We present a summary on the current status of two inversion algorithms that
are used in EARLINET (European Aerosol Research Lidar Network) for the inversion of data collected with EARLINET
multiwavelength Raman lidars. These instruments measure backscatter
coefficients at 355, 532, and 1064 nm, and extinction coefficients at 355 and
532 nm. Development of these two algorithms started in 2000 when EARLINET was
founded. The algorithms are based on a manually controlled inversion of optical
data which allows for detailed sensitivity studies. The algorithms allow us
to derive particle effective radius as well as volume and surface area
concentration with comparably high confidence. The retrieval of the real and
imaginary parts of the complex refractive index still is a challenge in view
of the accuracy required for these parameters in climate change studies in
which light absorption needs to be known with high accuracy. It is an extreme
challenge to retrieve the real part with an accuracy better than 0.05 and the
imaginary part with accuracy better than 0.005–0.1 or

On the basis of a few exemplary simulations with synthetic optical data we discuss the current status of these manually operated algorithms, the potentially achievable accuracy of data products, and the goals for future work. One algorithm was used with the purpose of testing how well microphysical parameters can be derived if the real part of the complex refractive index is known to at least 0.05 or 0.1. The other algorithm was used to find out how well microphysical parameters can be derived if this constraint for the real part is not applied.

The optical data used in our study cover a range of Ångström exponents and extinction-to-backscatter (lidar) ratios that are found from lidar measurements of various aerosol types. We also tested aerosol scenarios that are considered highly unlikely, e.g. the lidar ratios fall outside the commonly accepted range of values measured with Raman lidar, even though the underlying microphysical particle properties are not uncommon. The goal of this part of the study is to test the robustness of the algorithms towards their ability to identify aerosol types that have not been measured so far, but cannot be ruled out based on our current knowledge of aerosol physics.

We computed the optical data from monomodal logarithmic particle size
distributions, i.e. we explicitly excluded the more complicated case of
bimodal particle size distributions which is a topic of ongoing research
work. Another constraint is that we only considered particles of spherical
shape in our simulations. We considered particle radii as large as
7–10

The start of EARLINET (European Aerosol Research Lidar Network)
in 2000 marked the beginning of the development of inversion algorithms that
can be used for the retrieval of aerosol microphysical properties from Raman
lidar observations. Based on exploratory work

We followed two conceptual approaches. One methodology was developed at the Leibniz Institute for Tropospheric Research (TROPOS), Leipzig, Germany. The development of the methodology continues at the University of Hertfordshire (UH). The second method was developed at the University of Potsdam (UP) (Potsdam, Germany). Both methods in part follow the same basic mathematical concepts in the sense that they are true inversion algorithms. True inversion algorithms mean the following: (a) the underlying mathematical equations that connect the microphysical particle properties (which are the variables we are interested in) and the optical properties (which are the variables that are measured with lidar) are solved explicitly; (b) therefore we do not carry out forward computations, which are commonly referred to as Mie-scattering computations; (c) we do not use traditional look-up tables that contain an array of microphysical aerosol properties and the optical properties that belong to these microphysical properties; and (d) our approach neglects constraints that are used in forward computations, for example the need to prescribe the shape of the particle size distribution as input.

The advantage of our approach is that we can identify the share of fine-mode
and coarse-mode particles in particle size distributions, as the inversion
algorithms allow us to find approximate solutions of the underlying particle
size distributions. As with any other method, there exists plenty of
disadvantages, for example measurement errors have a direct impact on the
quality of the retrieval results. If measurement errors become too large, the
inversion algorithms will not be able to find reasonable solutions. The
inversion algorithms also respond strongly to systematic errors of the
optical input data. This means that if calibrations of the optical profiles are
not done carefully, or if optical data are faulty because of the incomplete
overlap between laser beam and field-of-view of the receiver telescope

With regard to work at TROPOS/UH, our algorithm development began on the basis
of data we obtained from the first truly operational multiwavelength Raman
lidar

The algorithm that was initially developed at TROPOS follows the concept of
Tikhonov's inversion with regularization

The retrieval of the CRI remains a major challenge in our work. The accuracy
requirements for the imaginary part of the CRI are considerable if we want to
obtain useful values (high accuracy and precision) of the single-scattering
albedo (SSA) which is one of the key parameters in climate change studies.

Nowadays we manage to obtain meaningful values of the SSA, but at the expense
of time-intensive, data-operator-controlled data analysis which involves a
careful evaluation of the inversion results (particularly of the CRI). We
learnt to retrieve the imaginary part to its correct order of magnitude if it
is less than 0.01

We analysed plenty of different aerosol types in the course of more than 15 years of measurements with multiwavelength Raman lidar. Still, a
statistically significant set of results for each aerosol type has not been achieved
because of the labour-intensive manual data analysis. Examples of aerosol
types we analysed involve urban/industrial pollution over Europe

We remain cautious with regard to the inversion of optical data that describe
non-spherical (mineral dust) particles, as to date we do not have a reliable
light-scattering model that allows us to describe light-scattering at
180

The Potsdam algorithm that was developed at UP is based on the concept of
using truncated singular value decomposition (TSVD) as regularization method,
(e.g.

