A new backscatter lidar forward operator was developed which is based on the distinct calculation of the aerosols' backscatter and extinction properties. The forward operator was adapted to the COSMO-ART ash dispersion simulation of the Eyjafjallajökull eruption in 2010. While the particle number concentration was provided as a model output variable, the scattering properties of each individual particle type were determined by dedicated scattering calculations. Sensitivity studies were performed to estimate the uncertainties related to the assumed particle properties. Scattering calculations for several types of non-spherical particles required the usage of T-matrix routines. Due to the distinct calculation of the backscatter and extinction properties of the models' volcanic ash size classes, the sensitivity studies could be made for each size class individually, which is not the case for forward models based on a fixed lidar ratio. Finally, the forward-modeled lidar profiles have been compared to automated ceilometer lidar (ACL) measurements both qualitatively and quantitatively while the attenuated backscatter coefficient was chosen as a suitable physical quantity. As the ACL measurements were not calibrated automatically, their calibration had to be performed using satellite lidar and ground-based Raman lidar measurements. A slight overestimation of the model-predicted volcanic ash number density was observed. Major requirements for future data assimilation of data from ACL have been identified, namely, the availability of calibrated lidar measurement data, a scattering database for atmospheric aerosols, a better representation and coverage of aerosols by the ash dispersion model, and more investigation in backscatter lidar forward operators which calculate the backscatter coefficient directly for each individual aerosol type. The introduced forward operator offers the flexibility to be adapted to a multitude of model systems and measurement setups.

In spring 2010, the Icelandic volcano Eyjafjallajökull erupted several
times. The emitted ash was found to be harmful for aircraft, and due to
uncertain information about spatial distribution and concentration of
volcanic ash, the European air space was closed for several days

Atmospheric chemistry models which allow for aerosol dispersion predictions
are, amongst others, COSMO-ART (Consortium for Small-scale Modeling, Aerosols
and Reactive Trace gases;

Lidar (light detection and ranging) is capable of providing information on
atmospheric particles with high temporal and spatial resolution. The most
basic lidar type is the backscatter lidar which measures the backscattered
signal intensity of a volume at a certain range. Comparing the data of such a
backscatter lidar that is operated in the UV with simulations of an
atmospheric chemistry model allows for the characterization of
transport and optical properties of aerosol particles near sources

While the number of sophisticated lidar instruments that provide
thermodynamic data

There are already several backscatter lidar forward operators available or in
development which are based on the calculation of the extinction coefficient.
The backscatter coefficient is then calculated assuming a given lidar
ratio

We designed a forward operator which is based on the distinct calculation of extinction and the backscatter coefficients in the model system. This forward operator can be adapted to particle-representing atmospheric model and backscatter lidar systems even using multiple wavelengths. It has the capability to calculate both the attenuated backscatter coefficient and the lidar ratio from model output data with a minimum set of external information. The name of the forward model is “backscatter lidar forward operator” (BaLiFOp).

In the following we explain the lidar principles and the theoretical
background for the backscatter lidar forward operator
(Sect.

The lidar principle is based on the emission of laser pulses into the
atmosphere and the measurement and analysis of the backscatter signals. The
received photon number per pulse

Instrument-dependent variables of the lidar equation are the wavelength

The range resolution is usually matched to the temporal resolution of the
data acquisition system by

Processes in the atmosphere are described by the backscatter coefficient

Elastic backscatter lidar systems do not allow for a separate measurement of

According to Eq. (

The total extinction coefficient

For isotropic scattering, the differential backscatter cross section is
derived from the scattering cross section

For non-isotropic scattering, a phase function

Molecule scattering and particle scattering are differentiated here, as the respective calculations depend on suitable physical theories and algorithms.

