# Statistical Analysis of 1996–2017 Ozone Profile Data Obtained by Ground-Based Microwave Radiometry

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}for decade 1996–2006 as compared to 4.58 g/m

^{2}for 2007–2017. Possible explanations of revealed offsets are proposed and discussed.

## 1. Introduction

## 2. Equipment, Measurements, and Data Processing

## 3. Statistical Analysis

#### 3.1. Statistical Parameters and Method of Errors Correction

_{2}(h)} of secondary retrieved profiles.

_{2}(h) = U(h) + ΔU(h) + δU(h) where ΔU(h) is the systematic error and δU(h) is random error with zero mean value. Then, one can obtain statistical parameters of the retrieval errors of the method itself—systematic errors $\mathsf{\Delta}U(h)=<{U}_{2}(h)-U(h)>$ and the dispersion of random errors ${\sigma}_{\delta U}^{2}(h)=<{[{U}_{2}(h)-U(h)-\mathsf{\Delta}U(h)]}^{2}>$. Here, we propose to use these parameters to clear (correct) some statistical parameters of initially retrieved ozone profiles U(h) from the methodical errors.

_{0}} via those obtained from the retrieved profiles {U}:

_{2}}. Then, it is possible to determine the same statistical parameters for {U

_{2}} as for {U}:

_{2}} can be determined. Then, we assume that these parameters are equal to corresponding error parameters of profiles {U}, retrieved by experimental data:

#### 3.2. Statistical Parameters of Retrieval Errors

#### 3.3. Probability Distributions of Ozone Variations in Two Decades (1996–2006 and 2007–2017)

#### 3.4. Mean and Variances of Retrieved Ozone Profiles in Separate Months of Two Decades (1996–2006 and 2007–2017)

_{2}are independent, so their effect is included at the correction of rms variances demonstrated in Figure 6, and appeared to be not significant.

#### 3.5. Inter-altitude and Time Covariance and Correlation Functions in Separate Months of Two Decades (1996–2006 and 2007–2017); Frequency Spectra of Time Covariance and Correlation Functions

_{0}have strong distortions in regions of small variances).

^{−1}(the Niquist frequency).

#### 3.6. Integral Statistical Parameters of Ozone Profiles of Two Decades (1996–2006 and 2007–2017).

^{2}for the decade 1996–2006 versus 4.58g/m

^{2}for 2007–2017.

#### 3.7. Comparison of Results with Other Research

## 4. Discussion

_{2}content), and temperature. It should also be mentioned that relative variations of the ozone concentration in the upper stratosphere and lower mesosphere related to solar activity are more significant than those of the corresponding integral ozone content. For example, it is reported that in the 23rd cycle, relative ozone changes between the solar minimum and maximum in the region of the Antarctic ozone hole amounted to 6.8–9.6% at altitudes of 22–31 km [31]. So, this effect could be considered as one of reasons for the decrease in variances in the 2007–2017 decade.

## 5. Conclusions

^{2}for the decade 1996–2006 versus 4.58g/m

^{2}for 2007–2017.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Inverse Problem of Microwave Radiometry Ozone Profiling

**K**= K(U, f, θ, h) is the nonlinear kernel of the equation; f is the frequency; and θ is the elevation angle of observations. Taking into account measurement errors, the Equation (A1) is solved by data vector ${T}_{B}^{\delta}={T}_{B}+\delta {T}_{B}$ with errors instead of the exact data vector ${T}_{B}$. To solve this equation, we used the iteration algorithm based on [16], where at each step of the iteration process, a solution is found by Tikhonov’s method of generalized discrepancy [17]. At the first step of iterations, an arbitrary continuous ozone profile U

_{1}(h) is substituted into the kernel of (A1); at the second step, the retrieved profile obtained at the first step is substituted into the kernel, and so on. The operator form of this iteration scheme for the nonlinear problem can be written as follows:

**K**

^{i}=

**K**(Ui, f, θ, h) and the model ozone profile is the first approximation U

_{1}(h), which can be chosen as an arbitrary function from ${\mathrm{W}}_{2}^{1}$. The data vector ${T}_{B}^{\delta}$ is measured with errors that satisfy

