The polar ionosphere is highly dynamic and contains plasma structures at a large range of scales e.g.. Measuring and observing the plasma, the structuring, and their spatiotemporal evolution is necessary to understand the underlying physics. It is thus connected to the larger context of how the ionosphere is coupled to the magnetosphere and the interplanetary space around us. One of the most powerful ground-based instruments to study ionospheric plasma are incoherent scatter radars (ISRs). Currently, there are six ISRs operational and one in construction (at the time of writing) in the high-latitude region in the northern hemisphere. Three of these are dish radars located in the northern Fennoscandia region and on Svalbard, and three are phased array radars located in Resolute Bay, Canada , and Poker Flat, Alaska in North America . The new, state of the art, phased array European Incoherent Scatter Radar 3D (EISCAT 3D) is currently under construction in Northern Norway . In contrast to dish radars, phased array radars are capable of volumetric measurements in multiple directions simultaneously see e.g.and references therein. This capability has proved itself to be a powerful tool for studying ionospheric structures such as polar cap patches and density “irregularities” e.g.. Electronic beam steering is used to achieve multiple beams covering the desired field of view and the temporal resolution is limited only by the time needed to achieve an acceptable signal-to-noise ratio in each beam. Depending on the given radar system, many factors affect the quality of the signal, and careful calibrations are needed for any radar.

We calculate a ratio of electron density measurements and use the distribution of ratios to find the optimal correction factor, GRDC. The ratio is given by 6Neratio=Net‾Nebeam, where Net‾ is the mean electron density of all beams at a timestep t ([1 × number-of-beams] – array), while Nebeam is the electron density measurement in a particular beam, and the 〈⋯〉 denotes averaging over at all timesteps of an experiment ([1 × number-of-timesteps] – array). The Neratio thus gives the ratios of the mean electron density measurement in all beams, at each time and altitude, to the electron density measurement in one beam. For every beam this gives one Neratio for every timestep of the experiment ([1 × number-of-timesteps] – array for every beam). Hence, we obtain a distribution of the ratios for each beam. We then make a Kernel density estimate of the distribution of these ratios for each beam. An example of this is shown in Fig. . Here a histogram (blue) of Neratio for Beam 27 at 250 km during the same experiment at RISR-N and RISR-C used for the example in Sect. . The Kernel density estimate is shown in red. The ratio at which the maximum probability density estimate is found is taken as the correction factor, GRDC, for that beam at that altitude. This calculation is then done for all beams at the altitude of interest. For this calculation, we can either use the electron density calculated from the fit to the ISR spectra, or we can use the electron density inferred from the backscattered power of the radar signal, NeFromPower. The difference between these options is discussed in Sect. . In the calculations, the altitude gate is found by the nearest-neighbor method. When doing these calculations on the electron densities calculated from the fitted Autocorrelation functions, these altitude gates are approximately 20 km while when using NeFromPower the altitude gates are approximately 3 km in the examples from AMISR data used below. The correction factor is used to scale the corresponding electron density measurement, Neorg, as: 7Necorr=NeorgGRDC to obtain the corrected electron density, Necorr.

An example of how the correction factors calculated using this method can vary with beam and altitude is shown for one example in Fig. . Here, the electron density used for the calculations is calculated from the backscattered power of the returned signal, NeFromPower, rather than from the fit to the ISR spectra. This electron density is measured at a higher range resolution. In the calculations, the altitude gates are found by the nearest-neighbour-method. The example in Fig.  shows this. These correction factors are calculated here for all beams, in 3 km horizontal altitude slices during an 11-beam (22 total) experiment from 12 to 14 October 2016 at RISR-N and RISR-C (see the third example in Sect. ).

The figure shows all beams along a longitudinal cut at the center meridional line of both radars. It is particularly clear from this figure that the two vertical beams from the two radars are not calibrated to match in the uncorrected electron density data as they have very different correction factors. It should be noted that this difference is also explained by the large error estimate (not shown) of these two beams.

