A standardized approach for the definition, propagation, and reporting of uncertainty in the ozone differential absorption lidar data products contributing to the Network for the Detection for Atmospheric Composition Change (NDACC) database is proposed. One essential aspect of the proposed approach is the propagation in parallel of all independent uncertainty components through the data processing chain before they are combined together to form the ozone combined standard uncertainty.

The independent uncertainty components contributing to the overall budget
include random noise associated with signal detection, uncertainty due to
saturation correction, background noise extraction, the absorption
cross sections of O

The ozone uncertainty budget is presented as much as possible in a generic form (i.e., as a function of instrument performance and wavelength) so that all NDACC ozone DIAL investigators across the network can estimate, for their own instrument and in a straightforward manner, the expected impact of each reviewed uncertainty component. In addition, two actual examples of full uncertainty budget are provided, using nighttime measurements from the tropospheric ozone DIAL located at the Jet Propulsion Laboratory (JPL) Table Mountain Facility, California, and nighttime measurements from the JPL stratospheric ozone DIAL located at Mauna Loa Observatory, Hawai'i.

The present article is the second of three companion papers that provide a
comprehensive description of recent recommendations made to the Network for
Detection of Stratospheric Change (NDACC) lidar community for the
standardization of vertical resolution and uncertainty in the NDACC lidar
data processing algorithms. NDACC (

Until now, there has been no comprehensive effort within NDACC to facilitate
a standardization of the definitions and approaches used to report vertical
resolution and uncertainty in the NDACC ozone lidar data products. To help
fill this gap, an International Space Science Institute (ISSI) international team of experts (

Our first companion paper (Part 1) (Leblanc et al., 2016b) is exclusively dedicated to the ISSI team recommendations for standardized definitions of vertical resolution. The present article (Part 2) provides a detailed description of the approach proposed by the ISSI team for a standardized treatment of uncertainty in the ozone differential absorption lidar (DIAL) retrievals. Another companion paper (Part 3) (Leblanc et al., 2016c) presents a similar approach for the standardized treatment of uncertainty in the temperature lidar retrievals.

Uncertainties in ozone DIAL measurements have been discussed since the early
development of the DIAL technique (Mégie et al., 1977). Early
publications dealt with the optimization of the wavelengths pairs for
tropospheric and stratospheric ozone measurements, taking into account the
measurement's error budget (e.g., Mégie and Menzies, 1980; Pelon and
Mégie, 1982; Papayannis et al., 1990). In the framework of the NDACC,
various groups have set up lidar instruments for the measurement of ozone
in the troposphere and stratosphere. They have generally described their
lidar systems with a detailed assessment of the measurement errors (e.g.,
Godin, 1987; Uchino and Tabata, 1991; McDermid et al., 1990; Papayannis et
al., 1990; McGee et al., 1991; Godin-Beekmann et al., 2003). In addition,
intercomparison campaigns set up in the framework of NDACC have assessed the
evaluation of lidar measurement uncertainties (see

The fundamentals of uncertainty with a metrological reference are briefly reviewed in Sect. 2. Based on these fundamentals, a standardized measurement model for the retrieval of ozone using the DIAL method is proposed in Sect. 3. Based on this model, detailed step-by-step expressions for the propagation of uncertainty through the ozone lidar algorithm are then provided in Sect. 4. In this section, quantitative estimates of each uncertainty component are provided in a generic manner whenever possible. Finally, two examples of uncertainty budgets taken from actual NDACC ozone DIAL measurements (nighttime measurement conditions) are provided in Sect. 5, followed by a short summary and conclusion. The reader should refer to the ISSI team report (Leblanc et al., 2016a) for aspects that are not fully described in the present article.

