Analytical Relationships Between Bulk Microphysical Parameters of Nucleation, Aitken, Accumulation and Coarse Mode Particles and Extinction and Backscatter Coefficients Measured with Lidar

(cid:2) We derive analytical relationships between bulk microphysical parameters of nucleation, Aitken, accumulation and coarse mode particles and extinction and backscatter coefficients measured at wavelengths 355, 532 and 1064 nm. The bulk parameters are represented by number concentration, mean (effective) radius, variance and complex refractive index. Analytical relationships hold true for arbitrarily shaped particles and complex refractive indices (cid:80) (cid:2)


INTRODUCTION
Retrieval of particle microphysical parameters from lidar measurements of extinction coefficients at 355 and 532 nm and backscatter coefficients at 355, 532 and 1064 nm is a complex task which can be solved in terms of parametric and direct, inversion (as, for example, least-squares approaches) and inversion with regularization methods. Direct methods of solving this task became popular recently. The direct approach uses a sample function, as for example a logarithmicnormally distributed function that approximates atmospheric particle size distributions. The most common case permits the use of up to 5 aerosol components [1]. In this contribution we further simplify this concept. We describe an arbitrary external mixture as superposition of 4 components, i.e. particles in the nucleation, Aitken, accumulation and coarse mode, respectively. We present standard algebraic equations that describe the interdependencies between backscatter and extinction coefficients and particle bulk parameters, i.e. number concentration, mean (effective) radius, variance and complex refractive index for each mode separately. In section 2 we present the methodology. In section 3 we carry out a numerical test with synthetic optical data. Section 4 summarizes our results.

METHODOLOGY
We aim to express the optical data (O) of each mode mo ( ) on the basis of its own particle microphysical parameters P mo ( eff,mode ), V mo , mo and mo where each number mode is approximated by a lognormal distribution, i.e.
Eq. (8) can be directly used for the analytical estimation of the coarse-mode extinction coefficient from particle microphysical parameters and conversely. The estimation mo Nu, Ai, Ac, Co, (9) wherer term true (O) is the true value. Eq. (8) neither depends on particle shape nor complex refractive index.

Nucleation mode and Aitken mode
We now consider the smallest particles lim 0.1 Pm. In this case the dimensionless size parameter is lim 2S lim /O | 1 at O=532 nm. We need to use the 1 st and 2 nd member of the series expansion that asymptotically approximates the kernels (O, , ) in Mie theory. The use of the 2 nd member of the series expansion provides us with a sufficiently accurate approximation, if the magnitude lim |1 [3]. This 2 nd order approximation allows us to find the kernels (O, , ) and one can show that   unknown parameters mo , P mo , V mo and mo . This system can be solved numerically, see Ref. [5], for the case of single modes and small particles in the Aitken mode. We stress that Eq. (14) and (15)  in Eq. (16) for these P mo and V mo still holds true.
In this case the estimation accuracy of the analytical expressions is better than H mo |60%.
In contrast to large particles, extinction and backscatter coefficients of small particles [see analytical expressions (14), (15) and (19), (20)] significantly depend on their complex refractive index. Still, these analytical expressions can be used for arbitrarily shaped particles because their size is much less than the measurement wavelengths used in lidar applications [3].

Intermediate size of particles
Our analysis of Eq. (8) and (14) shows that the extinction coefficient of large particles converges to their surface-area concentration; note that we can describe this parameter in terms of volume concentration normalized by effective radius, see Ref. [2]. The extinction coefficient of particles in the nucleation mode (3a) is proportional to volume concentration normalized by wavelength. The statistical analysis of our reference look-up table shows that the extinction coefficient at 532 nm and surface-area concentration are still linearly correlated but with a lower correlation coefficient D Ac (532) 0.57 Ac r60% = 0.57u4S Ac P Ac 2 exp(2ln 2 V Ac ) r60% (21) That means we can use the linear regression equation, but the estimation uncertainty grows to H Ac r60% in the worst case. Again, we note that Eq. (21) does not depend on the complex refractive index and particle shape; see Eq. (8).
The estimation accuracies of the analytical expressions that can be used for the extinction coefficient, see (8)  We consider the conditions (22) and determine preliminary (1 st order) estimates of the parameters 1 , P 1 , V 1 and 2 , P 2 , V 2 . Details on how this step is done exactly will be provided in a publication. We find the parameters for the 3 rd distribution which can be described by the following equations 3 1 2 (23a)

NUMERICAL TEST
We compare the extinction and backscatter coefficients (O) estimated with the analytical expressions (8), (14), (15), (19)-(21), (24) and computed with a Mie-scattering code [7]. Besides that comparison, we also compare the kernels (O, , ) that result from the use of Eq. (10) and (11) and which are described by the strict theory of Mie scattering [7]. We use for the comparisons lognormal size distributions with parameters P[0.001;2.5] Pm, V[1.35;2.55] and complex refractive indices in the intervals (2) to estimate the synthetic optical data. We split the parameter domain (P,V) into 5 regions (see Fig. 1b,c). We use Eq.  Fig. 1a and 1b, respectively. The estimation accuracy (9) for the extinction coefficient, i.e.
H D(532) at O 532 nm and 1.7-0.05 is shown in Fig. 1c. Fig. 1a shows that the kernels computed with the analytical expressions (dotted lines) and with the Mie-scattering code (solid lines) coincide for all particle sizes 0.1 Pm. This result shows why we obtain such a high accuracy of the estimated extinction (Fig. 1c)  (14), (15) can be used. We also find a high estimation accuracy of |H D(532) ||20% based on the correlation relationship used in region I. the correlation relationship (21) works, but the accuracy degrades up to 60%. Region III shows an estimation accuracy of H D(532) | -55% (overestimation).
This result indicates the limit where the use of the 2 nd order approximation of kernel functions still can be used. The region V where we use the combination of the correlation relationships and the 2 nd order approximation provides the accuracy to approximately |H D(532) ||50%.

CONCLUSION
We derive analytical expressions that allow us to estimate extinction and backscatter coefficients at wavelengths 355, 532 and 1064 nm. We derive these expressions and thus estimates of these coefficients based on the following approaches: a. 2 nd order approximation of kernel functions can be described by Mie-scattering theory for the case of small particles with 0.1 Pm (nucleation and Aitken modes), b. correlation relationships between extinction coefficient and surface-area concentration are used if particles are large and intermediate sizes, i.e. >0.1 Pm (accumulation and coarse modes), c. combination of approaches a and b in the case of small particles where |0.1 Pm (Aitken and/or accumulation modes). The expressions directly link the backscatter and/or extinction coefficients through the underlying particle bulk parameters such as number concentration, mean (effective) radius, variance and complex refractive index to standard algebraic equations. The expressions hold true for arbitrarily shaped particles and complex refractive indices R -I , where R !! I . Minimum uncertainties of the estimates of the parameters are 0-20% in the case of nucleation and coarse mode particles, respectively. Maximum uncertainties are 60% at worst in the case of Aitken and accumulation mode particles. We will show how these analytical expressions can be applied for estimating microphysical parameters and the contributions of nucleation, Aitken, accumulation and coarse mode particles to extinction and backscatter coefficients measured with lidar. A paper that describes the results is in preparation.