Improvements to climate model results in polar regions require improved
knowledge of cloud properties. Surface-based infrared (IR) radiance
spectrometers have been used to retrieve cloud properties in polar regions,
but measurements are sparse. Reductions in cost and power requirements to
allow more widespread measurements could be aided by reducing instrument
resolution. Here we explore the effects of errors and instrument resolution
on cloud property retrievals from downwelling IR radiances for resolutions
of 0.1 to 20 cm-1. Retrievals are tested on 336 radiance simulations
characteristic of the Arctic, including mixed-phase, vertically
inhomogeneous, and liquid-topped clouds and a variety of ice habits.
Retrieval accuracy is found to be unaffected by resolution from 0.1 to 4 cm-1, after which it decreases slightly. When cloud heights are
retrieved, errors in retrieved cloud optical depth (COD) and ice fraction
are considerably smaller for clouds with bases below 2 km than for higher
clouds. For example, at a resolution of 4 cm-1, with errors imposed
(noise and radiation bias of 0.2 mW/(m2 sr cm-1) and biases in
temperature of 0.2 K and in water vapor of -3 %), using retrieved cloud
heights, root-mean-square errors decrease from 1.1 to 0.15 for COD, 0.3 to
0.18 for ice fraction (fice), and 10 to 7 µm for ice
effective radius (errors remain at 2 µm for liquid effective radius).
These results indicate that a moderately low-resolution, surface-based IR
spectrometer could provide cloud property retrievals with accuracy
comparable to existing higher-resolution instruments and that such an
instrument would be particularly useful for low-level clouds.
Introduction
Knowledge of polar cloud properties is critical for understanding climate
change in polar regions. Polar regions are among the most rapidly warming
regions on Earth, with significant concurrent changes in cloud properties
that influence the amount of warming (Wang and Key, 2005) and indications
that sensitivity to clouds may increase in a warming Arctic (Cox et al.,
2015). Clouds have a strong influence on the polar surface energy budget
(Lawson and Gettelman, 2014; van den Broeke et al., 2017), influencing sea ice
loss (Francis and Hunter, 2006; Kay and Gettelman, 2009; Wang et al., 2019) and
Greenland ice melt (van den Broeke et al., 2017). Despite ongoing efforts to
improve cloud processes in climate models, the Intergovernmental Panel on
Climate Change (IPCC) finds that “clouds and aerosols continue to
contribute the largest uncertainty to estimates and interpretations of the
Earth's changing energy budget,” (Boucher et al., 2013). Improving the
representation of cloud processes in climate models requires observational
constraints, including ice and liquid water paths, particle size, and
thermodynamic phase (Komurcu et al., 2014; Winker et al., 2017). This is
particularly true for the polar regions, where clouds and cloud processes
are distinctly different from lower latitudes and present unique challenges
for modeling cloud radiative effects (Hines et al., 2004) and where
measurements are sparse.
Although ground-based observations in the polar regions are sparse,
measurements made during campaigns and at permanent field sites (e.g.,
Bromwich et al., 2012; Cox et al., 2014; Uttal et al., 2015, and references
therein; Lachlan-Cope et al., 2016; Silber et al., 2018) and from satellites
(e.g., L'Ecuyer and Jiang, 2010) have made important contributions to our
understanding of polar clouds. IR spectrometers are proven instruments for
remote sensing that have been part of many of these surface and
satellite-based measurements. Surface-based IR spectrometers are most
sensitive to the cloud base, providing an important complement to
satellite-based measurements. In particular, Atmospheric Emitted Radiance
Interferometer (AERI) instruments currently operate at Barrow
(1998–current), Eureka (2006–current), and Summit (June 2010–current): three
Arctic intensive observing sites (Uttal et al., 2015). In the Antarctic,
there have been only short-term surface-based IR spectrometer measurements,
including measurements made at Amundsen–Scott South Pole Station in 1992
(Mahesh et al., 2001) and 2001 (Rowe et al., 2008), at Dome C during Austral
summer 2003 (Walden et al., 2005) and 2012–2014 (Palchetti et al., 2015), and at
McMurdo (as part of the Atmospheric Radiation measurement (ARM) West
Antarctic Radiation Experiment, or AWARE; Silber et al., 2018). These
measurements are crucial, but represent only very sparse coverage of the
polar regions.
Because IR radiance measurements are passive, the energy requirements are
considerably lower than for active instruments such as lidar. Thus there is
the potential for portable, low-cost, autonomous IR spectrometers that could
be deployed to remote locations to make widespread IR radiance measurements
across the polar regions from which cloud properties could be retrieved.
Such measurements would be beneficial in a number of ways: First, they could
be used to fill gaps in satellite measurements. For example, cloud
properties were retrieved at Eureka from 2006 to 2009 from AERI measurements
made nearly continuously every ∼40 s (Cox et al., 2014).
By contrast, satellite overpasses are typically twice per day. Second,
surface-based measurements can be used to validate satellite-based
measurements. Finally, surface-based instruments are generally better at
characterizing clouds in the boundary layer. To demonstrate the feasibility
of such an instrument, the limitations of the retrieval given instrument
operational constraints and availability of ancillary data must first be
assessed.
In this paper, we explore the accuracy with which cloud properties could be
retrieved from a portable IR spectrometer, including optical depth,
thermodynamic phase, and effective radius. This paper builds on similar work
that explored the accuracy of cloud-height retrievals (Rowe et al., 2016).
One way to develop a robust, low-power portable spectrometer might be to
reduce the instrument resolution. Here we quantify cloud-property retrieval
accuracy as resolution becomes coarser, from 0.1 to 20 cm-1.
Cloud properties are retrieved from simulated downwelling radiance spectra
using the CLoud and Atmospheric Radiation Retrieval Algorithm (CLARRA). In
addition to retrieving cloud height (Rowe et al., 2016), CLARRA retrieves
cloud optical and microphysical properties from IR radiances using an
optimal inverse method in a Bayesian framework. Cloud property retrievals
are performed for simulated polar clouds with varying atmospheric thermal
and humidity structure, cloud optical depth (in the geometric limit,
hereafter COD), thermodynamic phase (including mixed-phase and supercooled
liquid), liquid effective radius, ice effective radius, ice crystal habit,
and cloud vertical structure. Mixed-phase clouds were simulated as an
external, homogeneous mixture of liquid and ice particles. We also examine
the sensitivity of retrieved results to noise and bias imposed on the
radiance as well as to errors in specified input parameters, especially the
atmospheric state and cloud height.
Simulated radiances
To test the effect of instrument resolution on the ability to retrieve cloud
properties from downwelling radiances, retrievals using CLARRA were
performed on a set of simulations. Using simulations rather than actual
measurements confers a variety of benefits: (1) the basic capability of the
model, in the absence of error, can be determined, setting a benchmark for
retrieval capability, (2) the effects of various sources of error (such as
noise, bias, or uncertainty in the atmospheric state) can be determined and
assessed independently, and (3) errors in the retrieved values are known and
thus can be compared to assess the uncertainty prediction from the CLARRA
model.
