The fact that polarisation lidars measure a depolarisation signal in liquid clouds due to the occurrence of multiple scattering is well known. The degree of measured depolarisation depends on the lidar characteristics (e.g. wavelength and receiver field of view) as well as the cloud macrophysical (e.g. cloud-base altitude) and microphysical (e.g. effective radius, liquid water content) properties. Efforts seeking to use depolarisation information in a quantitative manner to retrieve cloud properties have been undertaken with, arguably, limited practical success. In this work we present a retrieval procedure applicable to clouds with (quasi-)linear liquid water content (LWC) profiles and (quasi-)constant cloud-droplet number density in the cloud-base region. Thus limiting the applicability of the procedure allows us to reduce the cloud variables to two parameters (namely the derivative of the liquid water content with height and the extinction at a fixed distance above cloud base). This simplification, in turn, allows us to employ a fast and robust optimal-estimation inversion using pre-computed look-up tables produced using extensive lidar Monte Carlo (MC) multiple-scattering simulations. In this paper, we describe the theory behind the inversion procedure and successfully apply it to simulated observations based on large-eddy simulation (LES) model output. The inversion procedure is then applied to actual depolarisation lidar data corresponding to a range of cases taken from the Cabauw measurement site in the central Netherlands. The lidar results were then used to predict the corresponding cloud-base region radar reflectivities. In non-drizzling condition, it was found that the lidar inversion results can be used to predict the observed radar reflectivities with an accuracy within the radar calibration uncertainty (2–3 dBZ). This result strongly supports the accuracy of the lidar inversion results. Results of a comparison between ground-based aerosol number concentration and lidar-derived cloud-droplet number densities are also presented and discussed. The observed relationship between the two quantities is seen to be consistent with the results of previous studies based on aircraft-based in situ measurements.

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

The penetration depth of lidars into water clouds (100–300

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.

Note that in this work FOV refers to the

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

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.

The cloud model (i.e. representation) used in this work is a simple
but still useful model of cloud-base conditions

Lapse rates, in general, are usually defined to
be the negative of derivatives of different quantities with respect to
height. Note that in this work, for convenience, the LWC lapse rate
(

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

In this work,

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

The ratio between the volume mean radius and the effective radius (

Based on the results of LES modelling of stratocumulus

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

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

Example results MC calculations for a lidar wavelength of 355

As Fig.

Example results of MC calculations for a lidar wavelength of 355

Range of parameters used in the MC calculations.

Using our cloud model, MC runs were performed for various values of

In this work, we fix the lidar wavelength at 355

In Fig.

The MC calculations predict depolarisation profiles similar to those
observed in previous investigations

Figures

Example results of MC calculations for a lidar wavelength of 355

A fixed value of

Figures

Using

Normalised results from the application of Eq. (

Clearly Eq. (

As Fig.

Our prototype cost function (Eq.

Example results of a simulated 20

Normalised results from the application of Eq. (

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

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

The observation vector (

The forward model vector (

The elements of the error covariance matrix for the observations
(

Accordingly, for simplicity if we ignore the correlation in the
para and perp signals due to

In our procedure, we assign a priori estimates to the depolarisation calibration
parameters (

Once the cost function is minimised, the retrieved values of

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,

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 “

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)

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,

Simulated parallel and perpendicular attenuated backscatter signals
for a 355

Here the cloud-droplet number density was fixed to a value of
85

Virtual lidar and radar measurements corresponding to the track shown in
Fig.

An inversion procedure based on the minimisation of Eq. (

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

The logic behind this averaging strategy can be illustrated as follows.
In Fig.

From the investigations into the structure 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

Results of the retrieval applied to the simulated lidar data
along for two columns (corresponding to

Two sample inversion results corresponding to

Time series of inversion results as well as the true model values are
shown in Fig.

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 (

The bottom panel of Fig.

Equation (

For uni-modal size distributions of the type described by Eq. (

The continuous red lines in the bottom-panel of
Fig.

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

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

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.

The UV-depolarisation lidar at Cabauw is a commercial Leosphere
ALS-450 lidar operating at 355

An example of the type of observation that was selected for analysis is
presented in Fig.

As well as the lidar measurements, we also make use of the 35

Observed lidar and radar signals for 15 January 2011 at Cabauw from 16:00 to 18:00

Sample lidar and radar data as a function of altitude and time are
shown in Fig.

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.

Retrieved time series of

In spite of these two main differences, generally very good fits for
the first 100–150

As Fig.

Results from a second example time period corresponding to 4 January 2011 between about 18 and 19.7 h UTC are shown in
Fig.

By comparing the predicted and observed

As illustrated by the two specific example comparisons between the
observed radar reflectivity and that modelled using Eq. (

Instances with observed values of

Drizzle, on average, makes a substantial contribution to the
observed cloud-base region reflectivity for reflectivities above
about

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

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.

