1 Introduction

Meteor spectroscopy has been an active field of research in the Czech Republic since 1960. Initially, spectra were observed using an analogue camera system equipped with a rotating shutter located at Ondřejov. These cameras, with long focal distance, produced high-resolution spectra recorded on photographic plates and later on film. Despite the high quality of the recorded spectra, the cameras covered only part of the sky, the spectral range was limited to 3500-6600 Å, and the system proved to be expensive and impractical in several aspects. To overcome these limitations, a network of Spectral Digital Autonomous Fireball Observatories (SDAFO) was introduced in 2015 as part of the European Fireball Network.

These autonomous spectral observatories employ Canon EOS 6D cameras with removed UV-IR filters as their detectors. Mounted on the Canon EOS 6D cameras are Sigma 15mm f/2.8 fish-eye lenses, which provide a diagonal field of view of \(180^{\circ }\). Although these cameras offer lower resolution, they capture spectra within a wider spectral range, extending from 3800-9000 Å, reaching into the near-infrared region (NIR). Additionally, they enable improved observing conditions through a 30-second exposure time, a significant improvement over the full-night exposure times required by the analogue system. The observatories are now deployed at eleven stations. The analogue cameras remained in use until 2018, providing a three-year overlap with the digital cameras. This overlap made the comparison of commonly observed spectra between the two systems possible.

While the analogue cameras were equipped with a high-quality 600 grooves/mm glass grating, the digital cameras utilise budget 1000 grooves/mm plastic holographic gratings. Professional ruled gratings of the required size (diameter \(\sim 7~\textit{cm}\)) are unavailable for the digital cameras. Consequently, the gratings at the digital-camera stations are periodically changed to prevent issues arising from damage or accumulated impurities. For a more detailed explanation of both systems, refer to [1].

A comparison of spectra from the same meteor captured at different stations of the European Fireball Network revealed discrepancies in absolute and relative line intensities even after following all the calibration steps. We analysed several such multi-station cases and concluded that the observed differences indeed exist, and can not be attributed only to measurement uncertainties, as we initially thought and reported in [2].

Given such findings, in spite of devoting substantial consideration to the calibration steps, we decided to take a fresh approach and revise our steps from the very beginning. Results can only be as trustworthy as the methodology used to obtain them. The aims of this paper are to give a detailed report of our findings including the full methodology description with error estimates, as well as to raise awareness of the importance of proper data calibration and transparency of the used methodology in publications.

The original assumption that the fully calibrated spectra of the same meteor observed from different stations should be identical prompted a meticulous examination of the original images and the entire methodology used to obtain and analyse the spectra. Spectra with similar positions in the field of view (FoV) showed a tendency to produce more consistent relative line intensity ratios, while larger differences were observed for spectra located far apart. This behaviour raised suspicion about the flat-field, leading to a study conducted with an artificial light source in laboratory conditions.

The analysis of the laboratory spectra first showed that the previously determined flat-field must be revised, since the function is not axially symmetric as we previously thought. However, even after that, line intensities exhibited clear dependencies on the position of the spectral line in the FoV, as well as on its wavelength. Upon consulting similar findings [3], it was confirmed that the issue is indeed related to the flat-field. While the usual method of constructing it is correct for regular photometry, there are additional geometrical dependencies of intensity on the position of the spectrum in the FoV, which must be taken into consideration when working with slitless spectroscopy. We found no mention of this in any of the encountered meteor spectroscopy-related publications.

Other aspects of the calibration procedure include also the spectral sensitivity and extinction correction. Both of these aspects are affected by the newly-developed flat-field. Originally, both were treated simultaneously using measurements of the spectra of Venus and Moon captured on a clear night. From them, the spectral sensitivity was determined for the given extinction conditions at the time. Henceforth, when analysing a meteor spectrum, stellar photometry was used to determine a constant, wavelength-independent extinction coefficient that was used to scale the spectrum by a constant value.

Now, we developed an innovative approach using a simplified atmospheric model, which calculates the extinction curve assuming a given aerosol optical depth (AOD). The AOD, which is usually determined from meteorological measurements, is computed using the extinction coefficient determined from the stellar photometry. Using the determined AOD value, a wavelength-dependent extinction curve is computed. The procedure is performed on both the spectra used to determine spectral sensitivity and the observed meteor spectra, which allows for applying the spectral sensitivity and extinction corrections separately, with improved quality.

We used a meteor observed from several stations of the European Fireball Network to compare the previously used and newly developed methodology. The comparison showed a visible improvement in the correspondence of spectra captured from different stations in the part of the spectral range not influenced by the limitations of the equipment and new methodology.

2 A flat-field for slitless spectroscopy

2.1 Laboratory setup

We used a low-cost and affordable laboratory setup to calibrate the spectral cameras. We chose the fluorescent 15 W bulb with resolved spectral lines as a light source. This bulb was mounted in an ordinary lamp stand. The lamp’s shade opening was covered with cardboard, and only a small rectangular hole (slit) was cut in the cardboard, measuring 4.0 x 0.1 cm. We want to stress that this slit was used to shape the light source into a meteor-like form, and does in no way have a relation to slits used for slit spectroscopy. The same camera type (Canon EOS 6D) that is used in all SDAFO cameras was employed, as well as the same lens type (Sigma 15mm). The camera was mounted on the tripod several meters from the lamp and the spectral grating was placed in front of the lens.

