Lidar inversion with variable backscatter/extinction ratios: comment Martina Kaestner

University of Munich, Meteorological Institute, There-sienstrasse 37, D-8000 Munich 2, Federal Republic of Ger­many. Received 5 November 1985. 0003-6935/86/060833-03$02.00/0. © 1986 Optical Society of America. This Letter contains four aspects to the lidar equation: (1) Eq. (20) of Klett's1 paper is corrected; (2) application of a solution of the lidar equation to a

satellite lidar system is considered; (3) a discussion is presented on why the extinction coeffi­

cient and not the backscatter coefficient should be extracted from the lidar return signals; and

(4) a solution to the aerosol extinction coefficient is added directly derived from the lidar equation.

The lidar equation for molecules and aerosol particles reads

where P is the lidar return signal. The range dependence is often omitted for brevity of the formulas in this paper. All quantities are functions of the range r except the quantities CG = calibration constant and SR = lidar ratio with respect to air molecules [see Eq. (2)]. The quantity β is the backscat-tering coefficient, and σ is the extinction coefficient. The subscripts M (Mie) and R (Rayleigh) refer to the aerosol particles and molecules, respectively.

For the lidar equation, two solutions have been pro­posed1–3 which are applicable to atmospheres with low tur­bidity, where it is convenient to consider aerosol and molecu­lar scatterers separately. The Rayleigh signal must be considered to be a noise signal in the measuring signal. It can be calculated from an actual measured temperature-pressure profile or from an appropriate standard atmo­sphere. Both Klett1 and Fernald2,3 make use of a linear β-σ-relation to solve the lidar equation:

with S R = 8π/3, where S is the so-called lidar ratio σ/β. Fernald's solution can be obtained from Klett's by assuming SM to be a constant and not a function of the range r as Klett does. In Klett's paper1 a slip occurred in Eq. (20) which should be rewritten as

because the sign changes when changing the integral bounds. [Here Klett's terminology is used with S = ln(r2 • P), B R corresponds to 1 /S R , and Bp to 1 /S M , respectively.]

In practice the lidar return can only be measured for ranges greater than r0, where Γ0 is the range at which the transmitter and receiver beams overlap. The contribution

to the lidar equation (1) has to be estimated. If the lidar

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system is spaceborne, no difficulties will occur with the inte­gral amount I 0 , as there are with ground-based lidar systems. I0 is zero, and the lower integral boundary value may be replaced by r0 in Eq. (1) without introducing errors. More­over, for a satellite lidar r0 is the range, where the signal appears the first time beyond the noise signal. In principle, two solutions to the lidar equation are possible, the unstable one integrating from the lidar away to the far end and the stable one using Klett's method integrating from the far end to the lidar. Applying these methods to satellite lidar re­turns means that the boundary value for the unstable solu­tion is nearly identical to the known Rayleigh extinction coefficient σR, but the error propagation increases with de­creasing height above the ground. So this solution is more appropriate for a clear stratosphere when volcanic aerosol is absent and not for the lower troposphere or the more turbid planetary boundary layer. On the other hand, the stable solution is more appropriate for investigating the tropo­sphere, although the σM value of the ground layer σM(rm) has to be known or iterated.

Looking at the lidar equation (1), in principle, either the backscatter or the extinction coefficient can be evaluated. Regarding the measuring technique of a lidar system the backscatter coefficient seems to be appropriate. However, the backscatter coefficient gives only one single value of the whole scattering function. In the field of the aerosol re­search one goal describes the optical effects of aerosols in the radiation budget by optical parameters, such as the scatter­ing function, the extinction coefficient, or the refractive in­dex. The extinction coefficient, in particular, the aerosol extinction coefficient σM, is the unknown and wanted param­eter. The lidar return signals should be inverted to this more comprehensive parameter. It is directly related to the aerosol optical depth ΔM and aerosol transmittance TM::

Neither Fernald3 [Eq. (3) in his paper] nor Klett1 [Eq. (22) in his paper] derived their solutions explicitly for the aerosol extinction coefficient. Instead they were derived for the backscattering coefficient.

In this paper I will propose a solution for the extinction coefficient due to aerosols. In fact, the solution for σM can be inferred from the formulas for ΒM using the linear relation βM(r) = S M

– 1 ( r ) • Σ M( r ) [see Eq. (2)]. However, it will be

shown that σM can directly be derived from the lidar equa­tion. The stable solution of the lidar equation (1) for the aerosol extinction coefficient σM(r) is

with

where the index m refers to the value of the quantity at the range rm, e.g., SMm = SM(rm). The first summand of N(r) is related to an integration constant. It can be developed for the reference range rm, at which the aerosol extinction coeffi­

cient σM(rm), the boundary value, must be known a priori. An improvement of the extraction of an aerosol extinction profile will be achieved, if the boundary value σM(rm) is set to the calculated σM(rm-1) from the previous integration step. It means that the range rm diminishes toward r0. Beneath one boundary value it is assumed that the lidar ratio SM(r) is known from other available information. If not, it may be iterated according to Klett's1 proposal. 1 want to point out that for an iteration of SM(r) there is no need for the lidar to be calibrated, because changes in SM affect the slope of the logarithmic signal, which means d(lnP)/dr. [Note: d(InCG)/dr = 0.] With a well-calibrated lidar, it is also possible to iterate the boundary value, as Ferguson and Stephens4 and Mulders5 have outlined. Changes in the boundary value will shift the whole profile to lower or higher values throughout the whole atmosphere.

As stated before, the solution Eq. (6) is equivalent to Fernald's and Klett's solution, but I take the opportunity, in this Letter, to state that Eq. (6) can also be directly derived from the lidar equation (1). In the following, this approach will be sketched. First, the logarithm of the range-corrected lidar return signal is differentiated with respect to r to obtain the reduced form

Multiplying the whole equation by SM • β and replacing β by βR + βM and σ by σR + σM yield an inhomogenous Riccati differential equation in dσM(r)/dr. Applying the linear β-σ-relation [see Eq. (2)],

the Riccati differential equation is

with

The inhomogenous differential equation of second order is solvable by a first substitution, σM = V/2 – Q/4, leading to a quadratic normal form

with

By a second substitution V = 1/U + T the differential equa­tion can be integrated for U, the integration constant ap­pears, and the resubstitution

leads to the well-known solution for the aerosol extinction coefficient, e.g., Eq. (6).

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References 1. J. D. Klett, "Lidar Inversion with Variable Backscatter/Extinc-

tion Ratios," Appl. Opt. 24, 1638 (1985). 2. F. G. Fernald, B. M. Herman, and J. R. Reagan, "Determination

of Aerosol Height Distributions by Lidar," J. Appl. Meteorol. 11, 482 (1972).

3. F. G. Fernald, "Analysis of Atmospheric Lidar Observations: Some Comments," Appl. Opt. 23, 652 (1984).

4. J. A. Ferguson and D. H. Stephens, "Algorithm for Inverting Lidar Returns," Appl. Opt. 22, 3673 (1983).

5. J. M. Mulders, "Algorithm for Inverting Lidar Returns: Com­ment," Appl. Opt. 23, 2855 (1984).

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