Spectral Calibration for SO₂ Cameras with Light Dilution Effect Correction

Abstract

The detection ability of SO₂ cameras has been improved effectively, while the calibration is still the main factor that limits their measurement accuracy. This paper presents a nonlinear calibration theory by considering the effect of light dilution due to the path radiance as well as the dependence of plume aerosol on scattering wavelength. This new spectral calibration method is used to retrieve the SO₂ column density and emission rate of the Etna volcano. Results show that, compared with the DOAS calibration approach, the inversion error can be reduced by 13% if the new spectral calibration is adopted. The superiority of the proposed method will become more obvious for long-distance detection of optically thick plumes.

1. Introduction

SO₂ is one of the most important air pollutants. Despite the short service life of SO₂, it easily combines with atmospheric water vapor and forms sulfurous acid, oxidizing into acid rain quickly and efficiently, further damaging forest ecology and the acid balance of the water environment. In addition, SO₂ also sharply increases air toxicity and seriously affects human health. Research has shown that SO₂ stimulates the respiratory system and deteriorates pulmonary function, thereby weakening the immune system [1,2,3].

The main sources of SO₂ are the burning of fossil fuels and the eruption of volcanoes. The annual volcanic eruption process injects tens of millions of tons of SO₂ into the troposphere and upper stratosphere and causes huge, widespread, and long-term damage to the global ecological environment, further affecting economic development. As a common natural disaster, the hazard scope of volcanic eruption is extremely wide. The SO₂ produced by volcanic eruptions can cause fatal damage to commercial and military aircraft, even at a distance beyond 1000 km [4].

In reality, although the SO₂ emitted by volcanic eruption causes disasters and losses, it also serves as a crucial marker for monitoring volcanic activity. In the early stage of volcanic activity, magma breeds in the deep part of the earth and does not deform the surface. Thus, traditional seismic monitoring methods based on the pressure deformation principle of geography can hardly predict volcanic activity in the early stage. However, as an important product of volcanic activity, SO₂ escapes from the surface of the volcano during the gestation and migration of magma, suggesting that the changing rule of its content can shed insights into the prediction of volcanic eruptions [5]. In addition, monitoring the change in SO₂ column density in the crater offers important scientific significance for studying volcanic dynamics [6], volcanic degassing processes [7,8], spatial evolution of volcanic plumes [9], volcanic plume chemistry [10], and plume diffusion theories [11]. Thus, SO₂ is also called “the telegram in the deep part of the earth” by volcanic scientists.

Top international scientific research institutions have successively proposed optical remote sensing technologies and different mechanisms to monitor the emission of SO₂ from volcanoes. Examples include the Differential Optical Absorption Spectrometer (DOAS), Passive Multi-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) [12], Imaging Differential Optical Absorption Spectrometer (I-DOAS), open-path Fourier transform infrared spectrometers (open-path FTIR) [13], DOAS and Raman LIDAR, and thermal infrared cameras (nicAIR IIC) [14]. Among them, LIDAR is an active detecting technology and can accurately and quickly measure the SO₂ column density along the laser beam path through the volcanic plume, but it cannot observe the entire volume of plume emission [15]. DOAS and FTIR are superior because of their good portability, high spectral resolution, high sensitivity, and simple operation, but they are limited by spatial resolution and the availability of sample data. I-DOAS can visually display the column density information at any position in the field of view (FOV) via mechanical scanning to ensure the overall detection accuracy; however, due to the problem of low temporal resolution, its application in the remote sensing monitoring of volcanic plumes is seriously limited. IIC can simultaneously achieve high spatial resolution and high temporal resolution, but it is seriously affected by environmental interference and background radiation; moreover, due to the complex process of infrared radiation transmission, it is also limited by the problems of low detection sensitivity and large measurement error [16].

In recent years, the SO₂ camera, a new remote sensing method with obvious technical advantages in temporal resolution, spatial resolution, and accuracy, has achieved rapid development. This new technology has been quickly applied to the detection of volcanic activity and the study of the gas-dynamic structure of volcanic plumes since it was proposed [17,18]. With the rapid development of UV cameras and the gradual optimization of UV filter performance, the detection capability of SO₂ cameras has been effectively improved. In the last decade, the accuracy and reliability of SO₂ cameras have been further enhanced owing to the continuous development of plume transport velocity estimation theory and the gradual optimization of image correction methods [19,20,21,22]. However, the precise calibration of SO₂ cameras remains to be a problem limiting their measurement accuracy.