The results of this method are shown in the following three examples. The method presented here is applicable to any multi-point radar measurement. In anticipation of the EISCAT 3D radar , we utilize data from the AMISR phased array radar system to investigate the calibration technique described above. Hence, we have chosen three experiments, running different radar modes with different numbers of beams from the large RISR-N and RISR-C databases. All data shown here was downloaded from the online databases and error-filtered as suggested in the AMISR user manual . That is, electron density measurements smaller than or the same size as the error estimate, dNe, and data where the data quality parameters were poor, were excluded (set to Not-a-number). The error estimate and data quality parameters are normal data products provided for all RISR-N and RISR-C data. In our examples, we have used the electron density inferred from the fit to the ISR spectra, \FittedParams\Ne. Although the electron density from the backscattered power of the radar signal, \NeFromPower\Ne_NoTr, may also be used for this. To show the results of the calibration method, for each example (see Figs. –), we show the uncorrected electron density for all beams at one altitude in the top panel. The beams are ordered by magnetic latitude (y axis) so that all beams from RISR-N are in the upper half, while beams from RISR-C are shown in the lower half of the panel. The plots show uncorrected densities from the experiments for the times (x axis) where both radars were running simultaneously. The bottom panel of each figure shows the densities after the Ratio distribution correction. In Fig. , additional white lines and white shaded areas indicate the position of the terminator at the given altitude during the experiment. As the beams in this plot are organized only by their magnetic latitude, the terminator in these plots is jagged rather than a smooth line as it would be in a polar plot. Figures  and  do not have these, as the ionosphere in the field of view of the radars was continuously sunlit during these experiments.

The first example is from an imaginglp experiment run from 19 May 2019 22:01:09 UT to 23 May 2019 14:19:46 UT on RISR-N and RISR-C. Figure  shows the uncorrected and Ratio distribution corrected electron densities with a 5 min time resolution. This radar mode has 19 beams per radar (38 total), and at 250 km altitude the field of view is from 79.927 to 84.6578° North in magnetic latitude. In the upper panel, a calibration difference is clearly visible. The upper half, showing uncorrected electron density from RISR-N, is approximately 1×1011 m-3 larger than all observations from RISR-C, during the experiment, and significant beam-to-beam variations can be seen. This calibration difference could be problematic when comparing data from different beams. In the bottom panel, the corrected electron density, using the calculated correction factor, is shown. The small blue arrow on the right indicates beam 27, which is used as an example to show the Ratio distribution and Kernel density estimate in Fig. .

The second example is from a 22 beam (44 total) Convection67m radar mode experiment run on RISR-N and RISR-C from 3 May 2018 15:55:25 UT to 4 May 2018 14:59:45 UT. Figure  shows the electron density and the results of the correction from calculations at 200 km altitude with a 1 min time resolution. In the top panel, showing the uncorrected electron density from the fit to the ISR spectrum, there are nonphysical step changes between the electron densities in neighboring beams which are persistent throughout the experiment. That is, these differences are apparent for both high and low electron densities. These sharp transitions are likely caused by some calibration issue, as they are consistent throughout the dataset. Applying the calculated correction factor improves the electron density data and the instrumental effects of the density differences are reduced. After this calibration, the density structures moving through the field of view of the radars are still clearly visible.

The Worldday66 beam mode ran on RISR-N and RISR-C from 12 October 2016 21:27:07 UT to 15 October 2016 13:59:26 UT and 12 October 2016 21:41:27 UT to 14 October 2016 22:05:22 UT, respectively. Figure  shows the measured electron density at 200 km altitude from this experiment with a 1 min time resolution. This radar mode has 11 beams, so simultaneous observations with RISR-N and RISR-C give 22 beams in total with a field of view spanning from 75.5860 to 87.7° North magnetic latitude at 200 km altitude. In this figure, a difference in calibration is apparent, as RISR-C (bottom half) appears to consistently observe higher electron densities than RISR-N in the top panel. To separate this difference from a possible difference in illumination and thus ionization, the white lines in all panels indicate the position of the shadow terminator at this altitude. The shaded area is where the ionosphere at this altitude is in darkness. Additionally, it should be noted that three beams (beams 1, 12, 22 counting from the top) seem to differ significantly from the other beams in the radar. From the error of the fitted electron density, dNe (data product from AMISR, not shown), it is clear that those three beams also show the largest mean errors.

Here, the robustness and uncertainty of our method is investigated. In principle, the presented method is similar to the process performed before the data is made publicly available, in that a Magic constant or system constant is used to scale the measurements to calibrate for known characteristics of a given radar. However, here we calculate an additional correction factor from the electron density measurements themselves. For this, we can use either the electron density inferred from the fit of the ISR spectrum or the electron density as calculated from the backscattered power of the radar signal. Thus, we assess the ability of the method to provide a relative correction.