The definition of uncertainty recommended by the ISSI team for use by all NDACC lidar measurements is the combined standard uncertainty. It originates in the two internationally recognized reference documents endorsed by the Bureau International des Poids et Mesures (BIPM), namely the International Vocabulary of Basic and General Terms in Metrology (abbreviated “VIM”) (JCGM 200, 2008a, 2012), and the Guide to the Expression of Uncertainty in Measurement (abbreviated “GUM”) (JCGM 100, 2008b). These two documents and their supplements provide a complete framework for the treatment of uncertainty.

In a metrological sense (article 2.26 of the VIM) (JCGM 200, 2012), uncertainty is a “non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand based on the information used”. Measurement uncertainty includes components arising from systematic effects, as well as the definitional (or “intrinsic”) uncertainty, i.e., the practical minimum uncertainty achievable in any measurement. It may be a standard deviation or the half width of an interval with a stated coverage probability. The particular case of “standard uncertainty” is defined in article 2.30 of the VIM (JCGM 200, 2012), as “the measurement uncertainty expressed as a standard deviation”.

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

Correspondence between key values of coverage factor and level of confidence for two common probability distributions.

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:

The output quantity combined standard uncertainty

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

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.

In this section, a standardized measurement model for the retrieval of tropospheric and stratospheric ozone using the DIAL technique is constructed so that each input quantity introduced at one stage of the model is independent of the others.

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

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:

List of most commonly used ozone DIAL wavelength pairs.

For elastic (Rayleigh) scattering, the emitted and received wavelengths are
identical, yielding

Finally,

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
(

In this context, uncertainty components associated with particulate
extinction and backscatter (

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

The actual raw signal recorded the data files is a combination of laser light backscattered in the atmosphere, sky background light that can be parametrized by a constant offset, and noise generated within the electronics (dark current and possibly signal-induced noise) that can be parametrized by a linear or nonlinear function of time, i.e., altitude range.

Only channels operating in photon-counting mode are considered hereafter. For analog channels, uncertainty due to analog-to-digital signal conversion needs to be estimated. This estimation is highly instrument-dependent, and no meaningful standardized recommendations can therefore be provided. However, an example of the treatment of the analog detection uncertainty is provided for reference in the ISSI team report (Leblanc et al., 2016a).

In photon-counting detection mode, the recorded signals result from nonlinear transfer of the detected signals due to the inability of the counting electronics to temporally discriminate a very large number of photon-counts reaching the detector (“pulse pile-up” effect resulting in signal saturation) (e.g., Müller, 1973; Donovan et al., 1993). In the present work, we consider the most frequent case of non-paralyzable photon-counting systems (i.e., using “non-extended dead time”, Müller, 1973), which allows for an analytical correction of the pulse pile-up effect.

If

The ozone DIAL measurement includes detection noise, and it is desirable
to filter this noise whenever it is expected to impact the retrieved
product. The filtering process impacts the propagation of uncertainties, and
therefore should be included in the measurement model. For each individual
altitude

Given the above numerical signal transformations, a discretized version of
Eq. (14) can now be formulated as follows:

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:

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;

saturation (pulse pile-up) correction parameters (typically, photon
counters' dead time

background noise extraction parameters (typically, fitting parameters for
function

ozone absorption cross sections differential

Rayleigh extinction cross sections differential

ancillary air number density profile

absorption cross sections differential for the interfering gases

Number density profiles

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,

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

The expressions for the propagation of uncertainty presented in this section
are derived directly from the equations of our proposed standardized ozone
DIAL measurement model (previous section), and by systematically applying
the law of variance propagation described in Sect. 2 (Eq. 4). For clarity, throughout this section we will use the following
variable naming convention: each newly introduced output quantity

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

Ozone number density relative uncertainty (left) and ozone mixing ratio uncertainty (right) owing to detection noise for stratospheric ozone DIAL systems of varying performance, and for a 120 min integration time and 1 km vertical resolution. The systems' performance is measured as the altitude of 1 MHz count rate for both the ON and OFF channels signals. See text for details.

