Standard uncertainty is a particular case of the more general context of “expanded uncertainty”, which defines “an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand” (JCGM 100, 2008). Expanded uncertainty U is expressed as a multiple of the standard uncertainty u, the scaling factor being the “coverage factor” k so that U= ku. The value of k that produces an interval corresponding to a specified level of confidence requires detailed knowledge of the probability distribution characterized by the measurement result and its combined standard uncertainty. In measurement situations where the probability distribution characterized by a measurement and its uncertainty is approximately normal, and the effective degrees of freedom of u is of significant size (typically greater than 10), taking k=2 produces an interval having an approximate level of confidence p=95.5 %. Similarly, taking k= 3 produces an interval having an approximate level of confidence p= 99.7 %. Correspondence between several key values of k and p for the normal and rectangular probability distributions is reported in Table 1. The ISSI team recommended definition of standard uncertainty (k= 1) is commonly referred to as “1σ uncertainty”, which for a normal probability distribution, corresponds to an interval of confidence of approximately 68 %.

In complex measurement techniques such as lidar, the retrieved species profile depends on multiple instrumental and physical parameters (see Sect. 3), and the notion of “measurement model” needs to be introduced. In a metrological sense, a measurement model is defined as a “mathematical relation among all quantities known to be involved in a measurement” (VIM art. 2.48; JCGM 200, 2012). The measurement model can be written as follows: Y=fX1,X2,…,XN,, where Y is the output quantity in the measurement model (the retrieved species profile), and the Xn(n=1,2,…,N) are the input quantities in this model. The function f describing the measurement model can be written for individual values y of the quantity Y in a Taylor-expanded form: y=f(x1,x2,…,xN)=y0+∑n=1N∂y∂xnxn+12∑n=1N∑m=1N∂2y∂xn∂xmxnxm+13!∑n=1N∑m=1N∑l=1N∂3y∂xn∂xm∂xlxnxmxl+…. In the case of a small-disturbance approximation such as the estimation of measurement and retrieval uncertainty, the nonlinearity of the function Y is generally considered small enough so that the terms of order 2 and higher in the Taylor expansion can be neglected. This will be our assumption in the rest of this work, which leads to the first-order expression of the measurement model: y=f(x1,x2,…,xN)=y0+∑n=1N∂y∂xnxn. The true values of a model's input quantities are unknown. These quantities are characterized by probability distributions and should be treated mathematically as random variables (JCGM 100, 2008b). These distributions describe the respective probabilities of their true values lying in different intervals, and are assigned based on available knowledge. Each input quantity xn can therefore be assigned a standard uncertainty un, characterizing its distribution.

The output quantity combined standard uncertainty uy is defined in article art. 2.31 of the VIM (JCGM 200, 2012) as the “standard measurement uncertainty that is obtained using the individual standard measurement uncertainties associated with the input quantities in a measurement model”. Uncertainty components un can either be estimated by type A or type B evaluations. Both types of evaluation are based on probability distributions and the uncertainty components resulting from either type are quantified by variances or standard deviations (JCGM 100, 2008b). A type A standard uncertainty is obtained from a probability density function derived from an observed frequency distribution, while a type B standard uncertainty is obtained from an assumed probability density function based on the degree of belief that an event will occur, using best available knowledge. If some of the input quantities are correlated, covariances must be taken into account. In these conditions, the combined standard uncertainty is the estimated standard deviation associated with the result, and is equal to the positive square root of the combined variance obtained from all variance and covariance components using the “law of propagation of uncertainty” (art. 5.2 of the GUM, JCGM 100, 2008): uy2=∑n=1N∑m=1N∂y∂xn∂y∂xmcov(xn,xm)=∑n=1N∂y∂xn2un2+2∑m=1N-1∑n=m+1N∂y∂xn∂y∂xmcov(xn,xm). The covariance between two random variables X1 and X2 with estimates x1 and x2 is defined by cov(x1,x2)=x1-x1‾x2-x2‾‾. The horizontal bar symbolizing the mean is expressed as follows: x‾=1NP∑iP=1NPx(iP). Equation (4) can also be expressed in terms of correlation coefficient rnm instead of covariance: uy2=∑n=1N∂y∂xn2un2+∑m=1N∑n=1(n≠m)N∂y∂xn∂y∂xmrnmunum, with the correlation coefficients rnm defined as rnm=cov(xn,xm)unum. Correlations between input quantities cannot be ignored if present and significant. The associated covariances should be evaluated experimentally if feasible by varying the correlated input quantities (type A evaluation of covariance), or by using the pool of available information on the correlated variability of the quantities in question (type B evaluation of covariance). In type A evaluations, the default assumption is made that the distribution best describing an input quantity is a Gaussian distribution. When the uncertainty is evaluated from a small number of indication values, the corresponding distribution can be taken as a t distribution (JCGM 100, 2008b). For a type B evaluation, the only available information is that Xn lies in a specified interval [a, b]. In such a case, knowledge of the quantity can be characterized by a rectangular probability distribution with limits a and b. If different information is available, a probability distribution consistent with that information should be used (JCGM 104, 2009). In the case of ozone DIAL measurements, both types of evaluation are found, typically type A for the random uncertainty associated with detection noise and type B for all other uncertainty sources (see upcoming sections).