The physical quantity directly measured by SO₂ cameras is the SO₂ optical density (OD). This parameter requires additional equipment to obtain the conversion relationship between the OD of SO₂ and its column density prior to retrieving the SO₂ column density image. This process of determining the conversion relationship can be simply described as calibration. The two commonly used calibration methods for SO₂ camera systems are the gas cell and DOAS methods. The calibration cell method determines the linear relationship between the OD of SO₂ and its column density by inserting multiple SO₂ quartz cells with known amounts of SO₂ column density in the FOV [23]. This method has been the most straightforward and commonly used since several years ago. However, the frequent switching of calibration cells causes adverse effects on the real-time detection of SO₂ UV cameras. Furthermore, the reflection effect of the cell window causes non-negligible errors when determining the calibration function. For the DOAS calibration method, the SO₂ column density information at a certain point of the volcanic plume can be directly retrieved by measuring the OD, and then the conversion factor is determined. In contrast to the cell calibration method, DOAS can effectively reduce the influence of volcanic ash during SO₂ column density retrieval. However, obtaining the best matching relationship between the measurement position of DOAS and the corresponding pixels of the SO₂ camera is difficult during operation, further causing a large system error for the DOAS method.

This study first explains the working mechanism of the SO₂ camera, focusing primarily on the transmission characteristics of atmospheric ultraviolet radiation and then establishes the nonlinear model of SO₂ camera calibration under the effect of light dilution [24]. This dilution effect refers to the changes in spectral features during propagation due to the presence of path radiance arising from the scattering of solar radiation [25]. On this basis, a new spectral calibration method that can continuously calibrate the SO₂ camera in real time is proposed. Then, the new method is used to reprocess the Etna volcanic plume data collected by the Norwegian Air Research Institute with a SO₂ camera to more accurately retrieve SO₂ column density images, consequently enhancing the reliability of the SO₂ camera system.

2. SO₂ Camera

2.1. Theory

The SO₂ camera, which is based on realistic radiative transfer, consists of a UV-sensitive camera and two UV narrow-band filters [26,27,28]. The design scheme of using double narrow-band filters improves the anti-stray scattering interference ability of the SO₂ camera. The emitted plumes of volcanic eruptions, industrial chimneys, or ship exhaust not only contain SO₂, but also comprise abundant particulate matter, which can affect the measurement of SO₂ OD. Thus, the dual-channel SO₂ camera technology can eliminate this adverse effect and eventually improve the measurement accuracy [29,30].

Figure 1 shows the absorption cross-sections of SO₂ and O₃, the transmission curves of two UV narrowband bandpass filters at 310 and 330 nm, and the relative spectra of direct and scattered solar radiation. SO₂ exhibits a significant characteristic spectrum in the UV band within the region of 240–338 nm. For wavelengths shorter than 300 nm, the sunlight signal reaching the ground is almost completely absorbed by the ozone layer. Given the intensity of the solar-scattered signal and the wavelength distribution of the SO₂ absorption cross-section, the SO₂ camera generally selects a UV bandpass filter with a center wavelength of 310 nm and a transmission bandwidth of 10 nm as the signal channel. For this channel, particle (aerosol) scattering and SO₂ absorption reduce the intensity of scattered solar radiation. For the reference wavelength of 330 nm, on the other hand, the scattered solar intensity is affected only by the particle scattering. Thus, pure SO₂ OD can be obtained by determining the difference between the signal channel and reference channel.

On the basis of realistic radiative transfer, the 310 nm spectral channel can be expressed as:

$$I_{\mathrm{A}}=I_{\mathrm{A}0}\cdot \exp (-{\tau }_{S{O}_2}-{\tau }_m^{\mathrm{A}})$$

(1)

where $I_{\mathrm{A}0}$ and $I_{\mathrm{A}}$ represent the light intensity before and after passing through the plume, respectively, and ${\tau }_{{\mathrm{SO}}_2}$ and ${\tau }_m^{\mathrm{A}}$ are the OD of SO₂ and aerosol OD of the signal channel, respectively.

For the 330 nm channel, the scattered solar light is only subject to the extinction effect of particulate matter, i.e.:

$$I_{\mathrm{B}}=I_{\mathrm{B}0}\cdot \exp (-{\tau }_m^{\mathrm{B}})$$

(2)

where ${\tau }_m^{\mathrm{B}}$ is the aerosol OD of the reference channel. By synthetically handling the ODs of the two channels, the SO₂ OD can be obtained as:

$${\tau }_{{\mathrm{SO}}_2}=-\ln \frac{I_{\mathrm{A}}}{I_{\mathrm{A}0}}+\ln \frac{I_{\mathrm{B}}}{I_{\mathrm{B}0}}$$

(3)

After obtaining the OD of SO₂, the column density image of SO₂ can be retrieved based on the calibration relationship between the SO₂ OD and its column density.

