Download Cooling Rate Retrieval Paper Appendices

A.
Calculation of radiative heating/cooling rate profiles:
One-dimensional radiative heating/cooling rate profiles are a function of pressure for
each spectral interval. For circulation model calculations, the ultimate quantity of interest
is total shortwave heating rate profile and the total longwave cooling rate profile which
represent the integration of spectral heating/cooling rate profile information over the
3000-50000 cm-1 and 100-3000 cm-1 bands respectively. The value of the radiative
heating/cooling rate for a given pressure layer is directly proportional to the radiative flux
divergence for that layer:
 z  
dFnet  z 
C p  z 
dz
1
(A1)
where   z  is the spectral heating/cooling rate of the layer,  z  is the density of the
layer, C p is the heat capacity, and Fnet  z  is the net (upward-downward) flux for the
layer over the spectral interval denoted  . For the longwave, the heating rate profile is
generally negative, so the convention is to reverse the sign and use the term ‘cooling rate
profile’ instead.
The calculation of heating/cooling rate profiles is ubiquitous so a large amount of research
has been devoted towards addressing the accuracy and computational efficiency of these
values. For the clear-sky longwave conditions in local thermodynamic equilibrium, the
derivation of the net flux divergence follows from a solution to the Fundamental Equation
of Radiative Transfer because the source function is the Planck function. For upwelling
radiance, the solution is:


I  , z    B  z surf  T  , z surf , z   
z
zsurf
B  z '
T  , z' , z 
dz '
z '
(A2)
and for downwelling radiance, the solution is:
I  , z    B  z '

z
T  , z, z '
dz '
z '
(A3)
where I  , z  is the radiance over a spectral interval in a layer and viewing angle
cos    ,  is the surface emissivity, B is the Planck function,  z surf  is the surface
temperature, and T is the spectral transmittance between two layers. For the calculation
of heating/cooling rates, radiance calculations must be converted into flux:
F z    I  , z   d
1
(A4)
0
where the  superscript refers to separate upwelling and downwelling components. As
an analog to spectral transmittance, the concept of flux transmittance allows the
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representation of flux in similar terms as the solution to the radiative transfer equation in
terms of radiance without having constantly to declare the angular integration:
T f z ' , z    T  , z' , z   d
1
(A5)
0


F z    B  zsurf  T f zsurf ,z   
F z    B  z '

z
z
zsurf
B  z '
T f z, z '
dz '
z '
T f z',z 
dz '
z '
(A6)
(A7)
The definition of flux transmittance is such that
T f z ' , z1  z2   T f z ' , z1 * T f z1 , z2 
(A8)
Consequently, the net flux divergence is given by the derivative of the net flux
(upwelling-downwelling) and is given by:

T f z surf , z  
 
 B  z surf 
z


2 f

z


 T z ' , z dz ' 
2
 z  

 zsurf B   z '
C p  z  
z ' z

 

 2T f z, z '
dz ' 
 z B  z '
z ' z




(A9)
In practice cooling rate profile calculation is performed with an emphasis on
computational efficiency. With a given atmospheric state (temperature, H2O, CO2, O3,
CH4, N2O, and cloud optical depth profiles), band models programs are utilized to
estimate interlayer transmittance. This value is scaled empirically to yield flux
transmittance from which layer fluxes are derived. Radiance-to-flux conversion is
achieved formally and exactly through exponential integrals (Goody and Yung, 1988). In
practice, the computational expense of exponential integrals motivates the usage of
limited-point quadrature. Gauss-Jacobi quadrature is an optimal means for the
calculation of flux transmittance, and the utilization of three-point quadrature produces
little discrepancy as compared to complete angular integration. Many fast radiative
transfer models even use the diffusivity approximation (1-point quadrature) to calculate
flux transmittance. Quadrature methods tend to produce some error in the layer exchange
terms only with low optical depth bands. The discretized calculation of upwelling and
downwelling fluxes at layer boundaries (levels) is given by:



N 1

F zi    B  z surf  T f z surf , zi    B  zk 1 / 2 * T f z i , zk 1   T f z i , zk 
(A10)
k i
N 1


