Uncertainty of atmospheric microwave absorption model: impact on ground-based radiometer simulations and retrievals

Uncertainty of atmospheric microwave absorption model

Uncertainty of atmospheric microwave absorption model: impact on ground-based radiometer simulations and retrievalsUncertainty of atmospheric microwave absorption modelDomenico Cimini et al.

Domenico Cimini^{1,2},Philip W. Rosenkranz^{3},Mikhail Y. Tretyakov^{4},Maksim A. Koshelev^{4},and Filomena Romano^{1}Domenico Cimini et al. Domenico Cimini^{1,2},Philip W. Rosenkranz^{3},Mikhail Y. Tretyakov^{4},Maksim A. Koshelev^{4},and Filomena Romano^{1}

^{1}National Research Council of Italy, Institute of Methodologies for
Environmental Analysis, Potenza, 85050, Italy

^{2}Center of Excellence CETEMPS, University of L'Aquila, L'Aquila, 67100,
Italy

^{3}Massachusetts Institute of Technology, Cambridge, MA 02139, USA

^{4}Russian Academy of Sciences, Institute of Applied Physics, Nizhny
Novgorod, 603950, Russia

This paper presents a general approach to quantify absorption model
uncertainty due to uncertainty in the underlying spectroscopic parameters. The
approach is applied to a widely used microwave absorption model (Rosenkranz,
2017) and radiative transfer calculations in the 20–60 GHz range, which are
commonly exploited for atmospheric sounding by microwave radiometer (MWR).
The approach, however, is not limited to any frequency range, observing
geometry, or particular instrument. In the considered frequency range,
relevant uncertainties come from water vapor and oxygen spectroscopic
parameters. The uncertainty of the following parameters is found to dominate:
(for water vapor) self- and foreign-continuum absorption coefficients, line
broadening by dry air, line intensity, the temperature-dependence exponent for
foreign-continuum absorption, and the line shift-to-broadening ratio; (for
oxygen) line intensity, line broadening by dry air, line mixing,
the temperature-dependence exponent for broadening, zero-frequency line
broadening in air, and the temperature-dependence coefficient for line mixing. The
full uncertainty covariance matrix is then computed for the set of
spectroscopic parameters with significant impact. The impact of the
spectroscopic parameter uncertainty covariance matrix on simulated
downwelling microwave brightness temperatures (T_{B}) in the 20–60 GHz
range is calculated for six atmospheric climatology conditions. The
uncertainty contribution to simulated T_{B} ranges from 0.30 K (subarctic
winter) to 0.92 K (tropical) at 22.2 GHz and from 2.73 K (tropical) to 3.31 K
(subarctic winter) at 52.28 GHz. The uncertainty contribution is nearly
zero at 55–60 GHz frequencies. Finally, the impact of spectroscopic
parameter uncertainty on ground-based MWR retrievals of temperature and
humidity profiles is discussed.

Atmospheric absorption models are used to simulate the absorption and
emission of electromagnetic radiation by atmospheric constituents.
Atmospheric absorption models are thus crucial to compute radiative transfer
through the atmosphere (Mätzler, 1997; Saunders et al., 1999; Clough et
al., 2005; Buehler et al., 2005; Eriksson et al., 2011), which is needed to
simulate and validate passive and active remote sensing observations, such as
those from microwave radiometer (MWR) and radar instruments (Hewison et al.,
2006; Maschwitz et al., 2013). Absorption and radiative transfer models,
representing the forward operator for atmospheric radiometric applications,
are also exploited in physical approaches for the solution of the inverse
problem, i.e., the retrieval of atmospheric parameters from remote sensing
radiometric observations (Westwater, 1978; Rodgers, 2000; Rosenkranz, 2001;
Rosenkranz and Barnet, 2006; Cimini et al., 2010). Thus, absorption and
radiative transfer models, and their uncertainty, have general implications
for atmospheric sciences, including meteorology and climate studies.

Comparisons of different radiative transfer and microwave absorption models
have been performed to quantify the difference in calculated brightness
temperatures (T_{B}) and the agreement with ground-based, satellite,
shipborne,
and airborne radiometric observations (Westwater et al., 2003; Melsheimer et
al., 2005; Hewison, 2006a; Hewison et al., 2006; Brogniez et al., 2016).
However, the uncertainty affecting current microwave radiometric
observations is often comparable to the differences in radiative transfer
calculations, and thus clear and definite answers were not always
obtainable.

Absorption models are based on quantum mechanics theory and rely on
parameterized equations to compute atmospheric absorption given the
thermodynamic conditions and abundance of constituents (Rosenkranz,
1993). The spectroscopic parameters entering the parameterized equations are
determined through theoretical calculations or laboratory and field
measurements, and their values are continuously refined (Liebe et al., 1989;
Rosenkranz, 1998; Liljegren et al., 2005; Turner et al., 2009; Mlawer et
al., 2012; Koshelev et al., 2018). Review papers are published occasionally
to summarize the proposed modifications (Rothman et al., 2005, 2013; Gordon et al., 2017).
The absorption models described in
Rosenkranz (1998, 2017) are cited frequently in this paper
and are hereafter called R98 and R17, respectively. The review by Tretyakov (2016)
is also cited frequently, meaning Tretyakov (2016) and the references
therein.

The uncertainty affecting the values of spectroscopic parameters contributes
to the uncertainty of the simulated absorption, which in turn affects
atmospheric radiative transfer calculations. Thus, the uncertainty affecting
spectroscopic parameters contributes to the uncertainty of simulated remote
sensing observations and consequently to the uncertainty of remote sensing
retrievals of atmospheric thermodynamic and composition profiles (Boukabara
et al., 2005a; Verdes et al., 2005). This situation does not apply to
microwave radiometry only, but is general to all wavelength regions (Long
and Hodges, 2012; Alvarado et al., 2013, 2015; Connor et
al., 2016). However, it must be considered that the uncertainty affecting
different spectroscopic parameters may be correlated. Therefore, in addition
to the uncertainty affecting the single parameters, the full uncertainty
covariance matrix should be estimated to account for the correlation in
radiative transfer calculations and retrievals (Rosenkranz, 2005; Boukabara
et al., 2005b).

In the last decade, the Global Climate Observing System (GCOS) Reference
Upper-Air Network (GRUAN) has evolved from aspiration to reality (Bodeker et
al., 2015). GRUAN is now delivering reference-quality measurement of
essential climate variables (ECVs), for which the uncertainty contributions
are carefully evaluated. In addition to radiosonde observations (Dirksen et
al., 2014), ground-based remote sensing products are planned in GRUAN,
including from microwave radiometer (MWR) profilers. Most common ground-based
MWR profilers operate in the 20–60 GHz range to infer ECVs such as
tropospheric temperature and water vapor profiles and vertically integrated
water vapor and liquid water contents. MWR adds value to GRUAN by providing
redundant measurements with respect to radiosondes, but covering the complete
diurnal cycle at high (e.g., 1 min) temporal resolution. The various sources
of uncertainty for MWR retrievals have been reviewed in the framework of the
GRUAN-related GAIA-CLIM project (http://gaia-clim.eu/, last access: 1 May 2018, Thorne et
al., 2017). One such source is the spectroscopic parameter uncertainty,
which appears to be the least investigated among all (Maschwitz et al., 2013;
GAIA-CLIM, Gaps Assessment and Impacts Document (GAID) – G2.37, 2017). The premises above call for a thorough investigation
of the uncertainty affecting spectroscopic parameters entering current
microwave absorption models and their impact on MWR simulated observations
and retrievals. Focusing primarily on clear-sky retrievals, the main
constituents contributing to atmospheric microwave absorption in the 20–60 GHz range are water vapor and oxygen.

Thus, the main purpose of this paper is to introduce a rigorous approach for
quantifying the absorption model uncertainty. Although the approach is
general and not limited to any particular instrument, observing technique,
or frequency range, we demonstrate its use through the application to
ground-based microwave radiometer simulations and retrievals. The analysis
thus consists of the following four steps:

review recent work concerning water vapor and oxygen spectroscopic
parameters and their associated uncertainties;

perform a sensitivity study to investigate the dominant uncertainty
contribution to radiative transfer calculations;

estimate the full uncertainty covariance matrix for the dominant
parameters; and

propagate the uncertainty covariance matrix to estimate the impact on
MWR simulated observations and atmospheric retrievals.

Thus, the paper is organized as follows: Sect. 2 summarizes the equations
used in the considered microwave absorption model and defines their
parameters. Section 3 presents the results of the uncertainty sensitivity
study. Section 4 discusses the approach to estimate the uncertainty
covariance matrix. Section 5 presents the impact of spectroscopic uncertainty
on simulated downwelling 20–60 GHz T_{B} and on the associated
ground-based atmospheric temperature and humidity profile retrievals. Section 6 presents a summary, main conclusions,
and hints for future work. Finally,
the Appendix reviews recent updates to spectroscopic parameters in the
considered microwave absorption models.

Absorption happens when radiation travels through a dissipative medium. The
radiation intensity as a function of the path length l through the medium is
given by the Beer–Lambert–Bouguer law, $I\left(l\right)={I}_{\mathrm{0}}\cdot {e}^{-\mathit{\alpha}\left(\mathit{\nu}\right)\cdot l}$, in which I_{0} is the incident radiation intensity, I is the
transmitted radiation intensity passed through the medium, and α is
the absorption coefficient of the medium, which depends on the radiation
frequency ν. The absorption coefficient is a macroscopic parameter that
represents the interaction of incident electromagnetic energy with the
constituent molecules. Here we consider atmospheric absorption, and thus
α(ν) represents the absorption spectrum of the gas mixture
forming the atmosphere. The gas absorption spectrum is the sum of two
components: the resonant and nonresonant absorption. The resonant
absorption is a property of individual molecules; it occurs at certain
frequencies (absorption lines) associated, for example, with the change in
the angular momentum of the molecule (rotational transition) or the
oscillation frequency (vibrational transition). Nonresonant absorption
arises from the interaction of molecules with each other, i.e., due to the
nonideality of gas. Thus, the gas absorption coefficient can be expressed
as the sum of the resonance lines and the nonresonance absorption:

The following sections describe the resonant and nonresonant absorption
components and the parameterization as defined in the family of absorption
models considered here, i.e., R98 and R17 as well as others introduced in
Sect. 2.4. Therefore, the review presented here applies specifically to
this family of models. However, the approach presented in this paper can be
considered generally valid for any absorption model.

2.1 Resonant absorption

Resonant absorption is modeled by computing the contribution of each
significant absorption line (line by line). Following Rosenkranz (1993), the
power absorption coefficient at frequency ν for a specified molecular
species with n molecules per unit volume is given by

is the line-shape function, while the following line parameters refer to the
ith absorption line of the specified molecule: the center frequency
(ν_{i}), the half-width at half amplitude (Δν_{i}),
the integrated intensity at temperature T (S_{i}(T)), and the mixing
parameter (Y_{i}). Note that the summation in Eq. (2) only
includes i > 0, as negative resonances are included in the line-shape function,
and the zero-frequency transition (Debye absorption, which must be taken
into account in molecular oxygen), sometimes referred as to i=0, is treated
below. The line-shape function Eq. (3) considers the fact that in the case of two or
more lines contributing significantly to the absorption, there may be
non-negligible line mixing, in which case the resulting intensity of the
band cannot be calculated as a simple sum of isolated line profiles. Instead, the line-mixing coefficients Y_{i} account for the line-mixing effect in the first-order (in pressure) approximation suggested by
Rosenkranz (1975). A second-order expansion was later proposed by Smith (1981),
adding coefficients accounting for the mixing of line intensities and
shifting of line central frequencies.

