Linearity between MBS and IMD, ice radius r and ice number density n data, is a necessary and sufficient condition that also shows that IMD, r and n data, samples are exponentially distributed, see next section. In the following we will test this constraint. Figure a shows a scatter plot in a logarithmic frame for simultaneously measured MBS and IMD data. In order to test a linear relationship between x = MBS and y = IMD we introduce a general fit function described by a power law condition as 5y(x)=cxd⇔x(y)=(y/c)1/d with the two constants c (linear constant) and d (power constant). Only for d=1 we expect a perfect linear dependence between x and y. First of all, the logarithmic values of x and y do not yet have a high linear correlation (R=0.54), see Fig. a. Since the correlation coefficient R is unequal one, the two regression lines resulting from y(x):ln⁡x↦ln⁡y and x(y):ln⁡y↦ln⁡x differ from each other. This means that the best choice of a regression fit is determined by a straight line through the two regression points, which are defined by the means plus/minus standard deviations of logarithmic x and y data. Note that the positions of regression points also relate to the half-width of the angle that is spanned by the two regression lines y(x) and x(y). For our mean regression line we estimate d=0.873. The statistical error for d is Δd=±0.012 with a confidence level of 95 %, which indicates a significant nonlinearity. Hence we conclude that the pdf describing the distribution of IMD data is very likely not an exact exponential function and its cumulative distribution does not follow precisely a g-function description because the criterion of “linearity” is violated. In Fig. b we show a second example for the correlation between MBS and ice radius n. Again the correlation is about R=0.55, but now the power constant is even much smaller with d=0.497, which is far away from unity. Finally, we investigated the linearity between MBS and ice number density n where we find a weak negative correlation of R=-0.15 (not shown here). A best fit analysis yields a power value of d=-0.534, which again fails significantly the constraint of unity. Hence we conclude that ice radius and ice number density distributions should also not follow exponential (g-function) distributions.

In this section we first show some general characteristics of the new Z distribution. From these properties we derive conditions and constraints that will allow to estimate the specific values of the two free constants in Z, the scale parameter a and shape parameter b, for a given data sample.

First we show that Z is a correct pdf satisfying the normalization condition ∫0∞Zdz=1: ∫0∞Zdz=∫0∞a|b|zb-1e-azbdz=-|b|be-azb0∞=1. The definition range of Z(z;a,b) is z≥0, a>0, and b≠0 with Z(z<0)=0 and 10aZ=a|b|zb-1e-azb,ln⁡(Z)=ln⁡(a|b|)+(b-1)⋅ln⁡(z)-azb,b>0. For a negative b the distribution Z is described by 10bZ=a|b|1z|b-1|e-a(1/z)|b|,ln⁡(Z)=ln⁡(a|b|)+|b-1|⋅ln⁡(1/z)-a(1/z)|b|,b<0. The cumulative form of Z for b>0 is given by 11aZcum(z)=∫z∞Zdz′=e-azb,ln⁡(Zcum)=-azb,ln⁡(|ln⁡(Zcum)|)=ln⁡(a)+bln⁡(z). For b<0 we have to choose the cumulative calculation in reverse order starting the integration at zero. Naming the reverse cumulative with index zero as Zcum0 we get 11bZcum0(z)=∫0zZdz′=e-a(1/z)|b|=e-azb,ln⁡(Zcum0)=-azb,ln⁡(|ln⁡(Zcum0)|)=ln⁡(a)+bln⁡(z). Only the cumulative descriptions from Eqs. (11a) and (11b) allow us, in principle, to roughly estimate the constants a and b for a given data sample using the double-logarithmic functional dependence, whereas the direct logarithm of Z (Eqs. 10a, 10b) offers no possibility to solve for a and b. However, the method calculating the double-logarithmic cumulative is not recommended. Several numerical tests showed that a stable estimation of a and b from noisy data applying this double-logarithmic approach is an almost impossible task. Instead, we propose two different methods that rely on much more powerful principles (see next Sect. 5.2). Additionally, we have to take care of a possible negative value of b that can be only identified using Eq. (11b). In fact such a case occurs in the analysis of ALOMAR data. In Sect. 5.3 we will give an example where only a negative slope parameter describes the distribution of number density of ice particles.

