the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Information content of OCO-2 oxygen A-band channels for retrieving marine liquid cloud properties

### Graeme L. Stephens

Information content analysis is used to select channels
for a marine liquid cloud retrieval using the high-spectral-resolution oxygen
A-band instrument on NASA's Orbiting Carbon Observatory-2 (OCO-2). Desired
retrieval properties are cloud optical depth, cloud-top pressure and
cloud pressure thickness, which is the geometric thickness expressed in hectopascals.
Based on information content criteria we select a micro-window of 75 of the
853 functioning OCO-2 channels spanning 763.5–764.6 nm and perform a series
of synthetic retrievals with perturbed initial conditions. We estimate
posterior errors from the sample standard deviations and obtain ±0.75
in optical depth and ±12.9 hPa in both cloud-top pressure and cloud
pressure thickness, although removing the 10 % of samples with the highest
*χ*^{2} reduces posterior error in cloud-top pressure to ±2.9 hPa
and cloud pressure thickness to ±2.5 hPa. The application of this
retrieval to real OCO-2 measurements is briefly discussed, along with
limitations and the greatest caution is urged regarding the assumption of a
single homogeneous cloud layer, which is often, but not always, a reasonable
approximation for marine boundary layer clouds.

The oxygen A-band spans wavelengths with a wide range of absorption strength which can be exploited to determine photon path lengths and therefore retrieve cloud-top heights and potentially the within-cloud photon path, which is related to droplet number concentration and therefore cloud thickness. Meanwhile, cloud optical depth can be retrieved from reflectance in approximately non-absorbing “continuum” channels (Fischer and Grassl, 1991; Koelemeijer et al., 2001; Stephens and Heidinger, 2000). Such a retrieval that includes cloud geometric thickness or droplet number density would allow evaluation of model cloud physics (Bennartz, 2007). In addition A-band retrievals use reflected sunlight and so are physically independent of other common sources of cloud information such as longer wavelength infrared, which may misidentify cloud-top pressure in the presence of temperature inversions (Baum et al., 2012).

The photon path length of reflected sunlight is estimated by comparing
radiance between channels with different absorption characteristics. With
known absorption coefficients and similar scattering and reflection
properties between the channels, the photon path length is easily determined
from the Beer–Lambert law. This technique was first suggested as a way of
determining cloud-top altitude using the strong carbon dioxide (CO_{2})
absorption band near 2.0 µm with an atmospheric window near 2.1 µm
(Hanel, 1961). Subsequently the oxygen A-band near 0.76 µm
was proposed as it offers improved signal-to-noise ratio (SNR) and avoids overlap
with the 1.87 µm water vapour absorption band (Yamamoto
and Wark, 1961). It was noted that clouds are not “simple diffuse
reflectors” and that “absorption along the scattering paths within the
clouds must be considered”.

With a single measured ratio of two channels it is only possible to
determine the total photon path length and not distinguish between
above-cloud and within-cloud components, as this would mean obtaining two
pieces of information from a single measurement. One way of distinguishing
is to take multiple measurements from diverse viewing angles, as is done by
the Polarization and Directionality of the Earth's Reflectances (POLDER)
instrument series (Deschamps et al., 1994).
POLDER-3 has a “narrow” channel with a full width at half maximum (FWHM)
of 10 nm centred at *λ* = 763 nm, and a “wide” channel of FWHM 40 nm centred at 765 nm. Statistics of the inferred photon path from different
angles have been shown to be related to the cloud centroid pressure
(Ferlay et al., 2010), results of which have been tested
against CloudSat radar and Cloud-Aerosol Lidar and Infrared Pathfinder
Satellite Observations (CALIPSO) data (Desmons et al.,
2013). A more recent study used information content analysis based around
the characteristics of the Multiviewing, Multi-channel and
Multi-polarization Imaging (3MI) and the Multiangle SpectroPolarimetric
Imager (MSPI) instruments. This concluded that multi-angle measurements are
informative about cloud geometric thickness, particularly for clouds thicker
than 2–3 km (Merlin et al., 2016), which
notably excludes the marine stratocumulus regime.

Another proposal to obtain additional measurements that describe cloud
geometric thickness is to combine measurements from both the oxygen A-band
and B-band, such as those available from the Earth Polychromatic Imaging
Camera (EPIC) on the Deep Space Climate Observatory (DSCOVR). By considering
the sum and differences of the channel ratios it has been proposed that
cloud geometrical thickness can be retrieved when cloud optical depth (*τ*) is greater than 5 (Yang et al., 2013).

An alternative to multiple angles or additional bands is to measure more
channels in the A-band, as was done for the Scanning Imaging Absorption
SpectroMeter for Atmospheric Chartography (SCIAMACHY) on board ENVISAT
(Rozanov and Kokhanovsky, 2004), which when combined with the
Global Ozone Monitoring Experiment instruments (GOME and GOME-2) provides an
A-band record going back to 1995. Information content analysis based on GOME-2
characteristics, using a spectral resolution of 0.2 nm and assumed
signal-to-noise ratio (SNR) of 100 showed that two pieces of information could be
obtained (Schuessler et al., 2014). This study showed the
best performance when retrieving cloud-top height with either *τ* or
cloud fraction and reported that there was not sufficient information in
these assumed measurements to obtain cloud geometric thickness with
“satisfactory accuracy”.