For a model which is capable of distinguishing atmospheric gases such as
nitrogen, oxygen, argon, and water vapor, the molecule scattering calculation
could be performed for each individual gas type and molecule size using the
Rayleigh theory

Consequently, the molecule extinction coefficient

To calculate the scattering cross section

The scattering characteristics of larger particles are described by Mie's
solution of the Maxwell equations

As a rough estimate, the computational time increases by about 1 order of magnitude when using T-matrix instead of Mie scattering calculation routines and by another 2 orders of magnitude when using DDA instead of T-matrix. Another increase in computational time results from larger scatterers; i.e., an increase in the particle size results in an exponential increase in computing time. In this study, Mie scattering algorithms are therefore used to perform fast calculations. The effect of scattering by non-spherical particles is analyzed in a second step by T-matrix scattering calculations for several non-spherical particle shapes in the framework of sensitivity studies. This approach is required because the COSMO-ART volcanic plume simulation does not output any information about the particle shape distribution.

Mie scattering-related computations were performed using the
IDL (Interactive Data Language)
procedure “mie_single”, provided by the Department of Atmospheric, Oceanic
and Planetary Physics (AOPP), University of Oxford. Input parameters of the
procedure are the real part

As a warning, we would like to point out that the procedure changed its
definition of the backscatter efficiency: the 2012 release of mie_single
returns the so-called radar backscatter efficiency, which is

A major problem of discrete size distributions is the high sensitivity of the
optical cross sections to the particle size: a slightly different particle
radius may lead to quite a large change of the scattering properties. We
present in the following an approach to overcome this problem. Due to the
fact that naturally occurring particle size distributions are not discrete,
averaging the optical cross sections over certain size-intervals seems
straightforward. We will show that this approach indeed reduces the
problematic and unrealistic sensitivity significantly. If the model
represents only one type of particle, i.e., with a constant refractive index
but with discrete radii

The calculation of the effective values is performed for every discrete size
class

The forward-modeled total extinction coefficient and total backscatter
coefficient are the sum of the molecule and the particle extinction and
backscatter coefficients:

The two-way transmission

Within the forward operator, the two-way transmission is discretized by using the models' vertical layers as height increment and vertical resolution.

Even though the lidar ratio is not measured directly by current ACL systems,
the capability of simulating the lidar ratio for given scatterer types and
scatterer mixtures offers great potential for sensitivity studies but also
for comparison to research lidar systems such as Raman lidar. The forward-modeled total lidar ratio

To analyze the lidar ratio sensitivity independent of a models' particle size
class and type class configuration, we introduced the pure lidar ratio

Distribution and transport of volcanic ash over northwest Europe
sketched using georeferenced satellite images (Meteosat-9, Dust). After
georeferencing, the ash layers were retraced as colored polygons, where the
color of the polygons (yellow to red) represent consecutive time steps

The 2010 Eyjafjallajökull eruption was extensively analyzed by scientists
from many fields of research, resulting in a substantial knowledge base (see
ACP special issue “Atmospheric implications of the volcanic eruptions of
Eyjafjallajökull, Iceland 2010”). Ash layers were observed from a large
set of measurement instruments, allowing for tracking of the volcanic ash plume over Europe

In terms of dispersion modeling, such a volcanic eruption case has a
well-known aerosol source location. This feature renders the
Eyjafjallajökull eruption an important case study for aerosol dispersion
simulation models and respective validation methods

ACL networks are a valuable data source for analyzing the
vertical and horizontal structure of aerosol particles, model verification,
and data assimilation. A qualitative analysis of the Eyjafjallajökull ash
plume over Germany using observations from 36 ACL systems CHM15k manufactured
by Jenoptik (currently known as Lufft) was performed by

We used the NetCDF files with ACL raw data where one file contains the 24 h
measurement of one ACL station. From our analysis of the ACL
measurements from 14 to 16 April 2010, we identified six stations
where the volcanic ash plume was visible without being tainted by other
clouds or hidden by fog layers near the ground.
The received photon number per shot is calculated from

The equation for calculating the
attenuated backscatter coefficient from ACL-measured photon counts reads

Attenuated backscatter coefficient measurement from CALIOP used to
calibrate the ACL measurement during the Eyjafjallajökull eruption phase.
The volcanic ash plume is visible around 50.15

Unfortunately, the instruments provided no calibrated measurement data at
that time, so a linear calibration factor

As a last step, the high-resolution ACL data was gridded to the model's vertical resolution and to 15 min time steps. This also improved the signal-to-noise ratio of the ACL data.