The terms “systematic uncertainties” and “systematic errors”, widely used in the literature, are mathematically too ambiguous to be easily assimilated into the analytical expressions described in the GUM (JCGM 100, 2008) for the propagation of uncertainty. This terminology should be avoided and will therefore not be used here unless it explicitly refers to the terminology used in specific cited works. “Systematic component” refers to a component known to be present consistently in multiple samples of the same sampling population and owing to one or several well-identified systematic effects. For this reason a significant degree of correlation between measured samples is implied. It is only after reported systematic effects have been characterized by a randomized uncertainty component for each sample, and by a well-known correlation matrix within the sampling population, that they can contribute to the analytical implementation of the combined uncertainty budget. The term “randomize” here consists of computing the value of an uncertainty component arising from a systematic effect using a probability distribution obtained from a Type-B evaluation.

If an uncertainty component arising from a systematic effect cannot be randomized or if the covariance matrix within the sampling population cannot be computed, then this systematic effect cannot be accounted for in the uncertainty budget and it must be removed before measurements are made. If a systematic effect is reported as a nonzero (positive or negative) bias with the assumption that the value of this bias is known, then the measured samples must be corrected for this value before a combined uncertainty can be computed, and an uncertainty component characterizing the correction procedure must be introduced into the combined uncertainty budget. In order to preserve the full independence of a measurement, corrections for systematic effects must rely on the physical processes altering the measurement, and therefore must be applied to the input quantities Xi, not the output quantity Y.

The key aspect of the approach proposed hereafter is to carefully identify the independent input quantities impacting the ozone DIAL measurement model. Once all corresponding uncertainty components of systematic behavior have been randomized, applying the law of propagation of variance (Eq. 4) to multiple, independent uncertainty components allows for a standardized estimation of ozone combined uncertainty. The approach implies the replacement of a single, complex ozone DIAL measurement model by the successive application of multiple, simpler measurement sub-models. The sub-models consist of successive transformations to the raw lidar signals (e.g., saturation correction, background noise extraction, vertical filtering, see Sect. 3). At each sub-model level, standard uncertainty is evaluated in parallel for each independent uncertainty source introduced at the current or previous sub-model level. The final processing stage consists of combining all independent components together to obtain the ozone combined standard uncertainty.

To retrieve an ozone profile in the troposphere or stratosphere using the DIAL technique, we start from the Lidar Equation (e.g., Hinkley, 1976; Weitkamp, 2005). This equation in its most compressed form describes the emission of light by a laser source, its backscatter at altitude z, its extinction and scattering along its path up and back, and its collection back on a detector: P(z,λE,λR)=PL(λE)η(z,λR)δzz-zL2τUP(z,λE)β(z,λE,λR)τDOWN(z,λR), where

λE is the laser emission wavelength and λR is the receiver detection wavelength;

In the DIAL technique we consider the lidar signals measured at two different wavelengths, the light at one wavelength being more absorbed by the target species (here, ozone) than the light at the other wavelength (Mégie et al., 1977). Using the notation ON for the most absorbed wavelength, and OFF for the least absorbed wavelength, Eq. (9) can be rewritten for each of the emitted wavelength: PON(z)=PL(λ1)ηON(z)δzz-zL2τUP(z,λ1)β(z,λ1,λ2)τDOWN(z,λ2)POFF(z)=PL(λ3)ηOFF(z)δzz-zL2τUP(z,λ3)β(z,λ3,λ4)τDOWN(z,λ4). The emitted and received wavelength subscripts have been modified as follows:

λ1 and λ2 are the emitted and received “ON” wavelengths respectively;

For elastic (Rayleigh) scattering, the emitted and received wavelengths are identical, yielding λ1=λ2 and λ3=λ4. For inelastic scattering, the emitted and received wavelengths are different, and all four terms (two terms up and two terms down) are different (McGee et al., 1993). A list of most commonly used DIAL wavelength pairs for the measurement of stratospheric and tropospheric ozone is presented in Table 2 (see also Papayannis et al., 1990; Bösenberg, 1997).

