Introduction
The fact that a linear polarisation lidar will detect
a cross-polarised signal due to the occurrence of multiple-scattering in
liquid water clouds has been recognised since at least 1970
. Extensive field and laboratory observations
of the depolarisation of laser radiation in
water clouds have been made and various theoretical approaches have
been developed ranging from Monte Carlo-based (MC-based) models to
semi-analytic approaches; see for a review.
The penetration depth of lidars into water clouds (100–300 m) is
limited to what may be considered the cloud-base region; thus limiting
the region of the cloud where information can be directly
retrieved. However for semi-adiabatic cloud layers, number
concentration at cloud base and the rate of increase of the liquid
water content (LWC) strongly constrain the structure of the cloud as
a whole. The region of maximum supersaturation (above which no new
cloud condensation nuclei (CCN) are activated) is typically only a few
tens of centimetres to a few tens of metres above cloud base
and thus accessible, in general, to probing by
lidars. Thus any microphysical information potentially provided by
lidar observations will be of value for e.g. process studies involving
the quantification of aerosol–cloud interactions .
The general idea of using the depolarised return as a means to
determine water cloud microphysical properties, such as number
density, is not new and has been raised by several
authors. briefly raised the possibility and
presented the results of a double-scattering lidar model applied to
homogeneous (i.e. constant LWC and number density) clouds. More
recently, developed an inversion procedure based on the
constrained linear inversion of a double-scattering model of the
cross-polarised return applied to homogeneous clouds. Using
observations and MC models which include higher-order scattering, it
has also been noted that a tight correspondence exists between the
layer accumulated depolarisation ratio, layer integrated backscatter
and multiple-scattering factor . An
approach using (single field of view) depolarisation lidar has been
suggested by who, based on MC model results, noted
that for homogeneous water clouds that the depolarisation observed by
a lidar with a suitably large field of view (FOV) is expected to be,
to a good approximation, only a function of the optical depth.
In spite of the long history and the increasing understanding of the
relevant phenomenon, the use of depolarisation measurements to
retrieve cloud extinction and microphysical information appears to not
have seen widespread implementation. This may be due to the fact that
much of the theoretical work has focused on homogeneous clouds
(i.e. LWC and effective radius being constant with height) which are
not necessarily suitable models of actual clouds
. Another reason is the fact that while fast models
limited to second-order scattering are well established
, highly accurate general approaches taking into
account higher-order scattering and applicable to inhomogeneous
clouds are mainly limited to computationally costly MC approaches
(although some exceptions may exist
e.g.). Yet another, perhaps primary,
reason may be the shift in attention towards using multiple
FOV lidar observations
e.g. for which fast
and accurate forward models that treat scattering orders above 2 have
emerged in the past few years
e.g.. In spite of the
their apparent under-utilisation, the potential advantages of using the
depolarised lidar return in the context of water cloud lidar sensing
have been previously noted and it should
be noted that (single-view) depolarisation lidars, being of generally
simpler design, are much more common than multiple-FOV
systems. Thus a practical accurate depolarisation lidar water cloud
microphysical inversion scheme could potentially yield a large amount of
valuable data.
In this work we present a retrieval procedure using single
FOV
depolarisation lidars. The retrieval is based on
assuming that the cloud-base region can be characterised by
a quasi-linear (with height) LWC profile (i.e. constant LWC lapse rate)
and constant cloud particle number density. This set of assumptions
allows us to reduce the cloud variables to two parameters. In turn,
this enables the development of a fast and robust inversion procedure
using a look-up-table approach based on stored results from lidar
MC simulations.
The outline of the remainder of this paper is as follows. In
Sect. we present the cloud representation (model); we
employ and present and discuss the results of lidar
multiple-scattering MC calculations applied to our cloud
model. The lidar MC model is discussed in more detail in
Appendix . In Sect. we first describe the
basic inversion scheme based on the MC calculations and then describe the
extension of the scheme in order to include non-ideal effects such as
imperfect knowledge of lidar polarisation cross-talk. We then proceed
to demonstrate the function of the inversion scheme using simulated
lidar data based on large-eddy simulation (LES) cloud fields which
include areas of drizzle (Sect. ) and exhibit
realistic (e.g. variable) cloud structure. In Sect.
we demonstrate the application of the inversion scheme to various case
studies. The measurements in question were obtained at the Cabauw Experimental Site for Atmospheric Research (CESAR) multi-sensor atmospheric observatory in the central Netherlands
(www.cesar-observatory.nl). In particular, we present evidence to
support the accuracy of the inversion results by demonstrating the
consistency between observed values of cloud-base region radar
reflectivity compared with values of the reflectivity forward modelled
using the corresponding lidar-derived cloud parameters
(Sect. ). In
Sect. , we examine the values of the LWC produced by
the lidar inversion procedure and compare them with the corresponding
adiabatic values. Further, the results of a preliminary comparison
between lidar-derived cloud-base droplet number densities and
ground-based aerosol number density values are presented and discussed
in Sect. . The paper concludes with a summary of
the main points and findings.
Theory
Cloud model
The cloud model (i.e. representation) used in this work is a simple
but still useful model of cloud-base conditions . In
particular, we assume that cloud-droplet number density is constant as
is the altitude derivative (or lapse rate) of the liquid water content (Γl):
N(z)=N:z≥zb
and
LWC(z)=dLWCdz(z-zb)=Γl(z-zb):z≥zb,
where z is altitude and zb is the cloud-base altitude. Noting that for
droplets whose size is large compared to the wavelength of light involved,
α=2π〈r2〉 where α is the extinction
coefficient and we have
Reff=〈r3〉〈r2〉=32ρlLWCα,
where ρl is the density of liquid water and the brackets
denote averaging over the cloud particle size distribution.
If the LWC increases linearly with height above cloud base while
the number density remains constant, then the cloud-droplet effective radius profile has the following form
Reff(z)=Reff(zref)z-zbzref-zb1/3,
where z is the altitude and zref is some reference altitude The extinction coefficient profile can be found using
Eqs. ()–() leading to
α(z)=32ρl(z-zref)1/3ΓlReff(zref)(z-zb)2/3.
In this work, zref is set, somewhat arbitrarily, to be 100 m
above cloud base. Further in this paper, Reff,100 will
be used to denote the value of the effective radius at the reference altitude (i.e.
z-zb=100 m).
In order to link the effective radius and liquid water content to cloud-number
concentration it is necessary to specify the droplet size
distribution (DSD). Here we model the size distribution of the droplets using
a single-mode modified-gamma distribution :
dN(r)dr=NoRm1(γ-1)!rRmγ-1exp-r/Rm,
where Rm is the so-called mode radius, No is the total
number of particles in the distribution and γ is the shape
parameter. For this type of distribution
〈rn〉=(γ+n-1)!(γ-1)!Rmn,
where the brackets denote averaging over the size distribution. Thus
the relationship between the effective radius (Reff) and Rm is given by
Reff=〈r3〉〈r2〉=Rm(γ+2),
and the LWC is given by
LWC=No43πρl(γ+2)!(γ-1)!Rm3=No43πρlRv3,
leading to
Rv=(γ+2)!(γ-1)!1/3Rm,
where Rv is the volume mean radius.
The ratio between the volume mean radius and the effective radius (k) is an
important parameter for linking the cloud physical and optical
properties . From the preceding equations it can be seen that
k=Rv3Reff3=(γ+1)γ(γ+2)2.
Based on the results of LES modelling of stratocumulus
in this work we adopt a value of k equal to 0.75±0.15. Using
Eq. () this corresponds to a range of γ values between 5 and 14
with a k=0.75 corresponding to γ=9.