Spectra were obtained by moving the camera to different angles, so the spectrum of the bulb as well as the zero order changed their position in the FoV of the camera accordingly. These changes in the camera’s direction on the tripod caused repositioning of the lens front element and thus the geometry of the camera lens - slit - bulb was changing while rotating the camera on the normal three-way tripod head.

We found that, since the halogen bulb has no classical bulb shape but rather a helix shape observed from the top, the geometry changes resulted in different parts of the structured bulb being seen through the slit, causing variations in the absolute brightness of each spectrum. To eliminate this unwanted effect, we mounted the camera into the gimbal mount, keeping the front of the lens at a fixed position. Additionally, we placed a fabric from photographic soft boxes between the bulb and the slit. This setup ensured that any of the bulb’s structures were eliminated, creating a spatially even source of light. Consequently, while moving the camera on the tripod to observe the spectrum in different FoV positions, the same intensity of the light source was preserved.

We gave special attention to ensure correct exposure times. Before capturing calibration shots, exposures were carefully adjusted so that no R, G, or B pixel in any measured spectra, or in any measured zero order was saturated. Based on previous experiences, no measured pixel exceeded the value of 200 in the 8-bit format. Moreover, a condition was imposed, requiring the chosen exposure time to be long enough to eliminate the 50Hz flicker of the light bulb, meaning that the exposure was always at least 1/25s long.

To ensure consistency and rule out grating-related issues such as damage or imperfections, we used four different gratings, two of which are brand-new, one used and retired, and one only slightly used. We made several adjustments to the laboratory system until consistent measurements were achieved. The final correction utilised measurements from one of the brand-new gratings among the four, as all gratings exhibited the same behaviour. For this particular grating, we captured 90 images, which contained 108 usable spectra (some images had spectra on both sides).

Fig. 1
figure 1

A composite image of the laboratory spectra used for calibration. Images in the raw format are of higher quality

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A composite image of all laboratory spectra used for calibration is shown in Fig. 1. As was the case for the actual meteor camera, the grating was fixed so that the dispersion was approximately along the x-axis.

2.2 Constructing the flat-field

The construction of the slitless flat-field was divided into two separate components - a wavelength-independent and a wavelength-dependent one. The distinction is made purely mathematically, and we made no attempt to isolate individual dependencies or characterise the system in a theoretical fashion.

Once the laboratory spectra have been obtained using the method described in the previous section, the raw images in CR2 format are converted into linear-gamma 16-bit TIFF images. They are loaded into the FishScan software (not commercially available) developed by Jiří Borovička, where they are scanned to obtain the spectral information. The scan is performed by adjusting the measuring area (software slot) and a scanning path along the spectrum. The intensities of all pixels within the slot are summed along the scanning path and attributed approximate wavelengths with corresponding intensities and x and y positions of the slot in the image. The position of the zero order on the image is also saved. All positions are given in pixels, relative to the image centre. An example of a recorded laboratory spectrum is shown in Fig. 2.

Fig. 2
figure 2

The recorded spectrum (bottom) and scanned spectrum (top). Displayed wavelength values are expressed in Angstrom units

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The total zero order intensities were computed and plotted against their x and y positions in the image. The observed dependency was fitted with a polynomial function of the form:

$$\begin{aligned} I(x, y) = A x^2 + B y^2 + C x y + D x + E y + F. \end{aligned}$$
(1)

The obtained coefficients of the fit were scaled so that the maximal intensity I(x, y) would be equal to unity. The maximal measured intensity, as expected, almost perfectly coincided with the centre of the FoV.

The determined functional dependence is equivalent to the flat-field correction used in regular photometry, and in the continuation of the paper, it will be referred to as the wavelength-independent flat-field. It is displayed in Fig. 3. The correction was applied to all pixels in the image by dividing the original value by I(x, y) before scanning the spectrum.

Fig. 3
figure 3

A visualisation of the wavelength-independent flat-field

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Upon closer inspection of Fig. 3, it can be observed that the intensities decrease more gradually along the x-axis when moving away from the image centre, compared to to the y-axis. Moreover, the upper part of the image is consistently brighter than the lower part. The effect was tested with multiple cameras and gratings, and in all cases, it produced the same behaviour. We concluded that the asymmetry probably originates from the internal structure of the camera. In fact, this asymmetry is the reason for choosing a polynomial function of x and y positions of the zero order for the fit, rather than a function of the radius from the image centre, as was initially considered.

The subsequent steps of the analysis require the wavelength calibration of the spectra, which was done by comparing the known line positions of the fluorescent lamp with the observed line positions and correcting them using a polynomial function. An example of a lamp spectrum is provided in Fig. 2.