ΔσM is the Rayleigh cross section differential between the ON and OFF wavelengths computed along the beam path up to altitude z and back: ΔσM=σM(λ1)+σM(λ2)-σM(λ3)-σM(λ4). Nig is the number density of absorbing constituent ig, and Δσig is the absorption cross section differential of constituent ig along the beam path up to altitude z and back, and defined as Δσig(z)=σig(z,λ1)+σig(z,λ2)-σig(z,λ3)-σig(z,λ4). ΔαP is the extinction differential due to particles and computed along the beam path up to altitude z and back: ΔαP(z)=αP(z,λ1)+αP(z,λ2)-αP(z,λ3)-αP(z,λ4). This term depends strongly on the type of particulate matter, and is difficult to estimate for ozone lidar instruments that typically do not have dedicated aerosol channels (multi-wavelength, polarization, etc.).

Finally, Λη and Λβ are defined as Λη(z)=∂∂zln⁡ηON(z)ηOFF(z) Λβ(z)=∂∂zln⁡β(z,λ1,λ2)β(z,λ3,λ4), where ηON and ηOFF are the optical efficiencies of the ON and OFF channels respectively, including optical and spectral transmittance and geometric obstruction.

The ozone DIAL measurement model depends on the choice of the theoretical equations used as well as their implementation to the real world, i.e., after considering all the caveats associated with the design, setup, and operation of an actual lidar instrument. Equation (14) relates to the expected number of photons reaching the lidar detectors (PON and POFF), not the actual raw lidar signals recorded in the data files by a real instrument. Its practical implementation for the retrieval of ozone therefore requires, on one hand the addition of several signal correction procedures and numerical transformations that depend on the instrumentation, and on the other hand, the development of approximations and/or the adoption of assumptions aimed to reduce the complexity of the measurement model.

In this context, uncertainty components associated with particulate extinction and backscatter (ΔαP and Λβ terms in Eq. 14) will not be considered here. Their contribution is negligible in a cloud-free, “clean” atmosphere, which is mostly true for altitudes above 35 km (e.g., Godin-Beekmann et al., 2003), and in most cases of clear-sky, free-tropospheric ozone DIAL measurements for which the wavelength differential is small (Papayannis et al., 1990; McDermid et al., 2002). When present and non-negligible, the contribution of particulate extinction and backscatter is highly variable from site to site, time to time, and highly dependent on the nature and quantity of the particulate matter at the time of measurement. Rather different assessment methods exist (for a review, see e.g., Eisele and Trickl, 2005). Proposing a meaningful standardized treatment of this uncertainty component is therefore complex and beyond the scope of the present work. Similarly, uncertainty due to incomplete beam-telescope overlap correction (Λη term in Eq. 14) is instrument-dependent and often time-dependent for the same instrument. Therefore, no standardized formulation is provided here. However an example of treatment is provided in the ISSI team report (Leblanc et al., 2016a).

The detectors quantum efficiency and the effects of the data recorders (e.g., sky and electronic background noise, signal saturation) must be taken into account. Due to the diversity of lidar instrumentation, it is not possible to provide a single expression for the parametrization of these effects and obtain a unique, real-world version of Eq. (14) applicable to all systems. However, we will use standardized expressions that characterize the most commonly found cases, with the idea that the proposed approach for the propagation of uncertainty can be similarly applied to other cases.

Specifically, to transition from a theoretical to a real ozone DIAL measurement model, we will apply the following transformations.

For each lidar receiver channel, the actual raw signal R recorded in the data files is represented by a vector of discretized values rather than a continuous function of altitude range:z→z(k)andR(z)→R(k)fork= 1,nk.

Given the above numerical signal transformations, a discretized version of Eq. (14) can now be formulated as follows: NO3(k)=1ΔσO3(k)S(k)-ΔσMNa(k)-∑igΔσig(k)Nig(k). A product commonly derived from the lidar-measured ozone number density is ozone mixing ratio qO3. The transformation simply consists of dividing the lidar-measured ozone number density by the “best available” ancillary air number density: qO3(k)=1ΔσO3(k)S(k)Na(k)-ΔσM-∑igΔσig(k)qig(k). Most mixing ratio uncertainty components can be directly inferred from their number density counterpart. However, the known correlation between certain input quantities and air number density led us to provide explicitly, for each component, a formulation of both the number density and mixing ratio uncertainties. Specifically, in Eq. (25), we have used the absorbing constituents' mixing ratio qig instead of the number density Nig. In the rest of this work it will be assumed that either the mixing ratio or the number density, whichever quantity is independent of the air number density, should be used as input quantity. In particular, we will address the case of molecular oxygen (ig = O2) number density, which is fully correlated with air number density via its constant mixing ratio (qO2≈ 0.209).

Another important component of our ozone DIAL measurement model is the expression of the cross section differential (Eqs. 15–17), which has the following numerical implementation: ΔσX(k)=σX_1(k)+σX_2(k)-σX_3(k)-σX_4(k). The generic subscript X stands for O3 for ozone absorption cross sections, M for Rayleigh cross sections, and ig for absorption cross sections of the interfering gases. The subscripts 1 through 4 have the same meaning as in Eqs. (15)–(17).