COSMO-ART was set up by DWD in collaboration with Karlsruhe Institute of
Technology (KIT) for an ash-dispersion simulation of the volcanic emissions
during the eruptive phase of Eyjafjallajökull in spring 2010

Sketch of the particle size distribution represented by COSMO-ART for the Eyjafjallajökull dispersion simulation (red dots). The red lines with bars indicate the averaging margins that were defined for the calculation of effective optical cross sections.

For this study, the 78 h forecast was used, beginning on 15 April 2010,
00:00 UTC, which includes volcanic ash emission data starting from 14 April 2010,
06:00 UTC. Volcanic ash was represented by six discrete size classes with
aerodynamic diameters of 1, 3, 5, 10, 15, and 30

A detailed analysis of the emitted ash was performed by

Output variables of COSMO-ART used by the forward operator for the selected case study.

According to

Electron microscope images from the same study revealed that the volcanic ash
particles were sharp edged with a complex and asymmetric shape. The average
asymmetry factor was 1.8 for small particles (

The particle growth due to hygroscopic water coating was quantified to be about
2 to 5 % at a relative humidity of 90 %

The representation of the particles by the model is clearly simplified, so the effect of these simplifications on the scattering of laser light must be determined when applying the forward operator. For a lidar forward model, sensitivities of the backscatter cross section are critical because the received signal intensity is linearly coupled to the backscatter cross section and, consequently, to the attenuated backscatter coefficient.

Prior studies already showed the complexity of non-spherical scattering
calculations but there is no universal solution to the problem available.

Sensitivity of

The same as Fig.

It must be noted that these studies are required for most aerosol types as most naturally occurring aerosols are not perfectly spherical and even slightly non-spherical ellipsoids may have very different scattering characteristics compared to ideal spheres.

Look-up tables (LUTs) of Mie efficiencies and optical cross sections have been
created to reduce the effort spent on time-consuming scattering calculations. The
look-up tables have three dimensions: size parameter

The reasonable range of size parameters depends on the wavelength of the
lidar transmitters and the radius of occurring particles

As explained in Sect.

Relative errors of the effective extinction cross
section

Extinction cross section

A measure for the refractive index sensitivity of the effective optical cross
sections is given by Fig.

It is defined as the error of the optical cross sections if the reference
refractive index (

Settings of the T-matrix procedure for the particle shape sensitivity study. The parameters were kept constant during the study except the particle shape parameters (EPS and NP).

Extinction cross section spectrum for the reference particle
(sphere, dark grey line), six types of ellipsoids (EPS

The T-matrix calculations performed in this study are based on the FORTRAN code
for randomly oriented particles, written and provided by

The double-precision version of the T-matrix procedure was modified to
perform scattering calculations of multiple particle sizes automatically. In
addition, the procedure was extended by calculating and returning the
backscatter cross section

The same as Fig.

The same as Fig.

A list of T-matrix options we used for the particle shape sensitivity study
is shown in Table

In Figs.

No significant differences between the extinction cross section of spheres
and these ellipsoids were observable. The trend is, however, that cylindrically shaped
particles have a higher extinction cross section compared to ellipsoids.
Spheres have the lowest extinction cross section values over the whole
spectrum. Up to a volume-equivalent radius of 0.7

Regarding the backscatter cross section, there are significant differences
between the backscatter cross section of spheres and particles with other
shapes. Obviously, spheres are affected by interference effects which leads to
both fluctuating and oscillating values of the backscatter cross section,
while the backscatter cross section spectrum of other shapes is only weakly
fluctuating. As observed for the extinction cross section, the shape effect
becomes pronounced beginning at an equal-volume radius greater than
0.7

The particle shape effect on the pure lidar ratio is weakly pronounced for
small particle sizes (less than 0.75

Relative errors of the effective extinction cross section if
spherical particles are assumed but real particles have an elliptical
(NP:

The same as Fig.

A summary of the particle shape sensitivity study is shown in
Figs.

