The mesosphere and lower thermosphere (MLT) cover altitudes of approximately 70–120 km and serve as a key transitional region between the upper and middle atmosphere, which is influenced by forces from above, such as high-energy inputs from solar radiation, and from below through atmospheric waves . This region exhibits complex spatiotemporal dynamics primarily driven by atmospheric waves with different spatiotemporal scales, including gravity waves, tides, planetary waves, quasi-biennial oscillations, and interannual differences, that contribute to the variability in the lower region of the upper atmosphere. These phenomena are crucial for understanding the general circulation of the atmosphere, as the propagation and dissipation of these waves transport energy and momentum horizontally and vertically . The vertical coupling between the lower, middle, and upper atmosphere is strongly influenced by the prevailing wind fields, underscoring the importance of understanding the wind fields in the middle atmosphere . A comprehensive understanding of the MLT is essential to elucidate the wave-driven forcing of the upper atmosphere and the corresponding energy exchange between the upper and middle atmospheres. This wave-driven vertical coupling results in the cold summer mesopause temperatures, which can drop below 140 K at high latitudes, making the summer MLT the coldest place on Earth. The summer MLT deviates up to 100 K from the radiatively determined equilibrium temperature .

Meteoroids are extraterrestrial particles of cometary, asteroidal, or metallic composition that occasionally enter planetary atmospheres . Upon entry, they are referred to as meteors. As they penetrate the atmosphere, they produce an ionized trail due to frictional heating, a phenomenon known as meteor ablation in the MLT. The capability of meteor radars to detect specular or transverse scatter reflections from meteor trails enables continuous monitoring of the MLT, allowing for measurement of various atmospheric and meteor-related parameters, including mean winds and wind fluctuations (momentum flux), neutral air density variability, temperature, and pressure, as well as all parameters associated with the meteoroid origin such as the velocity, mass and source radiant (see e.g., ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; , among others).

During the past decades, Meteor radars have evolved into reliable and robust instruments for atmospheric measurements at the MLT, capable of continuous, automated, and remote operations, functioning uninterrupted as long as a power supply is available, regardless of weather conditions. A typical monostatic meteor radar measures horizontal winds with a temporal resolution of about 1 h, based on a limited number of meteor detections. However, increasing the number of meteors detected within a given spatial volume and time interval improves both accuracy and resolution of the wind measurement, as achieved with multistatic meteor radar networks. Such multistatic meteor radar networks can achieve temporal resolutions of a few minutes .

Over the past five years, several multistatic meteor radar networks have become operational, offering an unprecedented opportunity for observing atmospheric dynamics in three dimensions. These multistatic systems enable the observation of a standard atmospheric volume, which is ideal for inferring the spatial variability of three-dimensional (3D) winds by utilizing advanced statistical inversion methods and tomographic approaches, such as algebraic least squares fits based on the Volume Velocity Processing (VVP) introduced by , which was later applied in various studies (e.g., ; ; ; ) and more sophisticated Bayesian approaches such as the 3DVAR+DIV retrieval algorithms , other Bayesian implementations or machine learning .

The Volume Velocity Processing (VVP) retrieval method is an advancement of the earlier so-called Velocity Azimuth Display (VAD) method . Both approaches apply an appropriate fitting procedure to infer mean winds and to obtain a first-order approximation of the spatial variability of the flow field, under the assumption that these inhomogeneities can be linearly approximated . However, when the MLT winds are assumed to be arbitrary, this method encounters a limitation that can be overcome using the three-dimensional variational (3DVAR) algorithm implemented as part of the Agile Software for Gravity wAve Regional Dynamics (ASGARD) framework, demonstrating that non-linear error propagation, Gaussian weighting, and smoothness regularization can be applied to derive horizontal wind components . The 3DVAR resolves 3D wind fields across all grid cells, incorporating variable geographic and Cartesian coordinate grids. The 3DVAR+DIV method is an advancement of the 3DVAR algorithm , combining the radial velocity observations with the continuity equation. This method represents a transition from purely observation-driven parameter estimation to more physics- and model-based data analysis.

