A standardized approach for the definition and reporting of vertical resolution of the ozone and temperature lidar profiles contributing to the Network for the Detection for Atmospheric Composition Change (NDACC) database is proposed. Two standardized definitions homogeneously and unequivocally describing the impact of vertical filtering are recommended.

The first proposed definition is based on the width of the response to a finite-impulse-type perturbation. The response is computed by convolving the
filter coefficients with an impulse function, namely, a Kronecker delta
function for smoothing filters, and a Heaviside step function for derivative
filters. Once the response has been computed, the proposed standardized
definition of vertical resolution is given by

The second proposed definition relates to digital filtering theory. After
applying a Laplace transform to a set of filter coefficients, the filter's
gain characterizing the effect of the filter on the signal in the
frequency domain is computed, from which the cut-off frequency

Numerical tools were developed to support the implementation of these definitions across all NDACC lidar groups. The tools consist of ready-to-use “plug-in” routines written in several programming languages that can be inserted into any lidar data processing software and called each time a filtering operation occurs in the data processing chain.

When data processing implies multiple smoothing operations, the filtering information is analytically propagated through the multiple calls to the routines in order for the standardized values of vertical resolution to remain theoretically and numerically exact at the very end of data processing.

As part of the Network for the Detection of Atmospheric Composition Change
(NDACC, website:

To address these and other lidar retrieval issues, a group of lidar experts
formed an International Space Science Institute team of experts in 2011
(

The present article is the first of three companion papers that provide a comprehensive description of the recommendations made by the ISSI team to the NDACC lidar community for the standardization of vertical resolution and uncertainty. The present article (Part 1) is exclusively dedicated to the description of the proposed standardized vertical resolution. A second paper (Part 2) (Leblanc et al., 2016b) reviews the proposed standardized definitions and approaches for the ozone differential absorption lidars' uncertainty budget. The last paper (Part 3) (Leblanc et al., 2016c) reviews the proposed standardized definitions and approaches for the NDACC temperature lidars' uncertainty budget. Details that appear beyond the scope of the present three companion papers may be found in the ISSI team report (Leblanc et al., 2016a).

Though the ISSI team focus has been on the retrieval of ozone using the differential absorption technique (Mégie et al., 1977), and the retrieval of temperature using the density integration technique (Hauchecorne and Chanin, 1980; Arshinov et al., 1983), most recommendations made in the present and two companion papers can be followed for the retrieval of other NDACC lidar species such as water vapor (Raman and differential absorption techniques), temperature (rotational Raman technique), and aerosol backscatter ratio. One exception is when using an optimal estimation method (OEM) for the retrieval of temperature or water vapor mixing ratio as recently proposed by Sica and Haefele (2015, 2016), for which vertical resolution is implicitly determined from the full width at half maximum of the OEM's averaging kernels.

Vertical resolution, as provided in the lidar data files, is an indicator of the amount of vertical filtering applied to the lidar signals or to the measured species profiles. This filtering is applied in order to reduce high-frequency noise typically produced at the signal detection level. A higher vertical resolution means that the instruments are able to detect features of small vertical extent, while a lower vertical resolution implies a reduced ability to detect features of small vertical scale. Typically, vertical resolution is provided in a unit of vertical length (e.g., meter). Because the lidar signal-to-noise ratio strongly varies with altitude, the amount of filtering typically applied also varies with altitude, with more filtering applied at higher altitude ranges unless specific geophysical processes are investigated (e.g., gravity waves, stratospheric intrusions).