To assess the quality and robustness of the method, we present two approaches. In the first, we divide the uncorrected data from one experiment into smaller subsets of the whole experiment period. We then redo the calculations using the same method and compare the results to the results from using the whole dataset. Since we calculate the correction factor for every altitude and beam from the maximum of the distribution of Neratio for every beam, outliers, and enhancement, as well as very low measurements are effectively removed from the calculations. In Fig.  this can be observed as we see a wide range of Neratio measured within the same experiment. By taking smaller subsets of the whole dataset and comparing the results, an estimate is obtained of how well the filtering of outliers and enhancements performs. This is because one might get subsets only containing enhanced values or a very structured subset including multiple polar cap patches or similar auroral events. It also gives an indication of how robust the method is with regard to the length of a dataset and thus, the number of Neratio estimates needed for a good result. Figure  shows the corrected electron densities from the data presented in Fig.  when using subsets of 24, 12, 6, and 1 h. It is clear from this Figure that more “lines” or sharp transitions between beams (beams that appear to be differently calibrated from the rest) appear when using successively smaller subsets of data. The corrected electron densities get progressively worse when reducing the number of measurements used for the calculation. This is expected as a reasonable number of measurements are necessary to achieve a representative distribution of Neratio, to estimate the probability density and to find its maximum value. To quantify this quality of the result, we have recalculated the correction factor using 1000 randomly selected subsets of data and calculated the standard deviation and variance of G(RDC). This is shown in Fig.  where the green and purple shaded areas indicate the standard deviation and variance, respectively. The mean of the correction factor at each beam for every subset will converge toward the correction factor as calculated from the full dataset. This is because running many iterations of a randomly chosen 1 h subset of data will eventually cover the full experiment dataset and so the mean of all iterations will be the same as calculating the correction factors once from the full dataset. However, the standard deviation and variance indicate how sensitive the correction factor calculations are to the size of a subset of data compared to the full dataset.

The second approach used to assess the robustness of our method is by fitting a Gaussian function to the Kernel density estimate of the distribution of Neratio. The density function from the Kernel density estimate is not expected to resemble a Gaussian function as a whole, as electron density data generally contains data from transient events like polar cap patches, auroral events, or precipitation events that are not expected to be normally distributed throughout an experiment. The Neratio corresponding to the peak probability density indicates the most common ratio for the given beam at the given altitude for the current experiment. By fitting a Gaussian function to the Kernel density estimate in close proximity to this maximum, we can calculate the width of the Gaussian. This is used as a measure of the standard deviation of the Neratio distribution and leads us to an estimate of the uncertainty of the correction factor. Additionally, the standard error of the mean can be calculated. The mean is the centre of the Gaussian distribution, which corresponds to the Neratio value at which the peak of the probability density occurs. This is done for the three examples presented above in Figs. –. Figure  shows the calculated correction factors (y axis) using our method at the respective altitudes as presented in the Figures above. The purple line indicated the correction factors calculated when using the electron density from the fit of the ISR spectrum, while the red line indicated the correction factors when using the electron density calculated from the backscattered power of the radar signal. The standard deviation and standard error of the mean of the fitted Gaussian function are indicated for each beam (x axis), by the yellow and orange bars, respectively.

We present a method to improve the calibration of multi-beam ISR electron density data. The method calculates a correction factor, G, that is subsequently used to scale the electron density measurements. It is similar to the adjustments of the magic constant and system constant used by the EISCAT and AMISR radars, respectively (see Eq. ). The magic constant and system constant are inferred from knowledge of the radar system and the characteristics of the antenna pattern and radar parameters used for a given experiment. In contrast, the correction factors obtained from the method described here are determined from the multi-beam electron density data itself. For this, either the electron density from the ISR spectra fit or from the backscattered power of the signal could be used. Thus, our proposed method does not attempt to replace current calibration methods but rather adds a second-level calibration correcting for variable, unpredictable, and unaccounted variations in system gain. As such, this procedure should be considered as an integral part of the data-processing chain for every multi-beam ISR.

The method is based on the established Flat field calibration method used in imaging and photography see e.g.and references therein.. It uses a statistical approach, considering the distribution of the measurements, to estimate G. Here, the distribution of electron density ratios for each beam are found. Through a Kernel density estimate, we obtain the estimated correction factor, which is used to obtain corrected electron density measurements.