In the frequency range considered here (20–60 GHz), the line-mixing effect
is fundamental for understanding oxygen absorption, while it is
negligible for water vapor (Y_{i}≅0) (Ma et al., 2014). Then
for water vapor, the line-shape function reduces to the van Vleck–Weisskopf
profile:

The van Vleck–Weisskopf profile was demonstrated to fit experimental data
well on the 22 GHz line (Hill, 1986) and 183 GHz line (see Fig. 5 and related
references from Tretyakov, 2016); also, Koshelev et al. (2018) found that
speed-dependence effects amount to less than 1 % deviation with respect to
the van Vleck–Weisskopf profile near 22 GHz.

The van Vleck–Weisskopf profile can also be used for taking into account
zero-frequency transitions by letting ν_{0}=0 (Van Vleck, 1947).
All these transitions overlap each other and can be treated as a
single resonance line. This line in O_{2} may be included in the summation
of Eq. (2) as i=0, with ν_{0}=0, Y_{0}=0. However, a different
definition of line intensity must be used:

which has a finite nonzero value as ν_{0}→0. Thus, introducing
γ_{0} as the O_{2} zero-line half-width at half amplitude, this
absorption reduces to the following expression, which has the Debye line-shape factor (Rosenkranz, 1993):

Note that the line profiles (3, 4, 6) are valid only when the frequency
detuning satisfies $\left|\mathit{\nu}-{\mathit{\nu}}_{\mathrm{c}}\right|\ll {\left(\mathrm{2}\mathit{\pi}{\mathit{\tau}}_{\mathrm{c}}\right)}^{-\mathrm{1}}$, where τ_{c} is the finite duration
of molecular collision. Therefore, a way to model the line absorption is the
so-called line wing cutoff, i.e., assuming zero absorption at detunings
larger than a cutoff frequency. The value of the cutoff frequency proposed by
Clough et al. (1989), 750 GHz, is widely accepted and used in some
absorption models (R98; Clough et al., 2005). It should also be mentioned
that line profiles (3, 4, 6) take into account only the collisional
broadening mechanism and ignore additional line broadening related to thermal
molecular movement (Doppler broadening), which has a significant effect in
the considered frequency range only at very low gas densities (i.e.,
altitudes above 60 km). Fine effects of collisional narrowing of the resonance line,
due to speed dependence of absorbing molecule cross section or velocity-changing
collisions, are also ignored.

2.2 Nonresonant absorption

Nonresonant absorption accounts for the absorption characterized by
the smooth frequency dependence remaining after considering the effect of
resonant lines. The mechanism for nonresonant absorption arises from the
nonideality of atmospheric gases and corresponds to the absorption by
collisionally interacting molecules. At usual atmospheric conditions only
pair interaction is significant. This interaction during a finite time of
collision may lead to significant (either positive or negative) deviation of
resonance line far wings from the absorption calculated using profiles
(3–6). For each molecule, the sum of these deviations over all lines gives
absorption smoothly varying with frequency. Another component of
nonresonance absorption corresponds to molecular pairs (bimolecular
absorption). The latter can be further subdivided into three parts
corresponding to free molecular pairs, quasi-bound (metastable) dimers, and
true-bound (stable) dimers. All these absorption contributions also vary very
smoothly with frequency at atmospheric conditions due to either
the short lifetime of bimolecular state (free pairs and quasi-bound dimers) or
an extremely dense and collisionally broadened spectrum of loosely bound
molecular pairs (quasi-bound dimers and true-bound dimers).

To model nonresonance bimolecular absorption in the atmosphere, it
should be taken into account that pair interactions occur in any atmospheric
gases and their mixtures. For convenience, the treatment of atmospheric
nonresonance absorption is divided in two contributions, one deriving from
dry air and the other from water vapor.

The dry contribution is due to the interaction of dry air molecules with
each other. Only molecular nitrogen and oxygen are considered, as they
account for nearly 100 % of the atmospheric mixture and absorption.
Because of the dominant nitrogen contribution this component can be
approximately calculated in the considered frequency range as

where ${\mathit{\alpha}}_{{\mathrm{N}}_{\mathrm{2}}}\left(\mathit{\nu},T\right)$ is the absorption due to
N_{2}–N_{2} interactions and ε(νT) accounts for the
absorption due to O_{2}–O_{2} and N_{2}–O_{2} interactions,
considering N_{2} and O_{2} relative abundances and absorption
intensities (Boissoles et al., 2003).

Concerning the water vapor contribution to nonresonance absorption, despite
a general understanding of the physical nature (e.g., Shine et al., 2012;
Tretyakov et al., 2014; Serov et al., 2017), there are no sufficiently
accurate theoretical models for calculating the spectra of all necessary
components (especially in gas mixtures) and their temperature dependences.
Therefore, for practical purposes parameters of the observed nonresonant
absorption are determined using simple empirical models, which have not been
supported by accurate theoretical calculations and are based on experimental
data only (Tretyakov, 2016). The so-called continuum absorption is thus
empirically defined as the difference between the total observed absorption
and the calculated contribution of resonance lines:

Note that in such a definition the resulting continuum absorption contains
the nonresonant absorption as well as the unknown contribution from
resonance line far wings at frequency detunings exceeding the somewhat
arbitrary cutoff frequency introduced above.

2.3 Absorption model parameterization

The spectroscopic parameters appearing in the above equations may depend on
temperature (T) and pressure (P). Most experimental data on spectroscopic
parameters are obtained near room temperature, and thus tabulated values are
available at reference temperature T_{0} (usually 296 or 300 K).
Parametric functions are used to express the dependence on T and P in common
absorption models.

For the line intensity, the temperature dependence is given by the total
number of populated molecular states (the partition sum), which can be
calculated numerically (Gamache et al., 2017), and the population of
molecular energy levels corresponding to the transition. The latter is
calculated from the energy of the lower level and the frequency of the
corresponding transition. Thus, calling k the Boltzmann constant,
E_{low} the energy of the lower level, S(T_{0}) the intensity at the
reference temperature T_{0}, and introducing the so-called inverse
temperature ($\mathit{\theta}=\frac{{T}_{\mathrm{0}}}{T}$), the intensity is written as
(Rosenkranz, 1993)

where the temperature exponent n_{S} accounts for the temperature dependence
of the partition sum and differs for asymmetric (e.g., water vapor,
n_{S}≅2.5) and linear (e.g., oxygen, n_{S}≅2.0) molecules.

For pressure-broadened line coefficients, it is convenient to introduce
normalized coefficients relative to the reference temperature T_{0} and
independent of pressure. In general, experimental studies fit them to a
function of the form γ=γ(T_{0}) θ^{n}P, where γ(T_{0}) and n are constant coefficients. The
power function is generally suitable for atmospheric applications to account
for the temperature dependence of the above parameters as it works well
within ±50 K from T_{0}.

For water vapor absorption, the line width and the line center frequency are
differently affected in the case of broadening induced by water vapor (self-broadening, indicated by s) or by dry air (foreign broadening, indicated by
a). Thus, calling P_{w} and P_{d} the partial pressures of water vapor and
dry air and ${\mathit{\nu}}_{i}^{\mathrm{0}}$ the “zero pressure” transition frequency of
the ith absorption line, line broadening and shifting are written
respectively as

where γ_{i,s}, γ_{i,a} and
δ_{i,s}, δ_{i,a} are the self and foreign
parameters for broadening and shifting, respectively, at the reference
temperature T_{0}, and ${n}_{{\mathit{\gamma}}_{\mathrm{s}}}$, ${n}_{{\mathit{\gamma}}_{\mathrm{a}}}$,
${n}_{{\mathit{\delta}}_{\mathrm{s}}}$, and ${n}_{{\mathit{\delta}}_{\mathrm{a}}}$ are the temperature
exponents for line self-broadening, foreign broadening, self-shifting, and
foreign shifting. In R17, the ratio of shift to broadening (R_{i}) is used
as a parameter instead of the shifting parameter, e.g., ${R}_{i}={\mathit{\delta}}_{i}/{\mathit{\gamma}}_{i}$. This implicitly assigns the same temperature dependence to
broadening and shifting, which is done because of the absence of relevant
measurements for n_{δ}, although theory suggests that it could differ
from n_{γ} (Pickett, 1980).

Similarly, for oxygen it is convenient to introduce normalized broadening
(γ_{i}) and mixing (y_{i}) coefficients. In addition, the
water-to-air broadening (r_{w2a}) and mixing (${r}_{\mathrm{w}\mathrm{2}\mathrm{a}}^{\prime}$) ratios are
introduced for considering the broadening and mixing of oxygen lines induced
by water vapor. Line mixing depends on the off-diagonal elements of the
collisional interaction matrix, while the diagonal elements of that matrix
give the line width parameters. Therefore, both mixing and broadening depend
on the type of perturbing molecule, but because of the absence of
calculations and relevant measurements for ${r}_{\mathrm{w}\mathrm{2}\mathrm{a}}^{\prime}$, the model assumes
${r}_{\mathrm{w}\mathrm{2}\mathrm{a}}^{\prime}={r}_{\mathrm{w}\mathrm{2}\mathrm{a}}$. We believe that the possible systematic impact of this
assumption is smaller than other model uncertainties discussed in this
paper. Thus, the width and mixing coefficients are expressed as

where n_{a} is the temperature exponent for oxygen line broadening and
V_{i} represents coefficients introduced to account for the ${\mathit{\theta}}^{{n}_{\mathrm{a}}+\mathrm{1}}$
dependence (Liebe et al., 1992).

Line parameters that most significantly affect the line shape (e.g., ν_{i}, S(T_{0}), E_{low}, γ(T_{0}), and δ(T_{0})) can be
found in several spectroscopic databases, e.g., HITRAN (http://hitran.org/, last access: 1 May 2018; Gordon et al., 2017).

Concerning the water vapor continuum, it has been established (Liebe and Layton, 1987; Kuhn et al., 2002; Koshelev et al., 2011; Shine et al., 2012) that
the absorption can be represented as two terms corresponding to the
interaction of water molecules with each other (self-continuum component)
and the interaction between water molecules and air molecules (foreign-continuum component). In the frequency range considered here, the continuum
absorption depends quadratically on frequency (R98) and its temperature
dependence is described by a simple exponential function:

where we introduced the empirical numerical intensity coefficients for the self-induced
(C_{s}) and foreign-induced (C_{f}) water vapor continuum and their
respective temperature-dependence exponents (n_{cs}, n_{cf}).

For the dry continuum, Rosenkranz et al. (2006) proposed a
frequency-dependent factor f(ν) to fit the data calculated
by Borysow and Frommhold (1986), who modeled the bimolecular absorption for
N_{2}–N_{2} pairs. Calling C_{d} the intensity coefficient of the dry air
continuum and n_{d} the relative temperature-dependence exponent, the dry
continuum absorption is modeled as

2.4 Atmospheric absorption model in the 20–60 GHz range

In the frequency range considered here (20–60 GHz) and for tropospheric
conditions, atmospheric clear-air absorption is dominated by oxygen and water
vapor. Oxygen produces strong resonant absorption due to transitions in the
magnetic dipole spin-rotation band between 50 and 70 GHz. Collisional broadening
at increased pressures causes the 60 GHz band lines to blend together and at
pressures approaching atmospheric and higher the band absorption looks like
an unstructured composite feature spreading about ±10 GHz around 60 GHz,
with one line at 118.75 GHz. For water vapor, rotational transitions
of the electric dipole produce resonant absorption lines extending from the
microwave to the far infrared range, including lines near 22.235 GHz and
183.31 GHz. Since absorption lines are well separated, the line-mixing
effect is negligible (Y_{i}=0). In addition to line contributions,
water vapor absorption accounts for the continuum component, generally
divided into the self and foreign components. More details on the theory of
microwave absorption by atmospheric gases is given by Rosenkranz (1993).

