant parametric uncertainty through an otherwise deterministic system of equations. Drawback: Using gpc series expansion for the time-varying stochastic forcing terms is computationally expensive. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 3 / 24

5 INTRODUCTION MOTIVATION Main challenge: characterizing the uncertainty in the system states due to both parametric and temporal stochastic uncertainties simultaneously Approximate Solution to exact problem: Multiple-model estimation method, Monte Carlo (MC) methods. Exact solution to approximate problem: Gaussian closure, Equivalent Linearization, and Stochastic Averaging. Main Objective: develop analytical means to accurately characterize the state pdf of a linear system subject to initial condition uncertainty, white noise excitation and possibly non-gaussian parametric uncertainty. Uncertainty Marriage: handshake of Kalman filter with gpc. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 4 / 24

6 INTRODUCTION MOTIVATION Main challenge: characterizing the uncertainty in the system states due to both parametric and temporal stochastic uncertainties simultaneously Approximate Solution to exact problem: Multiple-model estimation method, Monte Carlo (MC) methods. Exact solution to approximate problem: Gaussian closure, Equivalent Linearization, and Stochastic Averaging. Main Objective: develop analytical means to accurately characterize the state pdf of a linear system subject to initial condition uncertainty, white noise excitation and possibly non-gaussian parametric uncertainty. Uncertainty Marriage: handshake of Kalman filter with gpc. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 4 / 24

7 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η METHOD 1: CONDITIONING FIRST ON UNCERTAIN PARAMETERS The conditional state pdf p(x Θ) is a normal distribution with mean µ and covariance Σ, i.e. p(x Θ) = N (x; µ, Σ). µ = A(Θ)µ + Bu (1) Σ = A(Θ)Σ + ΣA T (Θ) + G(Θ)QG T (Θ) (2) These conditional moment propagation equations are exact and depend only on the initial moments and the model parameters. The complete distribution of the original state vector x is: p(t,x) = p(t,x Θ(ξ ))p(ξ )dξ (3) = Ω Ω N (t,x; µ(t,θ), Σ(t,Θ))p(ξ )dξ (4) P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 5 / 24

8 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η POLYNOMIAL CHAOS Polynomial chaos is a term originated by Norbert Wiener in 1938, to describe the members of the span of Hermite polynomial functionals of a Gaussian process. According to the Cameron-Martin Theorem, the Fourier-Hermite polynomial chaos expansion converges, in the L 2 sense, to any arbitrary process with finite variance (which applies to most physical processes). This has been generalized by Xiu et al. (22) to efficiently use the orthogonal polynomials from the Askey-scheme to model various probability distributions. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 6 / 24

9 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η GENERALIZED POLYNOMIAL CHAOS: BASIC GOAL Approximate the stochastic system states in terms of a finite-dimensional series expansion in the infinite-dimensional stochastic space. The unknown coefficients are determined by minimizing an appropriate norm of the residual. The basis can be chosen for a given pdf, to represent the random variable with the fewest number of terms. For example, Hermite polynomials for Gaussian r.v. and Legendre polynomials for uniform r.v. The completeness of the space allows for the accurate representation of any random variable, with a given probability density function (pdf), by a suitable projection on the space spanned by a carefully selected basis. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 7 / 24

10 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η METHOD 1: p(t,x) = Ω N (t,x; µ(t,θ), Σ(t,Θ))p(ξ )dξ p(x Θ) = N (x; µ, Σ) µ = A(Θ)µ + Bu (5) Σ = A(Θ)Σ + ΣA T (Θ) + G(Θ)QG T (Θ) (6) Combining the state mean and covariance terms into a new state vector z R N where N = n(n + 3)/2 leads to ż = L(Θ)z + M(Θ)w (7) P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 8 / 24

