Ground-based remote sensing of atmospheric parameters is often limited to single station observations by vertical profiles at a certain geographic location. This is a limiting factor for investigating gravity wave dynamics as the spatial information is often missing, e.g., horizontal wavelength, propagation direction or intrinsic frequency. In this study, we present a new retrieval algorithm for multistatic meteor radar networks to obtain tomographic 3-D wind fields within a pre-defined domain area. The algorithm is part of the Agile Software for Gravity wAve Regional Dynamics (ASGARD) and called 3D-Var, and based on the optimal estimation technique and Bayesian statistics. The performance of the 3D-Var retrieval is demonstrated using two meteor radar networks: the Nordic Meteor Radar Cluster and the Chilean Observation Network De Meteor Radars (CONDOR). The optimal estimation implementation provide statistically sound solutions and diagnostics from the averaging kernels and measurement response. We present initial scientific results such as body forces of breaking gravity waves leading to two counter-rotating vortices and horizontal wavelength spectra indicating a transition between the rotational

The mesosphere/lower thermosphere (MLT) is of crucial interest to understand the vertical coupling between the middle and upper atmosphere. Internal atmospheric waves of various spatial and temporal scales drive the MLT dynamics and thus provide a considerable source of energy and momentum that is carried from the area of their origin to the altitude of their dissipation as shown by numerous experimental and theoretical studies

Our observational and experimental knowledge of GWs and their dissipation in the MLT has been acquired by remote sensing from ground-based and space-borne sensors, each of which have their own unique observational filters or sensor footprints

Ground-based instruments such as Rayleigh lidars and radars obtain high-resolution data of temperature, density and/or wind profiles at a single geographic location

Observations of spatially resolved winds are more challenging. Already

In this study, we present a 3D-Var retrieval algorithm which overcomes many of the limitations of previous analysis routines that retrieved the winds only in independent 2-D layers and required dense observational statistics or were based on the VVP method

The new algorithm was developed from scratch, is based on the optimal estimation approach

The new algorithm is applied to two meteor radar networks to demonstrate its performance using either only monostatic systems or a combination of monostatic and forward scatter passive receivers. The Nordic Meteor Radar Cluster (Nordic; see Table

Technical parameters of the Nordic Meteor Radar Cluster and CONDOR (ALO).

The second meteor radar network is called Chilean Observation Network De Meteor Radars (CONDOR) and employs a monostatic radar at the Andes Lidar Observatory (ALO), a passive receiver at the Southern Cross Observatory (SCO) and another passive receiver at Las Campanas Observatory (LCO). Based on these data sets, we demonstrate the utility of the technique with a series of case studies of body forces, vortices and other spatial features that are not observable by other techniques so far. Furthermore, we show a keogram analysis of a minor sudden stratospheric warming (SSW) in December 2019 and the latitudinal resolved tidal response. Finally, we demonstrate the possibility to obtain horizontal wavelength spectra as already outlined in

The paper is structured as follows. The Nordic Meteor Radar Cluster and CONDOR are described in Sect. 2. Section 3 provides a detailed presentation of the wind retrievals, the World Geodetic System 84 (WGS84) geometry, forward scatter solvers and the optimal estimation technique, which includes the concept of measurement response, variable geographic and Cartesian grids. The performance of the algorithm is demonstrated in Sect. 4, showing examples of body forces, horizontal wavelength spectra and keogram analysis. The results of the retrieval are discussed in Sect. 5, and conclusions are drawn in Sect. 6.

Meteor radar (MR) observations have become a standard tool to investigate MLT dynamics such as winds, tides, GWs and GW momentum fluxes

All MRs used in this study have very similar designs and operation principles, although they are manufactured by two different companies: ATRAD Ltd.