Equations (21)–(26) constitute our proposed standardized ozone DIAL measurement model. This model represents the set of equations adopted to estimate the standardized ozone uncertainty budget. The output quantity is ozone number density (left-hand side of Eq. 24) or mixing ratio (left-hand side of Eq. 25), while the input quantities are all the variables introduced on the right-hand side of Eqs. (21)–(22) and Eqs. (24)–(26). The input quantities' true values are unknown. These quantities' standard uncertainty must be introduced, then propagated through the ozone DIAL measurement model, and then combined to produce an ozone combined standard uncertainty profile.

Based on Eqs. (21)–(22), the instrumentation-related input quantities to consider in the NDACC-lidar standardized ozone uncertainty budget are the following:

detection noise inherent to photon-counting signal detection;

In order to limit the complexity of the standardization process, the contribution of uncertainty associated with the fundamental physical constants is treated differently from that of the other input quantities. Just like we did for standard uncertainty, we refer here to an internationally recognized and traceable standard for our recommendations on the use of physical constants, namely the International Council for Science (ICSU) Committee on Data for Science and Technology (CODATA, http://www.codata.org/), endorsed by the BIPM (Mohr et al., 2008). Within the CODATA, the Task Group on Fundamental Constants (TGFC) provides the scientific and technological communities a self-consistent set of internationally recommended values of the basic constants and conversion factors of physics and chemistry that can be found here: http://physics.nist.gov/cuu/Constants/index.html.

Our proposed approach ensures that there is indeed no propagation of uncertainty for fundamental physical constants. To do so, we truncate the CODATA-reported values to the decimal level where the CODATA-reported uncertainty no longer affects rounding. For example, the Boltzmann constant value reported by the CODATA is 1.3806488 × 10-23 JK-1 with an uncertainty of 0.0000013 × 10-23 JK-1. If we truncate to the value of 1.38065 × 10-23 JK-1, adding or subtracting its uncertainty does not modify the truncated value, and we therefore consider this value as “exact” (i.e., no uncertainty to be propagated). However, in the unlikely case that the uncertainty of a fundamental constant is of a similar order of magnitude as the uncertainty components identified in the previous paragraphs, this fundamental constant must be treated similarly to any other input quantities introduced in the measurement model; i.e., its uncertainty should be taken into account and propagated.

Random noise is inherently present in any physical system performing an actual measurement. In the case of the ozone DIAL measurement, it is introduced at the detection level, where the signal is recorded in the data files (raw signal R). The associated detection noise uncertainty is derived from Poisson statistics associated with the probability of detection of a repeated random event (Type A uncertainty estimation) (e.g., Measures, 1984). Using the subscript (DET) for detection noise, the uncertainty in the raw signal R owing to detection noise can be expressed independently for each altitude bin k and for each of the ON and OFF receiver channels as uRON(DET)(k)=RON(k)uROFF(DET)(k)=ROFF(k). This uncertainty component reflects purely random effects, and therefore implies no correlation between any of the samples considered. It is therefore propagated to ozone number density by consistently adding in quadrature the uncertainties of the individual samples used in the signal transformations. If we assume a non-paralyzable photon-counting hardware, it is propagated to the saturation and background noise-corrected signal P by applying Eq. (4) to the signal transformation equations (Eqs. 21–22) with no covariance terms: uPON(DET)(k)=PON(k)RON(k)2RON(k)uPOFF(DET)(k)=POFF(k)ROFF(k)2ROFF(k). It is finally propagated to the retrieved ozone number density NO3 and mixing ratio qO3 by applying Eq. (4) to the signal transformation equations (Eqs. 24–26) with no covariance terms: uNO3(DET)(k)=1ΔσO3(k)δz∑p=-nncp2(k)uPON(DET)(k+p)PON(k+p)2+uPOFF(DET)(k+p)POFF(k+p)2uqO3(DET)(k)=1Na(k)ΔσO3(k)δz∑p=-nncp2(k)uPON(DET)(k+p)PON(k+p)2+uPOFF(DET)(k+p)POFF(k+p)2. The order of magnitude of this uncertainty component depends on many factors including instrumentation, laser power, optical and electronic efficiencies, ozone abundance, integration time, and vertical resolution. Figure 1 shows this order of magnitude for typical NDACC stratospheric ozone DIAL systems. Depending on the lidar system considered, the Rayleigh backscatter channels (308/355 nm) may be optimized to measure in the upper stratosphere (high-intensity channels) or the lower stratosphere (low-intensity channels), or both. The results are presented as generically as possible in order to infer the order of magnitude of this uncertainty component for a wide range of system performances. Figure 1 shows cases representative of signals reaching a count rate of 1 MHz at six different altitudes (40, 35, 30, 24, 20 and 15 km) for both the ON and OFF channels. In addition to six Rayleigh backscatter DIAL pairs (solid curves, solid circles), one Raman backscatter pair (332/387 nm) is shown (open circles, dashed curves) with a typical counting rate of 1 MHz at 24 km for both the ON and OFF channels. The uncertainty values reflect a typical midlatitude climatological profile with ozone number densities increasing from 1018 to 5.1018 molec m-3 between 10 km and 24 km, and then decreasing from 5.1018 to 1017 molec m-3 from 24 to 50 km (which corresponds to a 8 ppmv mixing ratio peak at 34 km). All computations were made assuming 120 min lidar integration time, and constant 1 km vertical resolution following the standardized definition presented in our companion paper (Part 1, Leblanc et al., 2016). Shorter integration times or higher vertical resolutions would shift all curves towards the right (larger detection noise uncertainty), while longer integration times or degraded vertical resolutions would shift all curves toward the left (smaller detection noise uncertainty).

