A Minimum Variance Method for Lidar Signal Inversion

Transcription

1 468 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 31 A Minimum Variance Metho for Liar Signal Inversion ANDREJA SU SNIK Centre for Atmospheric Research, University of Nova Gorica, Nova Gorica, Slovenia HEIDI HOLDER Duke University, Durham, North Carolina WILLIAM EICHINGER Department of Civil an Environmental Engineering, The University of Iowa, Iowa City, Iowa (Manuscript receive 22 May 2013, in final form 30 October 2013) ABSTRACT A metho for the inversion of elastic liar ata is propose that etermines the value of the extinction coefficient to be use as a bounary conition, by minimizing the variance of the extinction coefficients in a elimite region. The metho works well for single-component atmospheres, where the elimite region contains aerosols from a single istribution of concentrations. For situations where there may be two or more istributions (e.g., a smoke plume in the ambient atmosphere), the spatial regions containing each istribution must be ientifie an treate separately. Examples of inversions are given, incluing an example of a range time scan showing the amount of shot-to-shot consistency that may be obtaine from the metho. 1. Introuction Despite the amount of effort that has been spent on the evelopment of liar signal inversion methos, extinction coefficients erive from liar signals are usually not reporte in the literature or use for quantitative analysis. Inversions from ifferent lines of sight (from multiangle liars) or from time averages (from staring liars) are processe inepenently using the techniques escribe below. For each inversion, a ifferent extinction coefficient at one en of the inversion range must be assume as a bounary conition. Even small errors in the assume coefficients propagate through the rest of the ata. A monostatic, single-wavelength liar system emits short laser pulses into the atmosphere an measures the power P of the light backscattere into the telescope as a function of istance r. For a nonabsorbing, scattering atmosphere, the liar return for singly scattere light is escribe by the liar equation (Kovalev an Eichinger 2004) Corresponing author aress: Anreja Susnik, Centre for Atmospheric Research, University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia. anreja.susnik@ung.si " ð # b(r) r P(r) 5 CT o r 2 exp 22 k(x) x, (1) r o where C is the liar system constant; T o is the transmission of the atmosphere to the minimum usable range, usually taken to be the istance to complete liar overlap (r 5 r o ); b(r) is the volume backscatter coefficient; an k is the volume extinction coefficient, incluing both molecular an aerosol extinction. Over the last three ecaes, a large number of liar inversion methos have been propose an examine. The first commonly use metho employs the Bernoulli solution that was evelope by Hitschfel an Boran (Hitschfel an Boran 1954) to obtain rain rates from raar returns. Known as the forwar solution, the extinction coefficients can be foun from the measure liar ata as S(r) 1/k k(r) 5 S(r o ) 1/k 2 2 ð, (2) r S(x) 1/k x k r o k o where S(r) 5 P(r)r 2 ; k o 5 k(r o ) is the assume (or measure) value of the extinction coefficient at the DOI: /JTECH-D Ó 2014 American Meteorological Society