Figure

Nordic Meteor Radar Cluster and CONDOR locations visualized with color-coded mean elevation. The maps were generated from etopo1 using the m_map package

Meteoroids entering the Earth's atmosphere are decelerated and heated by collisions with the ambient atmospheric molecules. Some of them, which have sufficient kinetic energy, are vaporized forming an ambipolar diffusing plasma column, which is called a meteor. MRs measure the line of sight or radial velocity of specular meteor trails that drift with the wind. The position of the meteor in the sky relative to the radar is determined by interferometry and typically given as azimuth and off-zenith angles. MRs have wide fields of view and thus large observation volumes, making them ideal scientific instruments to infer the spatial characteristics of atmospheric parameters.

Instantaneous winds are often derived by applying so-called all-sky fits

Very often MR horizontal winds are obtained under the assumption of

The residual bias for both networks is shown in Fig.

Histograms of the residual vertical wind bias for TRO, ALT, SOD, KIR, SVA and ALO MR.

The histograms are estimated from hourly winds and interpreted as mean residual bias due to the lack of a reliable validation possibility. We observed typical biases of a few cm s

Due to the large observational volume covered by the multistatic MR networks of the Nordic Meteor Radar Cluster and CONDOR, a realistic representation of the Earth's shape is required. Otherwise, large projection errors could result in significant biases. A good approximation of the Earth's shape is given by the WGS84 reference ellipsoid

Here, we briefly summarize how the WGS84 geometry is considered in the wind retrieval. All involved coordinate transformations are given in Appendix

transform geodetic radar coordinates into Earth-centered–Earth-fixed (ECEF) coordinates using transformation geodetic to ECEF (A1);

determine ECEF coordinates of meteor position using transformation ENU to ECEF (A3);

transform ECEF meteor position coordinates into the geodetic location of the meteor using transformation ECEF to geodetic (A2); and

transform observed line-of-sight velocity into the ENU coordinate frame at the meteor location using transformation ECEF to ENU (A4).

The implementation of the WGS84 Earth geometry turned out to be important for all meteor wind analyses, and it essentially reduces the projection and altitude biases of the observed meteors. In particular, at midlatitudes and high latitudes, these corrections are rather significant and lead to substantial improvements of the wind as well as the vertical profile of ambipolar diffusion and thus has an impact on the temperature estimates

Here, we briefly review the forward scatter position determination for multistatic meteor radar observations such as CONDOR. In ASGARD, we updated the algorithm to be more general and applicable to arbitrary meteor radar networks. In Fig.

CONDOR is located at the Andes and some stations are installed at considerable high altitude. The geodetic positions of the transmitter (Tx) at ALO (30.251944

In the next step, we solve the multistatic geometry and determine the Bragg vector, which lies within the plane spanned by the meteor, Rx and Tx locations. The angle

The Bragg vector is derived by transforming the Tx, Rx and meteor positions in ECEF coordinates and by computing the forward scatter angle

Finally, we compute the intersection point between the Bragg vector and the distance vector to perform a sanity check. In Fig.

Atmospheric tomography from multistatic meteor radar networks demands sophisticated mathematical approaches because the number of unknowns exceeds the number of observations. There are several strategies to solve such ill-posed and often ill-conditioned problems by assuming a certain smoothness of the solution or other constraints to cure the mathematical rank deficiency of the problem. Optimal estimation is a technique often applied for atmospheric remote sounding and is described in detail in

The optimal estimation technique makes use of Bayes' theorem and presents a general view to all solutions of an inverse problem

The 2D-Var/3D-Var wind retrievals are implemented using a posteriori PDF covariance

The forward model in the retrieval is given by the radial wind Eq. (

We also implemented an option to include longer correlations or polynomial structure functions, which satisfy

The tomographic retrieval domains can be user defined. Therefore, we implemented two geographic coordinate systems based on a Cartesian and a geographic grid. The Cartesian grid is represented by a geographic longitude and latitude, which defines a

Furthermore, we have to define voxels for each grid point. The default voxel is a cylinder with radius

In Fig.

