Bayesian inverse modeling for quantitative precipitation estimation (QPE)

Transcription

1 University Meteorological Institute Bonn Bayesian inverse modeling for quantitative precipitation estimation (QPE) Katharina Schinagl, 1 Christian Rieger, 2 Clemens Simmer, 1 Xinxin Xie, 1 Alexander Rüttgers, 2 Petra Friederichs 1 1 Meteorological Institute, University of Bonn, Bonn, Germany 2 Institute for Numerical Simulation, University of Bonn, Bonn, Germany July 4th, 2016, BCAM-IMUVA Summer School on Uncertainty Quantification for Applied Problems Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 1 / 19

2 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 2 / 19

3 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 2 / 19

4 Radar quantitative precipitation estimation (QPE) Picture from www. general-anzeiger-bonn. de (Thomas Franz) Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 3 / 19

5 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 3 / 19

6 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 3 / 19

7 Polarimetric radar radio waves are a series of oscillating electromagnetic fields horizontal orientation vertical orientation polarimetric radar (= dual-polarization radar) waves passing through clouds drop size distribution horizontal reflectivity Z H horizontal reflectivity Z V differential reflectivity Z DR specific differential phase K DP Picture from www. roc. noaa. gov/ wsr88d/ dualpol/ cross-correlation coefficient ρ HV Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 4 / 19

8 Radar error sources Picture from http: // www. eumetcal. org/ euromet/ english/ satmet/ s9320/ s htm partial beam blockage miscalibration attenuation systematic errors are corrected in the data random errors need to be included in the model Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 5 / 19

9 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 5 / 19

10 DSD and precipitation estimates Exponential DSD model [Marshall and Palmer, 1948] For D [D min, D max ], 0 < D min < D max < : N(Λ, N 0, D) = N 0 exp ( ΛD) [1/mm m 3 ] D equivalent volume diameter [mm] Λ slope parameter in [ 1 /mm] N 0 number concentration parameter in [ 1 /mm m 3 ] rain rate, after [Uijlenhoet and Pomeroy, 2001]: R = 6π 10 4 Dmax D min D 3 v(d)n(d)dd [mm/h] Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 6 / 19

11 Typical radar measurements after [Brandes et al., 2004]: horizontal reflectivity Z H = vertical reflectivity Z V = differential reflectivity D droplet diameter in [mm] N(D) drop size distribution 4λ4 Dmax π 4 K w 2 f HH (π, D) 2 N(D)dD [mm 6 /m 3 ] D min 4λ4 Dmax π 4 K w 2 f VV (π, D) 2 N(D)dD [mm 6 /m 3 ] D min Z DR = 10 log Z H Z V [db] f HH (π, D), f VV (π, D) back-scattering amplitudes of a drop at horizontal/vertical polarization f HH (D, 0), f VV (D, 0) forward-scattering amplitudes at horizontal/vertical polarization K w dielectric constant of water λ radar wavelength in [cm] Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 7 / 19

12 Typical radar measurements, ctd. specific differential phase K DP = 180λ π Dmax cross-correlation coefficient D min Re(f HH (0, D) f VV (0, D))N(D)dD [deg/km], Dmax fhh (π, D)f VV (π, D)N(D)dD D ρ HV = min Dmax f HH (π, D) 2 Dmax N(D)dD f VV (π, D) 2 N(D)dD D min D droplet diameter in [mm] N(D) drop size distribution f HH (π, D), f VV (π, D) back-scattering amplitudes of a drop at horizontal/vertical polarization D min f HH (D, 0), f VV (D, 0) forward-scattering amplitudes at horizontal/vertical polarization λ radar wavelength in [cm] Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 8 / 19

13 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 8 / 19

14 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 8 / 19

15 Bayesian estimation of DSD parameters data (Z H, Z V, Z DR, K DP, ρ HV ) unknown parameters (N 0, Λ) forward operators + error models prediction (R... at unobserved locations/time) Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 9 / 19

16 Inverse problem infer values of the parameters that characterize the system from given data measurement error Bayesian framework and MCMC methods for numerical solution Inverse problem X = R n state space u X unknown model parameter(s) Y = R q observation space y Y data η Y observational noise G : X Y forward operator y = G(u) + η y = G(u) e η (additive) (multiplicative) Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 10 / 19

17 Types of uncertainty uncertainty experimental (observational) uncertainty model uncertainty numerical uncertainty Bayes: prior sensitivity What error do we make by choosing a wrong prior? sensitivity to likelihood What error do we make by choosing a wrong likelihood? Inverse problem: parameter identifiability possible underdetermination condition of forward/backward operators Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 11 / 19

18 Bayesian estimation of DSD parameters, ctd. Bayes formula Λ, N 0 R p DSD parameters y R q data evidence P(Λ, N 0 y) P(y Λ, N 0 )P(Λ, N 0 ) multi-data mode, e.g. y = (Z H, Z DR, ρ HV ) with Z H R q, Z DR R q, ρ HV R q conditional independence of data P(Z H, Z DR, ρ HV Λ, N 0 ) = P(Z H Λ, N 0 )P(Z DR Λ, N 0 )P(ρ HV Λ, N 0 ) Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 12 / 19