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 10

Same as Fig. 1, but for tropospheric ozone systems. This time, integration time is 20 min and vertical resolution is 180 m. See text for details.

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

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:

If the ON and OFF channels share the same hardware, the apparatus is
considered identical for both channels, and the saturation correction
uncertainty should therefore be propagated to the retrieved ozone number
density and mixing ratio through the differentiation equation assuming full
correlation between the samples measured by the ON and OFF channels,
resulting in the following expressions:

Ozone number density relative uncertainty (left) and ozone mixing ratio uncertainty (right) owing to saturation correction for stratospheric ozone DIAL systems of varying performance. The systems' performance is measured as the altitude of 1 MHz count rate for both the ON and OFF channels signals. See text for details.

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).

Same as Fig. 3, but for tropospheric ozone systems. See text for details.

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

Because of the nature of the background noise correction (parameters

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):

If the ON and OFF channels share the same hardware, the background noise
correction uncertainty can be propagated to the retrieved ozone number
density and mixing ratio through the differentiation equation assuming full
correlation between the ON and OFF channels:

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

Ozone number density relative uncertainty (left) and ozone mixing ratio uncertainty (right) owing to background noise correction (linear fit) for stratospheric ozone DIAL systems of varying performance. The systems' performance is measured as the altitude of 1 MHz count rate for both the ON and OFF channels signals, using typical nighttime background noise conditions. See text for details.

Same as Fig. 5, but for typical tropospheric ozone channels.

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 “

In this case, the random component of the cross sections uncertainty

Equation (4) is applied to the DIAL equation (Eq. 24)
assuming no covariance terms from the cross section differential
Eq. (26). For elastic (Rayleigh) backscatter DIAL systems, the
corresponding component is propagated to ozone number density and mixing
ratio using

The cross sections' uncertainty component owing to systematic effects is not always present or reported. It is most often estimated by comparing several cross section datasets and observing biases between those datasets. The expression for the propagation of this component depends on the degree of correlation between the datasets used. Here we consider only two cases: when a unique source of cross section is used for all wavelengths (i.e., dataset originating from a single set of laboratory measurements), and when two independent cross section datasets are used for the ON and OFF wavelengths.

In the first case, it is assumed that the same dataset is used for the
absorption cross sections at all wavelengths. The systematic component of
the cross sections' uncertainty

Equations (47)–(54) show that the relative uncertainty in the retrieved ozone is directly proportional to the relative uncertainty in the ozone absorption cross section, which makes the latter the main source of uncertainty in the nominal region of the ozone DIAL method (Godin-Beekmann and Nair, 2012). Figure 7 shows, for several of the configurations just described and for several stratospheric and tropospheric ozone DIAL pairs, the ozone number density relative uncertainty as a function of the absorption cross section relative uncertainty. In all cases shown, it is assumed that all absorption cross sections have the same relative uncertainty. For stratospheric ozone DIAL pairs (308/355 and 332/387), the absorption cross section at the ON wavelength is much larger than that at the OFF wavelength, resulting in an ozone relative uncertainty mostly dominated by the absorption cross section uncertainty at the ON wavelength, and therefore leading to a one-to-one relationship (nearly diagonal straight line). For tropospheric ozone DIAL pairs (299/316, 289/299, 266/289, and 287/294), the absorption cross sections at the ON and “OFF” wavelengths are closer to each other. As a result, the curves depart slightly from the diagonal observed for the stratospheric pairs. A 1-to-1 relationship (diagonal) is also observed for the all-systematic case as a result of the linear combination of Eqs. (51)–(52).

Ozone relative uncertainty (%) as a function of absorption cross section relative uncertainty (%), assuming that all cross sections have the same relative uncertainty. Solid red, green, blue, and purple curves are used for cases of independent (random) datasets, and a dashed black curve is used for the case of full correlation between all cross sections (systematic).