11 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η METHOD 1: ż = L(Θ)z + M(Θ)w According to gpc z i (t, ξ ) Θ j (ξ ) L i j (Θ) = P z ir (t)φ r (ξ ) = z T i (t)φ(ξ ) (8) r= = P θ jr φ r (ξ ) = θ j T Φ(ξ ) (9) r= = P L i jr φ r (ξ ) = l T i jφ(ξ ) (1) r= M i j (Θ) L i jr = L i j(θ(ξ )),φ r (ξ ) M φ r (ξ ),φ r (ξ ) i jr = M i j(θ(ξ )),φ r (ξ ) φ r (ξ ),φ r (ξ ) where u(ξ ),v(ξ ) = Ω u(ξ )v(ξ )p(ξ )dξ = P M i jr φ r (ξ ) = m T i jφ(ξ ) (11) r= P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 9 / 24

12 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η METHOD 1: ż = L(Θ)z + M(Θ)w Substitution of the approximate expressions for x and Θ in moment equations leads to: e i (ξ ) =ż T i Φ(ξ ) N l T i jφ(ξ )z T q j (t)φ(ξ ) m T i jφ(ξ )w j (t) j=1 j=1 Galerkin projection leads to following deterministic equations: e i (ξ ),φ r (ξ ) = for i = 1,...,N and r =,...,P ċ = A p c + B p w (12) where, c(t) = [z T 1 (t),zt 2 (t),...,zt N (t)]t R N(P+1) is a vector of the gpc coefficients. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 1 / 24

13 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η METHOD 1: p(t,x) = Ω N (t,x; µ(t,θ), Σ(t,Θ))p(ξ )dξ The pdf of state vector x ( can be computed as ) follows: P p(x) = N t,x; z ir (t)φ r (ξ ) p(ξ )dξ (13) Ω Quadrature integration scheme can be used to evaluate the aforementioned integrals. Polynomial Chaos Quadrature (PCQ) [Dalbey et al., 28] r= The first two moments of the actual state vector x can be estimated analytically, as follows: P E[x i (t)] = E[N (t,x; z ir (t)φ r (ξ ))]p(ξ )dξ = µ i (t) (14) E[x 2 i (t)] = Ω Ω r= E[x 2 i (t) ξ ]p(ξ )dξ = P r= µ i 2 r (t) φ 2 r + Σ ii (t) (15) P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 11 / 24

14 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η METHOD 1: p(t,x) = Ω N (t,x; µ(t,θ), Σ(t,Θ))p(ξ )dξ The pdf of state vector x ( can be computed as ) follows: P p(x) = N t,x; z ir (t)φ r (ξ ) p(ξ )dξ (13) Ω Quadrature integration scheme can be used to evaluate the aforementioned integrals. Polynomial Chaos Quadrature (PCQ) [Dalbey et al., 28] r= The first two moments of the actual state vector x can be estimated analytically, as follows: P E[x i (t)] = E[N (t,x; z ir (t)φ r (ξ ))]p(ξ )dξ = µ i (t) (14) E[x 2 i (t)] = Ω Ω r= E[x 2 i (t) ξ ]p(ξ )dξ = P r= µ i 2 r (t) φ 2 r + Σ ii (t) (15) P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 11 / 24

15 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η Parametric Uncertainties Ω Stochastic Forcing [ p( t, x Θ( ξ ) ] = p( ξ) dξ Ω N Moment Propagation p x Θ ) ( 1 Physical Process Θ 1 Θ 2 Θ i Θ N 1 Moment Propagation Moment Propagation p x Θ ) ( i Ω p( x, t) Θ N Moment Propagation gpc Moment Propagation p x Θ ) ( N (a) Method 1: Conditioning first on uncertain parameters [Konda et al., 211] P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 12 / 24

16 PROPOSED APPROACH ẋ = A(Θ)x + B(Θ)u + G(Θ)η P h y s i c a l P r o c e s s Stochastic Forcing [ p( t, x η( ω) ] N = N( t, ω;, Q) dω Parametric Uncertainties Ω gpc p x η ) ( 1 η 1 Ω gpc η 2 η i η N 1 η N Ω Ω gpc gpc p x η ) ( i Moment propagation p( x, t) Ω gpc p x η ) ( N (b) Method 2: Conditioning first on Gaussian stochastic forcing [Konda et al., 211] P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 13 / 24