Given the lack of other sources, the theoretical positions of the spectral lines were found on [4]. The listed wavelengths were considered reliable, as using them, the wavelength calibration of the spectra was performed with smooth polynomial functions of small degrees. We did not review the line identifications provided by the author for the scope of this study. The author listed the assumed actual line positions and positions of the peaks. Only the positions of the peaks were used for wavelength calibration. In the event of the Wikimedia page being removed, the used data are shown in Table 1.

Table 1 A table of peak positions and line identifications for the used laboratory spectra [4]
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We divided the wavelength-calibrated spectra into 16 wavelength segments (Fig. 4). The total intensity of each segment was computed by integration, as the peak intensity would not be representative since the lines have variations imbued in various parts of the image. Some segments correspond to a single line, while others may include several lines. This approach was used in the case of mutually close lines to avoid small differences in wavelength calibration affecting the computed intensity. We resorted to the trial and error method to determine the segment boundaries for low signal-to-noise ratio lines, until we got the most consistent results.

Fig. 4
figure 4

Division of a laboratory spectrum into segments

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Each segment was represented with its average wavelength and the (x, y) position of that wavelength in the image. The spectra were divided into left and right, depending on their relative position in the image to the zero order.

We plotted the total intensity of each segment against the x and y positions of its average wavelength in the image, relative to the image centre. Here is a good place to elaborate on the aforementioned asymmetry. For the wavelength-independent flat-field, we didn’t make a distinction between the left and right spectra, since we only considered the zero-order position. However, when working with line positions, the need to consider them separately becomes evident. As shown in Fig. 5, both before and after applying the wavelength-independent flat-field, the x-symmetry is maintained in the data. Mirroring the x positions to \(-x\) for the left spectra while keeping the right spectra unchanged, results in an almost perfect overlap. Although the edges may initially appear different, they follow a 3D dependence which is more clearly visible in Fig. 7.

Fig. 5
figure 5

The intensity dependence on the position of the spectral line in the FoV at different wavelengths before (top) and after (below) applying the wavelength-independent flat-field. The left side shows the data before mirroring the left spectra, while the right side displays the data after mirroring

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Given that the upper part of the image is consistently brighter before applying the wavelength-independent flat-field, several other considerations were necessary. We determined that the used cameras do not automatically rotate the image after capturing it, and special care was taken to always work with originally oriented images to obtain the correct coordinates of the spectrum on the image. While this may seem like a trivial step, working with a \(180^{\circ }\) rotated image would yield incorrect results.

The following part of the section is challenging to explain in simple terms while keeping it clear and concise. To that end, several things have to be addressed once more before proceeding. The wavelength segment total intensity, which is the integrated intensity of the segment, will be referred to as segment intensity. When discussing a wavelength corresponding to a segment, it will refer to the average wavelength of the segment. Finally, the position of the segment on the image will refer to the position in the image corresponding to the average wavelength of the segment.

After applying the wavelength-independent flat-field correction and mirroring the x positions to \(-x\) for left spectra while keeping the right spectra unchanged, a clear dependence remained of the segment intensity on its x and y position in the FoV and its wavelength. For each segment, the dependence was described separately by the following polynomial function:

$$\begin{aligned} I_{\lambda }(x, y)&= A_0 + A_1 x + A_2 y + A_3 x y \nonumber \\&\quad + A_4 x^2 + A_5 y^2 + A_6 x^3 + A_7 x^2 y \nonumber \\&\quad + A_8 x y^2 + A_9 y^3 + A_{10} y^3 x^2 \\&\quad + A_{11} y^3 x + A_{12} x^3 y^2 \nonumber \\&\quad + A_{13} x^3 y + A_{14} x^3 y^3 + A_{15} x^2 y^2 \nonumber . \end{aligned}$$
(2)
Fig. 6
figure 6

The intensity dependence on the position in the FoV at \(\lambda = 4055 {\text{\AA }}\)

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Using the obtained factors, we calculated the intensity in the centre of the FoV, and divided all the factors by the obtained intensity:

$$\begin{aligned} I_{\lambda _i}^0(x, y) = \frac{I_{\lambda _i}(x, y)}{I_{\lambda _i}(0, 0)}; \quad i = 1,...,16. \end{aligned}$$
(3)

The intensity of a segment of a given wavelength in the centre of the FoV, \(I_{\lambda _i}(0, 0)\), was taken as a reference value. Analogous to segments, the same logic can be applied to discrete wavelengths. The fitted intensity thus represents the expected intensity of a spectral line of a given wavelength observed at a specific position in the FoV relative to the centre of the FoV. Once again, to distinguish between different aspects of the flat-field correction, this part will be referred to as the wavelength-dependent flat-field, since the obtained fit is different for each wavelength.

An example of the wavelength-dependent flat-field at \(4055 {\text{\AA }}\) is given in Fig. 6. The function used for the fit is meant to describe the dependence purely mathematically.