2 FEBRUARY 2014 S U SNIK ET AL. 469 FIG. 1. The propagation of the error in the extinction coefficient through the solution. The mile (black) line is the actual solution; the top an bottom lines (re an blue, respectively) represent errors in the initial conitions of 65%. The top (re) line solution goes through a singularity, eveloping nonphysical values of the extinction coefficient after the sign change at r 1.5 km. istance r o ;ankis a fitting constant [see Eq. (4)]. While wiely use (Barret an Ben-Dov 1967; Viezee et al. 1969; Davis 1969; Fernal et al. 1972; Collis an Russell 1976; Kohl 1978), this solution is unstable with respect to small errors in the assume bounary conition. An improve, stable version was first evelope by Kaul (1976) an later by Klett (1985b,a). Known as the backwar solution, this metho iffers primarily from the forwar solution in that the bounary value is taken at the far en of the liar ata, rather than at the beginning, an is written as S(r) k(r) 5 S(r f ) ð rf, (3) k(r f ) 1 2 S(x) x where from now on we assume k 1. Here r f enotes the farthest istance for which meaningful liar ata are available, an k(r f ) is the assume (or measure) value of the extinction coefficient at the istance r f. With both solutions, the assumption has been mae that the backscatter an extinction coefficients are relate by r b(r) 5 C(r)k k, (4) where it is customary to assume k 5 1 an the backscatterto-extinction coefficient, C(r),is assume to be constant at all ranges. This is equivalent to the assumption that the particulate type, composition, an size istribution is constant. Figure 1 shows the results from a forwar inversion an from initial estimates that are in error by just 5%. The Kaul Klett (Kaul 1976; Klett 1985b,a) solution is a significant improvement on the forwar solution, but it converges to the correct solution only FIG. 2. A histogram of the attenuation coefficients inverte from a single line of sight. This Gaussian-like istribution of extinction coefficients is mae from the ata shown in Fig. 5, which were obtaine from an inversion of the ata shown in Fig. 4. when the atmospheric extinction is sufficiently large to cause the integral in the enominator of Eq. (3) to be much larger than the other term. Neither metho is useful for a relatively clear atmosphere unless the extinction coefficients at the bounary are chosen with high accuracy. Since the quality of the inverte ata is epenent upon the choice of this extinction coefficient, it is critical that it be chosen carefully. While vertically staring liars often use Rayleigh extinction values at high altitue (Kovalev an Eichinger 2004; Fernal et al. 1972), some have suggeste using a value estimate from the slope metho (Kovalev 2006; Eberhar et al. 2007; Kunz an e Leeuw 1993), an at least one author has suggeste using nephelometer ata (Kovalev 2003); the evaluation of the efficacy of these methos is ifficult. This paper presents a methoology by which the extinction coefficients at the bounary may be selecte. 2. Description of the metho Our propose metho for inverting the ata minimizes the variance of the inverte liar signal in some esignate region of liar ranges [r j, r k ]. We are assuming that the aerosols in this volume of space originate from a single istribution that is, the size istribution an the istribution of the optical properties of the aerosols are not changing. While this has not been shown for aerosol concentrations, it is well istribute (temperature, water vapor concentration, win spee, etc.) (Stull 1988). This is a consequence of turbulent mixing in the atmosphere (Monin an Yaglom 2007a,b), an thus it woul be reasonable to expect that aerosol concentrations are normally istribute as well, at least in those parts of the atmosphere with active turbulence. The results of a successful inversion shown in Fig. 2 show a Gaussian-like

3 470 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 31 istribution for the extinction coefficients in the esignate region. The premise behin the inversion metho is that if the istribution of aerosol concentrations is isperse aroun some mean, then the minimum variance in the erive extinction coefficients will occur when the optimal bounary conition k o is selecte. Any error in the value of the bounary extinction coefficient use to invert the liar ata will introuce a systematic high or low bias in the calculate coefficients (see Fig. 1). Regarless of whether the bias is high or low, the systematic bias will increase the variance of the erive extinction coefficients. The minimum variance will occur when the correct extinction coefficient is use as the bounary conition if there are no other systematic errors (e.g., an erroneous backgroun subtraction value) an the contributions from noise an other ranom errors are evenly istribute above an below the average liar signal. The variance of the calculate coefficients in the esignate region is obtaine as where s 2 (k o ) 5 1 n å [k(r i ) 2 k] 2 i n å S(r 6 i ) ð i 4S(r o ) ri 2 k7, (5) S(r) r k o r o k 5 1 n å i k(r i ) (6) an the n 5 k 2 j subscripts inicate the use of iscrete values of the istance r in the interval [r j, r k ]. The forwar solution is use for several reasons. First, it is far more sensitive to errors in the initial estimate, with the result that the metho converges faster. Secon, the signal-to-noise ratio near the liar is much larger than at the extreme far en of the ata collection area. Last, the small size of the liar signal at long istances makes the value of S(r f ) sensitive to even small errors in the backgroun subtraction. Because of the importance of the terms, S(r o )/k o or S(r f )/k(r f ), in the enominator of the inversion equation, it is imperative that the ratio be as exact as possible. The fractional uncertainty of S(r) can be more than two orers of magnitue smaller at r o than at r f. Since the fractional uncertainty of the erive extinction coefficients is irectly proportional to the fractional uncertainty of the measure liar signal at the reference point, S(r o ) or S(r f ) (see, e.g., Bissonnette 1986), the use of a near-fiel reference reuces the uncertainty of the result by a large amount. The metho requires the esignation of the region [r j, r k ] over which the inversion will be one. This region is chosen on the basis of an expectation that the aerosol concentrations are the result of a single istribution with the same mean value an optical properties. For example, this is true for short vertical lines of sight well within the mixing layer; or for nearly horizontal lines of sight going through the homogeneous layer of precipitation; or in cases where aerosol scattering extensively ominates over molecular scattering. Then, an estimate k slp is mae for the average extinction coefficient in the esignate region using the slope metho. Uner most conitions, the final average is within a few percent of this value. Using this average extinction coefficient, k slp, a first estimate of k o can be foun from 2 3 k 0 o 5 1 n å S(r 6 o ) ð i 4S(r i ) ri 7 5 ; (7) 1 2 S(r) r k slp r o that is, using the backwar solution from Eq. (3), the average extinction coefficient k slp is propagate back to r o from all points in the esignate region an a mean is taken over all these values. The extinction coefficients k(r) at all ranges are then calculate with this bounary value k o 5 k 0 o using the forwar solution from Eq. (2), an the variance of this solution is then estimate from Eq. (5). Our algorithm to fin the value for the bounary conition that minimizes the variance thus starts with the estimate k slp from the slope metho yieling the first estimate of the bounary value k 0 o. Now, we take values that are 5% larger an smaller than k 0 o in an attempt to bracket the true solution, given as k min o 5 arg min s 2 (k o ). (8) k o With these three values we initiate a golen section search (Press et al. 2007), iterating until the ifference between the bracketing bounaries is less than 0.05%. We note that the forwar metho is extremely sensitive to the selection of k o ; a ifference as small as 0.2% can change the variance by more than a factor of 3. For an example of a typical epenency of the variance on k o, see Fig. 3. If the final solution is outsie of the originally estimate bouns (65% of the slope value), then the extinction coefficient istribution is likely to be a sum of more unerlying istributions with quite ifferent means.