Based on theoretical considerations and laboratory experimental data in the
1960s, the millimeter-wave propagation model (MPM) was developed for the
range from 20 GHz to 1 THz, including the 30 strongest water vapor lines, 44
oxygen lines, and an empirically derived water vapor continuum (Liebe and
Layton, 1987). This model was later revised, modifying the line
parameters (Liebe, 1989), the oxygen line coupling (Liebe et al., 1992), the
number of water vapor lines, and the continuum formulation (Liebe et al.,
1993; R98). More details on the differences between these, as well as other
absorption models, and the comparison with shipborne, aircraft, and
ground-based observations can be found in Westwater et al. (2003), Cimini et
al. (2004), Hewison (2006a), Hewison et al. (2006), and the references therein. The
above models are widely used and have been taken as references for the last
30 years. For example, the parameterized radiative transfer code RTTOV (Saunders
et al., 1999), widely used worldwide to assimilate satellite microwave
radiometer observations into weather models, is trained against calculations
made with the MPM87 (Rayer, 2001) and later modifications (Saunders et
al., 2017).

Appendix A gives a summary of the modifications to the R98 water vapor and
oxygen absorption models proposed in the open literature in the last 20 years
and subsequently imported in the current version of the model (R17). Here,
just to show the effects of the adopted modifications, Fig. 1 displays the
20–60 GHz downwelling T_{B} as computed with the R17 model and the
difference with respect to the reference R98 model. Six atmospheric
climatology conditions have been considered (tropical, midlatitude summer,
midlatitude winter, subarctic summer, subarctic winter, US standard).

Figure 1(a) Zenith downwelling T_{B} computed
using six reference atmosphere climatology conditions with the R17 model.
(b) Difference between T_{B} computed with the current and
reference versions (R17 minus R98) for the six atmosphere climatology
conditions. Note the features at 22 GHz, mainly attributable to the updated
line width (Payne et al., 2008), at 25–50 GHz due to the scaled continuum
(Turner et al., 2009), and at 50–55 GHz related to revised coefficients
for the 60 GHz band (Tretyakov et al., 2005).

The atmospheric absorption calculated from a model has in general a
nonlinear dependence on some spectroscopic parameters, as reviewed in
Sect. 2. With the assumption of small perturbations, however, one can
reasonably linearize that dependence for a given model:

where p is a vector whose elements are the parameters in the model,
having nominal value p_{0}; T_{B} is a vector of calculated
brightness temperatures at various frequencies using parameter values p,
while T_{B0} is calculated for parameter values p_{0}, and
K_{p} represents the model parameter Jacobian, i.e., the
matrix of partial derivatives of model output with respect to model
parameters p. It follows that the covariance matrix of T_{B}
uncertainties due to absorption model parameter is

where the symbol ⊤ indicates a transpose matrix. Thus, the full
covariance matrix of parameter uncertainties is necessary to compute the
uncertainty of calculated T_{B}, even for just a single frequency. The
values of spectroscopic parameters are determined in the spectroscopic
literature either theoretically or empirically from field and/or laboratory
experimental data and are thus inherently affected by uncertainty.
Spectroscopic parameters are affected by both random and systematic
uncertainties as a consequence of experimental noise and systematic errors.
Following the practice recommended by JCGM (2008), our analysis takes into
account the total (i.e., systematic and random) uncertainty of spectroscopic
parameters, which combine to contribute to the total uncertainty of
simulated T_{B}. If parameter values are determined with methods that
introduce correlation between them, their total uncertainty will also be
correlated. However, the spectroscopic literature provides at most the
uncertainty of individual parameters, not covariance.

Thus, this section presents a study of the absorption model sensitivity to
the uncertainty of spectroscopic parameters, with the purpose of identifying
the most significant contributions to the total uncertainty of modeled
downwelling T_{B}. A preliminary analysis is presented by Cimini et
al. (2017). For the identified relevant parameters, the full covariance
matrix is then estimated in Sect. 4. The approach is as follows. First, the
uncertainties affecting spectroscopic parameters are determined from
published literature or independent analysis. Then, each parameter (or
parameter type if known to be highly correlated) is investigated individually
by perturbing its value by ±1σ impact on the modeled downwelling
T_{B}. Six different climatologic conditions, as introduced in
Fig. 1, are considered to account for temperature, pressure, and humidity
dependences. Only parameters with 1σ uncertainty impacting the
modeled 20–60 GHz T_{B} for more than 0.1 K are considered in
Sect. 4 for an evaluation of their covariance.

3.1 Sensitivity to water vapor parameters

In the 20–60 GHz frequency range under consideration, only two resonant
lines (at 22 and 183 GHz) and the continuum contribute non-negligibly to
water vapor absorption. For the model parameters associated with these
absorption features, the uncertainties were either taken from the
spectroscopic literature or, where not available, were estimated from an
independent analysis of measurement methods. The resulting uncertainties, as
well as nominal values, for the water vapor parameters considered in this
sensitivity analysis are listed in Table 1.

Table 1List of water vapor parameters perturbed in the sensitivity
analysis.

For the resonant absorption, the following parameters are relevant: line
frequency (ν_{i}), intensity (S_{i}) and its temperature coefficient
(n_{S}), the lower-state energy (E_{low}), air and water broadening
(γ_{a} and γ_{w}) and their temperature-dependence exponents
(n_{a} and n_{w}), and the shift-to-broadening ratio (R_{i}). The
uncertainty estimates for most of these parameters are given by Tretyakov (2016)
within a review and expert assessment. The only exceptions are the
uncertainty estimates for γ_{a}, γ_{w}, and R_{i} at 22 GHz
taken from the more recent investigation of Koshelev et al. (2018) and the
uncertainty for n_{S}, which has been independently estimated within the
200–400 K temperature range as the maximal difference between numerical
calculation of the partition sums at various temperatures published by
Gamache et al. (2017) and their power approximation ${\mathit{\theta}}^{{n}_{\mathrm{S}}}$.

For the continuum absorption, four parameters are relevant, namely the self-
and foreign-induced intensity coefficients and their respective
temperature-dependence exponents (C_{s},C_{f}, n_{cs}, n_{cf}).
Uncertainties for C_{s} and C_{f} have been estimated considering that
R17 adopts values adapted from Turner et al. (2009), who also provide an
uncertainty estimate for the proposed multiplicative factors (0.79(18) and
1.11(10), respectively, for self and foreign coefficients). The uncertainties
for n_{cs} and n_{cf} are estimated to overlap, within
uncertainty,
the values given by Koshelev et al. (2011) based on laboratory measurements.
The resulting uncertainties (0.6 and 0.8, respectively) are more
conservative than those provided originally (Liebe and Layton, 1987; Liebe
et al., 1993).

The sensitivity analysis shows that among the 19 model parameters that
were perturbed by the estimated uncertainty (Table 1), only 6 impact the
modeled downwelling 20–60 GHz T_{B} for more than 0.1 K: C_{s},
C_{f}, n_{cf} and S_{i}, γ_{i,a}, R_{i} at 22 GHz. The
sensitivity of 20–60 GHz T_{B} to perturbations to these six parameters is
shown in Fig. 2. The impact of both positive and negative perturbations is
shown; their symmetry with respect to the zero line suggests that estimated
uncertainties represent small perturbations satisfying the linear assumption
in Eq. (17). These six parameters are considered in Sect. 4 for
an evaluation of their covariance. Although we note that Tretyakov (2016)
indicates larger uncertainty for n_{cs} at temperatures lower than 300 K,
it was found that even considering 5 times larger uncertainty (to cover within
uncertainty the value given for the range 270–300 K, i.e., 7.6(6)),
the impact remains small for the relatively cold climatology. Thus n_{cs}
is not considered for the analysis in Sect. 4.

Figure 2Sensitivity of modeled T_{B} to water vapor
absorption parameters. (a) Line intensity
(S_{i}) and air broadening
(γ_{i,a}) at 22 GHz.
(b) Shift-to-broadening ratio (R_{i}) at 22 GHz and
foreign-broadening temperature-dependence exponents
(n_{cf}). (c) Self-induced
(C_{s}) and foreign-induced
(C_{f}) broadening coefficients.
Solid lines correspond to negative perturbation (value − uncertainty),
while dashed lines correspond to positive perturbation (value + uncertainty).

Oxygen absorption includes the zero-frequency band, fine structure spectrum, and
pure rotational resonant transitions. The R17 model includes 49 oxygen
absorption lines, of which 37 are within the 60 GHz band, 1 is at 118 GHz
and the remaining 11 are in the millimeter to sub-millimeter range (200–900 GHz). Uncertainties
for the oxygen parameters were either retrieved from the spectroscopic
literature or, where not available, estimated from an independent analysis of
measurement methods.

For the resonant absorption, the following parameters are relevant: line
frequency (ν_{i}), intensity (S_{i}) and its temperature-dependence
exponent (n_{S}), the lower-state energy (E_{low}), air broadening
(γ_{a}) and its temperature-dependence exponent (n_{a}),
normalized mixing coefficient (y_{i}) and its temperature-dependence
coefficient (V_{i}), and the water-to-air broadening ratio (r_{w2a}).

The uncertainty estimates for most of these parameters are given by
Tretyakov et al. (2005). In particular, Tretyakov et al. (2005) provide
frequency uncertainty for 27 lines (N from 1 to 27, where N is the O_{2}
rotational quantum number). For the other lines, the maximum uncertainty
value has been assumed (i.e., 17 kHz), which is conservative with respect to
HITRAN.

Resonant line intensities and lower-state energies are taken from the HITRAN
2004 database (Rothman et al., 2005). Although newer calculations are
available in HITRAN 2016 (Gordon et al., 2017), the differences are within
the assumed uncertainty at 1 % and 0.25 %, respectively. The latter is a
rather conservative estimate, though its contribution turned out to be
irrelevant. Note that the 1 % uncertainty in O_{2} line intensities is
considered to originate mainly from the uncertainty of experimental
measurements of electronic transition band-integrated intensities, which were
used for intensity calculations of microwave lines. This uncertainty should
be correlated for all lines by the principle of determination and thus we assume
a single variable affecting all the lines. The uncertainty of the n_{S} value
for the 200–350 K temperature range was evaluated the same way as for water
vapor lines, i.e., comparing partition sum calculations by Gamache et al. (2017)
with their power-law approximation.

Values for oxygen line air-broadening and mixing parameters are taken from
Tretyakov et al. (2005). Line-broadening parameters are measured through
low-pressure laboratory experiments. Since individual lines are isolated at
low pressures, no correlation is considered between parameters of different
lines. Mixing parameters are determined at higher pressures, and their values
are correlated with the previously determined low-pressure parameters. So,
the line-mixing parameters are correlated with both themselves and the line
air-broadening parameters. Because of this relationship, consistency requires
that the number of considered line widths and the number of considered mixing
coefficients should be the same. Tretyakov et al. (2005) derived mixing
coefficients for lines with N from 1− to 33+ (34 in total), then
extrapolated to lines with N > 33 (i.e., four weak lines of the
60 GHz complex). Thus, we first investigated the impact of these remaining
four and the 11 rotational higher-frequency lines on 20–60 GHz
T_{B} by considering conservative and completely correlated uncertainty
estimates (10 % for line-broadening and 20 % for line-mixing parameters).
The impact was found to be negligible (< 0.1 K) and thus these
15 lines are not further considered in the following analysis. For the
remaining 34 lines (N from 1− to 33+), the uncertainty for line
air-broadening, mixing, and mixing temperature-dependence coefficients is
evaluated through the full covariance matrices, so their treatment is
postponed to Sect. 4.

