We performed extensive Monte Carlo (MC) simulations of
single-wavelength lidar signals from a plane-parallel homogeneous layer of
atmospheric particles and developed an empirical model to account for the
multiple scattering in the lidar signals. The simulations have taken into
consideration four types of lidar configurations (the ground based, the
airborne, the CALIOP, and the ATLID) and four types of particles (coarse
aerosol, water cloud, jet-stream cirrus, and cirrus). Most of the simulations
were performed with a spatial resolution 20 m and particle extinction coefficients

The proposed empirical model is a function that has only three free parameters and approximates the multiple-scattering relative contribution to lidar signals. It is demonstrated that the empirical model has very good quality of MC data fitting for all considered cases.

Special attention was given to the usual operational conditions, i.e. low
distances to a layer of partices, small optical depths, and quite narrow
receiver fields of view. It is demonstrated that multiple-scattering effects cannot be neglected when the distance to a layer of particles is about 8 km or higher, and the full RFOV is 1.0 mrad. As for the full RFOV of 0.25 mrad, the single-scattering approximation is acceptable; i.e. the multiple-scattering contribution to the lidar signal is lower than 5 % for aerosols (

It is well accepted that single-wavelength lidar signals from cloud or aerosol layers are affected by multiple scattering (MS) when the optical thickness is quite high, and/or the distance to a layer is large (see e.g. Winker and Poole, 1995; Bissonnette et al., 1995; and Winker, 2003). A large footprint of the receiver field of view (RFOV) is usually referred to as an intuitive justification of the multiple-scattering importance for signals of spaceborne lidars (see e.g. Winker and Poole, 1995, and Winker, 2003). For example, the Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP) has a footprint of about 90 m (Winker et al., 2010), which is “roughly two orders of magnitude larger than for ground-based or airborne lidars, due to the large distance from the atmosphere, allowing a much greater fraction of the multiply scattered light to contribute to the return signal” (Winker, 2003). It follows from Monte Carlo simulations of CALIOP signals that multiple scattering is of importance even though photon mean free paths are much larger than the footprint diameter, e.g. cirrus clouds or aerosol layers (see e.g. Winker, 2003).

If the distance to a cloud or an aerosol layer is low, the footprint is
rather small. For example, for a typical RFOV of 0.25 mrad and distance
to a layer of 8 km, the footprint diameter is 2 m; if the RFOV is 1.0 mrad, the
footprint diameter is 8 m. (Note that in this work RFOV refers to the full
angle.) If the distance to a layer is 1 km, e.g. an airborne lidar, the
footprint diameters become 0.25 and 1 m, respectively. Intuitively, one
may expect that the effect of multiple scattering on lidar signals can be
neglected with such low footprints and when the extinction coefficient of
turbid medium is quite low, for example, 1.0 km

A number of approximate models, i.e. non-Monte Carlo approaches to simulate lidar signals in multiple-scattering conditions, were developed from the 1970s to 2010s (see e.g. Bissonnette, 2005, and references therein; Eloranta, 1998; Hogan, 2008; and Hogan and Battaglia, 2008). A detailed analysis of those approaches is beyond the scope of this work. We only underscore that they are physically based; that is, some kinds of simplifications and/or approximations are employed, e.g. the time-dependent two-stream approximation (Hogan and Battaglia, 2008). Usually, the approximate models accept varying profiles or multiple layers of cloud and aerosol, and they are very fast as compared to Monte Carlo simulations. Moreover, the corresponding software, e.g. of the models by Eloranta (1998), Hogan (2008), and Hogan and Battaglia (2008), is freely available. At the same time, we believe that the accuracy level and the applicability bounds of the approximate models still need to be rigorously evaluated.

Some works devoted to Monte Carlo (MC) simulations of signals of
ground-based lidars were performed from the 1970s to 1990s (see e.g. Plass and
Kattawar, 1971; Kunkel and Weinman, 1976; Platt, 1981; Bissonnette et al.,
1995; and Ackermann et al., 1999). It was demonstrated that multiple scattering
affects lidar signals. At the same time, it should be mentioned that those
simulations were performed in conditions that were favourable for multiple
scattering: either with a high extinction coefficient (10 km

As for experimental data of ground-based or airborne lidars, it is common practice to assume that multiple scattering is negligible and can be ignored. Usually, that assumption is implicitly implied or mentioned with the relation to the following factors: a narrow RFOV, a small footprint, and a quite low value of the extinction coefficient. The only exception is cirrus clouds observed with a ground-based lidar; that is, the majority of works take into account multiple scattering employing one of the possible multiple-scattering functions (MSFs) (see the discussion in Appendix A) or models (see e.g. Nakoudi et al., 2021, and references therein).