Screen shot of the developed software and explanation of the post-processing procedure: selection of suitable grid points of CRI clustering along the diagonal domain (left), corresponding PSD and initial PDS (right), and tables with retrieved microphysical parameters for each selected CRI and mean values with standard deviation (bottom). The residual errors on the right hand side of the grid appear in ascending order from top to bottom on a logarithmic scale.

There is another challenge that needs to be considered. The CRI is actually
also unknown. In order to solve this problem, a grid of viable options for
the CRI (all combinations of real parts of CRI (RPCRI) and imaginary parts of
CRI (IPCRI)) is assumed. If this is not done, one additionally has to deal
with a non-linear problem. The CRI grid technique is very time consuming,
i.e. the computer runtime of the inversion process is very large.
Therefore, we developed a semi-automated software for spherical particles
which is able to run on a parallel processor machine

From a mathematical point of view, it is also very important (as a selection
criteria for an appropriate regularization method) to investigate the degree
of ill-posedness of the problem. Investigations show a moderate ill-posedness

The semi-automatic software was applied to measurement cases. We analysed
Raman lidar measurement data with three backscatter coefficients at 355, 532,
and 1064 nm and two extinction coefficients at 355 and 532 nm. From these
optical particle variables we retrieved microphysical particle properties of
different aerosol types. Successful retrievals have been made for
biomass burning and industrial pollution aerosols over Germany

We did not perform a direct comparison of the performance of the two algorithms (TROPOS/UH and UP) in this contribution as we are mainly interested in learning how we can meet demands for highly accurate single-scattering albedos. This demand will likely require a coordinated use of both algorithms in future rather than selecting one algorithm over the other algorithm in EARLINET. We carried out simulations with the main purpose of investigating what maximum accuracy and precision of microphysical parameters can be achieved (UP algorithm). The complex refractive index is the key to retrieving accurate values of single-scattering albedo and profiles of absorption coefficients. We therefore investigated if an accurate knowledge of the real part of the refractive index would significantly improve the retrieval of the imaginary part which then would feed into improved values of single-scattering albedo. We used the UH/TROPOS algorithm for this part of the study.

Section 2 presents main features of the two methodologies. Section 3 presents some simulation studies that illustrate the current status. Section 4 closes with a summary and outlook.

The microphysical properties are derived from solving Fredholm integral equations of the first kind

The backscatter and extinction kernel functions are denoted by

The kernel functions

Equation (

Disregarding the model error term we obtain

The investigated PSD

Neglecting the discretization error term we can express the measured optical
particle properties

Since the operator is ill posed, as mentioned above, the resulting linear
equation system Eq. (

The unknown vector of the weight coefficients

The TROPOS/UH algorithm and the UP algorithm use different regularization
methods. The TROPOS/UH algorithm uses Tikhonov regularization

With respect to the subsequently realized simulation study in which we apply simulated noise to the optical data, the following issue is considered. On the one hand, since noise or measurement errors are randomly distributed to the optical data one needs a reasonable sample size for the simulation study for the case of noisy data. On the other hand, the simulation studies are very time consuming. Thus, a compromise between reasonable sample size and runtime is needed. Therefore, we decided on 8–10 runs as sample size. Details of both algorithms are described in the next subsections.

Detailed descriptions of solving the modified version of Tikhonov's inversion
algorithm is explained in detail by

The number of base functions,

In the case of the algorithm developed at TROPOS, the base functions have
triangular shape on a logarithmic radius scale

We can solve Eq. (

Details on the appropriate choice of

Finding the solution requires the application of several constraints. These constraints stabilize the inversion problem and help us reject mathematical solutions that are physically not reasonable. We use the simplifying assumption of a wavelength- and size-independent complex refractive index of the aerosol particles.

The rationale for using a gliding inversion window is given by

We also discretize the CRI search space. In this contribution the real part

In that way we obtain

The simulations were carried out with synthetic data and with uncertainties added to the data. The main purpose of the simulation with erroneous data was to learn by how much the inversion products could deviate from the correct results for various error levels. We tried to answer this question by distorting the optical data such that extreme changes (distortions) of the spectra of the backscatter coefficients (at 355, 532, and 1064 nm) and the spectra of the extinction coefficients (at 355 and 532 nm) could be achieved. We assumed an uncertainty of 5, 10, and 15 % for the data points.

For example in the case of 15 % error, we added 15 % to the extinction
coefficient measured at 355 nm and we subtracted 15 % from the extinction
coefficient at 532 nm. We did the same for the backscatter coefficients at
355, 532, and 1064 nm. In that case six combinations of

Table 1 summarizes the parameters of the PSDs that were used in the simulation studies. Additional explanations are given in Sect. 3.1.

