AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-11-4273-2018Parameterizing cloud top effective radii from satellite retrieved values,
accounting for vertical photon transport: quantification and correction of the
resulting bias in droplet concentration and liquid water path retrievalsQuantifying and correcting the effect of vertical penetration assumptionsGrosvenorDaniel P.daniel.p.grosvenor@gmail.comhttps://orcid.org/0000-0002-4919-7751SourdevalOdranhttps://orcid.org/0000-0002-2822-5303WoodRoberthttps://orcid.org/0000-0002-1401-3828School of Earth and Environment, University of Leeds, Leeds, LS2 9JT, UKNational Centre for Atmospheric Science (NCAS), University of Leeds, Leeds, LS2 9JT, UKLeipzig Institute for Meteorology, Universität Leipzig, Leipzig, GermanyDepartment of Atmospheric Sciences, University of Washington, Seattle, USADaniel P. Grosvenor (daniel.p.grosvenor@gmail.com)20July20181174273428922December20174January20184January201818June2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://amt.copernicus.org/articles/11/4273/2018/amt-11-4273-2018.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/11/4273/2018/amt-11-4273-2018.pdf
Droplet concentration (Nd) and liquid water path (LWP)
retrievals from passive satellite retrievals of cloud optical depth (τ)
and effective radius (re) usually assume the model of an idealized
cloud in which the liquid water content (LWC) increases linearly between
cloud base and cloud top (i.e. at a fixed fraction of the adiabatic LWC).
Generally it is assumed that the retrieved re value is that at the
top of the cloud. In reality, barring re retrieval biases due to
cloud heterogeneity, the retrieved re is representative of
smaller values that occur lower down in the cloud due to the vertical
penetration of photons at the shortwave-infrared wavelengths used to
retrieve re. This inconsistency will cause an overestimate of
Nd and an underestimate of LWP (referred to here as the
“penetration depth bias”), which this paper quantifies via a
parameterization of the cloud top re as a function of the retrieved
re and τ. Here we estimate the relative re
underestimate for a range of idealized modelled adiabatic clouds using
bispectral retrievals and plane-parallel radiative transfer. We find a tight
relationship between gre=recloud
top/reretrieved and τ and that a 1-D relationship
approximates the modelled data well. Using this relationship we find that
gre values and hence Nd and LWP biases are higher for the
2.1 µm channel re retrieval (re2.1) compared to
the 3.7 µm one (re3.7). The theoretical bias in the
retrieved Nd is very large for optically thin clouds, but rapidly
reduces as cloud thickness increases. However, it remains above 20 % for
τ<19.8 and τ<7.7 for re2.1 and re3.7,
respectively. We also provide a parameterization of penetration depth in
terms of the optical depth below cloud top (dτ) for which the retrieved
re is likely to be representative.
The magnitude of the Nd and LWP biases for climatological data sets
is estimated globally using 1 year of daily MODIS (MODerate Imaging
Spectroradiometer) data. Screening criteria are applied that are consistent
with those required to help ensure accurate Nd and LWP retrievals.
The results show that the SE Atlantic, SE Pacific and Californian
stratocumulus regions produce fairly large overestimates due to the
penetration depth bias with mean biases of 32–35 % for re2.1
and 15–17 % for re3.7. For the other stratocumulus regions
examined the errors are smaller (24–28 % for re2.1 and
10–12 % for re3.7). Significant time variability in the
percentage errors is also found with regional mean standard deviations of
19–37 % of the regional mean percentage error for re2.1 and
32–56 % for re3.7. This shows that it is important to apply a
daily correction to Nd for the penetration depth error rather than
a time–mean correction when examining daily data. We also examine the
seasonal variation of the bias and find that the biases in the SE Atlantic,
SE Pacific and Californian stratocumulus regions exhibit the most
seasonality,
with the largest errors occurring in the December, January and February (DJF)
season. LWP biases are smaller in magnitude than those for Nd (-8
to -11 % for re2.1 and -3.6 to -6.1 % for
re3.7).
In reality, and especially for more heterogeneous clouds, the vertical
penetration error will be combined with a number of other errors that affect
both the re and τ, which are potentially larger and may
compensate or enhance the bias due to vertical penetration depth. Therefore
caution is required when applying the bias corrections; we suggest that they
are only used for more homogeneous clouds.
Introduction
Clouds have a major impact on Earth's radiative balance
and small changes in their properties are predicted to have large radiative
impacts e.g.. The amount of shortwave flux
reflected by fully overcast warm (liquid water) clouds for a given sun and
scattering angle, or the reflectance of a cloud, is primarily determined by
the cloud optical depth (τ), which in turn can often be characterized by
the liquid water path (LWP; the vertical integral of liquid water content)
and the cloud droplet number concentration (Nd). For a given cloud
updraft, Nd is determined by the number concentration and
physicochemical properties of aerosols. Thus, couching cloud reflectance in
terms of Nd links the cloud albedo to aerosol and microphysical
effects via the effect, making Nd a very useful
quantity to determine observationally. Nd can also influence cloud
macrophysical feedbacks via its control on rain formation
and stratocumulus cloud top entrainment
.
Satellite observations of clouds and Nd are immensely useful for
studying clouds, cloud–aerosol interactions and for model evaluation since
they afford large spatial and temporal coverage. A method to obtain
Nd from passive satellite observations e.g. from the
MODerate Imaging Spectroradiometer, MODIS; of τ and the
cloud droplet effective radius (re) for stratiform liquid clouds
has been previously demonstrated and is described further below. For more details see the
review paper on this technique, which also describes the
known sources of error. In cloudy environments, aerosol optical depth cannot
be retrieved from satellites, making cloud property observations such as
Nd and the cloud droplet effective radius (re) the only
useful indicator of the influence of aerosol on clouds. An advantage of using
Nd rather than re to study cloud–aerosol interactions is
that re is also determined by the cloud water content and thus is a
function of cloud macrophysical properties. Nd, in contrast, is
only weakly controlled by cloud macrophysics, allowing some separation of
microphysical and macrophysical effects.