To our knowledge, there exist no works where the applicability of the single-scattering approximation to lidar signals from low distances and low optical depths was thoroughly investigated. Such an investigation is one of the objectives of this work. It was performed using the Monte Carlo technique with special attention to quantitative data.

It follows from our extensive MC simulations that MS relative contribution to lidar signals has the same general behaviour as a function of the in-cloud penetration depth when plotted as a log-linear graph. That property is valid for a wide variety of particle properties, extinction-coefficient values, and lidar configurations (see figures in Sects. 5 and 6 below). Careful analyses of figures published in the literature confirmed that conclusion. The fact that a set of simulated data have the same general behaviour suggests the idea to search for a function which can provide a good fit to the data. Thus, the second objective of this work is to propose and test an empirical model which can be a simple and fast tool to compute multiple-scattering effects on lidar signals.

The organization of this paper is as follows. The methodology and conditions of our Monte Carlo simulations are presented in Sect. 2. Section 3 is devoted to the mathematical background and the analysis of some general features of multiple-scattering impact. Our empirical model of the multiple-scattering effect is discussed in Sect. 4. Section 5 is devoted to results of our MC simulations and fittings with the empirical model for cases of low distances and small optical depths. Section 6 is devoted to cases when the impact of multiple scattering is high, i.e. spaceborne lidars, high values of the extinction coefficient, and wide RFOVs. Some important methodological questions are discussed in Appendices A and B.

The principal tool to simulate lidar signals was the McRALI (Monte Carlo Radar Lidar) software developed at the Laboratoire de Météorologie Physique (Alkasem et al., 2017; Szczap et al., 2021). The software employs a forward Monte Carlo (MC) approach along with the locate-estimates method to simulate propagation of radiation (see e.g. Marchuk et al., 2013). McRALI is based on the 3DMCPOL model (Cornet et al., 2010). The polarization state of the radiation is computed using Stokes vectors and scattering matrixes of atmospheric compounds. It takes into account molecular scattering. In this work, the properties of the atmosphere were assigned according to the 1976 standard atmosphere (NOAA, 1976). McRALI is a fully 3D software; that is, values of the extinction coefficients, the single-scattering albedos, and the scattering matrixes are assigned in 3D space. Moreover, the mixture of different types of aerosols and/or clouds is allowed. The position of a lidar can be anywhere within or outside of the atmosphere; that is, spaceborne, airborne, and ground-based measurement conditions can be simulated. A user can assign a lidar beam direction, a RFOV, and a Stokes vector as well as a divergence of the emitted light. It was demonstrated in the work by Alkasem et al. (2017) that McRALI simulations are in good agreement with published results of lidar-signal modelling in multiple-scattering conditions.

Four lidar configurations were taken into consideration in this work. Two
configurations were monostatic coaxial zenith-looking lidars, i.e. the
ground-based (the altitude is

The other two configurations were spaceborne nadir-looking lidars. We call
the “CALIOP configuration” the lidar at an altitude of 705 km having the
RFOV 0.13 mrad and the EFOV 0.1 mrad. Only a wavelength of 0.532

The majority of our MC data were computed so that photons were integrated
over a range gate of 20 m; i.e. they correspond to photon counting mode.
Such a small value of the range gate was chosen with the aim to study multiple
scattering in detail regardless of the fact that it does not correspond to
real lidar systems. In other words, the spatial resolution of our data is 20 m. In order to assure good statistical quality of our Monte Carlo modelling,
each signal was simulated with

The simulations of this work were performed for four types of particles,
namely, a coarse-aerosol layer, a warm cloud, and two types of cirrus clouds.
A mixture of particles was not considered. Because Monte Carlo methods are
very time-consuming, our study was restricted to the case of the
plane-parallel homogeneous layer placed within the altitude