However, older theoretical work suggested that a spectral resolution of
better than 1 cm^{−1} (O'Brien and Mitchell, 1992) or
even 0.5 cm^{−1} (Heidinger and Stephens, 2000) is
required for an effective A-band retrieval that includes cloud geometric
thickness. In wavelength terms this is 0.03–0.06 nm, and is now achieved
by instruments carried by the Chinese Feng-Yun-3 series (most recently
FY-3D), the Japanese Greenhouse Gas Observing Satellite (GOSAT), the
European Sentinel-5 Precursor (Sentinel-5P, which carries the Troposphere
Measuring Instrument, TROPOMI) and NASA's Orbiting Carbon Observatory-2
(OCO-2).

This study considers OCO-2 and extends previous work that developed a lookup
table to retrieve cloud-top pressure and optical depth for single-layer
liquid clouds over ocean (Richardson et al., 2017). This
simple retrieval combined 20 of the 853 functioning A-band channels on OCO-2 into
two “super-pixels” or “super-channels” based on their O_{2} absorption.
The lookup tables were used for all locations and weather conditions and
were validated using collocated Moderate-resolution Imaging
Spectroradiometer (MODIS) and CALIPSO data (Taylor
et al., 2016). Here we develop an optimal-estimation-based retrieval
(Rodgers, 2000) for single-layer water clouds over oceans using
nadir-view OCO-2 measurements and subject it to several idealised tests.
This study's new contributions are (i) considering information content
aspects to select groups of channels rather than combined super-channels,
(ii) accounting for local meteorological conditions and (iii) adding cloud
pressure thickness to the retrieved state. We express cloud geometric
thickness in terms of hPa and refer to it as cloud pressure thickness with
the symbol Δ*P*_{c}. Our current analysis considers
aerosol-free cases as aerosols have not yet been properly implemented in our
modified cloudy-sky version of the radiative transfer model; this is an
avenue for future work and will be discussed in Sect. 5.

OCO-2 has 1016 A-band channels of which 853 function across all soundings with spectral sampling between 0.01 and 0.02 nm and a FWHM of 0.04 nm in wavelength, implying sufficient spectral resolution for geometric thickness retrievals. Low marine clouds are the primary cause of spread in net modelled cloud feedback (Bony and Dufresne, 2005; Zelinka et al., 2012), and we focus on these clouds, which complements the multi-angular retrievals from other sensors which appear to perform better for thicker clouds (Ferlay et al., 2010; Merlin et al., 2016).

OCO-2 is also promising as its SNR values commonly range from 300 to 800 in cloudy scenes and it flies in the A-Train constellation (L'Ecuyer and Jiang, 2010), allowing collocation with other sensors. Furthermore, its footprint size typically ranges from 1.2 to 2.3 km at nadir and compares favourably with both GOSAT (10.5 km diameter) and TROPOMI (7 km × 7 km), although its narrow swath of approximately 10 km is much reduced compared with TROPOMI's 2600 km.

Here we aim to develop a computationally efficient cloud retrieval for OCO-2 by selecting channels that contain the most information about the retrieved state properties, which speeds both the radiative transfer simulation and the optimal estimation calculations. In principle, the optimal channels may depend on the cloud case and on the across-track position of the measurement because the instrument line shapes (ILS) vary across the swath. Furthermore, neighbouring ILS overlap, so it is more computationally efficient to select neighbouring channels since the radiative transfer will already have been calculated for many of the relevant frequencies. We refer to the selection of neighbouring channels as a “micro-window” approach and use the OCO-2 Level 2 Full Physics Radiative Transfer Model (L2RTM; Boesch et al., 2015) with a set of representative atmosphere and liquid cloud states to select the optimal micro-window based on information content and posterior error criteria.

This approach aims to optimise a cloud property retrieval and due to
limitations related to the radiative transfer implementation and
computational burden, droplet size is not a retrieved property but
contributes to the posterior uncertainty. Above-cloud CO_{2} retrievals
have been found to require cloud droplet size for good accuracy
(Vidot et al., 2009) and therefore our current
implementation will not directly lead to above-cloud CO_{2} retrievals.

The paper is organised as follows: Sect. 2 describes the OCO-2 satellite measurements, radiative transfer model and general information content approach. Section 3 details the methodology specific to this paper, including the sample atmospheres, perturbations for determining covariance matrix components, the sequential channel selection procedure and information content and retrieval analysis. Section 4 reports the results of each of these cases, Sect. 5 discusses the results and describes how they will be applied in the real OCO-2 cloud retrieval, and Sect. 6 concludes.

The OCO-2 satellite orbits in a Sun-synchronous orbit as part of the A-Train
constellation (L'Ecuyer and Jiang, 2010). It follows a 16-day repeat cycle
with an Equator-crossing time near 13:30 local solar time in the ascending
node and follows the CloudSat and CALIPSO reference ground track. OCO-2 has
three viewing modes: a target mode for in-flight validation plus glint and
nadir modes for operational measurements. Currently the satellite alternates
nadir and glint orbits with some ocean orbits dedicated entirely to glint
mode. Here we use nadir soundings to allow future cross-comparisons with the
nadir-view instruments on CloudSat and CALIPSO. Several nadir orbits pass
over marine stratocumulus regions where OCO-2 offers unique value in terms of
determining cloud geometric thickness for clouds that are thick enough to
attenuate the CALIPSO lidar (Vaughan et al., 2009), and low enough that
CloudSat suffers significantly from surface clutter (Huang et al., 2012).
CloudSat measurements are further limited in terms of vertical resolution by
the radar bin size which is downsampled to 240 m (Stephens et al., 2008).
Currently, the main OCO-2 products are for column atmospheric CO_{2}
concentration (XCO_{2}; Crisp, 2008; Crisp et al., 2017; Eldering et al.,
2017; Osterman et al., 2016) and solar-induced fluorescence (SIF; Frankenberg
et al., 2014), which only use clear-sky soundings. Since any footprint that
is identified as possibly cloudy is not processed in the standard OCO-2
products, this work generates values from largely unused soundings.

