Journal of Applied Spectroscopy, Vol. 74, No. 4, 2007

REGAINING PROFILES OF EXTINCTION COEFFICIENTS OF OPTICALLY DENSE SCATTERING MEDIA IN LASER LOCATION MEASUREMENTS M. M. Kugeiko,* S. A. Lysenko, and S. V. Kvachenok

UDC 551.508

We suggest a procedure for regaining spectral values of the extinction coefficients ε(r) of optically dense scattering media in laser location measurements. Allowance is made for the contribution of multiple scattering to a recorded signal and its correction for the degree of change of the qualitative composition of the scattering medium. The procedure can increase the accuracy of regaining ε(r) and eliminate the "edge effect" at the end of the probed path. The latter is achieved by determining a calibration constant from the transparency value of the whole probing range, which is calculated from back-scattered signals corrected based on the constancy of the lidar ratio. We present an algorithm for calculating the correction coefficient. The efficiency of regaining profiles of ε(r) is estimated using the atmospheric situation of a pure atmosphere and an extended smoke cloud arising from forest fires as an example. Key words: dense medium, back-scattering, extinction coefficient, signal correction. Introduction. Theories and methods of remote laser atmospheric probing that have been developed until now are based, as a rule, on a single-pass scattering (SS) approximation. Effects of multiple scattering (MS) become important in the probing of optically dense media [1–3]. In the general case of determining spectral values of optical-locational characteristics (extinction coefficients and back-scattering) of optically dense media, the dynamic transfer equation must be solved using Monte-Carlo methods [4]. This problem is complicated because the size of the MS contribution depends on the optical characteristics and particle-size distribution of the cloud particles that a priori are unknown and, generally speaking, should be found from the signals themselves. MS noise can be excluded from the back-scattered signal and processed further by methods developed using the SS approximation if the optical media are not dense [5–7]. However, the interpretation of lidar measurements using the SS approximation is itself complicated in this instance because it requires the use of a priori information or assumptions about the investigated medium [8]. The equation of laser probing (ELP) considering MS is usually solved by representing it in a form that can isolate explicitly from the received signal the part that is due to SS. Such a recording enables the problem of regaining the profile of the extinction coefficient from the SS signal in the presence of MS noise to be examined. Considering this, the ELP without molecular extinction can be written as [2, 3]: r

⎫ ⎧ P (r) r = A [1 + δ (η)] gπ (r) ε (r) exp ⎨− 2 ∫ ε (x) dx⎬ , ⎭ ⎩ r 2

(1)

0

where P(r) is the back-scattered signal from point r; r0, the minimal distance from lidar at which full overlap of the field of vision of the receiver and the sent pulse is attained (corresponds to the end of the lidar shadow zone); A, the system apparatus constant; gπ(r) and ε(r), back-scattering indicatrices and extinction coefficient of the medium at point r; function δ(η) determines the ratio between MS and SS components of the lidar signal and depends in general on *

To whom correspondence should be addressed.

Belarussian State University, 4 Nezavisimosti Ave., Minsk, 220050, Belarus, e-mail: [email protected]. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 74, No. 4, pp. 522–527, July–August, 2007. Original article submitted January 19, 2007. 578

0021-9037/07/7404-0578 ©2007 Springer Science+Business Media, Inc.

r

the sighting geometry and parameter η = τθ ⁄ 2, where θ is the angular aperture of the receiver; τ =

∫ ε(x)dx

is the

0

optical thickness of the probing portion [0, r] [3]. As shown before [1, 2], the profile of ε(r) can be regained by the asymptotic signal method up to τ(r0, r) = 2–3 with satisfactory accuracy by probing optically dense media with a rather large energy potential, which enables signals to be obtained from portions with optical thickness τ(r0, r) = 3–4. For large optical thicknesses, a procedure consisting of the determination by nephelometry of the reference calibration value εk(r0) and the finding of the minimal value of the back-scattering indicatrix in an optically dense medium gπ(r0, r) at which infinite or negative values of ε(rmax) do not arise at the end of the path was proposed [9]. It satisfies the condition 2 −1

