scholarly journals Application of tomographic inversion in studying airglow in the mesopause region

1998 ◽  
Vol 16 (10) ◽  
pp. 1180-1189 ◽  
Author(s):  
T. Nygrén ◽  
M. J. Taylor ◽  
M. S. Lehtinen ◽  
M. Markkanen
Keyword(s):  
Gravity Waves ◽  
Random Noise ◽  
A Priori ◽  
Lower Thermosphere ◽  
Vertical Plane ◽  
Simulated Data ◽  
Altitude Range ◽  
Priori Information ◽  
Two Stages ◽  
Atmospheric Gravity

Abstract. It is pointed out that observations of periodic nightglow structures give excellent information on atmospheric gravity waves in the mesosphere and lower thermosphere. The periods, the horizontal wavelengths and the phase speeds of the waves can be determined from airglow images and, using several cameras, the approximate altitude of the luminous layer can also be determined by triangulation. In this paper the possibility of applying tomographic methods for reconstructing the airglow structures is investigated using numerical simulations. A ground-based chain of cameras is assumed, two-dimensional airglow models in the vertical plane above the chain are constructed, and simulated data are calculated by integrating the models along a great number of rays with different elevation angles for each camera. After addition of random noise, these data are then inverted to obtain reconstructions of the models. A tomographic analysis package originally designed for satellite radiotomography is used in the inversion. The package is based on a formulation of stochastic inversion which allows the input of a priori information to the solver in terms of regularization variances. The reconstruction is carried out in two stages. In the first inversion, constant regularization variances are used within a wide altitude range. The results are used in determining the approximate altitude range of the airglow structures. Then, in the second inversion, constant non-zero regularization variances are used inside this region and zero variances outside it. With this method reliable reconstructions of the models are obtained. The number of cameras as well as their separations are varied in order to find out the limitations of the method.Key words. Tomography · Airglow · Mesopause · Gravity waves

scholarly journals Retrieval of temperature profiles from CHAMP for climate monitoring: intercomparison with Envisat MIPAS and GOMOS and different atmospheric analyses

2007 ◽  
Vol 7 (13) ◽  
pp. 3519-3536 ◽  
Author(s):  
A. Gobiet ◽  
G. Kirchengast ◽  
G. L. Manney ◽  
M. Borsche ◽  
C. Retscher ◽  
...  
Keyword(s):  
High Altitude ◽  
Weather Prediction ◽  
A Priori ◽  
Michelson Interferometer ◽  
Altitude Range ◽  
A Priori Information ◽  
Climate Monitoring ◽  
Retrieval Scheme ◽  
Priori Information ◽  
Upper Boundary

Abstract. This study describes and evaluates a Global Navigation Satellite System (GNSS) radio occultation (RO) retrieval scheme particularly aimed at delivering bias-free atmospheric parameters for climate monitoring and research. The focus of the retrieval is on the sensible use of a priori information for careful high-altitude initialisation in order to maximise the usable altitude range. The RO retrieval scheme has been meanwhile applied to more than five years of data (September 2001 to present) from the German CHAllenging Minisatellite Payload for geoscientific research (CHAMP) satellite. In this study it was validated against various correlative datasets including the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) and the Global Ozone Monitoring for Occultation of Stars (GOMOS) sensors on Envisat, five different atmospheric analyses, and the operational CHAMP retrieval product from GeoForschungsZentrum (GFZ) Potsdam. In the global mean within 10 to 30 km altitude we find that the present validation observationally constrains the potential RO temperature bias to be <0.2 K. Latitudinally resolved analyses show biases to be observationally constrained to <0.2–0.5 K up to 35 km in most cases, and up to 30 km in any case, even if severely biased (about 10 K or more) a priori information is used in the high altitude initialisation of the retrieval. No evidence is found for the 10–35 km altitude range of residual RO bias sources other than those potentially propagated downward from initialisation, indicating that the widely quoted RO promise of "unbiasedness and long-term stability due to intrinsic self-calibration" can indeed be realised given care in the data processing to strictly limit structural uncertainty. The results thus reinforce that adequate high-altitude initialisation is crucial for accurate stratospheric RO retrievals. The common method of initialising, at some altitude in the upper stratosphere, the hydrostatic integral with an upper boundary temperature or pressure value derived from meteorological analyses is prone to introduce biases from the upper boundary down to below 25 km. Also above 30 to 35 km, GNSS RO delivers a considerable amount of observed information up to around 40 km, which is particularly interesting for numerical weather prediction (NWP) systems, where direct assimilation of non-initialised observed RO bending angles (free of a priori) is thus the method of choice. The results underline the value of RO for climate applications.