Here the word “filtering” is preferred to the word “smoothing” because it is more general and
applies to both smoothing and differentiation processes, the former process
being relevant to both temperature and ozone lidar retrievals, and the
latter process being relevant to the ozone differential absorption
technique. To optimize the useful range of lidar measurements, most lidar
signals or profiles are digitally filtered at some point in the retrieval
process. Over the years, NDACC lidar investigators have provided temperature
and ozone profiles using a wide range of vertical resolution schemes and
values, where the definition of vertical resolution appears to differ
significantly. The objective of the present work is not to recommend a
specific vertical resolution scheme, but instead to ensure that the
definition used by the data providers to describe their scheme is reported
and interpreted consistently across the entire network. The approaches and
recommendations in this article were designed so that they can be
implemented consistently by all NDACC lidar investigators and beyond (e.g.,
the Tropospheric Ozone Lidar Network, TOLNet, website:

Section 2 summarizes the basics of digital signal filtering, and provides a few examples of how vertical resolution can be expressed in terms of impulse response and digital filter cut-off frequency. Section 3 reviews a number of vertical resolution definitions used by the NDACC ozone and temperature lidar community. The results from Sects. 2 and 3 are used in Sect. 4 to recommend and detail two practical, well-known definitions of vertical resolution that can be easily linked to the underlying filtering processes. The numerical values of vertical resolution computed using these two definitions are compared for several types of digital filters. For the sake of completeness, a Supplement to the present paper provides additional characteristics of several commonly used smoothing and derivative filters.

Numerical tools were developed by the ISSI team to facilitate the implementation of the proposed standardized definitions. The tools consist of subroutines written in four scientific programming languages (IDL, MATLAB, FORTRAN, and Python) that can be inserted in the lidar investigators' data processing software in order to compute the numerical values of the standardized vertical resolution. The plug-in routines are available upon request from the corresponding author.

In this section we briefly review the mathematical background that allows us
to link vertical resolution to the lidar signal (or profile) filtering
process. Signal filtering for lidar data processing consists of either
smoothing, differentiating or smoothing, and differentiating at the same
time. To describe the filtering process, a signal

The signal filtering process at an altitude

The number of filter coefficients and the values of these coefficients
determine the actual effect of the filter on the signal. Three critical
aspects of the effect of the filter on the signal are (1) the amount of noise
reduction due to filtering, (2) the nature and degree of symmetry/asymmetry
of the coefficients around the central value which determines whether the
filter's function is to smooth, sum, differentiate, or interpolate, and (3) whether the magnitude of specific noise frequencies are being amplified or
reduced after filtering. In the particular case of an unfiltered signal
comprised of independent samples, and assuming that the variance of the noise
for the unfiltered signal is constant through the filtering interval
considered (

Example of the differing impact of two smoothing filters of
identical number of terms (

In the real world, we typically do not know the exact nature or behavior of the measured signal. Consider the example in Fig. 1: if the definition used to report vertical resolution in the data files were based on the number of points used by the filter, we would not be able to attribute the differences observed between the blue and red curves to a difference in the filtering procedure. We therefore need to find some analytical way to characterize a specific filter if we want to understand its exact effect on the signal and properly interpret features observed on the smoothed signal. We will see thereafter that it is indeed possible to determine the resolution of the filter by either quantifying the response of a controlled impulse in the physical domain, or by using a frequency approach and studying the frequency response of the filter.

The impact of a specific filter on the signal can be characterized by
computing the unit impulse response in the physical domain (usually called
the time domain in time series analysis). This can be done by using a
well-known, controlled input signal, e.g., an impulse, and by studying its
response after being convolved by the filter coefficients. Considering a
finite impulse response is equivalent to considering the output signal

Impulse response (left) and gain (right) for a digital filter
equivalent to fitting an unsmoothed signal with a polynomial of degree 1 or
2 using the least-squares method over an interval comprising

As in many signal processing applications, the frequency approach applied to
lidar signal filtering or lidar-retrieved profile filtering is a convenient
mathematical framework. It is a more abstract, but very powerful tool
allowing us to understand many hidden features of the smoothing and
differentiation processes. A succinct, yet clear discussion of the required
mathematical background is provided by Hamming (1989). Here, we will provide
a brief review of this background relevant to our applications.