F  zi    B   zk 1 / 2 * T f z i , zk 1   T f z i , zk 
k i
2
(A11)
Finite-difference derivatives of net flux produce flux divergence which is scaled to
produce the cooling rate profile. Evaluation of analytic derivatives with respect to
altitude for cooling rate profile calculation is generally avoided due to the inconsistency
between the vertical coordinate scheme needed for accurate radiative transfer calculations
and the Lipschitz condition [Jeffreys, 1988] which is necessary for derivative stability.
The band models are typically evaluated over the fewest bands while minimizing the
variation of the Planck function across the band and minimizing overlap between
absorptions from different species. For the longwave, 10-15 bands are generally
sufficient for reasonable accuracy for tropospheric and lower-stratospheric fluxes and
cooling rates
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B.
Linear Bayesian Retrieval:
There is a vast amount of literature devoted to appropriately balancing the information
derived from a measurement with prior understanding of the retrieval target. For detailed
discussions, see Twomey (1977), Hansen (1998), and Rodgers (2000). If we limit the
treatment of the retrieval to a first-order analysis which is the linear Bayesian retrieval, the
following analysis, summarized from Rodgers [2000], is sufficient:
The conditional probability distribution function (pdf) of the retrieval target x (i.e. the
profile of O3 or, for this study, the cooling rate profile) given the measurement y is
described by Bayes’ Theorem:
Px | y  
P(y | x) P(x)
P( y )
(B1)
where P (x ) is the prior pdf of the state x such that P (x)dx is the probability that x lies
within x, x  dx , P (y ) is the prior pdf of the measurement with similar meaning as
P (x ) , P(y | x) is the conditional pdf of y given x such that P(y | x)dy is the probability
that y lies in y, y  dy  when x has a certain value.
A linear retrieval problem starts with the linearization of the forward model Fx :
y  Fx  ε  Kx  ε
(B2)
where K is the Jacobian (sensitivity matrix) of the forward model with respect to the
retrieval target and ε is a description of the uncertainty.
If measurement error presents Gaussian statistics as it usually does:
 2 ln P(y | x)  y  Kx  Sε1 y  Kx   constant
T
(B3)
where the T superscript denotes matrix transpose, the -1 superscript denotes matrix
inverse, and S ε is the measurement covariance matrix which describes the estimated
uncertainty in the measurement. Assuming that the retrieval target also presents
Gaussian statistics:
 2 ln P(x)  x  xa  Sa1 x  xa   constant
T
(B4)
x a is the a priori description of the retrieval target and S a is the covariance matrix of
xa .
Substituting Eq. (B4) and (B3) into Eq. (B1) yields:
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 2 ln P(x | y)  y  Kx  Sε1 y  Kx   x  x a  S a1 x  x a   constant
T
T
(B5)
which is a quadratic form in x so we find that the retrieval statistics can be represented
by:
T
 2 ln P(x | y )  x  xˆ  Sˆ 1 x  xˆ   constant
(B6)
where x̂ is the retrieved quantity and Ŝ is the covariance matrix of that quantity. By
equating terms that are quadratic in x , we find that Ŝ is given by:
Sˆ 1  KT Sε1K  Sa1
(B7)
and by equating terms that are linear in x , we find that

xˆ  K T S ε1K  S a1
 K S
1
T
1
ε
y  S a1x a

(B8)
This result allows for the incorporation of both a priori knowledge and the measurement
and its uncertainty in order to derive an update to the retrieval target that is consistent with
what is known. It also produces an estimate of the a posteriori uncertainty in the retrieval
state which can be utilized in subsequent analysis using the retrieval results.
The result in Eq. (B7) and (B8) lend readily to information content analysis. The
information content of a measurement describes unidimensionally the amount of
understanding derived about a multidimensional quantity as a result of an imperfect
measurement. According to Shannon [1948], the information content can also be
described by the entropy of the probability distribution functions associated with the a
priori and a posteriori states. Given an assumption of Gaussian statistics for the quantity
of interest, the entropy, and thus the information content can be directly related to the
covariance matrix of the retrieval target:
S Pa  
1
log S a 
2 log 2
H  S Pf   S Pa 

ˆ 1
H  - 1 log S * S a
2
(B9)
(B10)

(B11)
where S is the entropy of a state associated with Pa and Pf , the probability distributions
of the retrieval target of the a priori and a posteriori states respectively. H is the
information content in bits and the vertical bars denote the calculation of the matrix
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determinant. Ultimately, this approach allows for the formal design of a retrieval system
that optimizes with respect to the retrieval target information content.
For the retrieval of profile quantities, it is useful to designate a method for determining the
vertical resolution of the retrieval. That is, it is expected that the retrieval will be
marginally sensitive to a perturbation of an infinitesimally thin layer associated with profile
being retrieved. The averaging kernel matrix is conventionally utilized to understand this
sensitivity and the entries in this matrix represent the sensitivity of the retrieved profile to a
perturbation in the true profile.
A
 xˆ
x
(B12)
In order to associate the averaging kernel with a gross metric of resolution, the concept of
the degrees of freedom of the signal is introduced as the number of statisticallyindependent quantities that can be retrieved for a profile retrieval.
dof  traceA
(B13)
Together, the degrees of freedom of the signal and the information content are useful for
characterizing and optimizing a retrieval system.
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C.
Cooling Rate Retrieval Methods:
Several papers have detailed methods for direct cooling rate retrievals using top-ofatmosphere measurements. Liou and Xue [1988] describe a relationship between radiance
measurements across a spectral band and the cooling rate profile in that band. The original
formulation for this cooling rate retrieval found that:


