Observations 3: Data Assimilation of Water Vapour Observations at NWP Centres

Transcription

1 Observations 3: Data Assimilation of Water Vapour Observations at NWP Centres OUTLINE: Data Assimilation A simple analogy: data fitting 4D-Var The observation operator : RT modelling Review of Radiative Transfer MW Clear sky RT: weighting functions Spectroscopy (v o,s, F) Line-by-Line to fast RT models Scattering RT Summary IR IR vs MW

2 Variational assimilation: A simple analogy - data fitting o Given some measurements, y(x), find the best model fit for model f(x,c) by adjusting coefficients in model (c) to achieve a best fit: Define a cost, J, that measures misfit of model to measurements eg RMS of y-f(x,c) Minimise the cost wrt c Involves computing the gradient of J wrt c o With prior knowlege of c,eg previous estimates of c with uncertainties U(c ), we can modify the cost to include two terms : misfit to model + misfit to prior estimates, something like : ( c c J ( c) = u( c ) ) ( y f ( x, c)) + u( y)

3 Data Assimilation: 4D-Var Data assimilation combines information from Observations 4DVAR A short-range background forecast that carries forward the information extracted from prior observations Error statistics Dynamical and physical relationships This produces the most probable atmospheric state (maximum-likelihood estimate)*** ***if background and observation errors are Gaussian, unbiased, uncorrelated with each other; all error covariances are correctly specified; model errors are negligible within the 1-h analysis window J(x) = (x background constraint b x) T B 1 (x b [ Ì (x)] x) + observation constraint T 1 [ y h(x) ] R [ y h(x) ] h(x) = h simulates the observations

4 Radiance Assimilation

5 Scattering in the MW: Why MW radiometers see through clouds and IR radiometers don t! Importance of scattering governed by size parameter: x=!r/_. Mie scattering important for x ~ 1 or greater. For water clouds, r~1_m, MW _ s ~-6 mm, -> x~1 - -> water clouds appear as homogeneous absorbers/emitters, -> scattering unimportant. For ice, rain or snow, r~1-1mm, x~1 -> scattering important.

6 Scattering: size parameter for clouds / hydrometeors (Adapted from Liou, An introduction to atmospheric radiation,. Table 5.1) A. Bodas-Salcedo

7 Microwave Sounders in operational systems F-16 SSMIS N-15 F-17 SSMIS T information from 5-6 GHz O absorption EOS-AQUA N16 N18 N19 N MetOp-A Q information from 183 GHz H O absorption, and window channels at (19,, 37, 89 and 15 GHz) Met Office ECMWF

8 Microwave data: some examples of measured radiances Brightness Temperature measurements obtained over ~1 hours by ( F-16 SSMIS ) 5.3 GHz 5.8 GHz 53.6 GHz. 15 GHz 183±7 GHz 183±5 GHz 54.4 GHz 183±1 GHz K 3K 15K 3K

9 Microwave data: some examples of measured radiances 1K 19 GHz (H pol) 3K Measurements obtained over ~1 hours by (F-16 SSMIS) 19 GHz (V pol) GHz (V pol) 37 GHz (H pol) 37 GHz (V pol)

10 Recap: Radiative transfer equation in clear skies S(ensor) di (over ds) Change in radiant intensity along a path is given by the sum of source and loss terms, in the clear sky case: emission and absorption, respectively: Hence: and: di emit = a B ds (emission) di abs = - _ a I ds (absorption) di = di emit + di abs = a (B-I) ds di/ds = _ a (B-I) (Schwarzschilds Equation)

11 Radiative transfer equation in clear skies Introducing optical path: and noting: τ ( s) = S s τ ( S) = β ( s') ds' a dτ = β ds a use integrating factor e - : di = I B dτ e τ d dτ di d Ie τ τ [ Ie ] τ = Be τ τ d τ [ Ie dτ I() = I( τ ) e = Be τ τ ] dτ = + τ τ Be Be τ τ dτ dτ after Petty (6)

12 Radiative transfer equation in clear skies I Radiance at sensor τ τ τ ( ) = I( τ ) e + Be d Radiance at start of path multiplied by path transmission τ Integral of source function along path multiplied by transmission between sensor and each point

13 Radiative transfer equation in clear skies Can write this in two other forms, firstly using: t = e τ, dt = e τ dτ, to get : or using : to get : I() dt I( S) = = = I( τ ) t( τ ) + dt ds I( s ds ) t( s ) + t(τ ) S s 1 Bdt. B( s) dt( s) ds ds Weighting function

14 Weighting functions k ν o ν 1 ν AMSU-A: 5-55 GHz O constant VMR(z) weighting function fixed. T-sounding linear H O - VMR(z) variable weighting function is state dependent. Q-sounding/analysis problem is NON LINEAR

15 The Microwave Spectrum Region covered by operational microwave sensors Moist Dry H O Rotation lines + Water vapour contiuum O Rotation Lines + Dry air continuum

16 Vibrational-Rotational (IR) / Rotational Spectroscopy (MW) Microwave absorption lines are normally due to rotational transitions. Differences in rotational energy levels are small in comparison to : Vibrational levels ( E ~ 1 5cm - 1 ) (IR spectroscopy) Electronic states ( E ~ 1,cm-1) UV/VIS spectroscopy E~ 1-5 cm -1 Dimer problem not a true bound state. potential function shallow - levels very close continuum-like absorption?