In this study, we present statistics from the Nordic Meteor Radar Cluster (NORDIC) observations covering the period from January 2021–June 2025, along with a Spherical Velocity Volume Processing (SVVP) method based on spherical geometry and a first-order approximation of the ellipsoidal shape of our Earth. We developed this novel algorithm to reduce biases in the estimated vertical winds, which are crucial for inferring reliable momentum fluxes and assessing the mean vertical motion due to the residual circulation. The SVVP algorithm targets the need for a computationally less expensive tomographic algorithm ideal for deriving climatologies or long time series of mean winds and first-order approximations of the wind field within a defined geographic domain, such as horizontal divergence, relative vorticity, stretching, and shearing deformation. We compare the newly developed SVVP with a previous VVP routine to highlight the importance of spherical geometry on derived parameters, including mean 3D winds, horizontal divergence, relative vorticity, stretching, and shearing deformation. We also perform a comparison to the Bayesian 3DVAR+DIV retrieval algorithm to investigate the temporal and spatial variability of the three-dimensional wind field in the MLT. The comparison focuses on statistical analyses, including an initial assessment of vertical winds, which are constrained by the continuity equation and horizontal divergence. This paper is structured as follows. In Sect. 2, we present the data description and statistical overview. Section 3 introduces the Volume velocity Processing on a sphere. Section 4 presents the error analysis and numerical robustness. Section 5 presents the results of wind climatology and statistical analysis. Section 6 provides the discussion. Finally, Sect. 7 summarizes our conclusions. We also provide supplementary material containing additional results for all radii and temporal scales of the VVP analysis.

In this study, we utilize observations from the Nordic Meteor Radar Cluster (NORDIC). A multistatic radar network of meteor radars located across Northern Scandinavia that consists of five monostatic meteor radars namely, Tromsø (69.59° N, 19.2° E; TRO), Alta (70.0° N, 23.3° E; ALT), Sodankylä (67.4° N, 26.6° E; SOD), Kiruna (67.9° N, 21.1° E; KIR), Svalbard (78.2° N, 16.0° E; SVA). However, for this study, we only use the mainland systems, and the Svalbard meteor radar was omitted. The period that is analyzed spans from January 2021–June 2025. Figure  and Table  illustrate the geographical locations of the four meteor radar stations and the technical parameters of the NORDIC Meteor Radar Cluster.

Figure  presents the two-dimensional (2D) horizontal distribution of meteors detected by the radar stations (indicated by color-coded dots) recorded over three days (1–3 April 2021), along with the altitude distribution for the entire study period. The scatter plot on the map illustrates the spatial distribution of meteor echoes, with colors representing the meteor echoes for each station: blue for Tromsø, purple for Alta, cyan for Kiruna, and magenta for Sodankylä. The stations exhibit overlapping coverage regions, with TRO, ALT, and KIR showing substantial overlap with each other compared to SOD, although SOD partially overlaps with KIR. The altitude distributions approximately show a Gaussian profile within the altitude range of 70–120 km, peaking around 90 km for all stations, with median altitudes of 90.8 km for TRO, 91.2 km for ALT, 90.0 km for KIR, and 89.6 km for SOD. Among all the stations, it is observed that the Alta radar station detected the highest meteor count, followed by Tromsø, Sodankylä, and Kiruna.

The altitudinal profile of the long-term changes in the detected meteor echo rates for all stations during the study period, along with the hourly distribution of meteors for 3 d (1–3 April), is illustrated in Fig. . A distinct inter-annual variation pattern is revealed, with a pronounced change in the number of meteor echoes with altitude and season. Meteor echoes are detected between the altitudes of 82 and 95 km and during the summer months for all the radar stations. The white lines represent gaps in the measurements of a system. In particular, the update in Kiruna in 2024 resulted in a gap, and a second gap occurred in 2025. The temporal distributions of selected days exhibit a similar trend, showing a diurnal variation, reaching maximum values in the morning (08:00–10:00 UTC) and minimum values in the evening (17:00–19:00 UTC).

The wind components in the MLT region are obtained using the 3DVAR+DIV algorithm with geographic grid coordinates , VVP, and the novel spherical VVP algorithm described in the next section. The retrievals have temporal resolutions of 15 min, 30 min, and 1 h, covering altitudes from 70–110 km with a vertical resolution of 2 km. The horizontal domain (radius) implemented in the new spherical VVP (SVVP) and VVP algorithm was varied from 200–400 km in steps of 50 km. Unless otherwise specified, the reference results presented in this study correspond to the 250 km horizontal domain for the VVP/SVVP method and a 30 min temporal resolution for both algorithms. For the 3DVAR+DIV retrievals, the Tikhonov regularization parameter (α) of 0.4 and 150 was used to assess their impact on the derived wind fields. The complete set of results is presented as Supplementary material.

In addition, we employ the JAGUAR-DAS whole neutral atmosphere reanalysis (JAWARA) data produced by the Japanese Atmospheric General Circulation Model for Upper Atmosphere Research-Data Assimilation System (JAGUAR-DAS) and the SE-WACCM-X version of the Whole Atmosphere Community Climate Model with thermosphere/ionosphere extension (WACCM-X) model for comparison with the SVVP and VVP output. Both datasets were extracted for the same longitude and latitude as the NORDIC Cluster and matched the altitude range for the VVP and SVVP Model output, as well as the time span of the data used in this study.

