Effects of systematic and random errors on the retrieval of particle microphysical properties from multiwavelength lidar measurements using inversion with regularization

In this work we study the effects of systematic and random errors on the inversion of multiwavelength (MW) lidar data using the well-known regularization technique to obtain vertically resolved aerosol microphysical properties. The software implementation used here was developed at the Physics Instrumentation Center (PIC) in Troitsk (Russia) in conjunction with the NASA/Goddard Space Flight Center. Its applicability to Raman lidar systems based on backscattering measurements at three wavelengths (355, 532 and 1064 nm) and extinction measurements at two wavelengths (355 and 532 nm) has been demonstrated widely. The systematic error sensitivity is quantified by first determining the retrieved parameters for a given set of optical input data consistent with three different sets of aerosol physical parameters. Then each optical input is perturbed by varying amounts and the inversion is repeated. Using bimodal aerosol size distributions, we find a generally linear dependence of the retrieved errors in the microphysical properties on the induced systematic errors in the optical data. For the retrievals of effective radius, number/surface/volume concentrations and fine-mode radius and volume, we find that these results are not significantly affected by the range of the constraints used in inversions. But significant sensitivity was found to the allowed range of the imaginary part of the particle refractive index. Our results also indicate that there exists an additive property for the deviations induced by the biases present in the individual optical data. This property permits the results here to be used to predict deviations in retrieved parameters when multiple input optical data are biased simultaneously as well as to study the influence of random errors on the retrievals. The above results are applied to questions regarding lidar design, in particular for the spaceborne multiwavelength lidar under consideration for the upcoming ACE mission.

ABSTRACT.-In this work we study the effects of systematic and random errors on the inversion 21 of multi-wavelength (MW) lidar data, using the well-known regularization technique, to obtain   Because of these challenges, the characterization of atmospheric aerosols is being made 56 through intense observational programs using remote sensing techniques. For example, NASA 57 has led several space-borne missions to study aerosol properties worldwide (e.g. the MODIS 58 instrument on the TERRA and AQUA platforms). However, satellite measurements possess 59 lower temporal resolution than ground-based systems. For example, the AERONET global 60 network [Holben et al., 1998] is providing large datasets of high temporal resolution ground-61 based aerosol measurements at more than 400 locations worldwide. But the aerosol retrievals by 62 AERONET and by many satellite platforms only provide column-integrated properties. By can be due to, for example, non-linearity of a photodetector or errors in calibration of the optical 97 data or the effect of depolarization due to optical imperfections in channels that are sensitive to 98 polarized light. From the methodological point of view, systematic errors can be caused by, for 99 example, errors in the assumed atmospheric molecule density profile, the selection of the 100 reference level (an "aerosol-free" region that may actually contain a small concentration of 101 particles), or the use of an incorrect extinction-to-backscatter ratio to convert backscatter lidar 102 measurements to extinction. 103 In general, we expect that systematic errors such as these can affect the retrieval. The aim 104 of this work, therefore, is to study the sensitivity of microphysical retrievals by the regularization 105 technique to systematic variations in the input optical data provided by the 3ß + 2α lidar 106 configuration. Particularly, we will focus on the study of bimodal size distributions widely found 107 in nature (e.g. Dubovik et al., 2002). We will show that the results obtained can also be used to 108 assess the sensitivity of the retrievals to random errors in a new way. The study involves 109 simulations based on three different bi-modal aerosol size distributions, one with a large 110 predominance of fine mode, another with slight predominance of coarse mode and the last one 111 with slight predominance of fine mode.