Once k has been specified No can be then be predicted from Γl and
Reff,100 using Eqs. () and ():
No=α10012πReff,100-21k,
where α100 is the extinction 100 m from cloud base.
Lidar MC calculations
Lidars (like radars) are time-of-flight active measurement
techniques. Pulses of laser light are transmitted into the
atmosphere and the backscattered signal is detected as a function of
time after each pulse has been launched. If only single scattering is
considered, the relationship between the detected
linearly polarised backscattered powers can be written directly as
P∥z=ct2=Cl,∥z2β∥(z)exp-2∫0zα(z′)dz′
and
P⟂z=ct2=Cl,⟂z2β⟂(z)exp-2∫0zα(z′)dz′,
where z is the range from the lidar, c is the speed of light, t
is the time-of-flight (so that z=ct/2), β∥ is the
range-dependent total (molecular + cloud + aerosol) parallel polarised
backscatter coefficient, β⟂ is the corresponding
coefficient for the perpendicular polarisation state,
Cl,∥ and Cl,⟂ the polarisation channel-dependent
lidar instrument constants and α is the range-dependent
extinction coefficient. The backscatter coefficients can be further
decomposed into the components corresponding to the molecular,
aerosol and cloud components:
β∥=β∥,m+β∥,a+β∥,c
and
β⟂=β⟂,m+β⟂,a+β⟂,c,
where the m subscripts denote the molecular contribution, a denotes
the aerosol contribution and c is used for the cloud
contribution. If the aerosols and cloud droplets being probed are spherical then
β⟂,a=β⟂,c=0 and
β⟂=β⟂,m=δmβ∥,m,
where δm is the molecular scattering linear depolarisation
ratio which mainly depends on the wavelength and spectral passband of
the lidar and is on the order of 0.2–0.4 % for typical passband
widths in the UV–VIS–NIR (ultra-violet–visible–near-inferred) wavelength range. Thus
under single-scattering conditions in water clouds, β⟂≪β∥. However with respect to lidar cloud measurements,
the multiple-scattering (MS) contribution to the signal can be many
times the single-scattering contribution. The occurrence of
multiple-scattering, in turn, may give rise to a perpendicularly
polarised return from clouds which is many order of magnitude greater
than that predicted from single-scattering theory
.
In order to calculate the polarised lidar backscatter, the
Earth Clouds and Aerosol
Radiation Explorer (EarthCARE) simulator (ECSIM) lidar-specific MC forward model
was used. ECSIM is a modular multi-sensor simulation framework
original developed in support of the EarthCARE but is flexible enough to be applied to
other instruments and platforms including upward
looking ground-based simulations. More information regarding the ECSIM
lidar MC model is given in Appendix .
Example results MC calculations for a lidar wavelength of 355 nm
corresponding
to semi-infinite clouds with a cloud base of 1.0 km with the values of
Γl and Reff at 100 m as indicated in the top-right of each panel while
γ=9. Here the lidar FOV is 0.5 mrad. As labelled in the bottom-left
panel, the right-most solid line in each
plot shows the parallel (para) range-corrected signal (RCS). The other solid
line shows the corresponding perpendicular (perp) RCS and the dashed-dotted
line shows the depolarisation ratio. Both the para- and perp-RCS profiles have
been normalised by the maximum para return.
As Fig. except for a lidar FOV of 2 mrad.
Example results of MC calculations for a lidar wavelength of 355 nm
corresponding to semi-infinite clouds with a cloud-base of 1.0 km for two
values of Γl as a function of Reff,100 for lidar FOVs of 0.5 and
2.0 mrad. Here, for each value of Reff,100, the para and perp attenuated
backscatter (ATB) values have been normalised by the maximum para return.
Range of parameters used in the MC calculations.
Parameter
Values
Cloud base (km)
0.5, 1.0, 2.0, 4.0
FOV (mrad)
0.5, 1.0, 2.0, 4.0
Reff,100 (µm)
2.0, 2.6, 3.3, 4.3, 5.6, 7.2, 9.3, 12.0
Γl (gm-3km-1)
0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0
Using our cloud model, MC runs were performed for various values of
Γl, Reff,100, different cloud-base heights and
different lidar field of views. The range of parameters used is given in
Table . Example results are shown in
Figs. and for a lidar
receiver FOV of 0.5 mrad and 2.0 mrad, respectively. The
laser divergence was fixed at 0.1 mrad and the wavelength is
355 nm. The results were not found to be sensitive (above the
1–2 % level) to the laser divergence so long as the laser divergence
was less than about half the receiver FOV. The MC calculations were run until
the estimated error level in the calculated depolarisation ratio was below
5 % for ranges below where attenuation has reduced the normalised
parallel return to a value below about 0.01, which, for a homogeneous cloud,
corresponds to an apparent OD of 2.3. Beyond this point it was judged that the
signal-to-noise (SNR) ratios of practical lidar measurements would be too
unfavourable to be exploitable. Results are shown for both the parallel and
perpendicular attenuated backscatters (ATB):
ATB∥(z)=z2P∥(z)
and
ATB⟂(z)=z2P⟂(z),
where P∥ and P∥ are the parallel and perpendicular
received powers, respectively.
In this work, we fix the lidar wavelength at 355 nm (tripled Nd:YAG
wavelength) since this corresponds to the wavelength of the
depolarisation lidar measurements we will eventually apply the theory
presented in this section to. We expect the results shown here to be
indicative of the behaviour at other wavelengths for the same FOV if
the Reff,100 variable is rescaled by the ratio of the wavelengths
and the LWC correspondingly adjusted to keep the extinction the same
(see Eq. ). This is due to the fact that cloud extinction
does not vary appreciably between 355 and 1064 nm and multiple-scattering effects generally scale with the effective angular width of
the forward-scattering lobe which, in turn, depends on the
λ/Reff ratio.
In Fig. it can be seen that for a FOV of
0.5 mrad that the maximum depolarisation reached in the
Reff(100m)=2µm cases is less than 0.2, while values of 0.4 are
reached in the case with Reff(100m)=8µm and Γl=1 gm-3km-1. In Fig. the general
pattern remains similar with depolarisation increasing with increasing
Γl and effective radius but, as expected, the depolarisation
ratios are correspondingly larger with the larger FOV. More example
results of the MC calculations are shown in continuous form in
Fig. . In all these examples the lidar laser
divergence was modelled as being Gaussian with a 1/e full width of
0.1 mrad.
The MC calculations predict depolarisation profiles similar to those
observed in previous investigations e.g.. Note here that the
clouds are effectively semi-infinite, that is, they have a cloud top at
infinity, this leads to the prediction of a generally increasing depolarisation
ratio profile with penetration into the cloud. Observations in thin water clouds
often reveal that the depolarisation ratio may exhibit a peak
which is associated with the penetration of the lidar signal to the cloud-top
region or beyond .
Figures – are
informative and show that the shape of the return signals and the
associated depolarisation ratio is a well-defined function of the LWC
and effective radius profile. However since the extinction profile
itself is a function of both the LWC and Reff profiles, the
variations shown in
Figs. – are the
result of changes in both the single-scattering return and the
associated multiple-scattering contributions. Using Eq. ()
it is possible to interpolate between the MC look-up-table entries to
examine how the signal and depolarisation ratio profiles behave as
a function of Reff,100, while the extinction profile is held
constant, thus isolating the effects of MS. Such an example is shown
in Fig. where the para, perp and
depolarisation profiles are shown for values of α100=5 km-1 and 10 km-1 (the extinction coefficient at 100 m
from cloud-base) as a function of Reff,100. If MS was not
occurring, there would be no variation present in the para profile as
Reff,100 changes and practically no perp signal would exist at
all. As it is, a clear dependence on Reff,100 is present in the
para and perp attenuated backscatters and in the depolarisation
ratios.