2.2. Light Dilution

The UV camera response is often linear in SO₂ column density and the differential optical depth when a standard Beer–Lambert absorption model is assumed. Therefore, the cell calibration method is generally used to retrieve SO₂ column density in cases where the camera is close to the plume and the amount of aerosol in the plume can be neglected. However, for volcanic monitoring, the SO₂ camera system is usually positioned at a point several kilometers away from the volcano to include the whole plume. In this condition, the scattered photons between the camera and plume come into FOV, which induces a systematic underestimation of the SO₂ retrievals.

SO₂ cameras with two filters allow for the compensation of aerosol attenuation. However, the Mie scattering of aerosols in the plume is strongly dependent on wavelength [31]. The aerosol OD of the two channels can be expressed as a numerical relation as:

$${\tau }_{\mathrm{m}}^{\mathrm{A}}=\mathrm{K}\cdot {\tau }_{\mathrm{m}}^{\mathrm{B}}$$

(4)

where $\mathrm{K}=\raisebox{1ex}{${\sigma }_{\mathrm{m}}^{\mathrm{A}}$}\!\left/ \!\raisebox{-1ex}{${\sigma }_{\mathrm{m}}^{\mathrm{B}}$}\right.$ is the ratio between the scattering cross-sections of the aerosol in the plume for the two different wavelengths.

By substituting Equation (4) into Equations (1) and (2), the SO₂ OD can be expressed as:

$${\tau }_{{\mathrm{SO}}_2}=-\ln \frac{I_{\mathrm{A}}}{I_{\mathrm{A}0}}+\mathrm{K}\cdot \ln \frac{I_{\mathrm{B}}}{I_{\mathrm{B}0}}$$

(5)

The scattering intensity of aerosol that varies with wavelength will cause a non-negligible error in the calculation of SO₂ OD. Moreover, the light dilution effect can cause large errors. When the distance between the camera and the measured target is sufficiently far, an incremental radiation I_D, which increases with the increase in distance L, will be produced due to the atmospheric scattering of photons between the plume and the instruments.

$$I_{\mathrm{D}}=I_0\cdot \left[1-\exp \left(-\epsilon \cdot L\right)\right]$$

(6)

where $\epsilon$ is the extinction coefficient, $I_0$ is the intensity of the sky background, and $I_{\mathrm{D}}$ is the incremental radiation.

According to Equation (6), the values of incremental radiation of the 310 and 330 nm channels are not equal. In consideration of the light dilution effect, the SO₂ OD should be written as [32]:

$${\tau }_{S{O}_2}=-\ln \frac{\left[I_{\mathrm{A}}\cdot \exp \left(-\epsilon \cdot L\right)+I_{\mathrm{DA}}\right]}{I_{\mathrm{A}0}}+\mathrm{K}\cdot \ln \frac{\left[I_{\mathrm{B}}\cdot \exp \left(-\epsilon \cdot L\right)+I_{\mathrm{DB}}\right]}{I_{\mathrm{B}0}}$$

(7)

3. OD Image Acquisition

The data set used in this paper was collected by Jonas Gliß et al. at 07:06 and 07:22 on 16 September 2015, using their self-developed SO₂ UV camera in Milo Town, about 10.3 km from Etna volcano plume source. The altitude of Milo Town and the crater is 803 and 3103 m, respectively [33]. The main instrumental information is summarized in

Table 1. Main instrument information.

Camera		DOAS
UV Camera	Hamamatsu C8484-16C	Spectrometer	Ocean Optics USB 2000+
On-band filter	Asahi UUX0310	Telescope	f = 100 mm, (f/4)
Off-band filter	Asahi XBPA330	Detector range	200–1100 nm
Camera resolution	1344 × 1024	pixel	2048
UV lens (focal length)	25 mm	Optical fibre	400 µm

.

The time for collecting picture information was short (about 15 min), so the solar zenith angle (SZA) did not change significantly, and it was considered that the calibration curve remained almost unchanged during the experiment. Figure 2a,d represent dark current images of channels 310 and 330 nm, respectively. The sky background and plume images corresponding to the two channels can be obtained through dark current correction, as shown in Figure 2b,c,e,f, respectively. The sky background images in both channels were taken by moving the camera FOV above the crater when no volcanic plume was detected, and the dark noise images were obtained after shooting the sky background with a lens hood on the back cover. The exposure time for A and B channel plume images was 33 and 8 ms, respectively.

Figure 3a,b present the ODs of 310 and 330 nm channels obtained by using the four-images (4-IM) method after obtaining their corresponding sky background and smoke plume images [24]. On the basis of the working principle of SO₂ cameras, the SO₂ OD image can be obtained by performing a subtraction operation between the OD images, which is shown in Figure 3c. Note that, in order to stress the details of the plume, the mountain areas have been masked in all images.