However, retrievals of Nd from space are still somewhat
experimental and there is a lack of comprehensive validation of the
retrievals and the assumptions required. There is a need to characterize and
quantify the associated errors; in this paper we focus on doing this for one
source of Nd error using a 1-year Nd data set for
stratocumulus clouds from MODIS.
The adiabatic Nd and LWP retrieval model and the vertical penetration depth bias
Nd and LWP are retrieved from passive satellite retrievals of
re and τ using an adiabatic cloud model that is described
below. However, as shown in and
, for a retrieval free from other error sources (e.g. those
due to cloud heterogeneity), the retrieved re is representative of
the re value lower down in the cloud due to the vertical
penetration of photons at the shortwave-infrared (SWIR) wavelengths used to
retrieve re. In contrast, the retrieved τ is comprised of
contributions from the extinction coefficient βext(h), where
h
represents height from cloud base, throughout the whole cloud profile:
τ=∫0Hβext(h)dh.
Here h=0 represents cloud base and h=H is cloud top.
βext(h) is defined as
βext(h)=π∫0∞Qext(r)r2n(r)dr,
where r is the droplet radius and n(r) is the droplet size number
distribution within a cloud unit volume such that
Nd=∫0∞n(r)dr. Qext(r) represents the
ratio between the extinction and the geometric cross section of a given
droplet and can be approximated by its asymptotic value of 2
since droplet radii are generally much larger than the
wavelength of light concerned (typically 0.6 to 0.85 µm) such that
the geometric optics limit is almost reached.
re and liquid water content (LWC) at a given height are,
respectively, defined as
re(h)=∫0∞r3n(r)dr∫0∞r2n(r)dr,
and
LWC(h)=4πρw3∫0∞r3n(r)dr,
where ρw is the density of liquid water. Combining
Eqs. () and () and inserting into
Eq. () gives
βext(h)=3Qext4ρwLWC(h)re(h).
To determine the form of re(h) in the above equation in terms of
L(h) and Nd(h) we can utilize the fact that the “k” value,
k=rvre3,
which is a measure of the width of the droplet size distribution (lower
values indicate wider distributions), has been shown to be approximately
constant in stratocumulus clouds . In this study we adopt a value of k=0.72, which is the
value assumed by the MODIS retrieval . rv is
the volume radius, defined as
rv(h)3=1Nd(h)∫0∞r3n(r)dr=3LWC(h)4πρwNd(h)=kre(h)3,
where we have used Eq. () to insert LWC and Eq. () to
write rv as a function of k and re. Now we utilize
the assumptions that Nd(h) is constant with height and that
LWC(h) is a constant fraction, fad, of the adiabatic LWC. The latter
equates to
LWC(h)=fadcwh,
where cw is the rate of increase of LWC with height
(dLWC/dz, with units kgm-4) for a moist
adiabatic ascent and is referred to as the “condensation rate” in
, or the “water content lapse rate” in
. See for a
derivation. cw is a constant for a given temperature and pressure.
Allowing these assumptions, using Eq. () to substitute for
re in Eq. () and combining with
Eqs. () and () we can write
τ*=∫0H*Qext3fadcw4ρw2/3Ndπk1/3h2/3dh=3Qext53fadcw4ρw2/3Ndπk1/3H*5/3.
At this stage, H* is any arbitrary height above cloud base and
τ* is thus the optical depth between the cloud base and that height.
H* can be expressed as a function of re(H*), k,
Nd and some constants by using Eqs. ()
and (). Then, given re(H*) and τ*,
Nd can be calculated as follows:
Nd=52πkfadcwτ*Qextρwre(H*)51/2.
Generally, when retrieving Nd it is assumed that the re
obtained from satellite is representative of that from cloud top, i.e.
re(H*)=re(H)e.g.. This would then mean
that τ* is the full cloud optical depth (τ) as retrieved by the
satellite and thus could be used in Eq. () above to obtain
Nd. However, since the re obtained by satellite is
actually equal to re(H*) then τ*<τ and thus
τ* should be used in Eq. () instead of the retrieved
τ; the problem lies in the fact that τ* is unknown. However, in
this paper we fit a simple function for τ* as a function of τ
based on radiative transfer (RT) modelling of a variety of idealized clouds.
Alternatively, Eq. () can be formulated using the retrieved
τ over the full cloud depth (setting τ*=τ) and the cloud top
re (setting re(H*)=re(H)). The problem then
becomes one of estimating re(H) from the retrieved
re(H*). Here we formulate a parameterization of re(H)/re(H*) as a function of τ. Note that either the τ or
re corrections should be applied to correct Nd, but not
both together.
Then we estimate the error introduced in Nd retrievals for 1 year
of MODIS data due to the usual assumptions of
re(H*)=re(H) and τ*=τ, on the assumption that
there are no other biases affecting the re retrieval. We label this
bias the “vertical penetration bias”.
The method of correcting re has the advantage over the τ
correction since it also allows a correction to the retrieval of LWP. LWP can
be estimated see e.g. using
LWP=59ρwre(H)τ.
For a corrected LWP the cloud top re and the retrieved (total)
τ values should be used. Since the retrieved re(H*) is
likely to be underestimated due to the vertical penetration depth bias, LWP
would otherwise be underestimated and the correct value can be obtained by
using the parameterized re(H) instead.