For the ozone number density relative uncertainty (left plot), the main feature is a nearly constant magnitude between 10 and 24 km associated with the gain of sensitivity resulting from the increase of ozone number density in the lower stratosphere, which compensates the loss of backscattered signal. Above 24 km, the exponential increase reflects the combined effect of the decrease in ozone number density and backscatter signal. In this latter region, the relative uncertainty increases by a factor of 20 every 10 km, as indicated by the black arrow. The thick long-dash black curve indicates the approximate location of the 1 MHz count rate as a function of altitude. Using this curve, the ozone relative uncertainty owing to detection noise can be estimated for any stratospheric ozone DIAL by simply starting from the known altitude of the 1 MHz count rate (located somewhere on the black curve), and then drawing a curve parallel to the existing colored curves. Note the factor of 2 between the Rayleigh and Raman backscatter channels' relative uncertainty curves for the same signal magnitude (blue solid curve and blue dashed curve respectively). The difference is due to a reduced sensitivity of the less-absorbing, longer Raman-shifted wavelengths. In terms of ozone mixing ratio (right plot), uncertainty owing to detection noise increases exponentially with altitude, the magnitude being multiplied by a factor of 10 every 10 km.

Figure 2 is similar to Fig. 1 but for typical tropospheric ozone DIAL systems. The uncertainty values shown correspond to a climatological ozone profile with number densities around 1018 molec m-3, i.e., 40–60 ppbv between the ground and 10 km height. Because the 289/299 nm DIAL pair is currently the most commonly used across NDACC, three different altitudes of 1 MHz count rate are shown for this pair. Three other pairs, namely 299/316, 287/294, and 266/289 nm are also shown at their typical magnitude (1 MHz count rate at 12, 5, and 2 km respectively). Not surprisingly, all 289/299 nm pairs show a similar uncertainty curve shape, and the uncertainty for the 266/289 nm increases at a much faster rate than the others due to the enhanced sensitivity and faster signal loss (large absorption).

This uncertainty component is introduced only for channels operating in photon-counting mode. If we consider a non-paralyzable counting hardware, the only input quantity to introduce is the hardware's dead time (sometimes called resolving time), which characterizes the speed of the counting electronics. The dead time τ and its uncertainty uτ are generally among the technical specifications provided by the hardware manufacturer (Type-B estimation). The associated saturation correction uncertainty is derived by applying Eq. (4) to Eqs. (21)–(22). Using the subscript (SAT) for saturation, it can be expressed independently for the ON and OFF channels as uPON(SAT)(k)=c2δzLPON2(k)uτONuPOFF(SAT)(k)=c2δzLPOFF2(k)uτ_OFF. The saturation correction uncertainty needs to be propagated through Eqs. (24)–(26). According to Eqs. (21)–(22), the same dead-time value is used at all altitudes for a given channel. When vertically filtering the signal, the saturation correction uncertainty is therefore propagated, assuming full correlation between neighboring altitude samples. Applying vertical differentiation (Eq. 24) therefore results in a linear combination of the samples' uncertainties identical to that applied to the samples' values. However, when combining the ON and OFF channels, two instrumental configuration cases need to be considered.

If the photon-counting hardware of the ON and OFF channels is different, the channels can be considered independent and the saturation correction uncertainty can be propagated to the retrieved ozone number density and mixing ratio through the differentiation equation (Eq. 24), assuming no correlation between samples measured in the ON and OFF channels (no covariance terms), thus resulting in the following expressions:uNO3(SAT)(k)=1ΔσO3(k)δz∑p=-nncp(k)uPON(SAT)(k+p)PON(k+p)2+uPOFF(SAT)(k+p)POFF(k+p)2uqO3(SAT)(k)=1Na(k)ΔσO3(k)δz∑p=-nncp(k)uPON(SAT)(k+p)PON(k+p)2+uPOFF(SAT)(k+p)POFF(k+p)2.