17 PROPOSED APPROACH ADVANTAGES One can compute the sensitivity of mean and covariance of conditional pdf p(t, x Θ) = N (t, x; z(ξ )) with respect to unknown parameter vector Θ(ξ ). Computational Cost: Method 1: n(n + 3)(P + 1)/2 simultaneous deterministic ODEs. Method 2: n(p + 1)[n(P + 1) + 3]/2 simultaneous deterministic ODEs. SMC: nn differential equations, N being number of Monte Carlo samples. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 14 / 24

18 NUMERICAL EXPERIMENT HOVERING HELICOPTER MODEL ẋ = Ax + Bu + B w u w, x = {u h,q h,θ h,y} T p 1 p 2 g p 5 p 1 A = p 3 p 4 1, B = p 6, B w = p 3 1 Four aerodynamic parameters are assumed to uniformly distributed with following bounds: p lb = [.49,.1,.126, ] T, pub = [.3,.25,2.394,.177 ] T Modeling errors due to unsteady flow, u w are assumed to be zero mean Gaussian white noise with a variance of σw 2 = 18( ft/s) 2. 4 u h 3 q h θ h 2 y nominal states time (s) (c) Hovering helicopter (d) Nominal states P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 15 / 24

19 NUMERICAL EXPERIMENT HOVERING HELICOPTER MODEL Mean of x Monte Carlo ref Method 1: PC of µ,σ Method 2: µ,σ of PC Mean of x CM 2 of x Monte Carlo ref Method 1: PC of µ,σ Method 2: µ,σ of PC CM 2 of x Mean of x 3 1 Mean of x CM 2 of x CM 2 of x time (s) time (s) time (s) time (s) (e) Propagation of mean (f) Propagation of variance CM 3 of x Monte Carlo ref Method 1: PC of µ,σ Method 2: µ,σ of PC CM 3 of x CM 3 of x Monte Carlo ref Method 1: PC of µ,σ Method 2: µ,σ of PC CM 3 of x CM 3 of x CM 3 of x CM 3 of x CM 3 of x time (s) time (s) time (s) time (s) (g) Propagation of 3rd central moment (h) 3rd central moment with 1 Monte Carlo runs FIGURE: Propagation of moments of the states [Konda et al., 211] P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 16 / 24

20 NUMERICAL EXPERIMENT HOVERING HELICOPTER MODEL 1 Monte Carlo reference 1 Method 1: PC of µ,σ 1 Method 2: µ,σ of PC x 1 Monte Carlo reference x 1 Method 1: PC of µ,σ x 1 Method 2: µ,σ of PC x 2 Monte Carlo reference x 2 Method 1: PC of µ,σ x 2 Method 2: µ,σ of PC x 3 Monte Carlo reference x 3 Method 1: PC of µ,σ x 3 Method 2: µ,σ of PC x x x 4 FIGURE: Histograms of the states [Konda et al., 211] P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 17 / 24

21 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION The BENT integral eruption column model was used to produce eruption column parameters (mass loading, column height, grain size distribution) given a specific atmospheric sounding and source conditions. BENT takes into consideration atmospheric (wind) conditions as given by atmospheric sounding data. Plume rise height is given as a function of volcanic source and environmental conditions. The PUFF Lagrangian model was used to propagate ash parcels in a given wind field (NCEP Reanalysis). PUFF takes into account dry deposition as well as dispersion and advection. Polynomial chaos quadrature (PCQ) was used to select sample points and weights in the uncertain input space of vent radius, vent velocity, mean particle size and particle size variance. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 18 / 24

22 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION The BENT integral eruption column model was used to produce eruption column parameters (mass loading, column height, grain size distribution) given a specific atmospheric sounding and source conditions. BENT takes into consideration atmospheric (wind) conditions as given by atmospheric sounding data. Plume rise height is given as a function of volcanic source and environmental conditions. The PUFF Lagrangian model was used to propagate ash parcels in a given wind field (NCEP Reanalysis). PUFF takes into account dry deposition as well as dispersion and advection. Polynomial chaos quadrature (PCQ) was used to select sample points and weights in the uncertain input space of vent radius, vent velocity, mean particle size and particle size variance. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 18 / 24