For the air-broadening temperature-dependence coefficient, R17 retains a
uniform value (0.8) for all lines (Liebe, 1989). We assume 0.05 uncertainty,
which covers more recent measurements from Makarov et al. (2008) and
Koshelev et al. (2016). Since R17 adopts the water-to-air broadening ratio
r_{w2a}, its value and uncertainty are respectively estimated as the mean
and standard deviation calculated by Koshelev et al. (2015) from a set of 19
measurements (N from 1 to 19).

For the zero-frequency absorption, two parameters are relevant: the
intensity (${S}_{\mathrm{0}}^{\prime}$) and broadening (γ_{0}) of the pseudo-line.
The intensity of the zero-frequency absorption is from the Jet Propulsion
Laboratory (JPL) catalogue (https://spec.jpl.nasa.gov/, last access: 1 May 2018; Pickett et al.,
1998). For the zero-frequency line broadening, consideration of the
measurements cited in Danese and Partridge (1989), as well as those of Ho
et al. (1972) and Kaufman (1967), lead us to assign an uncertainty of
50 MHz bar^{−1}
to the absorption model's value of γ_{0}=560 MHz bar^{−1} at 300 K.
Note that uncertainties in the intensity and broadening coefficients of the
zero-frequency component are negatively correlated because it is very
difficult to measure the broadening independently of the intensity for this
pseudo-line. This estimate based on the spread of published measurements
accounts for the combination of intensity and broadening uncertainties.

The sensitivity analysis shows that among the model parameters in Table 2,
which were perturbed by the estimated uncertainty, only the following impact
the modeled downwelling 20–60 GHz T_{B} for more than 0.1 K: S_{i},
γ_{a}, n_{a}, y_{i}, V_{i}, and γ_{0}. The sensitivity of
20–60 GHz T_{B} to perturbations to these parameters is shown in Fig. 3.
As for water vapor, the impact of positive and negative perturbations is
symmetric with respect to the zero line, suggesting that the linear
assumption is valid for the estimated uncertainties. Note that the
perturbation to S_{i} and n_{a} affects all lines simultaneously, while
the other resonant line parameters have been perturbed line by line. Although
for the present ground-based application the uncertainty of only a few lines
is relevant, we prefer to keep all 34 to make the calculation of the
parameter uncertainties more generally useful (e.g., for satellite
observations). Thus, the above six parameters (S_{i}, γ_{a},
n_{a}, y_{i}, V_{i}, γ_{0}) are considered in Sect. 4 for
an evaluation of their covariance. While for S_{i}, n_{a}, and γ_{0}
we consider three scalar parameters, for γ_{a}, y_{i}, and V_{i}
we consider 34 lines (N from 1− to 33+), leading to 34 coefficients for
each parameter type.

Table 2List of oxygen parameters perturbed in the sensitivity analysis.
^{*} Tables 1 and 5 from Tretyakov et al. (2005).

Figure 3Sensitivity of modeled T_{B} to
oxygen absorption parameters. (a) Line intensity
(S_{i}) and air broadening (γ_{i,a}). (b) Air-broadening
temperature-dependence exponents
(n_{a}) and nonresonant pseudo-line broadening
(γ_{nr}). (c) Mixing coefficients (y_{i})
and mixing temperature-dependence coefficients (V_{i}).
Note that the perturbation to S_{i}
and n_{a} affect all lines, while for the other
resonant line parameters we show the impact of the perturbation to just one
line ($N=\mathrm{25}-$) as an example. Solid lines correspond to negative
perturbation (value − uncertainty), while dashed lines correspond to positive
perturbation (value + uncertainty).

The sensitivity analysis of Sect. 3 shows that the absorption model
uncertainty on downwelling 20–60 GHz T_{B} is dominated by the uncertainty
on 6 spectroscopic parameters for water vapor and up to 105 parameters for
oxygen. For these parameters, we require the full covariance matrix of
parameter uncertainties to compute the uncertainty of calculated T_{B} at
any given frequency. This section summarizes the methods used to estimate the
uncertainty covariance matrix, including the off-diagonal terms giving the
covariance of each parameter with the others. Additional details can be found
in Rosenkranz et al. (2018) (abbreviated as R18 below). However, the
analysis here differs in three respects from the preliminary version in R18:
the method of estimating Cov(C_{f},C_{s}), the use
of a smaller uncertainty for γ_{0}, and the inclusion of
Cov(γ_{0},n_{a}), which was neglected in R18.

Although we use different methods to estimate covariances depending on how
the parameter values were measured, some general principles apply. If a set
of variables a_{i} has a causal dependence on another set of variables
b_{k},

to the uncertainty covariance of the a values. There may also be other
contributions to Cov(a).

A probability distribution can be conditional, and the uncertainty of one
parameter may be conditioned on an assumed value for a different parameter.
Sometimes reported values of a parameter or set of parameters have been
adjusted to fit measurements, while the experimenters considered other
relevant spectroscopic parameters as fixed. Now if we wish to include in our
analysis the uncertainty of one of the latter parameters (b) and it has a
covariance with a fitted parameter a, the influence of b on a will increase the
uncertainty of a above that which was found in the original experiment. That
increment of variance is also given by Eq. (21), which in the scalar case is
equivalent to

4.1 Uncertainty covariance matrix for water vapor parameters

Section 3.1 shows that for water vapor absorption six spectroscopic
parameters dominate the uncertainty of modeled 20–60 GHz T_{B}: three
related to the continuum (C_{s}, C_{f}, n_{cf}) and three to the
22 GHz resonant line (S_{i}, γ_{i,a}, R_{i}). Sections 4.1.1–4.1.3
describe the methods used to estimate the covariances of these six water
vapor spectroscopic parameters. Although the covariance matrix is the basic
object needed for calculation, Table 3 lists both the estimated covariances
of water vapor parameter uncertainties and the corresponding correlation
coefficients because the latter are more easily comprehended, being pure
numbers and normalized to the interval ($-\mathrm{1},\mathrm{1}$). The numerical values of the
full covariance matrix are also provided in the Supplement (in
ASCII and NetCDF formats).

Table 3Covariance (a) and correlation (b) matrices corresponding to
spectroscopic water vapor parameter uncertainties as derived in Sect. 4.
Note that C_{f} and C_{s} are evaluated at T_{0}=300 K, while
γ_{a} and S are evaluated at T_{0}= 296 K.

4.1.1 Covariance between water vapor line parameters

Intensity, width, and shift affect a line profile in different ways. But even
if the original spectroscopic measurements covered the line profile
adequately, a noticeable negative correlation between width and intensity
arises if both are simultaneously estimated from measured absorption. In the
present case, the only water line that survived the sensitivity screening for
the 20–60 GHz band is the one at 22.2 GHz; the intensity used here was
calculated independently from the width (Rothman et al., 2013), and the width
was measured without using that intensity (Payne et al., 2008). Therefore, we
consider errors in those two parameters to be uncorrelated. However, the
absorption model code under investigation here (R17) uses the aforementioned
ratio of shift to width ($R={\mathit{\delta}}_{\mathrm{a}}/{\mathit{\gamma}}_{\mathrm{a}}$, where
δ_{a} and γ_{a} are respectively the shift and width
coefficients). As shown in R18, that introduces a covariance between R and
γ_{a} of

where ${\mathit{\sigma}}_{{\mathit{\gamma}}_{\mathrm{a}}}^{\mathrm{2}}$ is the uncertainty variance of γ_{a},
and it corresponds to the small correlation of +1 % shown in Table 3
(positive because the nominal value of R is negative for this line).

4.1.2 Covariance between C_{f}, C_{s}, and other water vapor parameters

By definition, the water vapor continuum is the remainder after the
contribution of local resonant lines has been subtracted. Thus, if a line
width is revised, the continuum should also be revised to compensate for and
reproduce as well as possible the original brightness temperature
measurements of Turner et al. (2009) from which the continuum was derived. That
was done by adjusting the continuum coefficients C_{f} and C_{s} for use
with updated line parameters in R17. It should be the case no matter which
line is revised. If we separate the model parameters into continuum (con) and
line types, then as discussed in R18, the above statements are equivalent to
requiring that for each line separately, the covariance between the
continuum and line parameters and the line-parameter covariance matrix
satisfy

In order for the above equation to hold over a range of humidity, it should
apply to self and foreign gas effects separately. Both R and
γ_{a} apply to dry air, so we set Cov(${C}_{\mathrm{s}},R)=\mathrm{0}$ and
Cov(C_{s}, ${\mathit{\gamma}}_{i,\mathrm{a}})=\mathrm{0}$. On the other hand, line intensity
S
affects both components of the continuum, with resulting covariances; then
Eq. (24) can be solved for Cov(p_{con}, S) by making
${K}_{{p}_{\mathrm{con}}}\mathrm{2}\times \mathrm{2}$ (see R18). As shown in Table 3b, the
correlations of the continuum parameters with the 22 GHz line parameters are
very small because this is one of the weaker water lines. If our matrix had
included parameters for the 183 GHz water line, their covariances with the
continuum might well be significant.

Although n_{cf}, the continuum foreign-broadening temperature exponent, is
not a line parameter, it was held fixed by Turner et al. (2009) in fitting C_{f}
and C_{s} to the measured T_{B}. Therefore, any subsequent change in
n_{cf} should require a compensating change in C_{f}; hence, from
Eq. (24)

which turns out to produce a significant covariance (Table 3a). If
C_{f} is thus compensated for, C_{s} should not change, so
Cov(C_{s}, n_{cf}) =0.

4.1.3 Covariance between C_{f} and C_{s}

For the water vapor continuum, R17 adopts the multipliers proposed by Turner
et al. (2009) to the R98 parameter values of C_{f} and C_{s},
with small readjustments to accommodate the updated line widths in R17.
Turner et al. (2009) derived the multipliers by adjusting them to
fit ground-based radiometer measurements at 150 GHz. The simultaneous
fitting of two coefficients results in a correlation between them.

When brightness temperature measurement errors are uncorrelated, with
variance ${\mathit{\sigma}}_{n}^{\mathrm{2}}$, a least-squares fit (see, e.g., van der
Waerden, 1969; Stuart and Ord, 1991) results in the parameter-error
covariance matrix

where the subscript “m” is the index for the measurements of T_{B} and indexes i and j equal
1 for C_{f} or 2 for C_{s}; the derivatives are to be evaluated for each
atmospheric profile corresponding to T_{Bm} at the fitted values of
C_{f} and C_{s}. When the correlation coefficient ρ_{fs} between
C_{f} and C_{s} uncertainties is evaluated from Eq. (26), ${\mathit{\sigma}}_{n}^{\mathrm{2}}$ cancels, as does the determinant except for its sign, which in
this case is positive. Thus, for the simple case of the 2×2 matrix,

Although Turner et al. (2009) do not give the correlation coefficient, it can be
estimated from a simulation covering the same range of integrated
water vapor content, 0.37 to 2.76 cm. We used 12 values of humidity
distributed over this range in a subarctic summer model atmosphere,
yielding ${\mathit{\rho}}_{\mathrm{fs}}=-\mathrm{0.87}$, which is (presumably) approximately what
Turner et al. would have calculated. Then using the experimentally determined
uncertainties from Table 1, we have $\mathrm{Cov}\left({C}_{\mathrm{f}},{C}_{\mathrm{s}}\right)=-\mathrm{1.57}\times {\mathrm{10}}^{-\mathrm{19}}$ (which is ∼11 %
larger than previously estimated in R18 by means of an analogy with data
from Payne et al., 2011).