The set of simulated downwelling radiances is described in detail by Cox et
al. (2016) and by Rowe et al. (2016). The simulations are based on observed
Arctic atmospheric profiles and cloud properties meant to represent a
typical Arctic year, based on statistics from field observations (Cox et al.,
2016, and references therein; although designed for the Arctic, significant
overlap is expected for typical Antarctic atmospheric states, except perhaps
in winter in the interior, when the atmosphere is colder and drier). All
clouds were modeled as plane-parallel, single-layer clouds. Precipitable
water vapor (PWV) varied from 0.2 to 3 cm.
A base set of 222 simulated radiances was created for atmospheres with
vertically uniform clouds, using spheres for ice crystal habit (as well as
for liquid droplet shape). Cloud bases vary from 0 to 7 km, with about
70 % of clouds within the lowest 2 km and 30 % above; thickness varies
from 0.1 to 1.6 km; and temperatures vary from 225 to 282 K. Mixed-phase
clouds are modeled as externally mixed and span temperatures of 240 to 273 K. Cloud phase includes liquid-only (about one-sixth of cases), ice-only
(about one-sixth), and mixed-phase (about two-thirds). Statistics for
cloud properties are summarized in Table 1. (Statistics were generated for
lognormal distributions of COD and effective radii. Thus the standard
deviations were computed for the logarithms. For convenience, these were
converted to positive and negative linear deviations in Table 1.)
Statistics of cloud properties. Standard deviations were calculated
for the logarithms of cloud optical depth referenced to the geometric limit
(τg), effective radius of liquid (rliq), and effective radius
of ice (rice) because distributions for the logarithms were found to be
more Gaussian in shape; these standard deviations were converted to positive
and negative linear standard deviations for these quantities. The ice fraction,
fice, peaks strongly at both limits; thus no standard deviation is
provided.
A second set of simulated radiances was created for testing the effects of
cloud vertical inhomogeneity, including 23 cases from the base set for which
the cloud spanned multiple layers of the atmospheric model; these are
referred to as “diffuse.” Simulations were created for identical
conditions, including the total COD, except that clouds were modeled as
dense (physically thinner), inhomogeneous (the cloud was optically thicker
at the center and thinner at the upper and lower edges), or liquid-topped
(liquid cloud was confined to the uppermost layer, while ice cloud was
confined to the lower model layers).
A third set was created for testing the effect of ice habit on the
retrieval, including nine base cases having an ice COD greater than 0.5.
Simulations were created for identical conditions, except that single-scattering properties from different cloud habits were used: hollow bullet
rosettes, smooth plates, rough plates, smooth solid columns, and rough solid
columns (Yang et al., 2013).
The set of simulated spectra was created at monochromatic or perfect
resolution using the DIScrete Ordinates Radiative Transfer (DISORT;
Stamnes et al., 1988) model, with monochromatic gaseous optical depths created using
the Line-By-Line Radiative Transfer Model (LBLRTM; Clough et al., 2005) as
inputs. Spectra were then convolved with a sinc function to obtain sets of
spectra at resolutions of 0.1, 0.5, 1, 2, 4, 8, and 20 cm-1. These are
hereafter referred to as the “observed” spectra. Figure 1a shows a
spectrum at 0.5 cm-1 resolution, together with the clear-sky spectrum
for the same atmospheric conditions. Additional examples at 0.5 cm-1,
as well as at 4.0 cm-1, are given in Fig. 2 of Rowe et al. (2016).
(a) Clear- and cloudy-sky downwelling radiance (1 RU = 1 mW/(m2 sr cm-1)) for a typical polar atmosphere at a resolution of
0.5 cm-1. (b) Model errors (model–true) in downwelling radiances for
the clear-sky radiance shown in (a) (blue solid line), and box-and-whisker plots of model errors for all radiances, averaged in microwindows
(horizontal lines give the median, boxes give the 1st and 3rd
quartiles, and whiskers give the range).
CLoud and Atmospheric Radiation Retrieval Algorithm (CLARRA)
CLARRA retrieves cloud properties (cloud height and temperature, COD, ice
fraction, effective radius of liquid droplets, and effective radius of ice
crystals) from downwelling IR radiances, given knowledge of the atmospheric
state. As the first step in the retrieval, cloud heights are retrieved by
CLARRA as described by Rowe et al. (2016; see also references therein).
Alternatively, cloud heights can be input into CLARRA (e.g., from other
instrumentation, such as lidar, or from reanalysis models). Next, CLARRA
performs a fast preliminary retrieval to estimate cloud optical and
microphysical properties (Sect. 3.1). These are then used as first-guess
values in an iterative optimal nonlinear inverse retrieval (Sect. 3.2).
In preparation for running CLARRA, model atmosphere layer boundaries must be
chosen and the atmospheric profiles must be constructed (based on model and
measured data for the location and time of the downwelling radiance
spectrum). For this work, the same atmospheric profiles used to create the
simulated radiances are used (although errors are sometimes added). In
addition to uncertainty estimates for the observed radiance, the optimal
inverse retrieval requires a priori values for the optical and microphysical
properties and their covariance matrix. These can be taken from a
climatology or can be determined from the fast retrieval. In this work, the
statistics of the cloud properties used to create the simulated radiances
are used. Finally, the observed spectrum and associated covariance matrix
are needed (here, the simulated radiances with known errors are used). After
these preparations, CLARRA is run as follows.
Compute gaseous layer optical depths at monochromatic resolution.
Using the above and the temperature profile, calculate terms related to
emission and transmission by gases at the effective instrument resolution.
Retrieve cloud height (see Rowe et al., 2016), or alternatively input the
cloud height from another source.
Perform the fast retrieval that neglects scattering to get first-guess
optical and microphysical properties.
Perform the optimal iterative inverse method to retrieve cloud properties,
using the first-guess or previous iteration results, the a priori and covariance
matrix for the cloud properties, and the observed spectrum and its
covariance matrix.
Repeat step 5 until the result converges or a maximum number of iterations
is reached.
For step 1, gaseous layer optical depths are computed at monochromatic
resolution using LBLRTM. The cloud-height retrieval (step 3) was described
by Rowe et al. (2016). The fast retrieval (step 4), the optimal inverse
method (steps 5 and 6), and calculation of necessary terms (step 2) are
described below.
Fast preliminary retrieval
The preliminary retrieval provides a computationally fast estimate of cloud
properties. Cloud properties are retrieved from the absorption optical
depth, computed from the cloud emissivity, ignoring scattering. The fast
retrieval can be used to inform real-time decisions about measurements (e.g.,
duration of time to average spectra for noise reduction) as well as
providing estimates of cloud property statistics that can inform further
analysis. Cloud properties retrieved from the fast retrieval also serve as a
first guess for the iterative optimal inverse method described in the
following section, with the goal of enhancing performance by starting
iterations closer to the solution. Optionally, the fast retrieval results
can provide input statistics for the optimal inverse method (a priori means
and standard deviations). The description of the fast retrieval, below, can
be skipped without loss in continuity.