Aliasing: Any signal consisting of a finite number of equally spaced samples in
the physical domain is an aliased representation of a sine and cosine
function of frequency

In the case of lidar, the signal (or the ozone or temperature profile) is a
function of altitude range. The discretized independent variable is the
vertical sampling bin

Eigenfunctions and eigenvalues of a linear system: Any vector

Furthermore, any nonzero and non-unity matrix

Invariance under translation: The property of invariance under translation for the sine and cosine
functions implies a direct relation between the signal expressed in its
complex form and the eigenvalue

Using the above mathematical background, the filtered signal

The eigenvalue

We can express the transfer function more conveniently as a function of the
frequency

The maximum value

For a typical smoothing filter, the coefficients have even symmetry, i.e.,

For a derivative filter, the

With the complex notation of Eq. (8), the ideal vertical derivative of the signal can be written as follows:

The gain of the filter, then takes the following form:

Referring back to Eq. (1), the gain provides a quantitative measure
of the actual smoothing impact of the filter on the signal at a particular
location

Least-squares fitting is a well-established numerical technique used for
many applications such as signal smoothing, differentiation, and interpolation. The relation between the number and values of the filter coefficients
and the type of polynomial used to fit the signal can be found in many textbooks and publications (e.g., Birge and Weinberg, 1947; Savitsky and Golay,
1964; Steinier et al., 1972). In this paragraph we show that least-squares fitting with a
straight line and boxcar averaging are the same filter. We start with the simple case of
fitting five points with a straight line. We therefore look for the
minimization of the following function:

Now switching to the frequency domain and using Eq. (14), the
transfer function

If we were to consider an ideal low-pass filter with an infinite number of
terms, the theoretical transfer function would have values between 0 and 1,
representing the perfect gain of the filter (no ripples). The so-called
transition region corresponds to the region where we want the transfer function to drop from
a value of 1 at lower frequencies to a value of 0 at higher frequencies. The
width of the transition region is the bandwidth. We can define the cut-off frequency of a low-pass
filter as the frequency at which the transfer function equals 0.5. For most
low-pass filters this is at the center of the transition region. To design a
low-pass filter with the desired cut-off frequency

Impulse response and gain of low-pass filters using

The gain curves show that the transition region is narrower than that observed for the boxcar average filters, but the Gibbs ripples appear on both sides of the transition region. Just like for the modified least-squares fitting, we can reduce the magnitude of the Gibbs ripples by modifying the filter coefficients, specifically by applying additional weights to the filter coefficients, a process called “windowing”. Several examples of smoothing filters using Lanczos, von Hann, Hamming, Blackman, and Kaiser windows are provided for reference in the Supplement.

The simplest approximation of the derivative of a signal

Transfer function (TF) and gain of the central difference digital
filter. The gain (blue curve) is the transfer function (red curve) normalized
by 2

The filtering schemes or methods of several NDACC lidar investigators have
been reviewed and compared in previous works, e.g., Beyerle and McDermid (1999) and Godin et al. (1999). These studies concluded that vertical
resolution was not consistently reported between the various investigators.
Here we briefly review the filtering schemes or methods used by various
NDACC lidar investigators, and how vertical resolution is reported in their
data files, as of 2011. This review provided critical input to the ISSI team
to determine which definitions of vertical resolution are appropriate for use
in a standardized way across the entire network (see Sect. 4).

Observatoire de Haute-Provence (OHP, France), stratospheric ozone differential absorption lidar: A second-degree polynomial derivative filter (Savitsky–Golay derivative filter) is used (Godin-Beekmann et al., 2003). Vertical resolution is reported following a definition based on the cut-off frequency of the digital filter.

Table Mountain (California) and Mauna Loa (Hawaii) stratospheric ozone and temperature lidars operated by the Jet Propulsion Laboratory: Filtering is done by applying a fourth-degree polynomial least-squares fit (Savitsky–Golay derivative filter) to the logarithm of the signals for ozone retrieval. For the temperature profiles, a Kaiser filter is applied to the logarithm of the relative density profile. In both ozone and temperature cases, the cutoff frequency of the filter, reversed to the physical domain, is reported as vertical resolution (Leblanc et al., 2012).

