Analyses of in-situ hydrometeor measurements and what they imply - - PowerPoint PPT Presentation

Analyses of in-situ hydrometeor measurements

and what they imply about microphysical representations

Z.S. Haddad

Jet Propulsion Laboratory, California Institute of Technology

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Example: Morrison-Thompson scheme

• 5 species: CloudLiquid, CloudIce, Rain, Graupel, Snow
• Gamma distribution for each species x :

number of hydrometeors (per unit volume) whose diameter is between D and D + dD =

“intercept” “shape” “slope”

Nx Dµx e−ΛxD dD

Microphysics call: The model sends to the scheme (total number concentration, total mass concentration) for each species, along with the other prognostic variables, and the scheme then calculates conversion rates and amounts, and sends back to the model the new (number concentration, mass concentration) for each species

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Example: Morrison-Thompson scheme

• 5 species: CloudLiquid, CloudIce, Rain, Graupel, Snow
• Gamma distribution for each species x :

number of hydrometeors (per unit volume) whose diameter is between D and D + dD =

• For each species, µx is given a priori, and the scheme ingests

total number concentration NT and mixing ratio q and produces the corresponding Nx and Lx: and

Nx Dµx e−ΛxD dD

NT = Z Nx Dµx e−ΛxD dD = Nx Γ(µx + 1) Λµ+1

x

q = Z aDb Nx Dµx e−ΛD dD = a Nx Γ(µ + 1) Λµ+1

x

Γ(b + µ + 1) Λb

xΓ(µ + 1)

mass(D)

4

Example: Morrison-Thompson scheme

• 5 species: CloudLiquid, CloudIce, Rain, Graupel, Snow
• Gamma distribution for each species x :

number of hydrometeors (per unit volume) whose diameter is between D and D + dD =

• For each species, µx is given a priori, and the scheme ingests

total number concentration NT and mixing ratio q and produces the corresponding Nx and Lx: and

Nx Dµx e−ΛxD dD

NT = Z Nx Dµx e−ΛxD dD = Nx Γ(µx + 1) Λµ+1

x

q = Z aDb Nx Dµx e−ΛD dD = a Nx Γ(µ + 1) Λµ+1

x

Γ(b + µ + 1) Λb

xΓ(µ + 1)

5

Example: Morrison-Thompson scheme

• 5 species: CloudLiquid, CloudIce, Rain, Graupel, Snow
• Gamma distribution for each species x :

number of hydrometeors (per unit volume) whose diameter is between D and D + dD =

• For each species, µx is given a priori, and the scheme ingests

total number concentration NT and mixing ratio q and produces the corresponding Nx and Lx: and Nx = NT Λ1+µx

x

Γ(1 + µx) Λx = ✓a NT Γ(b + µx + 1) q Γ(µx + 1) ◆1/b

Nx Dµx e−ΛxD dD

6

Example: Morrison-Thompson scheme

• 5 species: CloudLiquid, CloudIce, Rain, Graupel, Snow
• Gamma distribution for each species x :

number of hydrometeors (per unit volume) whose diameter is between D and D + dD =

• For each species, µx is given a priori, and the scheme ingests

total number concentration NT and mixing ratio q and produces the corresponding Nx and Lx

• µCI = µR = µG = µS = 0;

µCL = f(NT) (Martin et al, 1994)

Nx Dµx e−ΛxD dD

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N (D) = N 0 Dµ e−ΛD q = 4 3

∫

π D 2 $ % & ' ( )

3

ρ N (D) dD = π 6 ρ Γ(µ +1) Dm

µ+1

(µ + 4)µ+1 N 0 σ m = (D − Dm)2 D3N (D) dD

∫

D3N (D) dD

∫

% & ' ' ( ) * *

1/ 2

= Dm µ + 4

Þ Assume G distribution: and let’s try to interpret the parameters in terms of physically meaningful quantities:

Dm = D D3N (D) dD

∫

D3N (D) dD

∫

= µ + 4 Λ

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q = 4 3

∫

π D 2 $ % & ' ( )