17 Broadening of atmospheric absorption / emission lines For most absorption / emission lines in the earths atmosphere, two broadening mechanisms are important: Pressure (collisional) broadening ( *) Doppler broadening If both are important (eg in IR) the line shape is given by the convolution of both the Voigt line shape, S, F

18 Line strengths (S) Nuclear spin degeneracy Isotopic abundance See Rothman et al 1987 S if ( T ) 8π 3hc gii a 36 [ 1 exp( c υ / T )] exp( c E / T ) R.1 3 = υif if i if Q( T ) partition function where :R if = < i M f> - transition dipole matrix element Note : S(T) Spontaneous emission term Boltzmann factor for ground state with energy E i = if no change in dipole moment involved in transition from i to j obtained through measurement, and available through published spectroscopic databases (eg HITRAN..)

19 Pressure Broadening (F) p T FWHM f ) ( 1 ) ( 1 ) ( ) ( 1 ), ( γ γ γ πτ γ γ ν ν γ π γ ν ν γ γ ν ν γ ν ν π ν ν = = = = Van Vleck and Weisskopf (1945) Lorentz line shape omits this factor Approximation valid near line centre at most microwave frequencies and higher Pressure broadening coefficient is obtained through measurement and is available from spectroscopic databases ~ 1-3 GHz. atm -1 mean time between collisions who cares? eg. GHz line widths

20 Doppler Broadening (F) g( ν, ν ) = σ = FWHM = πktν kt mc mc ν 8kT ln ν mc exp mc ( ν ν ) ktν Due to Doppler shift resulting from relative motion between observer and abs/emitting molecule Note line widths scale linearly with frequency

21 Doppler vs Pressure Broadening : IR vs MW Pressure broadening dominates in the MW, we can neglect Doppler broadening in most applications (not mesospheric sounding though)

22 Line-by-line to fast models (eg RTTOV) Use LbL to generate optical depths (_) in discrete layers, given inputs: _, _, S i - for each line in the spectral region of interest P, T, q for each layer, for a range of profiles (1 s) For each layer, find coefficients A, which solve (in a least squares sense) : y = A.x where (for a single frequency) : y = vector of simulated optical depths (1 N) x = matrix of predictors, constructed from p,t and q (M N) A= vector of coefficients (1 M) (M = no of predictors, N = no. of profiles)

23 Fast Models: eg RTTOV For a given input profile (x), generate optical depths (tau) in each layer, using an expression of the form: n τ = a i Pi Predictors are simple functions of input profile Fast: ~1mS per profile K, TL, AD generated from direct code Fast models now capable of : i= 1 n = number of predictors, typically ~ eg. T, q, T,. Scattering calculations in MW & IR allowing cloud/rain analysis Modelling trace gas effects (eg CO, CO, CH 4, O 3, N O)

24 Fast RT models (eg RTTOV) Version Function Input Output Direct K TL AD Forward calculation: generate T B from profile (x) Generate full Jacobian matrices (H) Generate increments in radiance (_T B ) from increments in profile variables Generate gradients of cost wrt profile variables from gradients of cost wrt T B. Profile (x) Channel specifications Observation geometry [as for direct] [as for direct] + increments in profile variables (_x) [as for direct] + normalised departures: R -1.d = R -1 (y-h(x b )) or R -1 (y-h(x b )-H_x) Radiances, usually as brightness temperatures (T B ) for all channels. Arrays containing H (= dt B /dx ) for all profile variables, and channels. Increments in radiances (_T B ) Gradients of cost wrt Profile variables.

25 RT with scattering Recall clear sky RT, where the change in radiant intensity along a path was given by the sum of sink (absorption) and source (emission) terms. Scattering contributes to both sink and source terms: S sink: di ext = - e Ids S source: di scat

26 RT with scattering di di 1 4π di = scat di = 4π = β Ids + β di( Ωˆ ) dτ ext βs 4π + di 4π + di p( Ω ˆ, Ωˆ ) I( Ω ˆ ) dω ds p( Ω ˆ, Ωˆ ) dω = 1 e emit a scat βs Bds + 4π then, dividing through by: = I( Ωˆ ) (1 ω ) B 4π ω 4π p( Ω ˆ, Ωˆ ) I( Ω ˆ ) dω ds 4π normalised phase function, represents scattering from angles into view direction dτ = β ds e p( Ω ˆ, Ωˆ ) I( Ω ˆ ) dω scattering albedo phase function

27 RTTOV-SCATT : A fast scattering model for MW RT Scattering albedo and phase function (asymmetry parameter for Henyey-Greenstein phase function) are pre-computed using Mie theory, and stored in look up tables. Look-up tables are indexed by hydrometeor amounts (for rain, snow, cloud water, cloud ice) See Appendix for details, also Bauer et al, QJRMS, 6, Vol.13, pp

28 Combined cloud-radiative transfer modelling Extinction [1/km] Rain Snow moist dry Graupel Cloud water Aggregates Cloud ice Frequency [GHz]

29 Summary 5.3 GHz 15 GHz MW data v. important for NWP! 5 GHz W. Fns span troposphere/ stratosphere & give info on T MW Spectroscopy & RT in clear skies is relatively simple (few lines) Fast RT models used in DA 5.8 GHz 53.6 GHz. 183±7 GHz 183±5 GHz 54.4 GHz 183±1 GHz 183 GHz channels are centred on H O line, channels selected to give vertical coverage. MW scattering RT more complex but Fast RT models now available K 3K 15K 3K

30 Summary Ocean surface is polarising, This gives ocean WS information 1K 19 GHz (H pol) 3K GHz (& 19/37/85) gives information on WV and cloud although vertical resolution is limited (wrt IR) Land surface emissivity is more difficult to model, this restricts use of data over land 19 GHz (V pol) GHz (V pol) 37 GHz (H pol) 37 GHz (V pol)