Data and methodsCalculation of τ and re corrections
In order to calculate
gre=re(H)re(H*),
and
dτ=τ-τ*,
we have performed re retrievals on idealized clouds using a similar
algorithm to that used for MODIS retrievals. We produced idealized clouds
that span a large range of stratocumulus-like clouds as represented by
combinations of Nd and LWP. We chose 41 values between
Nd=10 and 1000 cm-3 that were equally spaced in log
space and 91 values between LWP = 20 and 200 gm-2 spaced
equally in linear space. All of the possible combinations from this sampling
were used to sample the 2-D (Nd, LWP) phase space. For each
combination, discretized adiabatic model profiles following the form of those
described in Sect. (i.e. with a vertically
constant Nd and LWC that increases linearly with height) were
generated using cw=1.81×10-6kgm-4,
fad=0.8 and a vertical spacing of 1 m. The droplet size
distributions at each height were represented by a modified gamma
distribution with a k value of 0.72, i.e. representative of an effective
variance of 0.1. One-dimensional RT calculations, assuming
plane-parallel clouds, were performed on these profiles using DISORT
Discrete Ordinates Radiative Transfer Program;
radiation code in order to simulate reflectances at wavelengths of 0.86, 2.1
and 3.7 µm, matching those measured by MODIS to retrieve τ and
re over an ocean surface. Note that MODIS provides re
retrievals using both 2.1 and 3.7 µm wavelengths, which are
hereafter referred to as re2.1 and re3.7,
respectively. The MODIS re3.7 retrieval requires a correction to
account for the contribution to the observed radiance from thermal emission,
which is based on the observed 11 µm radiance
. We account for
this in our retrievals by removing the thermal contribution during the RT
calculation instead of via the 11 µm radiance, which should produce
a consistent end result. The RT calculations were performed assuming a black
surface, a clear atmosphere (i.e. gaseous absorption is neglected), using a
solar zenith angle (SZA) of 20∘ and a nadir viewing angle.
These reflectances were then used to retrieve τ and re values
using the bispectral method, as operationally
used by MODIS. To do so, a lookup table was built from reflectances
calculated for a range of clouds that were assumed to be plane-parallel in
nature, as assumed for the operational MODIS retrievals; i.e. these clouds
were uniform in the vertical and horizontal with infinite horizontal extent.
Again, a black surface and a k value of 0.72 were assumed along with the
same viewing geometry as for the RT calculations on the adiabatic clouds. A
fixed depth of 1 km was assumed with cloud base at an altitude of 1 km and
cloud top at 2 km, although the cloud depth has no major effect on the
reflectances generated for a given τ and re. gre was
then calculated using the retrieved and model top re values.
dτ was calculated by choosing the value from the model profile of
τ, as measured from cloud top downwards, that corresponded to the
location where the model profile re matched the retrieved
re.
Two-dimensional histogram of gre as a function of τ for a range
of clouds (see text) for the 2.1 µm re
retrieval (a) and the 3.7 µm retrieval (b). The
black line is the median gre in each τ bin after smoothing
over τ interval windows of 0.2. The white line is the fit to the mean
curve using Eq. ().
Figure a shows a 2-D histogram of gre
values as a function of τ for the 2.1 µm retrieval. It shows
that when plotted in this way gre forms a fairly tight relationship
with τ so that for a given τ only a small range of gre
values are possible. This suggests that the relationship can be parameterized
based upon a 1-D relationship fitted to these data with little loss of
accuracy. The median value of each τ bin is also plotted (after
smoothing over τ windows of 0.2) and this is the relationship used in
this paper. gre is seen to decrease with τ with a gradient
that decreases with τ. Similarly, Fig. shows
dτ vs. τ, which also shows a tight relationship that is suited to a
1-D parameterization. Fourth-order polynomial curves can be fitted (using the
least squares method) to the median value relationships that take the form
gre=a4τ4+a3τ3+a2τ2+a1τ+a0,
and
dτ=b4τ4+b3τ3+b2τ2+b1τ+b0.
Coefficients for the fitting curve (Eq. ) to
estimate the median gre value as a function of τ. The maximum
absolute error between the fit and the median line is also shown.
Coefficients for the fitting curve (Eq. ) to
estimate the mean dτ value as a function of τ. The maximum
absolute error between the fit and the mean line is also shown.
The coefficients of these fits are given in Tables
and along with the maximum errors for the fit (relative to the
mean or median line) for the range shown. The curves (white lines in
Figs. a and a) fit the mean
data well with maximum absolute differences of 0.001 and 0.09, respectively,
for the gre and dτ curves. However, there will be some
error when using this relationship (or the mean value relationship) due to
the spread in the gre and dτ values seen in the
underlying histograms.
Figures b and b show the
same results for the 3.7 µm retrieval. Again tight 1-D
relationships are suggested. Here, though, gre and dτ values
are lower for a given τ and the curves are steeper at lower τ
values, but flatten off much more rapidly. By τ=7.5 there is little
dependence of dτ on τ and dτ saturates at a mean value of
∼ 2.6. The fit estimates for the curves (Eqs.
and and Tables and ) again
match the actual curves closely with a maximum absolute error in
gre and dτ of 0.003 and 0.14, respectively.
As for Fig. except for dτ as a
function of τ and using Eq. () for the white
line.
MODIS data
For the MODIS data we use 1 year (2008) of MODIS Aqua data and follow a
similar methodology to that used in in order to
create a data set akin to the MODIS Level-3 (L3) product
. We processed MODIS
Collection 5.1 joint Level-2 (L2) swaths into 1∘×1∘
grid boxes. Joint L2 swaths are subsampled versions of the full L2 swaths
(sampling every fifth 1 km pixel) that also contains fewer parameters. We
process the data from L2 to L3 in order to allow the filtering out of data at
high SZAs and to provide both re2.1 and
re3.7 retrievals.
(a) Number of days in 2008 that fulfilled the criteria
required to be counted as a valid Nd retrieval. See the text for
details on the criteria. Various regions of interest are also denoted by the
boxes and numbers. (b) Mean optical depth for data set with
filtering criteria 1–7 applied (see text).
For this work we relax the screening methodology slightly from that used in
since here we are interested in the effects of the
vertical penetration Nd bias upon a more general global data set.
We applied the following restrictions to each 1∘×1∘
sample that goes into the daily average (since multiple overpasses per day
are possible) in order to attempt to remove some artifacts that may cause
biases:
At least 50 joint L2 1 km resolution pixels from the MODIS
swath that did not suffer from sunglint were required to have been sampled
within each grid box.
At least 80 % of the available (non-sunglint) pixels were required to
be of liquid phase based upon the “primary cloud retrieval phase flag”.
Analysis was only performed on these pixels. A high cloud fraction helps to
ensure that the clouds are not broken, since broken clouds are known to cause
biases in retrieved optical properties due to photon scattering through the
sides of clouds. Often retrievals of Nd are restricted to high
cloud fraction fields for this reason
and so we focus on such
data points here.