scholarly journals A method of combining coherence-constrained sparse coding and dictionary learning for denoising

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. V137-V148 ◽  
Author(s):  
Pierre Turquais ◽  
Endrias G. Asgedom ◽  
Walter Söllner
Keyword(s):  
Dictionary Learning ◽  
Seismic Data ◽  
Random Noise ◽  
A Priori ◽  
Seismic Signal ◽  
Noise Variance ◽  
A Priori Information ◽  
Seismic Section ◽  
Learned Dictionary ◽  
Priori Information

We have addressed the seismic data denoising problem, in which the noise is random and has an unknown spatiotemporally varying variance. In seismic data processing, random noise is often attenuated using transform-based methods. The success of these methods in denoising depends on the ability of the transform to efficiently describe the signal features in the data. Fixed transforms (e.g., wavelets, curvelets) do not adapt to the data and might fail to efficiently describe complex morphologies in the seismic data. Alternatively, dictionary learning methods adapt to the local morphology of the data and provide state-of-the-art denoising results. However, conventional denoising by dictionary learning requires a priori information on the noise variance, and it encounters difficulties when applied for denoising seismic data in which the noise variance is varying in space or time. We have developed a coherence-constrained dictionary learning (CDL) method for denoising that does not require any a priori information related to the signal or noise. To denoise a given window of a seismic section using CDL, overlapping small 2D patches are extracted and a dictionary of patch-sized signals is trained to learn the elementary features embedded in the seismic signal. For each patch, using the learned dictionary, a sparse optimization problem is solved, and a sparse approximation of the patch is computed to attenuate the random noise. Unlike conventional dictionary learning, the sparsity of the approximation is constrained based on coherence such that it does not need a priori noise variance or signal sparsity information and is still optimal to filter out Gaussian random noise. The denoising performance of the CDL method is validated using synthetic and field data examples, and it is compared with the K-SVD and FX-Decon denoising. We found that CDL gives better denoising results than K-SVD and FX-Decon for removing noise when the variance varies in space or time.

Recurrent high-speed solar wind co-rotating interaction region imprint on the ionosphere and atmosphere: GPS TEC variations and atmospheric gravity waves

2020 ◽  
Author(s):  
James M. Weygand ◽  
Paul Prikryl ◽  
Reza Ghoddousi-Fard ◽  
Lidia Nikitina ◽  
Bharat S. R. Kunduri
Keyword(s):  
Solar Wind ◽  
Gravity Waves ◽  
High Latitude ◽  
High Speed ◽  
Geomagnetic Storms ◽  
Coronal Holes ◽  
Lower Thermosphere ◽  
Ionospheric Currents ◽  
Atmospheric Gravity Waves ◽  
Atmospheric Gravity