Generally, the Z distribution has the ability to characterize many different types of distributions, see Fig. . Especially the shape parameter b determines the shape of the Z distribution describing nonlinear exponential, exponential, right-skewed, left-skewed, or symmetric curves. For 0<b<1 the pdf increase is nonlinear and exponentially accelerated to infinity as z approaches zero. For b=1 the pdf is exactly an exponential distribution having a positive finite value for z equal zero. For b>1, the function tends to zero as z approaches zero. When b is between 1 and 2, the function is right-skewed and rises to a peak quickly, then decreases for large z. When b has an approximate value between 3 and 4, the function becomes symmetric and bell-shaped like a normal distribution. Note that exact symmetry is given for a skewness equal to zero, which is true at z=3.60232. For b values larger than approximately 5, the function becomes again asymmetric changing the skewness to the left. For b<0, the function is skewed to the right and decreases steeply towards zero as z approaches zero. Note that Z is never negative and owns a local maximum described by the mode whenever b is negative or larger than 1. Finally we see that a double-logarithmic presentation of cumulative functions describes linear shapes with slope b, see Fig. j–l.

It is interesting to note that our new Z distribution is closely related to a more general Weibull distribution . Nevertheless there is a difference concerning the shape parameter b, which in our case is not only defined for positive values but also for negative values. Such a case is disregarded by a classical 2-D Weibull distribution.

Now we shortly summarize the mathematical descriptions of median, mode, mean, variance, and standard deviation parameters of Z for the case of a zero threshold. The calculations are described in detail in Appendix B for the general case of a nonzero threshold. Median: 12aν=ln⁡2a1/b. Mode: 12bη=b-1ab1/bforb>1,b<0;η=0for0<b≤1. Mean: 12cμ=Γb+1ba1b. Variance and standard deviation: 12dσ2=Γb+2ba2b-μ2,σ=Γb+2ba2b-μ2. The expressions of mean and variance use the gamma function Γ(t)=∫0∞xt-1e-xdx. Notice that the gamma function is defined for all real values of t except t=0 and all negative integer values of t. Note also that median, mode, mean, variance, and standard deviation parameters of Z coincide with those of an exponential distribution in the limit as b equals 1.

In this section we present two numerical methods to calculate the scale parameter a and shape parameter b describing the new Z distribution. First of all, since any measurement depends on a specific instrumental sensitivity, we have to introduce a threshold that we name zth. The remaining data sample consists of N observations zi with zi>zth. Then we calculate the mean m‾ and standard deviation s of data zi using Eq. (3), and also the median value m̃ from data zi.

Method (1): we investigate the corresponding theoretical moments from Z. In Appendix B we derive the theoretical mean μ (Eq. B5) and median ν (Eq. B3) for the Z distribution with a threshold constraint. Taking the estimates of mean m‾ and median m̃ from the sample as best proxies for the theoretical mean μ and median ν values of Z, we get the following equations: 13aν=ln⁡2a+zthb1/b⟶a=ln⁡2m̃b-zthb, 13bμ=-A|b|Γb+1b,azthbba1/b⟶0=m‾⋅ba1/b+A|b|Γb+1b,azthb. Note that the use of a threshold constraint involves the introduction of a scaling factor A=exp⁡(azthb), which is present in Eq. (13b). Inserting the algebraic term of a (right side of Eq. 13a into the right side zero-equation Eq. 13b) and using the threshold value of zth yields an equation only for b, which has to be computed iteratively. Once a numerical value of b has been estimated with a sufficient accuracy, we insert this b value into the upper right equation to get the numerical value for a.

We note that in classical statistics the method of moments determines a and b from the mean and variance equations. In principle this approach should be possible here too, but in practise the algebraic structure of the variance equation is too complicated, see Eq. (B6) in Appendix B. This means that the variance equation, if at all, is only iteratively solvable, whereas the use of the median equation offers an analytical transformation to a. Generally, we recommend to apply the proposed method using the mean and median equations. This straight-forward method is easy to program and produces reliable estimates of parameters a and b.

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 141a=-zthb+∑zibN,0=1b+a⋅ln⁡zth⋅zthb+∑ln⁡ziN-a⋅∑ln⁡zi⋅zibN. Interestingly, the left equation includes a term 1/N∑zib, which is the mean of the sample values weighted by power b, whereas the right equation includes the mean 1/N∑ln⁡zi of logarithmic data and 1/N∑zibln⁡zi. This shows a similarity to the computation of regression points used in Fig. . We insert a into the right equation that yields a unique equation for b, which again can be solved iteratively. Once b is fixed, the left equation allows us to determine a. In the following we will test our lidar data samples with these two procedures and we will show that both methods produce almost identical results.