After performing the fit on all 16 segments, 16 positional dependencies \(I_{\lambda _i}^0(x, y)\) were obtained for the average wavelengths of the segments, each characterised by 16 coefficients. The same number of coefficients as the number of segments is purely coincidental. A lower degree polynomial function could also have been used to describe the data. However, after different attempts, the selected function (2) provided the best results using the fewest parameters.

Thus far, the functional dependence on wavelength has yet to be determined and is only known for the average wavelengths of the used 16 segments. Several approaches may be taken to extend the description to an arbitrary wavelength.

One may be tempted to find a functional dependence of the individual computed coefficients on wavelength. For the mathematical approach used here, however, that would not yield a sensible solution, as the nature of the fitted surfaces is not a simple one, and the individual fit parameters do not follow any particular functional dependence on wavelength. A workaround is, therefore, needed.

To apply the correction to a spectrum using the coefficients computed from (3), the previous steps need to be repeated. Namely, the spectrum needs to be wavelength calibrated, the left spectra mirrored to \(-x\), and a wavelength-independent flat-field correction needs to be applied. Each spectral point is then characterised by its intensity, wavelength, and position in the FoV.

For each spectral point, 16 different intensity correction factors are computed at the (x, y) position of the point in the FoV according to (3) for the 16 different wavelengths with known coefficients. The computed factors are plotted against the corresponding wavelengths (average wavelengths of the segments used to determine the functional dependency) and fitted using a 3rd-degree polynomial function. This function describes the intensity dependence on wavelength for a point located at that specific position in the FoV. The desired intensity correction factor is then obtained for the wavelength of the current point using interpolation. Finally, the wavelength-independent flat-field corrected intensity is divided by the determined correction factor. The whole process needs to be repeated for each point of the spectrum separately, as the computed intensity correction factor depends on the position of the spectral line in the FoV.

While this may sound like a complicated and time-consuming computation, it is, in fact, a series of simple ones, which take only several seconds to compute for a computer of average specifications.

Fig. 7
figure 7

The intensity dependence on the position of the spectral line in the FoV at different wavelengths after the application of the wavelength-independent flat-field

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Figure 7 shows the segment intensity dependence on the position in the FoV for all the recorded spectra after the application of the wavelength-independent flat-field. The wavelength segment intensities are coloured according to their average wavelengths. Fits obtained for the different wavelength segments made using (2) are shown on Fig. 8.

Fig. 8
figure 8

The fitted intensity dependence on the position in the FoV at different wavelengths

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An additional, unavoidable issue is a lack of measurements at the edges of the FoV. The reason for it is a lack of visible spectral lines which can be used for wavelength calibration. The low coverage further worsens the reliability of the fit in those regions.

Fig. 9
figure 9

Maximal deviations of the measured intensities after the correction was applied

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As can be seen from Fig. 9, after applying the correction to the laboratory spectra, maximal deviations and uncertainties are highest in the expected areas for already discussed reasons. The plotted points were, once again, coloured in accordance with their wavelength. The correction is most reliable in the \(4900-6600~{\text{\AA }}\) region with standard deviations below \(5\%\) of the average measured values.

We also attempted an alternative one-step approach to the fit, which yielded comparable results. It can be achieved by directly fitting the intensities of each wavelength segment using (2) and (3) before the wavelength-independent flat-field is applied to the spectrum. This way, both flat-field components are applied at once.

However, we chose a two-component approach since the wavelength-independent flat-field needs to be applied in previous steps of the spectral analysis, namely, when performing stellar photometry. Moreover, the used approach is also more illustrative, as different effects are being corrected in different steps: the wavelength-independent flat-field corrects the lens and camera system vignetting, while the wavelength-dependent flat-field corrects the dependence of the measured spectrum brightness on the angle of incidence on the spectral grating.

2.3 Estimation of uncertainties

To determine measurement uncertainties, two sets of 10 images of the laboratory source were taken consecutively in identical conditions in two different parts of the FoV. Using the same wavelength segment division as in the previous parts of the analysis, the observed total intensities were computed for each of the segments. To represent the uncertainties, the standard deviation of the 10 measurements was divided by the average segment intensity, expressed in percentages, and plotted versus the average wavelength of the segment (Fig. 10).

Fig. 10
figure 10

Determined measurement uncertainties for a spectrum with the zero order near the centre (blue) and near the edge of the FoV (red)

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The uncertainties are larger for the measurements closer to the edge of the FoV. This is likely due to the strong distortion of the image due to the fish-eye lens. Additionally, as expected, segments with low signal-to-noise ratio have larger uncertainties. For brighter lines, the uncertainties are within \(2\%\), which is well within the expected uncertainty of real meteor spectra measurements.

2.4 Theoretical considerations

The influence of the grating itself on the absolute brightness of the spectrum, as well as on the brightness at different wavelengths, appears to be rather complex. After the wavelength-independent flat-field correction, the brightness of individual spectral ranges displayed neither a clear global maximum nor minimum. As shown in Fig. 7, brightness decreased with increasing x-coordinates, i.e., with an increasing angle of incidence when the camera with the grating was tilted in the x direction. The degree to which the intensity decreased also depended on the y-coordinate, i.e., on the angle of incidence in the y direction. The measured intensity for each spectral range was brightest at the highest tilt in the y direction and for a zero angle of incidence in the x direction.