NASA GSFC mobile ozone DIAL STROZ instrument (United States): For ozone, a least-squares fourth-degree polynomial fit derivative filter (Savitsky–Golay derivative filter) is used. The definition of vertical resolution in the NDACC-archived data files is based on the impulse response of a delta function, by measuring the FWHM of the filter's response. For the temperature retrieval (Gross et al., 1997), the profiles are smoothed using a low-pass filter (Kaiser and Reed, 1977), and a simple ad hoc step function is used to define the values of the vertical resolution.

Lauder (New Zealand) ozone lidar operated by RIVM (Netherlands): The definition of vertical resolution is based on the width of the fitting window used for the ozone derivation (Swart et al., 1994).

OHP and Réunion Island (France) tropospheric ozone DIAL: A second-degree polynomial least-squares fit (Savitsky–Golay derivative filter) is used to filter the ozone measurements. The vertical resolution is reported as the cut-off frequency of the corresponding digital filter.

Réunion Island (France) temperature lidar: A Hamming filter is applied to the temperature profile. The width of the window used is reported as the vertical resolution.

University of Western Ontario (Canada) Purple Crow Lidar: For climatology studies, the temperature algorithm applies a combination of three-point and five-point boxcar average filters or a Kaiser filter on the temperature profiles (e.g., Argall and Sica, 2007). Similar filters are used in space or time for spectral analysis of atmospheric waves (e.g., Sica and Russell, 1999). Filter parameters are reported in the data files that are locally produced and distributed to the scientific user community. Previously, files were distributed to users with the type of filter and full bandwidth of the filter. The variance reduction of the filter is folded into the random uncertainties provided. The product of the data spacing and the filter bandwidth gives the full influence of the filter at each point. With the development of a temperature retrieval algorithm based on an optimal estimation method, vertical resolution of the temperature profile is now available as a function of altitude (Sica and Haefele, 2015).

Tsukuba (Japan) ozone DIAL and temperature lidar: The algorithm uses second- and fourth-degree polynomial least-squares fits (Savitsky–Golay derivative filter). The vertical resolution is calculated from a simulation model that determines the FWHM of the impulse response to an ozone delta function. The FWHM is then mapped as a function of altitude. For temperature, a von Hann (or Hanning) window is used on the logarithm of the signal (B. Tatarov, personal communication, 2010).

Garmisch-Partenkirchen (Germany) tropospheric ozone DIAL operated by IFU: The algorithm initially used linear and third-degree polynomial fits (Kempfer et al., 1994), and then since 1996 a combination of a linear fit and a Blackman-type window (Eisele and Trickl, 2005; Trickl, 2010). The latter filter has a reasonably high cut-off frequency and does not transmit as much noise as the derivative filters used earlier at IFU (Kempfer et al., 1994). To report vertical resolution in the data files, a Germany-based standard definition of vertical resolution is used, following the Verein Deutscher Ingenieure DIAL guideline (VDI, 1999). This definition is based on the impulse response to a Heaviside step function. The vertical resolution is given as the distance separating the positions of the 25 and 75 % in the rise of the response, which is approximately equivalent to the FWHM of the response to a delta function. In the case of the ozone DIAL, the vertical resolution of both the Blackman-type filter used and the combined least-squares derivative plus Blackman filter. A vertical resolution of 19.2 % of the filtering interval was determined. For small intervals the latter value may change; i.e., the least-squares fit for determining the derivative is executed over just a few data points. For comparison, an arithmetic average yields a vertical resolution of 50 % of the filtering interval.

Transfer function (gain) of several smoothing (left) and
derivative (right) filters, all with exactly the same number of coefficients,

Finding transition regions at different frequencies means that the smoothing effect of the filters on the signal is different, even though the number of coefficients is the same. A vertical resolution definition based on the number of coefficients is therefore not reliable. Instead we need to choose a standardized definition based on objective parameters that are directly related to the effect a filter has on the signal. Two such definitions are proposed thereafter, definitions that are similar or closely related to the two remaining definitions identified in the present section.

The two definitions proposed here were chosen because they provide a straightforward characterization of the underlying smoothing effect of filters (see Sect. 2), and they appear to have already been used by a large number of NDACC investigators (see Sect. 3). The first definition is based on the width of the impulse response of the filter. The second definition is based on the cut-off frequency of the filter. Further justification for the choice of either definition is provided at the end of the present section.