OCO-2 functions in a pushbroom fashion with the footprint size dependent on
the viewing mode, but typically being 1.2–2.3 km. There are eight across-track
soundings, and each set of these is referred to as a frame in OCO-2
nomenclature. Within each sounding, measurements of reflected sunlight are
taken in the oxygen A-band, and weak and strong CO_{2} bands. The
CO_{2} bands are not considered in this analysis but do provide information on cloud
phase and droplet or particle size (Nakajima and King,
1990), and this information will be used when this retrieval is applied in
our observation-based study to identify likely liquid cloud cases.

The OCO-2 A-band instrument is a bore-sighted, imaging, grating spectrometer that measures 1016 channels spanning the wavelengths 759.2–771.8 nm. It is a flight spare from the original OCO mission and a number of focal plane array (FPA) elements have failed. 853 of the 1016 channels are available across all soundings and over 94 % of the damaged channels occur in the A-band continuum where there is redundancy, meaning little loss of information (Richardson et al., 2017).

This redundancy extends to the remaining undamaged FPA elements, meaning that fewer channels may be used to reduce the computational burden of a retrieval. The minimum number of channels required is equal to the number of elements in the retrieval state vector, provided that the channel responses to changes in the state vector properties contain orthogonal components. Therefore, for our desired retrievals of optical depth, physical thickness and cloud-top pressure, a single cloud retrieval requires at least three channels. The purpose of this study is to determine how many channels are required to cover a range of realistic cloud cases and to identify those channels.

A quirk of the OCO-2 instrument complicates this determination. The wavelength of channels varies slightly between across-track soundings, which means that the sampled oxygen absorption coefficient also varies. For this reason we separately analyse each of the eight frame sounding positions but will select a consistent micro-window of the same channels for each.

## 2.1 OCO-2 radiative transfer calculations

We use the OCO-2 Level 2 Full Physics Radiative Transfer Model (L2RTM) that
was developed for the OCO-2 XCO_{2} retrieval. Associated wrapper code
handles inputs such as interpolated ECMWF meteorological fields and accounts
for the OCO-2 satellite orbit, viewing geometry and instrumental response as
described in the OCO-2 data version 6 documentation (Boesch et al., 2015).
The radiative transfer is based on the LIDORT radiative transfer model with a
correction for the first two orders of scattering (Natraj and Spurr, 2007;
Spurr, 2006; Spurr et al., 2001) that fundamentally follows the eigenvector
approach to solving the radiative transfer equation (Flatau and Stephens,
1988). This model accounts for Earth's curvature for calculating atmospheric
path length of the incident and reflected solar beam but is otherwise
horizontally homogeneous. More details are provided in Spurr (2006) and
O'Dell (2010).

Although the L2RTM was designed for clear-sky XCO_{2} retrievals, it has been
validated in cloudy atmospheres by comparing OCO-2 observations with L2RTM
output assuming collocated MODIS and CALIPSO cloud properties
(Richardson et al., 2017). For homogeneous single-layer
liquid clouds over ocean, the root mean square error (RMSE) in continuum
channels was ±18 %, an overestimate of the model-only error as
this includes 3-D cloud effects, collocation error, parallax effects and
uncertainty in the MODIS and CALIPSO retrievals.

Clouds are implemented as follows: the atmosphere is defined on 20 levels, of which one is defined as the cloud centre, one as the cloud top and one as the cloud bottom. The cloud top is placed at the cloud-top pressure and the other cloud levels are equidistantly spaced to cover the cloud pressure thickness. An extinction coefficient is assigned to the centre level to result in the desired optical depth. Above the cloud the pressure levels are linearly interpolated from the cloud top to 1 Pa. Below the cloud they are linearly interpolated from the cloud bottom to the surface pressure. The level selected for the cloud centre is that whose pressure is closest to the cloud centre when linearly interpolated across the 20 levels from the surface pressure to 1 Pa. The L2RTM assigns extinction coefficients to layers by interpolating between levels, so a vertically homogeneous cloud layer is assumed.

Mie scattering computations are used within clouds using relevant
coefficients that are pre-calculated for gamma distributions of cloud
droplets based on a summary of low-cloud studies (Miles
et al., 2000). These values have only been pre-computed for integer values
of effective droplet size. This should not affect our results greatly since
our calculated uncertainties include a term spanning a range of droplet
sizes. Water surfaces at nadir are dark, and even in cloud-free cases there
is rarely sufficient SNR for the OCO-2 algorithm to attempt an XCO_{2}
retrieval. We assume a Cox–Munk surface reflectance function with the L2RTM
surface reflectance set to 0.10, but as we only use nadir view over ocean
there is little sensitivity to surface properties.

## 2.2 Optimal estimation and information content

We follow the principles of optimal estimation from Rodgers (2000), where a Bayesian retrieval combines an observation vector
** y** with a prior state vector

*x*_{a}and obtains a posterior state $\widehat{\mathit{x}}$. In our case the state vector consists of cloud-top pressure

*P*

_{top}, cloud pressure thickness Δ

*P*

_{c}and cloud optical depth

*τ*. This assumes that the observation can be related to the state by a linear forward model with some error

**:**

*ϵ*
where we refer to **K** as the Jacobian matrix as its
elements are ${K}_{i,j}=\partial {y}_{i}/\partial {x}_{j}$. Assuming Gaussian
distributions associated with *x*_{a} and
** y**, Rodgers (2000) shows that the best
estimate of the posterior state is

and its covariance matrix is

Here **S**_{a} is the prior covariance and
**S**_{ϵ} the observation covariance. From
Eq. (2) the posterior state $\widehat{\mathit{x}}$ is the prior
*x*_{a} plus an iteration that is based on the
difference between the observed and expected ** y** with
appropriate weighting for uncertainties. Equation (3) shows that the
posterior uncertainty $\widehat{\mathbf{S}}$ is reduced by an amount that depends on the
size of the Jacobian

**K**weighted by the observation uncertainty

**S**

_{ϵ}. Potential non-linearity in

*y***(**

*x***)**is addressed by iteration, with the linear expansion being determined about each iteration step.