The cloud emissivity is approximated as in Rowe et al. (2016):
ε=Robs-RclrBctc+Rc-Rclr,
where Robs is the observed radiance, Rclr is the clear-sky
radiance, Bc is the Planck function of cloud temperature, tc is the
surface-to-layer transmittance, and Rc is the surface-to-layer clear-sky
radiance. All terms must be at the effective instrument resolution (as will
be discussed in Sect. 3.3 and the Appendix).
The cloud reflectivity is ignored so that the cloud emissivity is assumed to
be 1 minus the cloud transmittance. The natural logarithm of the cloud
transmittance is the cloud absorption optical depth, which can thus be
calculated from quantities that are measured or can be calculated
independently of the cloud properties:
τa,obs=ln1-Robs-RclrBctc+Rc-Rclr.
The value of τa can also be calculated from the state variables:
COD (τg), ice fraction (fice), effective radius of liquid
(rliq), and effective radius of ice (rice),
τa=τg/21-ficeQa,liqrliq+ficeQa,icerice.Qa,liq and Qa,ice are the absorption efficiencies of liquid
and ice, determined from the extinction efficiencies Qe and the single-scatter albedos ω0. For ice
Qa,ice=Qe,icerice1-ω0,ice,
where Qe,ice and ω0,ice are determined for averages over a
lognormal distribution of particle radii corresponding to the effective
radius rice. For the fast preliminary retrieval, spheres were assumed
for ice, and single-scattering parameters for each particle radius were
calculated from Mie theory using the index of refraction of Warren et al. (2008), based on a temperature of 266 K. For liquid, single-scattering
parameters determined from temperature-dependent indices of refraction at
temperatures of 240, 253, 263, and 273 K were used (Rowe et al., 2013;
Zasetsky et al., 2005; Wagner et al., 2005). Letting T1 be the temperature
from this list that is closest to but lower than the cloud temperature and
T2 be the temperature closest to but higher than the cloud temperature,
Qa,liq is given as the weighted sum:
Qa,liq=w1Qe,liqrliq,T11-ω0,liqrliq,T1+w2Qe,liqrliq,T21-ω0,liqrliq,T1,
where w1=(T2-Tc)/(T2-T1) and w2=(Tc-T1)/(T2-T1).
The values of Qe,liq, Qe,ice, ω0,liq, and
ω0,ice, are pre-computed for the full range of possible
rliq and rice. The COD (τg) is retrieved by inverse
retrieval (using Eqs. 6 and 7 below, but with R replaced with
τa,obs, F replaced with Eq. 3, and γ=0).
Next, τa,obs is calculated for the retrieved τg and
for a variety of values of fice (0.2, 0.4, 0.6, 0.8), rliq
(integers between 5 and 30), and rice (even numbers between 10 and 50).
Calculating τa,obs for all combinations of these values is
computationally fast compared to other aspects of CLARRA. Finally, the
values of fice, rliq, and rice are selected that correspond to
the minimum absolute difference between τa,obs and τa.
Optimal nonlinear inverse method
The optimal nonlinear inverse method iteratively retrieves cloud properties
(COD, fice, rliq, and rice), using the results of the fast
retrieval as a first guess. The inverse method uses radiances from 400 to
600 cm-1 (allowing thermodynamic phase determination; Rathke et al.,
2002a) and from 750 to 1300 cm-1, which is sensitive to phase, COD, and
effective radius. Similar optimal nonlinear inverse methods have been used
to retrieve cloud properties from AERI instruments in the Arctic (Turner,
2005; Cox et al., 2014) and from satellite instruments (Wang et al., 2016; Poulsen et al., 2012; L'Ecuyer et al., 2019). Cloud
properties are retrieved from observed radiances averaged in microwindows
(see Table 2). The remainder of this section provides additional details
about the optimal nonlinear inverse method.
Microwindows used in the optical and microphysical cloud property
retrievals. The first column gives the central wavenumber, and the second column
gives the microwindow width for resolutions of 0.1 to 4 cm-1. For
resolutions of 8 cm-1, some microwindows were widened slightly so that
there was at least one point in the microwindow (a few were narrowed so that
there was only one point). Two sets of microwindows were used in this work:
a combination of those used by Rathke and Fischer (2000) and Mahesh et al. (2001),
indicated with superscripts R and M, and microwindows similar to those used
by Turner (2005), consisting of all wavenumbers in plain font (e.g., not
bold).
The inversion equation used here is the iterative Levenberg–Marquardt method
(Rodgers, 2000, and references therein),
xi+1=xi+1+γiSa-1+KiTSe-1Ki-1KiTSe-1R-Fxi-Sa-1xi-xa,
where x is the state vector, with a priori
xa and covariance matrix
Sa. The subscript i indicates the iteration number and
R is the observation, with covariance matrix
Se. F is the forward model (described
below), and the kernel (K) is the Jacobian matrix, computed
numerically by perturbing each state variable in turn and rerunning
F.
The Levenberg–Marquardt formulation is a hybrid of the Gauss–Newton
formulation and the method of steepest descent, with γ=0
defaulting to Gauss–Newton. As γ increases, Eq. (6) becomes more
heavily weighted towards steepest descent and convergence slows. Choosing
γ is difficult, as a large value of γ will slow the
retrieval. Here we start with γ=0. Each time the current
iteration causes the root-mean-square (rms) error between measurement and
forward model result to increase in magnitude by more than 1 RU, or by more
than double the current error, γ is increased (first to γ=1 and then) by a factor of 10; the retrieval is then repeated with the
new γ. After increasing γ, if a subsequent iteration does
not increase the rms error as described above, γ is decreased by a
factor of 10. Iterations are repeated until γ<0.01 or the
maximum allowed number of iterations is reached.
Error in the retrieved state variable is given by the covariance matrix
S=KTSe-1K+Sa-1-1.
Note that this equation applies only when γ=0. We find that our
criterion of γ<0.01 results in negligibly different
retrievals than for γ=0. Convergence is tested using
di2=xi-xi+1TS-1xi-xi+1≪n,
(Rodgers, 2000), where n is the length of x.
In this work, the observation R is derived from the simulated
spectra described in Sect. 2 by averaging radiances in microwindows between
strong gaseous emission lines. Microwindows used in this work for
resolutions of 0.1 to 4 cm-1 are shown in Table 2. They span 3–10 cm-1 and include at least one radiance (wavenumber spacing is
equivalent to resolution). For retrievals at 8 and 20 cm-1, the
closest measurement point to each central microwindow frequency was used.