Due to limited knowledge about the specific shapes of grooves or material structure constituting the holographic grating, the precise cause of this behaviour can only be hypothesised. As the gratings are holographic, it is reasonable to assume the grooves have a sinusoidal shape [5]. The decrease in brightness for the tilt in the x direction (perpendicular to the grooves) can be accounted for by the scattering of incident light, leading to a reduced efficiency of the grating at higher angles [5]. The behaviour of the grating when tilted in the y-axis is more perplexing, and the explanation is not immediately clear. Assuming a sinusoidal shape of the grooves, the apparent height of the sinus changes when the grating is tilted. As the angle of incidence increases from the position where the grating is perpendicular to the incident light, the apparent height h of the grooves rises, as shown in Fig. 11.

Fig. 11
figure 11

The apparent height of a sinusoidal profile for different incidence angles of the observed light

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However, the distance D between the grooves remains consistent. As highlighted in [5], the grating efficiency actually increases with a rising h/D ratio. This behaviour is also observed with all of the tested gratings.

Fig. 12
figure 12

The computed relative grating efficiency. Vertical dashed lines mark average wavelengths of the first and last wavelength segments

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Figure 12 depicts the relative grating efficiency across various spectra. Spectra recorded with a minor tilt of the grating in the x direction and a significant tilt range in the y direction were chosen. This meant that the effect of light scattering was comparable for those chosen spectra, demonstrating that in general, the grating efficiency decreased as the tilt of the grating (or angle of incidence) in the y direction diminished. For the spectra obtained with the laboratory setup, the maximum angle of incidence in the y direction was estimated to be \(43^{\circ }\). The grating efficiency is estimated based on the actual positions of the selected spectra and the correction function formulated for the wavelength-dependent flat-field. For the given position of the spectrum, the intensity was replaced by a flat profile and multiplied by the correction function. This simulation of how a theoretical spectrum with the same energy in all wavelengths would be observed using the same gratings can easily reveal the relative efficiency of the grating. This way it is not possible to derive the actual absolute grating efficiency, but this approach works for the relative comparison between each grating tilt and thus we call it relative efficiency. Normalisation was achieved by finding the highest value from all simulated efficiencies and assigning it a value of 100. The other curves were normalised based on this value. It’s noteworthy that for wavelengths above approximately \(7000~{\text{\AA }}\) the efficiency’s dependence on the angle of incidence isn’t compelling, which comes as no surprise since the behaviour of the flat-field for that region was extrapolated due to a scarcity of lines in laboratory spectra in that spectral region. The same conclusion holds for the efficiency estimate for the highest angle of incidence, as seen in Fig. 12. The number of spectra used for the wavelength-dependent flat-field was constrained at the edge of FoV, making the estimated grating efficiency somewhat uncertain.

3 Atmospheric extinction and spectral sensitivity

3.1 A joint approach

To obtain the instrumental spectral sensitivity curve, a light source with a known wavelength radiance is required. If a laboratory light source is not available, it is possible to use some astronomical object with a known spectrum. The spectrum of this source, captured by a given camera, is then divided by its true spectrum. The resulting curve is then the instrumental spectral sensitivity curve of the camera. When the captured spectrum is divided by a known spectrum corresponding to the spectrum of the object above the Earth’s atmosphere, the obtained curve of spectral sensitivity simultaneously includes a correction for the atmospheric extinction caused by Earth’s atmosphere in certain wavelength regions.

In FishScan, the extinction coefficient is computed from the photometry of stars (their zero order images). The measured star magnitude is

$$\begin{aligned} m = m_0 + \kappa X(z), \end{aligned}$$
(4)

where \(m_0\) is the catalogue star magnitude (outside Earth’s atmosphere), \(\kappa \) is the extinction coefficient, and X is the airmass depending on the star zenith distance z. In the joint approach, the extinction coefficient was taken as independent of wavelength and all the spectrum was scaled according to its value and the zenith distance of the meteor. Obviously, this approach was strictly correct only if the wavelength dependence of extinction was the same for the calibration object (e.g. Venus) and the meteor.

Since the spectra of even the brightest stars taken by the used spectral cameras have a low signal-to-noise ratio, the use of standard stars for the spectral sensitivity calibration is not possible. On the other hand the spectrum of the Sun, even with the shortest exposure times, is always overexposed. However, the Moon, Venus and Jupiter have a suitable brightness for this task. From several obtained spectra of these objects, taken at different times and at different altitudes above the horizon, curves of spectral sensitivity were derived and compared. The Venus reference spectrum was taken from [6], the Jupiter spectrum was taken from [7] and the lunar spectrum from [8]. In the case of the old SDAFO calibration, these curves exhibited significant differences, and it was difficult to evaluate the final spectral sensitivity curve.