Schematics summarizing the procedure that should be followed to compute the
standardized vertical resolution with a definition based on the impulse
response FWHM

The full width at half maximum (FWHM) of an impulse response, as introduced
in Sect. 2, is computed by measuring the distance (in bins)
between the two points at which the response magnitude falls below half of
its maximum amplitude. The NDACC lidar standardized definition of vertical
resolution proposed here is computed from the response

Define and/or identify the

Construct an impulse function of finite length

This function equals 1 at the central point (

This function equals 0 at all locations below the central point
(

Convolve the filter coefficients with the impulse function in order to
obtain the impulse response

Estimate the full width at half maximum (FWHM) of the impulse response

For a successful identification of the FWHM, the impulse response should have only two points where its value falls below half of its maximum amplitude, which is normally the case for all smoothing and derivative filters used within their prescribed domain of validity (see examples in Sect. 2 and in the Supplement). In the event that more than two points exist, the two points farthest from the central bin should be chosen in order to yield the most conservative estimate of vertical resolution.

Compute the standardized vertical definition

Figure 6 summarizes the estimation procedure. The unsmoothed signal
yields a FWHM of one bin. This result is derived by considering null
coefficients everywhere except at the central point (

Schematics summarizing the procedure that should be followed to compute the standardized vertical resolution with a definition based on impulse response when the signal or profile is filtered multiple times.

When several filters are applied successively to the signal, the response of the filter must be computed each time a filtering operation occurs, and vertical resolution needs to be computed only after the last filtering occurrence. The process can be summarized as follows: a first impulse response is computed with the first filtering operation. If no further filtering occurs, the impulse response is used to determine the FWHM and vertical resolution. If a second filtering operation occurs, the impulse response is used as input signal, and a second response is computed from the convolution of this input signal with the coefficients of the second filter. If no further filtering occurs, the second response is used to determine the FWHM and vertical resolution. If a third filtering operation occurs, the response output from the second convolution is used as input signal of the third convolution, and so on until no more filtering is applied to the signal. Vertical resolution is always computed from the final output response, i.e., after the final filtering operation. Figure 7 summarizes the procedure.

The cut-off frequency of digital filters is defined as the frequency at
which the value of the filter's gain is 0.5, typically located at the center
of the transition region between the passband and the stopband (see
Sect. 2). The NDACC lidar standardized definition proposed here
is computed from the cut-off frequency

Schematics summarizing the procedure that should be followed to compute the
standardized vertical resolution with a definition based on cut-off frequency

Define and/or identify the

Apply the Laplace transform to the coefficients to determine the filter's
transfer function and gain. For non-derivative smoothing filters, the
coefficients have even symmetry, i.e.,

For derivative filters, the coefficients have odd symmetry, i.e.,

For a successful cut-off frequency estimation process, the gain must be
computed with normalized coefficients

Estimate the cut-off frequency, i.e., the frequency

For a successful identification, the gain should have only one crossing with the 0.5 line, typical for the smoothing and derivative filters used within their prescribed domain of validity for lidar retrieval systems. In the event that several crossings exist, the frequency closest to zero should be chosen to ensure that the most conservative estimate of vertical resolution is retained.

Calculate the cut-off length

Compute the standardized vertical definition

Schematics summarizing the procedure that should be followed to compute the standardized vertical resolution with a definition based on cut-off frequency when the signal or profile is filtered multiple times.

In Sects. 4.1 and 4.2, we showed that, when using the proposed
definitions based on impulse response and cut-off frequency, the
standardized vertical resolution of an unsmoothed lidar signal (or profile)
is equal to the lidar sampling resolution. However, this equality between the
two definitions is not perfect for all filters. Here, we show that for most
filters, there is a well-defined proportionality relation between the two
definitions, but we also show that the proportionality factor depends on the
type of filter used. In the rest of this section, for convenience, we will
work with vertical resolutions normalized by the sampling resolution (unit:
bins). The results are therefore shown as cut-off width