Using radiances in microwindows minimizes the contribution by gases,
increasing sensitivity to cloud and reducing errors. However, due to the
finite resolution, gas emission from outside the microwindow is convolved
into radiances within the microwindow. For example, at a resolution of 0.1 cm-1, in a microwindow of 4 cm-1, the contribution from gaseous
absorption lines outside the microwindow will be minimal. As resolution gets
coarser, the gaseous absorption lines bordering the microwindow contribute
more and more, potentially decreasing sensitivity and increasing errors.
The state vector x is composed of COD, ice fraction, log of the
effective radius of liquid, and log of the effective radius of ice, so that
n=4. For the a priori (xa), means of the values of x used
to create the base set are used (Table 1). The covariance matrix
Sa is assumed to be diagonal, with diagonal elements based on
a standard deviation of about one-half the range of values; this is used
rather than using the standard deviations given in Table 1 to weight the
retrieval heavily toward the measurement rather than the a priori. The error
covariance matrix for radiance (Se) is assumed to be diagonal
with elements based on the model errors described in the next section and
the measured and simulated radiance errors due to any imposed errors, added
in quadrature. The first guess values (i=0) are determined from the fast
cloud property retrieval. The maximum number of allowed iterations was set
to 20 and the tolerance for convergence was set to d2<1. For
convenience, the result of the forward model acting on the retrieved state
vector is termed the retrieved radiance.
The forward model (F) is calculated by running DISORT with the
state variables and with effective-resolution gaseous optical depths
(described below). Other inputs to DISORT include the solar contribution,
surface albedo, temperature profile, and the Legendre moments that describe
the phase function, single-scatter albedo, and COD, which depend on the
state variables and cloud height. DISORT is run with 16 streams. Single-scattering properties were the same as for the fast preliminary retrieval.
Resolution and model errors
In this work, DISORT was used for both simulating the observed radiances
and for the forward model F. DISORT requires gaseous layer optical
depths, which are calculated more accurately for observed radiances compared
to those used in F. Gaseous layer optical depths computed by LBLRTM
are at monochromatic or perfect resolution and a fine wavenumber spacing,
and DISORT must be run for each wavenumber, after which the radiance must be
convolved to instrument resolution. This was done to simulate the
observations but is too computationally intensive for the iterative inverse
retrieval (i.e., for F). Instead, we develop a novel method for
producing effective-resolution gaseous layer optical depths (given in the
Appendix) so that DISORT need only be run for each microwindow.
Model errors arising from these differences are shown in Fig. 1b, as box-and-whisker plots of model errors for cloudy-sky radiances at 0.5 cm-1
resolution, in microwindows used in the cloud optical and microphysical
property retrievals. The errors were calculated as differences between
downwelling radiances calculated using the effective-resolution layer
optical depths (described in the Appendix) and monochromatic radiances
convolved with the instrument line shape (the radiance simulations described
in Sect. 2), and averaged in microwindows. At 0.5 cm-1 resolution,
median model errors are within ±0.02 RU (1 RU = 1 mW/(m2 sr cm-1)). For resolutions of 0.1 to 2 cm-1, all model errors are
within ±0.15 RU (figures for other resolutions are given in the
Supplement). For resolutions of 4 to 20 cm-1, model errors generally
increase with coarsening resolution, with maximum errors of -0.7 to 1.0 RU
at 20 cm-1 resolution (Supplement).
Another source of model error is related to the cloud-height retrieval. The
cloud-height retrieval also uses effective-resolution terms: the gaseous
radiance and the transmittance from the surface up to each possible cloud
layer (Rc and tc) and the clear-sky radiance (Rclr), described
in Rowe et al. (2016). Derivation of these quantities is given in the
Appendix. Model errors for a typical clear-sky radiance used in the cloud-height retrievals are also shown in Fig. 1b (solid blue curve); the error
shown is the difference between Rclr calculated in this work (as
described in the Appendix) and the monochromatic radiance from LBLRTM
convolved with the instrument line shape. As the figure shows, model errors
for clear skies are typically very low.
Imposed errors
To determine the impact of sources of error on the cloud property
retrievals, various errors were imposed on observed radiances, including
Gaussian noise (mean of 0.2 RU) and bias (±0.2 RU). In remote
locations, reanalysis datasets may be used for specification of the
atmospheric state. Wesslen et al. (2014) characterized temperature errors in
the European Centre for Medium-Range Forecasts (ECMWF) Interim
(ERA-Interim; Dee et al., 2011) as varying from -0.5 to 1 K. Rowe et al. (2016) found such errors to have a roughly equivalent effect on radiative
transfer calculations as a positive temperature bias of 0.2 K. Wesslen et
al. (2014) characterized water vapor errors to be 2 % to 10 %, with
lower biases in the first 3 km and higher biases above. Because water vapor
decreases rapidly with height, this was found to be roughly equivalent to a
water vapor bias at all heights of 3 % (Rowe et al., 2016). Thus, imposed
errors also included biases in the atmospheric temperature (±0.2 K)
and water vapor (±3 %). Higher biases in water vapor and
temperature were also tested (±10 % and ±1 K). Cloud optical
and microphysical properties were retrieved with these errors each imposed
in isolation, using both true cloud heights and cloud heights retrieved with
CO2 slicing as described in Rowe et al. (2016).
In addition to errors imposed in isolation, various combinations of the
above sources of errors were imposed on retrievals, as described in Sect. 5
below.
Results and discussionRetrieval overview
Use of the fast retrieval as a starting point for the inverse retrieval was
found to have a variety of benefits. The fast retrieval reduced rms errors
relative to the a priori: from 300 % to 6 % for τg, from 0.4 to 0.2
for fice, from 4.4 to 3.7 µm for rliq, and from 16 to 11 µm for
rice. This provided a first-guess for the inverse retrieval that was
closer to the solution, lowering retrieval errors slightly, modestly
increasing the number of cases that converged, and preventing convergence to
an incorrect solution for a few cases. Overall, the greatest improvement
from using the fast preliminary retrieval was reducing computation time; on
average, one fewer iteration was needed when the fast retrieval was used.
Figure 2 shows the inverse-retrieval trajectory, with iterations, for an
ice-only cloud with a COD of 0.89 and effective radius of 22 µm. The
retrieval trajectory is superimposed on error contours (root-mean-square
radiance differences). As the figure shows, the retrieval converged from the
first-guess value (red dot on right in each panel), to the minimum in four
iterations. Furthermore, the retrieval correctly converged to an ice-only
cloud, although the mean cloud temperature of ∼256 K falls
within the range of temperatures where mixed-phase clouds may occur.
Error contours for retrievals of ice effective radius
(rice), ice fraction, and cloud optical depth, as root-mean-square error
in radiances for an ice-only cloud. The retrieval trajectory (red line) and
results for each iteration (red dots) are superimposed on the contour
surface.