4. Calibration

The intensity of the scattering spectrum reaching the ground is determined by SZA, especially in the ultraviolet wavelength region where O₃ has a significant absorption (as shown in Figure 1). For different SZAs, the optical paths passing through the atmospheric ozone layer vary considerably, with different strengths of scattered light reaching the near-surface earth. This situation further differentiates the path lengths through the plume, eventually altering the ODs. Thus, precise real-time calibration for the SO₂ camera system must be ensured. At present, DOAS is the most commonly used calibration method due to its advantages in correcting for radiative transfer effects. However, it is quite difficult in engineering practice to obtain the best FOV correlation between DOAS and the UV camera, which inevitably causes systematic errors. Aiming to improve the calibration accuracy, we propose a new method of spectral calibration in this study.

4.1. DOAS Calibration

DOAS identifies gas components based on the narrowband absorption characteristics of gas molecules and infers the column density of trace gases according to the narrowband absorption intensity. The calibration curves for the DOAS method can be obtained by fitting the OD images on the same time series with the corresponding DOAS data in a linear function. This approach, which can reduce the impact of particles, gradually becomes the mainstream method of SO₂ UV camera calibration. However, the light dilution effect still affects the inversion accuracy, so it needs to be corrected. The DOAS calibration curves uncorrected and corrected by the light dilution effect are shown in Figure 4. During the calibration process, it is necessary to obtain the extinction coefficients of channels A and B, which are 0.0743 and 0.0654 km⁻¹, respectively, to correct the transmission of solar radiation from the plume to the cameras’ FOV. More detailed information on determining the calibration curve of DOAS is described by Gliß et al. [33]. Note that the calibration curve of the SO₂ camera obtained with DOAS is sensitive to the position, shape, and exact size of the DOAS FOV. This implies that the identification of the best correlation between SO₂ column density and its OD is very difficult, which leads to a nonignorable error between the measurement and the true value. Moreover, applying the specific point to the entire OD image is an inappropriate approach.

4.2. Spectral Calibration

On the basis of the measurement principle of the SO₂ UV camera and the theory of light dilution effect, a new method of personalized real-time calibration for all pixels of the camera is proposed for solving the calibration problem of SO₂ UV cameras. This method requires the incorporation of a spectrometric channel into the SO₂ imaging system for collecting the sky background spectra. Thereafter, the calibration curve of the SO₂ camera can be obtained based on the realistic radiative transfer.

For any pixel of the SO₂ UV camera, the signal intensity corresponds to the convolution integral of the camera quantum efficiency $Q(\lambda )$, the incident solar-scattered light intensity $L_o(\lambda )$, the SO₂ absorption cross-section $\sigma (\lambda )$, the SO₂ column density $C_{S{O}_2}$, and the transmission function $T(\lambda )$ of the UV bandpass filter [32]. On the basis of known filter transmittance and camera quantum efficiency (both can be obtained from the product delivery report), the calibration curve of the SO₂ UV camera at any time can be calculated in real time by measuring the spectral signal of the sky background with a spectrometer.

$$I_{\mathrm{A},\mathrm{B}}={\displaystyle {\int }_0^{\infty }{L}_0(\lambda )}\cdot \exp [-(\sigma (\lambda )\cdot C_{S{O}_2}+{\tau }_m(\lambda ))]\cdot T(\lambda )\cdot Q(\lambda )d\lambda$$

(8)

where $\lambda ={\lambda }_{\mathrm{A},\mathrm{B}}$ is the wavelength of the A or B channel.

Combining Equations (7) and (8), the functional relationship between the SO₂ OD and SO₂ column density can be obtained. Based on the study by Lübcke et al. [31] and combining the specific OD values of the A and B band channels, the scattering cross-section ratio K = 1.09 was chosen. Figure 5a,b show the spectral calibration curves under different particle ODs with the light dilution effect uncorrected and corrected, respectively. These calibration curves in Figure 5 were obtained through the constructed MODTRAN radiative transfer model [34], which only quantitatively explains the effect of optical dilution and particle optical thickness on the calibration curves. From the comparison of these two sets of calibration curves, it can be found that the relationship between SO₂ OD and its column density follows the Beer–Lambert approximation if the light dilution effect is neglected, which is in good agreement with the black line in Figure 4. The direct linear relationship is destroyed for distant or optically thick volcanic plumes, as the Beer–Lambert model does not account for complex radiative transfer. The larger the aerosol optical thickness, the lower the sensitivity towards SO₂. Moreover, the camera sensitivity also decreases with increasing SO₂ OD. This means that the column density of both SO₂ and aerosol in the plume must be taken into account if accurate calibration curves are desired.

