Backscatter Error Bounds for the Elastic Lidar Two-Component Inversion Algorithm

Abstract

Total backscatter-coefficient inversion error bounds for the two-component lidar inversion algorithm (so-called Fernald's or Klett-Fernald-Sasano's method) are derived in analytical form in response to the following three error sources: 1) the measurement noise; 2) the user uncertainty in the backscatter-coefficient calibration; and 3) the aerosol extinctionto-backscatter ratio. The following two different types of error bounds are presented: 1) approximate error bounds using first-order error propagation and 2) exact error bounds using a total-increment method. Both error bounds are formulated in explicit analytical form, which is of advantage for practical physical sensitivity analysis and computational implementation. A Monte Carlo approach is used to validate the error bounds at 355-, 532-, and 1064-nm wavelengths.

Authors’ response to the Reviewer#4’s comments

“…For instance, a k SM = 4.5 ± 1.4 m 2 g −1 is derived for fine smoke particles at 06:00 UTC (see Table 3). This value is in good agreement with that reported for Canadian forest fire smoke aerosols (Ichoku and Kaufman, 2005;Reid et al, 2005). However, a rather lower MEE value is obtained for the coarse-mode NS particles (k NS = 2.4 ± 0.5 m 2 g −1 ) at the same time.…”

“…Particular MEE values derived for smoke particles, k SM = 4.5 ± 1.1 and 1.9 ± 0.4 m 2 g −1 , are obtained at 06:00 and 14:00 UTC. These results would indicate that smoke plumes detected in the first scenario are predominantly composed of relatively pure fine biomassburning particles, with similar MEE values to those reported for Canadian boreal forest fire aged smoke particles (Ichoku and Kaufman, 2005;Reid et al, 2005). However, those observed in the second one would represent a mixed state of smoke particles with an enhanced coarse mode, thus decreasing their MEE.…”

“…Table 3 shows the AERONET parameters involved in the extinction-to-mass conversion (VC c,f , τ c,f ) at selected times for each aerosol case together with those typical particle densities Pd for each aerosol component. In particular, Pd values assumed for each type of aerosols are 2.60 g cm −3 for dust (Ansmann et al, 2012), 1.30 g cm −3 for smoke (Reid et al, 2005) and 0.92 g cm −3 for pollen (Platanus) particles (Jackson and Lyford, 1999;Zhang et al, 2014). For the other components, the particle density is obtained from the OPAC database .…”

Current Research in Lidar Technology Used for the Remote Sensing of Atmospheric Aerosols

“…This is detailed next, in the context of today’s standard form of the two-component elastic-lidar inversion algorithm. The backward form of the KF algorithm for the retrieval of the aerosol backscatter coefficient at ${\lambda }_0$ takes the form [39]:${\beta }^{aer}\left(R,{\beta }_m,{\overrightarrow{S}}^{aer},\overrightarrow{U}\right)=\frac{U\left(R\right)F\left(R,{\overrightarrow{S}}^{aer}\right)}{\frac{U_m}{{\beta }_m}+2true\underset{R}{\overset{R_m}{\int }}{S}^{aer}\left(v\right)U\left(v\right)F\left(v,{\overrightarrow{S}}^{aer}\right)dv}-{\beta }^{mol}\left(R\right),R\le R_m$ where $F\left(R,{\overrightarrow{S}}^{aer}\right)=\exp \left\{2\underset{R}{\overset{R_m}{{\displaystyle \int }}}\left[S^{aer}\left(u\right)-S^{mol}\right]{\beta }^{mol}\left(u\right)du\right\},U\left(R\right)=U^{2}P\left(R\right)$, is the range-corrected lidar return power, $S^{aer}\left(R\right)={\alpha }^{aer}\left(R\right)/{\beta }^{aer}\left(R\right)$ is the aerosol extinction-to-backscatter ratio (so-called aerosol lidar ratio), $S^{mol}=8\pi /3$ is the molecular lidar ratio, $R_m$ is the far-end calibration range, ${\beta }_m$ and $$…”

“…The works [45,46] show the importance of reducing the noise in U m by spatially averaging around the calibration range to reduce inversion error bars. A unified study on the impact of both random and systematic error sources and their spectral dependency can be found in [39]. The solution given by Equation (4) can also be found in a less formal, yet numerically equivalent, way taking into account that Equation (3) implies that ${\beta }^{aer}\left(R\right)={\beta }_m\frac{U\left(R\right)}{U_m}exp\left\{-2{{\displaystyle \int }}_R^{R_m}\left[{\alpha }^{aer}\left(x\right)+{\alpha }^{mol}\left(x\right)\right]dx\right\}-{\beta }^{mol}\left(R\right).$ Then ${\beta }^{aer}\left(R\right)$ can be found iteratively [47], starting with an initial guess of ${\alpha }^{aer}\left(R\right)$; this results in a first guess for ${\beta }^{aer}\left(R\right)$, which, using the assumed $S^{aer}\left(R\right)$ yields a refined ${\alpha }^{aer}\left(R\right)$ serving to calculate another iteration of ${\beta }^{aer}\left(R\right)$.…”