Adiabatic cloud-base liquid water lapse-rate values
(

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

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

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

Lidar-derived cloud-base number density (

A number of empirical relationships relating aerosol number concentration to
cloud-droplet number density for warm stratus clouds under different conditions
were compiled by

Line 3 corresponds to

Lines 2–4 were found by

The fit error bounds here were found using a bootstrap method

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

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

The ECSIM lidar MC model is similar in principle to the MC model
described by

In order to accurately model the behaviour of the lidar depolarisation
signal under multi-scattering conditions it is not sufficient to
specify the phase functions and polarisation-dependent backscatter
coefficient. It is necessary to specify the full-scattering
matrix. The relationship between the Stokes vectors of the scattered
and incident electromagnetic fields (with respect to the scattering
plane) can be written as

If the target scatterers are rotationally symmetric or randomly oriented
then this relationship is reduced to

The phase matrix is defined in terms of the scattering plane (i.e. the
plane defined by the incoming and outgoing scattered photon
paths). Thus the Stokes vector of the incoming radiation and the
resulting vector describing the scattered radiation must be rotated
with respect to the scattering plane. The Stokes vector resulting from
a photon scattered through an angle

Schematic representation of the various types of paths involving forward and backscatter events for second- and third-order scattering.

Linear and circular depolarisation profiles in a C1 cloud at
a distance of 2

Layer integrated integrated backscatter vs. layer integrated
depolarisation ratio. The symbols show the results of 532

The angles

A MC approach models the propagation of the laser photons in
a stochastic manner. Photons are, in effect, launched and propagated
within a graded extinction field. In a pure MC procedure,
photons are launched from their source (in this case the laser) with
an initial direction vector. The photon then travels a distance

In a pure MC approach photons are tracked until they are absorbed, detected or exit the simulation area. This approach is simple and accurate. However it is not very efficient as any given photon has a very small chance of being scattered back to the lidar receiver. Thus it is desirable to modify the basic MC approach to increase its computational efficiency.

In order to increase the efficiency of the calculation, the ECSIM MC model analytically calculates the amount of unscattered energy transmitted from the lidar present at each altitude and then proceeds to calculate the higher-order scattering by launching a number of appropriately weighted photon packets from each altitude bin (here 2000–5000 photons per altitude bin where are typically used). As the photon packets propagate and scatter, for each scattering event, the relative amount of signal received by the lidar is analytically calculated based on the packet weight, the optical thickness and distance between the event and receiver and the phase function of the scatterer in question together with the scattering angle back to the receiver. This contribution to the detected power is stored and in order to conserve energy the same amount of energy is then removed from the packet by appropriately reducing its weight.

In order to further increase the efficiency of the calculation, the
well-know technique of forcing scattering of the photon packets to
occur within a specified distance from the receiver axis (in this case
5 times the receiver field-of-view footprint) was implemented

The aforementioned approach is exact and orders of magnitude faster than a pure MC approach. However by invoking an approximation concerning the contribution of different photon packet paths, the computational efficiency may be improved further still.

Since cloud particles are usually large compared to visible
wavelengths their phase function is strongly peaked in the forward
direction

The ECSIM MC code has been compared to other lidar MC codes as with
generally excellent agreement being found. For example, for a number
of benchmark Cirrus cases the results of the ECSIM MC code was
compared with the results from the MYSTIC MC code

Simulated linear and circular lidar depolarisation
(

It has been previously noted that a robust and tight relationship between layer
integrated attenuated backscatter and layer integrated multiple-scattering
depolarisation ratio for water layers exists, particularly in the case of
space-based lidar

Based on the comparisons with other MC codes and independent analytical calculations the ECSIM lidar MC calculations are robust for both ground-based and space-based simulations of lidar multiple scattering in water clouds. The ability of the ECSIM model to replicate the relationship between integrated depolarisation ratio and integrated backscatter as observed by the CALIPSO lidar, in particular, is regarded as a strong “stress test” of the code since it involves a much wider range of angles and higher orders of scattering compared to ground-based simulations.

The good agreement between the observed cloud-base reflectivity values and those predicted on the basis of the lidar inversion results under apparently non-drizzling conditions is qualitatively supportive of the lidar results being accurate. In order to assess the realism of the lidar results in a more quantitative manner, we will examine the relationship between the observed cloud-base reflectivities and the values predicted on the basis of the lidar inversion results in more detail.

As a starting point, we will assume that for the cases we have
selected that the lidar results are representative of the cloud
properties, and that the drizzle water contents and number densities
are small compared to the cloud water content and number densities so
that the lidar-derived LWC is approximately equal to the total water
content. This allows us to compare the lidar-derived cloud-based LWC
values with those estimated using the radar reflectivity alone.
A comparison between the cloud-base liquid water contents retrieved by
the application of the lidar inversion procedure for the entirety of
our data set and the corresponding observed reflectivities is
presented in Fig.

Relationship between observed cloud-base reflectivity and
corresponding
cloud-base reflectivity predicted from the depolarisation lidar inversion
results. The lines correspond to three different

If we assume a model that takes into account the bi-modal structure of
the cloud droplets together with the drizzle droplets, then we can
carry out a more quantitative evaluation. Following

For this type of bimodal distribution,

If

In order to generate the theoretical

Values of

In general, a good match between the observed and theoretical

The aerosol size distribution measurements used in this paper were partly funded by European Union Seventh Framework Program (FP7/2007–2013) under grant agreement no. 262254.

The LES simulations used in this work were sponsored by the Dutch National Computing Facilities Foundation (NCF) who provided the use of supercomputer facilities.Edited by: U. Wandinger