Superior to the DOAS approach, the spectral calibration method could customize a personalized real-time calibration curve for each pixel, fully considering the influence of different SO₂ and particle content on camera sensitivity. Moreover, the spectral calibration method also takes the influence of incident angles on the effective transmittance of filters into account, which can significantly reduce the underestimation of the column density inversion caused by the nonparaxial effect.

In the experimental process, the actual calibration spectra were measured in real time using the spectrometer. In view of further comparing and analyzing the influence of the particle OD, the calibration curve data are fitted with the quadratic function ($C_{{\mathrm{SO}}_2}=a\cdot {({\tau }_{S{O}_2})}^2+b\cdot {\tau }_{S{O}_2}+c$). In addition, the coefficients a, b, and c of the real-time calibration curve obtained by the actual measurement spectral calibration method are constantly changing and, according to the theoretical analysis in Figure 5, it can be known that ${\tau }_m^{\mathrm{B}}$ also affects the results of the calibration curve, so there is some kind of relationship between the coefficients a, b, c, and ${\tau }_m^{\mathrm{B}}$. After several sets of measurements were calculated and analyzed, it was found that the coefficients a, b, and c were quadratically related to ${\tau }_m^{\mathrm{B}}$, i.e., $a=m_a\cdot {({\tau }_m^{\mathrm{B}})}^2+n_a\cdot {\tau }_m^{\mathrm{B}}+h_a$, $b=m_b\cdot {({\tau }_m^{\mathrm{B}})}^2+n_b\cdot {\tau }_m^{\mathrm{B}}+h_b$, and $c=m_c\cdot {({\tau }_m^{\mathrm{B}})}^2+n_c\cdot {\tau }_m^{\mathrm{B}}+h_c$, as shown in Figure 6. Thus, the accurate inversion of the SO₂ column density image under the light dilution effect should obtain the direct numerical relationships between the SO₂ column density and SO₂ OD and between the SO₂ column density and particle OD. However, the sensitivity of the instrument can significantly deviate from the linear relationship (i.e., Beer–Lambert approximation) of the optically thick plumes. The larger the particle OD value, the lower the instrument’s sensitivity towards SO₂. The slope of the calibration curves also increases with the widening distance between the SO₂ camera and the plume. Thus, $m_{a,b,c}$ should be calculated, with $n_{a,b,c}$ and $h_{a,b,c}$ corresponding to each pixel, to bring the coefficients into the calibration curve.

5. Inversion

5.1. Column Density Inversion

Due to the influence of the light dilution effect and different physical mechanisms, the SO₂ column density inversion results with calibration curves obtained from the DOAS and spectral methods will also be different. Figure 7 shows SO₂ column density images retrieved by these two calibration methods with and without the light dilution effect. It can be seen that the SO₂ column density images retrieved by DOAS (Figure 7a,b) and spectral (Figure 7c,d) calibration methods are consistent on the whole regardless of whether the light dilution effect is considered, which proves the scientificity of the spectral calibration method. Note that the scale range of the column density image corrected by light dilution is twice that of the uncorrected image. In addition, the point calibrated by the DOAS method is located in the center region of the image with the corresponding co-ordinates of (672, 512), as shown by the black cross in Figure 7a.

For a more intuitive expression of the difference with the inversion results retrieved by the two calibration methods, the column density values in the same column (200) of the column density images are selected and jointly presented in Figure 8. It is obvious that the inversion results are seriously underestimated without considering the light dilution effect. The result of the DOAS method is underestimated by about 2.2 times, while the underestimate for the spectral method is more obvious by about 2.5 times. Moreover, the column density curves retrieved by the two methods are more consistent under the condition of considering the light dilution effect with an average difference of about 10%. Furthermore, for areas with low particle concentration, their inversion results are in better agreement, with a difference of less than 4%. For values between the 500 and 930th row where aerosol optical thickness is relatively high, the SO₂ column density retrieved by the spectral calibration method is slightly lower than that retrieved by DOAS. This is because the spectral calibration method takes the wavelength dependence of particle scattering into account and, therefore, is essentially free of the influence of particulate matter in the plume.

5.2. Speed Acquisition

The inversion-derived column density images provide an effective visualization of the spatial distribution and temporal evolution of SO₂ plume within a two-dimensional domain. However, in practical engineering applications, there is often a greater focus on the variation in SO₂ gas emissions over time. Therefore, the column density image of the SO₂ plume needs to be converted to the SO₂ emission rate with the following equation [33]:

$$\mathrm{\Phi }(\mathcal{l})=f^{-1}{\displaystyle \sum _{m=1}^M{C}_{{\mathrm{SO}}_2}}\cdot v_{\mathrm{eff}}\cdot d_{\mathrm{pl}}\cdot \Delta s$$

(9)

where $\mathrm{\Phi }(\mathcal{l})$ is the SO₂ emission rate of the cross-section, $f$ is the focal length of the camera, $m,M$ is the position range of the plume cross-section, $v_{\mathrm{eff}}$ is the velocity component in the direction normal to the plume cross-section obtained by the optical flow method, $d_{\mathrm{pl}}$ is the distance from the camera to the plume, and $\Delta s$ is the integration step of the plume cross-section. From Equation (9), it can be seen that the gas velocity of SO₂ is another important factor affecting the accurate inversion of SO₂ emission rate. Note that the plume cross-section should be selected in the direction normal to the overall motion of the plume.