Effective optical cross sections of atmospheric gas molecules and
six volcanic ash size calculated for the ACL wavelength (

Time–height cross section of total backscatter coefficient, extinction coefficient, and two-way transmission, calculated by the forward model based on COSMO-ART output at Deuselbach station (western Germany). The vertical coordinates are given in kilometers above sea level (a.s.l.). The forward model used temperature, pressure, and volcanic ash particle data (no clouds, rain, fog, background aerosol, or other scattering objects). The two-way transmission is near 1 over clean-air conditions. Above ash layers, however, the two-way transmission has a value of only 5 %.

A list of effective extinction cross section and effective backscatter cross
section values for atmospheric gas molecules and for the six volcanic ash
size classes is shown in Table

Using the forward operator allows for plotting each variable of the lidar
simulation for analytic purposes (see Fig.

In comparison with Raman lidar measurements, both the maximum measured
extinction coefficient of

Point-data extraction of COMSO-ART output at the Deuselbach ACL
station; coordinate 1 is on 16 April 2010, 18:00 UTC, at a height of
1.9 km a.s.l. The individual backscatter coefficient

The same as Table

From output of the forward operator, the relative contribution to the total
signal and total mass density can be analyzed for each size class of
COSMO-ART, and total lidar ratio can also be calculated. This was done, for example, at two time–height coordinates: the first coordinate points to
model output from a coordinate inside the volcanic ash layer (Table

Regarding coordinate 1, the total backscatter coefficient is dominated by ash size classes 1, 2, and 3, while the signal contribution of classes 4, 5, and 6 is less than 5 % in total. The mass contribution is dominated by classes 3 and 4 while classes 2, 5, and 6 contribute 10 % of the total mass density. The total lidar ratio is 9.63 sr. Regarding coordinate 2, class 4 contributes about 68 % and class 6 about 30 % to the total backscatter coefficient. The mass contribution in coordinate 2 is also dominated by classes 4 and 6 but, in contrast to the backscatter coefficient, class 6 has a higher contribution to the total mass density than class 4. The total lidar ratio at this coordinate with predominately large particles is 46.53 sr.

General conclusions from this analysis about the relationship between
backscattering and mass, depending on particle size and wavelength, require
further investigation. For an application of the forward operator in this
study, however, there are two aspects to be mentioned: first, the
backscattering intensity inside the volcanic ash layer (coordinate 1) is
predominantly dependent on classes 1, 2, and 3, whose backscatter cross
sections are also overestimated by the forward operator due to the assumption
of sphericity (see Fig.

Attenuated backscatter coefficient of ceilometer (top) and forward
model (bottom) at Deuselbach station in Germany from 16 April 2010,
00:00 UTC, to 17 April 2010, 24:00 UTC, given in units of m

A comparison of ACL measurement and COSMO-ART simulation with an applied
forward operator at the Deuselbach ACL station in western Germany is shown in
Fig.

The qualitative comparison is currently limited to coordinates where the major fraction of scatterers are represented by both model and forward operator. There are, however, some scatterer fractions still missing in the present model runs for a comprehensive comparison: aerosol types other than volcanic ash such as anthropogenic emissions, mineral dust, soot, and pollen are not included, which leads to differences, especially in the planetary boundary layer. It is hard to predict yet whether the strong ACL signal in the planetary boundary layer is related to background aerosol extinction or errors of the COSMO-ART prediction. To further investigate this problem, future studies with several types of aerosols incorporated into the model are required.

A major purpose of the backscatter lidar forward operator is also performing quantitative comparisons of measurement and model output
data. Unfortunately, such a comparison is of limited validity in this case
study due to the unknown ACL calibration as noted in Sect.