The scattering matrixes were computed for the wavelengths 0.355
and 0.532

The single-scattering characteristics of ice particles were computed using
the improved geometric optics method (Yang and Liou, 1996); the particles
are assumed to be hexagonal ice crystals whose facets have deeply rough surfaces. As a consequence of the surface roughness, the scattering matrix of
ice particles has neither halo features (see e.g. Shcherbakov, 2013) nor
the delta transmission term (Yang et al., 2013). The size distribution of
particles was taken to be the gamma distribution. We have considered two
types of cirrus clouds that differ by the value of the effective diameter

Normalized phase functions: coarse aerosol – red lines, water cloud – blue lines, JS cirrus – black lines, Ci cirrus – green lines.

For subsequent discussions, we give in Table 1 parameters that have a
significant place in multiple-scattering theory. The effective diameter is
usually used to estimate the Fraunhofer diffraction angle

Integral parameters of the phase functions;

We use the following notations in this work. The function

The term “apparent attenuated backscatter” (see e.g. Chepfer et al.,
1999) is employed for lidar signals

A specific model of multiple scattering appears only when

The interpretation of MSF

Our Monte Carlo simulations provide the range-dependent lidar signals in the
single-, the double-, and the multiple-scattering conditions, that is,

It is instructive to see the double-scattering impact, especially when the
multiple-scattering effect is not high. Thus, we use the notations

Multiple-scattering contributions

Some important features of the multiple-scattering effect can be revealed when
lidar signals are simulated within a quite large range of the optical depth
in spite of limitations imposed by technical characteristics of receivers.
Figure 2 shows results of two cases that are quite distinguished in terms of
the configuration and particle properties. Both simulations were
performed with a spatial resolution of 20 m and a total number of photons
of

We used for the first case (Fig. 2a and b) the configuration of the MUSCLE
(MUltiple Scattering in Lidar Experiments) community (Bissonnette et al.,
1995; Winker and Poole, 1995). The distance to the water cloud C1 is low
(

The second case (Fig. 2c and d) deals with the configuration of the
Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP) (Winker et al.,
2010). The nadir-looking lidar is at an altitude of 705 km; the transmitter
has a wavelength of 0.532

To evidence the multiple-scattering effect, two functions are mostly used in
the literature, namely, the relative contribution of multiple scattering

The functions

It follows from our Monte Carlo simulations for different configurations
and/or particle properties that the computed functions

At large values of

The function

It is worthwhile to see how the MSF

Another function exists, which is somewhat similar to

The sensitivity of the parameterization to the adjusted parameters

In this work, all values of the fitting parameters

For the sake of brevity, we use the term “usual operational conditions”
(UOCs) when the distance from a lidar to a layer of particles is lower than
15 km, the RFOV

Figures 3 and 4 show the results of our MC simulations reported in terms of
the ratios

Multiple-scattering contributions

MC simulations of multiple-scattering functions

The cases of the low-value

It is seen in Fig. 3 that the function

Despite large shape variation in the ratios

Multiple-scattering contribution to lidar signals in
per cent (%) of the single scattering. The distance to the cloud base is

Multiple-scattering contribution to lidar signals in per cent (%) of the single scattering. The distance to the cloud base is 8 km; the lidar RFOV is 0.25 mrad. Values exceeding 5 % threshold are in bold font.

The applicability of the single-scattering approximation (SSA) to lidar
signals can be assessed on the basis of

In the subsequent discussion, we assume a 5 % threshold for the

The values exceeding that threshold are highlighted by the bold font in
Tables 2–3. In our opinion, the most important outcome is the fact that
the SSA has to be rejected in the cases of cirrus clouds when the RFOV is
1.0 mrad (see Table 2). Even with

As it was mentioned above, the majority of works take into account multiple
scattering employing one of the possible multiple-scattering functions (MSF).
Moreover, the simplified version

As for the MSF

Generally, there is much in common between all curves

It is seen in Fig. 4 that

As expected, our simulations confirm general properties of the multiple-scattering effect on lidar signals that can be found in the literature (see
e.g. Eloranta, 1998). Namely, the effect of multiple scattering as well as
the relative contribution of the third and higher orders of scattering
increase with increased extinction coefficient, in-cloud distance, and
receiver field of view. The proportion of light scattered within very small
angles, that is, within the forward-diffraction peak (see the inset in Fig. 1), is of upmost importance. That proportion is characterized by the angular width