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:

Ozone mixing ratio uncertainty as a function of Rayleigh cross section relative uncertainty (%), assuming that all cross sections used have the same relative uncertainty. Solid curves are used for cases of independent (random) cross section datasets, and dashed curves are used for the case of full correlation between all cross sections.

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 NO

For random effects and the Rayleigh backscatter case, we have the following:

Ozone mixing ratio uncertainty as a function of NO

Ozone mixing ratio uncertainty as a function of SO

Figure 10 is similar to Fig. 9, but for SO

An approach similar to that used for the other cross section differentials
can be used for the O

For random effects and the Rayleigh backscatter case, we have the following:

Ozone mixing ratio uncertainty as a function of O

Another source of uncertainty introduced in Eq. (25) is the a priori use of
ancillary NO

When the input quantity independent of air number density is the interfering
gas' number density

When the input quantity independent of air number density is the mixing
ratio of the interfering gas

Ozone mixing ratio uncertainty as a function of NO

The last input quantity to consider in our ozone DIAL measurement model is ancillary air number density. The air density is generally not estimated directly, but rather derived from air temperature and pressure. Here we provide expressions for the propagation of this uncertainty component for both cases, i.e., when air number density is considered the input quantity, and when temperature and pressure are considered the input quantities.

If the air number density

If number density is used as input quantity for the interfering gases'
profiles,

If mixing ratio is used as input quantity for the interfering gases'
profiles,

When using radiosonde measurements or meteorological analysis, the air
number density is typically derived from air temperature

If temperature and pressure are measured or computed independently, with
uncertainty estimates

If temperature and pressure are measured or computed independently, with
uncertainty estimates

If temperature and pressure are known to be fully correlated, and if number
density is used as input quantity for the interfering gases, the ozone
number density uncertainty owing to air number density will be written as follows:

If temperature and pressure are known to be fully correlated, and if mixing
ratio is used as input quantity for the interfering gases, the ozone number
density uncertainty owing to air number density will be written as follows:

Same as Fig. 12, but for interfering gas SO

Figure 14 shows the stratospheric ozone relative uncertainty (left) and mixing ratio uncertainty (right) as a function of the ancillary air number density, temperature or pressure uncertainty for typical midlatitude spring conditions. The solid curves represent the ozone uncertainty for each percent of air number density uncertainty, the dashed curves represent the ozone uncertainty for each degree of air temperature uncertainty, and the dotted curves represent the ozone uncertainty for each 0.1 hPa of air pressure uncertainty. The largest ozone uncertainty in the upper stratosphere is that owing to pressure. Figure 15 is similar to Fig. 14, but for tropospheric ozone DIAL systems. DIAL pairs using longer wavelengths (e.g., 299/316 nm) are more impacted than pairs using shorter wavelengths. Noteworthy, with current pressure–temperature measurement capabilities (typically 0.5 K and 0.1 hPa uncertainties), the lidar-retrieved ozone uncertainty owing to temperature is about 10 times larger than that owing to pressure uncertainty.

Stratospheric ozone relative uncertainty (left) and mixing ratio uncertainty (right) as a function of air number density, temperature, and pressure uncertainty, for typical midlatitude spring conditions. The solid curves represent the ozone uncertainty per percent of air number density uncertainty, the dashed curves represent the ozone uncertainty per degree of air temperature uncertainty, and the dotted curves represent the ozone uncertainty per 0.1 hPa of air pressure uncertainty.

Same as Fig. 14, but for tropospheric ozone DIAL systems.

Ozone DIAL instruments are most often designed with multiple signal intensity ranges in order to maximize the overall altitude range of the profile. Reduced signal intensity is achieved using neutral density filters or other optical systems attenuating the Rayleigh-backscattered signals, or using Raman backscatter channels, which typically are 750 times weaker than Rayleigh backscatter channels. Until now, our ozone DIAL measurement model referred to a single intensity range. We now provide an example of formulation for the propagation of uncertainty when the number densities for two intensity ranges are combined to produce a single profile. Combining individual intensity ranges into a single profile can occur either during lidar signal processing or after the ozone number density is calculated individually for each intensity range. Here we present the case of combining ozone number density after it was calculated for individual intensity ranges. The case of combining the lidar signals is presented in our companion paper (Leblanc et al., 2016b). The principles governing the propagation of uncertainty are the same in both cases.

