In this paper, we discuss the method for validation of random uncertainties in the remote sensing measurements based on evaluation of the structure function, i.e., root-mean-square differences as a function of increasing spatiotemporal separation of the measurements. The limit at the zero mismatch provides the experimental estimate of random noise in the data. At the same time, this method allows probing of the natural variability of the measured parameter. As an illustration, we applied this method to the clear-sky total ozone measurements by the TROPOspheric Monitoring Instrument (TROPOMI) on board the Sentinel-5P satellite.

We found that the random uncertainties reported by the TROPOMI inversion algorithm, which are in the range 1–2 DU, agree well with the experimental uncertainty estimates by the structure function.

Our analysis of the structure function has shown the expected results on total ozone variability: it is significantly smaller in the tropics compared to mid-latitudes. At mid-latitudes, ozone variability is much larger in winter than in summer. The ozone structure function is anisotropic (being larger in the latitudinal direction) at horizontal scales larger than 10–20 km. The structure function rapidly grows with the separation distance. At mid-latitudes in winter, the ozone values can differ by 5 % at separations 300–500 km.

The method discussed is a powerful tool in experimental estimates of the random noise in data and studies of natural variability, and it can be used in various applications.

The information about uncertainties of measurements is important in many data analyses: data averaging, comparison, data assimilation etc. The uncertainties are usually categorized into “random” and “systematic” (for more discussion, see von Clarmann et al., 2020).

For remote sensing measurements, the random component of uncertainty budget is estimated via propagation of measurement noise through the inversion algorithm (e.g., Rodgers, 2000). If the linear or linearized model is adequate, the Gaussian error propagation can be used for simplicity. In von Clarmann et al. (2020) the term “ex ante” is used for the uncertainty estimates by an inversion algorithm, as do we in our paper.

Ex ante uncertainty estimates might be incomplete: this might be due to incomplete or simplified models of the processes that describe the satellite measurements and/or unknown and unresolved atmospheric features. Another contributing factor might be the imperfect estimates of measurement uncertainties, as well as the uncertainties of external auxiliary data. Therefore, validation of theoretical (ex ante) uncertainty estimates is needed for remote-sensing measurements. For atmospheric measurements specifically, validation of random uncertainty estimates is not a trivial task because the measurements are performed in a continuously changing atmosphere.

This short paper is dedicated to a simple method, which allows simultaneous probing of small-scale variability on an atmospheric parameter and validation of random uncertainties in the measurements of this parameter.

The paper is organized as follows. Section 2 briefly describes the methodology of the analysis. In Sect. 3, we describe the TROPOspheric Monitoring Instrument (TROPOMI) total ozone data, which are used in our paper. In Sect. 4, we briefly explain the technical details of the computation of the structure function using TROPOMI data. The results and discussion are presented in Sect. 5. The summary (Sect. 6) concludes the paper.

In our work, we will exploit the concept of the structure function, which characterizes the degree of spatial dependence of a
random field

When using experimental (noisy) data for evaluation of variogram and structure
function, the difference of an atmospheric parameter in two locations is
defined not only by the natural variability of this atmospheric parameter,
but also by uncertainty of measurements. Therefore, with the spatiotemporal
separation

The schematic representation of the structure function estimated from noisy measurements.

This constitutes the principle of the proposed method: at very small separations, the estimation of the structure function will tend towards random error variance. This can be considered as an experimental random uncertainty estimate, ex post in the terminology of von Clarmann et al. (2020). The application of the structure function method requires many measurement points with different spatial and temporal separations, including very small separations, and these measurements should have nearly the same random uncertainties. For satellite measurements in limb-viewing geometry, such information is very limited. Nevertheless, several applications using this method have been published. Staten and Reichler (2009) applied this method to the validation of radio-occultation measurements by the Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC), which consists of identical instruments on board of six microsatellites. In their work, the authors evaluated two-dimensional structure function using the data from the beginning of COSMIC mission, when the satellites were in close orbits (and therefore measurements in close separations were found). An analogous method – evaluation of the one-dimensional structure function in polar regions (with transformation of temporal mismatch to spatial separation) – has been applied for validation of random uncertainty estimates of the MIPAS (Michelson Interferometer for Passive Atmospheric Sounding) and GOMOS (Global Ozone Monitoring by Occultation of Stars) instruments on board the Envisat satellite (Laeng et al., 2015; Sofieva et al., 2014). The one-dimensional structure function has been evaluated in Sofieva et al. (2008) using collocated temperature profiles by radiosondes at Sodankylä.