The same kind of study was done in view of the double-scattering
contribution. The main conclusion is that the empirical model fits well with the
functions

All but one simulation condition, that is, the input data from the
foregoing subsection, were used in our MC simulations for the airborne lidar.
Namely, we are dealing with the coaxial zenith-looking lidar that is at an
altitude of

Multiple-scattering contributions

MC simulations of multiple-scattering functions

As in Sect. 5.1, the results of our MC simulations are reported in
terms of the ratios

Multiple-scattering contribution to lidar signals in per cent (%) of the single scattering. The distance to the cloud base is 1 km; the lidar RFOV is 1.0 mrad. Values exceeding 5 % threshold are in bold font.

Multiple-scattering contribution to lidar signals in per cent (%) of the single scattering. The distance to the cloud base is 1 km; the lidar RFOV is 0.25 mrad. Values exceeding 5 % threshold are in bold font.

The percentage of the multiple-scattering relative contribution to lidar
signals is shown in Tables 4–5. Those values were computed using
corresponding

Again, we assume a 5 % threshold for the

Again, we provide

The general features of the ratios

These work data are limited to a set of cases because MC simulations are time-consuming. Some ideas about dependence of the MS relative contribution

The magnitude of MS contribution to lidar signals, i.e. the level of

The first idea that comes is to search for approximate relationships between

It follows from MC simulations of this work that

The effective diameter

In support of the approach above we obtained the following results. Optical
characteristics of sea-salt aerosol were computed at a wavelength of 0.532

It is well known that the phase function of ice particles depends not only
on the effective size but also on particle habit (see e.g. Yang et al.,
2013) and roughness of particle surface (see e.g. Shcherbakov et al.,
2006). The data library (Yang et al., 2013) provides reliable scattering, absorption, and polarization properties of ice particles in large spectral and size ranges, 11 ice crystal habits, and 3 surface roughness conditions (i.e. smooth, moderately roughened, and severely roughened). The
data library provides means to obtain the angle

Figures 7 and 8 show examples of the multiple-scattering effect on signals
of spaceborne lidars, i.e. the ratios

Multiple-scattering contributions

MC simulations of multiple-scattering functions

The features of all ratios

It is seen that our empirical model Eq. (10) fits well with the MC data in Fig. 7. The values of the free parameters

As is expected, the MSFs

The general features of the ratios

In the foregoing subsection, we studied the cases when the values of the
extinction coefficient were quite small (

Multiple-scattering contributions

Figure 9a and b show examples of the multiple-scattering effect on lidar
signals, i.e. the ratios

It is instructive to observe the behaviour of the MSF

The multiple-field-of-view techniques already have more than 4 decades of history in lidar measurements (see e.g. Allen and Platt, 1977; Bissonnette
et al., 2005; and Jimenez et al., 2020). Multiple-scattering impact is favoured
by increasing RFOV. Therefore, it is interesting to evaluate the performance
of the empirical model against MC simulations when the RFOV is quite wide.
The simulations were performed under the same conditions as in Sect. 5.1;
that is, for the ground-based lidars, the water cloud is within the altitude
range of 8–11 km, the extinction coefficient is 1.0 km

Multiple-scattering contributions

Figure 10a and b show examples of the multiple-scattering effect on lidar
signals, i.e. the ratios

We performed extensive Monte Carlo simulations of single-wavelength lidar
signals from a plane-parallel homogeneous layer of atmospheric particles.
The simulations have taken into consideration four types of configurations
(the ground based, the airborne, the CALIOP, and the ATLID) and four types
of particles (coarse aerosol, water cloud, jet-stream cirrus, and cirrus),
which have large difference in microphysical and optical properties. Most of
the simulations were performed with a spatial resolution of 20 m and a
particle extinction coefficient between 0.06 and 1.0 km

Such a large set of configurations and particle characteristics
covers a broad range of the multiple-scattering (MS) relative contribution
to lidar signals: from lower than 5 % to a factor of several thousand.
Despite the broad range of variations, all MS relative contributions have the
same general behaviour as a function of the in-cloud penetration depth when
plotted as a log-linear graph. At the near end, the function

The fact that the MS relative contribution can be fitted by a simple function for a large set of lidar configurations and particle characteristics is of importance by itself. It provides a new perspective on the problem of the radiative transfer related to lidar and radar measurements.

