# Comparative Spectral Analysis and Correlation Properties of Observed and Simulated Total Column Ozone Records

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## Abstract

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## 1. Introduction

## 2. Data and Methods

**Figure 1.**(

**a**) Empirical and simulated total column ozone (TO) time series for an equatorial location over the Pacific Ocean in Dobson units (DU). Black line belongs to the Bodeker Scientific record (label: Bd), red and blue denote CCM simulations by the LMDz-Reprobus (LMDz) and MRI models; (

**b**) Model time series over the Antarctic (see label). Note the different vertical scales.

**Figure 2.**Non-normalized power spectra for three different geographic locations (see legends) as a function of period. Note the logarithmic horizontal scales. Black lines with circles belong to the empirical Bodeker Scientific records (label: Bd), red and blue denote CCM simulations by the LMDz-Reprobus (LMDz) and MRI models. (

**a**) Equatorial location (Gabon, Africa); (

**b**) South Pacific Ocean; (

**c**) Québec, Canada.

#### 2.1. Spectral Weight Analysis

**Figure 3.**The high frequency tails of the spectra in Figure 2 on double logarithmic scales. Note that the period on the horizontal axes is given in units of day. The same labels, coloring and notations are used. Eleven-point running mean smoothing is applied to suppress high frequency noise.

#### 2.2. Detrended Fluctuation Analysis

**Figure 4.**Logarithm of DFA3 fluctuation function ${log}_{10}\left[F\left(n\right)\right]$ as a function of logarithmic window size ${log}_{10}n$ for three different geographic locations (see legends). Black lines belong to the empirical Bodeker Scientific records (label: Bd), red and blue denote CCM simulations by the LMDz-Reprobus (LMDz) and MRI models. (

**a**) Gulf of Guinea, near the equator; (

**b**) Indian Ocean, close to Christmas Island; (

**c**) Guo Dao, Xinjiang, China. Thin dashed lines indicate whole range fits for the LMDz data, thin straight lines in (b) and (c) denote the same for the restricted range (60 days–15 years), see text.

## 3. Results and Discussion

#### 3.1. Spectral Peak Maps

**Figure 5.**Geographic distribution of the spectral peak weights for the empirical Bodeker Scientific records (label: Bd) and CCM simulations by the LMDz-Reprobus (label: LMDz) and MRI models (label: MRI). The columns from left to right display the weights of semiannual, annual and QBO peaks, respectively. Note the different color-scale ranges for the QBO maps.

#### 3.2. DFA Exponent Maps

**Figure 6.**Geographic distribution of the DFA3 exponent values for the empirical Bodeker Scientific records (label: Bd) and CCM simulations by the LMDz-Reprobus (label: LMDz) and MRI models (label: MRI). The left column displays fitted values for the whole range of available data, the right column shows the results for the restricted range fits from 60 days to 15 years (from 1.78 to 3.74 on the log${}_{10}$ axis), see Figure 4.

**Figure 7.**Zonal mean values of the DFA3 exponents for the empirical Bodeker Scientific records (label: Bd) and CCM simulations by the LMDz-Reprobus (label: LMDz) and MRI models (label: MRI) as a function of geographic latitude. (

**a**) Fitted values for the whole range of available data; and (

**b**) restricted range fits from 60 days to 15 years (from 1.78 to 3.74 on the log${}_{10}$ axis), see Figure 4. Error bars represent the variability of exponent values along a circle of latitude.

**Figure 8.**Geographic distribution of the standard fitting error of DFA3 exponent values for the empirical Bodeker Scientific records (label: Bd) and CCM simulations by the LMDz-Reprobus (label: LMDz) and MRI models (label: MRI). The left column displays fitted values for the whole range of available data, the right column shows the results for the restricted range fits from 60 days to 15 years (from 1.78 to 3.74 on the log${}_{10}$ axis), see Figure 4. Note the different color scales at the two columns.