Over the past decade, Volume Velocity Processing (VVP) has become a more commonly used approach to assess spatial wind variability in multistatic meteor radar observations . Additionally, the basic idea of the VVP remains consistent with the original publications from . However, the implementation of the algorithm can become more complicated when describing larger domain areas , which can extend horizontally to several hundred kilometers. Here, we will outline the most recent version of our VVP implementation. The revised version builds on previous developments presented in , but is now generalized to spherical coordinates and incorporates higher-order corrections to account for the ellipsoidal shape of the Earth, and not only considers the WGS84 model for each measurement, but also the spherical geometry for the observation volume.

Recalling the original form of the VVP for plane geometry presented in , we can express the wind field by a Taylor expansion consisting of mean zonal, meridional, and vertical wind components and first-order deviations given by linear spatial derivatives; 1u(x,y)=u0(x0,y0)+∂u∂x⋅Δx+∂u∂y⋅Δyv(x,y)=v0(x0,y0)+∂v∂x⋅Δx+∂v∂y⋅Δyw(x,y)=w0(x0,y0). Here u, v, and w correspond to the zonal, meridional, and vertical wind components, respectively. The vertical wind is defined in the ENU (East-North-Up) direction above the WGS84 ellipsoid for the reference coordinate. The quantity represents the vertical motion of the entire cone segment spanned by the circle of influence and the width of the vertical layer (2 km default). The subscript 0 indicates the domain reference coordinate (x0,y0), which is a free parameter that can be arbitrarily selected. However, it turns out that the domain center appears to be a good choice. Thus, Δx=x-x0 and Δy=y-y0 measure the distance to any coordinate (x,y).

After a mathematical expansion and rearranging the horizontal derivatives, the first-order kinematic inhomogeneities of any linear or non-linear wind field can be approximated by the horizontal divergence, the relative vorticity, and the stretching and shearing deformation; u=u0+12∂u∂x+∂v∂y︸hor. divergence⋅Δx-12∂v∂x-∂u∂y︸rel. vorticity⋅Δy+12∂u∂x-∂v∂y︸stretching def.⋅Δx+12∂v∂x+∂u∂y︸shearing def.⋅Δy

2v=v0+12∂u∂x+∂v∂y︸hor. divergence⋅Δy+12∂v∂x-∂u∂y︸rel. vorticity⋅Δx-12∂u∂x-∂v∂y︸stretching def.⋅Δy+12∂v∂x+∂u∂y︸shearing def.⋅Δx.

The advantage of these higher-order kinematic terms is that they can be linked to other dynamic parameters, which are more intuitive for dynamic flow fields than the spatial derivatives alone. Furthermore, horizontal divergence is related to the vertical velocity, which is often more difficult to measure. Assuming incompressibility of the atmosphere permits reducing the continuity equation to the following expression ; 3div(u)=∂u∂x+∂v∂y+∂w∂z=0. The change in the vertical velocity Δw between two layers can then be derived by vertical integration of the horizontal divergence; 4Δw=∫z1z2div⋅vhdz. This vertical velocity profile depends on the boundary condition of the integral, which has to be determined separately. However, the method was often implemented for the data analysis of meteorological wind profilers in the troposphere. In these cases, the initial surface boundary condition for the vertical wind is zero. Furthermore, the Cartesian coordinates and plane geometry are sufficiently accurate for the small measurement domains typically covered by wind profilers, allowing for the reliable measurement of horizontal divergence and, thus, the inference of vertical winds.

However, multistatic meteor radar networks cover much larger domain areas extending over hundreds of kilometers, and, therefore, it is crucial to consider and represent the Earth's ellipsoidal geometry in the retrievals . In particular, vertical winds are very sensitive to projection errors. Tomographic retrievals, such as the 3DVAR and 3DVAR+DIV algorithms, approximate the Earth's ellipsoid shape by the World Geodetic System 84 (WGS84) reference ellipsoid . The horizontal divergence is inferred by calculating the distance between grid cells in zonal and meridional directions to estimate the spatial derivatives, accounting for the change in the Earth's radius with latitude. A similar method was implemented in a revised VVP algorithm dedicated to larger domain meteor radar networks presented in . Including the WGS84 geometry seems to be reliable up to scales of a few tens of kilometers. However, for much larger domains, the horizontal divergence and, thus, the vertical velocity, tend to produce more spurious results.