The only pixels used were those remaining after (2) for which the “cloud mask status”
indicated that the cloud mask could be determined, the “cloud mask
cloudiness flag” was set to “confident cloudy”, successful simultaneous
retrievals of both τ and re for the 2.1µm channel
were performed and the cloud water path confidence from the MODIS L2 quality
flags was designated as “very good confidence” (the highest level possible). This is a little different from the official MODIS L3 product
where a set of cloud products are provided that are weighted using the
quality assurance (QA) flags. Rather than weighting our L3-like product with the
QA flags we have simply restricted our analysis to pixels with the highest
confidence for water path.
The mean 1∘×1∘ cloud top height (CTH) is restricted to values lower than
3.2 km. This is done to avoid both deeper clouds for which Nd
retrievals are likely to be problematic due to the increased likelihood of a
breakdown of the assumptions required to estimate Nd, such as a
constant fraction of the adiabatic value for LWC and vertically constant
Nd, and increased retrieval issues due to cloud heterogeneity.
CTH is calculated from the MODIS 1∘×1∘ mean cloud
top temperature (CTT) and the sea surface temperature (SST) using the method
of . SST data were obtained from version 2 of the NOAA
Optimum Interpolation (OI) Sea Surface Temperature data set
(NOAA_OI_SST_V2) that provides weekly SST data at 1∘×1∘ resolution. This was interpolated to daily data on the assumption
that SST does not vary significantly over sub-weekly timescales.
The mean 1∘×1∘ SZA was restricted
to ≤ 65∘ following the identification of biases in the retrieved
τ, re and Nd at high SZAs
.
1∘×1∘ grid boxes were rejected if the maximum
sea-ice areal coverage over a moving 2-week window exceeded 0.001 %.
The sea-ice data used were the daily 1∘×1∘ version of
the “Sea Ice Concentrations from Nimbus-7 SMMR and DMSP SSM/I-SSMIS Passive
Microwave Data, Version 1” data set .
Only 1∘×1∘ grid points with mean τ>5 were considered
for the Nd data set due to larger uncertainties from instrument
error and other sources of reflectance error for τ and re
retrievals at low τ.
Following this screening, the 1∘×1∘ grid boxes
associated with each MODIS Aqua overpass were averaged into daily mean values
for ocean covered surfaces only. Figure a shows the
number of days from the year of data examined in this study (year 2008) that
fulfilled the above criteria and thus are likely to produce a good
Nd retrieval. Regions with high numbers of days where useful
Nd retrievals can be made have been selected for closer examination
in this study; they are listed in Table along with
information on the mean and maximum numbers of days of good data. The
permanent marine stratocumulus decks are among those selected, namely those
in the SE Pacific off the western coast of S. America (Region no. 1), in the
SE Atlantic off the western coast of southern Africa (Region no. 2), off the
coast of California and the Baja Peninsula (Region no. 3), in the Bering Sea
off the SW coast of Alaska (Region no. 6) and in the Barents Sea to the
north of Scandinavia (Region no. 8). These regions are where the highest
numbers of selected days occur with values ranging up to a maximum of 141
days (for the Bering Sea region). The Barents Sea region has the lowest
maximum number of days out of this group, reflecting the fact that
Nd retrievals cannot be made during a lot of the winter season in
this region due to a lack of sunlight. The Southern Ocean (Region no. 5) and
the NW Atlantic (Region no. 7) regions frequently produce stratocumulus,
although it is often associated with the cold sectors of cyclones and so its
location from day to day is more transient. These regions are also affected
by high SZAs in the winter seasons, which also restricts the
number of retrievals possible there. The East China Sea region (Region no. 4)
produces the lowest mean and maximum numbers of days since the stratocumulus
areas are mostly restricted to near the coast and occur mostly in the winter
season.
Nd was calculated for both re2.1 and re3.7
using Eq. () from the 1∘×1∘ daily mean
τ, re and CTT. This was done by using the retrieved τ
value in Eq. () along with both the retrieved re value
(i.e. assuming that re(H*)=re(H) as is often assumed
for Nd retrievals) and by estimating re(H*) using the
retrieved re along with the gre values that were
calculated as described above. This therefore gives Nd data sets
for the “standard” method and a corrected method, allowing the differences
between the two to be examined. A similar process was applied for the LWP
retrieval.
Results
Following Eq. (), the ratio between the uncorrected and corrected
Nd values can be shown to be
Nd (uncorrected)Nd (corrected)=re(H)re(H*)5/2=gre5/2.
Figure shows how the relative Nd bias varies
as a function of retrieved τ when using an re that has been
corrected using the gre from Fig.
(mean curve, black line). At τ=5 the relative error is 46 % for the
re2.1 retrieval and 28 % for the re3.7 retrieval. At
higher τ the errors reduce rapidly but remain above 10 % for the
re2.1 retrieval over the τ range shown. For the
re3.7 retrieval the relative error drops below 10 % for
τ>∼13. Thus, the overall degree of error due to this effect will be
determined by the distribution of τ for the regions of interest, which
we take into consideration here using MODIS data for a representative
Nd data set.
Alternatively, if the correction is formulated in terms of a correction to
τ we obtain
Nd (uncorrected)Nd (corrected)=ττ*1/2=ττ-dτ1/2.
The equation shows that, for a constant dτ, the relative Nd
bias due to an uncorrected τ value would increase with decreasing τ
as τ-dτ approaches zero.
The ratios of Nd values from the standard MODIS
calculation (using the retrieved re for re* in
Eq. ) to those from the corrected calculation (using the
corrected re for re* as calculated from the retrieved
value and gre; the gre values used are those shown by the
black line in Fig. ) vs. retrieved τ.
Figure b shows the time–mean τ for the data set
as filtered by criteria 1–7 above, i.e. to replicate the type of filtering
that would likely be performed for Nd retrievals.