Special attention was given to the usual operational conditions; i.e. when
the distance from a lidar to a layer of particles is lower than 15 km, the
RFOV

As for spaceborne lidars, the contribution of multiple scattering below 5 % is so exceptional that the single-scattering approximation should never be applied to data of such lidars.

Our simulations confirm general properties of the multiple-scattering effect on
lidar signals that can be found in the literature. Namely, the MS impact as
well as the relative contribution of the third and higher orders of
scattering increases with increased extinction coefficient, in-cloud
distance, and receiver field of view. The proportion of light scattered
within forward angles is of upmost importance. Our results suggest that the
angle

We computed the multiple-scattering function

Despite the fact that this work is limited to the cases of homogeneous
layers, we can propose two immediate application of our results. The
empirical model along with the parameters

The first application is that the set of data is used to compute profiles of apparent backscatter, which are employed to test inverse-problem algorithms. Therefore, a developer of an inverse algorithm can see its quality.

The second application is the following. As stated in the introduction, the accuracy level and the applicability bounds of the approximate models still need to be rigorously evaluated. Such an evaluation should be done in terms of the MS relative contribution, not in terms of apparent backscatter, because a model is devoted to simulating the MS effect. The evaluation should be done for a large range of experimental situations. Thus, the results of our work provide an easy way to begin the evaluation.

This work should be considered to be the starting stage of the model development
if needs of a practitioner are taken into account, especially when an
inverse problem is to be solved (see e.g. Voudouri et al., 2020). The next two stages have to be fulfilled: (i) development of an approach that predicts

It seems that the function

The utility of a multiple-scattering function (MSF) consists of the
possibility of considering effects of multiple scattering while dealing with
equations similar to the single-scattering lidar Eq. (1). In what
follows, the MSFs are written as functions only of the distance

Several approaches to define a MSF can be found in the literature. Similarly
to the transport approximation of the radiative transfer theory (see e.g.
chap. 17 of Davison, 1957), the constant factor

Another multiple-scattering function

Another way to consider the multiple scattering is used in the automated
algorithm of the Atmospheric Radiation Measurement programme's Raman lidar
(Thorsen and Fu, 2015); i.e. the MSF (in our notations

It is obvious that

Multiple-scattering functions.

It is a straightforward matter to transform Eqs. (A1) and (A2) into the
following forms:

An example of the multiple-scattering functions is shown in Fig. A1. The
MSFs are shown for the case shown in Figs. 5c and 6c; i.e. the homogeneous
cirrus cloud is within the altitude

Some misleading statements about properties of the MSFs can be found in the
literature. Thus, to complete this section, we note the following. The
functions

Figures 3 and 5 suggest that the multiple-scattering contribution

Equations (A7) and (A9) along with the first two terms of the series
expansion of the logarithm

When the multiple-scattering effect is ignored throughout the range of
distances from a lidar, it is enough to assign

It is seen from Eq. (7) that values of

We integrated Eqs. (1)–(2) with the step

Relative errors as functions of the optical thickness from the
layer base and the parameter

A typical example of relative errors

Multiple-scattering function

The data of Monte Carlo simulations are available from the corresponding author upon request.

The supplement related to this article is available online at:

VS developed the empirical model and Appendix A. FS and GM acquired funding. VS, FS, AA, GM, and CC contributed to developing the McRALI code, numerical simulations, and data treatment. VS wrote the manuscript with help from FS.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is part of the French scientific community EECLAT project (Expecting EarthCARE, Learning from A-train). The EECLAT community and research activities are supported by the National Center for Space Studies (CNES) and the National Institute for Earth Sciences and Astronomy (INSU).

This research has been supported by the National Institute for Earth Sciences and Astronomy (INSU grant).

This paper was edited by Alexander Kokhanovsky and reviewed by three anonymous referees.