Turner et al. (2009) held other parameters constant while adjusting the
continuum coefficients C_{f} and C_{s}. When we introduce
a variance of n_{cf} and its covariance with C_{f} (see
Sect. 4.1.2), then as discussed in reference to Eq. (22), a corresponding
increase by [Cov(C_{f},
${n}_{\mathrm{cf}})/{\mathit{\sigma}}_{{n}_{\mathrm{cf}}}$]^{2} to the experimentally determined variance of
C_{f} is required. That increases
${\mathit{\sigma}}_{{C}_{\mathrm{f}}}^{\mathrm{2}}$ from $\mathrm{3.09}\times {\mathrm{10}}^{-\mathrm{21}}$ to
$\mathrm{4.58}\times {\mathrm{10}}^{-\mathrm{21}}$, which is the value in Table 3a. However, when
Cov(T_{B}) is computed this increased variance will be
offset by the negative contribution of Cov(C_{f}, n_{cf}).
(Had n_{cf} been included in the least-squares fit, then it would have
been a 3×3 matrix, which would have produced a different result
originally.) Variance contributions from the 22 GHz line parameters are
negligible. The correlation coefficients in Table 3b were then computed using
the modified value of ${\mathit{\sigma}}_{{C}_{\mathrm{f}}}$.

4.2 Uncertainty covariance matrix for oxygen parameters

The sensitivity analysis in Sect. 3.2 shows that for oxygen absorption six
spectroscopic parameter types dominate the uncertainty of modeled 20–60 GHz
T_{B}: line intensity (S_{i}), air broadening (γ_{a}) and its
temperature-dependence exponent (n_{a}), normalized mixing coefficient
(y_{i}) and its temperature coefficient (V_{i}), and zero-frequency
broadening (γ_{0}). Parameters n_{a} and γ_{0} are scalar,
while γ_{a}, y_{i}, and V_{i} are vectors of 34 components (for
lines with N from 1− to 33+); although S_{i} is also a vector, its
percent of uncertainty is a scalar, thus leading to a 105×105 uncertainty
covariance matrix. Sections 4.2.1–4.2.4 describe the method used to estimate
the uncertainty covariance of these 105 oxygen spectroscopic parameters with
respect to each other. The numerical values of the full covariance matrix are
provided in the Supplement (both in ASCII and NetCDF formats).
Figure 4 depicts the resulting matrix as a color-scale image of sign-adjusted
correlation coefficients. For any two parameters p_{1} and p_{2} with
nominal values ${p}_{\mathrm{1}}^{\prime}$ and ${p}_{\mathrm{2}}^{\prime}$
and correlation coefficient ρ(p_{1},p_{2}), the
sign-adjusted correlation is defined as

If ${p}_{\mathrm{1}}^{\prime}$ and ${p}_{\mathrm{2}}^{\prime}$ have the same sign, ρ_{SA}(p_{1},p_{2}) reduces to
ρ(p_{1},p_{2}). If the signs differ, then ρ_{SA}(p_{1},p_{2}) has sign opposite to ρ(p_{1},p_{2}). If the standard deviations are small compared to the
nominal values, as is generally the case here, ρ_{SA}(p_{1},p_{2}) gives the correlation between the absolute values of the
parameters. ρ_{SA}(p_{1},p_{2}) can be negative, as is
the case for the relation between line intensities and the mixing
coefficients, which indicates that a positive error in intensities results
in underestimation of line mixing.

4.2.1 Covariance between oxygen line-broadening coefficients

Values for oxygen line air broadening are taken from Tretyakov et
al. (2005). They measured N_{2} broadening of O_{2} lines with
rotational quantum numbers N from 1 to 19 and self-broadening for N from 1 to
27 (the 1− line had previously been measured in Tretyakov et al., 2004). Uncertainties
of the measured line widths were estimated here by
considering the results of Tretyakov et al. (2005) and Koshelev
et al. (2016) together. Three sources were assumed to contribute to
the error budget: (i) the statistical uncertainty was determined from a
Padé approximation (Koshelev et al., 2016) of the N dependence of
γ_{a}, weighting all data by their respective 1∕σ;
(ii) a pressure gauge uncertainty of 0.25 %; and (iii) an uncertainty of 0.5 ^{∘}C
for the temperature sensors. The total uncertainty for each
line's air broadening was determined as the root sum of squares. Uncertainties
calculated for all lines with N≤19 are close to each other at
∼0.014 GHz bar^{−1}, so we use this value for all lines with N≤19. Even though the lines were measured separately by Tretyakov et
al. (2005), the pressure sensor and temperature sensor uncertainties contain
systematic components that (due to the same experimental setup) may have
introduced minor correlations between line widths. However, the broadening
parameter uncertainty originates mainly from the unknown baseline of the
apparatus. The work by Koshelev et al. (2016), in which different
sensors were used, confirmed that there was no noticeable bias in the earlier
measurements. This reasoning allows us to neglect potential correlations of
the measured line widths.

For the remaining lines, Tretyakov et al. (2005) extrapolated the broadening
coefficients by a straight-line graphical method, assuming a pivot value
(hereafter indicated with subscript ^{*}) such that

where ${N}_{\ast}=\mathrm{11}$ for N_{2} broadening and 17 for pure O_{2};
μ is the slope of the straight line and γ_{*}
averages the N− and N+ lines for N_{*}. The extrapolation
introduces correlations among those coefficients and between them and the
measurements with N > N_{*}, which were used to determine the
straight line, as discussed in detail in R18. Also, the uncertainties of the
extrapolated broadening coefficients increase with N up to a maximum of
0.032 GHz bar^{−1} at N=33. For the purpose of estimating covariances, the
extrapolation was modeled as though it was a formal linear regression. This
assumes that a straight line is the right extrapolation method, which seems
reasonable, although it cannot be tested because the very weak lines have not
been measured.

Figure 4 represents the sign-adjusted correlation coefficients as a color
image. The extrapolated coefficients (nos. 24–37 in Fig. 4) are strongly
correlated among themselves, although not perfectly. On the other hand, the
uncertainty of the zero-frequency broadening coefficient (no. 3) is assumed
to be uncorrelated with the line air-broadening uncertainties. Figure 5 shows
the γ_{a} values given by Tretyakov et al. (2005) and the associated
uncertainties as estimated above, together with the values and the
uncertainties of y and V, which are treated in the next two sections.

Figure 4Uncertainty matrix for oxygen absorption as a color-scale
image of sign-adjusted correlation coefficients (ρ_{SA}). See
Eq. (29) in Sect. 4.2 for the definition of ρ_{SA}. The
y axis label shows selected parameter indexes. The parameters are ordered
as follows: no. 1) S(300), no. 2) n_{a}, no. 3) γ_{0}(300), nos. 4–37) γ_{a}(300), nos. 38–71) y(300), nos. 72–105) V. The last
three parameter types are ordered following the O_{2}
rotational quantum number $N=\mathrm{1}-$, 1+, 3−, … 33−, 33+.

4.2.2 Covariance between oxygen line-mixing coefficients

Values for oxygen line-mixing coefficients are taken from Tretyakov et al. (2005), in which mixing coefficients
were determined from measurements made
near 1 atm of pressure and temperatures near 22–24 ^{∘}C
by an algorithm
that makes them dependent on the other parameters. Hence, uncertainties in
those other parameters contribute uncertainties to the mixing coefficients
as well as correlations with them. R18 shows that the estimation algorithm
can be represented in the form of a vector equation:

where y is the vector of normalized mixing coefficients defined by Eq. (13),
A is the matrix representing the linear estimation operation,
α is the vector of absorption measurements, and α^{b} is a
vector of absorption calculated from a baseline mixing coefficient set b.
Hence, applying Eqs. (20–21),

where I is the identity matrix, K_{y} and
K_{γ} are matrices of partial derivatives of baseline
absorption with respect to y and γ, respectively, and
σ_{S} is the fractional uncertainty in line intensities. The first
term above is the contribution of measurement noise with variance ${\mathit{\sigma}}_{\mathrm{noise}}^{\mathrm{2}}$ and the third and fourth terms represent the uncertainty
contributed by line widths and intensities in the derivation of the y values. In
Tretyakov et al. (2005), the baseline mixing coefficients were taken
from Liebe et al. (1992), who derived them by essentially the same
algorithm with very similar smoothing characteristics. Therefore, in the
second term of Eq. (32), the projection operator (I –
AK_{y}) should remove the variation of the mixing coefficients
obtained in Liebe et al. (1992), and the only part that will survive
is the original baseline, which is attributable to the coupling between the
positive-frequency resonances and the negative-frequency and zero-frequency
bands. In R18, the contribution of the second term in Eq. (32) is
estimated as $({\mathit{\sigma}}_{{\mathit{\gamma}}_{\mathrm{0}}}/{\mathit{\nu}}_{b}{)}^{\mathrm{2}}$ to each element of Cov(y), with ν_{b}= 40 GHz.

The mixing coefficient of the 1− line was measured separately in Tretyakov
et al. (2004), so it is not correlated with the others. Their estimated
uncertainty for its value is ${\mathit{\sigma}}_{y}(\mathrm{1}-)=\mathrm{0.01}$ bar^{−1}. The
y values measured at 295 K in Tretyakov et al. (2005) were adjusted to 300 K
using the temperature coefficients given by Liebe et al. (1992). However,
for the sake of simplicity that small correction was ignored here, and the
uncertainties of mixing coefficients at T_{0}=300 K are considered to be
the same as the measured coefficients. Hence, we assume no correlation
between the line-mixing coefficients at 300 K and the line-mixing temperature
coefficients, since they originate from different laboratories.

Table 4Uncertainty on simulated T_{B} (σ(T_{B})) at 14 HATPRO channel central frequencies
due to the uncertainty in O_{2} and H_{2}O absorption model parameters.
σ(T_{B}) is computed as the
square root of the diagonal terms of Cov(T_{B}), which was estimated considering the six
climatological atmospheric conditions introduced in Fig. 1.

4.2.3 Covariance between oxygen line-mixing temperature coefficients

The first-order line-mixing parameterization in R17 is given by Eq. (13). Table 5
of Tretyakov et al. (2005) lists coefficients a_{5} and a_{6} for each
line, a notation retained from Liebe et al. (1992). These are
related to the line-mixing coefficients as ${y}_{i}={a}_{\mathrm{5}}+{a}_{\mathrm{6}}$ and temperature
coefficients as V_{i}=a_{6}. Liebe et al. (1992) measured line mixing at
three temperatures and determined a_{6} by a linear regression versus
θ. We calculate the covariance matrix for the V values as

where x_{k} is the influence given by the regression to the mixing
coefficients at T_{k} in determining the V values (see R18). The baseline
b does not contribute to V because the three values of x_{k} sum to
zero. The first term in Eq. (33) is the measurement noise contribution.
Unlike the model parameters that are defined at 300 K, the V coefficients
depend on the value of n_{a}, and its uncertainty
${\mathit{\sigma}}_{{n}_{\mathrm{a}}}$ contributes the second term in Eq. (33); the
derivatives $\partial {V}_{i}/\partial {n}_{\mathrm{a}}$ were evaluated by
finite differences. The third term in Eq. (33) results from a comparison of
Liebe et al. (1992) to later work, which indicates that it contained some
systematic errors in intensities (generally ∼1 % or less) and in
line widths (typically ∼3.3 % smaller than those measured in
Tretyakov et al., 2005). The effect on V of those systematic errors,
${\mathit{\epsilon}}_{{V}_{\mathrm{sys}}}$, was also evaluated numerically, as described
in R18. We combine systematic and random errors in Eq. (33), as suggested by
JCGM (2008).