Outside the volcanic ash layer, the forward operator returns an attenuated
backscatter coefficient value of

Regarding the attenuated backscatter coefficient inside the ash layer,
however, the forward operator returns stronger signals inside the ash plume
as well as a lower transmission behind the ash plume compared to the ACL
measurement. The maximum value of the attenuated backscatter coefficient
returned by the forward operator (about

A backscatter lidar model capable of calculating both the extinction and
backscatter coefficients was introduced. Detailed studies concerning the
scattering properties of particles and molecules were performed. Instead of
assuming a lidar ratio for given particles, this forward operator allows for
calculating the scattering properties even for mixtures of different particle
types. Data of a COSMO-ART ash-dispersion simulation for the
Eyjafjallajökull eruption in 2010 were used to run the forward operator and
perform both qualitative and quantitative comparisons between the output of
the forward operator and measurement data of an automated ceilometer lidar
(ACL) system. A major challenge for setting up the forward operator for a
given scenario is the calculation of the effective extinction cross section

The atmospheric gas mixture was treated as a uniform mixture of

For particle scattering, the ranges
of particle sizes were selected
according to the volcanic ash classes used by COSMO-ART (six monodisperse
classes with diameters of 1, 3, 5, 10, 15, and 30

Due to uncertain refractive indices and shapes of the volcanic ash, sensitivity studies have been performed to analyze the impact of different particle types and shapes on the effective extinction and backscatter cross section and the pure lidar ratio. While the extinction cross section was only weakly sensitive to variable refractive indices and particle shapes, the backscatter cross section was strongly sensitive to both. However, the sensitivities reduce significantly when applying size-averaging algorithms. After averaging, the relative uncertainty of the effective backscatter cross section is up to 280 % within the defined range of refractive indices. This study also indicates the dependency of the forward operator on precise information about the particle's refractive index.

From the findings of

In the literature, we find measured lidar ratio values for volcanic ash between
40 sr and greater than 100 sr

The total lidar ratio calculated from COMSO-ART output at sample coordinates 1 and 2 resulted in values of 9.63 and 46.53 sr, respectively, which is – for the first coordinate – lower than the lidar ratio values of the Eyjafjallajökull ash plume measured by Raman lidar above Munich and Leipzig. From our analysis of the pure lidar ratio, we found an underestimation of the calculated lidar ratio for some size classes due to the assumption of spherical volcanic ash particles. However, the particle size class configuration of the model could also have a huge effect on the calculated lidar ratio values due to the ash size coverage and ash size class configuration. Therefore, the forward-modeled total lidar ratio in this scenario is not expected to exactly match the lidar ratio derived from measurements. Further investigation on this topic is required to optimize the particle size class configuration of models using monodisperse size classes and the representation of non-spherical particles in the forward operator in order to obtain a better representation of the total lidar ratio.

A time–height cross section comparison of ACL measurement and forward-modeled COSMO-ART output was shown. Similar structures were observed but some features were found at different times and heights. At the Deuselbach ACL station, some ash layer features were predicted quite precisely by the model, for example the time of arrival of the ash plume at about 06:00 UTC but vertically shifted by about 1.5 km. The ash plume intersection with the planetary boundary layer on 17 April 2010 at 03:00 UTC was simulated about 9 h too early on 16 April 2010 at 18:00 UTC. Fine structures of the ash layer were only observable in the simulation but not in the ACL data due to noise. Furthermore, the contribution of individual classes to the total backscatter coefficient and to the total mass density for two sample cases were analyzed.

The missing calibration coefficients of the ACL system required the definition of a
calibration constant

A comparison of the measured and forward-modeled volcanic ash-attenuated backscatter coefficient inside the volcanic ash plume led to the conclusion that the model-predicted ash concentration was too high which could potentially be resolved by reducing the model-predicted ash concentration manually by a given factor until the forward-modeled COSMO-ART predictions and ACL measurements are quantitatively similar. Such a reduction could be part of a simple particle data assimilation system helping to calibrate particle dispersion simulations before in situ measurements are available – assuming that the particles optical properties are known. It is therefore required to develop methods in the future which allow for fast determination of an aerosol type's refractive index range, shape and aspect ratio.

As aerosol dispersion processes are directly coupled to vertical and horizontal movements in the atmosphere, a comparison of forward-modeled and measured backscatter lidar profiles offers great potential for validating and improving the dynamic and thermodynamic components of an atmospheric chemistry model. For a model with variational data assimilation methods, the data assimilation system would select the prediction variation which best fits the atmospheric state provided by lidar measurements, resulting in continuous adaptation of the model prediction to the real-world situation.