P (r0) r0gπ (r0) εk (r0)

rmax

>2

∫

2 −1

P (r) r gπ (r) dr ,

r0

which follows from the general solution of the ELP [1–3]. This is equivalent to determining the value of the asymptotic signal enabling the profile of ε(r) to be obtained from infinite or negative values within the probing path. However, large errors in regaining values of ε(r), i.e, an "edge effect," typical of the literature methods [1–3], may arise in this procedure also at the ends of the probing path. Herein we propose a procedure that enables the accuracy of regaining ε(r) in optically dense media to be increased and the edge effect to be eliminated over the whole probing path [r0, rmax], i.e., at distances rmax, at which back-scattered signals are recorded. Method. A previous analysis [8] of the algorithms for calculating the profiles of ε(r) with complicated effects from all factors that hinder solution of the ELP indicates that it is preferable to know not the average indicatrix along the path but its relative path in optically dense multilayered media in order to increase the accuracy of regaining the profiles of ε(r) because the back-scattered signal can be corrected for the constancy of the back-scattered indicatrix. The physical meaning of the correction consists of alignment of the back-scattered signal at the boundary of different layers to a_ level such that it corresponds to the state with identical gπ in the layers [10]. Then, parameter q = _ gπ(r0, rb) ⁄ gπ(rb, rmax) must be established at the layer boundary, for example, at the point r = rb. The boundary of the media where the back-scattered signal is corrected for the drop of gπ(r) (change of qualitative composition of scattering medium) can be established by using the previously suggested [11] algorithm that consists of finding the maximum of the function 2

rmax

∫ P(x)x2dx where Im(r) =

r0

1−ξ

r

d ⎧ P (x) x ⎫⎪ Im (r) , D (r) = ⎬⎪ ⎨ 2 dx ⎩ 2Im (x) ⎭⎪r0 r

2

r

− ∫ P(x)x2dx, rmax is the distance of the probing medium in which the given signal-to-noise r0

ratio 0 < ξ < 1 is achieved. For the condition where the transparency _ _T2(rb, rmax) << 1 for portion [rb, rmax] and the back-scattering indicatrices [r0, rb] and [rb, rmax], equal to gπ1 and gπ2, respectively, are slowly changing or rapidly oscillating functions with small correlation radii, then, using the system of equations for I1 and I2: rb

2 2_ I1 = ∫ S (r) dr = 0.5AT0 gπ1 ⎡1 − T (r0, rb)⎤ , ⎣ ⎦

r0

rmax

I2 =

∫

rb

2 2 2_ S (r) dr = 0.5AT0 gπ2T (r0, rb) ⎡1 − T (r0, rmax)⎤ , ⎣ ⎦

579

where S(r) =

P(r)r2 , it is easy to obtain an expression for q on passing the boundary of layers rb: 1 + δ(η) _ 2 gπ1 T (r0, rb) q=_ = 2 gπ2 1 − T (r0, rb)

I1

.

(2)

I2

The integrated transparency of portion [r0, rb], T2[r0, rb], corresponding to a layer with stable and slightly changing gπ(r) and ε(r) values, is defined by the expression

⎡ rb T (r0, rb) = ⎢⎢ ∫ S (r) dr ⎢r ⎣ 1 2

⁄

2

r1

⎤ ∫ S (r) dr⎥⎥⎥ , where r1 = 0.5 (r0 + rb) . r0 ⎦

Making the correction to S(r) in portion [rb, rmax] for the constancy of the back-scattering indicatrix on the probing path and solving the system of equations for functionals I3, I4, and I5: r1

2 2_ I3 = ∫ Scor (r) dr = 0.5AT0 gπ1 ⎡1 − T (r0, r1)⎤ , ⎦ ⎣ r0

rb

2 2_ I4 = ∫ Scor (r) dr = 0.5AT0 gπ1 (r0, r1) ⎡1 − T (r0, rb)⎤ , ⎦ ⎣

(3)

r1

r

2 2 2_ I5 = ∫ Scor (r) dr = 0.5AT0 gπ1T (r0, rb) ⎡1 − T (r1, r)⎤ , ⎣ ⎦ rb

where Scor(r) = S(r), for r < rb and Scor(r) = qS(r), for r ≥ rb, r1 = 0.5(r0 + rb), it is easy to obtain the transparency of portion [rb, r] I5 ⎡ ⎤ 2 (I − I4)⎥ . T (rb, r) = ⎢1 − 2 3 (I4) ⎣ ⎦