In our OCO-2 cloud retrieval the state vector contains optical depth, cloud pressure thickness and cloud-top pressure while the observation vector is any subset of the 853 valid OCO-2 A-band channels. Using fewer channels reduces the computational burden, both in terms of the radiative transfer and for iterating the retrieval which would otherwise involve repeated inversion of 853 × 853 matrices.

It is common practice to select channels based on information content and/or degrees of freedom for signal (Chang et al., 2017; Mahfouf et al., 2015; Martinet et al., 2014; Rabier et al., 2002), and this approach has already been used in an oxygen A-band and B-band analysis for aerosol retrievals (Ding et al., 2016).

The information content is based on the concept of Shannon entropy and is
related to the volume of state space occupied by the probability
distribution *P* that represents our knowledge:

It is expressed in bits, which represents the number of binary digits required to represent the possible outcomes. A retrieval decreases the probability distribution volume, and this change in associated Shannon entropy (Shannon and Weaver, 1949) is the information content, IC, of the measurements:

In this case *S*(*P*_{0}) is the Shannon entropy associated with
the original probability distribution and *S*(*P*_{0}) the same value associated with the retrieved probability distribution. For
multivariate Gaussian descriptions of the probability distributions,
Rodgers (2000) shows that the information content of
measurements is

A related property is the degrees of freedom for signal *d*_{s}, which
represents the number of useful independent quantities in a measurement. It
may be thought of as how many different variables can be obtained from a
measurement, and with our three-component state vector we require a value
approaching three. It may be calculated from the prior and posterior state
covariances as follows:

Note the different order and inversion state of the covariance matrices
relative to Eq. (6). In our analysis we calculate IC, *d*_{s} and
posterior errors for continuous micro-windows of varying size and these
calculations require **S**_{ϵ} and
**S**_{a}. We assume prior covariances based partially on a
MODIS and CALIPSO cross-validation (Richardson et al.,
2017) and calculate the observation covariance
**S**_{ϵ} by perturbing atmospheric profiles. The
calculation of the covariances is described in Sect. 3.1 and the channel
selection approach in Sect. 3.2.

While theoretically three channels is sufficient to retrieve three state vector elements, it is not clear that the same three channels will apply in all cases. For example, while changes in cloud-top pressure of higher clouds may lead to strong responses in channels near line cores, light in these channels may be mostly absorbed by the time it reaches lower clouds, so less strongly absorbing channels will be preferred for lower clouds. Changes in absorption due to temperature or water vapour may also affect the relative response of radiances to cloud properties. For this purpose, we consider a variety of atmospheric and cloud properties.

Necessary observation covariances are derived by perturbing atmospheric
profiles and the IC, and *d*_{s} and posterior covariance are used to select
an optimal micro-window. Finally a retrieval is developed and tested on
cloudy atmospheres where the “truth” is assigned and pseudo-observations
and prior values are provided by sampling from the previously defined
covariance matrices.

For ease of presentation we restrict our analysis to three representative atmospheric states, three cloud heights (680, 750 and 850 hPa) and three cloud optical depths (5, 10 and 25). Together, this results in 27 combination cases. Effective droplet radius is assumed to be 12 µm, and cloud pressure thickness is determined from the cloud geometric thickness from a subadiabatic stratiform cloud model (Borg and Bennartz, 2007):

where *C*_{w} is the moist adiabatic condensate coefficient, for marine
stratocumulus we use 1.9 × 10^{−3} g m^{−4} (range given as
1–2.5 × 10^{−3} g m^{−4} from Brenguier,
1991) and LWP is the liquid water path, which is related to optical depth
*τ* and effective droplet radius *r*_{eff}:

where *ρ*_{w} is the density of water and *Q*_{ext} the area-weighted
mean scattering efficiency (Szczodrak et al., 2001),
which we take to be 2. This value is chosen as it represents the
large-particle limit for non-absorbing spheres (Herman,
1962) which is a reasonable approximation for cloud droplets in the oxygen
A-band. Cloud geometric thickness is converted to pressure thickness by
assuming that pressure decreases exponentially with altitude with a scale
height of 8 km. Note that the result of the combined Eqs. (8) and (9) comes from
an adiabatic cloud model in which the LWP increases linearly with height,
and differs by a factor of 5∕6 from the classic result derived for a
homogeneous cloud profile (Stephens, 1978). Neither
assumption is perfectly representative of reality, but the adiabatic profile
is expected to be more realistic and so it is used here.

For the representative atmospheric states, we select all collocated
soundings that are identified as single-layer liquid clouds by both MODIS
and CALIPSO during November 2015 and bin them according to absolute
latitude, in the ranges 0–20, 20–50 and
50–90^{∘}. The MODIS data are from product MYD06 at 1 km
horizontal resolution (Platnick et al., 2015)
and the CALIPSO data are from the 1 km resolution cloud layer product
01kmCLay (Vaughan et al., 2009). Within each bin the
collocated OCO-2 ECMWF-AUX meteorological profiles (including pressure,
specific humidity, temperature and wind speed) are averaged level by level. This
includes all meteorological inputs used by the L2RTM, such as pressure,
temperature, humidity and wind speed.