Example results of MC calculations for a lidar wavelength of 355 nm
corresponding to semi-infinite clouds with a cloud-base of 1.0 km for two
values of α100 as a function of Reff,100 for a lidar FOV of 1.0 mrad.
A fixed value of γ=9 was used to generate the results shown in
Figs. –. Other
simulations (not shown) conducted with γ=2 indicate that, for
FOVs ranging from 0.5 to 2.0 mrad, the values of the para and
perp signals and the associated depolarisation ratio change less than
10 % so long as Reff,100 is greater than 3 µm. For values of
Reff,100 of 2 µm the depolarisation ratio profile remains the
same within better than 10 %; however the shape of the normalised
para and perp returns past the peak para signal altitude can change by
up to 0.1 in absolute terms. This is likely not entirely due to
changes in the relative MS contribution but more to do with the fact
that for small effective radius values that the details of the phase
function itself becomes sensitive to the width of the distribution and
that even the approximate that α=2π〈r2〉 itself starts to
break down.
Information content: towards an inversion scheme
Figures – strongly
suggest (within the confines of our simplified model of cloud
structure) the possibility that microphysical information can indeed
be extracted from depolarisation lidar measurements. However it is
necessary first to examine the degree of uniqueness of the information,
i.e. how distinct are the signals corresponding to one distinct
(α100,Reff,100) pair from the set of all possible
observed signals corresponding to other (α100,Reff,100)
pairs. In order to do this, here we make use of the following simple
prototype cost function applied to our look-up-table results
χ2(α100,j,Reff,100,k;α100,Reff,100)=∑i=ibitB∥(zi;α100,Reff,100)-B∥(zi;α100,j,Reff,100,k)σB∥(zi;α100,Reff,100)2+B⟂(zi;α100,Reff,100)-B⟂(zi;α100,j,Reff,100,k)σB⟂(zi;α100,Reff,100)2,
where i is the altitude index with ib being the bottom and
it the effective layer top indices. The indices j and k refer
to the entries in the extinction and effective radius dimensions of
the look-up tables. Keeping in mind our goal of developing a practical
inversion algorithm and noting the fact that lidars are usually not
well calibrated in an absolute sense, Eq. () makes use
of the backscatters normalised by the maximum value of the parallel
attenuated backscatter on a profile-by-profile basis, that is, B∥ and B⟂ where
B∥(z)=ATB∥(z)max(ATB∥(z)),B⟂(z)=ATB⟂(z)max(ATB∥(z)),
and the σ terms in Eq. () represent the respective
uncertainties, which in relative terms for actual measurements, will
be in the range of a few percent above the immediate cloud bottom
region and increasing to a few tens of a percent with increasing
penetration into the cloud past the altitude of maximum return. Note
also that there is a implicit dependence of χ2,
B∥ and B⟂ on the cloud-base altitude and the
lidar FOV.
Using B∥ and B⟂ avoids the need for an
absolute calibration of the lidar and accounting for the transmission
between the lidar and cloud base. It is also useful to consider the
altitude range of the signals to treat. Taking into account that, with actual
observations, the below-cloud return will vary according to the
possible presence of below-cloud drizzle and varying aerosol loads
together with the finite SNR levels achievable as the lidar signal becomes
attenuated as it penetrates into the cloud, we limit the altitude range
to consider (zib-zit) according to the following
criteria:
zi≥zb
B∥(zi)≥10-2,
where here the value of the cloud base (zb) is known precisely.
Normalised results from the application of Eq. () for
a lidar wavelength of 355 nm as a function of α100,k and
Reff,100,k for two true values of α100 and
Reff,100 (as indicated) for a lidar FOV of 1.0 mrad and a cloud base of
1 km. The symbol is used to mark the location of the minimum of the cost
function.
Clearly Eq. () achieves its global minimum at the
point where the tabulated extinction and effective radius values match
the specified values of α100 and Reff,100
(i.e. χ2=0 at the point where α100,j=α100
and Reff,100,k=Reff,100). How well defined the global minimum
Eq. () is and if other local minima exist strongly
indicates the accuracy and precision we may expect in any inversion
procedure based on minimising Eq. () or similar
function. Normalised values of Eq. () (with σB∥ and σB⟂ proportional to B∥
and B⟂, respectively) for a lidar FOV of 1 mrad are presented
in Fig. . Here it can be seen that, as
expected, a well-defined global minimum exists where
α100,j=α100 and Reff,100,k=Reff,100 and that
the minimum is sharper for the smaller particle size cases. It can
also be seen that, in spite of the unique global minimum, that the
topology is complicated and that local minima exist. This indicates
that in any eventual practical inversion scheme care must be
taken so that the inversion scheme converges to the global minimum
rather than one of the local minima. It can also be seen that the
minima are less elongated along the effective radius axis for the
Reff,100=3µm than the Reff,100=9µm cases. This is
expected, since as the particle sizes increase, the associated forward-scattering lobe (which in the large particle limit contains 1/2 the
scattered energy) will eventually become much smaller than the lidar
FOV, leading to the decreased ability to distinguish between different
particle sizes since practically all the forward scattered light will
remain within the FOV. Results similar to those shown in
Fig. for a different FOV of 0.5 mrad are
shown in Fig. . Here it can be seen that,
when compared to the 1.0 mrad case, the minima associated with
the Reff=9.0µm cases are less elongated along the effective
radius axis when compared to the FOV = 1.0 mrad case. This is
a demonstration of the fact that smaller field of views allow more
sensitivity at larger particle sizes. The reason for this is similar
to the reason for the reduced sensitivity to larger particle sizes
discussed above in relation to Fig. .
As Fig. except for a receiver FOV of 0.5 mrad.
Effect of depolarisation calibration and FOV uncertainty
Our prototype cost function (Eq. ) does not depend
on the lidar backscatter signals being calibrated in a absolute sense;
however the para and perp channels must be calibrated in a relative
sense. Further, for many practical depolarisation lidars, a degree of
cross-talk between the two channels exist so that in practice one can
write
ATB∥=z21-δCP∥+δCP⟂
and
ATB⟂=Crz21-δCP⟂+δCP∥,
where δC is the polarisation cross-talk parameter and Cr is the
inter-channel depolarisation calibration constant .
Example results of a simulated 20 % error in the value of Cr are
shown in Fig. . Here it can be seen that the
location of the minimum can be shifted substantially by an error in
Cr with the effect being generally felt more by the effective
radius values. For practical lidar systems δc may be on the
order of a few percent or less; thus even a 50 % error in
the value of δc only produces a much smaller relative effect
than a 20 % misspecification of Cr. Roughly speaking, we
conclude that in order to be able retrieve Reff,100 to within
10 % Cr should be known to better than 5 %, while, for
typical cross-talk values, δc should be known to within
about 50 %.
Normalised results from the application of Eq. ()
as a function of α100,k and Reff,100,k for true values of
α100=10 and 30 km-1 and true values of
Reff,100=9µm with perturbed values of
Cr. For the left panels Cr has been set
20 % too low while in the right panels Cr has been
set 20 % too high. The white symbols show the location of the true
values of Reff,100 and α100 while the red symbols mark
the position of the actual cost-function minimum in each case.