3.2 Decoupling the effects

Atmospheric extinction is affected in a complex way by Rayleigh scattering, ozone absorption and diffusion by aerosols, which is described by a parameter called aerosol optical depth (AOD). To model it, precise meteorological measurements are required at the observing location. Considering a lack of such measurements or the availability of an appropriate atmospheric model, up until now, the problem of correcting the observed meteor spectra for atmospheric extinction was handled approximately, under the assumption that the observing conditions for spectra used for the calibration of spectral sensitivity do not significantly differ from those at the time of the meteor observation.

We made an effort to describe both effects individually and find a more precise way of handling the atmospheric extinction, starting by finding an atmospheric model compatible with the methodology devised so far. We found a freely accessible Python-based implementation [9] of a model which conformed with those requirements. The model computes three separate wavelength-dependent components of the atmospheric extinction, the total extinction being their sum.

The Rayleigh scattering component is computed analytically, following the formalism provided by [10]. The absorption by ozone molecules is also computed analytically, using coefficients given by [11]. The final component is diffusion by aerosols. To calculate it, a value of the AOD is needed, which can be approximated based on the weather conditions. However, such an approach would once again, yield only a rough estimate, possibly comparable to the methodology used thus far. To get a more precise estimate of the AOD and atmospheric extinction, a bit of backward engineering is required. Thankfully, even thought the AOD is wavelength-dependent, only its value at 550nm is required for the calculation. Further details regarding the computation and examples of results are provided in [12].

Since the visual magnitudes of stars are taken as the catalogue magnitudes when computing the extinction coefficient, the resulting extinction computed in the FishScan software is most representative for the wavelength of 550 nm, the effective wavelength of the visual band. Assuming this is the case, the value of the AOD in the model can be varied until an extinction curve is computed which corresponds at 550 nm to the extinction coefficient computed by FishScan. We stress that the computed extinction coefficient is significantly affected by applying the newly developed wavelength-independent flat-field.

Using such an approach, we may compute the extinction curve for both the measurements carried out on spectra used for determining the spectral sensitivity, and observed meteor spectra, which is exactly what the new method entails. The newly acquired spectra used for spectral sensitivity determination were calibrated using both components of the newly-developed flat-fields and multiplied by the transmission curve computed from the extinction curve. The spectral sensitivity factors which resulted from all calibration spectra obtained using this methodology proved to be much more consistent in comparison to the ones obtained using the previous approach. We obtained the resulting curve much more straightforwardly, by averaging individual curves and noise cleaning. Such a spectral sensitivity curve (Fig. 13) is now decoupled from the effect of the atmospheric extinction. We can now correct the observed meteor spectra by first applying the new spectral sensitivity correction. Then, using the extinction coefficient at \(550~\textit{nm}\) computed for the stellar observation, we construct the extinction curve, and from it the transmission curve. Finally, we divide the spectral sensitivity calibrated spectrum by the transmission curve to obtain a fully calibrated spectrum.

Fig. 13
figure 13

A comparison of the old and new spectral sensitivity curves

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3.3 Method comparison

As explained above, the originally computed spectral sensitivity also partially accounted for the atmospheric extinction, while the newly computed spectral sensitivity excludes the effects of atmospheric extinction from consideration. In spite of this fact, in this section we will refer to both curves as spectral sensitivity curves for simplicity.

Given that they represent different things, their direct comparison shown on Fig. 13 does not tell us much by itself. However, in an ideal case where both curves are constructed on spectra observed in same extinction conditions, the ratio of the new and old sensitivity curves should give us the extinction curve corresponding to those conditions.

In reality, however, the sensitivity curves were computed using multiple spectra of different sources in different parts of the FoV, with different flat-fields and under different observing conditions. That means that in a practical sense, the ratio will not correspond to any particular extinction curve. However, we do expect it to show a dependency which closely resembles an extinction curve, or a ratio of extinction curves.

For example purposes, we computed an extinction curve which matches the computed sensitivity curve ratio at 550 nm, assuming an altitude of the observing station of 500 m and a zenith distance of \(50^{\circ }\). Figure 14 shows their comparison.

Fig. 14
figure 14

A comparison of an atmospheric extinction curve and the ratio of the new and previous spectral sensitivity curve

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The overall shape of the dependency closely resembles the computed extinction curve as expected. However, some clear deviations are present, which can be attributed to the inaccuracy of the previous spectral sensitivity curve. This should come as no surprise considering that it was constructed from spectra to which a faulty flat-field correction was applied.

4 Results and discussion

At the very beginning, we must stress that, at the the time of writing this paper, absolute line intensities observed from different stations, even after all calibration steps, differ significantly. This aspect is currently under investigation and may be a topic of a future paper. All the compared spectra are scaled by hand to match best in the part of the spectrum with the highest spectral sensitivity.

To best visualise the comparison of spectra before and after the changes in methodology, simply plotting them all directly would be insufficient. Peak intensities are not representative and total intensities of profiles are difficult to compare due to different resolutions and profile shapes. In order to get around those issues, each spectrum is fitted with a synthetic spectrum. The synthetic profiles are computed using a Voigt function, which is characterised by a Gaussian factor determining the width of the profile, and a Lorentz function determining the shape of the wings. Once a spectrum is fitted and dependencies of the factors on wavelength are determined, the total intensity of each spectral line included in the fit is saved to a text file.