**Figure 9.**(

**a**) and (

**b**): Logarithm of DFA3 fluctuation function ${log}_{10}\left[F\left(n\right)\right]$ as a function of logarithmic window size ${log}_{10}n$ for two different geographic locations (see legends) over the Antarctic. Red and blue denote CCM simulations by the LMDz-Reprobus (LMDz) and MRI models, empirical data are not available; Red arrow on (

**b**) indicates the characteristic kink appearing everywhere for locations over the Antarctic; (

**c**) Power density spectra for the total column ozone anomaly records (annual cycle removed) simulated by the LMDz (red) and MRI (blue) models over an Antarctic location (see legend).

## 4. Conclusions

- Both the LMDz and MRI models provide an adequate reproduction of empirical data in the period of comparison (see Figure 1), however the latter seems to slightly overestimate total column ozone values.
- Spectral analysis reveals that both models underestimate high frequency (daily) variability (see Figure 3), MRI performs better from this point of view. Such deficiency might have a rather complex origin, because several dynamical processes (advection, diffusion and mixing) contribute to short time fluctuations.
- Spectral weight analysis indicates some anomalies in the dynamical core of both CCMs. The geographic locations of semiannual oscillations are shifted in the models. MRI is known to reproduce the QBO mode along the equator (albeit it is definitely weaker than in the empirical records), however it underperforms with respect of the annual component.
- The presence of quasi-biennial oscillations decreases the accuracy of DFA analysis (see Figure 4(a)). Band pass Fourier filtering can remove such spectral component (see [22]), however one should be careful, because the DFA method is based on residual (anomaly) analysis. Specifically, the standard Wiener filtering consists of two steps: firstly, the Fourier transform of a given time series is produced and the amplitudes are set to be zero or to some assumed (practically interpolated) base line values in the required frequency ranges, and secondly, an inverse Fourier transform produces the filtered time series. When this method is applied on a global scale, obviously the filtered components disappear everywhere, also from signals where QBO oscillations are not present. Care should be taken especially when the target is to detect long range correlations, because such procedure might distort the original properties of the time series. Too strong filtering can yield (geographically) inconsistent results as probably demonstrated in [35,36].Note that the low model values of high frequency variabilities are reflected by the fact that the initial parts of fluctuation functions remain systematically below the curve for empirical data (Figure 4).
- The DFA3 fluctuation functions are adequately reproduced by both CCMs (see Figure 6 and Figure 7). We do not claim that the curves represent pure power law behavior (see Figure 4); still the overall trends are close to linear on log-log scale, indicating the presence of long range correlations. (The appearance of clean power laws is not even expected, because many processes of various characteristic time scales contribute to the instantaneous value of any atmospheric variable at a given location.)
- Since the reproduction of correlation properties in the models is quite satisfactory in the overlapping geographic latitude band between −60${}^{\circ}$ and 60${}^{\circ}$, we can have a strong confidence in the prediction of both models that correlations are stronger over both polar regions. This observations is also supported by considering the role of polar vortices in the meridional transport processes.
- Classical methods of time series analysis have a limited applicability over the polar regions, especially over the Antarctic. Such strong nonstationary signals as represented by the total ozone level during the decades of ozone hole make any residual (anomaly) analysis very difficult. Standard methods to remove semiannual and annual oscillations fail when the amplitudes change strongly (see Figure 9), however an over-filtering might yield information loss about correlation properties.

## Acknowledgements

## Conflicts of Interest

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**MDPI and ACS Style**

Homonnai, V.; Jánosi, I.M.; Lefèvre, F.; Marchand, M. Comparative Spectral Analysis and Correlation Properties of Observed and Simulated Total Column Ozone Records. *Atmosphere* **2013**, *4*, 198-213.
https://doi.org/10.3390/atmos4020198

**AMA Style**

Homonnai V, Jánosi IM, Lefèvre F, Marchand M. Comparative Spectral Analysis and Correlation Properties of Observed and Simulated Total Column Ozone Records. *Atmosphere*. 2013; 4(2):198-213.
https://doi.org/10.3390/atmos4020198

**Chicago/Turabian Style**

Homonnai, Viktória, Imre M. Jánosi, Franck Lefèvre, and Marion Marchand. 2013. "Comparative Spectral Analysis and Correlation Properties of Observed and Simulated Total Column Ozone Records" *Atmosphere* 4, no. 2: 198-213.
https://doi.org/10.3390/atmos4020198