Figure

Input parameters of the particle size
distributions used in the simulation studies. We used monomodal PSDs with
mean radius 100 nm. We used PSDs normalized to one particle per cm

Examples of retrieved particle size distributions for the case of
the geometric standard deviation of 2.1, real part of 1.5, imaginary part of
0.01

The panel shows the results for moderately absorbing aerosols, i.e. the
imaginary part is 0.01

The left plot of each row (a, d, g, j) shows the results for the mean radius of
60 nm. The middle plot in each row (b, e, h, k) shows the results for the mean
radius of 140 nm. The right plot of each row (c, f, i, l) shows the results for
the mean radius of 300 nm. These three mean radii are equivalent to effective
radii of 0.23, 0.55, and 1.2

We see that the shape of the particle size distributions can be derived to some degree. The individual solutions in general exhibit similar features. The panel shows that we may not be able to derive the exact shape of the particle size distributions. The individual PSDs show peaks that are not exactly at the position of the peak of the true PSD. Nevertheless, if we average all individual PSDs, the mean solution is comparably close to the true PSD.

The Potsdam algorithm (UP) uses TSVD as a hybrid regularization method and
collocation with B splines

It is a well-known fact (e.g.

The linear equation system

The spline number

Figure

For the computation step, we have developed a parallel software that allows
us either to cope with the vast parameter space or to enable us to carry out
a more refined search for a solution. This part of the software is designed
such that it runs separately from the interactive set-up and evaluation step.
In this way it can be used for parallelized execution on a supercomputer or a
computer cluster. A master process splits the work task into small units and
delegates the calculations to any available worker process. Once a worker has
completed its task, it returns the results to the master. The current search
algorithm allows for what we describe as embarrassingly parallel processing
(i.e. it does not require any interaction between the workers and
therefore scales to a large number of workers

The screenshot in Fig.

It is obvious that including the wavelength- and size-independent complex
refractive index grid, (Fig.

As already mentioned in the introduction, a second regularization technique
has been included in the software for the purpose of comparison. This
regularization technique solves the linear equation system

Examples of colour-coded refractive index grids and PSDs for
noiseless input data. The rows correspond to different imaginary parts of the
CRI: 0, 0.005, 0.01 and 0.05. The first two columns contain fine-mode
particles with gsd

As noted above there is a strong connection between the distribution of the B-spline nodes and the quality of the reconstructed PSD. To take this
connection into consideration, the Páde algorithm adapts the nodes
according to certain rules. Indeed, it is easy to see how the PSD is strongly
smoothed out in areas in which only a few nodes exist. Strong slopes and
curvatures of the PSD require many nodes in
their vicinity for an accurate reconstruction. In order to account for this behaviour, the nodes
automatically slide towards radius intervals that have
larger weight in the PSD during the iteration process, in contrast to fixed non-equidistant Chebyshev
nodes. For more details we refer to

Forward calculations provide us with backscatter and extinction coefficients, which are subsequently used in the data inversion. Even in the noiseless case in which these input data are computed without explicitly considering measurement errors, we still obtain uncertainties because approximation and rounding errors are added to the coefficients, i.e. the input data that are used in the inversion are not truly noiseless. Even those small errors can be harmful for an ill-posed inversion problem.

For the post-processing procedure we manually select the best complex
refractive indices depending on the best PSDs. In Fig.

The selection procedure is easier for fine-mode particles (Fig.

For the real part 1.4 the diagonal structure more or less disappears for
fine-coarse- (Fig.

In this section we explain the selection procedure of CRI and PSD. The main
selection criteria are based on our knowledge of working with simulated and
experimental data for the case of a grid mesh of

First, if grid points of the CRI are located along a diagonal they can be
collected into one cluster as long as this selection results in a good
representation of the PSD (Fig.

Second, if grid points form a thin vertical or horizontal line and additionally provide a bad approximation of the solution, i.e. of the PSD, all data points along the vertical or horizontal line should be removed. Grid points that appear isolated at the end of a diagonal or an arbitrarily shaped region (we denote them as boundary points) may be removed.

Third, as a rule of thumb one should select at least 10 and at most 20 grid
points. If meaningful, one can use accumulative two or three mathematical norm grids
of the refractive index (as previously explained). If more than one cluster
exists and if these clusters have the same number of solutions it is
difficult to decide which cluster should be preferred. Changing the
mathematical norm grid may help in the decision making. For more details see

Finally, for fine-mode particles (

Furthermore, the use of the Páde iteration as regularization method often
leads to very good results with respect to the pattern of the obtained
refractive index grid. Therefore, the manually controlled selection process
of the CRI, as described above, often can be performed more easily with Páde
iteration than with TSVD regularization. This is especially true for the PSD
examples

In general grid points are not isolated, i.e. all points are located in
clusters. If a diagonal structure is absent, we know from experience that
meaningful solutions can be identified in arbitrarily shaped clusters too

Examples of PSDs for input data with 15 % noise: the rows correspond to different imaginary parts of the CRI: 0, 0.005, 0.01, and 0.05. The columns correspond to gsd: 1.5, 1.7, 1.9, 2.3. The real part of CRI is 1.5. The mean PSD, i.e. the solution, is shown as a solid red line.

