turner and ebell, 2013 doe/eu retrieval workshop, köln retrieval algorithm frameworks dave turner...
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Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Retrieval Algorithm Frameworks
Dave TurnerNOAA National Severe Storms Laboratory
Kerstin EbellUniversity of Cologne
DOE / EU Ground-based Cloud and Precipitation Retrieval Workshop
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Motivation
• Remote sensors seldom measure the quantity that is really desired
• So we must “retrieve” the quantity we desire from the observations that are made
• Often an ill-defined problem (i.e., there is usually not enough information in the observations)
• Classical analogy from Stephens 1991: “Remote sensing is like characterizing an animal from the tracks it makes in the sand”
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Tracks in the Sand
• What type of animal?• Large or small?• Young or old?• Male or female?• What color?
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Tracks in the Sand
• What type of animal?
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
From Observations to Geophysical Variables
Geophysical Variable(What we want to know)
Radiance or Backscatter
(What we observe)
Forward RT Model
Retrieval
X⌃Ym=Y+ε
F
X YX+δX
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
The Retrieval Challenge• Desire “observations” of geophysical variables to improve
our understanding of the Earth system• Remote sensing observations provide information about
the Earth system, but are not direct observations of the geophysical variables we desire
• Must “retrieve” the geophysical variables from the observations
• Typically is an ill-defined problem• Noise hinders the retrieval; so does resolution• Metadata (data about the data) can help constrain the
problem• Additional observations also help• Important to consider the uncertainties in the retrieved
quantities• Calibration, calibration, calibration
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Basic Retrieval Classes
• “Regression” methods– Linear, quadradic approaches, Neural networks, etc– “Tuned” to mean conditions; no guarantee that
retrieved profiles are consistent with observation– Computationally fast and always produces an
“answer”– Could be developed from
• Simulated observations• Collocated observations
• “Iterative” methods– Iterative, uses forward model and a first guess– Retrieved profiles are consistent with observation– Significantly slower than regression methods– Often case-specific error characterization is provided
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Example: Liquid Water Path
• Many MWRs observe around 23 and 31 GHz• These observations are sensitive to the LWP and the
amount of precipitable water vapor (PWV) in the column
• Because of the small size of cloud droplets with respect to the wavelength, the cloud droplets are in the Rayleigh scattering regime and thus the MWR observations are insensitive to cloud droplet size
• Observed signal is proportional to the third moment (i.e., <r3>) of the size distribution spectrum
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Microwave Spectrum
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Real Tb Observations
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Real Tb Observations: PWV
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Real Tb Observations: LWP
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
“Orthogonal”
PWV
LWP
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Retrieving LWP from the MWR (1)
• Purely a statistical method
• Can use historical dataset to determine coefficients ax
– Requires direct observations of LWP (e.g., from aircraft)
OR– Forward radiative transfer model
• Coefficients are site and season dependent• Fast and easy
€
LWP = a0 + a11Tb ,23 + a12Tb ,232 + a21Tb ,31 + a22Tb ,31
2
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Retrieving LWP from the MWR (2)
€
τ =lnTmr −Tbg
Tmr −Tsky
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟−τ dry
€
LWP = L1τ 23 + L2τ 31
Compute “opacity” τ at each frequency
• Also a purely statistical method• Again, use historical data or simulated data to
determine retrieval coefficients Tmr, τdry, Lx
• Advantages over other method:– Linear rather than quadratic– Less scatter than other method (i.e., better statistical fit)
• Coefficients (Tmr, τdry, L1, L2) are site/season dependent
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Clear Sky LWP Retrieval
Opacity Regression Retrieval
Liq
uid
Wat
er P
ath
[g
/m2 ]
Hour [UTC]
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Improved Regression Retrieval
• More information can often improve retrievals• J.C. Liljegren used surface meteorology to “predict”
the retrieval coefficients Tmr, τdry, L1, L2
• Removed the site and seasonal dependence• Improved accuracy of retrieved LWP€
L = a0 + a1Psfc + a2Psfcesfc + a3esfc2
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Improved Clear Sky LWP Retrieval
Improved Regression Retrieval
Liq
uid
Wat
er P
ath
[g
/m2 ]
Hour [UTC]
Opacity Regression Retrieval
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Iterative Retrieval
• Retrievals are used to ‘invert’ the radiative transfer• Regression approaches frequently will not agree
with the observation in a ‘closure study’• Iterative retrieval uses the actual forward model in
an iterative manner1. Start with first guess of atmospheric property of interest2. Compute radiance (obs) using forward model3. Compute computed “obs” with real observation, and modify