A new synthetic fit with unit total line profile intensities is then generated, using the desired Gaussian and Lorentz factor dependency on wavelength. For each spectrum, the total intensities are then loaded and the unit profiles are scaled to match the loaded total intensity. This way, the spectral profiles can be compared visually with the same shapes regardless of their original observed quality. It is important to note that this synthetic fit is purely mathematical, and in case of multiple overlapping lines in a profile, some differences in identification may arise between a well resolved spectrum and a spectrum of poor resolution, which would be avoided by using a physical model for the fit. These differences can be neglected from consideration, as they result purely from such simplified considerations. Only total profile intensities should be compared, regardless of the lines used for identification.

Such a comparison method is applied only when comparing multiple spectra of the same meteor captured from different stations. For a comparison of the same spectrum analysed by different methods, a direct comparison is plotted as the profiles are already of the same shape.

The meteor observed on 30th October 2022 at 22:16:40 UT from several stations of the European Fireball Network (Fig. 15) was used to test the newly implemented methodology. Good spectra of the same meteor captured in different parts of the FoV made it a prime candidate for the analysis.

Fig. 15
figure 15

Composite image of the meteor observed on 30th October 2022 at 22:16:40 from several stations of the European Fireball Network

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Depending on the position of a spectrum in the FoV and the measured atmospheric extinction, the effect of the new corrections varies significantly. This is demonstrated by comparing spectra \(210\_0\) and \(281\_0\) in different stages of the correction. For spectrum \(210\_0\) (Fig. 16) located near the centre of the image, the previous and new flat-field differ only slightly. The difference is much more pronounced for spectrum \(281\_0\) (Fig. 17) located near the edge of the FoV.

Fig. 16
figure 16

A comparison of the previous (red) and new (blue) flat-field applied to spectrum \(210\_0\)

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While relative line intensity ratios do not appear to be significantly affected for these particular spectra, one must not forget that the flat-field affects not only the line intensities, but also the computed atmospheric extinction coefficients. Switching to the new flat-field, the extinction coefficient for spectrum \(210\_0\) changed from 0.437 to 0.461 and for spectrum \(281\_0\) from 0.426 to 0.324. Calibrating the spectra for spectral sensitivity and atmospheric extinction using the previous (joined) and new (separate) approach showcases the differences.

For spectra obtained using the new and previous methodology, plotted on Figs. 18 and 19, differences in relative line intensities now become very apparent, especially at lower wavelengths where atmospheric extinction exhibits the strongest functional dependency.

Fig. 17
figure 17

A comparison of the previous (red) and new (blue) flat-field applied to spectrum \(281\_0\)

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Fig. 18
figure 18

A comparison of the relative line intensities obtained with the previous (red) and new (blue) analysis methodology applied to spectrum \(210\_0\)

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Fig. 19
figure 19

A comparison of the relative line intensities obtained with the previous (red) and new (blue) analysis methodology applied to spectrum \(281\_0\)

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The real question remaining is how reliable these newly introduced corrections are and whether they can be trusted. To answer this, a comparison of spectra from all analysed stations is presented on Fig. 20 using the previous methodology and on Fig. 21 using the newly devised methodology. Spectrum \(220\_1\) was omitted from the analysis due to poor focus. These are synthetic spectra with same profile shapes computed as described at the beginning of the section, after subtracting the continuum from the spectrum. It needs to be addressed that the continuum identification is manually approximated, since no physical fit is considered. While this approach may introduce additional uncertainty in relative line intensities, special care was taken to subtract it equally for spectra analysed with both methods. Any uncertainty introduced by such an approximation should therefore not affect the mutual correspondence of the spectra, which is the aim of this comparison. As in the previous plots, we stress that the spectra are scaled to match best in the part of the range with the best spectral sensitivity and only relative line ratios are considered.

Fig. 20
figure 20

A comparison of relative intensities of the analysed spectra of the meteor observed on 30th October 2022 at 22:16:40 UT from multiple stations, computed using the previously used methodology

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Fig. 21
figure 21

A comparison of relative intensities of the analysed spectra of the meteor observed on 30th October 2022 at 22:16:40 UT from multiple stations, computed using the newly devised methodology

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From the comparison, an obvious overall improvement is visible in the \(4000 {\text{\AA }} - 4500~{\text{\AA }}\) range after applying the new methodology. The observed differences between spectra from different stations can now be considered small enough to be attributed to measurement uncertainties which was clearly not the case for the previous results. The part of the spectral range below \(4000~{\text{\AA }}\) should not be considered reliable due to a strong continuum, low spectral sensitivity and an extrapolated new flat-field.

Relative line ratios are significantly affected by new corrections. Such differences are expected to produce a meaningful impact on the computation of relative elemental abundances. A physical fit of the spectrum is still in the development phase, so the severity of the effect has yet to be quantified.