&lt;p&gt;High-speed streams (HSS) from coronal holes dominate solar wind structure in the absence of coronal mass ejections during solar minimum and the descending branch of solar cycle. Prominent and long-lasting coronal holes produce intense co-rotating interaction regions (CIR) on the leading edge of high-speed plasma streams that cause recurrent ionospheric disturbances and geomagnetic storms. Through solar wind coupling to the magnetosphere-ionosphere-atmosphere (MIA) system they affect the ionosphere and neutral atmosphere at high latitudes, and, at mid to low latitudes, by the transmission of the electric fields [1] and propagation of atmospheric gravity waves from the high-latitude lower thermosphere [2].&lt;/p&gt;&lt;p&gt;The high-latitude ionospheric structure, caused by precipitation of energetic particles, strong ionospheric currents and convection, results in changes of the GPS total electron content (TEC) and rapid variations of GPS signal amplitude and phase, called scintillation [3]. The GPS phase scintillation is observed in the ionospheric cusp, polar cap and auroral zone, and is particularly intense during geomagnetic storms, substorms and auroral breakups. Phase scintillation index is computed for a sampling rate of 50 Hz by specialized GPS scintillation receivers from the Canadian High Arctic Ionospheric Network (CHAIN). A proxy index of phase variation is obtained from dual frequency measurements of geodetic-quality GPS receivers sampling at 1 Hz, which include globally distributed receivers of the RT-IGS network that are monitored by the Canadian Geodetic Survey in near-real-time [4]. Temporal and spatial changes of TEC and phase variations following the arrivals of HSS/CIRs [5] are investigated in the context of ionospheric convection and equivalent ionospheric currents derived from&amp;#160; a ground magnetometer network using the spherical elementary current system method [6,7].&lt;/p&gt;&lt;p&gt;The Joule heating and Lorentz forcing in the high-latitude lower thermosphere have long been recognized as sources of internal atmospheric gravity waves (AGWs) [2] that propagate both upward and downward, thus providing vertical coupling between atmospheric layers. In the ionosphere, they are observed as traveling ionospheric disturbances (TIDs) using various techniques, e.g., de-trended GPS TEC maps [8].&lt;/p&gt;&lt;p&gt;In this paper we examine the influence on the Earth&amp;#8217;s ionosphere and atmosphere of a long-lasting HSS/CIRs from recurrent coronal holes at the end of solar cycles 23 and 24. The solar wind MIA coupling, as represented by the coupling function [9], was strongly increased during the arrivals of these HSS/CIRs.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;[1] Kikuchi, T. and K. K. Hashimoto, Geosci. Lett. , 3:4, 2016.&lt;/p&gt;&lt;p&gt;[2] Hocke, K. and K. Schlegel, Ann. Geophys., 14, 917&amp;#8211;940, 1996.&lt;/p&gt;&lt;p&gt;[3] Prikryl, P., et al., J. Geophys. Res. Space Physics, 121, 10448&amp;#8211;10465, 2016.&lt;/p&gt;&lt;p&gt;[4] Ghoddousi-Fard et al., Advances in Space Research, 52(8), 1397-1405, 2013.&lt;/p&gt;&lt;p&gt;[5] Prikryl et al. Earth, Planets and Space, 66:62, 2014.&lt;/p&gt;&lt;p&gt;[6] Amm O., and A. Viljanen, Earth Planets Space, 51, 431&amp;#8211;440, 1999.&lt;/p&gt;&lt;p&gt;[7] Weygand J.M., et al., J. Geophys. Res., 116, A03305, 2011.&lt;/p&gt;&lt;p&gt;[8] Tsugawa T., et al., Geophys. Res. Lett., 34, L22101, 2007.&lt;/p&gt;&lt;p&gt;[9] Newell P. T., et al., J. Geophys. Res., 112, A01206, 2007.&lt;/p&gt;

scholarly journals Use of a Maximum Entropy Method as a Regularization Technique during the Retrieval of Trace Gas Profiles from Limb Sounding Measurements

2006 ◽  
Vol 23 (12) ◽  
pp. 1657-1667 ◽  
Author(s):  
J. Steinwagner ◽  
G. Schwarz ◽  
S. Hilgers
Keyword(s):  
Maximum Entropy ◽  
Random Noise ◽  
A Priori ◽  
Michelson Interferometer ◽  
Simulated Data ◽  
Optimal Estimation ◽  
Initial Guess ◽  
Entropy Method ◽  
Trace Gas ◽  
Regularization Techniques

Abstract The retrieval of trace gas profiles from radiance measurements of limb sounding instruments represents an inverse problem: vertical profiles of mixing ratios have to be extracted from sequences of horizontally measured radiances recorded by a spectrometer. Typically, these retrievals are plagued by random noise and systematic errors, necessitating the use of regularization techniques during inversion calculations. In the following, the use of selected maximum entropy operators as a regularization tool is discussed and their performance with conventional optimal estimation and Tikhonov-type regularization techniques is compared. The main gain with the proposed maximum entropy operators is that no a priori knowledge is needed; a reasonable initial guess profile is fully sufficient. The approach is verified by using simulated data of the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) instrument, an infrared Fourier transform spectrometer flown on the European Envisat mission.

scholarly journals Retrieval of nitric oxide in the mesosphere from SCIAMACHY nominal limb spectra