the first guess accordingly4. Repeat steps 2-4 until computed “obs” matches the real
observations (within uncertainties)
• Results will “close” with observations if retrieval converged
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Considerations• The forward model may have limited sensitivity to the
desired variable• Forward model may be highly non-linear, which affects
how the solution is found• Multiple solutions may exist for a given observation
(i.e., problem is ill-defined)• Uncertainties in the observations should be propagated
to the retrieved solution• Retrievals often use other data and/or assumptions
that may affect the retrieved solution; uncertainties in these parameters should also be propagated to the solution– Includes model parameters, which are often ignored
• Often only partial prior info on the solution is known
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Maximum a Posteriori (MAP)
• One of several iterative retrieval methods• Uses Bayes theorem• Incorporates a priori knowledge into the maximum
likelihood solution
€
P A B( ) =P B A( )P A( )
P B( )
Posterior =Likelihood x Prior
Normalizing Constant
A: the variable we desireB: the observation we have
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Estimating the Temperature Outside
Climatology
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Estimating the Temperature Outside
Obs with itsUncertainty
Climatology
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Estimating the Temperature Outside
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Estimating the Temperature Outside
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Estimating the Temperature Outside
Solution withits uncertainty
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Optimal Estimation - 1• Technique is an old one, with long history• Excellent book by Rodgers (2000)• Many good examples exist in literature• Assumes problem is linear and uncertainties are Gaussian
• However, the accuracies of the uncertainty in X is directly related to ability to properly define the covariance matrix of the observations Sε, which is a non-trivial exercise
• Key advantage is that uncertainties in the retrieved state vector X are automatically generated by method !
State vector
A priori
A priori’s Covariance “Obs” Covariance
Jacobian ObservationForward model
€
X n+1 = Xa + Sa−1 + KT Sε
−1K( )−1
KT Sε−1 Y − F X n
( ) + K X n − Xa( )[ ]
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Optimal Estimation - 2
• Linear– Forward model of the form y = K x– A priori is Gaussian
• Nearly linear– Problem is non-linear, but linearization about some prior state is adquate
to find a solution
• Moderately non-linear– Problem is non-linear, but linearization is adequate for error analysis but
not for finding a solution Many problems are like this
• Grossly non-linear– Problem is non-linear even within the range of the errors
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Optimal Estimation - 3• Moderately non-linear problems
– No general expression for locating optimal solutions as for linear and slightly non-linear problems
– Solutions must be found numerically and iteratively– Follow maximum a posteriori (MAP) approach and minimize the
cost function that is the sum of the “distance” between the observation and current calculation (weighted by observational covariance” and the distance between prior and current state weighted by prior covariance
– Numerical method is the Newtonian method to find successfully better approximations to the roots of the function g
€
J = y −F x( )[ ]TSe
−1 y −F x( )[ ] + x − xa[ ]TSa
−1 x − xa[ ]
€
g x( ) = ∂J∂x
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Newton Method
€
x i+1 = x i − ∇xg x i( )[ ]−1
g x i( )
From Wikipedia
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Optimal Estimation - 4
• Does not provide an explicit solution• Does provide a class of solutions and assigns a probability
density to each• We chose one state from the ensemble that is described
by the posterior covariance matrix
• Diagonal elements of provide mean squared error of• Off-diagonal elements provide information on the correlation
between elements of
€
ˆ S = Sa−1 + KT Sε
−1K( )−1
€
ˆ S
€
ˆ X
€
ˆ S
€
ˆ X
€
ˆ X
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Making It More Concrete: MWR Retrieval
€
X =PWV
LWP
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Y =Tb ,23
Tb ,31
⎡
⎣ ⎢
⎤
⎦ ⎥
€
K i , j =∂Fi
∂X j
X is the state vector Y is the observation vector
K is the Jacobian (2x2 matrix)F is the forward
radiative transfer model
€
Sε =σ Tb 23
2 0
0 σ Tb 31
2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
S is the covariance of the observations
€
X n+1 = Xa + Sa−1 + KT Sε
−1K( )−1
KT Sε−1 Y − F X n
( ) + K X n − Xa( )[ ]
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Even Better LWP Retrieval
Iterative Retrieval
Radiometric Uncertainty: 15 g/m2
Liq
uid
Wat
er P
ath
[g
/m2 ]
Hour [UTC]
Improved Regression Retrieval
Opacity Regression Retrieval
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
LWP Relative Uncertainty
• Radiometric uncertainty in MWR results in large relative uncertainty in LWP when the LWP is small
• Combine different observations to improve retrieval
Rel
ativ
e U
nce
rtai
nty
[%
]
Liquid Water Path [g/m2]
From the posterior
€
ˆ S
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Improving LWP Retrievals when the LWP is small
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Combined Infrared + Microwave Retrieval
€
Y =YMW
YIR
⎡
⎣ ⎢
⎤
⎦ ⎥=
Tb 23
Tb 31
I1
I 2
...