4.2.4 Covariance between different oxygen parameter types

The discussion in connection with Eqs. (20) and (21) indicates that
corresponding to the second, third, and fourth terms in Eq. (32) for
Cov(y), there must be uncertainty
covariances between the line-mixing coefficients of the 60 GHz band and the
line width and intensity parameters.

The negative signs in these equations originate because the computed
baseline absorption occurs with a minus sign in the determination of the
y coefficients. Likewise, corresponding to the second term of Eq. (33) for
Cov(V), there is an uncertainty covariance
between each V coefficient and n_{a}:

The value of γ_{0} was determined by Danese and Partridge (1989) from
radiometer measurements of the sky at a mountain site. Because the
atmospheric emission depends on the temperature profile, a covariance with
n_{a} results. We calculate a typical value for that site (White Mountain)
of ${K}_{{n}_{\mathrm{a}}}/{K}_{\mathit{\gamma}\mathrm{0}}=\mathrm{0.10}$ GHz bar^{−1}; thus, in analogy with Eq. (25),

corresponding to $\mathit{\rho}\left({\mathit{\gamma}}_{\mathrm{0}}{n}_{\mathrm{a}}\right)=-\mathrm{0.10}$. The increment of uncertainty variance for
γ_{0} due to Eq. (38) is 2 orders of magnitude smaller than the
value assigned to ${\mathit{\sigma}}_{\mathit{\gamma}\mathrm{0}}^{\mathrm{2}}$ and therefore negligible.

5 Uncertainty propagation to ground-based brightness temperature and retrievals

The uncertainty covariance matrices estimated in Sect. 4 for water vapor
and oxygen spectroscopic parameters are combined together to form
Cov(p), a 111×111 matrix. The two matrices
are combined block-diagonally, i.e., assuming no cross-covariances between
H_{2}O and O_{2} absorption model parameter uncertainties. Thus,
Cov(p) represents the uncertainty covariance
matrix of the H_{2}O and O_{2} absorption model parameters that were judged
relevant for downwelling T_{B} in the 20–60 GHz range. In this section,
Cov(p) is propagated to estimate its impact
on simulated downwelling T_{B} and ground-based temperature and humidity
retrievals.

5.1 Uncertainty on simulated brightness temperatures

The propagation of the absorption model parameter uncertainty to calculated
T_{B} is given by Eq. (18), which requires knowledge of
K_{p}, i.e., the Jacobian of calculated T_{B}
with respect to model parameters. The Jacobian K_{p} is
a n_{freq}×n_{par} matrix, where n_{freq} is
the number of frequency for which the T_{B} uncertainty should be
calculated and n_{par} is the number of considered parameters, 111 in our
case. Here we set n_{freq}=437, which includes 401 equally spaced
frequencies from 20 to 60 GHz (by 0.1 GHz increment), plus 36 corresponding
to the central frequencies of two widely deployed commercial MWRs, i.e., the
HATPRO (Rose et al., 2005) and MP-3000A (Ware et al., 2003). The Jacobian K_{p} has been estimated numerically by perturbing each
parameter individually by a small amount (corresponding to the parameter
1σ uncertainty). To represent different climatology conditions, six
realizations of K_{p} have been computed using the six
atmospheric climatology conditions introduced in Fig. 1. Thus, Cov(T_{B}) is computed from Eq. (18) using Cov(p) and K_{p} estimated as
above. Figure 6 reports σ(T_{B}), which is the
square root of the diagonal terms of Cov(T_{B}), for the whole 20–60 GHz range and for the six atmospheric
climatology conditions. Similarly, σ(T_{B})
values at the central frequencies of the two commercial MWRs are reported in
Table 4 (HATPRO, 14 channels) and Table 5 (MP-3000A, 22 channels).

Table 5As in Table 4 but at 22 central frequencies of MP3000-A channels.

Figure 5Oxygen line parameters as a function of rotational quantum number
N: line width γ_{a}(300) (squares), line mixing y(300)
(circles), and line-mixing temperature coefficients V (triangles). Error
bars indicate ±1σ uncertainties.

To appreciate the dominant contributions within the frequency range, the
different parameters have been grouped into seven types: intensity S (for both
O_{2} and H_{2}O), O_{2} line width γ_{a}, O_{2}
zero-frequency line width γ_{0}, O_{2} line mixing (y), O_{2}
line-mixing temperature dependence (V), H_{2}O continuum, H_{2}O line
width γ_{a}, and shift-to-width ratio R. The contribution of each
type to T_{B} uncertainty was estimated by propagating the uncertainty
covariance matrix reduced to the size of the parameters belonging to that type
only. Figure 7 shows the resulting contributions computed for the tropical
climatology conditions. We choose tropical conditions so that features at
22.2 GHz are evident above the continuum absorption.

Thus, looking at Figs. 6–7 and Tables 4–5, it seems convenient to discuss
the 20–60 GHz range in four parts: the proximity of the 22.2 GHz water vapor
line (20–26 GHz), the atmospheric window (26–45 GHz), the low-frequency
oxygen wing (45–54 GHz), and the opaque oxygen band (54–60 GHz). In the
following, the contribution dominance is inferred from Fig. 7, while the
typical values are inferred from Fig. 6 and Tables 4–5.

20–26 GHz: T_{B} uncertainty is dominated by uncertainty in water vapor
line width and shift coefficients, going from ∼0.3 K
(subarctic winter) to nearly 1.0 K (tropical).

26–45 GHz: T_{B} uncertainty is dominated by uncertainty in water vapor
continuum parameters, increasing with frequency from ∼0.4 to
1.2 K, with ∼0.2 K larger uncertainty in tropical with respect
to other climatology conditions.

45–54 GHz: T_{B} uncertainty is dominated by uncertainty in oxygen
line-mixing parameters (up to 2 K). Water vapor continuum, line-mixing
temperature dependence, and line intensity parameters also contribute to a
lesser extent (up to 1.0–1.2 K) at a respectively increasing frequency. The
total T_{B} uncertainty decreases with increasing temperature, which is
lower for tropical (up to 2.7 K) than for subarctic winter (up to 3.4 K)
conditions.

54–60 GHz: T_{B} uncertainty is below 0.5 K at 54–55 GHz and rapidly
approaches zero for frequencies above 55 GHz. In this very opaque region,
the contribution of absorption model parameters to simulated ground-based
T_{B} is negligible.

The qualitative conclusions above may sound somewhat obvious, at least to
microwave remote sensing experts. But the quantitative estimates are
unprecedented to our knowledge, especially in light of the evaluation of the
full uncertainty covariance matrix. One may wonder how high the
contribution of covariance matrix off-diagonal terms is. To evaluate it, T_{B}
uncertainty has also been computed considering Cov(p) as a diagonal matrix (i.e., all uncorrelated parameters). The
difference of σ(T_{B}) computed considering the
full uncertainty covariance matrix and a diagonal matrix is shown in Fig. 8.
The contribution of off-diagonal terms goes from −1.2 to 0.6 K. It mostly
affects the low-frequency oxygen wing, presumably due to line-mixing
parameters and their temperature dependence, with sharp gradients in the
46–52 and 52–54 GHz frequency ranges. It also affects the atmospheric
window, presumably due to water vapor continuum parameters, with a
contribution of the order of −0.3 to −1.0 K. This demonstrates that
off-diagonal terms cannot be neglected, especially in the uncertainty
characterization of the window and low-opacity channels of the HATPRO and MP3000-A
instruments.

Figure 6Zenith downwelling T_{B} uncertainty
(σ(T_{B})) due to the uncertainty in O_{2} and
H_{2}O absorption model parameters. Six climatological
atmospheric conditions (color coded) have been used to compute
K_{p}. σ(T_{B}) is computed as
the square root of the diagonal terms of Cov(T_{B}).

Figure 7Contributions to zenith downwelling
T_{B} uncertainty (σ(T_{B})) due to the
different types of O_{2} and H_{2}O
absorption model parameters. Tropical climatology conditions are
used here. The parameters are grouped into seven types: intensity S (for both
O_{2} and H_{2}O),
O_{2} line width γ_{a},
O_{2} zero-frequency line width γ_{0}, O_{2} line mixing
(y), O_{2} line-mixing
temperature dependence (V),
H_{2}O continuum, H_{2}O line width
γ_{a}, and shift-to-width ratio
R.

Finally, it shall be noted that the output of this analysis is
Cov(T_{B}), i.e., the full covariance matrix
of T_{B} uncertainties. A graphical representation of
Cov(T_{B}) is given in Fig. 9 for HATPRO
channels and US standard climatology. The resulting matrices computed for
HATPRO and MP3000-A channels and the six considered climatology are provided
in the Supplement.

Previous studies also reported values for σ(T_{B}) (Hewison et al., 2006; Hewison, 2007) and
Cov(T_{B}) (Hewison 2006b), though these
were estimated from relative T_{B} differences computed with a set of
absorption models available at that time. With respect to these values, we
report (i) smaller uncertainty at 20–30 GHz channels due to improved
accuracy of the 22 GHz line spectroscopic parameters and (ii) much
larger uncertainty at 50–54 GHz channels due to the consideration of
line-mixing parameter uncertainties, which likely canceled out partially in
the relative T_{B} difference approach used by Hewison (2006b, 2007).

5.2 Uncertainty on temperature and humidity retrievals

The uncertainty in absorption model parameters impacts the accuracy of
geophysical variables retrieved from radiometric observations through
inversion methods based on a forward operator. Here, the forward operator is
a radiative transfer model (RTM) relying on the spectroscopic parameters to
compute atmospheric absorption and emission and thus the measurable T_{B},
from atmospheric thermodynamical profiles. Examples of such inversion
methods are described in Cimini et al. (2006)
and include simulation-based regression, artificial neural networks, and the optimal estimation method (OEM).
The OEM is particularly suitable to investigate the uncertainty
contribution of spectroscopic parameters, as it allows one to perform an
assessment of the total statistical uncertainty, as well as of the forward
model parameter uncertainty (Rodgers, 2000). For example, it has been used
for a spectroscopic parameter sensitivity study for a
millimeter to sub-millimeter limb sounder instrument (Verdes et al., 2005) and
to estimate the impact of forward model parameters on the temperature
retrieval from a multiple-channel Rayleigh-scatter lidar (Sica and Haefele,
2015).

Thus, let us consider the OEM formalism. Following Rodgers (2000), the total
uncertainty covariance matrix of the retrieved atmospheric profile $\widehat{x}$
is

where Cov_{m} and Cov_{s} are
respectively the measurement and smoothing uncertainty covariance matrices,
while Cov_{p} is the model parameter uncertainty
covariance matrix. Cov_{p} is related to
Cov(p) through K_{p}, the
Jacobian of the forward model with respect to the parameters p, and the
sensitivity of the inverse method to the measurements (also called the
contribution function or gain matrix) ${\mathbf{G}}_{m}=\partial \mathbf{I}\left(\mathbf{m}\right)/\partial \mathbf{m}$ as

Assuming a linear Gaussian case as usual for ground-based radiometric
retrievals of atmospheric temperature and humidity profiles (Löhnert et
al., 2004; Cimini et al., 2006, 2010; Hewison, 2007) and calling
Cov(ϵ) and Cov(x_{a}) the covariance matrices of measurement and a priori
background uncertainty, the gain matrix is given by (Rodgers, 2000)

where K_{x} is the Jacobian of the forward model with respect to
the atmospheric state x. Finally, considering T_{B} as the measurements and
recalling Eq. (18), the model parameter uncertainty covariance matrix in Eq. (40) becomes

which contributes to the total profiling uncertainty as in Eq. (39). Note
that Cov(T_{B}) is the full spectroscopic
parameter uncertainty covariance matrix estimated in Sect. 5.1.
Accordingly, the combined uncertainty due to the O_{2} and H_{2}O absorption
model parameter is thus propagated into the retrieval space.