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The procedure that we used is the following: first the optical data consistent with the 113 three aerosol size distributions described above are generated using Mie theory. Then the optical 114 inputs are systematically altered to provide a known amount of systematic error in each of the 115 individual input data. The inversion code is run using both the biased and unbiased optical data  Where 'j' corresponds either to backscatter (ß) or extinction (α) coefficients, g j (λ i ) are the 127 corresponding optical data at wavelength λ i , n(r) is the aerosol size distribution expressed as the 128 number of particles per unit volume between r and r + dr, and K j,N (m,r,λ i .) are the number kernel 129 functions (backscatter or extinction) which are here calculated from Mie theory assuming 130 spherical particles and depend on particle refractive index 'm', particle radius 'r' and wavelength 131 'λ'. Finally, r min and r max correspond to the minimum and maximum radius used in the inversion.

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The size distribution in Equation 1 can be written in terms of surface (s(r) = 4πr 2 n(r)) or volume 133 (v(r) = (4/3)πr 3 n(r)) size distribution. The corresponding kernels are obtained by dividing 134 K j,N (m,r,λ i .) by 4πr 2 and (4/3)πr 3 respectively, and are thus given by: where K j,S (m,r,λ) and K j,V (m,r,λ) are the surface and volume kernel functions, respectively. for spherical particles can be found in the references [e.g. Bohren and Huffman, 1983].

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The regularization technique used here to solve equation 1 has been discussed extensively 145 elsewhere [e.g. Veselovskii et al., 2002Veselovskii et al., , 2004Veselovskii et al., , 2005] and thus we provide here only a brief 146 overview. The key point is identifying a group of solutions which, after averaging, can provide a 147 realistic estimation of particle parameters. Such identification can be done by considering the 148 discrepancy (ρ) defined as the difference between input data g(λ) and data calculated from the 149 solution obtained. The retrieval uses an averaging procedure that consists of selecting a class of number of solutions in arriving at the best estimate of the particle parameters.

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The inverse problem considered here is under-determined, so constraints on the inversion 156 are needed. We consider a set of possible values of the particle refractive index as well as a set of 157 possible radii within a certain size interval. In general, the retrieval result will depend on the 158 range of parameters considered: the larger the range, the higher the uncertainty of the retrieval as Where N t,i is the total particle number of the ith mode, ln[σ i ] is the mode width of the ith mode 173 and r i n is the mode radius for the number concentration distribution. The index i = f, c 174 corresponds to the fine mode and the coarse mode, respectively. In the retrieval procedure, the We consider three types of aerosol size distributions for the simulations which we call type I, 186 type II and type III. These size distributions are used to approximate real aerosol types found in 187 the atmosphere. All types use r f v = 0.14 µm, lnσ f = 0.4, r c v = 1.5 µm and lnσ c = 0.6. These mode 188 radii and widths are representative of those provided by Dubovik et al., (2002)    AERONET network provides refractive indices with very similar errors (Dubovik et al., 2000). 218 Thus, the range of refractive indexes proposed for the size distribution is enough to cover most 219 of the values obtained by AERONET (Dubovik et al., 2002).

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The regularization inversion is then performed on these data and we obtain the retrieved  Moreover, given that the longest wavelength measurement used here is 1064 nm, the technique 238 has reduced sensitivity to the coarse mode of the aerosol distribution. Thus, to stabilize the 239 retrievals, the maximum radius of the retrieval interval was set to 5 µm. Additionally, the kernel 240 functions for radius below 0.075 are very near to zero, and thus the minimum radius allowed was 241 set to 0.075 µm. The behavior of the kernel functions versus wavelength can be consulted, for 242 example, in Chapter 11 of Bohren and Huffman, 1983. 243 In the analysis that follows, we do not present results on the refractive index sensitivity 244 analysis. The reason for this is that we found that the retrieval of refractive index is very  were also performed here. Interestingly, the slopes of the linear fits of the extinction coefficients 306 present opposite signs to those determined for the retrieval of r eff , with positive values for α(355 307 nm) (a = 3.09 ± 0.12 for type I, a = 4.83 ± 0.22 for type II a = 3.05 ± 0.13 for type III) and 308 negative values for α(532 nm) (a = -2.78 ± 0.17 for type I, a = -4.09 ± 0.23 for type II and a = -309 2.61 ± 0.12 for type III). Therefore, we see in the retrieved results, for example, that to 310 compensate for a radius enhancement due to biased input data the retrieval tends to decrease 311 number density.  possesses different weights of fine and coarse mode.