Here, we introduce a new VVP based on local spherical coordinates to account for the latitude-dependent change of the Earth's radius. In contrast to previous routines, we outline in detail the basic steps and derive the key transformations that facilitate a comparison with the kinematic terms expressed in Cartesian coordinates, as shown in Eq. . Coordinates on a sphere are defined by a radius r, an azimuth ϕ, and a colatitude from the North Pole θ=90°-Φ for a given latitude Φ. For all further derivations, we will keep the mathematical convention. Furthermore, we define the zonal and meridional wind components as ϕ˙ and θ˙, respectively. The vertical velocity is given by r˙. The conversion to the well-known Cartesian vector winds in the ENU (East-North-Up) frame of reference is given by; 5u=rsin⁡θϕ˙v=-rθ˙w=r˙ Furthermore, we need to transform the spatial derivatives into spherical coordinates. It is worth noting that the derivative also acts on the basis vectors. Recalling that a geometric object (vector) is independent of the coordinate system, leading to the identity for any given wind vector in the form; 6u⋅ex+v⋅ey+w⋅ez=u=r˙⋅er-rθ˙⋅eθ+sin⁡θrϕ˙⋅eϕ. Here, (ex,ey,ez) is the basis vector in Cartesian coordinates and (er,eθ,eϕ) denotes the basis vector in spherical coordinates. Considering the additional contributions of the derivatives due to the basis vectors, we can write the spatial derivative terms of our wind vector in spherical coordinates as; 7∂u∂x⋅ex=∂ϕ˙∂ϕ⋅eϕ8∂u∂y⋅ey=(2ϕ˙⋅cos⁡θ+sin⁡θ⋅∂ϕ˙∂θ)⋅eθ9∂v∂x⋅ex=-1sin⁡θ∂θ˙∂ϕ⋅eϕ10∂v∂y⋅ey=(-∂θ˙∂θ-θ˙⋅cot⁡θ)⋅eθ Furthermore, the Δx and Δy terms have to be expressed in spherical geometry as well. We calculate the distance of the arc-segment between two geographic points using the WGS84 reference ellipsoid as implemented in ; 11Δx=rm⋅sin⁡θmΔϕ=rmsin⁡θm(ϕ-ϕ0)Δy=rmΔθ=rm(θ-θ0) Here rm and θm denote the mean Earth radius and mean zenith distance considering the reference geographic location in spherical coordinates (ϕ0,θ0) and any other geographic position expressed in zenith distance/co-latitude (ϕ,θ) (mathematical convention). These coordinates are related to the well-known geographic latitude by Φ=90-θ and ϕ corresponds to the longitude λ. We can now rewrite Eq. () in spherical coordinates and approximate the ENU vector wind (u,v,w) at geographic coordinates (long, lat) by the following set of equations; 12u(λ,Φ)=u0(λ0,Φ0)+∂ϕ˙∂ϕrmsin⁡θmΔϕ-sin⁡θ0⋅∂ϕ˙∂θrmΔθ+2u0⋅cos⁡θ0r0sin⁡θ0rmΔθv(λ,Φ)=v0(λ0,Φ0)-1sin⁡θ0∂θ˙∂ϕrmsin⁡θmΔϕ+∂θ˙∂θrmΔθ+v0⋅cot⁡θ0r0rmΔθw(λ,Φ)=w(λ0,Φ0). Here, we have already introduced some additional sign conventions. The colatitude is defined with the opposite sign compared to the meridional wind, and we replaced the angular velocity with the 3D Cartesian local vector winds in the ENU frame of reference. This step is not trivial, and we cross-checked all terms by comparing them to the standard VVP and by performing the derivation in latitudinal spherical coordinates. Transforming the VVP from Cartesian into spherical coordinates, which are more appropriate for the larger domain size typically covered by multistatic meteor radar networks, results in additional terms. The mean winds have an additional contribution due to the latitudinal extent of the observational domain. The resulting set of equations using latitude as a reference is given in Appendix . The meridional spatial derivatives for the zonal and meridional gradients depend on the mean zonal and meridional wind as well. The above-presented solutions apply to meteor radar networks at high, mid-, and low latitudes in the Northern and Southern hemispheres. However, there is a singular term in the meridional wind equation that results in two singularities, one at each pole. From Eq. (), we can also calculate the corresponding expression of the horizontal divergence (∇hu) and relative vorticity (∇×u)r in spherical coordinates; 13(∂u∂x+∂v∂y)=∇h⋅u=(∂ϕ˙∂ϕ+∂θ˙∂θ-v0⋅cot⁡θr0)14(∂v∂x-∂u∂y)=(∇×u)r=(1sin⁡θ∂θ˙∂ϕ-sin⁡θ⋅∂ϕ˙∂θ+2u0⋅cos⁡θr0sin⁡θ), and the stretching and shearing deformation; 15(∂u∂x-∂v∂y)=(∂ϕ˙∂ϕ-∂θ˙∂θ+v0⋅cot⁡θr0)16(∂v∂x+∂u∂y)=(1sin⁡θ∂θ˙∂ϕ+sin⁡θ⋅∂ϕ˙∂θ-2u0⋅cos⁡θr0sin⁡θ). Considering that line-of-sight velocities or radial velocities are always measured in the ENU frame of reference for each MR, we implemented a fitting scheme around the variables u0,v0,w0,∂ϕ˙∂ϕ,∂ϕ˙∂θ,∂θ˙∂ϕ, and ∂θ˙∂θ and calculated the higher-order kinematic terms from the obtained coefficients. A crucial aspect of the solver is the dependency of the gradient terms on the mean winds, or that the mean wind also contributes to the gradient. Our choice of the fitting coefficients in spherical coordinates avoids solving a nonlinear problem.