3

ρ N (D) dD = π 6 ρ Γ(µ +1) Dm

µ+1

(µ + 4)µ+1 N 0 Dm = D D3N (D) dD

∫

D3N (D) dD

∫

= µ + 4 Λ σ m = (D − Dm)2 D3N (D) dD

∫

D3N (D) dD

∫

% & ' ' ( ) * *

1/ 2

= Dm µ + 4 N (D) = N 0 Dµ e−ΛD

Þ Assume G distribution: and let’s try to interpret the parameters in terms of physically meaningful quantities:

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q = 4 3

∫

π D 2 $ % & ' ( )

3

ρ N (D) dD = π 6 ρ Γ(µ +1) Dm

µ+1

(µ + 4)µ+1 N 0 = πρ 24 N 0 Dm N (D) = N 0 Dµ e−ΛD

Þ Assume G distribution: Special case: exponential distribution, i.e. µ = 0

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If, in addition, the intercept N0 is assumed constant, this implies that

• Dm/q = constant, and therefore
• max(Dm)/min(Dm) = max(q)/min(q)

Dm = 24 πρN 0 q N (D) = N 0 Dµ e−ΛD

Þ Assume G distribution: Special case: exponential distribution, i.e. µ = 0

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If, in addition, the intercept N0 is assumed constant, this implies that

• Dm/q = constant, and therefore
• max(Dm)/min(Dm) = max(q)/min(q)

This is very dangerous, because it implies that in order to span a realistically large range of values of min(q) < q < max(q), we need to use a possibly unrealistic range of values of Dm

Dm = 24 πρN 0 q N (D) = N 0 Dµ e−ΛD

Þ Assume G distribution: Special case: exponential distribution, i.e. µ = 0

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If, in addition, the intercept N0 is assumed constant, this implies that

• Dm/q = constant, and therefore
• max(Dm)/min(Dm) = max(q)/min(q)

3.5 mm / 0.5 mm ≠ 10 g/Kg / 0.1 g/Kg (in the case of rain) 7 << 100 !!

Dm = 24 πρN 0 q N (D) = N 0 Dµ e−ΛD

Þ Assume G distribution: Special case: exponential distribution, i.e. µ = 0

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X = Rain rate X = Dm 100 (Xkwajex – Xcoare)/Xcoare Retrieved X from airborne radar data, using 2 forward operators: 1 from Kwajex microphysics and from 1 from COARE microphysics Kwajex = 10-minute averages, COARE = 1-minute averages

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N (D) = N 0 Dµ e−ΛD

Þ Assume G distribution: If, instead of assuming that the intercept N0 is constant, the assumption is made that L is constant: The above imply, regardless of whether µ is assumed constant or not: , i.e.

Dm = D D3N (D) dD

∫

D3N (D) dD

∫

= µ + 4 Λ σ m = (D − Dm)2 D3N (D) dD

∫

D3N (D) dD

∫

% & ' ' ( ) * *

1/ 2

= Dm µ + 4 Dm = Dm

2 /σ m 2

Λ ⇒ Dm σ m

2 = const

σ m = const Dm

0.5

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(assumption: L constant) Þ Unfortunately, in-situ observations say: σ m = const ⋅ Dm

1.5 ± noise

σ m ≠ const Dm

0.5

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(assumption: L constant) Þ Unfortunately, in-situ observations say: σ m = const ⋅ Dm

1.5 ± noise

σ m ≠ const Dm

0.5

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What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers

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What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers

Example of VHF spectra from two consecutive height bins

(Rajopadhyaya et al, J. Atmos. Ocean. Tech. 1998)

What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers

Example 1: 17 December 2005, convective initiation and trailing stratiform

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What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers

Example 2: 27 December 2005, stratiform with some mild & shallow convection

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What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers

Example 3: 21 March 2006, convection reaching beyond the 10km cut-off ceiling

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What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers
• 2. Joss-Waldvogel distrometer (ground)

25

What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers
• 2. Joss-Waldvogel distrometer
• 3. 2D-video distrometer

26

What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers
• 2. Joss-Waldvogel distrometer
• 3. 2D-video distrometer
• 4. Various airborne probes

27

What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers
• 2. Joss-Waldvogel distrometer
• 3. 2D-video distrometer
• 4. Various airborne probes

Main problem for 2, 3 and 4: sampled volume (< 10cm x 10cm x 1000m) is, at best, over 5 orders of magnitude smaller than model resolution (> 1000m x 1000m x 100m)

28

What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers
• 2. Joss-Waldvogel distrometer
• 3. 2D-video distrometer
• 4. Various airborne probes
• 5. Polarimetric ground radar (volume-aggregated effect)

ZHH = F1(N0, µ, L) ZVV = F2(N0, µ, L)

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What are these “in situ observations”?