A similar exercise was carried out to examine the sensitivity of the
results to the lidar FOV. It was found that, in general, a 15 % error
in the assumed lidar FOV leads to less than a 5 % error in the
extinction and/or effective radius. Since the FOV of lidar systems
are generally known better than a few percent, we consider this error
will generally be a secondary source of error in comparison with the
errors associated with the depolarisation calibration.
Inversion scheme
On the basis of the results presented in the previous section we
conclude that a practical inversion scheme is possible. That is, given
a measurement of B∥ and B⟂, useful estimates of
α100 and Reff,100 can, in principle, be produced by
finding the global minimum of Eq. () or similar
function. However on a practical level care should be taken in the
initialisation of the inversion scheme (due to the presence of
multiple local minima) and errors in the depolarisation calibration
(e.g. Cr and δc) should be taken into
account. Further, since the prototype cost function uses normalised
attenuated backscatters, the error in the normalisation should also
be accounted for. Accordingly, our practical inversion scheme
requires a more flexible functional form for our cost function. In
particular, we will use the following optimal-estimation
cost function:
χ2(x;xa,Se,Sa)=y-F(x)⊤Se-1y-F(x)+x-xa⊤Sa-1x-xa,
where x is the state vector, y is the observation vector and
F(x) is the forward model estimate of the observations;
Se is the observational error covariance matrix,
xa is a vector containing an a priori estimate of
the state vector and the a priori error covariance matrix is denoted by
Sa. As with the case with Eq. (),
the altitude limits of the summation are subject to the same conditions as
listed with Eq. (), with the additional constraint that
altitudes past the maximum of the observed depolarisation profile are not
considered. This is due to the fact that a sustained drop in the
depolarisation profile is expected to be associated with penetration into the
cloud-top region or beyond.
The observation vector (y) is composed of the observed elements of
B∥ and
B⟂ as defined by Eqs. () and ():
y=[B∥,1,B∥,2…B∥,nz,…B⟂,1…B⟂,nz].
The state vector (x) is defined as
x=ln(CN),Δzsin(ϕzp),ln(Cr),ln(δc),ln(Reff,100),ln(α100),
where CN is a factor introduced to account for any error in the
signal normalisation process, Δz is the range resolution of
the observations and sin(ϕzp) is a factor (constrained
by the use of the sine function to be between -1 and 1) used to account for
the uncertainty in assigning the altitude of the peak return (see
Step 1 in Sect. ) due to the finite vertical
resolution of the measurements.
The forward model vector (F(x)) is defined as
F(x)=CNmax(ATB∥)ATB∥,1…ATB∥,nz,ATB⟂,1…ATB⟂,nz,
where ATB∥ and ATB⟂ given by
Eqs. () and (), respectively, with
P∥ and P⟂ determined by interpolation using
the pre-computed look-up tables. Before interpolation in
α100 and Reff,100, the profiles are shifted in altitude
by an distance given by Δzsin(ϕzp) and then
binned to a vertical resolution matching the observations. The look-up
tables have been computed at a resolution of 5 m while, in this
work, the observations we will consider are at a resolution of 15 m.
The elements of the error covariance matrix for the observations
(Se) can be found by calculating the expectation value of the
difference between the observations and the optimal forward model fits:
Se,i,j=E(yi-Fi)(yj-Fj).
Accordingly, for simplicity if we ignore the correlation in the
para and perp signals due to δC, it can be shown that
Se,i,j=σCN2yiyj:1≤i≤nz,1≤j≤nz,i≠j=σyi2+σCN2yi2:1≤i≤nz,i=j=σCr2+σCN2yiyj:nz<i≤2nz,nz<j≤2nz,i≠j=σyi2+σCr2+σCN2yi2:nz<i≤2nz,i=j,
where σyi2 is the variance assigned to yi which
is estimated by averaging the observations themselves in time as
a function of altitude and σCN2 is the estimated
variance of CN which is similarly estimated from the
observations. σCr is the a priori uncertainty in the
depolarisation inter-channel calibration factor.
In our procedure, we assign a priori estimates to the depolarisation calibration
parameters (Cr and δC) and the normalisation factor CN, all
other factors are unconstrained by any explicit a priori. Thus non-zero
elements of the inverse of the a priori error covariance matrix are given by
Sa,1,1-1=σCNCN-2Sa,3,3-1=σCrCr-2Sa,4,4-1=σδCδC-2,
where we have assumed that the a priori estimates are all
uncorrelated. Here σδC is the assumed a priori
uncertainty in the depolarisation cross-talk factor. The Sa,2,2
term is zero since no a priori knowledge is assumed for the
ϕzp term in the state vector; however the term
sin(ϕzp) is still constrained by its very nature to be
between -1 and +1.
Once the cost function is minimised, the retrieved values of
α100 and Reff,100 can be used along with Eq. () to
find Γl, while No can be found via Eq. (). The
covariance matrix of the retrieved parameters
(CN,Cr,δC,α100,Reff,100) are found using standard
approaches e.g., and standard error propagation
techniques are then used to find the resulting error estimates for Γl and
No including the effects of the uncertainty in k.
The form of the cost function and state vector presented here was found to be
lead to rapid and reliable convergence, but should not be regarded as
definitive. The reader should be aware that other strategies may be more
appropriate, depending on the SNR of the observations and the
availability (or lack thereof) of useful a priori information. For example,
No and Γl could be used instead of
α100 and Reff,100. This would enable a priori estimates
of both No and Γl to be taken into account as
well as physical constraints, such as that the gradient of LWC, should
not be steeper than adiabatic. In our formulation, however it was found not to
be necessary to include a priori constrains on any state variables beyond
Cr, δc and CN.
Simulations: application to LES data fields
Visible MetoSat-SG Satellite image (top left) cloud optical
thickness (COT) field for an DALES simulation for the Cabauw
measurement site (bottom left) for the afternoon of 30 January 2011. Vertical extinction, LWC and effective radius slices
corresponding to the “y” = 5 km line indicated on the COT
plot (right panels). The red lines in the right panels indicate the
peak of the simulated lidar parallel attenuated backscatter while
the yellow lines indicate the cloud base returned by the retrieval
procedure.
In order to further develop and test the inversion procedure in
a manner which includes the effects of realistic cloud structure,
end-to-end simulations were conduced based on results from
LES model runs. In particular, output from the Dutch
Atmospheric LES model (DALES) was used. DALES uses
a bulk scheme for precipitating liquid-phase clouds. Condensed water is
separated into cloud water and precipitation. Cloud-droplet number
density is a prescribed parameter, while a two-moment bulk scheme is used to
treat precipitation . Temperature, pressure,
non-precipitable cloud water, precipitation water
content and precipitation droplet number density extracted from DALES
snapshots were used to create ECSIM scenes. ECSIM requires the
explicit specification of the cloud
(DSDs). The bulk scheme used in DALES does not provide explicit DSDs;
thus in order to build an ECSIM scene, it was necessary to impose
DSDs based on the LES output fields. For the precipitation mode
droplets the size distribution function embedded in the scheme of
was used. For the cloud droplets,
modified-gamma distributions (Eq. ) with a fixed value of
γ were assumed. Using the LES cloud LWC along with an imposed
value of No (which could be different from that assumed
internally in the LES model which in this case was 100 cm-3) together with
the assumed functional form of the size distribution then allows the DSDs to be
fully defined.