In addition to formulating the mathematical equations in spherical coordinates, we retain nonlinear error propagation for all coefficients and the generalized Tikhonov regularization, which was used in previous implementations . The nonlinear error propagation includes the line-of-sight velocity uncertainty from the recorded data, an angular uncertainty of 2° for the zenith and azimuth angles, and the temporal and spatial shear terms as introduced in , , and . The nonlinear error propagation is solved iteratively, as the mean wind directly scales the angular uncertainties. We apply up to 8 iterations to obtain a stable solution for the VVP and SVVP. More details are provided in Appendix .

The Tikhonov regularization also remains valid for the spherical-coordinate implementation in obtaining reliable vertical velocities. We calculate a predictor from the grid cells above and below a reference coordinate in altitude as well as in time, using a time step before and after. This regularization approach helps to cure ill-conditioned spatial sampling. However, the vertical wind shows a much better agreement between the direct Doppler-derived vertical velocity and the horizontal divergence-inferred vertical wind, providing confidence in the overall reliability of the implementation. The climatologies and long-term measurements presented in the results section are based on the Doppler-derived measurements, as a detailed analysis of the divergence-estimated vertical winds is beyond the scope of this paper. The divergence-estimated vertical winds require a lower or upper integration boundary, which is not straightforward to implement.

We also determine a mean seasonal or long-term vertical wind bias based on the considerations presented in . The underlying assumption is that multi-year or long-term vertical winds should have a vanishing mean. Considering that even a mean deviation in the vertical wind of 1 mms-1 over one year would result in enormous vertical motions of 31 kmyr-1. Such large vertical motions of air packages are not observed nor predicted by GCMs. In this study, we estimated the bias from even longer time series. Typical bias values are calculated for each altitude and range from 1–8 cms-1 and are subtracted to obtain the vertical wind speeds shown in the climatology and multi-year data. The altitude-dependent bias is estimated from; 17wbias(z)=1N∑tw(z,t), and subtracted from each altitude separately. In addition, we estimate the standard deviation, which corresponds more to the variability of the vertical wind rather than a statistical uncertainty of the mean. Furthermore, we can justify our bias correction by considering previous results. The most precise and accurate measurements of the vertical motion of the meteor layer are obtained by tracking the mean ablation altitude . These studies suggest that the middle atmosphere is shrinking by a maximum of 850 mperdecade. This translates to a mean vertical velocity of 2.6×10-6ms-1. Thus, it is reasonable to consider any substantial deviation from this value over a period of 5 years or from the climatological mean at each altitude as bias, which can be subtracted. This method is valid under the assumptions that the radars retain a constant sensitivity and that the statistical properties of the sporadic meteoroid population have no significant trend concerning the chemical composition, velocity, or mass of the meteoroids.

The bias correction is also more relevant for the Doppler-derived vertical velocity, which directly depends on the pointing accuracy. Considering typical wind speeds ranging from 10–100 ms-1 (planetary waves + tides + gravity waves) would introduce a vertical velocity of a few cms-1, assuming a pointing error of 0.1°. However, each system likely has an uncorrelated pointing error compared to the other radars in the network. Currently, we estimate an average difference in the source radiant position for different meteor showers (data not shown) of ±0.5° between the systems, which falls into the range of the applied bias correction.

SVVP retrievals are rather robust for the mean winds (u,v,w) concerning the number of radars in the network. There is almost no impact on those parameters when one of the systems is not operating. However, the gradient terms are rather sensitive to missing data in one section of the domain. Tromsø and Alta MR have a substantially higher number of detections (typically twice as many) compared to Kiruna and Sodankylä. As a result, the retrieved quantities require at least three operational radars, including Tromsø and Alta. A leave-one-station-out experiment resulted in a marginal degradation of the vertical wind magnitude comparable to the scaling behaviour between 15 and 30 min temporal resolution (See Supplement).

A more detailed diagnostic comparing planar and spherical geometry can be obtained by looking at the fitting residuals. Figure  shows the fitting residuals for two cases. The distance from the zero line of the residuals indicates a deviation from the model assumption. A perfect model would produce residuals with error bars that overlap the zero line. Linear trends or other features, such as a sinusoidal pattern, are often an indication that the model fit is not capturing all the information that is available in the data. The first case (upper panels) shows a typical situation where both fits exhibit a normal distribution around the zero line. However, VVP fits tend to produce more extreme values for some measurements, which mostly vanish in the SVVP implementation. Furthermore, the fitting errors for the mean wind components are often smaller using the SVVP geometry compared to the planar fit. The remaining standard deviation reflects the small-scale fluctuation inside the circle of influence, which is resolved by the 3DVAR+DIV, but acts as a noise source for the VVP and SVVP. As the standard deviation between VVP and SVVP exhibits only small differences, the overall errors are very similar for both model fits. The lower two panels c) and d) highlight a case where the first-order wind field approximation fails.

