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Stochastic Versus Uncertainty Modeling

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Stochastic Versus Uncertainty Modeling Introduction Analytical Solutions The Cloud Model Conclusions Stochastic versus Uncertainty Modeling Petra Friederichs, Michael Weniger, Sabrina Bentzien, Andreas Hense Meteorological Institute, University of Bonn Dynamics and Statistic in Weather and Climate Dresden, July 29-31 2009 P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 1 / 21 Introduction Analytical Solutions Motivation The Cloud Model Outline Conclusions Uncertainty Modeling I Initial conditions { aleatoric uncertainty Ensemble with perturbed initial conditions I Model error { epistemic uncertainty Perturbed physic and/or multi model ensembles Simulations solving deterministic model equations! P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 2 / 21 Introduction Analytical Solutions Motivation The Cloud Model Outline Conclusions Stochastic Modeling I Stochastic parameterization { aleatoric uncertainty Stochastic model ensemble I Initial conditions { aleatoric uncertainty Ensemble with perturbed initial conditions I Model error { epistemic uncertainty Perturbed physic and/or multi model ensembles Simulations solving stochastic model equations! P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 3 / 21 Introduction Analytical Solutions Motivation The Cloud Model Outline Conclusions Outline Contrast both concepts I Analytical solutions of simple damping equation dv(t) = −µv(t)dt I Simplified, 1-dimensional, time-dependent cloud model I Problems P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 4 / 21 Introduction Perturbed Physics Analytical Solutions Stochastic Model The Cloud Model Conclusions Conclusions Equation of Motion Small particle of mass m with velocity v(t) Subject to damping force F (t) = −µm · v(t) dv(t) = −µv(t)dt Solution of deterministic problem v(t) = v0 exp(−µt) P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 5 / 21 Introduction Perturbed Physics Analytical Solutions Stochastic Model The Cloud Model Conclusions Conclusions 1.0 'Perturbed Physics' 0.8 deterministic γ is random variable normal 0.6 uniform gamma v 0.4 dv(t) = −γv(t)dt: 0.2 Solutions for different γ 0.0 0.0 0.5 1.0 1.5 2.0 t 2 2 σ 2 γN ∼ N (µ, σ ) E[vN (t)] = v0 exp − µt + t 2p p sinh( 3σt) γU ∼ U µ ± 3σ E[vU (t)] = v0 exp(−µt) p 3σt µ2 σ2 −µt −µ2/σ2 γ ∼ Γ ; E[v (t)] = v 1 − Γ σ2 µ Γ 0 µ2/σ2 P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 6 / 21 Introduction Perturbed Physics Analytical Solutions Stochastic Model The Cloud Model Conclusions Conclusions Stochastic Model Damping from small-scale stochastic process dv(t) = −µv(t)dt + v(t)σξdt; ξt : Langevin force with E[ξt ] = 0 and Cov[ξs ; ξt ] = δ(t − s) Stratonovich calculus dvt = −µvt dt + σvt ◦ dWt Solution vt = v0 exp − µt + σWt (exponential Brownian motion) P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 7 / 21 Introduction Perturbed Physics Analytical Solutions Stochastic Model The Cloud Model Conclusions Conclusions Exponential Brownian motion σ2 E[v ] = v exp − µ − t t 0 2 h σ2 i Varv = v 2 exp − 2µ − σ2t − exp − 2µ − t t 0 2 Three realisation of an exponential Brownian motion with µ=2, σ2=1 Three realisation of an exponential Brownian motion with µ=2, σ2=3 2 6 det. solution det. solution mean value mean value 1.8 5 1.6 1.4 4 1.2 t t v 1 v 3 0.8 2 0.6 0.4 1 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 Time t Time t P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 8 / 21 Introduction Perturbed Physics Analytical Solutions Stochastic Model The Cloud Model Conclusions Conclusions Non-delta Correlated Gaussian Noise dv(t) = −µdt + v(t)t v(t)dt Solution Z t vt = v0 exp − µt + s ds 0 Noise t with D E[ ] = 0; Cov[ ; ] = exp(−kjt − sj): t s t 2k is an Ornstein-Uhlenbeck process solving p dt = −kt dt + DdWt P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 9 / 21 Introduction Perturbed Physics Analytical Solutions Stochastic Model The Cloud Model Conclusions Conclusions Non-delta Correlated Gaussian Noise D 1 − e−kt E[v ] = v exp − µt + t − t 0 2k2 k D = 2k D = k2 1.0 1.0 deterministic deterministic 0.8 det. normal 0.8 det. normal Expon. Brownian Expon. Brownian OU(k=0.01) OU(k=0.01) 0.6 OU(k=1) 0.6 OU(k=1) OU(k=100) OU(k=100) v v 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 t t P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 10 / 21 Introduction Perturbed Physics Analytical Solutions Stochastic Model The Cloud Model Conclusions Conclusions D = 2k D = k2 P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 11 / 21 Introduction Perturbed Physics Analytical Solutions Stochastic Model The Cloud Model Conclusions Conclusions 'Perturbed physics' E[v(t)] ! 