Retrievals using the base set of simulations indicate that the kernels are
typically sufficiently linear to converge on the solution, except for large
COD and effective radii. We find that the retrievals lose sensitivity to COD
between about 5 and 10 (see Sect. 5.2 below); in previous work retrieving
cloud properties from downwelling IR radiances in a similar wavenumber
range, cutoffs of 4 to 6 were used (Mahesh et al., 2001; Rathke, 2002a, b;
Turner, 2005). The retrievals were found to lose sensitivity to effective
radius above about 50 µm (see Supplement), which is in keeping with
Rathke and Fischer (2000) and Garrett and Zhao (2013), but differs from the
cutoff values of 25 µm used by Mahesh et al. (2001) and of 100 µm
by Turner (2005). In addition, when values approach these limits, the
retrieval was found to sometimes move away from the solution. To avoid this,
upper bounds were set for the COD (10) and effective radius (50 µm),
and the kernels were typically calculated for a step in the direction of
smaller COD and effective radius, that is, in the direction where
sensitivity is larger.
Nearly all retrievals converged to within the specified tolerance in
d2, with only zero to two cases failing to converge for any set of imposed
errors. Overall, convergence was achieved in a mean of four iterations (median
of three). At most two cases failed to converge within 20 iterations for any set
of imposed errors.
Retrieval errors
To determine the retrieval capability, errors in retrieved values are
examined in the absence of any imposed errors, where only model errors are
present. Table 3 shows errors in retrieved cloud properties (τg,
fice, rliq, and rice) for the base set of spectra, for spectral
resolutions of 0.1, 0.5, and 4 cm-1. Retrieval errors are shown for
different ranges of τg. For thin clouds (τg < 0.4), the low signal reduces sensitivity. For thick clouds (τg > 5), the spectrum begins to approach saturation, and sensitivity
to cloud optical and microphysical properties diminishes. Thus, both large
and small τg values can result in large errors in fice, rliq,
and rice (such increases are not seen for τg > 5
in Table 3 but occur when errors are imposed). By contrast, error in τg increases with increasing τg and is smallest for the
thinnest clouds. Based on these considerations, the ideal range for τg was identified as 0.4 < τg < 5. (To get a
sense of how common such clouds are, Cox et al. (2014) found that at Eureka,
Nunavut, in 2006–2009, clouds with optical depths of 0.25 to 6 accounted for
about 32 % of AERI measurements, 17 % when quality control procedures
and a PWV threshold of 1 cm were applied; in this work PWV is as high as 3 cm.) Unless otherwise specified, results will be presented for this range.
Retrieval errors for 0.4 < τg < 5 are overall
quite low, with magnitudes of errors in τg below 0.013, in
fice below 0.03, in rliq below 0.7 µm, and in rice below 4 µm. Overall, the table shows no trend in retrieval errors with
coarsening resolution for 0.4 < τg < 5.
Root-mean-square errors in retrieved cloud properties for base set
of spectra due to model error only (no errors imposed) for spectral
resolutions indicated. Errors are shown for cloud geometric optical depth
(τg), ice fraction (fice), effective radius of liquid
(rliq), and effective radius of ice (rice) for four ranges in
τg.
Retrieval accuracy was tested for two sets of microwindows. Set 1 consists
of 22 microwindows similar to those used by Turner (2005), indicated in
Table 2 in plain (non-bold) font; these were used in the retrievals
described below. Set 2 consists of the combined microwindows of Rathke et al. (2000) and Mahesh et al. (2001), indicated in Table 2 with superscripts R and
M (11 microwindows). Retrieval errors were found to be slightly lower for
set 1; therefore it is used in the remainder of this work. However,
differences were small (compare Table 4, described below, to Table S1 of the
Supplement), indicating that a smaller set of microwindows is likely
sufficient. Choice of optimal microwindows depends on noise level and
spectrally varying errors (e.g., due to errors in assumed profiles of
atmospheric water vapor and chlorofluorocarbons) and is therefore a
complicated but interesting topic for future work.
Errors in retrieved cloud properties for different imposed errors are given
in Table 4 for a spectral resolution of 0.5 cm-1 and τg
between 0.4 and 5. Magnitudes of imposed errors are given in the first
column except for cases of combined errors. Error combination (a) includes
noise of 0.2 RU, radiation bias of 0.2 RU, temperature bias of 0.2 K, and
water vapor bias of -3 %, and uses true cloud heights. Combination (b) is
the same but with opposite signs on biases. Combinations (c) and (d) are the
same as (a) and (b), respectively, but use retrieved cloud heights (similar
sets but with radiation biases of 0.5 RU are given in Table S2 of the
Supplement). Subsequent columns give the mean errors and the standard
deviations of the errors.
Errors in retrieved cloud properties (mean error and standard
deviation of error; SD; COD refers to cloud optical depth in the geometric
limit, rliq and rice are the effective radii of liquid and ice)
for various errors imposed on the observations (see text).
When true cloud heights are used, errors in τg are within ±0.2 for large biases imposed on the observed radiation, temperature, and
water vapor, (±1.0 RU, 1 K, and 10 %, respectively) or combined
errors, and within ±0.09 for smaller imposed biases (±0.2 RU,
0.2 K, and 3 %, respectively). Large imposed errors also lead to large
errors in fice, making it difficult to distinguish liquid and ice.
Errors in rice are typically 2 to 3 times larger than errors in
rliq. Mean errors reveal how biases in measured radiance, water vapor,
and temperature lead to biases in retrieved cloud properties. For example,
positive biases in observed radiances lead to negative biases in COD,
rliq, and rice, and positive biases in ice fraction, while the
reverse is true for negative biases in observed radiance.
When cloud heights are retrieved from the observed radiances (columns
labeled CO2 slicing and combined errors (c) and (d)), errors in cloud
height lead to biases in inferred cloud temperature. Biases in cloud
temperature cause errors that are spectrally flat. Because cloud emissivity
depends fairly linearly on τg, spectrally flat errors have a
large effect on τg. Furthermore, in the cloud-height retrieval
(CHR), the cloud is placed in the atmospheric model layer containing the
cloud height retrieved with CO2 slicing. This means that errors in COD
are also affected by the choice of atmospheric layering. One approach to
improving cloud temperature and optical depth is the geometric method of
Rathke et al. (2002b), for which the instrument would be designed to look at
multiple angles; this can also be used to examine the horizontal homogeneity
of clouds.
Additional work is needed to understand the effects of CHR errors on cloud
optical and microphysical property retrievals, for several reasons. First,
Rowe et al. (2016) found that CHR errors for CO2 slicing were most
sensitive to biases in observed radiance and temperature, with less
sensitivity to noise and biases in water vapor. By contrast, for an
alternate CHR method (MLEV) these sensitivities were found to be the
opposite. Since CHR errors translate into errors in retrieved COD,
it is important to choose the CHR method to use based on expected error
magnitudes. Second, Rowe et al. (2016; see, e.g., Fig. 7) found that CHR errors
generally decrease with increasing cloud signal, which should oppose the
tendency of optical and microphysical property retrieval errors to grow with
increasing COD. Finally, Rowe et al. (2016; Fig. 7) found that CHR
errors generally decrease with decreasing cloud height. Here we find
important consequences for retrievals of COD and fice. For example, when
errors are imposed (noise of 0.2 RU, radiation bias of 0.2 RU, temperature
bias of 0.2 K, water vapor bias of -3 %, and CHR errors in cloud height,
for spectra at 4.0 cm-1 resolution), comparing clouds with bases
above 2 km to those with bases below, rms errors in retrieved COD decrease
from 1.1 to 0.15, errors in fice decrease from 0.3 to 0.18, and errors
in rice decrease from 10 to 7 µm (errors remain at 2 µm for
the effective radius of liquid).