_{2}, the approximate solution converges to the exact solution in the metric ${\mathrm{W}}_{2}^{1}$ [17]. It is the main advantage of this method over other known methods. This means that, according to the Sobolev’s imbedding theorem, this approximate solution converges uniformly—i.e., in the metric C, where the maximum modulus is the norm. On the other hand, unlike well-posed problems, the rate of convergence here is not proportional to a decrease in $\delta $, but it is slower—as in all ill-posed problems. If the metric L

_{2}(square integrability) is used in the second term of (6), the convergence of the solution will also be in the metric L

_{2}[17].

_{0}is a fixed frequency.

- (a)
- Formation of the simulated ozone profile;
- (b)
- Computations of the spectrum of brightness temperature;
- (c)
- Adding of simulated Gaussian random errors with zero mean value $\mathsf{\Delta}{T}_{B}(f)$ and preset mean-square variations ${\sigma}_{T}^{2}{}_{B}(f)$;
- (d)
- Solving the inverse problem (9);
- (e)
- Comparison of the retrieved and simulated ozone profiles.

_{B}= 0.05 K is achieved in the LPI radiometer at the signal integration time of order 1 h.

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**Figure 1.**Profiles of ozone retrieval error parameters and ozone profile variations in 1996–2017: (

**a**) October ensemble; (

**b**) December ensemble. 1—systematic errors $\mathsf{\Delta}U(h)$; 2—rms random errors ${\sigma}_{\delta U}(h)$; 3—ozone rms deviations ${\sigma}_{U}(h)$ from the mean profile $<U(h)>$.

**Figure 2.**Inter-level covariance (

**a**) and correlation (

**b**) functions of retrieval errors for the December ensemble.

**Figure 3.**Retrieved day-time ozone profiles over Moscow. (

**a**) ensemble of 1996–2006 profiles; (

**b**) ensemble of 2007–2017 profiles.

**Figure 4.**Distributions of probability density of ozone profile variations. (

**a**) Ensemble of 1996–2006 profiles; (

**b**) ensemble of 2007–2017 profiles.

**Figure 5.**Seasonal variations of monthly mean ozone profiles averaged over decades 1996–2006 and 2007–2017 arranged from September (9th month) to May (5th month). (

**a**) Ensemble of 1996–2006; (

**b**) ensemble of 2007–2017.

**Figure 6.**Seasonal changes in monthly mean rms variances of the ozone profiles averaged over decades 1996–2006 and 2007–2017. Upper row: variances obtained from ensembles of retrieved profiles; bottom row: variances cleared from retrieval errors. Results are arranged from September (9th month) to May (5th month). (

**a**,

**c**) Ensembles of 1996–2007; (

**b**,

**d**) ensembles of 2007–2017.

**Figure 7.**Differences in monthly mean ozone profiles and their variances between two decades shown in Figure 5 and Figure 6: (

**a**) difference between monthly mean ozone profiles; (

**b**) difference between monthly mean rms ozone variances; (

**c**) rms sampling errors of the difference shown in Figure 7a. Results are arranged from September (9th month) to May (5th month).

**Figure 8.**Differences in relative monthly mean ozone profiles and their rms variances between the two decades, in percent: (

**a**) difference between monthly mean ozone profiles; (

**b**) difference between monthly mean rms ozone variances; (

**c**) rms sampling errors of mean ozone profiles difference. The results are arranged from November (11th month) to March (3th month).

**Figure 9.**Inter-altitude covariance functions for autumn (top row), winter (middle row), and spring (lower row) seasons calculated for decades 1996–2006 and 2007–2017.