This example is taken from the 60 min time resolution analysis and outlines the issue with two long temporal averages. The residuals indicate a sinusoidal oscillation likely related to a gravity wave moving through the domain. Again, the statistical uncertainties of the mean wind components for the SVVP are smaller compared to the planar implementation. The standard deviation from the zero line is more of an indication that more advanced retrievals could further reduce the scattering around the zero line. The fitting is implemented using an SVD solver (Fortran 77, adapted to MATLAB). The operational retrieval version does not store the SVD decomposition for each fit. Although the residuals indicate that there are cases where the first-order approximation starts to be no longer appropriate, it is expected that seasonal and climatological means hold. In particular, the 15 and 30 min temporal resolution data appear to be less affected compared to the 60 min temporal resolution. This also partially explains the scaling behaviour of the vertical winds with temporal resolution and spatial domain size, resulting in only partial sampling of wave periods or spatial scales.

However, to underline the robustness of the SVD, we present the normalized singular values for a single day in Fig.  to estimate also the condition number.

The condition number is given by the largest and smallest singular values. For single precision, the recommended truncation to avoid numerical noise affecting the fitting results is 1×10-5 . However, we have implemented the algorithm in double precision on an HPC. The singular values indicate that there is no rank deficiency for all fits that pass our quality control. The black dots represent mean values, and the grey shaded area visualizes the spread for all fits during one day.

The main benefit of the SVVP implementation is the error propagation, which permits estimating statistical uncertainties for all quantities. Both horizontal wind components show a peak of the Poisson distribution below 1 ms-1 and a width of less than 5 ms-1 in the tail. These statistical uncertainties are much smaller than the values obtained from monostatic all-sky fits with a similar retrieval setup. Monostatic analysis results in statistical uncertainties of 2–14 ms-1. Only the vertical winds require some additional considerations for the error analysis. The algorithm uses a Tikhonov regularization, which mathematically acts similarly to a Bayesian retrieval for quantities with very low measurement response. The Doppler vertical winds show such a low measurement response, often at the lower and upper altitudes of the meteor layer; thus, the solution is dominated by the Tikhonov regularization (implicit apriori).

Figure  shows normalized histograms of the statistical uncertainties for the zonal, meridional, and vertical winds (without regularization). The effect of the Tikhonov regularization is basically given by the Lagrange multiplier (α=1×10-2), which acts as an inverse covariance (a priori covariance) . The covariances for the gradient terms reflect a Poisson distribution similar to the zonal and meridional winds. The statistical uncertainties for all fitted gradients show similar peak values and width of the error distributions, indicating that all quantities are rather well-constrained by the network and distribution of the meteor detections inside the field of view.

The covariance for the gradient terms also provides a benchmark for how the SVVP or VVP can be used to reconstruct the spatially resolved wind field . The statistical uncertainty for the wind field increases with increasing distance from the reference point. This can be used as a threshold criterion for the largest accepted error for a given volume. Furthermore, the covariance permits extrapolating the statistical uncertainties for specific locations for dedicated comparisons such as Lidars, which often have a much smaller field of view .

In this section, we first present the seasonal and monthly mean wind variations and higher-order kinematic fields in the MLT region, based on the VVP plane geometry, spherical geometry (SVVP), their corresponding differences, and the 3DVAR+DIV method. We compare the SVVP and VVP model wind retrievals with the JAWARA and SE-WACCM-X datasets. We established a common data analysis pipeline to harmonize and reduce the data, removing the geophysical variability with periods shorter than a month. Mean winds, tides, and gravity waves are removed using the adaptive spectral filter . We compute 31 d average zonal, meridional, and vertical daily mean winds with a boxcar for each altitude. All algorithms leverage identical nonlinear error propagation and data curation to avoid impacts on the results due to data gaps or other data filtering. The comparisons solely show the impact of the different geometries and spatiotemporal filters (circle of influence). The seasonal comparisons are based on a 31 d window with periodic boundary conditions. A similar procedure was applied in a previous study . The median is computed when 40 % of all possible measurements for a time-altitude bin are available, but a reliable monthly mean is only obtained with a data coverage of 80 % or not more than 11 missing days per month. This threshold is considered the minimum required by the World Meteorological Organisation (WMO) for climatologies. All comparisons are conducted based on a 30 min temporal resolution, with a radius of 250 km for the SVVP, as a baseline. The radius of 250 km for the SVVP appeared to provide the best overlap between all four MR without including large blank spaces in the domain (see Fig. ).