Root-mean-square error in retrieved cloud properties as a function
of resolution, where rliq is the effective radius of liquid and
rice is the effective radius of ice, for cases with and without imposed
error, as described in the text.
Errors in retrieved cloud properties are shown as a function of resolution
from 0.1 to 20 cm-1 for clouds with bases below 2 km, in Fig. 3.
Errors are shown for base cases with no imposed error and for a combination
of imposed errors: noise of 0.2 RU, radiation bias of 0.2 RU, temperature
bias of 0.2 K, water vapor bias of -3 %, and CHR errors in cloud height.
No trend is seen in retrieval errors for resolutions of 0.1 to 4 cm-1,
after which errors increase. For clouds with bases above 2 km, errors are
larger for optical depth and ice fraction (Fig. S4 of the Supplement),
and trends with resolution are similar but less pronounced. (Scatter plots
of true vs. retrieved cloud properties are given in Figs. S5 and S6 of the
Supplement.) Based on these trends, an instrument resolution of 4 cm-1 seems to be a good compromise for reducing resolution while
avoiding increases in retrieval errors. For example, at 0.5 cm-1 (for
clouds at all heights), rms retrieval errors are 0.6 for COD, 0.2 for
fice, 3 µm for rliq, and 8 µm for rice; at 4 cm-1
they are nearly the same (0.6, 0.2, 2, and 8 µm, respectively).
Retrieval error covariance matrix
Discussion of errors so far has focused on actual retrieval errors, which
can be calculated because simulated data were used as the observation set.
For real measurements, error analysis relies on the covariance matrix
S, which in turn depends on the kernels and covariance
Se (Eq. 8); Se is calculated by adding
measurement and forward model errors in quadrature; model errors are
determined from errors in water vapor or temperature profiles. Here we
determine how well S represents retrieval errors. For unbiased,
normally distributed errors, the diagonals of S should correspond
to the 68 % confidence interval. We can test this by comparing retrieval
errors to the diagonal of S. This is complicated by the fact that
S is not constant but depends on x (because the kernels
depend on x). Thus for each retrieved x, the absolute
error was divided by the square root of the appropriate diagonal element of
the corresponding S. For Gaussian errors, this ratio should be
< = 1 for 68 % of retrievals (and < = 2 for 95 % of
retrievals). In the absence of imposed error, only 52 % to 63 % of
retrievals had a ratio within 1 (for Se based on model
errors). The lowest model errors are likely underestimates since it is
unlikely all sources of error in the forward model were captured. A minor
increase in model error (0.03 RU) gave values between 68 % and 77 %.
However, the error distributions were found to decrease more slowly than
Gaussians, with only 78 % to 87 % of errors (rather than 94 %) falling
within the second standard deviation indicated by S.
For imposed noise of 0.2 RU, only 52 % to 58 % of retrievals were found to
have a ratio within 1, suggesting that model errors are amplified in the
presence of error. This is likely because away from the correct solution,
the estimate of S is incorrect. Increasing the contribution of
noise to Se by 30 % accounted for this, resulting in values
of 65 % to 70 %.
S was found to provide a poor indication of retrieval errors due to
biases in radiance, temperature, water vapor, or cloud height. This is
likely because the inverse retrieval is based on an assumption of unbiased,
normally distributed errors. For biases in radiance and water vapor and for
errors in cloud height, S is particularly nonrepresentative for
COD, for which only 11 % to 25 % of cases fall within 1 standard deviation
for S (for other properties the range is 36 % to 78 %). Biases in
temperature affect S similarly for COD, fice, rliq, and
rice (range of 48 % to 66 %). This underscores the importance of
removing bias errors from measurements whenever possible to ensure that
S provides the best possible representation of errors.
Cloud vertical inhomogeneity and ice habit
Errors in retrieved cloud properties (from spectra at 0.5 cm-1
resolution) due to failing to capture cloud vertical inhomogeneity are shown
in Table 5. For the upper set of cases shown in the table, errors were not
imposed and true cloud heights were used. In performing the retrieval the
correct cloud base and top were used, but the cloud was assumed to be
vertically homogeneous in terms of COD and phase; thus the cloud model is
accurate for dense and diffuse clouds but not for inhomogeneous or
liquid-topped clouds. This emulates a measurement where the cloud base and
top are known from an ancillary instrument such as a lidar. As expected,
therefore, errors are similar for dense and diffuse clouds. For
inhomogeneous clouds, which are thinner at the upper and lower edges, errors
are slightly larger for τg. The largest retrieval errors are
found to be for liquid-topped clouds, particularly for τg and
fice, for which errors are about 5 times as large. These errors are
large because the cloud heights are effectively wrong for the liquid and ice
layers of the cloud. A lidar that can classify phase would allow reduction
of these errors down to the level seen for other cloud types. The
enhancement of errors in liquid-topped clouds relative to other cloud types
disappears when errors are imposed on the observations (imposed noise of 0.2 RU, radiation bias of 0.2 RU, temperature bias of 0.2 K, and water vapor
bias of -3 %; see the last two sets of cases in Table 5). This is true
when true cloud heights are used (middle set) and when they are retrieved
(lowest set). (Similar trends are found when the radiation bias is increased
to 0.5 RU, as shown in the Supplement.)
Radiance errors (1 RU = 1 mW/(m2 sr cm-1)) for
different methods of approximating the radiance at a resolution of 0.5 cm-1. Approximate radiances are computed using mean perfect-resolution
layer optical depths R(〈Δτ̃〉),
mean surface-to-layer perfect-resolution transmittances R(〈t̃〉), or mean surface-to-layer transmittances after
convolution to the instrument R(〈t〉); averages are over
0.5 cm-1(a) or over microwindows (b). Approximate radiances are
compared to simulated radiances at 0.5 cm-1 resolution (Rclr),
which are averaged over microwindows in (b).
Root-mean-square errors in retrieved cloud properties for
vertically varying clouds: cloud optical depth (COD), ice fraction
(fice), and effective radii of liquid and ice (rliq and
rice). For the upper set of cases, errors were not imposed on
observations (error =n) and true cloud heights were used. The middle set
of cases includes imposed errors with true cloud heights (error =y),
while the lowest set includes imposed errors with retrieved cloud heights
(error =y*; see text).