## 3.1 Calculation of observation covariances

For simplicity we assume that the components of
**S**_{ϵ} are independent and consider
error contributions from instrumental uncertainty
**S**_{I} and that introduced by uncertainty in
the temperature profile **S**_{T}, humidity
profile **S**_{q} and effective
droplet radius **S**_{reff} such that

In reality, the temperature and humidity uncertainties are likely to be
correlated, but this simplifies the calculation and allows unique
attribution of covariance sources. The matrix **S**_{I} is
a diagonal matrix, so averaging over more channels reduces the total
posterior uncertainty even if the Jacobians are not independent. Its
elements are equal to the square of the instrumental uncertainty, which
depends on the radiance.

For **S**_{T} and **S**_{q} we follow the approach of
Chang et al. (2017) and perturb the
tropical, mid-latitude and high-latitude atmospheric profiles 2000 times
for temperature and humidity separately with uncertainties based on 1 km
resolution AIRS validation results (Divakarla et al.,
2006). For temperature we add a uniform perturbation to each level with a
value sampled from a zero mean (*μ*) Gaussian distribution with standard deviation
(*σ*) of ±1.5 K. For specific humidity we sample from a zero
mean Gaussian distribution with a standard deviation of unity, then scale this value
based on pressure level. The scaling is equivalent to ±20 % of the
initial specific humidity at the surface, increasing linearly to ±50 % of the layer values at 250 hPa and remaining at ±50 % for
levels with lower pressure. The calculation was also performed with 2000 perturbations applied to *r*_{eff} by sampling from a lognormal distribution
that approximates the effective radius distribution reported by MODIS for
our November 2015 low cloud cases. This lognormal fit has an arithmetic mean
of 12.0 µm, but after excluding values outside the 4–30 µm
retrieved by MODIS, the arithmetic mean is 12.6 µm and 5–95 % of
the values fall within 7.5–19.4 µm. We choose *r*_{eff} = 12 µm in our default retrieval as we are restricted to integer values by the
available L2RTM Mie scattering tables, and based on its similarity to the
full distribution mean.

For each set of perturbations we simulated the A-band spectra for cloud
optical depths of 5, 10 and 25 and solar zenith angles (SZAs) of approximately
30, 45 and 60^{∘} with a cloud-top pressure of
850 hPa. We calculate covariances at a single value of *P*_{top}, but the
convergence of our synthetic retrieval tests across a range of true
*P*_{top} values shows that we obtain reliable results regardless.

The output spectra are provided for each of the eight different instrument line shapes associated with the eight different OCO-2 across-track sounding positions.

For each set of 2000 perturbed outputs, we estimated the covariance matrix
elements, *S*_{i,j} where *i*, *j* refer to channel indices, as follows:

where the sum is over the *N* = 2000 spectra of radiance *I*, which are
individually referred to using the index *k*. In this case $<{I}_{i}>$ and
$<{I}_{j}>$ are the sample mean radiances in the relevant channels *i* and *j*.

## 3.2 Channel selection

Equations (3) and (6) state that we can determine the information content and posterior error covariance from the prior covariance, observation covariance and Jacobians. Our aim is to select the optimal micro-window of consecutive OCO-2 channels to provide a retrieval that efficiently reduces the posterior state error.

We use the L2FP radiative transfer model to simulate OCO-2 spectra for marine
liquid clouds of *τ* at 5, 10, 25 and *P*_{top} at 680, 750,
850 hPa, for each of the three meteorological cases described in Sect. 3.1 and
for each of the eight across-track sounding positions. In each case, the
solar zenith angle is 45^{∘} and the Jacobians for *τ*,
*P*_{top} and Δ*P* are determined by finite differencing. The
relevant observation covariance is that determined for the same sounding
position, region and optical depth in Sect. 2.2 at SZA = 45^{∘}.
Prior covariance is assumed to be diagonal, equivalent to an error of 1.5 in
*τ*, of 60 hPa in *P*_{top} and of 7.5 hPa in Δ*P*. Our
*τ* prior error comes from applying the ±18 % error in simulated
radiance for homogeneous clouds when provided with MODIS optical depth
(Richardson et al., 2017). Our *P*_{top} uncertainty is from the
standard deviation of the differences between OCO-2 and CALIPSO
*P*_{top} when using a simple lookup table for OCO-2, which we intend
to use for the OCO-2 prior. The Δ*P* uncertainty is similar to the
±20 % error associated with Eq. (8) for clouds of cloud fraction
> 0.8 reported in Bennartz (2007).

We consider the information content IC, and the three diagonal elements of the
posterior covariance matrix **S**_{x}. The information content
accounts for non-diagonal terms in the posterior covariance, allowing an
objective best selection, while the diagonal elements allow more intuitive
interpretation of the magnitude of the posterior uncertainty. We refer to
these using the symbol *σ* with a relevant subscript, such that
${\mathit{\sigma}}_{\mathit{\tau}}^{\mathrm{2}}={S}_{\mathit{\tau},\mathit{\tau}}$, where *S*_{τ,τ} is the element
of the covariance matrix corresponding to the *τ*−*τ* covariance. Note
that we present the square root of this value, i.e. *σ*.

This approach represents a sample of 27 unique cloud–meteorology cases across the eight different sets of OCO-2 instrument line shapes, resulting in 216 total cases. When selecting the optimal micro-window for retrievals, it is necessary to select not just its location but also its size (i.e. number of neighbouring channels within the micro-window).

