In order to evaluate the radar echoes detected by two different systems in terms of strength, underlying backscatter mechanism and frequency, the received signals were converted to radar volume reflectivity η. Radar volume reflectivity is defined as the power which would be scattered if all power were scattered isotropically with a power density equal to that of the back-scattered radiation, per unit volume and per unit incident power density . In other words, it describes how much power a small volume scatters evenly in all directions relative to the incoming radar power. This leads to the following expression: 1η=Pr128π22ln⁡(2)r2PtGtGrλ2eθ[1/2]2cτ where r is the range to the scatterers; Gt and Gr denote the one-way gains of the transmitting and receiving antennas, respectively; θ[1/2] represents the effective half-power half-width of the combined transmit–receive antenna beam; λ is the radar wavelength; e is the system efficiency, primarily accounting for losses within the antenna feed system; Pt and Pr are the transmitted peak power and the received signal power, respectively; c is the speed of light; and τ is the effective pulse width . The factor 2ln⁡(2) serves as a correction term that compensates for the non-uniform antenna gain across the half-power beamwidth .

The conversion of the received signal to an absolute quantity, namely radar reflectivity, requires precise calibration of the radar system, which serves as the fundamental basis for the quantitative interpretation of radar experiments conducted with ALWIN and MAARSY. Detailed descriptions of two calibration techniques routinely employed for these systems are provided in . All system-dependent parameters in Eq. () can be concentrated into a single system constant, csys. In this paper only vertical sounding is considered. Consequently, the radar reflectivity η becomes a function solely of the height of the scatterers, z, and the absolute value of the received signal power, Pz. 2η=Pz⋅csys⋅z2

The PMSE datasets used in this study consist of radar volume reflectivity measurements averaged over 5 min and 300 m, both for ALWIN and MAARSY. To ensure data reliability and remove outliers, PMSE occurrences were identified and flagged within the datasets. A PMSE event was defined as an increase in radar reflectivity above the detection threshold that persisted for at least 20 min, corresponding to four consecutive averages within a single range gate.

Figure  shows a histogram of the total number of PMSE events as a function of volume reflectivity for the entire data set (1999–2025), as well as the contributions from ALWIN (1999–2008) and MAARSY (2010–2025). The histograms for each year were also determined separately and then averaged for all years (right panel in Fig. ). Differences in minimum signal detectability, which controls the left-hand slope of the distributions in both panels in Fig.  are primarily related to antenna array size and peak transmitted power. In addition, variations in radar experiment configurations, particularly changes in receiving antenna setups and the number of coherent integrations used during specific campaigns, also contribute to different detectability . Variations in other system parameters were corrected for using the system factor in Eq. () and through receiver calibration.

The mean volume reflectivity distribution shown in Fig.  spans from the radar detection limits up to maximum values of approximately 10-9 m⁻¹ for both ALWIN and MAARSY. For ALWIN, the distribution peaks at about 7.9×10-15 m⁻¹, whereas the peak for MAARSY occurs at a lower value of 1.6×10-15 m⁻¹. The volume reflectivity quantiles are listed in Table . For ALWIN, the weakest 1 % of detected echoes (Q0.01) exhibit reflectivities of η≤1.3×10-16 m⁻¹. In contrast, the corresponding MAARSY value Q0.01 is substantially lower, with Q0.01=7.9×10-18 m⁻¹ reflecting the radar's superior sensitivity.

To obtain an occurrence rate (OR) analysis of PMSE which is as unbiased as possible with respect to seasonal and diurnal variability, and in order to allow for a qualitative comparison with other studies e.g. a lower threshold was applied which is defined as follows (see also Fig. ): 3ηmin⁡=10-15⋅z2(85km)2m-1 which takes into account the height dependence of η as introduced in Eq. () (z = height in km). A detailed analysis of the occurrence rates from ALWIN and MAARSY is shown in Fig.  where a common threshold of η≥ηmin⁡ has been applied. The average seasonal occurrence rate (blue curve in the top panel) is consistent with, and directly comparable to, the results reported in previous studies e.g.. PMSE with volume reflectivity η≥ηmin⁡ were observed, on average, between day 136 (15 May) and day 241 (28 August). The standard deviations associated with the start and end dates are 6.36 and 4.51 d, respectively, yielding an average PMSE season length of approximately 104 d. During the core period from mid-June to mid-July, the average occurrence rate is about 83 %. The earliest onset of the PMSE season at Andøya was recorded on day 126 (2 May 2018), while the latest onset occurred on day 155 (4 June 2002). The earliest seasonal termination was observed on day 230 (18 August 2010), and the latest on day 249 (6 September 2022). These values are summarised in Table .