Root-mean-square errors in retrieved cloud properties, assuming a
spherical ice habit, for ice clouds of varying habit (first column): cloud
optical depth (COD), ice fraction (fice), and effective radius of ice
(rice). For the upper set of cases, errors were not imposed (error =n) and true cloud heights were used. The middle set of cases includes
imposed error with true cloud heights (error =y), while the lowest set
includes imposed errors with retrieved cloud heights (error =y*; see
text).
Errors in retrieved cloud properties (from spectra at 0.5 cm-1
resolution) due to assuming a spherical ice habit are shown in Table 6. The
first column of the table shows the true ice habit. The upper set of data
has no other imposed errors, while the lower two sets have the same imposed
errors as for vertically varying clouds. Retrieval error in rliq is not
shown because clouds were mainly ice. In the absence of imposed errors,
compared to spheres, the increase in error is greatest for τg,
for which errors increase by an order of magnitude or more. This large
increase suggests that errors in habit mainly bias the magnitude rather than
spectral shape of the cloud emissivity. Overall, errors are the smallest for
solid columns. However, differences in errors based on assumed ice habit
diminish when errors exist in observations and cloud heights are retrieved
(bottom set). Thus, using a realistic ice habit can minimize errors, but
this becomes less important when cloud height is also retrieved.
Conclusions
This work explores the capability of a low-resolution IR spectrometer for
retrieving cloud properties in polar regions. To this end, the CLoud and
Atmospheric Radiation Retrieval Algorithm (CLARRA) was used to retrieve
cloud properties (height, following Rowe et al., 2016, as well as COD, ice
fraction, effective radius of liquid, and effective radius of ice) from
simulations of surface-based IR downwelling radiances, to determine the
effect of instrument resolution on accuracy. CLARRA includes a method for
calculating gaseous transmission and emission terms at the effective
instrument resolution, minimizing model errors. A fast-forward retrieval
rapidly retrieves preliminary cloud optical and microphysical properties,
which then serve as inputs into an optimal nonlinear inverse method. Cloud
properties were retrieved from 222 simulated radiances based on atmospheric
and cloud conditions characteristic of the Arctic, with additional tests of
sensitivity to cloud vertical inhomogeneity and ice habit.
Sensitivity studies for vertically varying clouds indicate that, in the
absence of observational errors, errors in retrieved cloud properties are
highest for liquid-topped clouds that are assumed to be homogeneously
mixed phase (relative to clouds that are dense, diffuse, or inhomogeneous
vertically). However, in the presence of errors in observations, the gap in
retrieved cloud-property errors between liquid-topped clouds and other cloud
structures disappears. Future work is needed to assess errors when multiple
cloud layers are present. For different ice habits, sensitivity studies
indicate that use of a reasonable guess for the ice habit can help minimize
errors, but these differences become minor in the presence of observational
errors.
Retrieval accuracy was determined as a function of resolution for model
errors, CHR errors, and a variety of imposed observational errors, including
random noise as well as biases in the measured spectrum and atmospheric
state. In the absence of imposed errors, errors in retrieved cloud
properties were found to be 0.007 for COD, 0.03 for fice, 0.7 for rliq, and 3 µm for rice (0.5 cm-1 resolution; COD
between 0.4 and 5). In the presence of imposed errors, errors in retrieved
COD and ice fraction were found to be strongly affected by bias errors in
cloud height, which in turn are high when the CHR is used. Furthermore, CHR
errors typically decrease with decreasing cloud base height (Rowe et al., 2016), with consequences for optical and microphysical property retrievals.
For example, for a combination of errors including noise of 0.2 RU,
radiation bias of 0.2 RU, temperature bias of 2 K, water vapor bias of
-3 %, and CHR errors (at 4.0 cm-1 resolution), comparing clouds with
bases above 2 km to those with bases below, the rms error decreases from 1.1
to 0.15 for COD and from 0.3 to 0.18 for fice, pointing to a strong
potential for retrievals of low clouds.
Retrieval errors were found to be fairly invariant to resolution up to about
4 cm-1, after which accuracy declined. For example, at 0.5 cm-1
resolution, for the combination of errors given above, rms retrieval errors
(for clouds at all heights) are 0.7 for COD, 0.2 for fice, 3 µm for
rliq, and 8 µm for rice. At 4 cm-1 these errors are
similar (0.6, 0.2, 2, and 8, respectively). Taken together, this lack of
sensitivity to resolution indicates that a moderately low-resolution
(∼4 cm-1) surface-based IR spectrometer could provide
cloud property retrievals with accuracy comparable to existing higher-resolution instruments. Furthermore, these retrievals would be particularly
useful for low-level clouds, for which accuracy is likely to be highest.
Code availability
Simulated radiances at monochromatic resolution (Cox et al., 2015) are
available by email to the corresponding author. Computer code is available
at Bitbucket
(https://bitbucket.org/{4e9c3a2c-5ac5-40f9-a7e4-0c9578f88b21}, Rowe, 2019),
including repositories containing Python computer code
(runDisort_py)
and MATLAB/Octave computer code (runDisort_mat)
for creating cloudy-sky spectra using DISORT (Stamnes et al., 1988). See also
Rowe et al. (2013, 2016).
Approximations for cloud-height retrievals
To solve the radiative transfer equation in LBLRTM and DISORT (Stamnes et al., 1988), the atmosphere is divided into model atmospheric layers and the
approximation is made that the Planck function varies linearly with optical
depth through the layer (Wiscombe et al., 1976; Clough et al., 1992). In the
absence of scattering, the downwelling radiance from a layer at a given
wavenumber is approximated as
ΔR̃L=∫τ̃L-1τ̃LB̃τ̃e-τ̃secθdτ̃,
where the tildes indicate monochromatic, or perfect, resolution (all
quantities with tildes depend on wavenumber), τ is defined as the
vertical optical depth from the surface up to some height (e.g., within layer
L), τL-1 is from the surface to the layer bottom, and τL is from the surface to the layer top. (Parentheses are used here and
below to indicate dependence.) B̃ is the Planck function and
θ is the viewing angle from zenith. Note that the formulation here
differs from that of Clough et al. (1992); here, RL, τL, and
the transmittance, tL (defined below) are defined from the bottom of the
model atmosphere (e.g., from Earth's surface) to the top of layer L.
Quantities that are for layer bottom to top only are indicated with a delta.
Using these conventions means that Eq. (A1) represents the radiance from
layer L that is transmitted by the atmosphere below to the surface. The
viewing angle is included explicitly here so that τ refers to the
vertical optical depth.
The surface-to-layer-top transmittance depends on the optical depth,
t̃L=exp(-τ̃Lsecθ).
The linear-in-optical depth approximation for B allows the integral to be
solved, yielding
ΔR̃L=-B̃Lt̃L+B̃L-1t̃L-1-ΔB̃LΔt̃LΔτ̃L,
where B̃L-1 and B̃L are the Planck functions of the
temperature at the lower and upper boundaries of layer L, and the deltas
indicate the change across the layer. (Note that ΔR̃L is
calculated slightly differently in LBLRTM, following Clough et al., 1992; the
two methods give similar results.)