Image of 3D-Var retrieved wind fields obtained above the Nordic Meteor Radar Cluster using a Cartesian grid

The difference plot shows the variability between both retrievals and provide an estimate on the spatial variance due to the different grids. We obtained a variance of about 20 m s

Furthermore, there is a major dissimilarity between the integral (line-of-sight) irradiance observations from satellites or ground-based radiometers and the differential meteor observations, which have a well-defined location. This has an important implication for the interpretation of the width of the averaging kernel for a given grid cell and retrieved quantity in the 3D-Var retrieval. An increased width of the averaging kernel reflects an interpolation/extrapolation of information from more remote grid cells. From Fig.

In Fig.

Panels

For the Nordic Meteor Radar Cluster, an example of the observed 3D-Var wind field and the corresponding measurement response is shown in Fig.

The same as Fig.

ASGARD offers several possibilities for the a priori state vector. However, considering the optimal estimation technique and the properties of the forward model, which is linear in all coefficients, the final impact of the retrieved parameters appears to be almost independent of the choice of the a priori state vector for all grid cells. This is a major benefit compared to previous retrievals presented in

There are three options available. The first option is a zero-wind state vector for all wind components. The second option is a mean wind vector for the horizontal wind and

Gravity wave dynamics is a vital topic of research. The 3D-Var wind retrievals allow us to study new aspects of spatially and temporally resolved GWs and their interaction with the mean flow.

Recently, there have been several studies on secondary GW generation due to such body forces using a mechanistic model above the Andes and Antarctic Peninsula

Figure

Body force of a breaking gravity wave above CONDOR. Panels

The spatial and vertical dimensions of the vortices and the forcing region are in good agreement with the theoretical work of

The four panels show the measurement response for the zonal and meridional wind components and both altitudes of the body force event above CONDOR.

A major benefit of multistatic observations is the imaging capabilities of large-scale features such as vortices, which remain hidden in monostatic measurements. CONDOR is located at a scientifically interesting region between the low latitudes and midlatitudes. We very often observe strong regional-scale shears in the zonal and meridional wind components associated with vortices. Figure

The two panels present color-coded 3D-Var retrievals of zonal and meridional winds. The black arrows highlight a large vortex above the Andes observed on 14 March 2020.

Besides the GW dynamics, the 3D-Var retrieval also captures spatial information of the regional effects of larger-scale dynamics such as sudden stratospheric warmings or atmospheric tides and planetary waves. The 3D-Var retrieval results in a significant increase of the available information within the domain area. For a temporal resolution of 1 h and a vertical resolution of about 2 km between 70–110 km altitude, we obtain about 480 images per day. If we analyze the wind fields at 10 min resolution, which was occasionally achievable, we generate 2880 images per day. However, for atmospheric tides and planetary waves, it is sufficient to look at certain cross sections of the domain area. This can be achieved by keogram analysis, which provides information about the wind dynamics at the temporal evolution for a given longitude and altitude. Here, we present altitude–time–wind (ATW) plots, which either show the mean winds over the domain area or at a specific longitude and latitude.

The two left panels show altitude–time–wind (ATW) plots for the zonal and meridional color-coded wind strength integrated over the Nordic domain from 20 November to 15 December 2019. The two right panels highlight the zonal and meridional wind strength for the grid cell located around geographic latitude (69.79

In Fig.

Panels

The keograms in Fig.

The four panels show color-coded latitudinal keograms of zonal and meridional winds for 22

In Fig.

The four panels show a ATW measured with CONDOR and the corresponding keogram for the altitude of 92 km. The data were recorded between 1 and 14 March 2020.

Another way to investigate the GW activity from the 3D-Var retrievals is to make use of horizontal wavelength spectra, as presented in

Horizontal wavelength spectra are estimated from the 3D-Var retrievals by computing Lomb–Scargle periodograms

Estimated horizontal wavelength spectra for the Nordic Meteor Radar Cluster. The gray points are individual spectral observations, the thin black line is the daily mean spectrum, the cyan and green lines represents fits to the respective wavelengths scale, and the blue and red lines are idealized slopes for planetary waves (PWs) and GWs.