The altitude distributions of the PMSE occurrence rates of individual years and their average value are also shown in Fig. . They are characterized by a nearly symmetric, Gaussian-like shape. The maximum value of the mean altitude distribution occurs at an altitude of 84.7 km with a standard deviation of 412 m.

The mean diurnal, altitude-resolved variation of PMSE occurrence (see Fig. ) reveals a region of enhanced occurrence between 83 and 87 km. This region emerges shortly before midnight LT and persists until approximately 15:00 LT. During the afternoon hours, the occurrence frequency exhibits a pronounced decline. The daily variation in PMSE occurrence for the individual years as shown in the top panel of Fig.  reflects the measurement time; that is a value of 100 % per range gate means that an echo has been recorded in the same range gate during one of the 5 min daily time intervals on all days with available measurements. The mean diurnal PMSE occurrence in the top panel of Fig.  exhibits a pronounced pattern, with large variability around a mean value of approximately 20 % between midnight and 17:00 LT. During the morning hours, the mean diurnal cycle shows a weak maximum at 05:09 LT. This feature is not well defined, as indicated by the large spread of the individual annual maxima. All yearly mean curves show a clear maximum between 11:00 and 15:00 LT, with an average peak occurring around 12:59 LT. The curves show a pronounced minimum between 17:00 LT and midnight. The 26-year mean occurrence rate decreases sharply during this interval, reaching about 10 % at 20:39 LT, before increasing again to approximately 20 % by around 01:30 LT. Potential reasons for this variation will be discussed in a later section.

In order to investigate long-term trends of PMSE appearance over Andøya, the start and end dates and the resulting length of the PMSE season were determined. The upper panel of Fig.  illustrates the start and end dates of the PMSE seasons from 1999 to 2025, derived from daily PMSE occurrence rates exceeding the threshold of ηmin⁡ (see Eq. ). Both the onset and termination of the PMSE season exhibit pronounced interannual variability, with particularly late onsets observed in 2002 and 2010. Linear trend analysis reveals a negative trend for the season onset (-0.22 d yr⁻¹) and a positive trend for the season termination (0.24 d yr⁻¹). These opposing trends, most notably the progressively earlier onset of the PMSE season, also impact the overall season duration (see Fig. ).

For the study of the long-term behaviour of PMSE occurrence rates, a procedure previously applied by and later by was applied. From the daily PMSE occurrence rates with a signal strength of η≥ηmin⁡ mean values were calculated for each year for the period from 1 June to 31 July. The data set was extended by using SOUSY measurements for the years 1994–1997, as described above. The extended data set comprises a total of 30 years of measurements between 1994 and 2025 in Andøa over a period of 32 years, making it the longest measurement series of its kind worldwide.

The extended time series of mean PMSE occurrence rates (OR-15) is shown in the middle panel of Fig. . The dashed black line indicates a positive linear trend of 0.28 % yr⁻¹. Corresponding mean values (June/July) of the solar Lyman-α (Ly_α) radiation and the geomagnetic Ap index are also shown. Solar Ly_α radiation is the dominant ionisation source (via ionisation of nitric oxide) in the undisturbed ionospheric D region, whereas the geomagnetic Ap index may serve as an indicator of precipitating high-energy particle fluxes . The correlation coefficients between the mean PMSE occurrence rates and the corresponding Ly_α and Ap index averages are r=0.32 and r=0.39, respectively, indicating that PMSE occurrence depends on both solar activity and high-energy particle fluxes.