Once a scene was created, the ECSIM lidar and radar forward models
were applied to generate time series of simulated observations. The
ECSIM lidar and radar forward models both simulate the effects of the
respective virtual instrument footprints, sampling rate and instrument
noise levels (for more information see the ECSIM Models and Algorithm
Document, ). An example of a DALES-derived
cloud optical thickness (COT) field along with vertical slices corresponding to
the 355 nm extinction, LWC and Reff fields taken along the indicated
path is shown in Fig. . The scene corresponds to
a overcast stratocumulus deck with a degree of drizzle present. The LES
model was driven with boundary conditions corresponding to the
meteorological situation surrounding the CESAR measurement site
in the central Netherlands (52∘ N, 5∘ W)
on 30 January–1 February 2011. More details
concerning the meteorological context of this scene can be found in
. The scene shown here corresponds to
a snapshot at 16:00 UTC on the 31 January.
Simulated parallel and perpendicular attenuated backscatter signals
for a 355 nm depolarisation lidar with a FOV of 1 mrad. Also shown are the
corresponding linear depolarisation ratio and the radar reflectivity (Ze).
Here the cloud-droplet number density was fixed to a value of
85 cm-3. The scene has a horizontal resolution 50 m and
a vertical resolution of 10 m. The LWC panel shows the total
(cloud + precipitation/drizzle) water. Here the drizzle water content is
much lower the cloud water content and contributes little to the
extinction. However the presence of drizzle is clear in the effective
radius panel, particularly below the cloud base.
Virtual lidar and radar measurements corresponding to the track shown in
Fig. are shown in Fig. . Here a 355 nm
depolarisation
lidar with a field of view of 1 mrad was simulated along with the observed
radar reflectivity corresponding to a 35 GHz cloud-profiling radar with a pulse
length of 20 m and a simulated antenna diameter of 1.25 m. It can be
seen that the depolarisation ratio increases from cloud-base and decreases
sharply above cloud top, although it is quite noisy in this region. It can also
be seen that while the lidar measurements are apparently not strongly
influenced by the presence of drizzle the simulated radar signals are.
This is, of course, expected since the radar reflectivity is proportional to
the sixth moment of the hydrometer size distribution so that the radar
reflectivity is strongly impacted by the presence of even small numbers of
drizzle-sized droplets (see Eq. ).
Inversion procedure
An inversion procedure based on the minimisation of Eq. () was
developed and tested using the scene described above and other similar
DALES-derived scenes. The steps of the full procedure are outlined below.
Step 1: Averaging and binning of data
The altitude of the peak observed parallel lidar attenuated backscatter is
found for each profile. Each profile is shifted in altitude so that the peaks
match and then the desired number of profiles are averaged. The
uncertainties (the σyi2s) are estimated by evaluating the
corresponding variance profiles.
The logic behind this averaging strategy can be illustrated as follows.
In Fig. it can be seen that the altitude of the peak
return is not constant. Further, even in these simulations the cloud
base can be difficult to unambiguously define due to the variations in
cloud altitude and the presence of sub-cloud drizzle. When dealing
with real observation the additional complicating factor of the
presence of growing aerosol particles may also complicate the
determination of the effective cloud base. In our procedure, we
largely avoid the need to very accurately identify cloud
base
directly from the observation by using the peak of the observed
parallel lidar attenuated backscatter as our reference. The minimum
altitude considered in the inversion procedure is based on a threshold
of Bpara=0.05 (which likely overestimates the true cloud base
but largely avoids drizzle and aerosol effects), while an estimate of
the true cloud base can be produced as a by-product of the fitting
procedure determined by the optimal fit to the observations.
Step 2: Initialisation of minimisation procedure
From the investigations into the structure of Eq. ()
we know that spurious local minima in our cost function likely
exist. For this reason it is necessary to specify an appropriately
close initial guess when numerically minimising Eq. (). It
was found that a simple grid search of 10–15 values of α100
between 1 and 30 km-1 and Reff,100 between 3 and
12 microns with the values of CN,Cr,δC set to their
respective a priori values was appropriate in order to find a suitable
initial guess for the minimisation procedure.
Step 3: Minimisation of Eq. ()
A two step method to minimise the cost function was implemented in a robust
manner. First we apply the gradient-free Nelder–Mead simplex algorithm
. Then, as an additional convergence check and
to improve the accuracy of the minimisation, the simplex algorithm results
were
followed by an application of Powell's algorithm
. Finally, as described in
, after convergence the curvature matrix around
the minimisation point was numerically evaluated and the resulting covariance
matrix of the retrieved parameters was found.
Inversion results
Results of the retrieval applied to the simulated lidar data
along for two columns (corresponding to x=2.0 and 2.5 km). Here the
black lines are the simulated observations at a vertical resolution
of 15 m and a horizontal resolution of 400 m while the
corresponding depolarisation ratio is given by the green line. Here
Cr was set to 1.0 and the depolarisation cross-talk parameter
(δC) was set to 0.3. The red lines are the fits to the
parallel and perpendicular attenuated backscatter and the blue line
is the corresponding fit depolarisation ratio.
Two sample inversion results corresponding to x equal to 2.0
and 2.5 km are shown in Fig. . Here it can be seen
that a very good match between the simulated observations and the results of
the retrieval procedure are obtained. The results shown here correspond to
a horizontal averaging of 0.2 km which corresponds to averaging across
five
consecutive simulated lidar profiles. It is interesting to note that that
the simulated signals bear a striking similarly to actual observations
extending even to the qualitative appearance of the signals above cloud top
.
Time series of inversion results as well as the true model values are
shown in Fig. . In this set of trials (which
contain the results presented in Fig. ), the
assumed error in Cr was set to 5 %, and for δC 20 % and
the a priori values were set to match the true values. The SNR of the
lidar signals themselves are functions of the signal strength but are
generally in the range of 20 to 40 for the case depicted here. It can
be seen that the agreement between the retrieval results for
α100 and Reff,100 as well as the derived variables
Γl and N is generally within 10 % or better on
a profile-by-profile basis.
Results of the retrieval applied to the simulated lidar data along
with the radar reflectivity simulated using the lidar results. Here the
black lines show the retrieval results with the grey bands indicating the
estimated 1-sigma uncertainty range. The red lines show the true values
extracted directly from the LES-derived model fields. The light-blue line in
the LWC panel indicates the value of the adiabatic (Γl) slope at cloud base. The
light-blue line in the reflectivity panel indicates the true reflectivity
levels if the contribution of the drizzle mode is removed.
The bottom panel of Fig. shows the radar reflectivity
corresponding to a level 100 m above the retrieved cloud base. In order to
predict the radar reflectivities corresponding to the lidar retrieval results we
note that the the relationship between radar equivalent reflectivity (Ze)
and LWC can, by rearranging Eq. (22) of ,
be written as
Ze=LWCρl48π|K||Kw|2ReffReff′4,
where |K| is the dielectric factor for water which is temperature
and frequency dependent and |Kw| is a reference value of |K|
corresponding to a fixed reference temperature. For our purposes at
35 GHz, |Kw| is fixed to a value 0.964. Reff′ is the so-called
lidar–radar effective radius and for spheres is defined as
Reff′=〈r6〉〈r2〉1/4.
Equation () can be re-written to emphasise the role played by the ratio
of
Reff′ to Reff. If we define
Rr≡Reff′Reff,
then we have
Ze=LWCρl48π|K|Kw|2Rr4Reff3.
