RETRIEVAL OF THE TROPOSPHERIC AEROSOL MICROPHYSICAL CHARACTERISTICS FROM THE DATA OF MULTIFREQUENCY LIDAR SENSING

Regular lidar measurements of the vertical distribution of aerosol optical parameters are carried out in Tomsk (56 0 N, 85 0 E) since April, 2011. We present the results of retrieval of microphysical characteristics from the data of measurements by means of Raman lidar in 2013. Section 2 is devoted to the theoretical aspects of retrieving the particle size distribution function ) (r U (SDF) assuming a known complex refractive index m (CRI). It is shown that the coarse fraction cannot be retrieved unambiguously. When estimating ) (r U and m together (section 3), the retrieved refractive index is non-linearly related to the optical coefficients and the distribution function, which leads to appearance of different, including false values of m . The corresponding ) (r U differs only slightly, so the inaccuracy in m does not essentially affect the retrieval of the distribution function.


INTRODUCTION
At present, lidar systems are applied for routine observations in the European Aerosol Research LIdar NETwork (EARLINET) [1], the Asian Dust NETwork (AD-Net) [2], and the lidar network in CIS countries (CIS-LiNet) [3].At night, they allow three backscattering coefficients including the free troposphere.The spectral set of the optical coefficients and the errors of their estimation allow us to proceed to reconstruction of the vertical distribution of the microphysical aerosol characteristics.

RETRIEVAL OF THE SIZE DISTRIBUTION FUNCTION UNDER THE ASSUMPTION OF KNOWN REFRACTIVE INDEX
The determination of the particle size distribution function ) ( r U (SDF) for the known complex refractive index m (CRI) is reduced to the inversion of a system of linear algebraic equations.There is a problem for the lidar data: the parameters of the fine fraction are retrieved well for any regularization matrix, but the parameters of the coarse fraction are estimated ambiguously.The stabilizer order in fact specifies the range of the correct retrieval of the coarse aerosol mode parameters.This in turn leads to biased estimates of coarse U .This clearly expressed dependence of the solution obtained on the restrictions on its smoothness is primarily caused by insufficient information, since for the coarse aerosol mode the spectral dependence of the optical coefficient becomes weak, and they contain in fact only two independent values.(2) can be found in the following form: where  is the regularization parameter, and Q is the smoothing matrix., see [4]).At the same time, the size distribution function of fine particles is retrieved well.
The statistical regularization method gives more accurate results: where u   2) is rewritten in the form where , The solution of ( 5 can be considered as an upper boundary of the correct estimation of the mean radius of the coarse fraction (and of the SDF as a whole and ) from lidar measurements.However, the statistical regularization method allows application of the plausible ) ( r U (plausible here implies the SDF retrieved previously from lidar or other measurements) for estimation of u W .The results are presented in more details for 462 empirical; models obtained at the Zvenigorod AERONET site in 2011-2012 [4].

SIMULTANEOUS RETRIEVAL OF THE SIZE DISTRIBUTION AND REFRACTIVE INDEX
Although in practice m and U(r) are determined in parallel, there is a principle difference: CRI is a parameter of the kernel functions of the system of Fredholm's equations (1) and is not linearly related neither with the optical coefficients nor the distribution function.It makes the problem more complicated, because there are no standard methods for solving the non-linear inverse problems.It is not easy even to answer the question: is it always possible to estimate m from lidar     2 3  measurements?The study of possible errors does not completely answer this question, because the errors can be caused both by the quality of the inversion algorithms and by efficiency of the used a priori data, and so on.
In general form, the problem is reduced to minimization of the discrepancy functional:     .However, possible ambiguity in the determination of the coarse particle parameters and possible false refractive index estimation have to be taken into account.
used by us allows the errors to be decreased down to  5%    and  2%    ,

Figure 1 .
Figure 1.Examples of the SDFs reconstructed for two models; stabilizers of the zero (Q = Q 0 ), first (Q = Q 1 )and second orders (Q = Q 2 ) were used.

Figure 1
Figure 1 demonstrates the results of estimating the size distribution function by the Tikhonov method (3).The biased estimates of coarse U are predetermined by the a priori choice of the stabilizer order that in turn can lead to different values both on the abscissa and the ordinate, in particular, it fixes the geometrical mean radius of coarse particles is the sought vector of weight coefficients, and u W W ,  are the covariance matrices of the errors, respectively.The matrix 1  u W is non-diagonal, and the preliminary estimation u W can make it possible to consider the data on the occurrence of the two aerosol modes (in fact, on the presence of several extrema), thereby cutting off a considerable number of physically unjustified solutions in the regularization stage.Let us approximate the coarse k Eq. ( ) for the coefficients b i is reduced to a solution of the system of three linear equations d b B    , where the components of the matrix B and the vector d  have the following forms: , of Eq. (1)) and on the other hand on the same coefficients ) ( m g calc j , simulated for the all possible values of the real real m and imaginary image m parts of the refractive index.Application of iterative procedures like in [6] is unfounded, because a consequence of the undetermined data used is the solution dependence on the initial approximation.It is worthwhile to use the relative error in Eq. (6), because the coefficients  and  can differ by two orders of magnitude. of the area lead to the appearance of the false local minima.Thus, in the presence of errors, multiple values of the refractive index on the   image real m m , plane correspond to the unique set of the values of the optical coefficients and distribution function too, and the problem of retrieving m is undetermined even for a known SDF.

Figure 2 .
Figure 2. Different discrepancy functionals at the known SDF.