Multistatic meteor radar networks are increasingly being established worldwide. These networks also cover much larger domain areas and have rather different layouts, considering the combination of passive and active radars. The Nordic Meteor Radar Cluster is unique in that it only includes monostatic radars with good overlap in the network , although passive receiver stations can be added in the future. Multistatic networks also offer the possibility to implement different mathematical approaches to infer 3D winds, such as the Bayesian 3DAVR+DIV retrieval or algebraic least-square fits based on different implementations of the VVP .

The 3DVAR+DIV retrieval solves the ill-posed and ill-conditioned tomographic problem for a pre-defined geographic domain. The main advantage of the method is the retrieval of arbitrary wind fields and small-scale features or non-linearities in the wind field. Considering the entropy definition introduced by and , we calculate the Shannon measurement response, an additional diagnostic to estimate the mixing of a priori information and contributions from the measurements on the final solution of the state vector (see Fig. ).

In this study, we used the measurement response as a proxy to determine the most reliable circle of influence for the SVVP. This diagnostic also reveals correlations inside the domain volume due to the implemented mathematical smoothness constraint. These correlations are estimated by averaging kernels, a standard diagnostic in many remote sensing applications . An increased measurement response is important to ensure that each fitted parameter can be estimated from the observations. This is, in particular, worth considering for mathematically ill-conditioned problems. Due to the random occurrence of meteors inside the measurement domain, there might be situations where not all three wind components can be determined with the same precision and accuracy. Typically, ill-conditioned parameters appear to be inflated. The VVP and SVVP are based on least-squares fits and approximate the spatial variability by expanding the wind field into a Taylor series. The coefficients are solved by an algebraic solution to a linear set of equations. The advantage of the VVP/SVVP is the reduced computational resources compared to a 3DVAR+DIV retrieval. In this study, we present a novel implementation of the VVP in spherical coordinates (SVVP) and approximate the ellipsoidal shape of the Earth for iso-surfaces based on WGS84 coordinates . There have been previous implementation attempts of the VVP in spherical coordinates by mixing Cartesian and spherical coordinates . However, our resulting equations differ substantially in the gradient terms. A comparison of the VVP in plane geometry and the newly developed SVVP shows remarkable differences in the higher-order kinematic terms, such as horizontal divergence, relative vorticity, and stretching and shearing deformation. The horizontal wind components exhibit smaller but still noticeable deviations. In particular, the horizontal divergence shows a characteristic altitude variation in plane geometry with positive values above 90 km during the entire year.

Such a vertical structure also leads to biases in vertical winds derived from horizontal divergence, resulting in climatological vertical wind speeds of several ms-1, which seem physically unrealistic for a seasonal value . The implemented SVVP shows a clear seasonal pattern and no systematic altitude structure for the horizontal divergence, which also results in much more realistic seasonal vertical winds in the order of several cms-1. The SVVP algorithm performs best on seasonal time scales when the underlying assumptions are statistically valid. However, on short time-scales, the algorithm might not capture non-linear gradients induced by certain gravity waves or bores . Furthermore, the gradient terms and mean winds are physically only meaningful for the selected reference point.

The vertical winds calculated in this study were debiased following information theory . The most robust estimate of the vertical motion over longer periods is provided by the meteor ablation altitude and reaches values of a few hundred meteors per decade . Therefore, it is reasonable to subtract this mean vertical wind from the climatology and the long-time series to remove potential offsets in the network due to the alignment of the antennas or possible phase offsets in the interferometry. These bias corrections are almost constant with altitude and in the order of 1–8 cms-1 for diameters of 200 and 250 km in the SVVP analysis. Larger domains tend to produce also larger and altitude-dependent bias corrections.

We also performed a sensitivity study, defining circles of influence around the selected reference coordinates of 200–400 km in radius with a step size of 50 km. Furthermore, we analyzed all the measurements collected by the Nordic Meteor Radar Cluster from 2021–2024 using temporal resolutions of 15, 30, and 60 min to determine the impact of temporal averaging. For the comparison, we selected a temporal resolution of 30 min as a reference. The temporal resolution had almost no influence on the derived horizontal wind components and resulted in a correlation of 0.96 for the zonal and 0.91 for the meridional wind.

However, the vertical wind shows clear scaling behavior, with larger values at the lower temporal resolution and smaller values in the 15 min analysis. Spectral analysis of horizontal and vertical winds (see Fig. ) shows that all retrievals indicate a -5/3 slope for the gravity wave range.

Furthermore, lidar observations from McMurdo have shown that the largest vertical velocities occur at the smallest scales . However, due to the sparser sampling, these larger vertical velocities start to be undersampled at the 15 min temporal resolution. This is, in particular, the case during times with fewer meteor detections (afternoon time in UTC), which results in an underestimation of the retrieved vertical winds. The horizontal mean winds are less affected and indicate only a rather small effect (see Fig. ). A temporal resolution of 60 min already suffers from smearing effects. Fast small-scale gravity waves (ducted gravity waves) have substantial phase speeds and can propagate up to 150–200 km within one hour through the domain, resulting in sampling issues that inflate the observed vertical winds (opposite effect to the 15 min retrieval). Currently, the 30 min temporal resolution appears to be the most robust retrieval considering sufficient sampling and temporal resolution.