To make this problem tractable, we select micro-windows of the following
size: 5, 10, 25, 50, 75, 100, 150, 200 and 500 neighbouring channels. For
each of these possible sizes we calculate IC, *d*_{s} and the
diagonal posterior error terms for every overlapping micro-window of that
size. For example, the 853 individual OCO-2 channels allow 849 overlapping
five-channel micro-windows, for which we determine the information content
values for each of the 216 cases.

For each size of micro-window we choose the one with the highest mean
information content across the 216 cases. While this may result in a
different location for each size of micro-window, the location is fixed for
an individual case; i.e. the five-channel micro-window consists of the same five channels in all 216 cases. We select the optimal micro-window size as that
with > 80 % of the 500-channel IC, optical depth posterior
*σ*_{τ,τ} better than ±0.05 and a posterior of better
than ±1 hPa in the pressure terms ${\mathit{\sigma}}_{{P}_{\mathrm{top}},{P}_{\mathrm{top}}}$ and
*σ*_{ΔP,ΔP} for all 216 cases. These
thresholds are by nature subjective and arbitrary.

## 3.3 Theoretical retrieval test case

We perform synthetic retrievals with known true cloud cases in mid-latitude
meteorology and a 45^{∘} solar zenith angle. For each cloud case we
perform 50 retrievals using a 12 µm droplet size and the prior cloud
state is sampled from Gaussian distributions with *σ*_{τ} of
±30 %, ${\mathit{\sigma}}_{{P}_{\mathrm{top}}}$ of ±60 hPa. Cloud pressure
thickness is calculated from Eq. (8) with LWP from Eq. (9), and in the
optimal estimation a prior *σ*_{ΔP} of ±25 % is assumed.
The atmospheric humidity and temperature profiles are perturbed by sampling
from the same distributions used to derive the covariance matrices in
Sect. 3.2 and the observed spectrum in each case is generated by taking the
simulated spectrum from the “truth” case and perturbing it by sampling from
the relevant covariance matrix that has been scaled for the cloud properties
according to Sect. 3.2. The squared OCO-2 radiance uncertainties are added to
the diagonal elements of the observation error covariance matrix with no
cross correlation. We use the standard OCO-2 version 7 uncertainties, and SNR
increases as the radiance in a given channel increases. The median SNR for an
individual spectrum ranges from just over 400 for the *τ* = 5 cases to
around 700 for the *τ* = 25 cases. The single-channel SNR reaches a
minimum of 72 in an absorption band channel in a *τ* = 5 case, and a
maximum of 763 in a weakly absorbing channel in a *τ* = 25 case.

Forty true cloud cases are used with five of each case where optical depth
ranges from 5 to 40 in increments of 5 and cloud-top pressure is randomly
selected to be between 680 and 900 hPa and rounded to the nearest 10 hPa. The
prior cloud properties are assumed to be unbiased and so are randomly sampled
from a Gaussian distribution with a mean equal to the truth and a standard deviation
equal to the prior errors above. Each synthetic retrieval begins with a
separate prior, and the prior is also used as the first guess. The retrieval
attempts assume *r*_{eff} = 12 µm but the true *r*_{eff} is allowed to
vary and is randomly sampled from a literature summary of marine
stratocumulus results, scaled to ensure a mean value of 12 µm
(Miles et al., 2000). The *r*_{eff} distribution
effective variance is fixed in each case in order to use the pre-calculated
scattering properties used with the L2RTM code, but given the wide range of
effective mean values considered, it is not expected that allowing the
effective variance to change would greatly affect the results.

For each of the 50 perturbed prior states and observation spectra, we perform a standard 10-iteration optimal estimation retrieval (Rodgers, 2000) using the Gauss–Newton solution to optimise each step. These retrievals are done using the 75-channel micro-window selected following Sect. 3.3. The sample means and standard deviations are then compared with the known true state and indicate the theoretical performance of the micro-window retrieval.

Results are presented here for the first sounding position, which is
leftmost when facing northwards along track during the ascending node. Our
conclusions are not affected by changing the sounding position. For
illustration, we select the case of SZA = 45^{∘}, *τ* = 10
and *P*_{top} = 850 hPa and then present the square root of the diagonal
components of covariance matrices for temperature, humidity and effective
radius in Fig. 1. This shows both the absolute
and fractional uncertainty in the radiance due to each factor. Droplet size
dominates, consistently contributing near 3 % of the radiance, although
the temperature uncertainty contributes up to 1.5 % in the darker
absorption channels.

Figure 2 shows the full covariance matrices for each component using the same mid-latitude meteorology, cloud properties and SZA as Fig. 1. The strongest and most consistent positive cross-correlations occur for the effective droplet size.

While the overall patterns are similar for different cloud optical depths, solar zenith angles or regional meteorology, the absolute values of the covariance matrices change. A retrieval requires an estimate of the error covariance that is relevant for the given measurement, but these matrices are computationally intensive to prepare, and storing and accessing a large number of them would make the retrieval less efficient. We will therefore use a single set of retrieval matrices, one for each across-track sounding position, and then scale the matrix to account for changes in solar zenith angle, meteorology and optical depth.

Figure 3 shows the relationship between the
observation covariance matrix excluding the instrumental term
**S**_{I} for *τ* = 10, SZA = 30^{∘} and
*τ* = 25, SZA = 60^{∘} with mid-latitude meteorology. Only
the upper-diagonal elements of each matrix have been plotted to avoid
duplication and values are scaled by ${\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{2}}$, where *μ*_{0}=cos *θ*_{SZA}. There is a linear relationship between the
two matrices meaning that one may be reconstructed from the other. The
results are similar for tropical and high-latitude cases, and for all
soundings.