For satellite measurements in nadir-viewing geometry, the smallest separation is usually defined by the ground pixel size of an instrument, and the application of the structure function method looks very attractive: measurements with small spatiotemporal mismatch can be found in all locations and in all seasons. However, we are not aware of the application of the structure function method for validation of random uncertainty estimates from nadir-viewing satellite instruments. In our paper, we use total ozone measurements by TROPOMI on board Sentinel-5P, which has a very fine spatial resolution, for the illustration of the structure function method.

The TROPOMI satellite instrument on board the Copernicus Sentinel-5
Precursor (S5P) satellite was launched in October 2017 (

The data are available in near-real-time, offline and reprocessing streams. In our studies, the Level 2 offline data product of the total ozone column (TOC) is used. This product relies on the operational implementation of the GODFITv4 algorithm, used for producing total ozone climate data records from many nadir-viewing sensors (GOME (Global Ozone Monitoring Experiment), SCIAMACHY (SCanning Imaging Spectrometer for Atmospheric CHartographY), GOME-2, OMI (Ozone Monitoring Instrument), OMPS (Ozone Mapping and Profiler Suite)) with excellent performance (Garane et al., 2019; Lerot et al., 2014). Total ozone columns are derived using a non-linear minimization procedure of the differences between measured and modeled sun-normalized radiances in the ozone Huggins bands (fitting window: 325–335 nm).

The total ozone product includes an estimate for the random uncertainty associated with each observation. The latter is simply obtained by the propagation through the inversion solver of the radiance and irradiance statistical errors provided with the measurements (in Level 1 products). In addition to the instrumental noise, some pseudo-random errors (i.e., systematic errors varying rapidly at short spatiotemporal scales, “model errors” in the terminology of von Clarmann et al., 2020) may be present in the data due to imperfect corrections for the presence of clouds in the probed scene. In order to limit this term, our main analysis will focus on the clear-sky conditions. We use the operational TROPOMI cloud product (Loyola et al., 2018) to select ozone data with cloud fraction smaller than 0.2.

Figure 2 shows typical TROPOMI clear-sky total ozone column observations in one orbit. Typical values of random uncertainties (Fig. 2, center) range from 0.5 to 2 DU. As shown in Fig. 2 (right), the measurement points in a certain latitude band are performed nearly at the same time, for one orbit.

In our analyses, we selected the TROPOMI Level 2 clear-sky total ozone data
in several broad latitude bands (60–90

The computation of structure function requires finding the differences in
ozone and the corresponding spatial separation (i.e., distance in latitude
and longitude) between every pair of data pixels. Theoretically it could be
achieved by considering one point and comparing it with the rest of
observations, then moving to another point and again comparing it with all
other observations. However, owing to the very high spatial resolution of
TROPOMI and thus an extremely large amount of observation points even for
one orbit, such an operation is very demanding computationally. To ensure
numerical efficiency, the algorithm is simplified while preserving the
underlying principle: instead of using all observations we consider
a sufficiently large amount of observations. For each orbit and for each
latitude band, we create a set of

Illustration of structure function in July 2018 and other
associated parameters, for latitude 30–60

After computing the average of the squared difference in ozone and spatial
separation for each orbit, the monthly-averaged structure functions are
created. The monthly average is based on 400–450 structure functions from
individual orbits, so in total 800–1000 million data pairs are used for
evaluation of monthly averaged structure functions. The smallest bin for
evaluation of the structure function is 5 km

Figure 3a shows the example of the structure
function evaluated for July 2018 in the latitude band 30–60

The structure functions evaluated in different latitude bands and seasons
are shown in Fig. 4. Color represents

Structure function (expressed as

The obtained morphology of ozone variability is quite expected: it is overall much smaller in the tropics than at middle and high latitudes, where it has a pronounced seasonal cycle. At mid-latitudes in winter and spring, the ozone variability is very strong, even for small separations. Except at high northern latitudes in winter and spring, the structure functions are anisotropic with a stronger variability in the latitudinal direction.

Structure function (in DU) for different latitude bands (columns) and months (rows), with the focus on small separations. Colored circles at the origin indicate the uncertainty estimates (ex ante) given by the inversion algorithm.