Thus ΔRL is the radiance from the layer that makes it to the
surface. The total (clear-sky) radiance is the sum of all the layer
radiances. To match instrument resolution, the clear-sky radiance needs to
be convolved with the instrument line shape S,
Rclrν=∫-∞∞∑LΔR̃L(ν̃)S(ν,ν̃)dν̃,
where the dependence on wavenumber has been included explicitly. Equation (A4)
can also be calculated directly by running LBLRTM and convolving with the
S (typically a sinc function). We will use Rclr calculated in this manner
to test the remaining approximations.
In practice the integral need only be performed over the small wavenumber
region characterized by the width of S (typically a sinc function). Switching
the order of the sum and the integral, we have
Rclrν≈∑LΔRL(ν),
where
ΔRL(ν)≡∫-∞∞ΔR̃L(ν̃)S(ν,ν̃)dν̃.
In addition to Rclr, the cloud-height retrieval (Rowe et al., 2016)
requires the gaseous radiance from the surface up to each possible cloud
layer (Rc), which can also be calculated from ΔRL,
Rc≈∑L=1cΔRL.
Finally, the cloud-height retrieval requires the transmittance of the
atmosphere below the cloud (tL; in Rowe et al., 2016, it is referred to as
tc) at the effective instrument resolution. Examining Eqs. (A1)–(A6) shows
that it is more accurate to convolve the Planck function multiplied by the
surface-to-layer transmittance. Thus we define the effective transmittance
from the surface to a layer as
tLν≡∫-∞∞B̃Lν̃t̃Lν̃Sν,ν̃dν̃/BL(ν).
To summarize how these approximations are used for the cloud-height
retrieval, first gaseous layer optical depths Δτ̃L
are computed using LBLRTM. Next, Δτ̃L values are summed from
the surface up to each layer to get τ̃L. Equation (A2) is then
used to calculate t̃L, and Eqs. (A3) and (A6) are used to
calculate ΔRL. Equation (A5) is used to calculate
Rclr, and Eq. (A7) is used to calculate Rc for each model
layer that could contain cloud (for cloud heights within layers, terms are
interpolated). Equation (A8) is used to calculate tL.
Approximations and model error for cloud optical and
microphysical property retrievals
Retrieval of optical and microphysical cloud properties requires
effective-resolution layer optical depths, ΔτL, as input
into the DISORT radiative transfer code. One method to create the set of
ΔτL might be to reduce the resolution of the layer
optical depths. However, the above equations suggest that a more accurate
method would be in terms of transmittances. Inserting Eq. (A2) into Eq. (A8)
and breaking up the integral gives
ΔRLν=-∫-∞∞B̃Lt̃LSν,ν̃dν̃+∫-∞∞B̃L-1t̃L-1Sν,ν̃dν̃-∫-∞∞ΔB̃LΔt̃LΔτ̃LS(ν,ν̃)dν̃.
The first two terms on the right-hand side of this equation have the same
form as the integral in Eq. (A8) and can be replaced with
-BLtL and BL-1tL-1. Thus it makes sense to create the set of
ΔτL using tL (noting that the third term in Eq. A9
also includes the monochromatic layer gaseous optical depth and thus
represents a source of error).
Due to ringing, tL can be greater than 1 or less than 0, resulting in
optical depths outside physical bounds. To minimize ringing, transmittances
were averaged over small spectral regions between strong emission lines, or
microwindows (Table 2). Observed radiances are therefore also averaged over
microwindows. (Note that it might be more accurate to average the term in
brackets in Eq. A8; an alternate option would be to use an apodization
function rather than a sinc function in Eq. A8 to reduce ringing; these
are both interesting topics for future work.) Following this, transmittances
below 10-40 and above 1 were modified such that 10-40<=tL<=1.
Finally, layer optical depths are calculated from tL. For the first layer,
Δτ1≡-log(t1).
For subsequent layers, the optical depths of all layers below must be
subtracted.
ΔτL≡-log(tL)-∑x=1L-1Δτx.
The advantage of approximating radiances using layer optical depths derived
from surface-to-layer transmittances convolved to the instrument resolution
is shown in Fig. 4. Errors for this convolved-transmittance (CT) method are
compared to errors for radiances calculated using optical depths derived
from averaged monochromatic surface-to-layer transmittances (or from
averaged monochromatic layer optical depths). In Fig. 4a averages are over
the wavenumber spacing (no averaging is needed for the CT method, for which
the spectra are already at the appropriate wavenumber spacing). Figure 4b shows
errors for averages over microwindows. Errors are determined by comparison
with Rclr calculated as in Eq. (A4), using LBLRTM and then convolving
to the desired resolution (0.5 cm-1 here) and (in b) averaged over
microwindows. Errors are reduced significantly by the CT method, relative to
the other approximations. In microwindows (Fig. 4b), errors are within 3 or
30 RU for the other methods, whereas for the CT method they are <=0.01 (for this example, the range for cloudy cases used in this work is
shown in Fig. 1); thus the CT method represents a significant improvement.
Finally, it is worth noting that errors at instrument resolution are also
fairly low (Fig. 4a). This is shown here for reference only, and is not used
in this work, but has the potential for use in a cloud-height retrieval that
includes scattering, using DISORT.
To summarize the approximations used for the cloud optical and microphysical
property retrievals, the set of effective-resolution gaseous layer optical
depths needed for running DISORT is calculated as follows. The first few
steps are the same as for the cloud-height retrieval: Δτ̃L values are computed using LBLRTM and these are summed from the surface to
each layer to get τ̃L, and Eq. (A2) is used to calculate
t̃L. Next, Eq. (A8) is used to calculate tL, which is then
averaged over microwindows and bounded to be between 10-40 and 1. Equations (A10) and (A11) are then used to calculate ΔτL. Since
DISORT is run at single precision, serious errors can result for very small
input optical depths; thus ΔτL was increased as needed
such that ΔτL>=10-5.
The supplement related to this article is available online at: https://doi.org/10.5194/amt-12-5071-2019-supplement.
Author contributions
SN calculated Legendre moments and single-scatter albedo from
single-scattering parameters. CC led creation of simulated
spectra used in this work. VPW conceived of the idea and provided
guidance. PR performed all other calculations and wrote the
paper with input from all authors.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
Computer code was
written in the Python computing language and figures were created using
Matplotlib. We are grateful for help with Python coding from Daniel
Neshyba-Rowe and for advice on running DISORT from Istvan Laszlo.
Financial support
This research has been supported by the National Science Foundation, Division of Arctic Sciences (grant no. 1108451), the National Science Foundation, Office of Polar Programs (grant no. 1543236), and the National Science Foundation, Division of Chemistry (grant no. 1807898).
Review statement
This paper was edited by Andrew Sayer and reviewed by Bryan A. Baum and D. D. Turner.
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