In this paper we present a new description of statistical probability density functions
(pdfs) of polar mesospheric clouds (PMCs). The analysis is based on observations of
maximum backscatter, ice mass density, ice particle radius, and number density of ice
particles measured by the ALOMAR Rayleigh–Mie–Raman lidar for all PMC seasons from 2002
to 2016. From this data set we derive a new class of pdfs that describe the statistics of
PMC events that is different from previous statistical methods using the approach of an
exponential distribution commonly named the

First studies of probability distributions of polar mesospheric clouds
(PMCs) were reported by

In the following years many observational PMC analyses of seasonal statistics have been
published frequently using the

The

This paper makes an attempt to investigate in more detail the statistics of probability
density functions (pdfs) of PMC climatology for various ice parameters. In the following
we analyze a PMC data record of maximum backscatter, ice mass density, ice particle
radius, and number density from the period 2002–2016 measured by the ALOMAR
Rayleigh–Mie–Raman (RMR) lidar. From the analysis of these ALOMAR data, we derive a new
class of pdfs of PMC distributions that, as we will show, modifies and improves the
exponential (

The data set obtained by the ground-based RMR lidar, located at the Arctic station ALOMAR
(69

In this paper we will analyze the climatology of all ice seasons from 2002
until 2016 merging all 15 seasons to one data record. Within this combined
data set we then get a total number

In general, the seasonal climatology of PMC events with measured ice parameters, such as
integrated backscatter, maximum backscatter, column ice mass, albedo, or ice mass
density, has been supposed to follow an exponential distribution that we name

The general form of the exponential distribution

A statistical analysis of ice parameters has to take into account the aspect
of specific sensitivities of different instruments. For example the ALOMAR
lidar is generally sensitive to a backscatter signal larger than a threshold
about 2–

For a threshold of zero (

For a given data sample

We investigate the frequency distribution of MBS at 532 nm in units of

Figure

Now we investigate other ice parameters from the ALOMAR data set with respect to
exponential distributions, namely the frequency distributions of IMD in units of
mg m

We summarize that ice mass density, ice radius, and ice number density do not
follow an exponential distribution in contrast to maximum backscatter. In the
following section we will show that this is reasonable and is based on the
fact that a functional link between MBS and the other data sets of IMD,

Linearity between MBS and IMD, ice radius

In this section we will present the major part of the new statistical approach in order to describe frequency distributions of different PMC parameters.

There exists a general mathematical method (“integration by substitution”) that
provides the opportunity to transform between pdfs with different statistical variables.
This is done by the following procedure: assuming a given pdf

We apply this method for two ice parameters, namely MBS with variable

In a next step we introduce, in an arbitrary manner, a third ice parameter named

In this section we first show some general characteristics of the new

First we show that

Generally, the

It is interesting to note that our new

Now we shortly summarize the mathematical descriptions of median, mode, mean,
variance, and standard deviation parameters of

In this section we present two numerical methods to calculate the scale parameter

Method (1): we investigate the corresponding theoretical moments from

We note that in classical statistics the method of moments determines

Method (2): we also present a second method using a maximum likelihood
approach, see Appendix C. The parameters are again calculated from two
equations (Eq. C5) with

Frequency distributions and

Applications of the

In contrast to MBS, the

The sample of ice number density shows a completely different behavior with a slope
parameter that is negative with

Same as Fig. 5, but parameters

Now we repeat the analysis using method 2. Figure

In the derivation of the

We already showed a linear dependance in the logarithmic frame using linear regression
(LR) for maximum backscatter and ice particle radius, see Fig.

Now the

Our new proxy method (

We summarize that we present a new method in order to construct artificial data samples
provided

In the next section we will discuss the power law assumption (Eq. 5) and the
physical meaning of the shape parameter

In this section we discuss some theoretical aspects of the power law dependence on ice
parameters in order to validate the justification of Eq. (5). We use again the assumption
as already discussed in Sect. 2 that at the altitude of maximum brightness (MBS) and ice
mass density (IMD) there exists in the real atmospheric background an ice particle
distribution that is perfectly Gaussian-distributed (

Furthermore, we assume from the analysis of lidar observations (three-color measurements)
that the relation of mean radius and variance is according to

We compute the mass of a spherical ice particle with radius

But the new form of the

In this study we present a new method to describe statistical probability density functions (pdfs) for different ice parameters of PMC. We analyze a climatology of ice seasons from 2002 until 2016 as measured by the ALOMAR lidar. From this data set we derive ice cloud parameters of maximum backscatter, ice mass density, ice radius, and ice number density whose occurrence frequencies are investigated with respect to exponential distributions. We show that only maximum backscatter follows an exponential distribution, whereas ice mass density, ice radius, and ice number density frequencies fail to fit satisfactorily to an exponential distribution. The reason for these deviations from exponential behavior is based on the fact that these ice parameters are not linearly dependent on each other.

We introduce a new probability density distribution (

Perhaps the most important application of the new method is the possibility to construct
unknown data sets for different ice parameters that approximate true data to a high
degree. We show in Sect. 6.1 that a linear regression analysis in a logarithmic data
frame offers a good opportunity to approximate data provided that a pair of data samples
exists that allows for the calculation of power law coefficients

The ALOMAR lidar data are available at:

When considering a threshold (

In the following all quantities take into account a threshold

Probability density function

For a single observation, the likelihood function

UB drafted the paper. All authors reviewed the paper and interpreted the data. JF and GB provided the lidar data from ALOMAR.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Layered phenomena in the mesopause region (ACP/AMT inter-journal SI)”. It is a result of the LPMR workshop 2017 (LPMR-2017), Kühlungsborn, Germany, 18–22 September 2017.

We appreciate the financial support from the German BMBF for the ROMIC/TIMA project. We thank Gary E. Thomas for very helpful and stimulating contributions, discussions, and review. The publication of this article was funded by the Open Access Fund of the Leibniz Association.

This paper was edited by Martin Dameris and reviewed by two anonymous referees.