As an example of the spectroscopic contribution to profiling uncertainty we
apply the approach described above to HATPRO channels (as in Table 4),
specifically (i) seven K-band channels (22.24 to 31.40 GHz) and (ii) seven
V-band channels (51.26 to 58.0 GHz), to compute the impact on specific
humidity and temperature profile retrievals, respectively. For the sake of
result reproducibility, simple diagonal Cov(ϵ) and Cov(x_{a}) matrices are
assumed here, with reasonable values resembling typical matrices adopted in
ground-based microwave profiling (Martinet et al., 2015; Martinet et al.,
2017). Specifically, we assume a constant uncertainty for T_{B}
measurements ($\mathbf{Cov}\left(\mathit{\u03f5}\right)={\mathit{\sigma}}_{{T}_{\mathrm{B}}}^{\mathrm{2}}\mathbf{I}$, with ${\mathit{\sigma}}_{{T}_{\mathrm{B}}}=\mathrm{0.5}$ K) and a
priori temperature profile ($\mathbf{Cov}\left({\mathit{x}}_{\mathrm{a}}\right)={\mathit{\sigma}}_{T}^{\mathrm{2}}\mathbf{I}$, σ_{T}=1.5 K), while also
assuming a decreasing-with-height uncertainty for a priori specific
humidity profile ${\mathit{\sigma}}_{Q}\approx {\mathit{\sigma}}_{Q}\left(\mathrm{0}\right){e}^{-z/H}$ (where z is
height in kilometers, σ_{Q}(0)=3.2 g kg^{−1}, and H=4 km). The
a priori background x_{a} and Jacobian K_{x} are
defined on 101 pressure levels, from 0.005 to 1050 hPa. These levels are
selected to be denser close to the surface (34 levels below 2 km),
specifically for downwelling radiative transfer calculations. The vertical
spacing of the adopted levels is given in De Angelis et al. (2016).

Figure 8Difference between σ(T_{B}) as computed
considering the full uncertainty covariance matrix and its diagonal matrix
(i.e., off-diagonal terms are set to zero).

Figure 9T_{B} uncertainty covariance matrix
due to O_{2} and H_{2}O absorption
model parameter uncertainty at HATPRO channels for US standard
climatology. Numbers in the table are in K^{2}, while
the color scale is in
log_{10}(K^{2}).

Figure 10Uncertainty in temperature retrievals from ground-based
MWR due to the uncertainty in O_{2} and
H_{2}O absorption model parameters. The observation
vector considered here consists of T_{B} at the 14
HATPRO channels. Six climatological atmospheric conditions (color coded)
have been used to compute K_{b} and
K_{x}. The square roots of the diagonal
terms of Cov_{p} are shown. 101 pressure
levels from 0.005 to 1050 hPa are used here. These levels have been selected
specifically to be denser close to the surface (34 levels below 2 km). The
vertical spacing of levels is given in Fig. 1 of De Angelis et al. (2016).

The square roots of Cov_{p} diagonal terms are shown in
Figs. 10 and 11 for temperature and specific humidity profiling,
respectively. Note that these uncertainty profiles shall be considered just
as relative, as they depend upon the vertical grid spacing and the choice of
Cov(ϵ) and
Cov(x_{a}). Nonetheless, Figs. 10 and 11
show that the contribution of absorption model uncertainty to the profile
retrieval uncertainty is generally not negligible. For temperature,
the absorption model contributes less near the surface and more in the upper
atmosphere; these are respectively the direct consequences of negligible
uncertainty for O_{2} opaque channels (55–58 GHz) and significant
uncertainty for O_{2} transparent channels (50–55 GHz). Above 3 km, the
impact increases for colder and drier conditions. Though less clearly, this
also holds below 3 km for all but tropical conditions, which show a peak
around 2 km. This is due to the fact that lower V-band channels (51–52 GHz)
gain sensitivity to boundary layer temperature as moisture increases. These
channels are the most affected by absorption model uncertainty (Fig. 6 and
Table 4) and thus contribute to larger temperature uncertainty in the lower
layers. For specific humidity, the absorption model contribution to
uncertainty simply increases with increasing moisture. This is a direct
consequence of increasing K-band T_{B} uncertainty corresponding to
increasing moisture, as seen in Fig. 6. Values are particularly high for
relatively drier climatology (e.g., arctic); this is simply a consequence of the
assumed a priori σ_{Q}, which is typical of midlatitude
climatology. Reducing σ_{Q} by a factor of 10 (to be closer to values
for dry climatology), the uncertainty profile would be reduced roughly by the
same factor.

With respect to the absorption model parameter contribution in Figs. 10
and 11, the uncertainty due to measurement noise (i.e., the diagonal terms of
Cov_{m}) is of comparable magnitude, though with
different vertical shape and little dependence on climatology (not shown).
Note that in the actual retrieval process, the contribution of absorption
model parameter uncertainty to the total profiling uncertainty can be
equivalently treated as Cov_{p} or as adding an
absorption model term to the measurement uncertainty, i.e.,
$\mathbf{Cov}\left(\mathit{\u03f5}\right)+{\mathbf{K}}_{\mathrm{p}}\mathbf{Cov}\left(\mathit{p}\right){\mathbf{K}}_{\mathrm{p}}^{\top}$ (Rodgers, 2000).

Radiative transfer models have general implications for atmospheric sciences,
including meteorology and climate studies. Atmospheric absorption modeling is
a key component of radiative transfer codes, which are extensively used for
the retrieval of atmospheric variables and the assimilation of radiometric
observations into NWP. Uncertainties in atmospheric absorption models thus
contribute to the uncertainty of atmospheric retrievals and observations vs.
background comparison. The analysis above shows a viable approach to quantify
the uncertainties of atmospheric absorption modeling and the impact on
radiative transfer calculations and atmospheric retrievals. The approach
relies on the estimation of the full covariance matrix of parameter
uncertainties, which is necessary to compute the uncertainty of calculated
T_{B} at any given frequency. The approach is general and not limited to
any particular instrument, technique, or frequency range. The approach can be
applied to any absorption model and it can be easily extended to
other frequencies and observation geometry (e.g., from satellite). To demonstrate
its use quantitatively, we apply this approach to a widely used microwave
absorption model (R17, Rosenkranz 2017), focusing on the 20–60 GHz frequency
range commonly exploited for atmospheric remote sounding by ground-based MWR
profilers.

We have summarized the modifications made in the last 20 years to a
reference absorption model (Rosenkranz, 1998), leading to the current version
of the model R17. We reviewed the spectroscopic literature searching for
uncertainty estimates affecting the spectroscopic parameters entering the
absorption model code. In the considered frequency range, atmospheric
absorption is dominated by water vapor and oxygen. The associated parameters
and their uncertainties are reported in Tables 1 and 2, respectively, for
water vapor and oxygen absorption. We performed a sensitivity analysis by
perturbing each parameter by its estimated uncertainty and quantifying the
impact on simulated T_{B} for six climatology conditions. The uncertainty
of the following parameters is found to impact 20–60 GHz T_{B}
calculations by more than 0.1 K in any of the considered climatologies.
Concerning water vapor absorption, these are self- and foreign-continuum
absorption coefficients, line broadening by dry air, line intensity,
the temperature-dependence exponent for foreign-continuum absorption, and the line
shift-to-broadening ratio. Concerning oxygen absorption, the dominating
parameters are line intensity, line broadening by dry air, line mixing,
the temperature-dependence exponent for broadening, zero-frequency line
broadening in air, and the temperature-dependence coefficient for line mixing. Thus,
from the initial set of 319 considered parameters, 111 are retained for
further analysis (6 for water vapor and 105 for oxygen). For the retained
parameters, we estimated the full uncertainty covariance matrix, i.e.,
including parameter uncertainty variances and cross-covariance between
uncertainties of different parameters. Since the spectroscopic literature
provides at most the uncertainties of individual parameters, but not the
covariance between them, the off-diagonal terms of the uncertainty covariance
matrix had to be estimated by investigating the possible correlation between the
methods used to retrieve the parameter values. The full uncertainty
covariance matrix (111×111) as estimated is provided in the Supplement.

Then, the contribution of the spectroscopic parameter uncertainties, including
the covariance between them, to the uncertainty of simulated downwelling
20–60 GHz T_{B} is calculated for six climatology conditions using the
estimated uncertainty covariance matrix (Fig. 6). Dividing the 20–60 GHz
range into four parts, typical T_{B} uncertainties are (i) ∼0.3 K
(subarctic winter) to nearly 1.0 K (tropical) at 20–26 GHz, (ii) ∼0.4 to 1.2 K with additional ∼0.2 K uncertainty in tropical
conditions at 26–45 GHz, (iii) up to 3.4 K inversely proportional to temperature
at 45–54 GHz, and finally (iv) below 0.5 K at 54–55 GHz rapidly approaching zero for
frequencies above 55 GHz. The dominant uncertainty contributions are water
vapor line width and shift at 20–26 GHz, water vapor continuum at
26–45 GHz, and oxygen line mixing at 45–55 GHz; finally, absorption model
uncertainty becomes negligible at 55–60 GHz. Despite the fact that these qualitative
conclusions may sound obvious, at least to microwave remote sensing experts,
the quantitative estimates are unprecedented to our knowledge, especially in
light of the evaluation of the full uncertainty covariance matrix. It is
shown that off-diagonal terms affect the low-frequency oxygen wing,
presumably due to covariance of line-mixing parameters and their temperature
dependence, but also the atmospheric window, presumably due to covariance of
water vapor continuum parameters. The total contribution depends upon
frequency and ranges from −1.2 to 0.6 K, demonstrating that off-diagonal
terms cannot be neglected, especially in the uncertainty characterization of
window and low-opacity channels.

The resulting uncertainty on simulated T_{B} is also calculated at the
channels of two of the most common commercial MWRs, i.e., HATPRO and MP3000-A.
The computed Cov(T_{B}) values, of which one
example is shown in Fig. 9, are provided for the two instruments and for
the six climatology conditions in the Supplement. These
matrices may be directly exploited as the additional observation uncertainty
related to absorption model in any retrieval and data assimilation procedure
exploiting either of the two instruments. Just to give an example, the
absorption model uncertainty is propagated to ground-based MWR retrievals,
showing its impact on retrieved temperature and humidity profiles for the
six climatology conditions (Figs. 10 and 11). It is shown that the
contribution of absorption model uncertainty to the profile retrieval
uncertainty depends on climatology (increasing temperature uncertainty with
decreasing average temperature, increasing humidity uncertainty with
increasing moisture), and it is generally not negligible, though the actual
values depend on retrieval settings (such as a priori information and vertical
spacing, among others).

Finally, let us underline the fact that the presented uncertainty quantification
contributes to a better understanding of the total uncertainty affecting
radiometric products, thus reducing the chances of systematic errors in NWP
data assimilation and observation-derived climate trends. Note that the
presented uncertainty covariances of spectroscopic parameters are generally
valid, while the T_{B} sensitivity analysis and uncertainty
quantifications
are strictly valid only for the ground-based geometry and the considered
frequency range. Future work may include the application of the proposed
approach to higher frequencies and upwelling T_{B}, requiring a new
sensitivity analysis. Further modification to the R17 absorption model may
be considered to account for recent findings from spectroscopic laboratory
experiments (e.g., inter-branch coupling suggested by Makarov et al., 2013,
temperature exponent n_{a} suggested by Koshelev et al., 2016,
consideration of the speed dependence of the collisional relaxation effect
influencing diagnostic line profiles as shown in Koshelev et al., 2018). In
addition to uncertainties of parameters within a given absorption model,
other errors can be contributed by approximations made in formulating the
model, such as the H_{2}O continuum formulation or neglect of higher-order
line mixing in O_{2}. Those uncertainties would need to be treated by a
different analysis.