Figure 4 is similar to Fig. 3 but for typical tropospheric ozone DIAL systems. At an altitude range larger than 3 km, the relative uncertainty is divided by 2 every 1 km, while the mixing ratio uncertainty is divided by 3 every 2 km. Values above 10 % (6 ppbv) are found only at the very bottom of the profiles, when the signal dynamic range increases dramatically (near-range measurements).

At far range, backscattered signal is too weak to be detected and any nonzero signal reflects the presence of undesired skylight and/or electronic background noise. This noise is typically subtracted from the total signal by fitting the uppermost part of the lidar signal with a linear or nonlinear function of altitude B. A new uncertainty component associated with the noise fitting procedure must therefore be introduced. Here we provide a detailed treatment for the simple case of a linear fit. It can be easily generalized to many other fitting functions. The linear fitting function takes the form: B(k)=b0+b1z(k) For many well-known fitting methods (e.g., least-squares), the fitting coefficients bi can be calculated analytically together with their uncertainty ubi and their correlation coefficient rbi,bj (Type A estimation) (Press et al., 1986). Using the subscript (BKG) for background noise, the background noise correction uncertainty is expressed independently for the ON and OFF channels by applying Eq. (4) to the signal transformation equations (Eqs. 21–22), and using the linear form of Eqs. (21)–(22), for B, we obtain uPON(BKG)(k)=ub0_ON2+ub1_ON2z2(k)+2z(k)ub0_ONub1_ONrb0,b1_ONuPOFF(BKG)(k)=ub0_OFF2+ub1_OFF2z2(k)+2z(k)ub0_OFFub1_OFFrb0,b1_OFF. The above two equations can be derived analytically for any fitting function for which the fitting method allows for the proper estimation of the fitting parameters' covariance matrix (e.g., least-squares and singular value decomposition).

Because of the nature of the background noise correction (parameters bi independent of altitude), the approach used for the propagation of saturation correction uncertainty can also be used for the propagation of background noise correction uncertainty. In other words, the following is true.

If the data acquisition hardware of the ON and OFF channels is different, the background noise correction uncertainty can be propagated assuming no correlation between the ON and OFF channels (no covariance terms):uNO3(BKG)(k)=1ΔσO3(k)δz∑p=-nncp(k)uPON(BKG)(k+p)PON(k+p)2+uPOFF(BKG)(k+p)POFF(k+p)2uqO3(BKG)(k)=1Na(k)ΔσO3(k)δz∑p=-nncp(k)uPON(BKG)(k+p)PON(k+p)2+uPOFF(BKG)(k+p)POFF(k+p)2.

The order of magnitude of the propagated ozone uncertainty owing to background noise correction depends on many factors, including the relative magnitude of the ON and OFF signals with respect to noise being subtracted, and the slope of the signal-induced noise if signal-induced noise is present. Figure 5 (respectively Fig. 6) shows one example of this magnitude and its change with altitude for stratospheric (respectively tropospheric) ozone DIAL pairs with a constant background noise extracted. In this case, the coefficient b1 is set to zero, and the only uncertainty is that associated with the fitting parameter's uncertainty ub0. The rate at which uncertainty increases with altitude in this case is simply determined by the signal slope differential (for example, sharp increase for the tropospheric pair 266/289 nm compared to the pair 299/316 nm, as shown in Fig. 6).

The above case (constant noise) and the case of noise having a well-known, mild constant slope are the simplest cases to deal with, for which the only uncertainty component to consider is that owing to the fitting parameters. In the presence of non-negligible signal-induced noise, the slope of the noise is no longer constant with altitude, and the background correction becomes much more uncertain. The uncertainty associated with nonlinear fits is typically larger than that associated with a linear fit, but most importantly, the actual altitude dependence of the signal-induced noise is usually unknown, and an additional uncertainty component that cannot be quantified accurately should be introduced. For this reason, it is strongly recommended to design lidar receivers in such a way that no signal-induced noise is present at all.

Uncertainty owing to the ozone absorption cross section differential is computed by applying Eq. (4) to the DIAL equation (Eq. 24). The actual magnitude of this uncertainty can be very different depending on the type of backscatter (Rayleigh or Raman), and depending on the source of ozone absorption cross section used Eq. (26). Temperature-dependent ozone absorption cross sections values originate from various published works by spectroscopy groups around the world (e.g., Gorshelev et al., 2014; Serdyuchenko et al., 2014; Bass and Paur, 1984; Bogumil et al., 2003; Chehade et al., 2013; Daumont et al., 1992; Brion et al., 1998; Burrows et al., 1999). These groups usually provide at least one type of uncertainty estimate associated with the cross section values. Occasionally, they provide separate components owing to systematic and random effects. If present, these two components are not introduced and propagated similarly. To account for this distinction, the subscripts “R” (for random) and “S” (for systematic) will be used thereafter whenever needed. Expressions for the ozone uncertainty owing to the absorption cross section differential are now provided for four common cases.

