This study suggests a stochastic model for time series of daily zonal (circumpolar) mean stratospheric temperature at a given pressure level. It can be seen as an extension of previous studies which have developed stochastic models for surface temperatures. The proposed model is a combination of a deterministic seasonality function and a Lévy-driven multidimensional Ornstein–Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the deseasonalized temperature model is an order 4 continuous-time autoregressive model, meaning that the stratospheric temperature is modeled to be directly dependent on the temperature over four preceding days, while the model’s longer-range memory stems from its recursive nature. This study is based on temperature data from the European Centre for Medium-Range Weather Forecasts ERA-Interim reanalysis model product. The residuals of the autoregressive model are well represented by normal inverse Gaussian-distributed random variables scaled with a time-dependent volatility function. A monthly variability in speed of mean reversion of stratospheric temperature is found, hence suggesting a generalization of the fourth-order continuous-time autoregressive model. A stochastic stratospheric temperature model, as proposed in this paper, can be used in geophysical analyses to improve the understanding of stratospheric dynamics. In particular, such characterizations of stratospheric temperature may be a step towards greater insight in modeling and prediction of large-scale middle atmospheric events, such as sudden stratospheric warming. Through stratosphere–troposphere coupling, the stratosphere is hence a source of extended tropospheric predictability at weekly to monthly timescales, which is of great importance in several societal and industry sectors.
<p>The MADEIRA project (Middle Atmosphere Dynamics: Exploiting Infrasound Using a Multidisciplinary Approach at High Latitudes) is a four-year basic research project finishing in the spring 2023, funded by the Research Council of Norway.&#160;</p> <p>Its primary objective has been in elucidating the 30-60 km altitude range over large spatial scales using wind and temperature constraints from infrasound data collected at Arctic stations. Wave propagation modelling and infrasound interpretation from well-characterized sources provide remote atmospheric sensing. These data are more continuous in space and time than from many other direct measurement techniques. An aim has been to constrain high-top atmospheric models and explore stratosphere-mesosphere coupling with meteor radar wind measurements sampling the 70-100 km altitude range in combination with the infrasonic data. Another ambition has been to develop real-time diagnostic tools for the stratospheric polar vortex circulation and extreme events like Sudden Stratospheric Warmings.&#160;</p> <p>Thanks to this project, the research teams have got the opportunity to explore several aspects and building blocks related to infrasound-based middle atmospheric probing and to work towards an assimilation of such datasets into atmospheric models. This paper reviews key research output from the project and highlights accomplishments in the domains of, e.g.: Tropospheric and stratospheric cross-wind estimation using infrasound from explosions; Assimilation of atmospheric infrasound data to constrain tropospheric and stratospheric winds; Atmospheric wind and temperature profile inversion in an ensemble model context; Microbarom radiation and propagation model benchmarking; Speeding up infrasound transmission loss estimation using deep learning; Probing internal middle atmospheric gravity waves; Using a machine learning and stochastics-founded model to provide near real-time stratospheric polar vortex diagnostics. </p> <p>This project has included several high risk / high gain components and we highlight results that maybe could be labelled as unexpected successes, but we also discuss challenging research obstacles that occurred in our journey.</p>
<p>This study suggests a stochastic model for time series of daily zonal (circumpolar) mean stratospheric temperature at a given pressure level. It can be seen as an extension of previous studies which have developed stochastic models for surface temperatures. The proposed model is a combination of a deterministic seasonality function and a L&#233;vy-driven multidimensional Ornstein&#8211;Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the deseasonalized temperature model is an order 4 continuous-time autoregressive model, meaning that the stratospheric temperature is modeled to be directly dependent on the temperature over four preceding days, while the model&#8217;s longer-range memory stems from its recursive nature. This study is based on temperature data from the European Centre for Medium-Range Weather Forecasts ERA-Interim reanalysis model product. The residuals of the autoregressive model are well represented by normal inverse Gaussian-distributed random variables scaled with a time-dependent volatility function. A monthly variability in speed of mean reversion of stratospheric temperature is found, hence suggesting a generalization of the fourth-order continuous-time autoregressive model. A stochastic stratospheric temperature model, as proposed in this paper, can be used in geophysical analyses to improve the understanding of stratospheric dynamics. In particular, such characterizations of stratospheric temperature may be a step towards greater insight in modeling and prediction of large-scale middle atmospheric events, such as sudden stratospheric warming. Through stratosphere&#8211;troposphere coupling, the stratosphere is hence a source of extended tropospheric predictability at weekly to monthly timescales, which is of great importance in several societal and industry sectors.</p>
<p>A stochastic model for daily-spatial mean stratospheric temperature over a given area is suggested. The model is a sum of a deterministic seasonality function and a L&#233;vy driven vectorial Ornstein-Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the model is an order 4 continuous-time autoregressive (CAR(4)) process, derived from data analysis suggesting an order 4 autoregressive (AR(4)) process to model the deseasonalized stochastic temperature data empirically. In this analysis, temperature data as represented in ECMWF re-analysis model products are considered. The residuals of the AR(4) process turn out to be normal inverse Gaussian distributed random variables scaled with a time dependent volatility function. In general, it is possible to show that the discrete time AR(p) process is closely related to CAR(p) processes, its continuous counterpart. An equivalent effort is made in deriving a dual stochastic model for stratospheric temperature, in the sense that the year is divided into summer and winter seasons. However, this seems to further complicate the modelling, rather than obtaining a simplified analytic framework. A stochastic characterization of the stratospheric temperature representation in model products, such as the model proposed in this paper, can be used in geophysical analyses to improve our understanding of stratospheric dynamics. In particular, such characterizations of stratospheric temperature may be a step towards greater insight in modelling and prediction of large-scale middle atmospheric events like sudden stratospheric warmings. Through stratosphere-troposphere coupling, this is important in the work towards an extended predictability of long-term tropospheric weather forecasting.</p>
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