For uni-modal size distributions of the type described by Eq. ()
the ratio of the lidar–radar effective radius to the normal effective radius is given by
Rr=(γ+5)(γ+4)(γ+3)(γ+2)31/4,
which varies between 1.13 for γ=9 and 1.28 corresponding to
γ=3. Thus for uni-modal distributions there is
a well-constrained relationship between reflectivity and the product of
the water content and the cube of the effective radius. However it is
well known that this is not the case in general if even small amounts
of drizzle are present e.g.. In particular, the
value of Rr yielded by Eq. () represents a lower limit
and multi-modal distributions can yield much higher values
. This will be considered in more detail in
Sect. and Appendix .
The continuous red lines in the bottom-panel of
Fig. show the true total reflectivity of the
drizzle and cloud droplets combined, while the light-blue line shows
the contribution of just the cloud droplets. It can be seen that the
reflectivity predicted by the lidar results is a consistently better
match to the cloud-only reflectivity. This is expected due to the lack
of sensitivity of the lidar signals to the presence of optical thin
drizzle. This result implies that it will be useful to compare the
radar reflectivity derived from the lidar inversion results to actual
observation (as will be done in Sect. ). Agreement, however can only be expected in non-drizzling conditions. Cases where the
observed Ze is greater than the predicted reflectivity levels may
indicate the presence of drizzle. However cases where the observed
Ze are less than the reflectivity levels predicted on the basis
of the lidar inversion results via Eq. () are not physical
and would indicate a problem with the observations or the inversion
procedure (e.g. convergence to the wrong minimum) or with the radar
calibration.
As well as the results directly presented here, several other trials were
conducted using the same scene but with the fixed cloud-number density set to
lower and higher values as well as trials where the a priori values of Cr
and δC were perturbed. It was generally found that the results were
largely insensitive to errors in δC but errors in Cr were
important. For example, it was found that a 5 % error in Cr coupled to
a similar a priori error estimate couple leads to errors in Reff,Γl and N
of 10–15 %. Runs were also conducted where the assumed lidar FOV was changed
from the true value. For example, if the true FOV was 1 mrad but the look-up-tables corresponding to 0.5 mrads were used to conduct the inversion then it was found that Reff was
overestimated by a factor of about 20–25 %, while Γl was overestimated by
about a factor of 27–30 %, leading to an underestimation of N by close to
a factor of 2.
Application to Cabauw observations
In this section, we describe the application of the depolarisation
lidar inverse procedure to a substantial number of instances of actual
observations. The inversion procedure was applied to about 150 selected periods ranging in time from a few 10s of minutes to several
hours of boundary layer (BL) stratus clouds observed at the
CESAR measurement site in the central Netherlands using
a depolarisation lidar operating at 355 nm. In particular, cases from
May 2008 (coinciding with the European Integrated project on Aerosol, Cloud, Climate, and Air Quality Interactions (EUCARI) impact campaign,
www.atm.helsinki.fi/eucaari/.) as well as cases from January and July
2010 were selected. The observational data used in this study are
freely available from the CESAR database
(http://www.cesar-database.nl/).
The actual data record of UV-depolarisation lidar observations is much
more extensive than the limited number of cases presented here; however the
immediate aim here is not to conduct an exhaustive analysis of the
results but to demonstrate the consistency and realism of of the
depolarisation inversion results. A more extensive application and
analysis is intended to be the focus of future work.
Illustration of the case selection criteria. Here all three of the
boxed areas satisfy the conditions of being well-defined stratus water layers.
However only the green outlined region appears to be connected to the
surface. The data consist of measurements made using the ALS-450 system at Cabauw.
Measurements and case selection
The UV-depolarisation lidar at Cabauw is a commercial Leosphere
ALS-450 lidar operating at 355 nm which has separate parallel and
perpendicular channels. The system has been in operation at Cabauw
since mid-2007 with breaks in the record ranging from weeks to several
months. The data was acquired with a vertical resolution of 15 m and
a temporal resolution of about 30 s. The depolarisation
inter-channel calibration factor and the corresponding cross-talk
parameters were estimated using the method described in
. The values of Cr and δC
were found to be stable between instrument servicing which occurred
between intervals ranging from a few months to a year. However within
certain periods the cross-talk (δC) appeared to vary
quasi-diurnally by up to 50 % (possibly linked to the temperature of
the unit). The field of view of the lidar was found to be stable
between servicing. The FOV of the lidar system was estimated by
fitting an overlap function to lidar signals acquired during selected
cloudless periods with low well-mixed BL aerosol burdens in
a procedure similar in nature to those described in
. The overlap model used was produced by
convolving Eq. (7.72) of with a Gaussian
function in order to model the effects of an divergent emitted laser
beam. The resulting overlap model is a function of the separation of
the transmitter and receiver optical axes, the effective beam and
receiver diameters as well as the effective beam divergence and
receiver FOV. The separation between the emission and receiver optical
axes and the beam and receiver diameters were found by physically
making measurements on the device itself. The fits then yielded
estimates of the effective beam divergence and the receiver FOV. As
was the case with the Cr parameter, the FOV was found to be
stable between instrument services and, depending on the particular
time interval, the FOV was found to vary between about 0.5 and
1.5 mrad.
An example of the type of observation that was selected for analysis is
presented in Fig. . It is our intention to focus on
well-defined warm cloud layers. Further, as will be presented and discussed
later in Sect. , we wish to compare our derived cloud-number density estimates to aerosol number concentration measurements made near
the surface. Thus we further limit our focus to layers that appear to be
physically linked to well-mixed boundary layers. In Fig. all
three of the boxed regions are well-defined stratus layers. However the
higher altitude regions are clearly above the top of the boundary layer as
indicated by the sharp gradient in lidar signal present at about 2.4 km.
As well as the lidar measurements, we also make use of the 35 GHz
lidar observations at Cabauw. The cloud radar is a vertically pointing
Doppler radar with a vertical resolution of 89 m and a temporal
resolution of approximately 15 s. Further details of this system are
given in . For the periods involved in this study
the radar reflectivity calibration uncertainty is thought to be in the
range of 2–3 dBZ.
Examples
Observed lidar and radar signals for 15 January 2011 at Cabauw from 16:00 to 18:00 UTC.
Sample lidar and radar data as a function of altitude and time are
shown in Fig. for 15 January 2011 from 16:00 to 18:00 UTC. Here a stratus layer is present with the cloud base varying
between 0.75 and 0.85 km. The lidar data have a vertical resolution of
15 m and a temporal resolution of 30 s. The corresponding
normalised attenuated backscatter as a function of distance from the
altitude of the peak parallel return (binned to a temporal resolution
of 3 min) as well as two sample inversion results are shown in
Fig. . By comparing the lidar data shown in
Fig. against that shown in
Fig. it can be seen that the profile-to-profile
variation is indeed reduced. The sample fit results indicate that the
observed signal profiles largely conform to our expectations based on
the look-up-table values themselves and the LES simulation-based
results discussed earlier (e.g. those presented in
Fig. ); however some differences may be
noted. First, the cloud base is generally not as sharply defined in
the actual measurements as in the LES-based simulations. One possible
reason for this is presence of drizzle, especially likely in the
earlier profile which is below an area of elevated radar
reflectivity. Another likely factor is the existence of small-scale
variability at scales finer than the resolution of the LES
simulations. Still another reason may be due to the presence of
not-quite activated but still strongly growing aerosol present
just below cloud base. Another notable difference between the LES
simulation-based results and the observation is that the observed
depolarisation ratio above 100–150 m from cloud base is often (but not
always) less than expected on the basis of the look-up-table
calculations and the LES-based simulations. This is presumably due to
the departure of the real-cloud structure from our assumption of
constant LWC slope and constant N (due to e.g. the effects of
mixing at cloud top). That this observed behaviour is linked to slight
non-linear effects in the lidar signal detection can also not be
strictly ruled out.