The UP algorithm was tested with the same examples (Table 1) as the
TROPOS/UH algorithm. The selection of the best CRI grid points, i.e. the
mean CRI, is always strongly connected with the corresponding PSD, i.e. we
look for similar shapes of the PSD. The retrieved mean PSD solution,
Figs.

Figure

In summary, the retrieval of the PSD in the case of noiseless data is
excellent with regard to retrieval accuracy and retrieval precision. Only for
almost all real and imaginary parts in the coarse-mode case and for
fine-coarse-mode cases with real parts around 1.4 and imaginary parts of 0

Figure

The retrieved PSDs of fine-mode particles with gsd

For the combination of real parts 1.5 and 1.6 and

In summary, if 15 % noise is added to the input data, the PSD can still be retrieved quite well for fine-coarse-mode particles in the case of real parts 1.5 and 1.6 in combination with all imaginary parts.

Table

In our study we consider particles of radius below 500 nm as fine-mode particles and particles above 500 nm as coarse-mode particles. We did not investigate explicitly bimodal particle size distributions. All tests were done with monomodal particle size distributions. Nevertheless we think that the choice of our particle size distributions still allows us to infer conclusions on the performance of our algorithms with respect to what is usually denoted fine-mode fraction and coarse-mode fraction of bimodal particle size distributions, even if the investigated size distributions are only monomodal. The sensitivity of the algorithm toward particle size depends on the underlying Mie-scattering efficiencies for single particles, and this dependence does not change in dependence of the modality of the particle size distribution.

Effective radii of 0.15 and 0.2

With regard to the complex refractive indices we tested three real parts, i.e. 1.4, 1.5, and 1.6. In our opinion these values cover a realistic range of real parts that can be expected for atmospheric particles. Values of around 1.4 describe highly refractory particles. Sea salt belongs to this class of particles. The value of 1.5 can be used to describe industrial pollutants, for example sulfuric acid. Soot has a high real part of 1.8, but it is usually not found in pure form. Thus, if soot mixes with other aerosol components the real part reduces. We estimate that 1.6 is a representative value for pollution particles, for example biomass burning particles that contain some amount of black carbon.

The imaginary part varies over several orders of magnitude. Our goal in our
software development is that we can find the correct value at least to
within

An accuracy of approximately

In the first step we used the UP algorithm to test if we are able to derive
the imaginary part to within

We had to restrict our simulations to a few imaginary parts because the
inversion algorithms are manually operated and the data analysis is time
consuming. We selected a few imaginary parts that were meant to give us a
reasonable overview on the retrieval performance if particles are
non-absorbing (imaginary part

Input parameters of the size distributions used in the simulation
studies. We used monomodal PSDs with mean radius 100 nm and 5 different geometric standard deviations. We used PSDs
normalized to one particle per cm

One point that must be considered in these simulations is the fact that, with
regard to experimental conditions, we likely will not find all possible
combinations of the particle size parameters (effective radius, real, and
imaginary part of the complex refractive index) listed in
Table

Table

The extinction-related Ångström exponents largely cover the range of values we found from measurements of extinction coefficients at 355 and 532 nm. We regularly measure extinction-related Ångström exponents of 1–2 in regions that are affected by anthropogenic pollution. Maximum values that have been measured are as high as 2.5, but we did not test this scenario in our study.

Values below 1 describe large particles in the coarse-mode fraction of
particle size distributions. The most likely candidate of an aerosol type
with extinction-related Ångström exponents below 1 is mineral dust.
However, we cannot simulate with reasonable confidence optical data that
describe mineral dust particles. Until now we could not identify a
light-scattering model that would allow us to model trustworthy values of
particle backscatter coefficients, i.e. scattering at 180

We also considered the case of extinction-related Ångström exponents around 0. Again, this scenario most likely occurs if large mineral dust particles are present. However, large sea-salt particles may also show extinction-related Ångström exponents of around 0 and for that reason we simulated such cases as well. We point out that it is not unlikely to find extinction-related Ångström exponents slightly less than 0, as shown in Table 2. It is unclear if negative values can be resolved by Raman lidar measurements in view of realistic measurement errors of 10–20 %.

The critical point in this table is the lidar ratios. Several values are quite clearly very unusual. Some lidar ratios are considerably higher than 100 sr, and some values are considerably lower than 20 sr.

There remains the question of whether it is justified to simulate scenarios in which lidar ratios considerably exceed 100 sr or drop below 20 sr. We believe that we should consider such extreme outliers in at least a few studies for two reasons. First, we can test the robustness of our algorithms for such extreme cases. The second point is that the underlying microphysical properties do not seem to be completely out of range of numbers we can expect for atmospheric particles. It is simply the combination of specific values of particle size distribution and CRI that creates these outliers of lidar ratios.