**Figure 10.**Inter-altitude correlation functions for autumn (

**top row**), winter (

**middle row**), and spring (

**lower row**) seasons calculated for decades 1996–2006 and 2007–2017.

**Figure 11.**Time covariance functions for autumn (

**top row**), winter (

**middle row**), and spring (

**lower row**) seasons calculated for decades 1996–2006 and 2007–2017.

**Figure 12.**Spectrum of covariance functions for autumn (

**top row**), winter (

**middle row**), and spring (

**lower row**) seasons calculated for decades 1996–2006 and 2007–2017.

**Figure 13.**Time correlation (autocorrelation) functions for autumn (

**top row**), winter (

**middle row**), and spring (

**lower row**) seasons calculated for decades 1996–2006 and 2007–2017.

**Figure 14.**Spectrum of correlation functions for autumn (

**top row**), winter (

**middle row**), and spring (

**lower row**) seasons calculated for decades 1996–2006 and 2007–2017.

**Figure 15.**Profiles of ozone integral mean and rms variances: (

**a**) 1, 2—profiles of ozone integral mean $<{U}_{0\text{}1996-2006}^{\mathrm{int}}(h)$ and $<{U}_{0\text{}2007-2017}^{\mathrm{int}}(h)$; 3, 4—profiles of ozone rms variances ${\sigma}_{0\text{}1996-2006}^{\mathrm{int}}(h)$, ${\sigma}_{0\text{}2007-2017}^{\mathrm{int}}(h)$; (

**b**) 1, 2—profiles of ozone rms variances ${\sigma}_{0\text{}1996-2006}^{\mathrm{int}}(h)$, ${\sigma}_{0\text{}2007-2017}^{\mathrm{int}}(h)$ (the same as in Figure 15a).

**Figure 16.**(

**a**) 1—altitude profile of the difference Δ<U

_{0}

^{int}> between integral mean ozone profiles for ensembles of 2007–2017 and 1996–2006; 2—profile of corresponding rms sampling errors δΔ<U

_{0}

^{int}>; (

**b**) 1—profile of the difference Δσ

_{0}

^{int}between integral ozone variances for ensembles 2007—2017 and 1996–2006; 2—profile of the difference between corresponding noncleared variances Δσ

^{int}.

**Figure 17.**Inter-altitude integral mean covariance functions of ozone profiles: (

**a**) ensemble of 1996–2006; (

**b**) ensemble of 2007–2017; (

**c**) difference $\mathsf{\Delta}{B}_{0}^{\mathrm{int}}(h)={B}_{0\text{}2007-2017}^{\mathrm{int}}(h)-{B}_{0\text{}1996-2006}^{\mathrm{int}}(h)$

**Figure 18.**Inter-altitude integral mean correlation functions of ozone profiles: (

**a**) ensemble of 1996–2006; (

**b**) ensemble of 2007–2017.

**Figure 19.**Cleared integral mean time-covariance functions of ozone profiles: (

**a**) ensemble of 1996–2006; (

**b**) ensemble of 2007–2017.

**Figure 20.**Cleared integral mean time-correlation functions of ozone profiles: (

**a**) ensemble of 1996–2006; (

**b**) ensemble of 2007–2017.

**Figure 21.**The 23rd and 24th cycles of solar activity: 1—monthly mean number of solar spots; 2—smoothed line.

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**MDPI and ACS Style**

Gaikovich, K.P.; Kropotkina, E.P.; Rozanov, S.B. Statistical Analysis of 1996–2017 Ozone Profile Data Obtained by Ground-Based Microwave Radiometry. *Remote Sens.* **2020**, *12*, 3374.
https://doi.org/10.3390/rs12203374

**AMA Style**

Gaikovich KP, Kropotkina EP, Rozanov SB. Statistical Analysis of 1996–2017 Ozone Profile Data Obtained by Ground-Based Microwave Radiometry. *Remote Sensing*. 2020; 12(20):3374.
https://doi.org/10.3390/rs12203374

**Chicago/Turabian Style**

Gaikovich, Konstantin P., Elena P. Kropotkina, and Sergey B. Rozanov. 2020. "Statistical Analysis of 1996–2017 Ozone Profile Data Obtained by Ground-Based Microwave Radiometry" *Remote Sensing* 12, no. 20: 3374.
https://doi.org/10.3390/rs12203374