Table lists the regional means of these time–mean
values along with the regional means of the standard deviations of τ
over time. It shows that the mean τ values of the tropical and
subtropical regions are generally lower than those at higher latitudes. The
East China Sea, Barents Sea, NW Atlantic and Southern Ocean regions exhibit
the highest mean τ values out of those examined and so should be
expected to show the lowest Nd biases due to the vertical
penetration effect. The SE Atlantic region (and the region to the west of
Africa in general) show low τ and can be expected to give high
Nd biases. Table also lists the fraction
of days for which τ≤10 (fτ≤10). τ=10 is the value
above which Nd biases drop below 31 % for the 2.1 µm
retrieval and below 14 % for re3.7 according to
Fig. .
Thus fτ≤10 indicates the fraction of days for which daily
Nd biases will be greater than 31 % for that channel. The
values in the table indicate that even in the least affected region (Barents
Sea) this will occur for 21 % of the days. For the SE Atlantic and SE
Pacific region the percentages rise to 69 % and 53 % of the days,
respectively. Thus, the vertical penetration depth Nd bias is
prevalent in all regions for which Nd data sets are likely to be
used and particularly so in the subtropical stratocumulus regions where
Nd retrievals have been widely used and studied.
Regional statistics for the various marine stratocumulus regions
shown in Fig. . Shown are the mean and maximum number
of days that fulfill the screening criteria in order to be considered as
useful Nd retrievals; the regional means and standard deviations
(σ) of the time-averaged optical depths (τ) for the screened data
set; and the regional mean of the fraction of days for which τ≤10
(fτ≤10), which is calculated using only data from grid points
for which the number of days with Nd data was ≥ 15.
Regional means of the predicted time–mean percentage biases in
Nd and LWP due to the vertical penetration depth error and regional
means of the relative (percentage) standard deviations (over time) of the
percentage Nd and LWP biases (i.e. regional means of the values in
Fig. and the equivalent for LWP). Bias results are shown
for both the 2.1 and the 3.7 µm re retrievals.
The overall bias is now estimated using 1 year of actual MODIS data in order
to obtain a realistic distribution of τ values. However, it should be
noted that the data set used is deliberately filtered in order to only retain
data points that are likely to give useful Nd data, namely low
liquid clouds with extensive 1∘× 1∘ cloud
fractions, i.e. predominately stratocumulus. This is done in order to assess
biases for the types of clouds that Nd data sets will typically be
used to study.
Figure shows a map of the mean percentage biases and
Table gives the regional means of the values in the map.
Considering firstly the biases for the re2.1 retrieval, the biases
are highest in the tropics and subtropics. The regional mean bias is
34.5 % for the SE Atlantic region (Region no. 2), which is the
stratocumulus region that seems to suffer the most. The biases are a little
lower for the other major stratocumulus regions; e.g. for the SE Pacific
region (Region no. 1) and the Californian region (Region no. 3) the mean
biases are 32 %, although the biases increase further west, where the
dominant cloud regime tends to shift towards trade cumulus clouds. The
remaining regions all have mean biases of 24–28 %. The Barents Sea
region (Region no. 8) has a value of only 23.7 %, representing the
stratocumulus region with lowest mean bias. These results indicate higher
τ values for the clouds in the East China Sea, Southern Ocean, Bering
Sea, NW Atlantic and Barents Sea regions relative to the Californian and SE
Pacific stratocumulus regions, with the SE Atlantic region exhibiting the
lowest τ values. This is confirmed by the mean τ values shown in
Table . The biases for the re3.7
retrieval display the same spatial patterns as for re2.1, but are
significantly lower; the mean value in the region with the maximum bias (SE
Atlantic, Region no. 2) is 17 % and in the region with the lowest
bias (Barents Sea, Region no. 8) it is 10 %.
The regional mean LWP biases are also listed in Table .
They are negative since an re underestimation from the vertical
penetration effect leads to an LWP underestimate (see
Eq. ). The biases are also smaller in magnitude than for
Nd due to the smaller sensitivity of LWP to re inherent
in the latter equation. They are anticorrelated with the Nd biases
such that the region with largest Nd bias (SE Atlantic) has the
largest negative LWP bias of -11.1 %. The smallest magnitude bias
occurs in that Barents Sea region (-8 %).
Maps of the annual mean percentage error for uncorrected
Nd retrievals using a year (2008) of daily MODIS data that have been
filtered to select data points in which Nd retrievals are
favourable and therefore most likely to be used for Nd data sets
(see text for details). The left plot shows the results for the
re2.1 retrieval and the right for the re3.7 retrieval.
As for Fig. except showing the relative (as a
percentage) standard deviation of the percentage Nd bias over time,
i.e. σ% bias/% bias‾.
It is also useful to know how variable the biases are from day to day for a
given point in space since this will determine how useful the application of
a single offset bias correction might be for correcting Nd biases
for daily data. Figure shows the time variability of the
bias in the form of the relative standard deviations (over time) of the
percentage Nd biases. It reveals that the percentage bias in
Nd generally has a larger relative standard deviation at latitudes
above around 40∘ with values typically ranging up to around
30–50 % (of the mean percentage Nd bias) for the
2.1 µm retrieval. Relative variability is greater for the
3.7 µm retrieval, perhaps due to the much lower mean percentage
errors. Some of the selected regions show more variability than others, in
particular the Barents Sea and East China Sea regions.
Table gives the regional means of the relative standard
deviations revealing values that range from approximately 20 to 40 % of
the mean percentage biases for the 2.1 µm retrieval and
30–60 % for the 3.7 µm one. This shows that the application of
a single annual mean offset bias correction is likely to lead to fairly large
biases for the Nd estimates for individual days for regions where
the mean Nd errors are significant. If daily data are used to
determine relationships between cloud properties and Nd without
correcting for the biases examined here then significant variability in
Nd might be introduced that may affect those relationships via
non-linear effects.
Figure shows how the percentage Nd biases
change with season for the re2.1 retrieval only. Interestingly, the
highest biases tend to occur in the DJF season for the SE Pacific and SE
Atlantic stratocumulus regions, indicating that τ values are lower in
DJF for those seasons. The September–October–November (SON) season also generally produces higher biases
than March–April–May (MAM) and June–July–August (JJA) for those regions, particularly for SE Pacific. For the East
China Sea region the biases are lower in SON and DJF seasons than in the
other seasons. We note that there are little data in this region for JJA since
there are few low-altitude clouds with large regional liquid cloud fractions
there in this season. Either the other regions do not show a large amount of
seasonal variability or Nd data are only available for part of the
year due to a lack of sunlight in the winter months.