The solar- and geomagnetically induced components were removed using a simple twofold regression analysis. Mean PMSE occurrence rates, denoted as ORmod, were estimated as a function of Ly_α and the geomagnetic Ap index: 4ORmod=a+b⋅Lyα+c⋅Ap

The estimated values ORmod were then subtracted from the observed occurrence rates OR-15: 5ΔOR=OR-15-ORmod

The resulting residuals, ΔOR, are shown in the lower panel of Fig.  and exhibit a clearly positive linear trend of 0.3 % yr⁻¹.

To ensure consistency with previous studies by and recently by , an additional regression analysis using a threefold formulation, 6ORmod=a+b⋅Lyα+c⋅Ap+d⋅t, was applied to the mean values of Ly_α, Ap, and PMSE occurrence rates OR-15. In this formulation, the partial regression coefficient d (in % yr⁻¹) directly represents the long-term trend in PMSE occurrence and can directly be compared with the trend derived from the residual PMSE occurrence rates ΔOR obtained from the twofold regression analysis (Eqs.  and ). In addition, the methods already used in were also applied, which are based on Eqs. () and () and only consider the influence of Ap, while Ly_α is neglected and vice versa.

The results from the five applied methods are summarised in Table . The first trend (0.28 % yr⁻¹), shown in the middle panel of Fig. , was obtained directly from the PMSE occurrence rate OR-15 without accounting for the effects of Ap or Ly_α. The second (0.33 % yr⁻¹) and third (0.25 % yr⁻¹) trends were derived after removing only the geomagnetic contribution (OR(Ap)) or only the Ly_α contribution (OR(Ly_α)), respectively. The fourth trend (0.30 % yr⁻¹), depicted in the bottom panel of Fig. , was estimated after eliminating both solar and geomagnetic influences (OR(Ly_α, Ap)) using Eqs. () and (). The fifth trend (0.35 % yr⁻¹) resulted from the threefold regression analysis (OR(Ly_α, Ap, t)) described in Eq. (). In summary, all trends reported in Table  are positive and cluster around 0.3 % yr⁻¹.

In this section, potential trends of radar reflectivities, η, are discussed. More precisely, the temporal behaviour of yearly mean η-values, ηz, at the maximum of the distribution shown in Fig. , namely at 84.0 km, are investigated. As can be seen in Fig. , the mean of all yearly ηz-values is 10-13.52=3.02×10-14 (here, and in the following, all η-values are in units of m⁻¹). We will later use the highest and lowest values in the entire time series as an indication of the spread of ηz. The corresponding values are ηzhi=10-13.17=6.76×10-14 from year 2012, and ηzlo=10-13.83=1.48×10-14 from year 2020, respectively. There is a very weak negative trend in Fig.  with a slope (on the logarithmic values) of m=-0.0064±0.0039 [(m⁻¹) yr⁻¹], which is not significant. Standard deviations in each year determined on the logarithmic values are also shown. Typical values are 0.5, i.e., the variation of mean reflectivies within a year is on the order of a factor of 3.

A potential correlation of ηz with solar activity, namely with the solar flux at 10.7 cm (F10.7), was investigated. As can be seen in Fig. , there is a relevant positive correlation (correlation coefficient r = 0.46). Since electron densities in the PMSE region are expected to increase with increasing solar activity, and since radar reflectivity depends on electron density, one would ostensibly expect a positive correlation, as observed. It should be noted, however, that the dependence of η on electron densities is rather complicated since it involves further parameters, such as the gradient of electron densities. This topic will be investigated in more detail in the future.

The study was extended to other altitudes close to the height of maximum reflectiviy. It turns out that the results presented above are representative for the entire height range close to the maximum reflectivity.