Normalised parallel and perpendicular attenuated backscatters as
a function of altitude from the peak of the observed parallel backscatter profile
(left panels) corresponding to the data shown in Fig. .
Example fit results are shown on the right for 16.49 and 16.92 UTC.
Retrieved time series of Reff,100, Γl and N
for 15 January 2011
from about 16.4 to 17.1 UTC. The light-blue line in the Γl plot
indicates the adiabatic limit at cloud base. The black line in the Ze
panel shows the reflectivity predicted by Eq. () corresponding to
the first 100 m radar bin fully above the estimated cloud base while the
red line shows the corresponding actual radar observed value. The radar
calibration uncertainty (not indicated) is thought to be in the range of 3 dBZ.
In spite of these two main differences, generally very good fits for
the first 100–150 m from cloud base are found. Time series of the
inversion results corresponding to Fig. are shown
in Fig. . Here it can be seen that Reff,100′
appears to have been fairly constant at about 4 µm and is
retrieved with an estimated error of about 30 %, while the Γl
values are generally about 40 % of the adiabatic value. The cloud-number concentration values are fairly constant with an average value
of about 400 cm-3 and an estimated uncertainty on the order of
25 %. A comparison between the reflectivity predicted using the lidar
inversion results using Eq. () and the observed values is
shown in the middle right panel of Fig. . In order
to conduct the comparison, the radar data were binned to the same time
resolution as the lidar inversion results. To avoid the effects of
partially filled radar bins near the cloud base, for each inversion
time step, the altitude limits corresponding to the first radar height
bin fully above the cloud base returned by the inversion procedure
were found. These altitude limits were then used to average the lidar
predicted Ze. Here it can be seen that, similar to the LES-based
results, e.g. bottom panel of Fig. , the
results are generally within a few dBZ of each other with the
observations higher by about 2–3 dBZ. This bias is consistent with
the presence of low amounts of drizzle. Given the uncertainty in the
radar calibration one can not be conclusive but the fact that the
agreement between 16.9 and 17.0 UTC is in the region with the lowest
reflectivities is also suggestive of drizzle being the cause of the
offset. Past 17.0 UTC, the lidar results predict more reflectivity
than was observed. This is not physical and points either to
a problem in the lidar retrieval for this time period or, which is
considered more likely in this case, that here partially filled radar
bins likely could not be avoided. This is based on the fact that for
this time period the cloud was likely physically thinner than 200 m
which is equivalent to about 2 radar pixels in height.
As Fig. except for data corresponding to 4 January 2011.
Results from a second example time period corresponding to 4 January 2011 between about 18 and 19.7 h UTC are shown in
Fig. . Here it can be seen that retrieved
parameters are roughly in the same range as the results shown in the
previous figure; however in general the estimated uncertainties in
the retrieved quantities are more variable and generally larger. This
may be linked to the fact that the lidar observations contain more
profile-to-profile variability than the previous case or the fact that
drizzle is more prevalent in this case. This is evident by examining
the reflectivity panel along with the panel in which the comparison
between the lidar derived and observed Ze is shown. The regions
of detectable reflectivity present below the lidar-derived cloud
base and the occurrence of reflectivities values above -25 dBZ are both
indicative of the presence of drizzle.
By comparing the predicted and observed Ze values for this case
it can be seen that good agreement between the lidar-derived Ze
values and the actual radar observations is present past about 11.25 UTC, which is associated with cloud-base region reflectivities below
about -35 dBZ. For earlier time periods the observed radar
reflectivity is substantially higher than the lidar predicted
values. These periods are associated with cloud-base reflectivities
above -30 to -35 dBZ which are known to be associated with the
presence of drizzle at cloud base e.g..
Comparison with cloud radar reflectivity measurements
As illustrated by the two specific example comparisons between the
observed radar reflectivity and that modelled using Eq. ()
presented in the previous section, the observed reflectivity values
are often apparently impacted by the presence of drizzle. This notion
is explored in a more quantitative fashion in Appendix where
we use a bi-modal cloud and drizzle size distribution representation together with
Eqs. ()–() applied to the full 3-month
set of cases. As discussed in detail within Appendix , it was
found that the lidar predicted Ze values are largely consistent with
the full set of co-located radar observations (see
Fig. ). In particular it was found that
Instances with observed values of Ze below those expected
from the application of Eq. () are rare.
Drizzle, on average, makes a substantial contribution to the
observed cloud-base region reflectivity for reflectivities above
about -35 dBZ.
The relationship between the observed and predicted
reflectivities are broadly quantitatively consistent with those
predicted using a bi-modal size distribution model where the ratio
of the drizzle mode number density to the cloud-droplet number
density is on the order of 10-4 to 10-1 and values of LWC in
the range of 0.05–0.1 gm-3, respectively.
At this point in time, due to the lack of an independent means of
assessing the drizzle contribution to the reflectivities, the
comparison between the lidar predicted and observed cloud-base region
reflectivities can not be taken as definitive validation of the lidar
results. However it can be robustly stated that the lidar results are
indeed physically consistent with the observed radar reflectivities.
LWC near cloud base
Adiabatic cloud-base liquid water lapse-rate values
(Γl,a) and the
corresponding lidar-derived values (Γl). The thin-dashed line
represents
the one-to-one line (adiabatic fraction = 1), while the thicker solid
and dashed lines show the observed relationship based on the
chi-square mean observed fraction and corresponding uncertainty.
In addition to the comparison with the radar observations, an other
independent evaluation criteria to judge the realism of the lidar
results is the comparison of the lidar-derived Γl values
with the corresponding adiabatic values (Γa). Using
temperature and pressure data for Cabauw extracted from atmospheric
analyses, the adiabatic liquid water mixing ratio lapse rate was
calculated for the times and cloud-base altitudes of the lidar
observations. A comparison between the adiabatic values and the
observed values are shown in Fig. . Here it can be seen
that the lidar observations imply a cloud-based adiabatic fraction of
0.451±0.007. Only a few observations approach the adiabatic
limit and none exceed it in a statistically significant manner. It can
be noted that the sub-adiabatic fraction values seen here are within the
range of previous in situ-based observations
e.g. which were
interpreted to be largely the result of entrainment at cloud
base. Further, it is interesting to note that recently a new method of calculating cloud droplet number concentration in the cloud-base region was proposed by
. This leads to the finding that the ratio of
supersaturation to the liquid water mixing ratio at the altitude of
maximum supersaturation should be universal. That is, at the height
of maximum supersaturation, which for stratus clouds is reached within
a few 10s of metres from cloud base, the adiabatic fraction should be
independent of updraught velocity or number density.
predicted that this universal value should be
equal to 0.44 (see Eq. 11 of ). The value of
0.44 compares very favourably with our finding of 0.451±0.007,
although more work and consideration would have to be done to properly
judge the significance of this result.
Comparison with Scanning
Mobility Particle Sizer aerosol measurements
At this point we feel that enough confidence in the depolarisation
lidar-derived products has been accumulated so that a preliminary
comparison between aerosol number concentrations and lidar-derived
cloud-base number concentrations is feasible. As well as the remote-sensing equipment, Cabauw also hosts a number of in situ probes
including a Scanning Mobility Particle Sizer (SMPS) instrument which
measures aerosol size distributions between diameters of 10 and
470 nm. As described in , the SMPS instrument
is housed in the basement of the Cabauw meteorological tower but the
instrument is connected to a laminar flow sampling tube with an inlet
at 60 m elevation so that the sampled air is expected to be more
representative of the BL as a whole. Loss of some particles on the
sampling tube walls does occur but this has been corrected for and,
for the measurements used here, is not expected to be a significant
source of uncertainty.