Percentage of simulation cases that result in given retrieval accuracy.

We carried out the inversions as described in the methodology section. We
tested how well we can retrieve the parameters of interest under favourable
circumstances. In this study we define favourable circumstances as the
situation in which we have approximate knowledge of the real part of the
complex refractive index. This latter assumption is based on a recent study
by

We also tested our algorithm under the assumption of comparably unfavourable measurement error scenarios. We distorted each optical data point by its maximum value of either 5, 10, or 15 %. In that regard, errors of 15 % represent the worst case scenario in this study. We did this distortion without considering the possibility that data points may be correlated to each other and thus error bars may also not be independent of each other. We assume that the inversion of such ”extremely” distorted backscatter and extinction spectra would result in microphysical parameters that also deviate to a maximum value from the correct values. This assumption of course has the flaw that the inversion is a non-linear problem. That means an extreme distortion of optical input data may not necessarily need to lead to a maximum deviation of the retrieved microphysical parameters from their true values. However, we believe that we will learn more about this type of error analysis in this first attempt and that we can refine it in future studies.

Left panels: TROPOS/UH algorithm retrieval results for

Figure

According to

Under this assumption of known refractive index, the other parameters can be
derived accordingly. The results are shown as squares in Fig.

Surface area concentration on average shows exceptionally high accuracy. The
precision (the uncertainty bars reflect the statistical noise in terms of
1 standard deviation) is high compared to the error bars we obtain for
effective radius and volume concentration. We find one outlier (see Fig.

Volume concentration in general shows the same features as effective radius, i.e. the retrieval accuracy is generally good for particles in the fine-mode fraction. We find some outliers for the intermediate cases and the coarse-mode case, mainly for real parts of 1.4.

The real parts follow from the application of the methodology suggested by

The imaginary part can be found within

The noteworthy point and a new result compared over previous studies is that
our simulations give us an impression of the likely value of the
overestimation of the imaginary part. If the true imaginary part is less than
0.01 we may overestimate the imaginary part by 0.005

Figure

The microphysical properties do not differ significantly from the results if the real part is retrieved to 0.1 accuracy and if we take account of the overall uncertainties of our inversion results. That is, any attempt to further improve the real part retrieval may not necessarily result in significant improvement of other data products and we consider this an important outcome of our study.

With regard to single-scattering albedo, Fig.

The retrieved single-scattering albedo on average can be derived to within

With regard to the impact of measurement errors of the optical data on our
inversion results, we carried out numerous simulation studies in the past

In this study we made the first step toward refining our error analysis. We used 5, 10, and 15 % noise for each optical data point. We wanted to test by how much the microphysical parameters could shift for a given measurement uncertainty. In this study we are mainly interested in the likely maximum shift of the mean value of each data product. We try to provide a first answer to that question by what we call extreme error estimation. We distorted the backscatter and extinction spectra to their extremes and took the average of the eight extreme error runs as the final inversion result.

Another idea behind this approach of using maximum distortion is to find out what could possibly go wrong in data inversion if we do not know the error model that describes the uncertainty bars of the optical input data. For example we need to know how systematic and statistical error are computed from the Raman lidar data. Knowledge of the systematic and the statistical uncertainty would allow us to compute the uncertainty bars of the microphysical parameters in a more refined way. We need to know if a Gauss-like error distribution function sufficiently well describes the error bars of the optical input data, or if a different error distribution function is more appropriate. We emphasize that error distribution functions contain information that can be used in data inversion as additional pieces of constraint that might help us to improve the inversion results.

Our procedure needs to be taken with caution. It is just the first step
toward dealing with the concept of error analysis in a more concise way. For
example, we applied the noise for each data point individually, i.e. we did
not consider that errors of the individual data points of a given
3

In the following we only discuss the results for 15 % error (right panel
of Fig.

This effect of the loss of accuracy is obvious for effective radius. The uncertainties could become unacceptably large for some of the retrieved effective radii if we do not use other types of information to constrain the results or if we do not know the mathematical function that describes the error bars.

Left panels: TROPOS/UH algorithm results for

Surface area concentration remains within a reasonable range of uncertainty if we use 50 % deviation from the true values as benchmark. The retrieval error remains rather well behaved for the two PSDs that describe fine-mode particles, but also the other types of PSDs, i.e. the transition type and the coarse-mode type still deliver useful results.

Volume concentration shows reasonable results (we again use 50 % retrieval
error as benchmark when we talk about useful results) for all imaginary parts
except the value 0.05

The imaginary part is our main target of future studies as it allows us to
derive light absorption coefficients of particles. We find that the
uncertainty (variation) of the imaginary parts that we obtain from the
inversion of erroneous data to large part does not differ from the results we
obtain from the inversion of error-free data. The exception is high
imaginary parts (0.05

We find on average that single-scattering albedo is underestimated, regardless of the true value of the single-scattering albedo. We do not see a significant difference of the quality of the results (accuracy and precision) of single-scattering albedo at 355 nm and single-scattering albedo at 532 nm.