A single profile covering the entire useful range of the instrument is
typically obtained by combining the most accurate overlapping sections of
the profiles retrieved from individual ranges. The thickness of the
transition region typically varies from a few meters to a few kilometers,
depending on the instrument and on the intensity ranges considered. Assuming
that the transition region's bottom altitude is

Because of its random nature, ozone uncertainty owing to detection noise for
the combined profile is obtained by adding in quadrature (no covariance
terms) the detection noise uncertainties of the individual ranges:

Having reviewed and propagated all the independent uncertainty components considered in our ozone DIAL measurement model, we can combine them into a single total uncertainty estimate.

If number density is used as input quantity for the interfering gases, the
combined standard uncertainty of retrieved ozone number density and mixing
ratio can be written as follows:

Similarly, the total combined ozone density (or mixing ratio) uncertainty can be used to characterize a single profile, but should not be used for the combination of “dependent” profiles (for example a climatology computed from multiple profiles measured by the same instrument). Instead, uncertainty components owing to systematic effects in altitude and/or time must be separated from components owing to random effects. Typically, uncertainty owing to detection noise will always be added in quadrature, while for other components, knowledge (type A or type B estimation) of the covariance matrix in the time and/or altitude dimension(s) will be needed. For this reason, it is recommended that a trace of each individual component is always kept together with the combined standard uncertainty.

Input quantities and their uncertainty used to compute the ozone uncertainty budget presented in Figs. 16 and 17.

The uncertainty components discussed in the previous section were quantitatively reviewed, for most cases, in parametric form, so that the order of magnitude of each component could be estimated for a wide range of instrument performance. Here we provide two actual examples using existing measurements from the Jet Propulsion Laboratory (JPL) tropospheric ozone DIAL at the NDACC site of Table Mountain Facility (California), and the JPL stratospheric ozone DIAL at the NDACC site of Mauna Loa Observatory (Hawai'i). In these two examples, the input quantities' uncertainty estimates are taken from the JPL in-house data processing software used to process the routine JPL lidar data archived at NDACC. A list of those input quantities and their uncertainty is compiled in Table 3.

Figure 16 shows the full ozone uncertainty budget for a 2 h
measurement obtained on 13 March 2009 from the JPL stratospheric ozone DIAL
located at Mauna Loa Observatory, Hawai'i. The ozone number density
uncertainty budget is on the left (in %), the ozone mixing ratio
uncertainty budget is on the right (in ppmv). All components previously
identified are present except the three components associated with
absorption by SO

Example of ozone relative uncertainty (left) and mixing ratio uncertainty (right) budget computed for the JPL stratospheric ozone DIAL located at Mauna Loa Observatory (Hawai'i) using the standardized approach presented in this work (nighttime measurements).

Same as Fig. 16, but for the tropospheric ozone DIAL system located at the JPL Table Mountain Facility (California) (nighttime measurements).

After optimal combination of all three DIAL pairs, the ozone number density
standard uncertainty results mainly from three components, namely, Rayleigh
extinction cross section differential (dark blue curve) at the bottom of the
profile, ozone absorption cross section differential (dark green curve) in
the middle of the profile, and detection noise (red curve) at the top of the
profile. For the derived ozone mixing ratio (right plot), the uncertainty
component associated with the a priori use of ancillary air pressure (light
blue curve) becomes abruptly important above 30 km as a result of the
transition between the a priori use of radiosonde measurement (