(4)

Because the stability of the solution of the ELP (1) is greater the greater the a priori information about remote portions of the path [3, 9], it is desirable that for I5 in Eq. (4) r → rmax, i.e., the lidar calibration constant must be determined from the transparency of the whole probing path. The solution of the ELP relative to the profile of the extinction coefficient is written as [3, 11]:

Scor (r)

ε (r) =

rmax

2 2

∫

1 − T (r0, rmax) r 0

r

,

(5)

Scor (x) dx − 2 ∫ Scor (x) dx r0

where T2(r0, rmax) = T2(r0, rb)T2(rb, rmax). Because the difference between the integrals in the denominator of Eq. (5) is always positive [because rmax ≥ r and T2(r0, rmax) < 1], the solution of the ELP Eq. (5) is stable and does not lead to meaningless values of ε(r). It is also noteworthy that this procedure assumes that the conditions relative to Eq. (2) are fulfilled, which in the general case may not be. So, the transparency T2(r0, rb) can be determined only if additional measurements are used. One of these algorithms for determining the transparency T2(r0, rb) of a pure and weakly cloudy portion of the path with additional measurements made by a nephelometer has been published [11]. The essence of the algorithm 580

[11] consists of selecting the value of the portion transparency so that the value ε(r) at the origin of the lidar shadow zone (point r = 0) coincides with that measured using nephelometry upon interpolation of the profile of the extinction coefficient obtained for this portion with the selected transparency. From the system of functionals (3) for r = rmax, it is easy to show that the transparency of portion T2(r0, rb) established by this method can be used to determine the transparency of the whole probing path T2(r0, rmax): I5 ⎤ I5 ⎡ 2 2 . T (r0, rmax) = T (r0, rb) ⎢1 + ⎥− + I I I 3 4⎦ 3 + I4 ⎣

(6)

The function δ(η) must be known in addition to the calibration constants in order to obtain the profile of ε(r) in optically dense media. There exist today very few effective methods for establishing this function. An algorithm for processing lidar signals that enables the optical thickness τ and the profile of ε(r) of the cloudy layer at 1.5 ≤ τ ≤ 6 to be determined for reception of a set of signals corresponding to various angular reception apertures has been described [12, 13]. However, researchers in practice usually equip receivers with a single field of vision. In this instance known equations that approximate the results of accurate calculations or experimental measurements of aerosol formations of a similar type performed earlier by an iterative procedure must be used to estimate the contribution of MS. Let us examine one of the suggested variations [8]: ⎧

⎫

Iδ (τ)J = a ⎨⎩exp ⎡bτ (rb, r)⎤ − 1⎬⎭ , ⎦ ⎣

(7)

where a and b are constants (a > 0, b > 0) and Iδ(τ)J is an approximation of the numerical function δ(η) that depends only on optical thickness τ(rb, r). Application of similar approximations allows any level of MS for handling the lidar signal to be selected by choosing the corresponding constants. The only requirement imposed on these approximations for obtaining a stable solution of the ELP is a regular increase of Iδ(τ)J with increasing optical thickness. It was assumed for the previous modeling [8] that the MS level is the same as in cirrus clouds. The estimate of MS made before [14] was used. According to these data, the factor [1 + Iδ(τ)J] ≈ 3 for τ = 1.37 for lidar with a receiver field of vision of 1.6 mrad operating at 532 nm. This value was taken as the reference and it was also assumed that the MS increases substantially with further increase of the optical thickness. The highest value of this factor was proposed as 20 for the maximum probing distance. This level of MS is attained for a = 0.6 and b = 1. The function Iδ(τ)J depends on the optical thickness and, therefore, on the profile of the extinction coefficient. However, the latter cannot be determined without knowing Iδ(τ)J. In this instance, an iterative procedure must be developed. It is assumed that Iδ(τ)J = 0 before starting the iterations. This enables the initial profile of ε(r) and, therefore, τ(r) to be obtained by the proposed procedure that includes Eqs. (2)–(6). The first estimate of the function (1) (1) Iδ(τ)J = 0 can be obtained by this. The estimate is used in the next iteration to calculate ε (r). The condition for exiting the cycle for modeling ⎪ rmax ⎪ (i+1) ⎪ ⎪ ε (r) dr ∫ ⎪ ⎪ ⎪ r0 ⎪ −4 ⎪1 − r ⎪ ≤ 10 max ⎪ ⎪ (i) ⎪ ε (r) dr ⎪ ∫ ⎪ ⎪ r0 ⎪ ⎪ is reached after 6–8 iterations. Estimation of the Effectiveness. The effectiveness of the suggested procedure is estimated by numerical modeling. The atmospheric situation that was previously described [8] is examined. The probing path includes two adjoining portions with qualitatively different optical properties, i.e., the pure atmosphere and an elongated smoke cloud arising from forest fires with boundary rb ≈ 1 km. The indicatrices of back-scattering of smoke and cloud are assumed to be constant and equal to 0.05 and 0.03 sr–1. Figure 1 shows the back-scattered signal calculated for the given profiles ε(r) and gπ(r) (solid line). Like before [8], path portion [r0, rb] is proposed to be homogeneous. MS does not 581