I n
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
€
F =FMW
FIR
⎡
⎣ ⎢
⎤
⎦ ⎥
€
X n+1 = Xa + Sa−1 + KT Sε
−1K( )−1
KT Sε−1 Y − F X n
( ) + K X n − Xa( )[ ]
€
Sε =Sε MW 0
0 Sε IR
⎡
⎣ ⎢
⎤
⎦ ⎥
Forward models FMW and FIR need to be consistent!
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
LWP Relative Uncertainty
• Combining the infrared and microwave significantly reduces the relative uncertainty in LWP for small LWP clouds
Rel
ativ
e U
nce
rtai
nty
[%
]
Liquid Water Path [g/m2]
From the posterior
€
ˆ S
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
The Classification Problem
• Many retrieval algorithms are only applicable for certain types of clouds (e.g., liquid only stratiform, ice-cloud only)– Running incorrect retrieval method often grossly violates the
assumptions in the retrieval, leading to huge errors
• Need automated methods to classify the cloud conditions at a given time– Allows the correct retrievals to be performed
• Classification algorithms provide discrete (vs. continuous) output
• There is (and will always be) uncertainty in the sky classification; how to capture this uncertainty and propagate it into the retrieval uncertainty?
• Simulaneously retrieve classification and cloud prop’ties?
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
What is Sε ?• Uncertainty in the observations and forward model• Treated as the sum of two covariance matrices
• Instrument covariance matrix Sy can be difficult to determine– How to quantify this matrix? – How does it depend on conditions?
• Forward model parameter uncertainties in Sb
– Virtually every forward model has some tunable parameters that have some uncertainty – “unknown knowns”
– Many forward models make other assumptions that we may not realize which have uncertainties – “unknown unknowns”
• Often KbSbKbT is orders of magnitude larger than Sy
€
Sε = Sy + KbSbKbT
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
An Example for SyQuantifying the Noise in the AERI Radiance Observations
Applying NF reduces random error 4x but introduces some correlated error
Unfiltered
PCA Filtered
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
An Example for SbQuantifying the Impact of other Trace Gases on AERI Retrievals
Spectral region used for H2O Profiling
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Summary
• Retrieving geophysical variables from observations is a non-trivial process– Error sources include random noise in the obs, bias in obs or
forward model, retrieval technique applied, small sensitivity, etc.
• Adding information typically improves the retrieval– Reduces noise using ‘redundant’ channels– Improves accuracy when more sensitive channels are added– Try to add channels that are “orthogonal”– Allows additional variables to be retrieved
• Forward model uncertainty and parameters important• Defining prior and observational covariances non-trivial
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Let’s Remove the Human Component of the Error!
Turner and Ebell, 2013 DOE/EU Retrieval Workshop, Köln
Good Outcome for WorkshopMy Opinion Anyway
• Quantifying Sε for the different instruments typically used for
cloud / precipitation retrievals– Quantifying Sy
– Identifying the important (tunable) forward model parameters
– Quantifying Sb
• Quantifying Sa for the different geophysical variables
– 1-sigma uncertainty in the atmospheric variable we desire
– Between variables a and b
– Between different height levels i and j
• In the matrices S?, both the diagonal and off-diagonal elements are important!