Among the various techniques available to obtain the velocity field of SO₂ gas, the optical flow method has emerged as the leading algorithm for estimating gas diffusion motion owing to the notable advantages in computational speed and accuracy. The Farnebäck optical flow method is a pyramidal iterative approach that estimates motion by starting with coarse features and then gradually refining them until the optical flow of each pixel across the entire image is obtained [35]. The Farnebäck optical flow method was shown to be effective in terms of operational speed as well as computational accuracy in the study by Peters et al., 2015 [36], so the Farnebäck optical flow method was next used to estimate the velocity field of SO₂ gas. Figure 9 shows the SO₂ column density image at 07:07 on 18 September 2015, along with information on the plume velocity calculated using the Farnebäck optical flow method (as depicted by the black arrows). The optical flow method utilizes the OD acquired from the 310 nm band channel to estimate the displacement vector fields; each displacement vector field contains 20 × 20 pixels. The vector length of each display was amplified tenfold to enhance plume motion details.

5.3. Emission Rate Inversion

After obtaining the velocity field of SO₂ gas, the SO₂ emission rate can be determined by inverting Equation (9), combined with the SO₂ column density image of Section 5.1 inversions, as shown in Figure 10. Specifically, $f=25 \mathrm{mm}$, $d_{\mathrm{pl}}=10.3 \mathrm{km}$, $\mathsf{\Delta }s=4.65 \mathsf{\mu }\mathrm{m}$. As depicted in the figure, the SO₂ emission rates obtained through the spectral calibration method and the DOAS calibration method are consistent over time, with an error margin of no more than 15%, which provides strong evidence for the scientific validity of the spectral calibration method.

To further illustrate the differences between the two methods, a linear fit will be made to the inversion results of the SO₂ emission rates of the two calibration methods. The resulting fitting equation is shown in Figure 11, which reveals that the equation for both methods is Y = 0.87 × X and the R-squared value is 0.9987, demonstrating the feasibility of using spectral calibration. Conversely, the DOAS calibration method produced a fitting coefficient of less than 1 when compared to the spectral calibration method, likely due to OD of the particles, which cannot be effectively corrected using the DOAS approach. In addition, the DOAS calibration method can directly obtain the spectral information, so it can more comprehensively consider the influence of radiation transmission on the measurement results and is, therefore, more effective in eliminating errors caused by the light dilution effect. However, this method suffers from the problem of difficult spatial matching, which leads to systematic errors. The spectral calibration method may be inferior to the DOAS calibration method in eliminating the light dilution effect, though it does not have the problem of systematic errors due to spatial matching. Therefore, some errors are inevitable for the inversion results obtained from these two methods.

6. Discussion

In this paper, a spectral calibration method for nonlinear calibration is proposed, focusing on correcting the specific effects of light dilution and aerosol extinction and comparing it with the DOAS calibration method to verify the scientific validity of the spectral calibration method. The comparison of the results of the inversion of the two calibration methods is shown in Figure 10, and it can be seen that there is a relatively good consistency between the two but there is still an error of 13%. The main reasons are analyzed as follows: first, the DOAS calibration method uses a spectrometer to directly measure the SO₂ column density within the plume, and the light dilution effect can be well taken into account by modeling multiple scattering. However, the DOAS calibration method requires matching the SO₂ column density at a point in the plume measured by the spectrometer with the SO₂ OD measured by the SO₂ camera at the same spatial location, which usually has the problem of difficult matching and is prone to systematic errors. In contrast, the spectral calibration method is based on a single scattering model to correct for the light dilution effect. Although this effect is slightly underestimated, no difficulty is found in matching the FOV. Thus, it is conceived that the matching accuracy is better than that of the DOAS calibration method. However, there are some errors in the accuracy of the monitoring distance and extinction coefficient during the calculation, which will further have some influence on the spectral calibration method. Meanwhile, the spectral calibration method is based on the assumption that the SO₂ OD is a quadratic function of the SO₂ column density, which may be invalid when the concentration of carbon black particulate matter in the plume is particularly large [37]. Ultimately, under the combined influence of the above factors, there is a large error in the SO₂ emission rate resulting from the inversion using the two calibration methods.