Fig. 1. Numerical modeling of a back-scattered signal from a smoke cloud: correction of the signal for the degree of change of the qualitative medium composition for seven iterations (ordered from below to above) (a), regaining the profile of the extinction coefficient (b); model profile (1) and profiles of ε(r) regained in the first and last iterations (2 and 3).

Fig. 2. Numerical modeling of a noisy back-scattered signal from a smoke cloud: correction of the signal for the degree of change of the qualitative medium for seven iterations (ordered from below to above) (a), regaining the profile of the extinction coefficient (b); model profile of ε(r) and that regained after seven iterations (1 and 2). occur on it, i.e., δ(η) = 0, because of the small extinction coefficient. The last supposition is confirmed by previous calculations [15] that indicate that the effect of smoke on the contribution of MS appears only at rather large reception angles and probing optical thicknesses. Figures 1 and 2 show the modeling results. Back-scattering indicatrices corrected for change and the contribution of MS to lidar signals for seven iterations (dashed lines) are plotted in Fig. 1a. It can be seen that the relative change of the lidar signal obtained after seven iterations (upper dashed line) at the smoke–cloud boundary corresponds accurately with the relative change of the indicatrix at this boundary. Figure 1b illustrates regaining the extinction co582

efficient profile by the described algorithm. The average uncertainty along the path of the regaining for this ideal case, where there is no high-frequency noise in the back-scattering signal, is 9%. Figure 1b suggests that even at the ends of the probing path rmax = 6 km, there is no edge effect whereas for the analogous situation in the literature [8], the given and regained profiles of ε(r) begin sharply to diverge even with r = 4 km. Figure 2 shows results of regaining the same profile of the extinction coefficient as in Fig. 1 but with highfrequency noise in the lidar signal (Fig. 2a), which limits the maximum probing distance to ≈5 km. This in turn decreases the accuracy of Eq. (2). Random deviations in the range of 5% also appear in the aforementioned values gπ(r). In this instance the average uncertainty of regaining ε(r) for portion [0.5, 5 km] is 20%. It is noteworthy that the proposed solution of the ELP requires the following conditions to be satisfied. The back-scattering indicatrix in the layers is a slowly changing or rapidly oscillating function with a small correlation radius. It is necessary to know the ratio of the MS and SS components of the lidar signal as a function of distance or optical thickenss (tables [3] defining this ratio can be used). The optical thickness of the cloud is such that the maximum probing distance determined by the signal-to-noise ratio is located within it. The last condition can be destroyed for the case where the cloud (smoke) is spatially limited and the probing pulse penetrates through the cloud into the pure atmosphere with a small back-scattering coefficient. In this instance the signal weakens to the noise level not by the exponential weakening in Eq. (1) but by a decrease of the back-scattering coefficient upon passage of the pulse into the pure atmosphere. Then the optical thickness of the cloud can be insufficient for Eq. (2) to be valid and this method will give elevated values of ε(r). Conclusion. The suggested procedure for regaining spectral values of ε(r) for optically dense media that considers MS in the recorded signal and correction of the signal for the level of change of the qualitative composition of the scattering medium allows the accuracy of the determination of ε(r) to be increased and the edge effect at the end of the probing path to be eliminated.

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