Seasonal mean percentage Nd biases for the re2.1
retrieval only.
Discussion
There are some caveats to the results that we presented here that we now
discuss. We have shown that, theoretically, the effect of retrieving a lower
re than the cloud top re can be corrected for by simply
replacing re with the parameterized cloud top version or by
removing dτ from the observed τ. However, this rests upon the
parameterizations being valid across all of the cloud types relevant for the
Nd and LWP data sets. The relationships are based on the retrieved
re for a range of clouds, although only for a nadir viewing angle
and a SZA of 20∘.
showed that dτ has some dependence on viewing geometry and so the
consideration of a wider range of view and SZAs should ideally
be made.
In addition, a liquid water condensation rate (cw) value of
1.81 × 10-6 kg m-4 was assumed for the model adiabatic
clouds, which corresponds to a cloud temperature of 278 K at a pressure of
850 hPa. In reality, cloud temperatures and hence cw will vary,
mainly as a function of cloud temperature. We have performed sensitivity
tests using a value of 1.0 × 10-6 kg m-4, which
corresponds to a cloud temperature of 262 K. This is likely to be close to
the coldest temperature attained by boundary layer clouds over the oceans
that are coupled to the surface, which are generally the types of clouds for
which the droplet concentration retrievals are applied. The results (not
shown) reveal mean differences (across the τ values tested) in the mean
dτ line (i.e. the white line in Fig. a) of
4.2 % (the maximum difference was 10 %) for the 2.1 µm
retrieval and 3.6 % (maximum of 7.6 %) for the 3.7 µm one.
For gre the differences were of the opposite sign and much smaller,
with mean differences in the median gre line of -0.45 % (the most
negative difference was -0.6 %) for the 2.1 µm retrieval and
-0.3 % (-0.66 % was the most negative difference) for the
3.7 µm one. Therefore, the effect of cw changes is
relatively minor. These results would also apply for equivalent changes in
the cloud adiabaticity (i.e. the value of fad).
The modelling of the idealized clouds and the correction rests on the
assumption that re increases monotonically with height within the
cloud (following the adiabatic assumption), but there is some suggestion that
the development of precipitation-sized droplets might lead to larger droplets
being preferentially found below cloud top
.
However, found that MODIS retrievals of re
performed on model-generated clouds were not significantly affected by the
presence of precipitation. Also, during the VOCALS field campaign in the SE
Pacific region, aircraft observations showed that re generally did
increase with height up to cloud top , indicating
that this is not a problem at least for the near-coastal clouds tested.
Further offshore the likelihood of precipitation increases as clouds become
more cumulus-like and so for those clouds the issue may be greater and hence
more caution should be exercised when interpreting the results presented here
for such regions.
Evaporation effects related to entrainment also have the potential to reduce
re, Nd and LWC near cloud top and hence negate some of
the assumptions upon which the Nd retrievals rest. However, we
argue that the entrainment effect upon re is likely to be minimal
for two reasons: firstly, the evidence suggests that for stratocumulus clouds
extreme inhomogeneous mixing occurs at cloud top, which reduces the LWC and
Nd, but does not change re. Secondly, the results of
indicate that entrainment occurs within
approximately the first 0.5 optical depths from cloud top on average; the
penetration depths calculated here are considerably larger than this for
reasonably thick clouds (Fig. ). The effect of the
reduced Nd and LWC within the entrainment zone is not so clear-cut;
this would negate the assumption of a vertically constant Nd and
monotonically increasing LWC used to formulate the total τ. However,
given the likely small τ contribution from the entrainment region
relative to the total τ, this effect is likely to be small.
It is also clear that the suggested correction for the vertical penetration
effect should only be applied to the retrievals of Nd with
consideration of other bias sources. These other potential error sources are
numerous and include re biases due to subpixel heterogeneity
, 3-D
radiative effects , assumptions regarding the degree
of cloud adiabaticity fad in Eq. ;, the choice of k value assumed
constant;, the assumption of a vertically uniform
Nd, the assumed droplet size distribution shape and width
, viewing geometry effects
and
upper-level cloud and aerosol layers
. These errors have the potential to bias Nd in a way that
opposes the positive bias expected from the vertical penetration effect such
that the overall biases may cancel out. Indeed, the largest source of error
in Nd is likely that from re biases given the sensitivity
of Nd to re in Eq. (). MODIS re has
generally been shown to be biased positively compared to aircraft
observations , which
would lead to a negative Nd error when taken alone. Thus, the
application of the correction described in this paper in isolation has the
potential to enhance any negative bias in Nd caused by a positive
re bias.
Our paper quantifies the vertical penetration bias in isolation to the other
effects mentioned above. It should be questioned, though, whether the
presence of cloud heterogeneity and other effects somehow prevent the effects
of the vertical stratification from influencing the retrieved re,
making it irrelevant. This could be a potential explanation for why it is
often observed that re2.1 is larger than re3.7 in contrast to the direction expected from
adiabatic clouds given the vertical penetration effect, since it is known
that subpixel heterogeneity effects tend to cause a positive re2.1
bias relative to re3.7. We argue, though,
that the vertical stratification effect occurs in addition to other effects
(e.g. heterogeneity) with the latter cancelling out and often exceeding the
former such that the positive difference between re2.1 and
re3.7 would be even larger without the vertical stratification
effect. The cancellation of biases may also explain why VOCALS aircraft
measurements tended to show that re2.1
and re3.7 were very similar.
We also note that there are many situations when the expected result due to
vertical stratification of re does occur (i.e.
re3.7>re2.1), as demonstrated in
and Fig. . This
shows ratios between re3.7 and re2.1 for an example MODIS
scene in the SE Pacific stratocumulus region. Ratios using the uncorrected
MODIS re values are shown, which shows that the ratio exceeds 1
for most of the stratocumulus cloud region (the clouds that adjoin the coast)
with ratios ranging from around 1.1 to 1.2. In the more broken clouds the
ratio is less than 1, which is likely a result of cloud heterogeneity.