Figure 11As in Fig. 11, but for specific humidity retrievals.

Uncertainty covariance matrices for the spectroscopic
parameters considered here, as well as the resulting T_{B} uncertainty
covariance matrices for HATPRO and MP3000A channels, are available as a
Supplement to this paper. The
absorption model by Rosenkranz (2017) is available as a FORTRAN 77 code at
https://doi.org/10.21982/M81013 (Rosenkranz, 2017). Older versions, including the one
used here (15 May 2017), are available at
http://cetemps.aquila.infn.it/mwrnet/lblmrt_ns.html (last access: 23 October 2018).

The following two sections review the set of modifications to the R98
model for water vapor and oxygen absorption,
respectively, proposed in the open
literature in the last 20 years and subsequently imported in the current R17
version of the model.

A1 Water vapor

The R98 model uses 15 water vapor lines, similar to the strongest lines used
in MPM89, while the other 15 lines have been omitted as they were judged to
have a negligible impact. For the water vapor continuum absorption, the model
combines the foreign-broadened component from MPM87 with the self-broadened
from MPM93, increased by 15 % and 3 %, respectively, to compensate for
the line truncation at cutoff frequency (±750 GHz). This model is
still maintained and there have been several modifications since the 1998
version.

Since 2003, the model has included the pressure line shift mechanism
investigated by Tretyakov et al. (2003) and Golubiatnikov et al. (2005). For
the 22.23 and 183.31 GHz absorption lines, the only two relevant for the
frequency range under study here, the main modifications are the adoption of
the air-broadened line widths determined in Payne et al. (2008) using
ground-based radiometric measurements, leading to −5.1 % and
+4.5 % line width change, respectively. The −5 % modification to
the 22.23 GHz line width was already proposed by the independent
investigation of Liljegren et al. (2005). Other modifications for the 22.23
and 183.31 GHz absorption lines are for line intensity (+0.3 % and
+0.5 %, i.e., from HITRAN 1992 to 2012 update), the temperature
exponent of air broadening (+10 % and +20 %, respectively), and
the self-broadened line width (+0.8 % and −1.0 %), while the
temperature exponent of self-broadening only changed for the 22.23 GHz line
(+64 %).

Figure A1C_{s} vs.
C_{f} for the R98 model (+) and its modification by Turner et
al. (2009) (×), with uncertainty contours. Note the different scales
on the two axes.

Parameters for higher-frequency lines (321–916 GHz) were modified according
to different sets of spectroscopic measurements (Colmont et al., 1999;
Podobedov et al., 2004; Koshelev et al., 2007; Golubiatnikov et al., 2008;
Koshelev, 2011; Tretyakov et al., 2013), leading to modifications in
air-broadened line width (order of 1 %–15 %), the temperature exponent of
air broadening (2 %–5 %), and self-broadened line width (1 %–9 %). Other line
parameters are from the HITRAN 2012 database (Rothman et al., 2013).

Concerning the water vapor continuum, the main modifications follow the
results of Turner et al. (2009) suggested by an analysis of ground-based
observations at 150 GHz. The suggested adjustments to the two components of
the water vapor continuum in the R98 model are in opposite directions (i.e.,
increasing the contribution from the foreign-broadened component while
decreasing the contribution from the self-broadened component). Figure A1
plots C_{s} vs. C_{f} for the R98 model and its
modification by Turner et al. (2009) with their respective uncertainty
contours. These uncertainties are conditioned on the nominal values of
n_{cs} and n_{cf}, which are the same in both models. The
uncertainty ellipse for Turner et al. is drawn using the correlation
coefficient of −0.87 found in Sect. 4.1.3. Note that the details of
continuum and resonant absorption are inextricably related in any model,
meaning that the empirical definition of the continuum (Eq. 8) implies that
the parameters must be used only with exactly the same resonance absorption
they were defined with. Thus, the adjustment factors were recomputed in 2015
accounting for the resonant line adjustments discussed above, leading to
+9.8 % and −21.1 % change from R98 in air-broadened and
self-broadened coefficients, respectively. The results of Turner et
al. (2009) are indirectly supported by the analysis of Payne et al. (2011).
In fact, Payne et al. (2011) developed adjustment factors for the MT_CKD
water vapor continuum model (Clough et al., 2005; Mlawer et al., 2012), which
agree within the stated error bars with those given in Turner et al. (2009)
for the same MT_CKD model. The results of Turner et al. (2009) also seem
supported by independent investigations based on satellite observations in
the 10.7 to 89 GHz range (Wentz and Meissner, 2016) and around the 183 GHz
line (Bobryshev et al., 2018).

More recently, two papers presented further modifications to the spectroscopy
underlying microwave remote sensing of atmospheric water vapor, i.e.,
Tretyakov (2016) and Koshelev et al. (2018). Tretyakov (2016) presents a
historic review, discussing in chronological order the measurement and
analysis that lead to estimates of spectroscopic parameters for the water
vapor absorption continuum and resonant lines near 22 and 183 GHz.
Tretyakov (2016) also provides an expert assessment of the best estimate for
the spectroscopic parameter values and their uncertainty based on the
analysis of all the available data. These parameter values provide the best
fit of the absorption model to the available data, taking into account the
measurement errors reported by the authors and the probabilities of possible
systematic errors. In almost all cases, with the exception of the 22 GHz
line self-broadening, the estimated parameter values agree within uncertainty
limits with those given in HITRAN, though in most cases HITRAN uncertainty
estimates are more conservative. Concerning the water vapor continuum
absorption, Tretyakov (2016) finds that the adjustments to R98 proposed by
Turner et al. (2009), based on zenith-looking ground-based radiometric
observation, lead to a worse fit to the laboratory and field (parallel to
Earth-surface path) measurements, particularly noticeable in the self
component. However, Fig. A1 shows that the model uncertainties have
appreciable overlap. Finally, Koshelev et al. (2018) present laboratory
measurements devoted to refining the 22 GHz line-shape parameters. Koshelev
et al. (2018) suggest line width values within the uncertainty of those given
by Tretyakov (2016), though with smaller estimated uncertainty by a factor of
∼3 (air broadening) and ∼10 (self-broadening). Similarly, the
air-broadening shift parameter agrees with that of Tretyakov (2016)
with an estimated uncertainty reduced by a factor of ∼3. Conversely, the uncertainty of the self-broadening shift parameter is
reduced by a factor ∼1.5, and the values from Tretyakov (2016)
and Koshelev et al. (2018) do not fit within the stated uncertainty.

A2 Oxygen

The R98 model adopts the same oxygen line parameters as given in MPM92,
except for sub-millimeter frequencies for which frequency and intensity are taken
from the HITRAN 1992 database (Rothman et al., 1992). Other differences with
respect to MPM92 are the temperature dependence (1∕T) for 118.75 GHz line
width, with the temperature dependence of sub-millimeter line widths being equal to
that of lines in the 60 GHz band (e.g., $\mathrm{1}/{T}^{{n}_{\mathrm{a}}}$, with n_{a}=0.8).
Concerning the line-mixing model, the MPM and the R98 model exploit
first-order mixing with coefficients derived by the method given in
Rosenkranz (1988). The following modifications have been implemented in R17.

The line intensities are from the HITRAN 2004 database (Rothman et al.,
2005). The zero-frequency line intensity is from the JPL catalogue
(https://spec.jpl.nasa.gov/; Pickett et al., 1998). The line central
frequencies and width coefficients for the 60 GHz band are taken from
Tretyakov et al. (2005), who report measurements for precise broadening and
central frequencies of fine structure lines and a revision of line-mixing
coefficients. The effect of different values for the 60 GHz line parameters
on MWR simulations and retrievals was shown to be significant both for
ground-based (Cadeddu et al., 2007) and satellite (Boukabara et al., 2005a,
b; Rosenkranz, 2005) observations. In particular, Cadeddu et al. (2007) show
that the parameter values proposed by Tretyakov et al. (2005) lead to better
agreement with two independent datasets of ground-based MWR observations than
those found in HITRAN (Rothman et al., 2005; Hoke et al., 1989) and also that
these modifications are essential to reduce the clear-sky bias in the
liquid–water path retrievals.

The line width and line-mixing coefficients for the 118 GHz line are taken
from Tretyakov et al. (2004), who report results of laboratory
investigations of the pressure-dependent parameters of the single 118 GHz
line. The sub-millimeter line widths are from Golubiatnikov and Krupnov (2003),
except the one at the 234 GHz line that comes from Drouin (2007).

Makarov et al. (2011) proposed a model for the 60 GHz absorption band based
on the second-order line-mixing expansion of Smith (1981), showing an
improved fit of observed absorption profiles between 54 and 65 GHz, but this
model is not adopted in R17. In fact, during this analysis, significant
absorption differences (∼10 %) were found in the band wings (e.g.,
∼50–53 GHz) comparing calculations made with Makarov et al. (2011)
line-mixing coefficients against original measurements from Liebe et
al. (1992). This was attributed to systematic errors in O_{2}
concentration of the order of 0.5 %–1.5 % in the 245–335 K
temperature range. Dmitriy S. Makarov, Philip W. Rosenkranz, and
Mikhail Y. Tretyakov are currently working on a revised second-order model
(Makarov et al., 2018).

For the dry continuum, R98 only considered the N_{2}–N_{2} contribution
with a pure ν^{2} dependence. This is a particular case of Eqs. (7) and
(15), with $\mathit{\epsilon}\left(\mathit{\nu},T\right)=\mathrm{0}$ and f(ν)=1. This was revised (Rosenkranz et al., 2006) by fitting f(ν) as in Eq. (16) through the data of Borysow and Frommhold (1986) and
including the N_{2}–O_{2} and O_{2}–O_{2} bimolecular absorption
with a constant value for ε suggested by Pardo et al. (2001)
and later by Boissoles et al. (2003). The latter is used in R17.

In order to consider the broadening of oxygen lines by water vapor with
little modifications to the original model, R17 adopts the mean value of the
water-to-air broadening ratio suggested by Koshelev et al. (2015).

More recently, Koshelev et al. (2016) report measurements of line widths and
their temperature exponents for 12 oxygen lines (rotational quantum
number N ranging from 1 to 19). The fixed value of the temperature exponent
(n_{a}=0.8) adopted in the MPM and the R98–R17 models fits the value
reported in Makarov et al. (2008) for the 1− line (0.785(35)) but falls
outside the mean value (0.765(11)) reported by Koshelev et al. (2016). This
suggests that the temperature exponent values suggested by Koshelev et al. (2016),
or their mean value, could be adopted to increase the accuracy of
absorption modeling.

DC and PR designed the research,
contributed to data processing and analysis, and wrote the original
manuscript. MYT, MAK, and FR provided advice and contributed to data
analysis. All the co-authors helped to revise the
manuscript.

This work was partially supported by the EU H2020 project GAIA-CLIM
(Ares(2014)3708963, project 640276). Mikhail Y. Tretyakov and Maksim A. Koshelev acknowledge
state project no. 0035-2014-009. Domenico Cimini acknowledges the useful
advice from Stefan Bühler, Richard Larsson, and Oliver Lemke in the early
stage of the analysis.

Edited by: Jui-Yuan Christine Chiu
Reviewed by: Vivienne Payne and two anonymous referees

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