## 4.1 Micro-window selection

Figure 4 shows IC and *d*_{s} spectra using micro-windows consisting
of 5, 75 or 200 OCO-2 channels. Also shown are the posterior errors in cloud
properties taken from the square roots of the diagonal components of
**S**_{x}.

In this cloud case (mid-latitude, *τ* = 10, *P*_{top} = 850 hPa), the
greatest information content comes from selecting channels near absorption
features and avoiding the far wings of the A-band where only optical depth
is reliably retrieved, as these channels have little O_{2} absorption and
so are uninformative about photon path length. Otherwise, the five-channel
micro-window is most sensitive to its placement within the spectrum:
information content varies from 4.4 to 9.4 bit depending on the
micro-window's location.

Micro-windows that contain fewer channels are more sensitive to changes in
the instrument line shapes and cloud conditions. For example, for the
five-channel micro-window in Fig. 4, the
best-performing channel has an information content of 9.4 bit. However, for
a different cloudy case, *τ* = 25, *P*_{top} = 680 hPa, and for
sounding position 8 instead of 1, the information content is reduced to 6.0 bit. This is a substantial loss relative to the best possible micro-window
for that cloud case, which has 8.4 bit of information.

To assess the relative trade-offs between increased speed and decreased
performance we take the micro-window with the highest mean information
content across all cases. We then plot the central value and full range of
the 216 values for each selected micro-window size in
Fig. 5, along with our chosen thresholds as
dashed lines in each panel. The median case in the 50-channel micro-window
passes our IC threshold and in all cases passes the *τ* uncertainty
threshold, but it has multiple cases that fail the *P*_{top} and
Δ*P*_{c} thresholds. By contrast, the 75-channel micro-window
containing the OCO-2 channels 353–426 (indices counting from one for the
full 1016 OCO-2 L1bSc channels) consistently satisfies our *P*_{top} and
Δ*P*_{c} criteria and reduces the full wavelength range from
759.2–771.8 to 763.5–764.6 nm.

Figure 6 shows an example cloudy scene spectrum
simulated for OCO-2 and highlights the chosen 75-channel micro-window in
red. Also shown is an approximated GOME-2 spectrum based on the MetOp-B
instrument characteristics (Munro et al., 2016). We
approximate the ILS using Gaussian instrument line shapes, taking the 0.21 nm spectral sampling from Table 1 and FWHM of 0.50 nm from Table 2 of Munro
et al. (2016). While OCO-2 spectra allow three independent pieces of information
to be obtained (see the reported *d*_{s} in the figure caption) our
calculations agree with previous work that the GOME-2 resolution only
provides approximately two (Schuessler et al., 2014).
Consistent with older theoretical work (Heidinger and Stephens, 2000; O'Brien and
Mitchell, 1992) this analysis supports the case the high spectral
resolution of OCO-2 leads to additional information about cloud geometric thickness.

## 4.2 Theoretical retrieval case

Example synthetic retrieval iterations using the 75-channel micro-window are
shown in Fig. 7 for *τ*=10 and *τ*=25 cases, and convergence typically occurs within a few iterations. Lines are
coloured according to their *χ*^{2} values and it is clear that these are
larger for cases where the result settles away from the true state. The
posterior sample standard deviations are presented in
Table 1 for the full samples and for cases where we
filter the results by excluding the 10 % of cases with the highest *χ*^{2} in each case. The greatest effect of filtering by *χ*^{2} is to
reduce the uncertainty in the cloud-top pressure and cloud pressure
thickness from 12.9 to 2.9 and 2.5 hPa respectively. The mean
standard deviation in the *τ* retrieval is ±0.75 across all
cases, but this is inflated by a large value in the *τ* = 35 cases.

OCO-2 O_{2} A-band spectra are rich in information about cloud properties.
Continuum channels with little absorption respond strongly to cloud optical
depth, while the radiance in absorption bands is dominated by photon path
length, which increases with cloud-top pressure or cloud pressure thickness.
A channel's response to cloud properties depends largely on its oxygen
absorption coefficient (Fischer and Grassl, 1991; Koelemeijer et al., 2001;
Stephens and Heidinger, 2000), and since many channels have similar
absorption coefficients there is redundant cloud information in OCO-2
spectra.

We ultimately selected 75 neighbouring channels as containing the majority of the cloud information. Observation covariance matrices were developed based on uncertainty related to the atmospheric temperature and humidity profiles, in cloud droplet effective radius and instrumental uncertainty. These covariances depend on the meteorological profile, solar zenith angle and cloud properties. Additionally, instrument line shapes vary across the OCO-2 swath, so a separate covariance matrix is required for each of the eight across-track OCO-2 footprints. Fortunately, when cloud or meteorological properties change, the covariance matrix elements tend to be approximately linearly related, so an arbitrary covariance matrix can be reconstructed from a known case. There is greater spread in the reconstructed humidity component but this contributes a small fraction of the total covariance, which is dominated by uncertainty in the droplet radius whose component is well reconstructed.

Using 75 channels substantially reduces the retrieval processing time
relative to the 853 available channels, and its usefulness was demonstrated
in a set of eight synthetic test cases where a known cloud case was retrieved.
In our perturbed tests the retrieval typically converged within two iterations, although a few cases converged on a local optimum instead of
approaching the truth. Fortunately, these cases can generally be identified
from the associated *χ*^{2}, indicating that when this approach is
applied to real OCO-2 data, it may be possible to flag cases where there is
less confidence in the retrieval.