2017 ◽  
Vol 10 (1) ◽  
pp. 209-220 ◽  
Author(s):  
Stefan Bender ◽  
Miriam Sinnhuber ◽  
Martin Langowski ◽  
John P. Burrows
Keyword(s):  
Nitric Oxide ◽  
Climate Models ◽  
A Priori ◽  
Lower Thermosphere ◽  
Ultra Violet ◽  
Retrieval Algorithm ◽  
Altitude Range ◽  
Mesosphere And Lower Thermosphere ◽  
Scanning Imaging ◽  
Nominal Mode

Abstract. We present a retrieval algorithm for nitric oxide (NO) number densities from measurements from the SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY (SCIAMACHY, on Envisat) nominal limb mode (0–91 km). The NO number densities are derived from atmospheric emissions in the gamma bands in the range 230–300 nm, measured by the SCIAMACHY ultra-violet (UV) channel 1. The retrieval is adapted from the mesosphere and lower thermosphere mode (MLT, 50–150 km) NO retrieval (Bender et al., 2013), including the same 3-D ray tracing, 2-D retrieval grid, and regularisations with respect to altitude and latitude.Since the nominal mode limb scans extend only to about 91 km, we use NO densities in the lower thermosphere (above 92 km), derived from empirical models, as a priori input. The priors are the Nitric Oxide Empirical Model (NOEM; Marsh et al., 2004) and a regression model derived from the MLT NO data comparison (Bender et al., 2015). Our algorithm yields plausible NO number densities from 60 to 85 km from the SCIAMACHY nominal limb mode scans. Using a priori input substantially reduces the incorrect attribution of NO from the lower thermosphere, where no direct limb measurements are available. The vertical resolution lies between 5 and 10 km in the altitude range 65–80 km.Analysing all SCIAMACHY nominal limb scans provides almost 10 years (from August 2002 to April 2012) of daily NO measurements in this altitude range. This provides a unique data record of NO in the upper atmosphere and is invaluable for constraining NO in the mesosphere, in particular for testing and validating chemistry climate models during this period.

scholarly journals Formation of Multilayered Sporadic E under an Influence of Atmospheric Gravity Waves (AGWs)

Atmosphere ◽  
2020 ◽  
Vol 11 (6) ◽  
pp. 653
Author(s):  
Goderdzi G. Didebulidze ◽  
Giorgi Dalakishvili ◽  
Maya Todua
Keyword(s):  
Phase Velocity ◽  
Wind Velocity ◽  
Drift Velocity ◽  
Gravity Waves ◽  
Lower Thermosphere ◽  
Metallic Ions ◽  
Vertical Wavelength ◽  
Vertical Drift ◽  
Sporadic E ◽  
Atmospheric Gravity

The formation of multilayered sporadic E by atmospheric gravity waves (AGWs), propagating in the mid-latitude lower thermosphere, is shown theoretically and numerically. AGWs with a vertical wavelength smaller than the width of the lower thermosphere lead to the appearance of vertical drift velocity nodes (regions where the ions’ vertical drift velocity, caused by these waves, is zero) of heavy metallic ions (Fe+). The distance between the nearest nodes is close to the AGWs’ vertical wavelength. When the divergence of the ion vertical drift velocity at its nodes has a minimal negative value, then these charged particles can accumulate into Es-type thin layers and the formation of multilayered sporadic E is possible. We showed the importance of the ions’ ambipolar diffusion in the formation of Es layers and control of their densities. Oblique downward or upward propagation of AGWs causes downward or upward motion of the ion vertical drift velocity nodes by the vertical propagation phase velocity of these waves. In this case, the formed Es layers also descend or move upward with the same phase velocity. The condition, when the horizontal component of AGWs’ intrinsic phase velocity (phase velocity relative to the wind) and background wind velocity have same magnitudes but opposite directions, is favorable for the formation of the multilayered sporadic E at fixed heights of the sublayers. When the AGWs are absent, then horizontal homogeneous wind causes the formation of sporadic E but with a single peak. In the framework of the suggested theory, it is shown that, in the lower thermosphere, the wind direction, magnitude, and shear determine the development of the processes of ion/electron convergence into the Es-type layer, as well as their density divergence. Consideration of arbitrary height profiles of the meridional and zonal components of the horizontal wind velocity, in case of AGW propagation, should be important for the investigation of the distribution and behavior of heavy metallic ions on regional and global scales.

scholarly journals Rapid quantitative pharmacodynamic imaging with Bayesian estimation