23 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION The BENT integral eruption column model was used to produce eruption column parameters (mass loading, column height, grain size distribution) given a specific atmospheric sounding and source conditions. BENT takes into consideration atmospheric (wind) conditions as given by atmospheric sounding data. Plume rise height is given as a function of volcanic source and environmental conditions. The PUFF Lagrangian model was used to propagate ash parcels in a given wind field (NCEP Reanalysis). PUFF takes into account dry deposition as well as dispersion and advection. Polynomial chaos quadrature (PCQ) was used to select sample points and weights in the uncertain input space of vent radius, vent velocity, mean particle size and particle size variance. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 18 / 24

24 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION TABLE: Eruption source parameters based on observations of Eyjafjallajökull volcano and information from other similar eruptions. Parameter Value range PDF Comment Vent radius, b, m Vent velocity, w, m/s Mean grain 2 boxcars: size, Md ϕ and Uniform Measured from radar image of summit vents Range: 45- Uniform M. Ripepe, Geneva, Switzerland, , presentation Multi- Modal Uniform Woods and Bursik (1991), Table 1, vulcanian and phreatoplinian. A. Hoskuldsson, Iceland meeting 21, presentation σ ϕ 1.9 ±.6 Uniform Woods and Bursik (1991), Table 1, vulcanian and phreatoplinian. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 19 / 24

25 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION % Change in Concentration Height = 2km Height = 2 4km Height = 4 6km Number of Particles FIGURE: Concentration (52N 13.5E) vs. Number of PUFF Particles P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 2 / 24

26 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION nash Conc P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 2 / 24

27 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION FIGURE: SEVIRI Data: Ash Top Height (16th April, 21) P. SINGLA LAIRS.ENG.BUFFALO.EDU 21 / 24

28 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION FIGURE: PCQ Runs: Ash Top Height (Mean) (16th April, 21) P. SINGLA LAIRS.ENG.BUFFALO.EDU 21 / 24

29 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION FIGURE: PCQ Runs: Ash Top Height (Std. Dev.) (16th April, 21) P. SINGLA LAIRS.ENG.BUFFALO.EDU 21 / 24

30 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION FIGURE: SEVIRI Data: Ash Top Height (15th &16th April, 21) P. SINGLA LAIRS.ENG.BUFFALO.EDU 22 / 24

31 PRELIMINARY RESULTS ICELAND VOLCANO (EYJAFJALLAJÖKULL) ERUPTION FIGURE: PCQ Runs: Ash Top Height (15th & 16th April, 21) P. SINGLA LAIRS.ENG.BUFFALO.EDU 22 / 24

32 CONCLUDING REMARKS WHAT NEEDS TO BE DONE Efficient hybrid Bayesian approach is developed for the accurate determination of uncertainty propagation in linear dynamic models with Parametric, initial condition uncertainties, and driven by additive white Gaussian noise (AWGN) process. Can be extended for nonlinear systems using Kolmogorov equation. The uncertainty due to stochastic forcing is propagated using mean and covariance propagation equations and that due to uncertain model parameters using polynomial chaos. The moment propagation equations are exact only for white Gaussian stochastic forcing in linear dynamic models, the polynomial chaos approach can be used for any probability distribution of model parameters. Proposed approach is less computationally demanding than the standard Monte Carlo methods. P. SINGLA LAIRS.ENG.BUFFALO.EDU 23 / 24

33 CONCLUDING REMARKS WHAT NEEDS TO BE DONE Efficient hybrid Bayesian approach is developed for the accurate determination of uncertainty propagation in linear dynamic models with Parametric, initial condition uncertainties, and driven by additive white Gaussian noise (AWGN) process. Can be extended for nonlinear systems using Kolmogorov equation. The uncertainty due to stochastic forcing is propagated using mean and covariance propagation equations and that due to uncertain model parameters using polynomial chaos. The moment propagation equations are exact only for white Gaussian stochastic forcing in linear dynamic models, the polynomial chaos approach can be used for any probability distribution of model parameters. Proposed approach is less computationally demanding than the standard Monte Carlo methods. P. SINGLA LAIRS.ENG.BUFFALO.EDU 23 / 24