Previous aircraft-based studies have found correlations between
aerosol number density and cloud-droplet number concentration. For
example, by using number concentrations of aerosols measured with an
Passive Cavity Aerosol Spectrometer Probe (PCASP) (which measures
particles with diameter between 0.13 and 2 µm) and co-mounted
Forward Scattering Spectrometer Probe (FSSP) cloud-droplet
measurements, were able to demonstrate statistically
significant relationships between the aerosol and cloud-droplet
measurements. The observed relationship between the lidar-derived
cloud-number densities Nd and the tower-based SMPS measurements
is shown in Fig. . Here, following
, the aerosol number concentrations shown are
representative of particles with diameter greater than 50 nm. This was
done to be consistent with the earlier data upon which the previous
empirical relationship are based. The aerosol concentrations were also
adjusted for the difference in air density between the ground and
cloud base by assuming that the aerosol number density mixing ratio is
conserved.
Lidar-derived cloud-base number density (Nd) and SMPS
ground-based number density of aerosols with radii greater than
0.025 µm adjusted for density at the cloud-base altitude
(Na). The symbols follow the same conventions as
Fig. . The light-blue lines (labelled 1–5)
represent previously defined independent empirical relationships
based on in situ measurements made with the aid of aircraft. Line-1
corresponds to Eq. (), 2 corresponds to Eq. (),
3 corresponds to Eq. (), 4 corresponds to Eq. ()
and 5 corresponds to Eq. () The grey lines (6) show the
fit to the points produced using Eq. () along with the 1-sigma error
bands.
A number of empirical relationships relating aerosol number concentration to
cloud-droplet number density for warm stratus clouds under different conditions
were compiled by and . In Fig. lines
1–5 represent independent relationships draw from previous aircraft based work
and are listed in . Here line 1 corresponds to
log10(Nd)=0.257log10(1.22Na)+1.95,
which was originally found by . Line 2 corresponds to
Nd=-765.5+395.71log10(Na).
Line 3 corresponds to
Nd=-698.4+356.61log10(Na)
and line 4 corresponds to
Nd=-382.15+215.83log10(Na).
Lines 2–4 were found by . Line 5 is given by
Nd=-27.9+0.568Na-2.1×10-4Na2
and was originally found by . Line 6 represents the
relationship found in this present work via chi-square fitting and is
given by
Nd=(-547±8)+(291±4)log10(Na).
The fit error bounds here were found using a bootstrap method
. All of the above relationships are
roughly consistent with each other and with the findings of this
present study. That they are similar, even though the aerosol chemical
composition may have been different between the different studies and
time periods, may be due to the fact that, so long as the aerosol is
not hydrophobic, size plays a dominant role in determining
whether a given aerosol particle can act as a CCN or not
. The fact that the results obtained using the
lidar-derived cloud-base cloud-number concentrations and tower-based
measurements aerosol measurements are consistent with earlier
completely independent studies strongly supports the validity of the
lidar inversion results and further supports the notion that under
apparently well-mixed BL conditions (as assessed by evaluation of the
lidar backscatter measurements) the tower-based aerosol number
concentration measurements are indeed representative of the BL as
a whole.
Conclusions
In this work a novel method for determining cloud-base properties by
exploiting the signature of multiple scattering on depolarisation
lidar signal was developed. The method is novel yet firmly based on
older established ideas and principles. The inversion procedure has
not been evaluated against direct measurements (e.g. coincident
in situ measurements). However even at this arguably preliminary
stage, we have a high degree of confidence in the results. This
confidence is based on the following considerations:
There is a rather direct connection between the variables determining the relevant
lidar radiative transfer (e.g. Extinction and effective cloud particle radius) and the cloud physical parameters of interest (e.g. liquid water content and cloud droplet number concentration).
Application of the method to LES generated clouds shows that,
within reasonable limits, the method is robust to deviations from
strict adiabatic cloud structure, the presence of drizzle and
variations in cloud-base altitude.
Under low-reflectivity conditions, where the reflectivity
contribution of the drizzle droplets can be neglected, it has been
demonstrated that lidar results can be used to predict the observed
radar reflectivity within the uncertainty of the radar
calibration. Under general circumstances, where drizzle is present
the range of bi-modal size distribution parameters required for
consistency between the lidar and radar measurements are well within
the range of accepted values.
Cloud-base LWC values were found to never exceed the adiabatic
limit by an amount outside of the respective error
estimates. Further, the observed average cloud-base adiabatic
fraction is consistent with the range of previous observations.
The results obtained by comparing the polarisation lidar-derived cloud-base cloud-droplet number concentrations with
tower-based aerosol number concentrations yields a relationship
consistent with completely independently derived relationships based on
previous in situ aircraft-based measurements.
The evaluation examples presented in this work represent a small
fraction of the data available from the Cabauw site. A more extensive
application of the method to the Cabauw data should be
conducted. Additional, further validation work (possibly involving the
use of in situ cloud measurements from, for example, the EUCARI/IMPACT
campaign) should be carried out. In this work, we
have used results from a commercial UV depolarisation lidar that was
not developed with this application in mind. Future developments could
be imagined involving the optimisation of instrument parameters
(e.g. wavelength, FOV, dynamic range) directed towards the
implementation of the method developed here including the possible
integration of multiple fields-of-views. Further, it should be noted
that, depending mainly on the instrument FOVs, the methods described
in this work may be applicable to a large body of existing lidar
measurements made by depolarisation lidars operating in the UV as
well as in the visible (e.g. 532 nm) wavelength ranges.
A key variable lacking in the examination of the relationship between
the cloud-droplet number concentrations and the aerosol number
concentrations is knowledge of the characteristics of the vertical
velocities at cloud base. Such information may be difficult to
reliably extract from radar Doppler observations (as indicated by the
almost constant presence of drizzle at cloud base) but could be
reliably supplied by Doppler lidar measurements. Future studies
involving paired Doppler and depolarisation lidars are thus
recommended.
The technique described in this work is specific to the case of upward looking
terrestrial depolarisation lidars. The larger footprints involved and the
change from viewing cloud top instead of cloud bottom means that the specific
technique used in this work is not applicable to spaceborne lidars. If a
technique similar to the one presented in this work were applicable to
spaceborne lidars then the global information so obtained could be very
valuable, so the matter is worth considering.
The characteristics of the depolarisation return from water clouds has been
successfully exploited using CALIPSO lidar observations as a means to
determine cloud phase , but it is unclear at this point if the
approach used here could be usefully adapted to the case of spaceborne
lidars. In order for this to occur, a suitable model of cloud-top conditions
to serve the analogous role of the simple cloud-base model used in this work
would have to be identified or formulated. Second, extensive simulations
including, ultimately, the effects of expected noise levels, would have to be
carried out to determine what, if any, cloud microphysical information may be
recoverable. In particular, the larger footprints associated with space-based
lidars lead to relationships between such quantities as the integrated
backscatter and the integrated depolarisation ratio which
seem to be weakly dependent on the cloud microphysics when compared to the
case with ground-based observations. Nevertheless, CALIPSO lidar observations
and related simulations do suggest that microphysical information may indeed
be recoverable (see Fig. and the related discussion) but
further dedicated work would have to be carried out to establish this.