Figure

The main results were already discussed in the context of
Fig.

If we introduce optical data errors, the quality of the inversion results
worsens. Our approach of creating an extreme distortion of the optical
spectra gives us some insight into the maximum uncertainties that may occur.
We find that effective radius and volume concentration may exceed 50 %
uncertainty. Surface area concentration in most cases stays within

Retrieval results for effective radius (first row), surface area
concentration (second row) and volume concentration (third row). The first column
corresponds to noiseless data. The second column corresponds to data with
15 % noise. The last column shows the relative retrieval error for
increasing noise level of the input data: 0, 5, 10, 15 % for gsd 1.7 as an
example. Each subfigure

Accumulative distribution of inversion results for UP algorithm

Retrieval results for real part of CRI (first row) and imaginary part
of CRI (second row). First column corresponds to noiseless and second to
15 % noisy input data. The last column shows the relative retrieval error with
increasing noise level of the input data: 0, 5, 10, 15 % for gsd 1.7 as an
example. Each subfigure

Relative error of the retrieval results for the SSA 355 nm (first row) and
SSA 532 nm (second row). The first column corresponds to noiseless and second to
15 % noisy input data. The last column shows the relative retrieval error
with increasing noise level of the input data: 0, 5, 10, 15 % for gsd 1.7 as
an example. Each subfigure

Accumulative distribution of inversion results for TROPOS/UH
algorithm (left panel) and UP algorithm (right panel). TROPOS/UH algorithm:
results are shown for the constraint that the real part can be extracted to
an accuracy of 0.05 (solid thin line) and 0.1 (solid thick line), according
to

The simulation results of the UP algorithm are shown in
Figures

First, we evaluate the effective radius retrieval and the retrievals of total
surface area and volume concentration. In the case of noiseless input data,
for almost all fine-mode particle examples (

For almost all fine-coarse-mode and coarse-mode particles (

The total surface area and volume concentrations are two very stable
microphysical parameters. In almost all cases these parameters can be
retrieved to better than 15 %. With regard to surface area concentration
there are only two outliers, i.e. for

Second, we evaluate the retrieval of the CRI. The retrieval of the real part
is very stable. In particular for 1.5 (Fig.

The imaginary part has a larger variation if the particles are more
absorbing (Fig.

The relative error is less than 55 % except for eight outliers which for most part have an imaginary part of 0.01i. At the moment the reason for this behaviour is unknown.

In summary, with respect to the imaginary part, 83 % of all examples show
retrieval errors below 50 %, and 94 % of all investigated imaginary parts
show retrieval errors below 70 % (Fig.

In the case of noisy input data (15 %), the relative error of the effective
radius is monotonically decreasing with increasing real part of the CRI,
i.e. for real part 1.4 the relative error is less than 60 %, for 1.5 it is
less than 40 %, and for 1.6 it is less than 30 %. For real part 1.4, the
effective radius is overestimated in all examples of investigated PSDs. For
the real parts 1.5 and 1.6, this overestimation still occurs in the case of
the fine-mode particles (Fig.

The surface area concentration is the most stable parameter in the case of
inversion of noisy data. The retrieval error is less than 20 % (1 outlier)
with respect to the real part 1.4. In the case of real parts of 1.5 and 1.6,
the retrieval error is less than 15 % (two outliers). The parameter is slightly
overestimated for fine-coarse- and coarse-mode particles (Fig.

The volume concentration is also slightly overestimated, but in contrast to
surface area concentration this overestimation is for the most part only
happening for fine-mode particles (Fig.

In contrast to the noiseless data case, the real part of CRI 1.4 is always
underestimated (1 outlier) in the case of noisy data (Fig.

With regard to the imaginary part of the CRI, we find that the error bars are
very large for strong light-absorbing particles (precision is low), i.e.
imaginary parts are 0.03

As already mentioned before, if we want to obtain an absolute accuracy of
approximately

We evaluate the accuracy of single-scattering albedo for the case of
noiseless and noisy optical data. As expected from the absolute retrieval
error of the imaginary part of the CRI – which is larger for strong
light-absorbing particles – the single-scattering albedo at 355 and 532 nm
shows more or less the same behaviour in the noiseless and noisy cases respectively (Fig.

With respect to 15 % error of the optical data, we found that in 100 % of all
examples the retrieval errors are below 12 % for the single-scattering albedo
at 355 and 532 nm. The retrieval errors are well below 6 % for 87 % of all
examples at the wavelength 355 nm and for 88 % of all examples at the
wavelength 532 nm (Fig.