1 for t ! 1 for all parameter distributions with non-positive support Stochastic model Exponential Brownian motion: 2 2 I Non-stable solutions (σ > 2µ), increasing variance (σ > µ) OU noise: I Noise induced drift increases with 1=k (D = 2k) I 'Perturbed physics' solution for k ! 0 (D = 2k) 2 I Exponential Brownian motion for k ! 1 (D = k ) P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 12 / 21 Introduction Model Description Analytical Solutions Perturbed Physics The Cloud Model Stochastic Parameterization Conclusions 1-dimensional Cloud Model I Cylinder - constant radius I Horizontally homogen in constant environment I Prognostic equations for w(z), T (z), qv (z), qc (z), qr (z) Fig: Dynamics of Multiscale Earth Systems, C.Simmer(Eds.) P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 13 / 21 Introduction Model Description Analytical Solutions Perturbed Physics The Cloud Model Stochastic Parameterization Conclusions Koeln 04.07.1994 RR=11,723mm 10 g/kg 9 4 8 0.6 7 3 6 2 0.2 5 0.4 2 2 Hoehe in km 4 1.8 3 1.6 1.4 mm/h 1.2 1 1 2 0.8 40 0.4 0.2 30 1 20 10 0 0 0 0 10 20 30 40 50 60 70 80 90 Zeit in min P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 14 / 21 Introduction Model Description Analytical Solutions Perturbed Physics The Cloud Model Stochastic Parameterization Conclusions 1 @(ρw) 2 0 = − − u~ ρ @z R „ « @w @w 2 2 2 Tv − Tv;e = −w + α jwe − wj(we − w) + (w − w~ )~u + g − (qc + qr )g @t @z R R Tv;e @T @T 2 2 2 = −w + α jwe − wj(Te − T ) + (T − T~ )~u − Γ w + SS @t @z R R d T @qv @qv 2 2 2 dqv = −w + α jwe − wj(qv;e − qv ) + (qv − q~v )~u − @t @z R R dt @qc @qc 2 2 2 dqc = −w + α jwe − wj(qc;e − qc ) + (qc − q~c )~u − @t @z R R dt @qr @qr 2 2 2 dqr @r qr @(ρVR ) = −w + α jwe − wj(qr;e qr ) + (qr − q~r )~u − + V + @t @z R R dt R @z ρ @z | {z } | {z } | {z } | {z } advection turbulent entrainment dynamical entr. sources P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 15 / 21 Introduction Model Description Analytical Solutions Perturbed Physics The Cloud Model Stochastic Parameterization Conclusions Micro-physical processes after Kessler (1969) ... Fig: von der Emde, Kahlig (1989) P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 16 / 21 Introduction Model Description Analytical Solutions Perturbed Physics The Cloud Model Stochastic Parameterization Conclusions ? Condensation: dqv ? qv − qvws 1 − ? = 2 · for qv > qvws dt ? "Lc qvws ∆t qv ! qc VC 1 + 2 cpRaT ? dqc ? Conversion: − ? = kc (qc − qc;0) for qc > qc;0 dt ?CR q ! q AU 1 −3 c r qc;0 = ρ , kc = 10 ? dqc ? π −(3+βr ) − ? = Eαr n0;r qc · λ · Γ(3 + βr ) dt ? 4 Accretion: CRAC qc ! qr E = 1 collection efficiency, αr , βr ;n0;r : Marshall-Palmer (3+βr ) 4 λ−(3+βr ) = ρqr πρps n0;r P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 17 / 21 Introduction Model Description Analytical Solutions Perturbed Physics The Cloud Model Stochastic Parameterization Conclusions Perturbed Physics kc ! kc (1 + η) E ! E (1 + η) 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 rain in mm 6 rain in mm 6 5 5 4 4 3 3 2 2 1 Conversion 1 Accretion 0 0 −1 0 1 2 −1 0 1 2 P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 18 / 21 Introduction Model Description Analytical Solutions Perturbed Physics The Cloud Model Stochastic Parameterization Conclusions Perturbed Physics 0.05 Conversion 0.12 Accretion 0.10 0.04 0.08 0.03 0.06 density density 0.02 0.04 0.01 0.02 0.00 0.00 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 rain in mm rain in mm Introduction Analytical Solutions The Cloud Model = Conclusions P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 19 / 21 Introduction Model Description Analytical Solutions Perturbed Physics The Cloud Model Stochastic Parameterization Conclusions Stochastic Parameterization (solved with Milstein Scheme) kc ! kc (1 + ξt ) E ! E (1 + ξt ) 0.08 Conversion 0.05 Accretion 0.04 0.06 0.03 0.04 density density 0.02 0.02 0.01 0.00 0.00 11.64 11.68 11.72 11.76 11.8 11.2 11.4 11.6 11.8 12 12.2 12.4 rain in mm rain in mm P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 20 / 21 Introduction Analytical Solutions The Cloud Model Conclusions Conclusions and Outlook I 'Perturbed physics' ensembles do not represent realizations from real process I Stochastic parameterization may provide realistic distributions I Solutions strongly depend on covariance function of noise (in time and in space) I Stochastic parameterizations should be derived from microphysical processes P.Friederichs, M.Weniger, S.Bentzien, A.Hense Stochastic versus Uncertainty Modeling 21 / 21.
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