Applications of the Z distribution to ALOMAR data of maximum backscatter (x), ice mass density (y), ice particle radius (r), and ice number density (n) are shown in Fig. . Note that thresholds have been computed from the regression functions (Eq. 5) described in Sect. 3.2 on the basis of xth=3×10-10 m-1 sr-1, resulting in yth=22 mg m-3, rth=22.3 nm, and nth=662 cm-3. The values of scale parameter a and shape parameter b have been calculated with the method of mean and median equations (method 1). Then the theoretical curves of Z and theoretical values of mean, median, mode, and standard deviation have been calculated by inserting the values of a, b, and threshold zth into Eqs. (B1)–(B6). Obviously the pdf Z sometimes has no simple exponential shape, which is the case for ice mass density, ice radius, and ice number density. As we see in Fig.  all Z-pdf curves (in blue) match the original data histograms with a high accuracy. The relative error is in a range of about 6 %–10 % except that ice number density has a relative error of 15 %. When we compare the mean, median, mode, and standard deviation derived from the theoretical distribution and corresponding estimates from data samples, we see a precise coincidence of mean and median values. Not surprising this is due to the fact that the parameters a and b have been computed by the mean and median method, which guarantees the preservation of mean and median values. Nevertheless standard deviation and mode also always show a good agreement within the error range. A closer look to the maximum backscatter distribution shows that MBS data are almost perfectly exponentially distributed with b=0.931, which is not too far away from b=1 for an exact exponential pdf. As we had already shown, see Sect. 3.1.1, MBS data are very likely exponentially distributed, now the Z-distribution analysis confirms this result. Hence we conclude that the commonly used exponential (g-function) analysis might only be a reasonable statistical method in the case of analyzing MBS lidar data.

In contrast to MBS, the Z distribution of IMD shows a function that converges rapidly to zero for small IMD values. The distribution is described with b=1.355, which significantly deviates from b=1 for a precise exponential function. Note that the mode of the data sample at 40 mg m-3 differs from the theoretical mode of 23 mg m-3 because of a relatively high statistical noise in the data. But mean, median, and standard deviation values agree almost perfectly. Similar to IMD, the ice radius distribution indicates a significant nonexponential behavior with b=1.833. The distribution converges to zero as the radius approaches zero. The curve is skewed to the right and has a maximum at r=25.8 nm, which differs only slightly from the mode of the data sample at r=27.8 nm. Again mean, median, and standard deviation values agree almost perfectly.

The sample of ice number density shows a completely different behavior with a slope parameter that is negative with b=-0.819. The physical meaning is that the parameter ice number density is negatively correlated with all other ice parameters. For example, large ice numbers n correspond to small ice radii, IMD and MBS values. As a consequence this leads to a threshold of n in the reverse direction, that is from large values to small values defined by n<nth=662 cm-3. One can see this feature in the right tail of Z(n) plotted as a dashed curve, see Fig. d. The reverse behavior is also present for small values of n. Small values of n are measured for very bright PMC events with large MBS that have small occurrence rates. Therefore, the number of small ice particles has a relatively high uncertainty due to their low occurrence frequency, and it is this statistical error that produces some deviations from the fit curve to the data in the range of n=0–80 cm-3. We note that the numerical procedure computing the pair (a,b) from the method of mean/median (Eqs. 13a, 10b) has automatically detected the existence of a negative slope parameter b without any a priori information.

Now we repeat the analysis using method 2. Figure  summarizes the (a,b) values and statistical moments calculated from the method of maximum likelihood estimators. As can be seen the maximum likelihood approach computes almost identical results for all ice parameters. We have also added in Fig. a–c the histogram bars (in black) for all data being smaller than the threshold. Please keep in mind that the calculation of theoretical distribution curves is based exclusively on data larger than the threshold. Hence, decreasing or increasing a threshold will change the specific values of a and b. Figure d shows the ice number density distribution where we have added in the histogram (in black) all data being larger than the threshold. Again, also the maximum likelihood method has automatically detected the existence of a negative slope parameter b for the ice number density distribution.

In the derivation of the Z distribution we used the assumption that all ice parameters of maximum backscatter x = MBS, ice mass density y = IMD, ice particle radius r, and ice number density n are connected with one another by the power law given in Eq. (5). In Sect. 5.3 we showed that the Z pdf describes with a high accuracy each distribution of these ice parameters, which, in turn, means that indeed there exists at least an approximative power law between ice parameters. We discuss a suitable justification of this power law relation in more detail in Sect. 6.2. In the following we will show that the use of a Z distribution allows us to construct artificial unknown data samples of various ice parameters that approximate true data to a high degree. We think that such an application is one of the most beneficial outcomes from the new Z-distribution approach. We explain the numerical procedure by the help of a practical example.