For the dual-channel UV imaging system, the intrinsic factors of the system and the monitoring conditions can also cause errors in the inversion results. When using the SO₂ camera for experiments, other light from the atmosphere (especially short-wave ultraviolet and blue light) entering the optical path can cause interference with the quality of the images and, thus, affect the accuracy of the SO₂ column density inversion. To better study the distribution and variation of SO₂ column density in the smoke plume, filters must be placed to allow only specific wavelengths of light to enter the camera. However, filter placement can significantly affect the accuracy of SO₂ inversion, especially at the edges of the image. When the filter is placed behind the lens, undesirable variations in maximum transmittance and central bandpass wavelength at the edge of the image can be greatly reduced. In the following sections, we analyze in detail the effect of differences in quantum efficiency between the two channels and the sensitivity of the off-axis system on SO₂ inversion, using filter placement behind the lens as an example.

6.1. Quantum Efficiency

Quantum efficiency refers to the photoelectric conversion efficiency, which is the ratio of the number of photons received by the camera to the actual number of electrons produced. The higher the quantum efficiency, the more effectively the camera can capture smoke plume signals, thus improving the detection sensitivity and measurement accuracy of SO₂. The dual-channel UV camera uses two channels with different wavelengths for measurement, which can eliminate some interfering factors in the atmosphere and improve the accuracy of SO₂ column density inversion. However, the quantum efficiency of the two channels may not be the same, as shown in the spectral response curve of Figure 12a, where the quantum efficiency varies with wavelength. Therefore, the effect of the quantum efficiency of the two channels must be evaluated and corrected before conducting experiments. The central wavelengths of the filters used in this experiment were 310 and 330 nm, with bandwidths of 10 nm. Figure 12b shows the impact of the difference in quantum efficiency between the two channels on the SO₂ inversion error under such conditions, with the error for SO₂ inversion results not exceeding 0.7% despite the 20% difference in quantum efficiency between the two channels. In the experiment, the relative difference in the quantum efficiency of the camera used was about 4.5% for the two channels, with the error of the corresponding inversion results for SO₂ not exceeding 0.2%, making it negligible.

6.2. Off-Axis System Sensitivity

Off-axis system sensitivity usually refers to the imaging quality sensitivity at a point off the optical axis in an optical system due to system errors. At the off-axis point, the light is no longer perpendicular to the optical axis, causing the angle of optical incidence on the filter to shift, thus changing the effective transmittance and central wavelength of the filter, which will affect the SO₂ UV camera sensitivity.

Although, ideally, all light is incident vertically, this condition is not always satisfied in an optical imaging system. As a result, the effective transmittance of the filter is not consistent with the manufacturer’s specifications, and the effect of the angle of incidence of the light needs to be considered. The effective bandpass center ${\lambda }_C$ can be expressed as [20]:

$${\lambda }_C\approx {\lambda }_F{\left[1-{\left(\frac{n_{\mathrm{air} }}{n_F}\right)}^2{\sin }^2(\theta )\right]}^{1/2}$$

(10)

where $n_{\mathrm{air} }$ is the air refractive index, $n_F$ is the filter refractive index, and $\theta$ is the angle of incidence. When the angle of incidence increases, the effective bandpass center ${\lambda }_C$ decreases, i.e., it is shifted toward the short-wave direction.

In the experiment, the filter is positioned behind the lens as shown in Figure 13. This practice converges the light passing through the lens on the filter to effectively reduce undesirable variations at the edges of the image. Nevertheless, the angle of incidence may still impact the central wavelength, effective transmission bandwidth, and transmittance of the filter, even when placed behind the lens. Therefore, a thorough examination of these effects is necessary to pinpoint potential error sources in the inversion of SO₂.

When the filter is placed behind the lens, in order to specifically obtain the effective transmittance of the device, the angle of incidence can be expressed in terms of the position where the light intersects with the lens, using polar co-ordinates in the form of the filter mark and finding the angle of incidence $\theta (r,\phi )$; at this point, $\theta (r,\phi )$ is:

$$\theta (r,\phi )=\arctan \left(f^{-1}{\left(f^2{\tan }^2(M\alpha )+r^2-2fr\tan (M\alpha )\cos \phi \right)}^{1/2}\right)$$

(11)

where r is the distance from the light shining on the filter to the center of the filter, $\phi$ is the direction relative to the x, $\alpha$ is the viewing angle, and $M$ is the magnification.