However, it would be expected that Nd retrievals would not be
applied to such clouds. The figure also shows the ratios calculated using
re3.7 and re2.1 values that have been corrected using the
gre factors. If the differences between re3.7 and
re2.1 were caused by vertical stratification alone and our
parameterization were correctly predicting the cloud top re for both
MODIS channels, then this ratio should be equal to 1. This is the case for
the clouds close to the coast, indicating that our parameterization is working
well for these clouds. The ratios are a little higher than 1 further north
and west (around 1.05–1.08) indicating that either our parameterization is
not working correctly for these clouds or other factors are causing
relative differences between re3.7 and re2.1 (e.g.
subadiabaticity, cloud heterogeneity). Figure
shows the percentage of pixels for which re3.7>re2.1 for
90 days of MODIS SE Pacific observations divided into four different
heterogeneity bins. Heterogeneity is characterized by the Hσ
parameter , which is the standard deviation of the
250 m resolution 0.86 µm reflectance (R0.86) divided by the
mean R0.86. It is clear that for many regions relative re
values that are consistent with an adiabatic profile occur more than 50 %
of the time, particularly when the cloud heterogeneity is low. This suggests
that it may be possible to use Hσ to determine the situations in
which the bias correction is more applicable. However, it is hard to
definitively prove our argument within the scope of this study, particularly
for more heterogeneous regions, since it would likely require computationally
expensive 3-D RT modelling of known cloud fields (e.g. from
LES models), followed by re and τ retrievals.
Ratios of re3.7 to re2.1 for a MODIS snapshot
scene from the Southeast Pacific stratocumulus region from 16 June 2015.
(a) Using uncorrected re values. (b) Using
re values that have been corrected using the parameterizations for
gre (for both re3.7 and re2.1). A ratio of 1
is expected for the plot on the right if the relative differences between
re3.7 and re2.1 are caused by vertical stratification
alone and if the parameterization is correctly predicting the relative
differences.
The percentage of pixels for which re3.7>re2.1 for
90 days (January, February and March of 2008) of 0.1∘ resolution
MODIS Collection 6 observations for the SE Pacific stratocumulus region. Only
single layer liquid clouds are included and data points have been filtered to
exclude τ<5 and partially cloudy pixels. The four panels are for four
different bins of the heterogeneity parameter (the standard deviation of the
250 m resolution 0.86 µm reflectance divided by the mean
reflectance) with bin ranges labelled in square brackets above the panels and
the x and y axes in degrees longitude and latitude, respectively.
actually demonstrated that MODIS Nd
agreed rather well with Nd from aircraft for the SE Pacific region
despite a fairly large positive re bias; this was thought to be due
to the fortuitous cancellation of (for Nd) the re bias
with biases in the k parameter and fad. However, the agreement
between aircraft and MODIS Nd seen in
would deteriorate if a correction for the Nd bias due to the
penetration depth effect discussed here were also applied.
Table indicates that the result would be a MODIS
Nd underestimate of around 32 % (average for SE Pacific, Region
no. 1) for the 2.1 µm retrieval, assuming perfect initial
agreement. This indicates that another Nd bias may have been
operating in order to give the good observed agreement.
The MODIS retrieval uses reflectances from both a visible and a SWIR wavelength channel, with the former being primarily
determined by τ and the latter by re. However, a bispectral
retrieval is used and so there is also some sensitivity of the retrieved
τ to the SWIR reflectance, which will be representative of the
re below cloud top due to the vertical penetration effect. This,
combined with the fact that the MODIS forward retrieval model assumes a
vertically uniform cloud, will result in the retrieved τ being biased
relative to the real value (assuming the real cloud has an adiabatic
profile). Figure shows the difference between the
retrieved and model profile τ; the bias is negative and smaller in
magnitude than 5 % for the 3.7 µm retrieval. They are slightly
larger for the 2.1 µm retrieval, but still lower in magnitude than
5 %, except at re≲ 7 µm. Although it should
be noted that some of this bias may be due to other causes related to the
inconsistencies between the vertically uniform and adiabatic models rather
than the re vertical penetration bias. Since the retrieved
Nd is proportional to the square root of τ, this will lead to
small Nd biases. Biases in LWP will be similar to those in τ
since LWP is proportional to τ, but re biases are still likely
to dominate (e.g. see Fig. ). Thus, we have not
pursued this further.
In this paper we have only considered retrievals over the ocean, although
retrievals over land for τ and re are available for MODIS.
MODIS surface albedo uncertainties are likely to be much higher over land
than over the oceans since the surface
albedo is much more variable over land. In addition, cloud masking is more
difficult over land, particularly over non-vegetated surfaces, transitional
areas between desert and vegetated surfaces and above high-altitude regions
. We have ignored land regions in order
to avoid such complications and also because stratocumulus clouds are more
prevalent over ocean regions . However, the
results shown in this paper may still apply over land. The results of
and their Fig. 14 suggest that
surface albedo uncertainties are more important at lower optical depths
(≲ 5) and for the 2.1 µm retrieval (relative to the
3.7 µm one). Thus, for thicker clouds and the 3.7 µm
retrieval land surface albedo issues may be less problematic.
Percentage τ bias (retrieved vs. actual value from the input
model profile) as a function of τ and re.
Finally, we note that the thermal emission correction for the MODIS
re3.7 (see Sect. ) retrieval has some
uncertainty that should be considered; the uncertainty for this is included
(combined with other uncertainties) in the MODIS Collection 6 pixel level
uncertainty products . It is possible that effects
additional to those included, such as cloud heterogeneity and surface
heterogeneity, may further increase the uncertainty beyond that
estimated in the MODIS products, but these are currently not well documented.