An approach similar to that used for the ozone absorption cross section differential uncertainty can be used for the Rayleigh extinction cross section differential uncertainty by applying Eq. (4) to the DIAL equation (Eq. 25) and the cross section differential equation (Eq. 27). Analytical expressions of Rayleigh scattering based on atmospheric composition usually provide better cross section estimates than laboratory studies, e.g., Bates (1984); Eberhard (2010); Bucholtz (1995). Using an analytical expression to compute Rayleigh extinction cross sections is equivalent to considering the case of a single-source component (namely, the analytical function), therefore implying full correlation between all values. Under this assumption, the Rayleigh extinction cross section differential uncertainty propagated to ozone number density and mixing ratio can be written for Rayleigh and Raman backscatter channels: uNO3(ΔσMS)(k)=Na(k)uσM_1(S)(k)+uσM_2(S)(k)-uσM_3(S)(k)-uσM_4(S)(k)ΔσO3(k)uqO3(ΔσMS)(k)=uσM_1(S)(k)+uσM_2(S)(k)-uσM_3(S)(k)-uσM_4(S)(k)ΔσO3(k). When cross section uncertainties owing to only random effects are used and for Rayleigh backscatter channels, the Rayleigh extinction cross section differential uncertainty uNO3(ΔσMR) propagated to ozone number density and mixing ratio can be written as follows: uNO3(ΔσMR)(k)=2Na(k)ΔσO3(k)uσM_1(R)2(k)+uσM_3(R)2(k)uqO3(ΔσMR)(k)=2uσM_1(R)2(k)+uσM_3(R)2(k)ΔσO3(k). For Raman backscatter channels, this uncertainty component can be written as follows: uNO3(ΔσMR)(k)=Na(k)ΔσO3(k)uσM_1(R)2(k)+uσM_2(R)2(k)+uσM_3(R)2(k)+uσM_4(R)2(k)uqO3(ΔσMR)(k)=uσM_1(R)2(k)+uσM_2(R)2(k)+uσM_3(R)2(k)+uσM_4(R)2(k)ΔσO3(k). Equations (56), (58), and (60) show that for a specific DIAL pair, the lidar-retrieved mixing ratio uncertainty is directly proportional to the relative uncertainty in the Rayleigh cross section. Figure 8 shows, for several tropospheric (left plot) and stratospheric (right plot) ozone DIAL pairs, the ozone mixing ratio uncertainty as a function of the Rayleigh cross section relative uncertainty. In the figure, it is assumed that all Rayleigh cross sections have the same relative uncertainty value. A cross section relative uncertainty of 100 % leads to an ozone number density uncertainty which has a magnitude that is equal to the error that results from neglecting the Rayleigh extinction correction. For a particular value of Rayleigh cross section relative uncertainty, the DIAL pairs with longer wavelengths (e.g., 299/316 for tropospheric systems, and the Raman pair for stratospheric systems) yield larger ozone mixing ratio uncertainties. Similar behavior was reported for the aerosol interference (Völger et al., 1996; Eisele and Trickl, 2005).

Once again, an approach similar to that used for the ozone absorption and Rayleigh cross section differentials can be used for the absorption cross section differential of the interfering gases. The resulting uncertainty components owing to random and systematic effects and propagated to ozone number density and mixing ratio can be written for NO2 and SO2 (ig = NO2, SO2). The particular case of absorption by O2 in the Herzberg and Wulf bands region is presented in the next paragraph.

For random effects and the Rayleigh backscatter case, we have the following: uNO3(ΔσigR)(k)=2Nig(k)ΔσO3(k)uσig_1(R)2(k)+uσig_3(R)2(k)uqO3(ΔσigR)(k)=2qig(k)ΔσO3(k)uσig_1(R)2(k)+uσig_3(R)2(k). For random effects and the Raman backscatter case, we have the following: uNO3(ΔσigR)(k)=Nig(k)ΔσO3(k)uσig_1(R)2(k)+uσig_2(R)2(k)+uσig_3(R)2(k)+uσig_4(R)2(k)uqO3(ΔσigR)(k)=qig(k)ΔσO3(k)uσig_1(R)2(k)+uσig_2(R)2(k)+uσig_3(R)2(k)+uσig_4(R)2(k). For systematic effects, if using a single dataset for both ON and OFF wavelengths, for both Rayleigh and Raman backscatter, we have the following: uNO3(ΔσigS)(k)=Nig(k)ΔσO3(k)uσig_1(S)(k)+uσig_2(S)(k)-uσig_3(S)(k)-uσig_4(S)(k)uqO3(ΔσigS)(k)=qig(k)ΔσO3(k)uσig_1(S)(k)+uσig_2(S)(k)-uσig_3(S)(k)-uσig_4(S)(k). For systematic effects, if using two different datasets for ON and OFF wavelengths, for both Rayleigh and Raman backscatter, we have the following: uNO3(ΔσigS)(k)=Nig(k)ΔσO3(k)uσig_1(S)(k)+uσig_2(S)(k)2+uσig_3(S)(k)+uσig_4(S)(k)2uqO3(ΔσigS)(k)=qig(k)ΔσO3(k)uσig_1(S)(k)+uσig_2(S)(k)2+uσig_3(S)(k)+uσig_4(S)(k)2. This time the ozone mixing ratio uncertainty is proportional to the relative uncertainty in the cross section and to the mixing ratio of the interfering gas. Figure 9 shows, for several tropospheric (left plot) and stratospheric (right plot) ozone DIAL pairs, the expected ozone mixing ratio uncertainty per part-per-billion of NO2, and as a function of the NO2 cross section relative uncertainty. In the figure, it is assumed that all NO2 cross sections have the same relative uncertainty. A cross section relative uncertainty of 100 % is equivalent to neglecting NO2 absorption. DIAL pairs with longer wavelengths yield a larger ozone mixing ratio uncertainty due to the large NO2 cross section values in the UV region. In “normal” NO2 background conditions, the relative impact of NO2 absorption on retrieved ozone remains very small for both tropospheric and stratospheric ozone systems.