34 CONCLUDING REMARKS WHAT NEEDS TO BE DONE Efficient hybrid Bayesian approach is developed for the accurate determination of uncertainty propagation in linear dynamic models with Parametric, initial condition uncertainties, and driven by additive white Gaussian noise (AWGN) process. Can be extended for nonlinear systems using Kolmogorov equation. The uncertainty due to stochastic forcing is propagated using mean and covariance propagation equations and that due to uncertain model parameters using polynomial chaos. The moment propagation equations are exact only for white Gaussian stochastic forcing in linear dynamic models, the polynomial chaos approach can be used for any probability distribution of model parameters. Proposed approach is less computationally demanding than the standard Monte Carlo methods. P. SINGLA LAIRS.ENG.BUFFALO.EDU 23 / 24

35 CONCLUDING REMARKS WHAT NEEDS TO BE DONE We were able to predict ash footprint with good confidence even though source uncertainty is large. False positives are large. Predicted average mass loading is of similar magnitude assuming a 1-km thick down wind plume as suggested by CALIPSO data. Convergence in concentration value is slow. The PCQ approach uses BENT and PUFF as black-box models Any other models for ash dispersion can be included in the uncertainty analysis. Effect of uncertainty in wind data still needs to be analyzed. Uncertainty Marriage Question arise: Can we estimate the distribution of source parameters using Satellite imagery or other sensory data? Likelihood functions: validating the sensor data. uniqueness of the parameters. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 24 / 24

36 CONCLUDING REMARKS WHAT NEEDS TO BE DONE We were able to predict ash footprint with good confidence even though source uncertainty is large. False positives are large. Predicted average mass loading is of similar magnitude assuming a 1-km thick down wind plume as suggested by CALIPSO data. Convergence in concentration value is slow. The PCQ approach uses BENT and PUFF as black-box models Any other models for ash dispersion can be included in the uncertainty analysis. Effect of uncertainty in wind data still needs to be analyzed. Uncertainty Marriage Question arise: Can we estimate the distribution of source parameters using Satellite imagery or other sensory data? Likelihood functions: validating the sensor data. uniqueness of the parameters. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 24 / 24

37 CONCLUDING REMARKS WHAT NEEDS TO BE DONE We were able to predict ash footprint with good confidence even though source uncertainty is large. False positives are large. Predicted average mass loading is of similar magnitude assuming a 1-km thick down wind plume as suggested by CALIPSO data. Convergence in concentration value is slow. The PCQ approach uses BENT and PUFF as black-box models Any other models for ash dispersion can be included in the uncertainty analysis. Effect of uncertainty in wind data still needs to be analyzed. Uncertainty Marriage Question arise: Can we estimate the distribution of source parameters using Satellite imagery or other sensory data? Likelihood functions: validating the sensor data. uniqueness of the parameters. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 24 / 24

38 CONCLUDING REMARKS WHAT NEEDS TO BE DONE We were able to predict ash footprint with good confidence even though source uncertainty is large. False positives are large. Predicted average mass loading is of similar magnitude assuming a 1-km thick down wind plume as suggested by CALIPSO data. Convergence in concentration value is slow. The PCQ approach uses BENT and PUFF as black-box models Any other models for ash dispersion can be included in the uncertainty analysis. Effect of uncertainty in wind data still needs to be analyzed. Uncertainty Marriage Question arise: Can we estimate the distribution of source parameters using Satellite imagery or other sensory data? Likelihood functions: validating the sensor data. uniqueness of the parameters. P. SINGLA (PSINGLA(@)BUFFALO.EDU) LAIRS.ENG.BUFFALO.EDU 24 / 24

Uncertainty Evolution In Stochastic Dynamic Models Using Polynomial Chaos

Noname manuscript No. (will be inserted by the editor) Uncertainty Evolution In Stochastic Dynamic Models Using Polynomial Chaos Umamaheswara Konda Puneet Singla Tarunraj Singh Peter Scott Received: date