We already showed a linear dependance in the logarithmic frame using linear regression (LR) for maximum backscatter and ice particle radius, see Fig. b. Hence we can compute artificial ice radius proxies r̃i, named as LR proxy of true data ri, as a function of MBS-data xi from the regression power law function (Eq. 5) with r̃i=cxid and with power law coefficients c=13.509 and d=0.497. Figure a shows a comparison between LR proxy and original ice radius data where we test the identity of the two data samples. The correlation coefficient is the same as shown in Fig. b with R=0.55. Mean and median values of proxy and original data are almost identical, and a regression analysis shows a perfect identity (c=1.000 and d=1.000). Now we calculate the frequency histogram of LR proxy ice radii, see Fig. b, and compare the histogram with the original Z distribution of ice radii already shown in Fig. c. We find that the LR proxy approximates the mean, median, mode, and standard deviation values of the original Z distribution with an relative error of 9.5 % comparable to the original error of 9.1 %. We conclude that a linear regression analysis of logarithmic data offers a good opportunity to approximate data, provided that a pair of data samples exists that allows for the calculation of power law coefficients c and d from regression methods.

Now the Z-distribution approach offers a more general possibility to derive artificial data samples without any knowledge of correlation and regression coefficients. Indeed we will show that results from the Z approach are very close to results from a regression analysis. Again, our goal is to approximate ice radius data from a given maximum backscatter data sample. But now we suppose that no data of ice particle radii r exist, hence any correlation and regression analysis is not possible. First, we assume that a data sample of x = MBS of number N exists and also its Z distribution Z(x,ax,bx) with ax=0.140 and bx=0.931 is well known, see Fig. a. Secondly, we assume that we know a priori the form of the Z distribution Z(r,ar,br) of ice radius r, e.g., with values of parameters ar and br from Fig. c (ar=1.269×10-3, br=1.833). Please keep in mind that such information about scale and shape parameters of the ice radius distribution could be also provided from independent satellite measurements that are capable of measuring ice particle radii, e.g., AIM-SOFIE.

Our new proxy method (Z proxy) requires the following transformations. We first transform the xi values (i=1,…,N) into the z domain with xi↦zi:z=axxibx followed by a second transformation with zi↦ri:ri=(zi/ar)1/br resulting in 15xi↦zi↦ri:ri=ax⋅xibxar1/br=c⋅xid with c=axar1/br,d=bxbr. Note that the derivation of c and d in Eq. (15) is based on the same mathematical steps when we developed the Z distribution from Eqs. (8) to (9). Inserting the ax, bx, ar, and br values into Eq. (15) determines the power law coefficients for Z proxy r̃ with c=12.994 and d=0.508. These values do not exactly coincide with c and d values obtained from the regression method, see above, but the identity test between Z proxies and true ice radii shows a very good coincidence, see Fig. c. Again, mean and median values of proxy and original data are practically identical, and a regression analysis shows an almost perfect identity (c=1.095 and d=0.980). Finally we calculate a frequency histogram of Z proxies, see Fig. d, and find a good agreement between proxies and true pdf. Mean, median, and standard deviations of Z-proxy data correspond perfectly to original ice radius data, and the relative error has now even decreased to 9.2 %.

We summarize that we present a new method in order to construct artificial data samples provided Z-descriptions of these data sets exist. By means of a consecutive arranging of ice parameters starting at a given data sample, this method allows us to construct any artificial data sample within (x,y,r,n). This method can also be applied to other data sets, e.g., ice parameter measurements from satellite observations. For example, a data sample of ice water content (IWC) obtained from satellite measurements might be analyzed in terms of a Z distribution estimating the scale and shape parameters a and b of the IWC distribution. This would allow us to establish a connection between satellite IWC data to lidar data samples (x,y,r,n) through Eq. (15), hence the satellite IWC data could be transferred to lidar maximum backscatter, ice mass density, ice particle radius, and ice number density. Vice versa the knowledge of a satellite IWC Z distribution would allow us to transform lidar observations into IWC proxies and compare these with the original IWC observed by the satellite. We think that our proposed transformation method could be very helpful to connect different ice parameter data from different instruments, either from satellite observations or ground-based measurements. We also think that this new approach might be important in trend analysis of PMC.

In the next section we will discuss the power law assumption (Eq. 5) and the physical meaning of the shape parameter b, which might be introduced as a new trend variable in the analysis of PMC long-term changes.

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 (Ni) as Ni(τ)=1σi2πexp⁡-12τ-riσi2. We also assume that the geometric shapes of these ice particles are spheres with ice radii τ with mean radius ri and variance σi2. Again index i=1,…,N relates to the ith measurement in a given data sample of number N. Ni is normalized to ∫0∞Ni(τ)dτ=1. When we assume an ice number density of ni particles per cubic centimeter we get the expression ∫0∞ni⋅Ni(τ)dτ=ni. Note that mean ice radii ri and ice number densities ni are elements of our lidar data climatology, which we have introduced in Sect. 2.