2015 ◽  
Author(s):  
Jonathan M. Koller ◽  
M. Jonathan Vachon ◽  
G. Larry Bretthorst ◽  
Kevin J. Black
Keyword(s):  
Random Noise ◽  
A Priori ◽  
Simulated Data ◽  
Previous Method ◽  
Proof Of Concept ◽  
Computationally Efficient ◽  
Primate Brain ◽  
Simulation Testing ◽  
Novel Method ◽  
Improved Accuracy

ABSTRACTWe recently described rapid quantitative pharmacodynamic imaging, a novel method for estimating sensitivity of a biological system to a drug. We tested its accuracy in simulated biological signals with varying receptor sensitivity and varying levels of random noise, and presented initial proof-of-concept data from functional MRI (fMRI) studies in primate brain. However, the initial simulation testing used a simple iterative approach to estimate pharmacokinetic-pharmacodynamic (PKPD) parameters, an approach that was computationally efficient but returned parameters only from a small, discrete set of values chosen a priori.Here we revisit the simulation testing using a Bayesian method to estimate the PKPD parameters. This improved accuracy compared to our previous method, and noise without intentional signal was never interpreted as signal. We also reanalyze the fMRI proof-of-concept data. The success with the simulated data, and with the limited fMRI data, is a necessary first step toward further testing of rapid quantitative pharmacodynamic imaging.

scholarly journals Retrieval of temperature profiles from CHAMP for climate monitoring: intercomparison with Envisat MIPAS and GOMOS and different atmospheric analyses

2007 ◽  
Vol 7 (1) ◽  
pp. 3229-3268 ◽  
Author(s):  
A. Gobiet ◽  
G. Kirchengast ◽  
G. L. Manney ◽  
M. Borsche ◽  
C. Retscher ◽  
...  
Keyword(s):  
High Altitude ◽  
Weather Prediction ◽  
A Priori ◽  
Michelson Interferometer ◽  
Satellite System ◽  
Altitude Range ◽  
A Priori Information ◽  
Climate Monitoring ◽  
Retrieval Scheme ◽  
Priori Information

Abstract. This study describes and evaluates a Global Navigation Satellite System (GNSS) radio occultation (RO) retrieval scheme particularly aimed at delivering bias-free atmospheric parameters for climate monitoring and research. The focus of the retrieval is on the sensible use of a priori information for careful high-altitude initialisation in order to maximise the usable altitude range. The RO retrieval scheme has been meanwhile applied to more than five years of data (September 2001 to November 2006) from the German CHAllenging Minisatellite Payload for geoscientific research (CHAMP) satellite. In this study it was validated against various correlative datasets including the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) and the Global Ozone Monitoring for Occultation of Stars (GOMOS) sensors on Envisat, five different atmospheric analyses, and the operational CHAMP retrieval product from GeoForschungsZentrum (GFZ) Potsdam. In the global mean within 10 to 30 km altitude we find that the present validation observationally constrains the potential RO temperature bias to be <0.2 K. Latitudinally resolved analyses show biases to be observationally constrained to <0.2–0.5 K up to 35 km in most cases, and up to 30 km in any case, even if severely biased (about 10 K or more) a priori information is used in the high altitude initialisation of the retrieval. No evidence is found for the 10–35 km altitude range of RO bias sources other than those potentially propagated downward from initialisation, indicating that the widely quoted RO promise of "unbiasedness and long-term stability due to intrinsic self-calibration" can indeed be realized given care in the data processing to strictly limit structural uncertainty. The results demonstrate that an adequate high-altitude initialisation technique is crucial for accurate stratospheric RO retrievals and that still common methods of initialising the involved hydrostatic integral with an upper boundary temperature or pressure value derived from meteorological analyses is prone to introduce biases from the initialisation data to the retrieved temperatures down to below 25 km. Above 30 to 35 km, GNSS RO delivers a considerable amount of observed information up to around 40 km, which is particularly interesting for numerical weather prediction (NWP) systems, where direct assimilation of non-initialized (a priori-free) observed RO bending angles is thus the method of choice. The results underline the value of RO for climate applications.

scholarly journals An improved method using adaptive smoothing for GNSS tomographic imaging of ionosphere