19 Model setup (data likelihood) modeling of random errors Z H = G ZH (Λ, N 0 ) ɛ Z V = G ZV (Λ, N 0 ) ɛ Z DR = G ZDR (Λ, N 0 ) ɛ K DP = G KDP (Λ, N 0 ) + ɛ 1 [db] 1 [db] 0.1 [db] 1 [deg/km] Z H = G ZH (Λ, N 0, t) = 4λ4 Dmax π 4 K w 2 f HH (π, D) 2 N(D)dD [mm 6 /m 3 ] D min this term and the respective variants for Z V, Z DR, K DP and ρ HV have to be evaluated very often efficient strategy needed lookup-tables simplifications: we assume the data to be only rain, from summer, low in the beam, below the brightband Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 13 / 19

20 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 13 / 19

21 COSMO-DE numerical weather prediction model GME COSMO-DE COSMO-EU Picture from S. Theis, DWD Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 14 / 19

22 Using the COSMO data goal: estimate the DSD parameters Λ and N 0 Λ, N 0 unknown for real world-data (except very occasional measurements) we go into the simulated environment in COSMO validate model against values there 3D polarimetric radar forward operator in COSMO (Xinxin Xie, MIUB) uncertainties in the simulated environment as estimate of uncertainties in the real world Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 15 / 19

23 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 15 / 19

24 Working with the COSMO-data 1 COSMO-DE data for two days: June 6th 2011 and June 22nd 2011 qnr, qr, qc, qi, qs, qg, qh, qv, t, ps 2 compute DSD parameters N 0 and Λ from COSMO-DE output 3 from N 0, Λ and t get polarimetric variables Z H, Z V, K DP, ρ HV radar forward operator, t-matrix method 4 choose suitable subset for computations Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 16 / 19

25 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 16 / 19

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50 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 17 / 19

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76 Outline 1 Introduction 2 Polarimetric radar Introduction to polarimetric radar Radar forward formulas 3 Modeling Bayesian inverse problem approach COSMO-DE numerical weather prediction model 4 Data sets 22nd June, th June, Summary and Outlook Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 18 / 19

77 Outlook up to now: experiment with generated test data prototype algorithm evaluation of available data for June 6th and June 22nd next steps: use COSMO-based data as input to MCMC improve efficiency of calls to radar forward operator (lookup-tables) Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 19 / 19

78 University Meteorological Institute Bonn Thank you for your attention! Brandes, E. A., Zhang, G., and Vivekanandan, J. (2004). Comparison of polarimetric radar drop size distribution retrieval algorithms. Journal of Atmospheric and Oceanic Technology, 21(4): Marshall, J. S. and Palmer, W. M. K. (1948). The distribution of raindrops with size. Journal of meteorology, 5(4): Uijlenhoet, R. and Pomeroy, J. (2001). Raindrop size distributions and radar reflectivity? rain rate relationships for radar hydrology. Hydrology and Earth System Sciences Discussions, 5(4): Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 19 / 19

79 scattering amplitudes for the forward formulas we need f HH (0, D) forward-scattering amplitude f VV (0, D) forward-scattering amplitude f HH (D) backward-scattering amplitude f VV (D) backward-scattering amplitude forward scattering is for θ = 0 and backward scattering for θ = 180 these are entries of the scattering matrix, e.g. forward scattering matrix ( S f s f = 11 s22 f ) ( ) fvv (0, D) f s21 f s22 f = VH (0, D) f HV (0, D) f HH (0, D) scattering amplitudes depend on incident wave (angles, frequency...), drop (shape, diameter, refractive index) Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 1 / 3

80 Computation via t-matrix method scattered wave can be written as [ E sca] ν exp (ikr) Eφ sca = S(n sca, n R inc ) [ E inc ν E inc φ ] (1) write inc and sca fields in vector spherical functions E inc (R) = E sca (R) = n=1 n=1 n m= n n m= n [a mn RgM mn (kr) + b mn RgN mn (kr)] (2) [p mn RgM mn (kr) + q mn RgN mn (kr)], R > r 0 (3) Tmatrix gives relation between coefficients of inc and scattered field [ ] [ p T 11 T = 12 ] [ ] a q T 21 T 22 b T matrix depends only on the properties of the scattering particle (shape, size, refractive index, orientation...) and is independent of the directions of scattering compute Tmatrix once, then use it for computation of different scattering matrices S Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 2 / 3 (4)

81 Numerical computation of T-matrix extended boundary condition method (EBCM) vector spherical functions also for internal field E int (R) = m r refractive index also linear relation n n=1 m= n [c mn RgM mn (m r kr) + d mn RgN mn (m r kr)] (5) [ ] a = b [ Q 11 Q 12 Q 21 Q 22 ] [ ] c d elements of Q are two-dimensional integrals over particle surface which are solved numerically (depend on particle properties) T = RgQ [Q] 1 formulas for RgQ and Q in literature T-matrix method is analytically exact, numerical convergence depends on Q computation time depends (mostly?) on particle size and aspect ratio (6) Katharina Schinagl Meteorological Institute of the University of Bonn (MIUB) 3 / 3