Proposed standardized definitions for vertical resolution and uncertainty in the NDACC lidar ozone and temperature algorithms – Part 1: Vertical resolution

Abstract

Abstract.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 z = δz × H FWHM , where δz is the lidar's sampling resolution and H FWHM is the full width at half maximum (FWHM) of the response, measured in sampling intervals.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 f C , defined as the frequency at which the gain equals 0.5, is computed. Vertical resolution is then defined by z = δz/(2f C ). Unlike common practice in the field of spectral analysis, a factor 2f C instead of f C is used here to yield vertical resolution values nearly equal to the values obtained with the impulse response definition using the same filter coefficients. When using either of the proposed definitions, unsmoothed signals yield the best possible vertical resolution z = δz (one sampling bin).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.

Effective resolution concepts for lidar observations

“…Finally it is worth mentioning that, within the lidar community, other approaches rooted in the numerical derivative problem have been proved to be effective, and there are other methods providing alternative and reasonable ERes definitions (Leblanc et al, 2012;Pornsawad et al, 2012;Trickl, 2010;Shcherbakov, 2007;Eisele and Trickl, 2005;VDIVerein Deutscher Ingenieure, 1999). For this reason a more exhaustive comparison with other approaches to the evaluation of the ERes will be likely done in future in the frame of EARLINET activities.…”

“…13 allow linking both the approaches provided for the ERes estimation to the cut-off frequency. The cut-off frequency associated with the frequency response of a smoothing filter can be used for estimating the effective resolution (Godin, 1987;Godin et al, 1999;Masci, 1999;Beyerle and McDermid, 1999;Leblanc et al, 2012). For this reason, considering a generic low-pass filter L(p), the following equation is valid:…”

“…Incidentally, it should be noted that since a smoothing filter damps effectively only high frequencies and since it is common to deal with white noise, the low-frequency portion of the noise is still present in the smoothed signal, for example in the form of long-wave ripples (Gans, 1992). Moreover, a link is established between the ERes estimated with each of those two approaches and the corresponding cut-off frequency definition, as discussed in Sect 3.4 (Godin et al, 1999;Masci, 1999;Beyerle and McDermid, 1999;Leblanc et al, 2012).…”

“…Moreover, this procedure, also known as the "step function" method, has been already tested in the frame of the first EAR-LINET algorithm intercomparison . There are alternative approaches to estimate the ERes such as the analysis of the smoothing effect on the full width at half maximum (FWHM) of a finite impulse (Leblanc et al, 2012), the response to a Heaviside step function (VDIVerein Deutscher Ingenieure, 1999; Eisele and Trickl, 2005;Vogelmann andTrickl, 2008, Trickl, 2010), or the application of the regularization theory to lidar data processing (Pornsawad et al, 2012;Shcherbakov, 2007). However, for SG filters, the "step function" method apparently shows some ambiguous results as can be seen in the examples reported in Fig.…”

“…The primary aim of this paper is to study the problem of the effective resolution (ERes) estimation for lidar products using a different approach with respect to those already presented in the literature (Godin at al., 1999;Beyerle and McDermid, 1999;Shcherbakov, 2007;Trickl, 2010;Leblanc et al, 2012;Pornsawad et al, 2012). A parametric study of several smoothing filters is performed, and the results provide recommendations on the optimization of lidar signal processing.…”