Vertical velocities are also retrieved from the 3D-Var algorithm. However, we are lacking an independent source for validation, and thus we consider them as an additional quality control rather than a geophysical measurement. The measurement responses for the inferred vertical velocities are rather low, implying that this parameter requires larger correlation lengths compared to the horizontal wind components. We use a priori covariances of 5–7 m s

The histograms shown in Fig.

Histograms of the residual vertical velocity bias derived from the 3D-Var retrieval for three successive days and integrated over the full 3-D domain area.

Finally, we demonstrate the applicability of the 3D-Var algorithm to retrieve large-scale wind fields and meteorological wind maps in an area spanning from the Arctic and south to central Norway. Therefore, we expanded the domain of a geographic grid from 61 to 81

This demonstration retrieval was performed with data from December 2012. During this time, a unique configuration of MRs was operational in the Nordic countries and on Svalbard. In addition to the Nordic Meteor Radar Cluster described previously, the Alta MR was in operation on Bear Island

Figure

Meteorological wind maps obtained from the 3D-Var algorithm using a geographic grid and all MRs involved in ARISE.

When examining the measurement response for such retrievals, shown in Fig.

Measurement responses for the large-scale retrieval visualized with next-neighbor correlation corresponding to about 120 km resolved scales.

Optimal estimation is an established mathematical technique to retrieve atmospheric quantities and results in statistically sound solutions to inverse problems based on a priori knowledge. During the past years, several multistatic meteor radar networks have been deployed. However, these observations are often analyzed using least squares techniques to infer horizontal inhomogeneities in the wind fields

More recently, initial analyses applying inverse theory were presented by

Another benefit of the 3D-Var retrieval is the possibility to derive the measurement response and the averaging kernels. Both quantities provide essential information about the inverse problem and the reliability of the retrieved quantities and how much information is mixed in from either the a priori distribution or due to remote correlations. We presented some examples of measurement responses for CONDOR and the Nordic Meteor Radar Cluster that apparently indicate that not all wind components can be retrieved within the domains assuming just a next-neighbor correlation with equal reliability reflected by a reduced measurement response. The measurement response for forward scatter systems or multistatic passive links is much less homogeneous compared to monostatic systems. This is understandable in the context of the forward scatter ellipse, which basically results in a large area with a small or almost negligible measurement response for the horizontal winds in the region between the Tx and Rx, whereas the horizontal winds are difficult to retrieve only directly above a monostatic radar. The measurement response for CONDOR contains two separated patches of increased measurement response for the zonal and meridional components, which reflects the alignment of the passive receivers in the north and south directions.

The ASGARD software permits us to use different retrieval grids for the 3D-Var, which can be arbitrarily defined, and use either a Cartesian spacing following the Earth's surface or a geographic grid that is regular in longitude and latitude. This flexibility simplifies potential comparisons to other observations or model outputs for validation. Furthermore, large domains are possible that cover large parts of the globe. Also other data sets can be easily included, even if there is no direct overlap between the observation volumes.

Secondary waves and their resulting forcing on the MLT are an emerging topic

Furthermore, we derived horizontal wavelength spectra similar to those presented in

Finally, we want to mention some additional quality controls that are included in the retrieval. We carefully evaluated all systems for potential range offsets that correspond to altitude deviations between the radars within a network. For the Nordic Meteor Radar Cluster, we performed seasonal analyses and looked for systematic deviations of characteristic seasonal structures such as the summer wind reversal in the zonal wind. A similar attempt was pursued for CONDOR but at a much shorter timescale of 14 d by comparing the vertical diurnal tidal patterns between all systems. The interferometry and alignment of the antennas were validated by estimating the vertical wind biases. Some MRs indicate some skewness in the vertical wind histograms, pointing at some systematic differences. However, all systems observed a residual vertical wind bias on the order of a few cm s

In this study, we present a new 3D-Var retrieval algorithm of the ASGARD software, which is designed to infer 3-D tomographic winds from multistatic observations. The retrieval is applied to two meteor radar networks – CONDOR in Chile in the Andes and the Nordic Meteor Radar Cluster. The 3D-Var retrieval is based on an optimal estimation approach and Bayesian statistics. Thus, the retrieved winds are statistically sound solutions considering the a priori knowledge.

