Assessment of Feasibility of Reconstructing Microphysical Characteristics of Mist with the Use of White-Light Lidars at Short Paths
The further investigations are aimed at assessment of reconstruction feasibilities for lognormal distribution functions with a half-width larger than 0.01 pm. A disadvantage of the genetic algorithm consists in the rather long search for unknown parameters. In addition, the conditions for its termination cannot always be selected unambiguously. We have considered standard approaches to solution of the inverse problem of reconstruction of the particle size distribution functions, as well as approaches based on solution of the problem of optimization from the parameters of the distribution function and scaling parameters:
where a is the signal scale parameter, b is the coefficient of contribution to scattering from artificial aerosol at an angle of 174.5° (in fact, it accounts for the ethylene glycol particle concentration and the relative contribution in comparison with the molecular and aerosol atmospheric components), r and g are the vectors of modes and half-widths of the Gauss distribution densities, M is the coefficient of normalization to the unit distribution density, Д74 5(r, A;) is the coefficient of scattering at the corresponding angle for the ethylene glycol particle with the radius r, вштКА') is the total coefficient of scattering by aerosol and molecular atmosphere, (, h) is the extinction coefficient, Д is the depth of aerosol formation, (3(X) = /и ф(г',M,а,Г,g)fi(r,X)dr' + f3atm(A;) is the extinction in a cloud, Ie(A) and IS(A) are the backscattered signal spectrum and the supercontinuum spectrum coming to the detector (results from multiplication by transmittance), ваш(ХЬ is the background atmospheric extinction. In general, the considered gradient methods and regularization methods failed to provide a positive result. Then we have proposed other approach, which is considered below.
If the model transmittance or transmittance calculated by model data is used in Equation 5.68, then, with the standard aerosol and molecular model, this equation can be simplified and transformed into a system of equations for wavelengths A. The coefficients aj and а2, accounting for the contribution of the background (model) scattering component in the direction of 174.5° and the contribution of the scattering by mist play the role of scaling coefficients
The particle size distribution function is lognormal and depends on two parameters (Figure 5.22):
Figure 5.22 Examples of lognormal distribution functions for mists with different parameters g.
The parameter determining the particle number density N is taken into account in the scaling coefficient a1, and therefore it is taken equal to unity.
We propose the following approach: A sample of pairs of radii rm and parameters of width of the lognormal function g is created (see Equation 5.70). For every pair, the mist scattering coefficients are calculated for different wavelengths. Then the calculated coefficients are substituted into Equation 5.71
Then, search for the coefficients a1 and a2 is performed. It is obvious that they can be found through solution of the overdetermined system of linear Equations 5.69 with two unknowns.
Among the obtained coefficients, positive pair a1 and a2 are selected at solution of system (5.69). Then among them, we select the pair and the parameters of the lognormal function rm and g, at which functional (5.71) is minimal. It is obvious that the accuracy of solution depends on previously calculated (3174 5(, rm, gj) and noise present in the spectral signal, as well as errors in determination of the spectral profile of model scattering coefficients.
In the case of consideration, we have carried out the study for modes of lognormal distributions from 0.2 to 2 pm in the range of radii from 0.0001 to 4 pm and for the parameter g from 1.01 to 1.3. For matching of half-widths as functions of
296 ? Optical Waves and Laser Beams in the Irregular Atmosphere the modal radius, the parameter g is specified as 1 + alrm, where a varies from 0.01 to 0.3. Thus, 5000 readouts for radii from 0.001 to 4 pm are created along with 50 ones for the parameter a. Enumeration of all versions takes about 15 min in a 2.5 GHz processor. A disadvantage associated with the computation time can be readily eliminated through parallelizing of algorithms and narrowing of the number of previously calculated Д74 5(, rm, gj). Although, as the experience shows, the decrease of the number of steps in radii more often can lead to other results, because the problem to be solved is ill-posed and allows multiple solutions. A possible way is to take a certain set of separated solutions providing a minimum as optimal solutions and then to search for the more accurate solutions in their vicinity.
Then, we have analyzed the possibilities of reconstruction of the distribution functions with allowance for the different level of noise in a signal.
Figure 5.23a shows the relative contributions to scattering, at which the guaranteed reconstruction of the particle size distribution is possible at a noise level of 10% for the parameter a equal to 0.03, 0.1, and 0.2. As the width of the distribution function increases, it is necessary to increase the contribution from mist to scattering. For the parameter a equal to 0.2, the scattering by mist should exceed the scattering by clear atmosphere two and more times. For narrow widths of the distribution functions, the situation is better, when the order of scattering can be the same or even several times smaller. This is caused by the fact that for narrow widths the scattering spectrum has high-frequency singularities, whereas at an increase of the width the scattering at a wavelength is a slightly varying function.
Figure 5.23b shows possible errors of reconstruction of the distribution mode for the noise level of 10%—20% at different widths of the lognormal function. Thus, for example, for the 0.2-pm mode the reconstruction 0.34 pm, while for 0.5 pm, it gives 0.6 and 0.49 pm. In the previous case, the reconstruction from experimental data by the genetic algorithm yielded the 0.6-pm mode for particles with a diameter of 1 pm, which is in agreement with the obtained results. For
Figure 5.23 Ratio of the mist contribution to scattering to the scattering by clear atmosphere (a) and deviations of the reconstructed modal radii from preset model radii (b).
larger radii, more accurate results were obtained, but near the mode of 1.4 pm multiple solutions are obtained.