Conclusions
We have described and quantified a positive bias in satellite retrievals of
cloud droplet concentration (Nd) and liquid water path (LWP) that
make use of the adiabatic cloud assumption to estimate these quantities from
satellite-observed cloud optical depth (τ), effective radius
(re) and cloud top temperature. We term the bias the “vertical
penetration bias”. It arises due to the well-documented vertical
penetration of photons with wavelengths in the shortwave-infrared range into
the upper regions of clouds, so that re retrievals are
representative of values some distance below cloud top
rather than being those at
cloud top as assumed by the Nd and LWP retrievals. Here we
quantified the optical depth as measured from cloud top downwards, dτ,
at which the retrieved re equaled the actual re for
adiabatic clouds covering a large range of total cloud optical depths and
Nd values. We showed that knowledge of dτ allows a corrected
Nd to be calculated by subtracting dτ from the observed τ
and using that in the Nd retrieval instead of τ. We
characterized dτ as functions of τ for the 2.1 and
3.7 µm re retrievals (re2.1 and
re1.6, respectively) and found that a 1-D relationship approximates
the modelled data well. dτ increases with τ and is larger for
re2.1 than for re3.7 and so the vertical penetration
Nd bias affects retrievals based on re2.1 more than those
using re3.7. Similarly, we also parameterized the true cloud top
effective radius (re(H)) as a function of the retrieved
re and τ, allowing both a corrected Nd and LWP to be
calculated by using re(H) instead of the retrieved value. Both the
dτ and re correction methods give similar results for the
Nd retrievals suggesting that the latter is preferable since it
also allows for a correction to LWP. However, for some applications it may be
useful to be able to parameterize dτ.
We quantified the vertical penetration Nd bias for 1-year
Nd and LWP data sets. The corrections presented here suggest that
Nd and LWP errors will increase as the τ value of the cloud
scene gets lower. For many regions that are considered trustworthy for
Nd and LWP retrievals (typically stratocumulus regions), there are
high frequencies of low τ values and the Nd biases are
significant. For example, for the SE Pacific and SE Atlantic regions clouds
with τ≤10 (for which Nd errors are expected to be
≥ 31 % for re2.1 and ≥ 15 % for
re3.7) occur, respectively, 53 and 69 % of the time on average.
The mean re2.1 vertical penetration Nd biases for these
regions were 32 and 35 %, respectively. Out of the stratocumulus regions
examined, these two were the worst affected. For re3.7 the
Nd biases were much smaller; for example, mean biases for the SE
Pacific and SE Atlantic regions were 15 and 17 %, respectively.
Nd biases were predicted to be worse for the tropical and
subtropical regions than for higher latitudes. The time variability of the
biases were also examined and were shown to be significant (regional mean
standard deviations of 19–37 and 32–56 % for re2.1 and
re3.7, respectively). This indicates that long-term averages of the
vertical penetration Nd bias corrections are not useful for
correcting Nd data over short timescales (e.g. daily Nd
data). We also examined the seasonality of the Nd biases and showed
that, for the stratocumulus regions, generally the DJF season was worst
affected, followed by SON.
LWP biases were of a lower magnitude than those for Nd and were
negative. The largest biases were again for the SE Atlantic region where the
mean bias was -11.1 % and the smallest for the Barents Sea region
(-8 %). Biases were also lower when using re3.7 with a
maximum (most negative) bias of -6.1 %.
We caution that the correction for the Nd and LWP vertical
penetration biases presented here should only be considered in combination
with corrections for other biases that affect τ and re.
suggest a correction for the subpixel heterogeneity
bias effect, but corrections may not currently exist for all biases and it is
likely that some unidentified biases still exist. Therefore, we recommend that our correction is currently only applied to homogeneous cloud scenes in
order to minimize possible entanglements with biases resulting from
heterogeneity effects, which are not accounted for. Such conditions can be
obtained by limiting retrievals to associated heterogeneity (Hσ)
values (available in MODIS MYD06 Collection 6 products) to less than about
0.1. Otherwise Nd and LWP biases could be made worse, for example,
in situations where the fortuitous cancellation of opposing errors leads to
initially small Nd errors. The latter was suspected to have
occurred for the comparison between MODIS Nd retrievals and in situ
aircraft observations as presented in . We showed
that the SE Pacific, which is the region examined in that study, had a mean
vertical penetration depth error of 32 %, suggesting that another
unidentified Nd bias may have been operating in order to give good
agreement.
Previous studies have shown that re3.7 is less prone to biases due
to subpixel averaging
. Thus,
combined with the work presented here, this supports the conclusion that
re3.7 likely represents a better choice for use in Nd and
LWP retrievals.
For future work, it is recommended that additional characterization of
dτ and gre is performed for a range of viewing geometries in
order to ensure that the results presented here are robust for all cloud
retrievals. The use of 3-D radiative transfer calculations and simulated
retrievals upon known LES model fields would also be useful for investigating
how heterogeneity effects might interact with the vertical penetration
effects. Further investigation into how the presence of precipitation affects
our assumptions and results is also warranted.
The data set is built from publically available MODIS
Level-2 data (Collection 5). The MODIS data were obtained from NASA's Level 1
and Atmosphere Archive and Distribution System (LAADS
http://ladsweb.nascom.nasa.gov/, ). NOAA_OI_SST_V2 data (SST data)
were
provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their website
at http://www.esrl.noaa.gov/psd/.
Other data sets used in this article are available upon request to the
corresponding author.
DPG developed the concepts and ideas for the
direction of the paper, performed the error calculations using the MODIS
data, produced the figures and wrote the manuscript. OS performed the
retrievals upon the idealized adiabatic clouds. All authors provided
additional input and comments on the paper during the paper writing process.
The authors declare that they have no conflict of interest.
Acknowledgements
Daniel P. Grosvenor was funded by both the University of Leeds under Paul
Field and from the NERC funded ACSIS programme via NCAS. Odran Sourdeval was
funded by the Federal Ministry for Education and Research in Germany (BMBF)
in the High Definition Clouds and Precipitation for Climate Prediction
(HD(CP)2) project (FKZ 01LK1503A and 01LK1505E). Robert Wood's
contribution was supported on NASA award number NNX16AP31G. Edited by: Andrew Sayer Reviewed by: Zhibo
Zhang and one anonymous referee
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