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As the regularization scheme used here computes the size distribution using the range of 369 permitted radii of 0.075 -5 µm, the fine mode part of the distribution (but not the coarse mode) 370 is completely covered by this inversion window, and thus we study fine mode volume radius and 371 fine mode volume concentration. Table 1 also shows the sensitivities of these two parameters to

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The sensitivity tests applied to the different sets of data have shown linear dependencies.

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The data presented in Table 1    additive. We also performed the regularization retrieval with the new set of data affected by two 452 or more simultaneous biases, called "simulated deviations". Later we computed the differences 453 in the microphysical properties based on the slopes given in Table 1 and those actually retrieved 454 running the code with the new biased optical data and characterized the differences. Using this 455 procedure, we generated for each absolute value of bias a statistical dataset that includes many 456 different configurations of the different optical channels. Those datasets are analyzed using Box-

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Whisker diagrams as shown in Figure 4 for the effective radius.  Table 1. Furthermore, very similar additive properties were found for 474 aerosol type III (graph not shown for brevity). Therefore, for the bimodal size distributions used 475 here that cover most of those size distributions obtained by AERONET, we conclude that the 476 results of Table 1 can be reliably used to calculate the deviations in retrieved quantities due to 477 multiple simultaneously biased input data. 478 We take this result to be an indication that, as mentioned earlier, the solutions found by

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Using the same procedure as for 15% random error, for V, r fine and V fine . In the same way, Table 3 reports the means of the deviation of every 553 microphysical property for varying amounts of random uncertainty in the input data. As 554 mentioned above, the departures of these deviations from zero indicate that random uncertainties 555 in the input optical data can induce varying amounts of systematic bias in the retrieved 556 properties. This effect is found more with the type II aerosols that possess a higher fraction of 557 larger particles. Such a population is more likely to have different slopes in Table 1

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[Insert Table 2 here] 563 [Insert Table 3 here] 564 Müller et al., [1999a,b] and Veselovskii et al., [2002,2004] studied 10% random 565 uncertainties in the optical data in the 3β + 2α lidar configurations by introducing random errors 566 in the optical data and running the regularization code repeatedly. These studies reported that the 567 retrieved uncertainties were on the order of 25% for r eff , V and S, 30% for r mean and 70% for N.

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These values are quite similar to those reported in Table 2 for our computations of 10% random Veselovskii et al., [2002,2004]. The method shown here for assessing the sensitivity of 571 retrievals to random errors is generally consistent with these earlier results but permits the 572 influence of varying amounts of random error to be studied. It also permits the influence of 573 random errors in different input optical channels to be quantified. We will now apply this 574 capability to the problem of instrument specification. presented clearly indicate, however, that for most quantities it is uncertainties in the extinction 588 coefficients that need to be constrained more carefully than those in the backscattering data.  Table 1 can be used to assess which channels would benefit most from decreased uncertainty      Table 1. The 'x' axis represents the difference between microphysical parameters with no errors in the input optical data and those affected by random errors in the optical data. Random errors were simulated by a normal distribution centred at zero and with standard deviation of 15%. The random number generator is initialized at different values for each of the 5 optical data used in the 3β + 2α lidar configuration. The mean value of the deviation between the microphysical parameter affected by random error and that unaffected by random error is included in the legend.  1: Percentage deviations in the aerosol microphysical properties as a function of systematic errors in the optical data ε. Particularly, the slopes 'a' of the linear fits Y = aX are presented, where 'X' is the systematic bias in the optical data and Y is the corresponding deviation in the microphysical properties. All these fits presented linear determination coefficient R 2 > 0.90. For the cases when there is a difference in slope between positive and negative biases in the input data, the slopes relating to the positive biases are indicated by (p) while those associated with negative biases are indicated by (n).