The absolute values reported by the Raman lidar systems at a wavelength of
1064 nm agreed within
the measurement uncertainties and expected natural differences in the sampled
air mass with the results of the forward operator; see
Sect.

There are, however, some error sources remaining: first, there are only
molecules and the six volcanic ash classes represented while background
aerosol is missing completely. Second, the ACL calibration is of limited
precision. Third, the contribution to the attenuated backscatter coefficient
of ash size classes 4, 5, and 6 is relatively low even though these classes
carry a large portion of the mass. This relationship depends on the ACL's
wavelength. In our case of a wavelength of 1064 nm, the sensitivity is
highest for particles with a diameter smaller than about 10

In conclusion, further investigation in scattering calculations of non-spherical particles is recommended to get more realistic optical cross sections for the forward operator. A decrease in uncertainties related to the forward operator can be achieved by refractive index measurements at the exact ACL wavelength. Refractive index measurements are a basic aspect of the forward operator as the optical cross sections can only be calculated if the aerosols' refractive index is known precisely. The model – and consequently the forward operator – must represent more aerosol types, especially background aerosols, mineral dust, sea salt, and soot, as missing extinction near the ground may cause the forward operator to overestimate the attenuated backscatter coefficient value from layers behind. Additionally, qualitatively more scatterer size classes are required to also represent the fine fraction and very large particles in the atmosphere. One approach for a better representation of the natural size spectrum of aerosols is the use of continuous number-size distributions which are aggregated from multiple distribution functions (“modal” approach). This already includes the size averaging which is necessary for monodisperse size distributions. Furthermore, the model delivers exact information about the outer margins, i.e., the number density of the fine and the extreme coarse fraction which is currently not reproduced by model and forward operator in the selected case study.

As many ACL devices are operating proprietary firmware, the manufacturers
have to be sensitized to data quality and reproducible measurement
calibration. Therefore, it is required that calibration is performed
automatically and transparently. In future lidar measurement networks, the
number of HSRL systems and Raman lidar
systems could potentially increase and allow for the assimilation of
extinction coefficient and backscatter coefficient directly. Activities are ongoing to collect,
homogenize, and distribute observations within an international framework using
present automated lidar systems.
Observation projects such as EARLINET

The uncertainties in both modeling and measurements will require sophisticated data assimilation algorithms not only for typical atmospheric variables but also for aerosol optical properties. Also a very good first guess of model simulations with respect to aerosol particles will be necessary so that more sources, types, and sinks can be included. Within its priority project KENDA (Kilometer-Scale Ensemble Data Assimilation) the COSMO Consortium has developed an ensemble Kalman filter for data assimilation on the convective scale. It has been used operationally by MeteoSwiss and DWD since March 2017. An advantage of the ensemble data assimilation system is that the assimilation can be carried out based on the pure forward operator, and that it is not necessary to calculate derivatives of the forward operator or the adjoint tangential model for carrying out data assimilation. Also, it naturally introduces model increments for all variables where some dynamic covariance is observed from the underlying ensemble model runs. DWD aims to test the assimilation of ACL data into the COSMO-ART model based on BaLiFOp.

The data for this paper can be made available upon request from the authors Armin Geisinger (armin.geisinger@uni-hohenheim.de), Andreas Behrendt (andreas.behrendt@uni-hohenheim.de), and Volker Wulfmeyer (volker.wulfmeyer@uni-hohenheim.de).

The authors declare that they have no conflict of interest.

The present study was part of the research project 50.0356/2012 funded by the German Federal Ministry of Transport and Digital Infrastructure (BMVI, prior BMVBS). We furthermore acknowledge the contributors to COSMO-ART, to the ceilometer network, to the IDL procedure mie_single, and to the T-matrix codes we used as a basis for our study. We are also thankful for helpful discussions with Cristina Charlton-Perez, Ina Mattis, Werner Thomas, and Frank Wagner. Furthermore, we would like to acknowledge the travel support and very interesting discussions within the framework of the European COST (Cooperation in Science and Technology) action towards operational ground-based profiling with ceilometers, Doppler lidars, and microwave radiometers for improving weather forecasts (TOPROF). Edited by: Andrew Sayer Reviewed by: two anonymous referees