Furthermore, we assume from the analysis of lidar observations (three-color measurements) that the relation of mean radius and variance is according to σi=0.4ri for ri<37.5 nm and σi=15 nm for ri≥37.5 nm . This assumption has also been applied in the analysis of AIM/SOFIE-CIPS PMC satellite data . In order to simplify calculations we apply this relation σi=0.4ri also for ri larger than 37.5 nm. We now investigate the question of which backscatter lidar and ice mass signals result from such an ice distribution.

We compute the mass of a spherical ice particle with radius τ as 4/3πρiceτ3 with density of ice ρice=932 kg m-3. The backscatter signal from a single ice particle is calculated as aLτ5.8 with the lidar constant aL=1.5×10-11 m2. Then the maximum backscatter xi and ice mass density yi are estimated by an integration of the radius distribution from zero to infinity as xi=aLn∫0∞τ5.8Ni(τ)dτ=aLn∫0∞τ5.810.4ri2πexp⁡-12τ-ri0.4ri2dτyi=43πρicen∫0∞τ3Ni(τ)dτ=43πρicen∫0∞τ310.4ri2πexp⁡-12τ-ri0.4ri2dτ assuming a constant number density n of ice particles. Only the integral of yi is analytically computable with a solution in which the error function defined by the integral erf(x)=2/π∫0xexp⁡(-t2)dt is part of the solution: xi=43πρicen3750ri3erf5τ-ri8ri-2125πri25τ2+25riτ+33ri2exp⁡25τ4ri-25τ2+ri28ri20∞. The integral for xi includes the term τ5.8 that arises from Mie-scatter theory for light scattering of a wavelength of 532 nm (ALOMAR RMR lidar) at spheres in a range of radii with 1–100 nm. The exponential value of 5.8 approximates exact Mie-scatter calculations with a relative error less than 0.5 % in this radii range. Unfortunately, the integral can only be solved analytically if the exponent is an integer number as 5 or 6. Nevertheless, we are able to solve this integral by means of numerical methods with the specific exponent of 5.8. In a next step, we construct analytical approximations fx(ri) and fy(ri) for both integral solutions using a typical value of n=200 cm-3 with fx(ri)=xi=a1aLnri5.8,fy(ri)=yi=a243πρicenri3. The linear constants a1 and a2 with values a1=4.20 and a2=1.47 are optimal dimensionless parameters. fy approximates the analytical solution of yi with a relative error less than 0.7 % in the range ri=[0,45 nm], and less than 1.2 % in the range ri=[45,70 nm]. A precise solution of xi resulting from numerical methods of integration is approximated by fx with a relative error less than 0.3 % in the range ri=[0,45 nm], then the relative error increases linearly to a maximum error of 5 % at ri=70 nm. We find that the solutions fx and fy approximate the general power law condition p=cqd (Eq. 5) inside a small error range. Hence these analytical examples show that MBS is a function of ice radius proportional to ∼r5.8 (d=5.8), the same is also true for IMD (∼r3, d=3). It also follows that MBS and IMD are consequently connected through a power law condition with MBS ∼ IMD5.8/3 (d=5.8/3=1.93).

But the new form of the Z-distribution technique opens up whole new perspectives for the validation of the analytical examples based on the ALOMAR lidar data samples. We transform the z distribution of IMD into the MBS domain using Eq. (15) with yi↦zi:zi=ayyiby and zi↦xi:xi=(zi/ax)1/bx that gives 16x=ay⋅ybyax1/bx=c⋅yd with c=ayax1/bx,d=bybx. We insert into Eq. (16) the values of ax=0.140, bx=0.931, ay=4.321×10-3, and by=1.355 from Fig. a and b and get c=0.023 and d=1.46. The power constant (d=1.46) derived from the shape parameters bx and by of the Z-distribution analysis of real ALOMAR IBS and IMD data is significantly different from the power estimate (d=1.93) belonging to the analytical example that necessitates various assumptions, e.g., Gaussian-distributed ice particles at the height of maximum backscatter, constant ice particle number, or spherical shape of ice particles. Hence, we conclude that the determination of shape parameters b from a Z-distribution analysis of observational data therefore provides a qualitative indication of the actual microphysical state that controls real ice formation processes. This leads to the idea that as a future task long-term changes in PMC formation might be characterized by potential long-term changes in b that indicate long-term changes in atmospheric background conditions and microphysical ice constraints of ice formation.