In the case of good/ideal basic (first-level) calibration, this additional calibration method scales all electron densities with a factor of G=1 in each beam. If there are unaccounted variations the method contributes to a near ideal calibration. Additionally, for the case where the electron density in one beam is accurately calibrated, e.g. with a plasma line measurement, all calibration factors should then be scaled such that the calibration factor of that beam is G=1. Lacking a beam with such accurate calibration, there will be one system-wide calibration scaling uncertainty. The calibration method presented here is also useful for future experiments and future radars like EISCAT 3D, where data volume is an issue, as it would reduce the need for a plasma line measurement in every beam.

One notable result of our method is observed in Fig. , where it is clear that different correction factors calculated from the raw electron density (/NeFromPower/Ne_NoTr for AMISR data) vary with range within the same beam. The received signal power in the radar should decrease by a factor of 1R2 with range only. There should be no range dependence in the correction factor. It is unclear why this is observed.

A possible explanation could be that there is a Faraday-rotation of the signal as it propagates through the ionospheric plasma and scatters back. If the polarization of the transmitted signal is only circular in the bore-sight directions, which would be the case for a phased-array radar transmitting at peak power, off-boresight beams would have slightly elliptical polarization. For example, assuming a parallel magnetic field component of 52 000 nT, a total electron content of 10 TECu, the Faraday rotation would be approximately 72° for a 440 MHz EM wave. Radars, such as Irkutskt ISR do observe Faraday rotation effects in power for off-boresight beams , where the transmitted polarization is elliptic, and the received polarization ends up not being matched to the transmitted polarizations

However, investigating this in full depth should be the objective of a follow-up study. If there are remaining structures in the electron densities, as shown in Figs. –, from the fits of the ion-line spectrum, this procedure can be applied to those electron-densities such that those structures are reduced, regardless of the reason for such systematic structures to appear.

In Sect.  it was shown that there is sometimes a need to compensate for irregularities and nonphysical step-changes in the calibration of multi-beam electron density data from ISRs. Additionally, there is no way to determine which radar beam is the “correct” one without a separate, independent measurement. The presented examples show that our method improves calibration and data usability. This is especially valuable for research where quantitative, inter-beam comparisons of electron density changes are of interest, e.g., studies of plasma patches, irregularities, and turbulence. Although there is a general need for accurate independent measurements for optimal calibration, we argue that by applying this method, the electron density measurements in all pointing directions within one experiment become inter-calibrated. An effort was made to analyze the quality and robustness of the method in Sect. .

One challenge of this method is considered in Fig. . Although it can efficiently be applied to any multi-beam ISR experiment, it is somewhat sensitive to the length of the dataset. That is, to acquire a good Kernel density estimate of the ratio distribution, an adequate number of measurements is required. Figure  illustrates this as it is clear that the results of the correction become progressively worse when reducing the number of measurements. This means that there is a minimum number of measurements needed to acquire a reasonable result. The required number of measurements changes depending on the experiment and data resolution. However, the result illustrated in Fig.  indicates that a 12 h experiment duration produces a noticeable improvement in the RISR experiment used as an example.

Calibration of ISR is a complex and difficult task, especially when independent measurements of the electron density are only available in one pointing direction or not at all. We have proposed a method for the correction of electron density measurements from multi-point ISRs. This method is intended to add to the current efforts that are made for the calibration of multi-point radars and can increase data usability and quality by considering variable, unaccounted and, unpredictable variations in system gain. It is based on the well-established Flatfield correction method in imaging and photography. We exploit the similarity between independent measurements in separate pixels in one image sensor and multi-beam radar measurements. The correction factors are estimated based on a Kernel density estimate of the distribution of Neratio calculated from all measurements of an experiment. This is subsequently used to estimate the correction factor. The method is robust and efficient for the examples presented here. It is sensitive to the length of the experiments when these are well below 12 h. The errors and robustness of the method were estimated through two different approaches. Accurate independent measurements are needed to determine absolute values of electron density and an optimal calibration. However, we argue that by applying the presented method, the electron density measurements in all pointing directions within one experiment become intercalibrated. Thus, this second-level calibration is especially valuable for studies of plasma patches, irregularities, turbulence, and other research where inter-beam changes of electron density are of interest.

The presented method is strictly based on the observed electron density data and requires no additional input. It is efficient and its application on existing and future data is straightforward. Additionally, it requires minimal user input and is not expected to increase the demand for computing power. Thus, we suggest that it be implemented and applied as a second-level calibration in future calibration schemes of multi-point radars where any calibration is required. This will improve the calibration and data accuracy of phased array incoherent scatter radar measurements and thus overall data usability and quality.