4 FEBRUARY 2014 S U SNIK ET AL. 471 ambient air. In principle, any number of istributions can be hanle by this metho, but each must be ealt with iniviually an the values in each line of sight must accurately represent the istribution as a whole. The regions of space where each istribution lies must be ientifie an then inverte separately. Of serious concern are situations in which there are two ientifiable istributions that are not sufficiently ifferent to enable separation of the regions of space where each applies. In short, the metho can be summarize as follows: FIG. 3. Typical epenence of the forwar-solution variance on input parameter k. The initial approximations from the slope metho an from Eq. (7) are also shown with the re an blue arrows, respectively. The black arrow enotes the final location of the variance minimum. Because of the use of the classical forwar solution without separating the effects of molecules an aerosols, the metho will work for single-component atmospheres in which there is only a single extinction coefficient istribution present. Implicit in this is the assumption of a single extinction-to-backscatter ratio for this istribution. There are a number of papers that iscuss the uncertainty associate with the assumption of a single extinction-to-backscatter ratio over an extene region of space (Klett 1985a; Bissonnette 1986; Sasano et al. 1985; Sasano an Nakane 1984; Hughes et al. 1985; Kaestner 1986). While this boy of work applies to the current metho, we note that attention to the istribution of extinction coefficients will mitigate this problem. Different extinction-to-backscatter ratios will prouce istributions with ifferent mean values. If the ratios are significantly ifferent, then the presence of two istributions shoul be evient from a histogram of the extinction coefficients. The metho can be applie when two or more istributions are present. First, it is applie on an entire ataset to etermine an initial estimate of the extinction coefficients. A histogram of the istribution of these coefficients is mae an use to etermine the number an nature of the unerlying istributions. If the istribution is similar to that in Fig. 2, then further processing is not require. If two or more istributions are present, then the spatial extent of each of them must be etermine. An example of a two-istribution atmosphere woul be one in which there was a plume from some source that was emitting nearby. The extinction coefficients from the plume will have their own istribution, but with a ifferent mean value an with than that of the fin the value for k slp using the slope metho; fin the initial bounary value for k 0 o by taking the mean of back-propagate k slp from all points in the esignate region, Eq. (7); bracket the solution with minimal variance, then the interval of 6 5% aroun k 0 o is taken; use a version of the bisection algorithm to fin the minimum iteratively, Eq. (8). 3. Examples The metho was teste on synthetic liar ata, an we were able to etermine the extinction coefficients to the accuracy specifie by the convergence criterion, even with the aition of ranomly istribute noise. Since the metho shoul work best for single-component atmospheres, in which there is only a single istribution of aerosol concentrations present, the metho was teste on ata from situations where aerosol scattering ominates over molecular scattering. A consierable amount of elastic liar ata was available, obtaine uring a series of rainstorms in the fall of The ata were taken by the University of Iowa s miniature scanning liar, operating at mm. The relative insensitivity to molecular scattering at near-infrare wavelengths, couple with the scavenging of particulates by the rain an intensity of the rain, makes these ata a classic example of a single-component atmosphere. The liar was place on the fifth floor of the IIHR Hyroscience an Engineering builing, looking near horizontally (0.768 elevation) towar an area where rain gauges were installe. The ata from several nearly horizontal lines of sight that were likely to meet the single istribution were selecte at ranom for application of the metho. One example of the range-correcte ata is shown in Fig. 4. These ata represent the average of 25 laser pulses (mae over 0.5 s), taken nearly parallel to the groun. The region from 300 to 900 m was chosen as the area over which the variance woul be minimize by fining the appropriate value for k o at the beginning of the liar