0
 z K  , z dz   I     I  
(C1)
where K  , z  is a cooling rate weighting function that is
K  , z   C p  z  T  , z ,  
(C2)
by ansatz where  refers to a wavenumber interval spanning an entire band (~ 100 cm-1),
 refers to higher spectral resolution measurements (~ 1 cm-1), and T  , z,   refers to
the transmittance function between height z and outer space at viewing angle  .
From Eq. (C1) the term I   refers to spectrally band-averaged radiance measured at a
viewing angle of  which is very similar to the angle used for the diffusivity
approximation. The coefficients  and  are variables that depend on the ratio of
band-averaged transmittance to spectral transmittance. This paper determined that the
cooling rate retrieval is indeed feasible and numerically stable for cooling arising from
the rotational bands of water vapor (far infrared), but many of the practical details
associated with instrumentation viewing geometry requirements, spectral coverage, and a
comparison between standard methods and direct retrieval methods were left to future
research.
Feldman et al [2006] utilized some of the methods described in Liou and Xue [1988] to
retrieve the CO2 v2 band cooling rate from AIRS radiance data. This work found that,
with a variety of assumptions based on cross-track scene correlation and empiricallyderived covariance matrices (see Mlynczak and Mertens, 1998), an optimal estimation
theory approach to cooling rate retrievals using the AIRS instrument finds that the lower
stratospheric cooling arising from the CO2 v2 band.
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D.
Conversion from spectral radiance to spectral flux
Satellite-borne remote sensing observations necessarily measure radiance in order to
achieve a reasonable spatial resolution. Wide field-of-view measurements that would
actually measure top-of-atmosphere flux smear out the spatial heterogeneity while
providing little information relevant to energy balance questions. However, circulation
models calculate and utilize fluxes (and heating/cooling rates) extensively and calculate
them with spatial resolution of several kilometers. Numerous efforts have been
undertaken to address the practical conversion of top-of-atmosphere radiance
measurements to high spatial-resolution top-of-atmosphere fluxes in order to address
radiation balance issues. The Earth Radiation Budget Experiment and its successor the
Clouds and Earth Radiant Energy System have been estimating broadband (1 shortwave,
2 longwave radiometers) top-of-atmosphere flux measurements for over two decades
utilizing an advanced viewing geometry that observes the same footprint at several
viewing angles.
The central problem with the conversion from radiance measurements to flux is at odds
with practical issues related to viewing geometry. The formal integration of radiance to
produce flux is given by:
F  
2
0
 /2

0
I  ,  sin  cos   d d
(D1)
Since longwave radiance tends to vary minimally in the azimuthal direction, the
conversion from radiance to flux is simplified as:
F   I   d 
1
(D2)
0
where   cos . Formal numerical quadrature generally takes place through the use of
interpolating functions. Newton-Cotes formulae are utilized where values of the
integrand are measured at regularly-spaced integrals whereas Gaussian quadrature are
8
typically more accurate and can be utilized where the integrand can be evaluated at
specifiable points throughout the interval of integration. Unfortunately, both methods of
quadrature generally require evaluation of the integrand at points throughout most of the
interval of integration and this requires extreme viewing angles from a satellite-borne
platform.
Therefore, a variety of methods have been developed for this brand of quadrature that
essentially requires extrapolation at one integration end-point. Work by (SOURCES)
have developed angular distribution models using empirically-derived, scene-specific
anisotropic factors which are multiplied by the broadband radiance measurements to
yield fluxes. The anisotropic factors are given by:
 * I  ,  
Fˆ 
R  ,  
(D3)
where F̂ is the broadband (5-50 μm) IR flux estimated from broadband radiance
radiance measurements at zenith and azimuth angles  and  respectively and R is the
scene-specific angular distribution model which is derived empirically and the different
scenes are classified according to surface type and cloud cover. These methods generally
introduce only small errors into the calculation of fluxes and have allowed for large-scale
atmospheric energetics studies. Estimated error introduction is less than 4 W/m2 for total
OLR given proper scene type separation.
Deriving spectral fluxes from TOA measurements, as is required by this research, has
received much less attention, though it is arguably of great importance in the
determination of whether models, which are correctly calculating the outgoing longwaveand shortwave-radiation (OLR and OSR respectively), are correctly describing the
atmospheric state necessary to achieve OLR and OSR fluxes (SOURCE). In this case, a
statistical approach to the estimation of TOA spectral fluxes is warranted. According to
Huang et al (2006), estimation of tropical atmospheric fluxes from radiance
measurements using the IRIS-D instrument can be performed according to the following:
Fˆ  s1, I  s2,
(D4)
where s1, and s2, are regression coefficients. Total OLR error introduction using this
approach is estimated at 2 W/m2.
9