• 1. Co-located 50 MHz + 0.92 GHz zenith-pointing profilers
• 2. Joss-Waldvogel distrometer
• 3. 2D-video distrometer
• 4. Various airborne probes
• 5. Polarimetric ground radar (volume-aggregated effect)

ZHH = F1(N0, µ, L) ZVV = F2(N0, µ, L) Kdp = F3(N0, µ, L; coefficients of oblateness relation)

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Why do the correlations matter? The model M is a complex (nonlinear) function which simulates the evolution of the state variables in time: But in reality M is very non-linear in µ – and yet models typically use one set of constant values (representing the mean), and ignore error bars (let alone any more realistic representations of the joint uncertainty in the components of µ) In particular, how the uncertainty in xt depends on µ is difficult to represent

dxt = M(xt) dt + N · dbt dxt = M(xt; µ) dt + N · dbt

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Why do the correlations matter? The model M is a complex (nonlinear) function which simulates the evolution of the state variables in time: But in reality M is very non-linear in µ – and yet models typically use one set of constant values (representing the mean), and ignore error bars (let alone any more realistic representations of the joint uncertainty in the components of µ) In particular, how the uncertainty in xt depends on µ is difficult to represent (mainly because you cannot put in a single “average” value

• f µ and expect the solution M(x,µ) to represent the average forecast)

dxt = M(xt) dt + N · dbt dxt = M(xt; µ) dt + N · dbt

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Why do the correlations matter? The model M is a complex (nonlinear) function which simulates the evolution of the state variables in time: But in reality One way to make it easier to quantify, because µ is large-dimensional, is to try to “reduce its dimensionality”, by assuming that some of its components are constant. But assuming µ2, µ3, … , µ100 constant while keeping µ1 variable is nonsense if (µ1, µ2) are correlated, etc. A straightforward way to stay consistent is to replace µ with an equivalent set of uncorrelated parameters

dxt = M(xt) dt + N · dbt dxt = M(xt; µ) dt + N · dbt

35

Why do the correlations matter? The model M is a complex (nonlinear) function which simulates the evolution of the state variables in time: But in reality A straightforward way to stay consistent is to replace µ with an equivalent set of uncorrelated parameters To do this, you have to know the joint behavior of the (µi, µj) – you don’t get to impose / invent a joint behavior!

dxt = M(xt) dt + N · dbt dxt = M(xt; µ) dt + N · dbt

36

Why do the correlations matter? The model M is a complex (nonlinear) function which simulates the evolution of the state variables in time: But in reality A straightforward way to stay consistent is to replace µ with an equivalent set of uncorrelated parameters To do this, you have to know the joint behavior of the (µi, µj) – you don’t get to impose / invent a joint behavior! Of greatest initial interest is the portion of the joint behavior that can be observed, or that has an effect on observables.

dxt = M(xt) dt + N · dbt dxt = M(xt; µ) dt + N · dbt

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q = 4 3

∫

π D 2 $ % & ' ( )

3

ρ N (D) dD = π 6 ρ Γ(µ +1) Dm

µ+1

(µ + 4)µ+1 N 0 Dm = D D3N (D) dD

∫

D3N (D) dD

∫

= µ + 4 Λ σ m = (D − Dm)2 D3N (D) dD

∫

D3N (D) dD

∫

% & ' ' ( ) * *

1/ 2

= Dm µ + 4 N (D) = N 0 Dµ e−ΛD

Þ Assume G distribution: One conclusion is: Must account for correlations between the parameters

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N (D) = N 0 Dµ e−ΛD

Þ Assume G distribution: One conclusion is: Must account for correlations between the parameters Another is: Must quantify the joint behavior of all static parameters and dynamic (process) parameters Because it is impossible to observe all simultaneously, and because many are impossible to observe at all, need to answer the converse: given instantaneous observations (of rainrate, liquid and ice water paths, radiative fluxes), how mutually ambiguous are the microphysical parameters?