Regarding the absolute deviation and the cases of noiseless input data, we
found an underestimation of the SSA at 355 nm (and similarly for SSA at 532 nm)
for non- and weak light-absorbing particles in the intermediate and
coarse mode. For non- and weak light-absorbing particles in the fine-mode, the
accuracy limit of

We complete our evaluation by showing how the retrieval errors increase with
respect to increasing noise level. For that purpose we use one example, i.e.
fine-mode particles with gsd

Although the surface area concentration is a very stable parameter in the
retrieval process. Surprisingly, Fig.

The retrieval of the real part of the CRI (Fig.

We summarize the status of two manually operated data inversion algorithms
(TROPOS/UH algorithm and UP algorithm) that are used to derive microphysical
parameters of atmospheric particles. The optical input data for these
algorithms are collected with EARLINET multiwavelength (3

We tested the algorithms ability to derive the investigated particle
parameters as accurately as possible, if the optical input data are error
free and the real part of the complex refractive index is known to 0.1 or 0.05 uncertainty. This latter assumption was introduced in our
simulation study in view of results published by

We find that the effective radius of the PSDs in the fine-mode, the intermediate case (fine- and coarse-mode particles), and the coarse-mode can be retrieved well. Accuracy usually is better than 25 % in all cases, though we notice outliers. Surface area concentration can be retrieved with an accuracy of approximately 10 %. Statistical errors (precision) is just about large enough to include the true values too. In most cases we slightly underestimate the true values. Volume concentration can be retrieved well for the fine-mode, intermediate case, and coarse mode. Accuracy is better than 20 % except for six outliers out of 75 investigated cases.

The real part was retrieved to either 0.1 or 0.05 uncertainty according to
the methodology by

It seems that the accuracy of the retrieved imaginary part in general is
better than 30 % for all investigated cases if the true imaginary part is

If the imaginary part is less than 0.01 we can merely decide if the mean
value is

Single-scattering albedo can be derived to approximately 0.05 in many of the
investigated cases. Accuracy is in part better than 0.05, but if we include
statistical errors we find deviations up to 0.1. These results however depend
on the constraint of the real part. We find that if we can constrain the
real part to 0.05, single-scattering albedo can be derived to better than
0.05 as long as its true value is

We finally notice that it may not be so important for the retrieval quality
of effective radius, and surface area and volume concentration if the real
part cannot be constrained to better than 0.1. Effects are more pronounced
with regard to the imaginary part, though accuracy nearly always stays within

With regard to our extreme error computations we find that results for the
most part show an accuracy better than 50 %. In many cases the accuracy is
significantly better than 50 %. We find outliers. We do not have sufficient information
from this limited set of simulations that allows us to identify a pattern
that could explain when the outliers occur. We believe that the inversion
method is robust enough to provide microphysical size parameters with

With regard to the imaginary part we notice that on average the accuracy
worsens with increasing noise level. We find more cases of overestimation
and underestimation of the true imaginary part compared to the case
of correct optical data. Results slightly differ if the real part is
constrained to 0.05 or 0.1. However, if we take only results for
which the true imaginary part is larger than 0.01, we still find that the
retrieved imaginary parts are within

Single-scattering albedo seems to be significantly underestimated compared to
the results for single-scattering albedo in the case of noiseless optical
data. In particular we notice a wider scatter of values of single-scattering
albedo around the true values. In contrast, in the case of noiseless optical
input data, the results for single-scattering albedo remain rather well
confined within a narrow band of

We summarize here our results of the UP algorithm for 15 % noisy input data.

The total surface area concentration is the most stable parameter within the
microphysical retrieval procedure, i.e. 99 % of all examples are below
20, and 85 % are below 15 % retrieval error (Table

We found for the imaginary part of the CRI that for non- (0

The evaluation statistics show that the relative retrieval errors of
effective radius and total volume concentration are prominently larger for
the real part 1.4. The retrieval of the imaginary part of the CRI has
significantly larger error bars for strongly light-absorbing particles of
0.03

For fine-mode particles with gsd

As expected from the absolute retrieval error of the imaginary part of the
CRI, which is larger for strong absorbing particles, the single scattering
albedo shows more or less the same behaviour for 355 and 532 nm. For the
single scattering albedo, the relative errors stay below 12 % in 100 % of all examples. In more detail, 87 % of all examples for 355 nm and 88 %
of all examples for 532 nm are well below 6 %. With respect to the absolute
error we achieve an accuracy limit of

Figure

With regard to the UH/TROPOS algorithm we see that, if we have knowledge of the real part, approximately 50 % of all solutions for effective radius deviate by less than 5–8 % from the true results. We find that 75 % of all solutions have uncertainties of less than 15 %. Each solution deviates less than 50 % from the true results.

If uncertainties of the optical data are 15 %, only 50 % of all solutions have errors less than 50 %. The uncertainties of the other 50 % of the solutions may be considerably larger. We note that this is an extreme estimate, as we distorted the optical backscatter and extinction spectra to their maximum.

In the case of surface area concentration, 75 % of all simulated cases deviate less than 10 % from the true results, regardless of whether the real part is known to 0.05 or 0.1 accuracy. If we introduce uncertainties, we find that 75 % of all solutions deviate less than 20 % from the true results.