Figure 10 shows, for the smoothing filters introduced in Sect. 2 and in the Supplement, the correspondence between the standardized vertical resolutions (in bins) computed using the cut-off frequency and the impulse response, for full widths comprised between 3 and 25 points. The black solid circle at coordinate (1,1) indicates the vertical resolution for the unsmoothed signal (or profile). The gray horizontal and vertical dash-dotted lines indicate the highest possible vertical resolutions for the impulse-response-based and cut-off frequency-based definitions respectively. The gray dotted straight lines indicate the result of the linear regression fits between the two definitions, and the numbers at their extremities are the values of the slope for three of the four types of filters used. There is no proportionality between the two definitions for the low-pass filters (diamonds) because the cut-off frequency is prescribed for this type of filter. Note that the factors of 1.2 and 1.39 do not correspond to the ratio of 1.0 that is assumed for the unsmoothed signal. Very similar conclusions can be drawn for the derivative filters, as demonstrated by Fig. 11 (which is similar to Fig. 10 but for the derivative filters introduced in Sect. 2 and in the Supplement).

Figure 12 is similar to Fig. 10, but this time shows the period after the filters were convolved with the windows introduced in the Supplement. The windows change the proportionality constant between the two definitions, but this constant appears to be approximately the same for a given window, specifically around 1.04 for Lanczos, 1.0 for von Hann, 0.92 for Blackman, and 1.0 for Kaiser (50 dB). Table 1 summarizes the proportionality constants for all filters and all windows introduced in Sect. 2 and in the Supplement.

Comparison between the cut-off frequency-based and the impulse-response-based standardized vertical resolutions for several smoothing filters introduced in Sect. 2 and in the Supplement. The numbers at the end of the dotted straight lines indicate the proportionality constant (slope) between the two definitions for three of the four types of filters used. There is no such proportionality for the low-pass filter (prescribed cut-off frequency).

Same as Fig. 10, but for derivative filters (Sect. 2 and the Supplement).

Proportionality factor between the impulse-response-based and the cut-off frequency-based definitions of vertical resolution for the filters and windows introduced in Sect. 2 and in the Supplement.

Same as Fig. 10, but for the filters being convolved with the four windows introduced in the Supplement.

Figure 13 shows, for the filters introduced in Sect. 2 and in the Supplement, the correspondence between the two proposed standardized vertical resolutions (in bins) and the number of filter coefficients used (full widths comprised between 3 and 25 points). The dashed gray line represents unity slope (i.e., one bin for one filter coefficient), and the numbers at the end of the red and blue dotted straight lines indicate the slope of the linear fit applied to the paired points for each definition. As expected for a boxcar average, the impulse-response-based definition yields a vertical resolution (in bins) that is equal to the number of terms used (see Fig. 2). This is a particular case for which reporting vertical resolution using the number of filter terms yields a result identical to the impulse-response-based standardized definition. Note that for low-pass filters with a prescribed cut-off frequency, the vertical resolution does not depend at all on the number of filter terms used (right-hand plot).

Correspondence between cut-off frequency-based (red) and impulse-response-based (blue) vertical resolution (in bins), and the number of filter coefficients used (full width), for three filters introduced in Sect. 2. The dashed gray line represents unity slope (i.e., one bin for one point), and the numbers at the end of the red and blue dotted straight lines indicate the slope of the linear fit applied to the paired points for each definition.

Same as Fig. 13, but this time for the period after convolution by a von Hann window.

Figure 14 is similar to Fig. 13, but this time shows the period after convolution by a von Hann window. Except for the low-pass filter, there is a factor of approximately 2 between the number of terms used by the filter and the vertical resolution for both definitions. Figure 15 is similar to Fig. 13, but for three selected derivative filters.

Proportionality factor between the number of filter coefficients (full width) and vertical resolution based on cut-off frequency (in bins) for the filters and windows introduced in Sect. 2 and in the Supplement.

Proportionality factor between the number of filter coefficients (full width) and vertical resolution based on impulse response FWHM (in bins) for the filters and windows introduced in Sect. 2 and in the Supplement.