Figure 5 shows the structure functions for selected
latitude zones (the same as presented in Fig. 4),
but with the focus on small separations, for January 2019 and July 2018. In
Fig. 5, the colored circles near the origin
indicate the mean (for the corresponding latitude zone and month) ex ante
uncertainty estimates in the pairs with small separation distances. We
observe that in the regions of small (20

The distribution and statistical parameters of experimental uncertainty
estimates using the structure function method (ex post) and the theoretical
uncertainty estimates provided by the inversion algorithm (ex ante) in the
tropics and at mid-latitudes are shown in Fig. 6.
The individual values of the structure function and ex ante uncertainties
(black dots in Fig. 6) are selected for small
separations: 20

The distributions of experimental uncertainty estimates using the structure function method (ex post: magenta and red color) and the theoretical values by the inversion algorithm (ex ante: cyan and blue color) in the tropics and at mid-latitudes. Black dots show individual values, circle is median, horizonal dash is mean, thick vertical lines span over 16th–84th inter-percentile range, thin vertical lines span over 5th–95th inter-percentile range of the uncertainty estimates.

The structure function method is also a powerful tool for detecting
non-accounted pseudo-random errors. To demonstrate this, we compare in
Fig. 7 the structure functions in the tropics for
TROPOMI ozone data in clear-sky and cloudy conditions (cloud fraction

One-dimensional structure functions in latitude and in longitude separations (color lines), which are computed from a two-dimensional structure function like in Fig. 4 by averaging over longitude–latitude separations from 0 to 20 km. The symbols at zero separations indicate ex ante uncertainty estimates.

It is quite evident that the structure function method can be applied to any dataset in which data with different separation distances can be found. The approach might especially be useful for other remote sensing measurements in nadir-looking geometry, which have fine horizontal resolution. The datasets should not be necessarily remote sensing measurements. The structure function can also be applied, for example, to modeled data by a chemistry-transport model, in order to estimate numerical noise in the model.

The analyses performed in our paper have shown that the structure function method – i.e., the evaluation of rms differences as a function of increasing spatial separation – is a powerful tool, which allows quantification of random noise in the data. The limit at zero mismatch provides the experimental estimate of the random uncertainty variance. In our paper, we applied the structure function method to validate the TROPOMI clear-sky total ozone random uncertainty estimates by the inversion algorithm. We found that the latter are very close to the experimental ones provided by the structure function method, in the regions of small total ozone natural variability. This indicates adequacy of the TROPOMI random error estimation.

At the same time, the structure function method provides the detailed
information about the natural variability of the measured parameter. For
TROPOMI total ozone, we have analyzed the structure functions in different
seasons and latitude zones. We found the expected results: the overall
variability is the smallest in the equatorial region, and the largest
variability is at middle and high latitudes in winter and spring. At these
locations and during these seasons, the rms of ozone differences grows rapidly with the
separation between measurements achieving

The structure function method is also a powerful tool for detecting non-accounted pseudo-random errors. In the paper, we have demonstrated this by comparing the structure functions and theoretical uncertainty estimates for TROPOMI ozone measurements in clear-sky and cloudy conditions.

The structure function method discussed in the paper can be equally applied to other remote sensing measurements or atmospheric model data.

The Level 2 total ozone column datasets are available at

VFS and HSL have performed the analyses and wrote the major part of the text. CL, FR and DGL are the developers of the TROPOMI total ozone inversion algorithm. All the authors (VFS, HSL, CL, FR, DGL and JT) contributed to writing the paper.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Towards Unified Error Reporting (TUNER)”. It is not associated with a conference.

The authors thank EU/ESA/DLR for providing the TROPOMI Sentinel-5P Level 2 products used in this paper. The work of Viktoria F. Sofieva, Hei Shing Lee and Johanna Tamminen was supported by the ESA-funded project SUNLIT and the Academy of Finland, Centre of Excellence of Inverse Modelling and Imaging. The work of Fabian Romahn and Diego G. Loyola for the development of TROPOMI retrieval algorithms and processors has been supported by DLR (S5P KTR 2472046). Viktoria F. Sofieva thanks the TUNER team for the useful discussions.

This research has been supported by the ESA-funded project SUNLIT; the Academy of Finland, Centre of Excellence of Inverse Modelling and Imaging (decision 336798); and DLR (grant no. S5P KTR 2472046).

This paper was edited by Doug Degenstein and reviewed by two anonymous referees.