In the case of volume concentration we find that 50 % of all solutions deviate less than 10 % from the true values, 75 % of all cases deviate less than 20 % from the true values, and in 90 % of all simulated cases the retrieval error is less than 50 %. If we assume uncertainties of 15 % for the optical data, volume concentration may deviate less than 20 % from the true value in 50 % of all simulation cases.

With respect to the imaginary part the deviation is less than approximately
0.006 in 75 % of all cases. If measurement errors are included the deviation
is

Single-scattering albedo at 355 and 532 nm can be derived to better than
0.04 in 75 % of all cases if the real part is known to 0.05–0.1. In 90 %
of all simulation cases we can retrieve single-scattering albedo to better
than 0.06. If we introduce measurement errors the uncertainty of
single-scattering albedo is

The next paragraphs describe the retrieval results of the UP algorithm. With regard to the noiseless data case, we find that approximately 50 % of all solutions for the effective radius deviate by less than 5, and 90 % of all solutions deviate by less than 14 % from the true values. Each solution deviates less than 25 % from the true values. In case of uncertainties of 15 % of the optical data, only 50 % of all solutions have errors less than 25, and 85 % of all solutions stay below 50 % deviation.

The total surface area concentration proves to be the most stable parameter as already mentioned. That is, 75 % of all simulated cases have uncertainties less than 8, and 95 % of all cases deviate less than 15 % from the true values. Even in the data case of 15 % noise, 85 % of all investigated cases deviate by less than 15 % from the true values, i.e. the data error is not amplified during the retrieval process.

Concerning the noiseless data case of the total volume concentration, we find that 50 % of all examined cases deviate less than 5 % from the true results, and 90 % of all cases have less than 15 % uncertainty. If we assume 15 % optical data error, the qualitative behaviour of the uncertainty behaviour of volume concentration is similar to the uncertainty behaviour of effective radius. Only 47 % of all retrieved solutions show errors less than 20 %, and 90 % of all solutions deviate less than 50 % from the true values.

With respect to the real part of the CRI we find that the differences between
the retrieved and the true values in the noiseless data case are

With regard to the imaginary part we observe that there is no big difference
between the noiseless and noisy data case. On average the deviation between
retrieved values and true values is

The single-scattering albedo at 355 and 532 nm shows a similar qualitative
behaviour as the imaginary part of the CRI, namely, there is only a quite
small difference between the noiseless and noisy data case. On average the
deviation is

Table

We think that we can achieve a retrieval accuracy of the size parameters of 20 %. Any better value seems unrealistic in view of measurement errors and errors caused by the inversion method. A retrieval accuracy of 0.005 of the imaginary part may be possible. We furthermore target a retrieval accuracy of 0.05 for single-scattering albedo.

We note that both algorithms have advantages and drawbacks. The simulation
studies use the same size distributions and complex refractive indices but
follow different aims in testing each algorithm. Therefore, a truthful
comparison between the algorithms is not possible. Nevertheless, it is
remarkable that the surface area concentration is found to be the most stable
microphysical parameter in the retrieval process. For example, in the noiseless
optical data case and the case of surface area concentration, a deviation of
less than 20 % compared to the true result is achieved in 95 % (for TROPOS/UH) and 97 % (for UP)
of all investigated cases. Moreover, for the case of 15 % noise
of the optical data, volume concentration deviates less than 20 % from the
true values in 48 % (for TROPOS/UH) and 47 % (for UP) of all simulated cases (Table

Future development work of the TROPOS/UH algorithm will focus on deriving fine mode and coarse mode particle parameters separately. We want to investigate to what extent a wavelength-dependent complex refractive index of aerosol particles influences the quality of the retrieval results. The reason for this study is that we assume a wavelength-independent refractive index in our retrievals. We want to investigate whether we can derive the single-scattering albedo with less uncertainty if the true value of SSA is above 0.9.

In future the Potsdam group will improve the presented software using the two regularization methods, namely TSVD and iterative Páde method in parallel to utilize advantages of both methods simultaneously.

Currently the Potsdam group is investigating a microphysical retrieval procedure via regularization for non-spherical particles, in particular, spheroidal particles, i.e. oblate and prolate particles. In future a user-friendly software tool for non-spherical particles could be developed.

This work has been supported since 2000 under Grant No. EVR1-CT-1999-40003
(EARLINET project) of the Environment Program of the European Union and Grant
No. RICA-025991 (EARLINET-ASOS project) of the 6th Framework EU program. The
work has been supported partially by the European Union Seventh Framework
Program for research, technological development, and demonstration under grant
agreement No. 289923 – ITaRS and by DAAD project “Ostpartnerschaften” of
Potsdam University. Finally, this work received partial funding from the
European Union's Horizon 2020 research and innovation programme under
grant agreement No. 654109 (Actris).
Edited by: A. Ansmann
Reviewed by: three anonymous referees