Same as Fig. 13 but for selected derivative filters and windows from Sect. 2 and the Supplement.

The factors between the vertical resolutions (in bins) and the number of filter coefficients are compiled in Tables 2 and 3 for the cut-off frequency-based and the impulse-response-based definitions, respectively.

In this section, it was shown that each recommended definition of vertical resolution yields its own numerical values; i.e., for the same set of filter coefficients, the reported standardized vertical resolution will likely have two different numerical values depending on the definition used. Unfortunately, there is no unique proportionality factor between the two definitions that could be used for all digital filters in order to obtain a unified homogenous definition yielding identical values. However, after reviewing this homogeneity problem, the ISSI team concluded that both definitions should still be recommended because the computed values remain close, specifically within 10 % if using windows and within 20 % if not using windows, and because each definition is indeed useful for specific applications. For example, the cut-off frequency-based definition is particularly useful for studies of gravity waves from lidar temperature measurements because it can provide, through the transfer function, spectral information that can help interpret quantitative findings on the amplitude and wavelength of lidar-observed waves. This type of information is not available when using the impulse-response-based definition. On the other hand, the impulse-response-based definition is widely used in atmospheric remote sensing, and it provides information in the physical domain similar to that provided through the averaging kernels of optimal estimation methods (e.g., microwave radiometer measurement of ozone or temperature).

When archiving the ozone or temperature profiles, reporting values of
vertical resolution using a standardized definition such as

If the data provider chooses to report standardized vertical resolution
information based on the impulse response definition, the complete vertical
resolution information should include the following:

a vector

a two-dimensional array of size

a vector

metadata information clearly describing the nature of the reported vectors and arrays.

a vector

a two-dimensional array of size

a vector

metadata information clearly describing the nature of the reported vertical resolution vector, frequency vector, and two-dimensional gain array.

a vector

a two-dimensional array of size

a vector

a vector

a two-dimensional array of size

a vector

metadata information describing clearly the nature of all reported vectors and arrays.

Numerical tools were developed and provided to the NDACC principal investigators (PIs) in order to
facilitate the implementation of the network-wide use of the proposed
standardized definitions. These tools consist of easy-to-use plug-in
routines written in IDL, MATLAB and FORTRAN, which convert a set of filter
coefficients into the needed standardized values of vertical resolution
following one or the other proposed definitions. The tools are written in
such a way that they can be called in a lidar data processing algorithm each
time a smoothing and/or differentiating operation occurs. The routines can
handle multiple smoothing and/or differentiating operations applied
successively throughout the lidar data processing chain, as described in
Sects. 4.1 and 4.2. The routines are available on the
NDACC lidar working group website (

The routine “NDACC_ResolIR” computes vertical resolution values with a definition based on the FWHM of the filter's impulse response. When the routine is called for the first time in the data processing chain, the sampling resolution and the coefficients of the filter are the only input parameters of the routine. The routine convolves the coefficients with an impulse (delta function for smoothing filters and Heaviside function for derivative filters) to obtain the filter's impulse response, and then identifies the full width at half maximum (FWHM) of this response. The response and the value of vertical resolution are the output parameters of the routine. The product of the response full width by the sampling resolution is performed inside the routine. When a second call to the routine occurs (second smoothing occurrence), the vertical resolution output from the first call is no longer used. Instead, the response output from the first call is used as input parameter for the second call, together with the sampling resolution and the coefficients of the second filter. The input response is convoluted with the coefficients of the second filter to obtain a second response. The routine identifies the FWHM of this new response. Once again the vertical resolution is computed inside the routine by calculating the product of the new FWHM and the sampling resolution. The new response and the new vertical resolution are the output parameters of the routine after the second call. The procedure is repeated as many times as needed, i.e., as many times as a smoothing or differentiation operation occurs.