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Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• Mon. Wea. Rev. 2010, 2012, 2014, 2016
• Avoid Gaussian assumption
• Use Lagrangian single-column version of Tao’s GCE
• Idealized simulation: track perturbations of base profiles of

potential temperature and moisture, driven by time-varying profiles of vertical motion and water vapor tendency

• Resulting convection does simulate two precip morphologies:

collision-coalescence in convective type, melting of snow and graupel in stratiform type

• Similar to variational data assimilation, given O, obtain the

conditional pdf of µ using Bayes: p(µ|O) = p(O|µ) p(µ)

• Unlike variational data assimilation, in order to avoid Gaussian

assumption, use MCMC

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Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• Mon. Wea. Rev. 2010, 2012, 2014, 2016

How does MCMC work? repeat the following until you obtain a sufficiently large ensemble:

• 1. Generate a candidate new parameter value µ? randomly from

the current parameter value µc, using a distribution pap(µc, µ?)

• 2. Run the forward model M( • , µ?) to produce forecast x
• 3. Compare H(x) with observation O
• 4. If the fit is good, keep µ? as a new member of the ensemble,

update µc=µ?, and go back up to 1.

• 5. Otherwise, reject µ? and go back up to 1.

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Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• Mon. Wea. Rev. 2010

For the observations, a subset of the following is used:

• Precipitation rate (PCP)
• Liquid water path (LWP)
• Ice water path (IWP)
• Top-of-the-atmosphere ShortWave radiative flux (OSR)
• Top-of-the-atmosphere LongWave radiative flux (OLR)

Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• Mon. Wea. Rev. 2010

time-height cross sections of model output

40

Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• MWRev. 2010

Joint PDFs of parameter pairs

when the obs were PCP, LWP, IWP only

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Note:

• Graupel fall speed parameters
• Rain intercept

Joint PDFs of parameter pairs

when the obs were PCP, LWP, IWP, OSR, OLR

Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• MWRev. 2010

42

Note:

• Graupel fall speed parameters
• Rain intercept
• Snow intercept and density

Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• MWRev. 2012

Joint PDFs of parameter pairs

when the obs were radar profiles

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Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• MWRev. 2016

3D model, conditioned on surface rain, LWP, IWP, OSR, OLR Storm observed on 23 February 1999 in Rondonia, Brazil

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2km res 250m res 65 km

Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• MWRev. 2016

3D model, conditioned on surface rain, LWP, IWP, OSR, OLR

44

Hollow histogram: unconstrained ensemble Filled histogram: MCMC ensemble (parameters constrained by obs)

Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• MWRev. 2016

3D model, conditioned on surface rain, LWP, IWP, OSR, OLR

44

Hollow histogram: unconstrained ensemble Filled histogram: MCMC ensemble (parameters constrained by obs)

Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• MWRev. 2016

3D model, conditioned on surface rain, LWP, IWP, OSR, OLR

44

Hollow histogram: unconstrained ensemble Filled histogram: MCMC ensemble (parameters constrained by obs)

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Vukicevic+Posselt+VanLier-Walqui: Markov Chain Monte Carlo sims

• Mon. Wea. Rev. 2012:

Conditions on measured (synthetic) radar reflectivity

• Mon. Wea. Rev. 2014 (Van Lier-Walqui):

Conditions on measured (synthetic) radar reflectivity, but instead

• f sampling joint distribution of parameters, assume that each

process has a multiplicative efficiency and sample the joint distribution of the multiplicative factors

• Mon. Wea. Rev. 2014 (Posselt):

Conditions on supposed retrieved precip etc as in 2010, but allowing multiple truths (perturb to generate uncertainty)

• Mon. Wea. Rev. 2016 (Posselt):

Conditions on supposed retrieved precip etc as in 2010, but with Tao’s 3D model – look at three coarse layers at three coarse stages