Comparing different circles of influence for the Nordic Meteor Radar Cluster reveals the best consistency for radii of 200 and 250 km, with a tendency towards systematic deviations at increasing radii. In particular, the horizontal divergence, and consequently the vertical wind, appears to be inflated for larger circles of influence compared to the smaller domains with 200–250 km radius. Only the 200–250 km circle is truly multistatic; when we increase the volume even more, not all wind components are equally well determined in all parts of the domain. This becomes clearer when looking at the measurement response from the 3DVAR+DIV shown in Figs.  , and .

Only the 200 and 250 km circles of influence exhibit an almost homogeneous spatial distribution in the measurement response, whereas the 400 km circle indicates areas with dark blue color or lies even outside the pre-defined 3DVAR+DIV domain. Furthermore, when the domain size is increased too much, additional projection errors due to the elliptical shape of the Earth become significant, which are not yet accounted for in the derived equations but are crucial for determining reliable vertical winds. Vertical winds are also more sensitive to the overall statistics inside the domain; therefore, the upper and lower edges of the meteor layer are problematic due to the lower number of meteor detections, and thus the vertical winds are only trustworthy between 85–95 km altitude.

Another important aspect of the SVVP and also the 3DVAR+DIV algorithm is their applicability to other meteorological networks, such as weather radars or multistatic Lidar wind measurements in the boundary layer. Both algorithms require information about the range, time, angle of arrival, and line-of-sight velocity. The domains can be fairly flexible, adapted to specific needs. However, both algorithms can get computationally expensive when applied to very large data sets or very high-resolution domains, e.g., a weather radar network with a domain grid of 1–2 km spatial resolution and a few hundred meters vertical spacing between layers. However, it could be beneficial to resolve small-scale updrafts in thunderstorms or other small-scale dynamics that are often hidden or unresolved in standard data analysis.

Accurate estimation of three-dimensional wind fields in the mesosphere and lower thermosphere (MLT) is crucial for advancing our understanding of atmospheric dynamics and variability. However, it has posed a longstanding significant challenge in atmospheric science, particularly regarding the reliable retrieval of vertical wind. The MLT region is influenced by radiative forcing from both above (space) and wave-driven dynamics from the middle atmosphere, posing challenges for GCMs to reproduce a realistic mean state of the dynamics . Neutral winds in the MLT play a vital role in mediating the transport of energy and momentum, thereby coupling the upper and lower atmosphere and shaping the global circulation patterns.

In this paper, we present 3D spatially resolved wind fields in the MLT using advanced retrieval techniques. Specifically, we introduce a novel SVVP method in spherical coordinates to improve the physical consistency of wind retrievals, and we compare its performance against the conventional VVP method in plane geometry and the 3DVAR+DIV method. Utilizing meteor radar observations spanning from 2021 to summer 2025 from the Nordic Meteor Radar Cluster, we analysed data statistics, seasonal mean wind characteristics, and inter-method correlations. Wind retrievals were performed with temporal resolutions of 15, 30, and 60 min and radii ranging from 200–400 km in steps of 50 km. The 30 min temporal resolution was used as the reference to evaluate the impact of temporal averaging. All retrieval methods reproduced the expected climatological pattern of mean winds; however, correlation analyses revealed some discrepancies among methods. The 3DVAR+DIV approach better captures small-scale variability and is more suitable for arbitrary wind fields, whereas the VVP/SVVP methods produce a smoother wind field and are computationally less expensive. Notable differences emerged in the wind components and the higher-order kinematic variables between the plane and the spherical implementations of the VVP. The horizontal wind components showed smaller but still systematic deviations. In particular, the horizontal divergence in the plane geometry indicated a characteristic altitude variation with positive bias throughout the year, pointing to systematic issues. The SVVP exhibited a clearer seasonal pattern and no systematic altitude structure, which also resulted in more self-consistent seasonal variations in vertical winds in the order of several cms-1, which falls within the range of theoretical expectations from SE-WACCM-X (grid size comparable to the 3DVAR+DIV retrievals) and JAWARA. Our findings highlight the importance of a correct geometric representation of the implemented retrieval and temporal averaging to ensure sufficient statistics to obtain reliable MLT winds. The newly improved SVVP method demonstrates enhanced physical consistency, although an absolute validation requires alternate observations to verify the obtained results. It provides a robust framework for future investigations of MLT dynamics, including wave-mean flow interactions and momentum coupling processes in the middle/upper atmosphere for longer time series. In contrast, the 3DVAR+DIV retrieval is beneficial for high-resolution case studies.