5 472 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 31 FIG. 4. The light-colore line is the average value of the extinction coefficients calculate using the propose metho. The arkcolore line is the liar signal, whose slope on a semilog plot can be use to estimate the average extinction coefficient. ata. The solution is shown as the centerline in Fig. 1 an also in Fig. 5. Note that the solution, when extene all the way to 1900 m still provies reasonable (albeit increasingly noisy) values for the attenuation coefficients. The average value of the calculate extinction coefficients represents a slope that is shown in Fig. 4, an which is 0.72% ifferent from the value obtaine from a least squares application of the slope metho. The retrieve istribution of extinction coefficients from this single line of sight is shown in Fig. 2. This istribution is remarkably Gaussian, consiering the small sample size. The metho was applie to what the authors call time omain (TD) scans. These scans point the liar along a single line of sight an show ata along this line of sight with time. While the analysis of each line of sight inepenently prouces attenuation coefficients that are remarkably similar for each time interval, the metho as FIG. 6. An example of S function from liar ata along a single line of sight with time. implemente was to analyze an entire file (representing in these cases, a 15-min perio of time) as a whole. An average attenuation coefficient was calculate for the entire ataset, an separate values of k o were etermine using Eq. (7) for each of the lines of sight. This shoul not be one if there is reason to suspect that the backscatter-to-extinction ratio is changing with time. An example of time epenence of the S function from liar ata along a single line of sight is shown in Fig. 6. The resulting extinction coefficients obtaine with this metho are shown as Fig. 7. In this figure, arker values represent large values of the extinction coefficient (regions in which it is raining more intensely) an lighter values represent small values (regions in which the rain is less intense). No smoothing has been carrie out on the ata. The effects of noise can be seen as increasing fuzziness in the ata at long ranges, especially in regions more istant than 1000 m from the liar. FIG. 5. The attenuation coefficients obtaine from an inversion of the ata shown in Fig. 4. These attenuation coefficients are the source of the istribution shown in Fig. 2. The two solutions aroun the variance minimum (minimum 6s) are shown in gray. FIG. 7. The results of an application of the minimal-variance metho to the liar ata in Fig. 6. The esignate region where the variance of the extinction coefficients has been minimize was between 300 an 900 m.

6 FEBRUARY 2014 S U SNIK ET AL Conclusions A metho for the inversion of elastic liar ata has been propose that etermines the value of the extinction coefficient use as the bounary conition by minimizing the variance of the erive extinction coefficients in a elimite region. The metho works well for single-component atmospheres, where the elimite region contains a single aerosol concentration istribution. For situations where there may be two or more istributions (e.g., a smoke plume in the ambient atmosphere), the spatial regions containing each istribution must be ientifie an treate separately. Perhaps most encouraging, the high egree of shot-to-shot consistency that can be obtaine from the metho is shown in the inversion of the TD scan. This suggests that a metho coul be foun to properly invert multiimensional liar scans. There are several ifficulties with the metho. Each istribution of particulates must be ientifie from the ata, the physical locations of each istribution must be ientifie an then coefficients must be calculate for each line of sight for each istribution. Implicit in the metho is that the average value of the attenuation coefficients oes not change significantly in time or space. Thus, the metho is not suite for situations in which the aerosol concentration changes monotonically for example, liar scans in the vertical irection where the concentration ecreases with altitue. The metho will generate a solution that attempts to make the extinction coefficients have the same mean at all altitues rather than ecreasing as they shoul. Last, the metho nees to bemoifietoworkwithmultiplecomponentatmospheres. This is by far the most general case. Not only are low to moerately turbi atmospheres the most common, but none of the conventional inversion routines works well for such cases. A moification of this metho coul provie a means to aress these types of ata. Acknowlegments. A. S. wishes to thank Martin O Loughlin for his careful reaing of the manuscript. The research of A. S. was supporte by the Ministry of Higher Eucation, Science, an Technology of Slovenia, an the Slovenian Research Agency. REFERENCES Barret, E., an O. Ben-Dov, 1967: Application of the liar to air pollution measurements. J. Appl. Meteor., 6, Bissonnette, L., 1986: Sensitivity analysis of liar inversion algorithms. Appl. Opt., 25, Collis, R. T. H., an P. B. Russell, 1976: Liar measurement of particles an gases by elastic backscattering an ifferential absorption. Laser Monitoring of the Atmosphere, E. D. Hinkley, E., Springer-Verlag, Davis, P., 1969: Analysis of liar signatures of cirrus clous. Appl. Opt., 8, Eberhar, W., P. Massoli, B. McCarty, J. Machol, an S. Tucker, 2007: Comparison of liar an cavity ring-own measurements of aerosol extinction an stuy of inferre aerosol graients. Eos, Trans. 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