Our idealised posterior errors of ±0.75 in optical depth and better
than ±3 hPa in cloud-top pressure and cloud pressure thickness are
based on assuming that convergence can be identified from the *χ*^{2}
values, and that the cloud is single-layered and horizontally homogeneous
within the OCO-2 field of view of approximately 1.4 km × 2.2 km. This
is a reasonable approximation in marine stratocumulus decks, where the
typical length scale of variability in LWP can be 10–30 km
(Wood and Hartmann, 2006), but will be violated in many low-level cloud cases such as at the edges of the stratocumulus–trade cumulus
transition.

In addition, the assumption of a single scattering layer is commonly broken: multi-layered clouds are ubiquitous (Li et al., 2015), although for overlying cirrus it may be possible to identify and flag many of these cases based on the inferred distribution of photon path lengths from A-band measurements (Min et al., 2004). Alternatively, since OCO-2 flies in the A-Train it would also be possible to use other sensors such as CALIPSO (which is now leaving the A-Train) or MODIS to identify multi-layer cloud cases, or scenes in which there is heavy aerosol loading. Cases of heavy aerosol loading are most common over the Namibian stratocumulus region with common occurrence in June–July–August (JJA) and a peak in September–October–November (SON). A combination of CALIPSO, CloudSat and International Satellite Cloud Climatology Project (ISCCP) data imply that in the SON Namibian stratocumulus region, approximately one-third of low clouds have overlying aerosol, and approximately half of these cases are smoke (Devasthale and Thomas, 2011; Winker et al., 2010). Scattering layers overlying a marine cloud tend to reduce the effective retrieved cloud layer pressure due to the reduced mean path length of those photons reflected from the overlying layer (Vanbauce et al., 1998). Assessment of aerosol effects will be necessary in future work.

It was also assumed that the clouds will be reliably identified as liquid,
and that a constant effective droplet size may be assumed. Droplet size
variance has been included in terms of the observation covariance, but this
limits our retrieved posterior covariance. Cloud identification is
relatively simple for nadir A-band reflectance measurements over ocean, as
for most solar zenith angles the surface is dark and cloudy scenes may
simply be identified when reflectance exceeds some threshold. The OCO-2
instrument also carries weak and strong CO_{2} band spectrometers, and with
ice absorbing more strongly than water in the near infrared we will be able
to use well-known retrieval principles to obtain cloud phase (Nakajima and King, 1990).

Our assumptions mean that the true error of a cloud retrieval based on OCO-2 will be larger than that reported here, but our results suggest that the use of a 75-channel micro-window is justified as the basis of an OCO-2 cloud retrieval for marine liquid cloud properties.

The OCO-2 satellite carries an O_{2} A-band spectroradiometer with high
spectral sampling. Our analysis supports that this spectral sampling is
sufficient to, in principle, allow determination of the optical depth,
cloud-top pressure and geometric pressure thickness of clouds. It has been
demonstrated that observed OCO-2 spectra respond largely as expected to
changes in cloud optical depth and cloud-top pressure (Richardson et al., 2017), but that study did not use modern
Bayesian techniques. Such techniques account for relevant conditions such as
line broadening due to local meteorology, and they also account for prior
information and cross-correlation between the responses of individual
channels.

Here we report that the OCO-2 A-band spectra contain much redundant information as a number of channels experience similar oxygen absorption. After accounting for observational errors associated with uncertainty introduced by meteorology, cloud droplet size and instrumental error, it was found that with a micro-window of 75 continuous channels, most of the information from the full 853-channel spectrum is retained. In a perfectly linear theoretical case, posterior error in cloud-top pressure and cloud pressure thickness were reduced below ±1 hPa and optical depth below ±0.05.

Using perturbed synthetic tests, the majority of cases approached the known
truth and the full sample posterior errors averaged ±0.75 in optical
depth and ±12.9 hPa in *P*_{top} and cloud pressure thickness. Cases that
converged to a state away from the truth could generally be identified by
their large *χ*^{2} values, and removing the 10 % of worst cases
reduced the posterior sample standard deviation in *P*_{top} and Δ*P*_{c} to ±2.9 and ±2.5 hPa.

These results apply in an ideal theoretical case of a uniform single-layer liquid droplet cloud, and retrieval errors will be larger in reality where these assumptions do not apply. However, violations of these assumptions such as real-world cloud heterogeneity will likely have a similar effect on both the full spectrum and on our selected 75-channel micro-window. We therefore propose that these assumptions do not affect our primary conclusion regarding the relative performance of our optimised retrieval versus a more intensive, full-spectrum retrieval.

The OCO-2 science data are available online from the NASA Goddard GES DISC at https://disc.gsfc.nasa.gov/datasets/OCO2_L1B_Science_V8r/summary (OCO-2 Science Team et al., 2017). The radiative transfer code is available from github at https://github.com/nasa/RtRetrievalFramework.

The authors declare that they have no conflict of interest.

The research described in this paper was performed
at the Jet Propulsion Laboratory, California Institute of Technology,
sponsored by NASA. Mark Richardson was funded by the OCO-2 and CloudSat projects. Mark Richardson
would like to thank James McDuffie and Jussi Leinonen for providing
radiative transfer and optimal estimation code assistance, plus Matt Lebsock, Annmarie Eldering, Mike Gunson, Chris O'Dell,
Tommy Taylor, Heather Cronk and Aronne Merrelli for helpful technical discussions.

Edited by: Alexander Kokhanovsky

Reviewed by: four anonymous referees

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