To investigate the relationship between potential trends of PMSE and background conditions, the crucial parameters impacting η are recapitulated in the following. The main idea is that in the presence of neutral air turbulence and charged ice particles, the spectrum of electron density fluctuations at the radar Bragg scale is enhanced so that significant backscatter is created. There are several models describing this effect. In this paper a model going back to is applied, which has frequently been used in PMSE studies see, for example,where also alternative models are discussed. In the Batchelor model the radar reflectivity η(k) at the radar Bragg wavenumber, k, is given by:

7η(k)=8π3⋅fαqRiPr⋅re2⋅ϵνωB2⋅Me2⋅k-3⋅exp⁡-q⋅(ηKol⋅k)2Sc8Me=ωB2Neg-dNedz-NeHn where Me is the reduced potential refractive index gradient see.

The parameters in these equations are discussed in more detail in the references given above. The values used here (typical for the PMSE region) are as follows: fα=2; q=2; α=0.83; Ri = 0.81 = Richardson number; Pr = 1 = Prandtl number; g=9.81 [m s⁻²]; re=2.82×10-15 [m] = classical electron radius; ωB=2π/(5×60)∼0.02 [1 s⁻¹] = Brunt-Väisälä frequency; Hn=7000 [m] = neutral density scale height; ν=0.8 [m² s⁻¹] = kinematic viscosity; ϵ=50 [mW kg⁻¹] = turbulent energy dissipation rate; ηKol=(ν3/ϵ)1/4 = Kolmogorov microscale. For the Schmidt number, Sc, the approximation Sc=6.5⋅rice2 (rice = ice particle radius in nm, see ) is used. Furthermore, the electron density, Ne, and the electron density gradient dNe/dz are needed. In this paper the values Ne=1×109 [m⁻³] and dNe/dz=2.4×105 [m⁻⁴] are used which are compatibel with the empirical reference model FIRI and fit the observations from MAARSY.

In Fig.  spectra of reflectivities for various ice particle radii, i.e., various Schmidt numbers, are shown. It is obvious from this plot that the presence of ice particles is needed to create significant radar echoes, i.e., the spectrum for r=0 nm (Sc=1) gives much too small values for η, whereas even relatively small radii (but larger than approximately 5 nm) can already explain MAARSY observations.

There is a long-standing discussion on whether or not there is a trend in ice particle radii and/or number densities, and in related phenomena such as noctilucent clouds . One might expect that long-term observations of PMSE could contribute to this discussion since they rely on the presence of ice particles. However, PMSE also depend on other parameter, e.g., on turbulence and electron density. This paper concentrates on the sensitivity of radar reflectivity on ice particle radii and electron densities.

Figure  shows radar reflectivities as a function of ice particle radii (and Schmidt numbers) for three values of electron density and a given electron density gradient (see insert in that Figure). In all cases the model results are consistent with MAARSY, in particular if the range of values observed in the entire time period (ηzlo, ηzhi, see Fig. ) and the natural variabilty shown in Fig.  are taken into account.

As can be seen in Fig. , reflectivities are nearly independent of ice particle radii for rice larger than ∼4-5 nm. This has important implications: even small ice particles are sufficient to produce PMSE (as long as they are larger than 4-5 nm), and the intensity of radar reflection is independent of ice particle size in this case. This is consistent with numerous observations that PMSE may be present in the absence of NLC (note that NLC require ice particles with radii larger than ∼15-20 nm). It also means that PMSE are not well suited to study trends of ice particles if these particles exceed a size of 4–5 nm. Fig.  also demonstrates that the insensitivity of η to rice is valid for a rather large range of electron densities, namely 1 × 10⁸–1 × 10¹⁰ m⁻³, at least for our choice of the electron density gradient, dNe/dz. It should be noted, however, that the dependence of η on Ne and dNe/dz is rather complicated which requires a careful investigation and is beyond the scope of this study.

Note that our conclusions are only valid for VHF radars such as MAARSY (53.5 MHz). The radar Bragg scale of higher frequency radars such as the EISCAT VHF radar (224 MHz) is larger compared to MAARSY and the spectra are indeed sensitive to ice particle radii and Schmidt numbers (see dashed line in Fig. ). In fact, some studies have used the wavelength dependence of the ice particle radii impact on radar reflectivity to deduce microphysical parameters .