Figure 10 is similar to Fig. 9, but for SO2. The ozone mixing ratio uncertainty owing to SO2 cross section uncertainty is almost negligible for stratospheric DIAL pairs (Higgins band) because of the weak SO2 absorption in this region compared to that of ozone. The impact of SO2 absorption on retrieved ozone is therefore negligible except in the case of heavy SO2 loads (i.e., 100 ppbv or above).

An approach similar to that used for the other cross section differentials can be used for the O2 absorption in the region of the Herzberg and Wulf bands (Fally et al., 2000). This interfering absorption only impacts DIAL measurements using wavelengths shorter than 294 nm. In addition, the impact depends on the position of the laser line with respect to the position of the individual Herzberg lines. When the lines are coincident and the resulting absorption non-negligible, the expression of uncertainty for this component owing to random and systematic effects and propagated to ozone number density and mixing ratio can be formulated in the same manner as the other interfering gases, with the exception that the O2 mixing ratio qO2 is assumed as a well-known constant (qO2∼0.209).

For random effects and the Rayleigh backscatter case, we have the following: uNO3(ΔσO2R)(k)=2qO2Na(k)ΔσO3(k)uσO2_1(R)2(k)+uσO2_3(R)2(k)uqO3(ΔσO2R)(k)=2qO2ΔσO3(k)uσO2_1(R)2(k)+uσO2_3(R)2(k). For random effects and the Raman backscatter case, we have the following: uNO3(ΔσO2R)(k)=qO2Na(k)ΔσO3(k)uσO2_1(R)2(k)+uσO2_2(R)2(k)+uσO2_3(R)2(k)+uσO2_4(R)2(k)uqO3(ΔσO2R)(k)=qO2ΔσO3(k)uσO2_1(R)2(k)+uσO2_2(R)2(k)+uσO2_3(R)2(k)+uσO2_4(R)2(k). For systematic effects, if using a single dataset, for both Rayleigh and Raman backscatter, we have the following: uNO3(ΔσO2S)(k)=qO2Na(k)ΔσO3(k)uσO2_1(S)(k)+uσO2_2(S)(k)-uσO2_3(S)(k)-uσO2_4(S)(k)uqO3(ΔσO2S)(k)=qO2ΔσO3(k)uσO2_1(S)(k)+uσO2_2(S)(k)-uσO2_3(S)(k)-uσO2_4(S)(k). For systematic effects, if using two different datasets for ON and OFF wavelengths, for both Rayleigh and Raman backscatter, we have the following: uNO3(ΔσO2S)(k)=qO2Na(k)ΔσO3(k)uσO2_1(S)(k)+uσO2_2(S)(k)2+uσO2_3(S)(k)+uσO2_4(S)(k)2uqO3(ΔσO2S)(k)=qO2ΔσO3(k)uσO2_1(S)(k)+uσO2_2(S)(k)2+uσO2_3(S)(k)+uσO2_4(S)(k)2. Equations (69)–(76) show that the ozone mixing ratio uncertainty owing to O2 absorption is directly proportional to the relative uncertainty in the O2 cross section. Figure 11 shows, for several tropospheric ozone DIAL pairs, the expected ozone mixing ratio uncertainty as a function of the O2 cross section relative uncertainty. In the figure, it is assumed that all O2 cross sections have the same relative uncertainty value. A cross section relative uncertainty of 100 % is equivalent to neglecting O2 absorption. As already mentioned, the discrete Herzberg absorption lines are very narrow and the effective cross sections depend strongly on the position of the laser line with respect to those lines. Therefore Fig. 11 is shown as an example, only knowing that the effective cross sections may differ greatly from one tropospheric ozone lidar instrument to another.