The routine NDACC_ResolDF computes vertical resolution values with a definition based on the cut-off frequency of a digital filter. When the routine is called for the first time in the data processing chain, the sampling resolution and the coefficients of the filter are the only input parameters of the routine. The routine applies a Laplace transform to the coefficients to obtain the filter's gain, and then identifies the cut-off frequency. The inverse of the doubled cut-off frequency is multiplied by the sampling resolution to obtain the vertical resolution. The gain and the vertical resolution are the output parameters of the routine. When a second call to the routine occurs (i.e., a second smoothing operation occurs), the cut-off width output from the first call is not used anymore. Instead, the gain output from the first call is used as input parameter for the second call, together with the sampling resolution and the coefficients of the second filter. The product of the input gain and gain computed from the second filter is the new gain from which the routine identifies the cut-off frequency. A new vertical resolution is obtained by multiplying the inverse of the newly computed doubled cut-off frequency by the sampling resolution. The new gain and the new vertical resolution are the output parameters of the routine after the second call. The procedure is repeated as many times as needed, i.e., as many times as a smoothing or differentiation operation occurs.

The standardization tools became available in summer 2011. They were distributed to several members of the ISSI team for testing and validation. Using simulated lidar signals and a series of Monte Carlo experiments, their implementation was validated for several NDACC ozone and temperature lidar algorithms. Several examples of this validation are provided in the ISSI team report (Leblanc et al., 2016a). Ideally, an NDACC-wide implementation should follow. The implementation will not be considered complete until the vertical resolution outputs of all contributing data processing software have been quantified and validated following the same procedure as that described in Leblanc et al. (2016a).

In the present work, we recommended using one or two standardized definitions of vertical resolution that can unequivocally describe the impact of vertical filtering on the ozone and temperature lidar profiles. The coefficients of the filter used in the vertical smoothing operation are chosen by the lidar investigator, and therefore constitute the key information for the derivation of vertical resolution using a standardized definition.

The first standardized definition recommended for use in the NDACC ozone
and temperature lidar algorithms is based on the width of the response to a
finite-impulse-type perturbation. The response is computed by convolving the
filter coefficients with an impulse function, namely, a Kronecker delta
function for smoothing filters and a Heaviside step function for derivative
filters. Once the response has been computed, the standardized definition of
vertical resolution proposed by the ISSI team is given by

The other recommended definition relates to digital filtering theory. After
applying a Laplace transform to a set of filter coefficients, we can derive
the filter transfer function and gain, which characterize the effect of the
filter on the signal in the frequency domain. A cut-off frequency value

The ISSI team developed numerical tools to support the implementation of
these definitions across the NDACC lidar groups. The tools consist of
ready-to-use “plug-in” routines written in IDL, FORTRAN, MATLAB, C

Over the years, NDACC lidar PIs have been providing temperature and ozone profiles using a wide range of vertical resolution schemes and values, and these values were reported using different definitions. Here we did not recommend using a specific vertical resolution scheme, but instead we recommended using standardized definitions of vertical resolution that can be used consistently across lidar observation networks. The proposed approach was designed so that the standardized definitions can be implemented easily and consistently by all lidar investigators (e.g., NDACC, TOLNet). Though the recommendations apply to the retrieval of ozone by the differential absorption technique and temperature by the density integration technique, they can likewise apply to the retrieval of other NDACC species such as water vapor (Raman and differential absorption techniques), temperature (rotational Raman technique), and aerosol backscatter ratio with the exception of the optimal estimation method (OEM) for the retrievals of temperature and water vapor recently proposed by Sica and Haefele (2015, 2016), for which vertical resolution is determined from the FWHM of the OEM's averaging kernels.

In our two companion papers (Leblanc et al., 2016b, c), the ISSI team provided recommendations on the standardized treatment of uncertainty for the NDACC ozone and temperature lidars (Part 2 and Part 3 respectively). It is anticipated that the widespread use of the standardized definitions and approaches proposed in our three companion papers will significantly improve the interpretation of atmospheric measurements, whether these measurements are made for validation purposes (e.g., comparison of correlative measurements) or scientific purposes (e.g., studies of vertical structures observed in the measured profiles).

The data 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 report 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 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. The team would also like to acknowledge J. Bandoro for his help in the design of the MATLAB filtering tools. Edited by: H. Maring Reviewed by: M. J. Newchurch and one anonymous referee