The one peculiarity which has yet to be explained are spectra \(203\_0\) and \(203\_1\) captured from the same station. They are mutually consistent in most of the spectral range including the sodium doublet at \(5890~{\text{\AA }}\), yet the intensity of the said sodium doublet is inconsistent with the rest of the observing stations. This might be due to a non-considered real physical effect possibly having to do with the optical depth of the meteor plasma as observed from a different incidence angle. Such a consideration might be supported by the fact that the meteor was observed from that station with the lowest radiant distance. For other stations, the radiant distance was in the \(70^{\circ }\) to \(105^{\circ }\) range, while for station 203, it was at only \(15^{\circ }\). At present, however, no clear conclusion is reached regarding the observed discrepancy.

5 Conclusions and future goals

We developed a new methodology of digital meteor spectra reduction, involving changes to the flat-field, spectral sensitivity and atmospheric extinction. The new flat-field was constructed for the \(4000~{\text{\AA }} - 7100~{\text{\AA }}\) wavelength range by fitting the behaviour of spectra of a known fluorescent light-source with high-degree polynomial functions. A simple atmospheric model was applied to construct an extinction curve by using an extinction coefficient at 550 nm computed from stellar photometry. Such considerations were applied to the observed calibration spectra of Venus, Jupiter and the Moon in order to characterise the spectral sensitivity of the instrument.

We performed an extensive study on spectra of a meteor captured by multiple stations of the European Fireball Network on 30th October 2022 at 22:16:40 UT. One spectrum was omitted from the analysis due to poor focus. All other spectra were analysed using the previous and new methodology. A comparison at different stages of calibration revealed that spectra nearest to the centre of the FoV are least affected by the changes and the effect becomes more prominent when approaching the edge of the FoV. We also compared all the captured spectra from different stations, analysed by the old and new methodology. The newly introduced corrections proved to result in a clear improvement of correspondence of the spectra throughout the portion of the spectral range covered by them. Relative line ratios were affected in a significant way. The effect of the changes to the computation of relative elemental abundances is yet to be quantified. Station 203 exhibits a peculiar behaviour which may be linked to a different physical effect.

While we introduced an obvious improvement with the new methodology, there are several issues which need to be addressed. The most important one is that, given the complexity of the problem, it is not possible to reliably extrapolate the flat-field correction values outside of the range covered by laboratory data. To cover the full observational range, several laboratory sources need to be used and analysed accordingly. In order to get reliable results, the signal-to-noise ratio of the laboratory spectra should be as high as possible without saturating any of the lines or the zero order. Geometrical restrictions also apply to the real and laboratory observations, preventing a reliable wavelength calibration for spectra at the edge of the FoV. Namely, a certain number of well-resolved lines is needed for calibration, covering a wider range of wavelengths. For spectra at the edges of the FoV, the number of lines is insufficient, so only a few measurements can be made, resulting in larger uncertainties for the fit in that region. The low spectral sensitivity at wavelengths below \(4000~{\text{\AA }}\) makes the Calcium lines present there highly uncertain. At wavelengths above \(6000~{\text{\AA }}\), the signal-to-noise ratio of the laboratory spectra used for the flat-field is low. Additional uncertainties may rise from continuum identification, which is hard to model and varies between stations. In addition, absolute line intensities differ between observations due to a still unknown reason.

Taking all of these considerations into account, the new corrections are currently valid and reliable in the \(4000 - 6000~{\text{\AA }}\) range, as we confirmed from the multi-station comparison. This part of the spectral range includes most of the spectral lines relevant to a physical study. Using additional known sources the range may be extended in the future. It also appears that relative line intensities of lines at higher wavelengths are not as strongly affected by the newly determined corrections, so an approximate study of those lines may still be possible using previous methodology.

Given the current limitations imposed by the new calibration methodology, we may expect to still have uncertainties for lines of neutral magnesium (3829.35 Å, 3832.30 Å) and singly ionised calcium (3933.67 Å, 3968.47 Å) at the beginning of the spectral range, especially if they lie far from the centre of the image. The same can be expected for lines of oxygen (7771.94 Å, 7774.17 Å, 7775.39 Å), and nitrogen (8680.24 Å, 8683.39 Å) at the end of the spectral range.

However, we expect to see a significant improvement in the reliability of the iron lines, which are well observable in the 4800 Å - 5500 Å range, lines of neutral magnesium (5167.32 Å, 5172.68 Å, 5183.60 Å), neutral sodium (5669.80 Å, 5675.70 Å), and possibly lines of singly ionised silicon at (6347.10 Å, 6371.36 Å).

An improvement may be expected also for lines of other elements, such as manganese or chromium. However, they are usually of lower intensity and are more affected by the signal-to-noise ratio or continuum subtraction approximation.

We stress that the magnitude of these effects is specific to the used equipment and needs to be properly determined experimentally if physically relevant conclusions are meant to be drawn from the obtained results. Future plans involve addressing the issues and limitations of the newly developed methodology, creating a physical fit of the spectra and quantifying uncertainties.