Figure 17 shows the full ozone uncertainty budget for a 2 h
measurement obtained on 18 November 2009 by the tropospheric ozone DIAL
located at JPL Table Mountain Facility (TMF), California. Once again, the
ozone number density uncertainty budget is on the left (in %), the ozone
mixing ratio uncertainty budget is on the right (in ppmv). The TMF lidar
samples air mostly above the boundary layer so the components associated
with absorption by SO

The present article is the second of three companion papers on the recommendations made to the NDACC lidar community for the standardization of vertical resolution and uncertainty in their lidar data processing algorithms. Here the focus was on the ozone DIAL uncertainty budget. The definition of uncertainty recommended to be used for all NDACC lidar measurements is combined standard uncertainty, as defined by the BIPM (JCGM 200, 2012; JCGM 100, 2008). In the approach proposed here all the individual, independent uncertainty components are propagated in parallel through the data processing chain. It is only after the final signal transformation is applied (i.e., leading to the actual values of ozone number density and mixing ratio) that the individual uncertainty components are combined together to form the combined standard uncertainty, the primary and mandatory variable of the newly proposed NDACC-standardized ozone DIAL uncertainty budget.

The individual uncertainty components identified by the ISSI team comprise
the random noise associated with signal detection, uncertainty due to
saturation correction, background noise extraction, the absorption
cross sections of O

The introduction and step-by-step propagation of each single uncertainty component through the ozone data processing chain was thoroughly reviewed by the ISSI team and detailed here. The validity of the approach and correctness of the recommended expressions were quantitatively verified using simulated lidar signals and Monte Carlo experiments. The details of these experiments are given in the ISSI team report (Leblanc et al., 2016a). The objective was not to estimate the magnitude of each uncertainty contribution, but to verify that the propagation expressions provided in Sect. 4 were theoretically correct and properly implemented.

Every source of uncertainty should be reported in the NDACC-archived metadata file. Providing quantitative information on the ancillary datasets used is also highly recommended. Whether or not using the NDACC-standardized uncertainty budget approach, the best estimate of the ozone combined standard uncertainty must be reported in the NDACC-archived data files. In addition, individual standard uncertainty components that contribute to the ozone combined uncertainty should be reported in the NDACC-archived data files whenever possible.

Typically, NDACC ozone lidar profiles are given as a function of altitude and for an averaging time period ranging between a few minutes and several hours. For each reported uncertainty component, the systematic or random nature of the underlying effects associated with this component should be reported in both altitude and time dimensions. When using multiple NDACC-archived ozone or temperature lidar profiles, for example, to produce a climatology, each reported uncertainty component must first be computed separately based on the expected systematic or random behavior of the process associated with it, and only after that, be combined.

Because each lidar instrument is unique, not all sources of uncertainty have been identified or reviewed in this paper. For unidentified sources, as well as uncertainty owing to analog detection, overlap correction, and particulate backscatter and extinction corrections mentioned earlier but not treated, the NDACC lidar investigators should use the same generic approach as that used for the sources identified and treated here, and should add those components to the uncertainty budget following the same definitions, methodologies, and propagation principles. It is advised that dedicated working groups be formed in the near future to address the standardization of the treatment of these uncertainty components.

The recommendations and approaches proposed by the ISSI team for ozone and temperature in the present paper and the other two companion papers can be largely extended to water vapor and aerosol.

The data used to produce the figures shown here are not publicly available. However, they can be obtained by contacting the first author at thierry.leblanc@jpl.nasa.gov.

This work was initiated in response to the 2010 call for international teams of experts in Earth and Space Science by the International Space Science Institute (ISSI) in Bern, Switzerland. It could not have been performed without the travel and logistical support of ISSI. Part of the work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under agreements with the National Aeronautics and Space Administration. Part of this work was carried out in support of the European Space Agency VALID project. Robert J. Sica would like to acknowledge the support of the Canadian National Sciences and Engineering Research Council for support of the University of Western Ontario lidar work. Edited by: H. Maring Reviewed by: two anonymous referees