The possibility to perform 3-D tomographic retrievals at the MLT on different geographic or Cartesian grids provides more flexibility for the validation of the obtained winds or to combine and extend the domain areas. Furthermore, the results presented indicate that the algorithm is fast enough to run continuously and still to permit the analysis of small-scale structures. The keogram analysis and the ATW plots are suitable to investigate the large-scale dynamics and to provide a quick overview of the network performance.

The 3D-Var retrievals are capable of resolving counter-rotating vortices that might be related to body forces, which is demonstrated in a case study of a westward-propagating GW above the CONDOR network. The algorithm resolved the two counter-rotating vortices and the acceleration direction of the body force. The spatial scales (horizontal and vertical) for the body force event correspond to the expected theoretical results from previous studies. We were also able to observe large-scale vortices of about 300 km in diameter, which would remain otherwise undiscovered.

Due to the optimal estimation implementation of the 3D-Var, it is possible to compute the measurement response for each time and altitude bin. This information is required to identify potential holes or gaps within existing networks that can be closed by installing additional passive systems or monostatic meteor radars. The measurement response of monostatic radars is optimal to ensure unbiased wind inversion and the most isotropic measurement response. Passive multistatic links are ideally used in combination with monostatic radars. Finally, it is worth considering that passive multistatic links alone show an anisotropic measurement response caused by the shape of the forward scatter ellipse that could lead to biases.

The Appendix includes all geometric coordinate transformations that are used in the paper and are also presented in

Radars often are connected to a GPS receiver, which provides the geodetic coordinate in longitude (

The reverse transformation from ECEF to a geodetic longitude (

An object observed at a distance

The conversion of ECEF to ENU coordinates is given by the line-of-sight vector from the radar towards the meteor in the frame of the ENU coordinates at the geodetic location of the meteor. If the line-of-sight velocity vector is observed at a certain azimuth (az) and zenith (ze) angle relative to the radar, it has a different azimuth az

The data are available upon request. Please contact Alexander Kozlovsky (alexander.kozlovsky@oulu.fi) for the Nordic Meteor Radar Cluster and Alan Liu (LIUZ2@erau.edu) for CONDOR to obtain the 3D-Var retrievals.

GS developed ASGARD and the 3D-Var algorithm. AL, AK and QZ supported the development and implementation of the software at CONDOR and the Nordic Meteor Radar Cluster. JK, EB, CH, MT, SN, PE, RH, NM and ML provided meteor radar data from the networks and supported the implementation of the software. All authors contributed to the editing of the manuscript.

Gunter Stober, Juha Vierinen and Jorge L. Chau submitted a patent on multistatic meteor radar observations using combined pulsed and CW transmitters.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Gunter Stober is a member of the Oeschger Center for Climate Change Research (OCCR) at the University of Bern. Most of this software was developed during a sabbatical at the University of Western Ontario. Gunter Stober is grateful to Peter Brown and Robert Sica for the support and discussions during this visit. The authors also want to thank Chris Adami and Ian Reid from ATRAD Ltd., and Brian Fuller and Adrian Murphy from Genesis Ltd. for valuable discussions about some technical aspects of the deployed radars and software. Gunter Stober thanks Andreas Dörnbrack from DLR Oberpfaffenhofen for valuable discussions and helpful comments.

This research has been supported by the STFC (grant no. ST/S000429/1 to Mark Lester) and the Schweizerische Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant no. 200021_200517/1 to Gunter Stober). The Esrange meteor radar operation, maintenance and data collection were provided by the Esrange Space Center of the Swedish Space Corporation. The 3D-Var retrievals were developed as part of the ARISE design study (

This paper was edited by Bernd Funke and reviewed by two anonymous referees.