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1. Algorithm Theoretical Basis
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The European Aerosol Research Lidar Network, EARLINET, was founded in 2000 as a research
project for establishing a quantitative, comprehensive, and statistically significant database for the
horizontal, vertical, and temporal distribution of aerosols on a continental scale. 

ACTRIS/EARLINET stations are typically able to retrieve aerosol optical properties, such as extinction and backscatter coefficients,
lidar ratio, optical depth, and the Angstrom exponent if Raman lidar signals are available. In cases when only elastic lidar signals are used,
backscatter and a backscatter-related Angstrom exponent are derived. 

The calculation of all these quantities is done after the cloud masking procedure and it is therefore attributed exclusively to aerosol particles. 

 
1.1 Physical meaning of the retrieved properties  
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When laser radiation with power :math:`P_L` at wavelength :math:`\lambda_L` is sent into the atmosphere, part of the radiation is backscattered.
The optical power :math:`P(\lambda,\lambda_L,z)` of the backscattered radiation received from the distance :math:`z` at wavelength :math:`\lambda` depends
on atmospheric composition through two parameters: the backscattering coefficient and the extinction coefficient, and is described by the lidar equation: 

.. math::
   P(\lambda,\lambda_L,z) \sim \frac{P_L}{z^2}\beta(\lambda,\lambda_L,z) \exp \left[-\int_0^z \alpha(\lambda,\xi) d\xi \right] \exp \left[ -\int_0^z \alpha(\lambda_L,\xi) d\xi \right]
   :label: eq_lidar

The backscattering coefficient :math:`\beta` is the fraction of incident radiation backscattered for unitary solid angle and for unitary length [m-1sr-1]. 
It depends on the kind of scattering process and on both  emission (:math:`\lambda_L`) and detection (:math:`\lambda`) wavelength. 

It is due to contributions of both molecules (m) and particles (p) of atmosphere: 

.. math::
   \beta(\lambda,\lambda_L,z)=  \beta_m(\lambda,\lambda_L,z)+\beta_p(\lambda,\lambda_L,z)  
   :label: eq_beta

The extinction coefficient is defined as the energy flux reduction per unitary path [m-1].  

It gives a measurement of the energy loss of the laser beam in the atmosphere. 

It is due to contributions of both molecules (m) and particles (p) of atmosphere deriving from both the scattering (s) and absorption (a) processes: 

.. math::
  \alpha(\lambda,z)=\alpha_{m,a}(\lambda,z)+\alpha_{m,s}(\lambda,z) + \alpha_{p,a}(\lambda,z)+\alpha_{p,s}(\lambda,z) 
  :label: eq_alpha

The extinction coefficient integrated over a spatial path provides the optical depth:  

.. math::
  \tau(\lambda,z)=\int_0^z  \alpha(\lambda,\xi)d\xi
  :label: eq_tau
 
The Raman configuration allows for the retrieval of the range-resolved particle lidar ratio. The lidar ratio is defined as the ration between the
particle extinction coefficient and the particle backscatter coefficient: 

.. math::
   S(\lambda,z)=\frac{\alpha(\lambda,z)}{\beta(\lambda,z)}
   :label: eq_lr

It is a parameter strongly related to the microphysical properties of the aerosols: shape, size distribution, chemical composition,
relative humidity. Unlike :math:`\alpha` and :math:`\beta`, :math:`S` doesn’t depend on atmospheric aerosol load, but only on aerosol type. 

The combination of the particle extinction at different wavelengths allows for the calculation of the Angstrom exponent:

.. math::
   k_{\alpha}(\lambda_1,\lambda_2,z)=\frac{\ln \left[ \frac{\alpha(\lambda_1,z)}{\alpha(\lambda_2,z)} \right] }{\ln \left[ \frac{\lambda_2}{\lambda_1} \right]}
   :label: eq_kalpha
 
This quantity is size dependent assuming larger values for smaller particles and ranges between -1 for very big particles and 4 for molecules.  

Similarly to the Angstrom exponent, the backscatter related Angstrom exponent can be calculated as:

.. math::
   k_{\beta}(\lambda_1,\lambda_2,z)=\frac{\ln \left[ \frac{\beta(\lambda_1,z)}{\beta(\lambda_2,z)} \right] }{\ln \left[ \frac{\lambda_2}{\lambda_1} \right]}
   :label: eq_kbeta
 
As for the Angstrom exponent this quantity is size dependent assuming larger values for smaller particles. However, it has to be noted that it is
even more sensitive than Angstrom exponent to the size of the particles, because the backscatter itself is more size-related than the extinction coefficient.
 

 

1.2 Basic concepts for the retrieval of aerosol optical properties  
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ACTRIS/EARLINET is mainly based on Raman lidar stations, i.e. lidars equipped with elastic channel (detection channel at the same wavelength of transmitted laser beam) and an additional channel for detecting the N2 Raman-shifted signal. This additional channel allows the direct measurement of the aerosol extinction (Ansmann et al., 1990). This means having the capability of independent retrieval of extinction and backscatter coefficient in good signal-to-noise ratio conditions, using the retrieved extinction in the elastic lidar equation reported above. Whenever this is not possible an assumption about the relationship between extinction and backscatter is needed for solving the lidar equation affecting the overall uncertainty of the aerosol backscatter coefficient. Within ACTRIS/EARLINET, aerosol extinction profiles are reported only when the Raman channel capability is used and therefore only with direct assessed measurement of the extinction coefficient profile.  

Solving the N2 Raman lidar equation involves a derivative respect to the range of the logarithm of the signal. This procedure is complex from mathematical point of view and needs for specific smoothing approaches. Within EARLINET many efforts have been done for comparing the different suitable procedures (Pappalardo et al., 2004): the linear fit has been identified as the most appropriate one. Two options are available the weighted and not weighted linear fit.

For what concerns the aerosol backscatter coefficient profiles, the SCC can provide aerosol products in a flexible way choosing from a set of possible pre-defined analysis procedures: it enables the retrieval of particle backscatter coefficients with the elastic technique by using both the Klett method (Klett, 1981; Fernald, 1984) and the iterative algorithm (Di Girolamo et al., 1995), but also the computation of particle backscatter coefficient profiles after the Raman method (Ansmann et al., 1992).   

Statistical errors are calculated starting from the statistical errors affecting the lidar detected signals: the statistical errors affecting the optical properties can be calculated using the Monte Carlo or error propagation law. The provided errors do not include the uncertainties related to the assumption made in the retrieval algorithms like: the uncertainty to the atmospheric molecular profile, the wavelength dependence of the extinction, the absence of aerosol in the backscatter calibration range and more relevant the lidar ratio values assumption in the elastic backscatter method and the calibration of the depolarization channels. The quantification of the resulting overall error is still under investigation and object of studies within ACTRIS/EARLINET. The current approach is to reduce as much as possible such errors improving the quality assurance procedures together with the calibration centre and selecting the best possible approaches (e.g. intensifying the scheduling of the depolarization calibration procedures).