PLoS ONE ◽  
2021 ◽  
Vol 16 (5) ◽  
pp. e0250613
Author(s):  
Rushang Jia ◽  
Xumin Yu ◽  
Jianping Xing ◽  
Yafei Ning ◽  
Hecheng Sun
Keyword(s):  
A Priori ◽  
Simulated Data ◽  
Reconstruction Technique ◽  
Electron Content ◽  
Observation Data ◽  
A Priori Information ◽  
Improved Method ◽  
Adaptive Smoothing ◽  
Total Electron ◽  
Priori Information

Global navigation satellite system (GNSS) is a well-established sensors in the recent ionosphere research. By comparing with classical meteorological equipments, the GNSS application can obtain more reliable and precious ionospheric total electron content (TEC) result. However, the most used GNSS ionospheric tomography technique is sensitive to a priori information due to the sparse and non-uniform distribution of GNSS stations. In this paper, we propose an improved method based on adaptive Laplacian smoothing and algebraic reconstruction technique (ALS-ART). Compared with traditional constant constraints, this method is less dependent on a priori information and adaptive smoothing constraints is closer to the actual situation. Tomography experiments using simulated data show that reconstruction accuracy of ionospheric electron density using ALS-ART method is significantly improved. We also use the method to do the analysis of real observation data and compare the tomography results with ionosonde observation data. The results demonstrate the superiority and reliability of the proposed method compared to traditional constant constraints method which will further improve the capability of obtaining precious ionosphere TEC by using GNSS.

Rapid intensification of tropical cyclones: Vortex waves seeded by aurorally-generated atmospheric gravity waves?

2020 ◽  
Author(s):  
Lidia Nikitina ◽  
Paul Prikryl ◽  
Shun-Rong Zhang
Keyword(s):  
Solar Wind ◽  
Tropical Cyclones ◽  
Gravity Waves ◽  
Lower Thermosphere ◽  
Lower Atmosphere ◽  
Rapid Intensification ◽  
Mean Flow ◽  
Atmospheric Gravity Waves ◽  
Convective Bursts ◽  
Atmospheric Gravity

&lt;p&gt;Convective bursts have been linked to intensification of tropical cyclones [1]. We consider a possibility of convective bursts being triggered by aurorally-generated atmospheric gravity waves (AGWs) that may play a role in the intensification process of tropical cyclones [2]. A two-dimensional barotropic approximation is used to obtain asymptotic solutions representing propagation of vortex waves [3] launched in tropical cyclones by quasi-periodic convective bursts. The absorption of vortex waves by the mean flow and formation of the secondary eyewall lead to a process of an eyewall replacement cycle that is known to cause changes in tropical cyclone intensity [4]. Rapid intensification of hurricanes and typhoons from 1995-2018 is examined in the context of solar wind coupling to the magnetosphere-ionosphere-atmosphere (MIA) system. In support of recently published results [2] it is shown that rapid intensification of TCs tend to follow arrival of high-speed solar wind when the MIA coupling is strongest. The coupling generates internal gravity waves in the atmosphere that propagate from the high-latitude lower thermosphere both upward and downward. In the lower atmosphere, they can be ducted [5] and reach tropical troposphere. Despite their significantly reduced amplitude, but subject to amplification upon over-reflection in the upper troposphere, these AGWs can trigger/release moist instabilities leading to convection and latent heat release. A possibility of initiation of convective bursts by aurorally generated AGWs is investigated. Cases of rapid intensification of recent tropical cyclones provide further evidence to support the published results [2].&lt;/p&gt;&lt;p&gt;References&lt;/p&gt;&lt;p&gt;[1] Steranka et al., Mon. Weather Rev., 114, 1539-1546, 1986.&lt;/p&gt;&lt;p&gt;[2] Prikryl et al., J. Atmos. Sol.-Terr. Phys., 2019.&lt;/p&gt;&lt;p&gt;[3] Nikitina L.V., Campbell L.J., Stud. Appl. Math., 135, 377&amp;#8211;446, 2015.&lt;/p&gt;&lt;p&gt;[4] Willoughby H.E., et al., J. Atmos. Sci., 39, 395&amp;#8211;411, 1982.&lt;/p&gt;&lt;p&gt;[5] Mayr H.G., et al., J. Geophys. Res., 89, 10929&amp;#8211;10959, 1984.&lt;/p&gt;

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