The effective transmittance $T(\lambda ,\alpha )$ of the device is obtained by integrating the maximum transmittance of the filter $T_C$, the Gaussian bell curve G over the effective aperture radius R, and all relative directions $\phi$.

$$T(\lambda ,\alpha )=\frac{T_0}{T_F\pi R^2}\cdot {\displaystyle {\int }_0^R{\displaystyle {\int }_0^{2\pi }{T}_C}}(\theta (r,\phi ))\cdot G\left(\lambda ,{\lambda }_C(\theta (r,\phi )),{\sigma }_F\right)rd\phi dr$$

(12)

where $T_0$ is the normalization factor and $T_F$ is the nominal maximum transmittance.

Figure 14 illustrates the relative absorption cross-section of SO₂ in the presence of a filter placed behind the lens, along with the effective transmittance curves of the two channels at viewing angles of 0°, 5°, and 10°. As shown in the figure, the transmittance of the filter reaches its maximum value within its designated wavelength range, with a rapid decrease observed outside this range. Furthermore, the angle of incidence affects the transmittance of the filter. When the light is not incident vertically, the actual transmittance deviates from the standard curve provided by the manufacturer, and the effective bandpass center also shifts. At normal incidence, the transmittance of the effective bandpass center is highest and exerts minimal impact on the overall transmittance. However, as the angle of incidence varies, the effect of the effective bandpass center on transmittance increases, with effective bandpass center moving towards shorter wavelengths.

Variations in maximum transmittance and effective bandpass center can have a significant impact on the experimental outcomes. Increasing viewing angles lead to a gradual reduction in maximum transmittance, which, in turn, decreases the signal-to-noise ratio at the detector edge. Additionally, the effective bandpass center shifts towards shorter wavelengths, altering the system sensitivity to SO₂ and ultimately influencing the inversion results. Given that both factors are influenced by variations in angle of incidence resulting from converging light through the lens, changes in viewing angle can serve as an indicator of SO₂ sensitivity. With the filter placed behind the lens and the center wavelength of 310 and 330 nm, respectively, the SO₂ error with the change in view angle is shown in Figure 15. The figure shows that the impact on SO₂ inversion is minimal at the center of the camera’s sensing surface due to near-vertical incidence, but the error gradually increases with increasing viewing angles. However, even at the image edge, where the error reaches 3%, it remains within acceptable limits.

7. Conclusions

This study introduces the dual-channel SO₂ UV camera and its application to volcanoes, illustrates the working principle of obtaining SO₂ column density images, analyzes the light dilution effect on SO₂ column density inversion, and focuses on the physical mechanism and implementation method of the new spectral calibration method. Moreover, on the basis of the DOAS and the new spectral calibration methods, the SO₂ column density images are retrieved. The comparative analysis demonstrates the great advantages of the new calibration method in ensuring accuracy and operability, especially for long-distance detection.

Then, after introducing the working principle of the dual-channel SO₂ UV camera and the spectral characteristics of SO₂ absorption cross-section and solar-radiation scattering in the atmosphere, the reasons for selecting the 310 and 330 nm bands are clarified. On the basis of the realistic radiative transfer, the SO₂ OD of the Etna volcanic plume is obtained. By analyzing the principles of the DOAS calibration method, this study expounds the reasons why the calibration method gives misleading results in SO₂ remote monitoring. On the premise of synthetically accounting for the influences on camera sensitivity and calibration curves due to the light dilution effect and wavelength dependence of particle scattering, a new spectral calibration method is proposed. This method takes into account the underestimate of the inversion result caused by the nonparaxial effect of the filters and can achieve personalized real-time calibration for each pixel of the camera systems. Furthermore, the reason why the calibration curve obtained by this new method presents nonlinear characteristics is also highlighted. A quadratic function relationship exists between the calibration coefficient and the particle OD. The instrument sensitivity towards SO₂ decreases significantly with increase in particle OD. The comparison between the SO₂ images retrieved by the two calibration curves suggests that the new calibration method is more rigorous and scientific than the DOAS method. Moreover, its superiority becomes more obvious for long-distance detection of optically thick plumes. In addition, compared with the DOAS calibration approach, the inversion error of the emission rate can be reduced by 14% if the spectral calibration is adopted due to the fact that the new method is essentially free of the influence of particulate matter in the plume.

Despite the major advantages of the new calibration method, its novelty necessarily means that many algorithmic and instrumental problems remain unresolved. For instance, aerosol scattering and absorption are difficult to accurately distinguish in practical settings, and the calibration errors caused by light-absorbing aerosols are extremely hard to predict or estimate. The general impact of aerosol absorption on camera sensitivity reduces the average penetration depth of solar radiation scattering that passes through the plume. In addition, the effects of atmospheric visibility and thick clouds will lead to a much larger uncertainty in SO₂ column density inversion. Therefore, more sophisticated approaches must be proposed to confidently calibrate SO₂ cameras with higher accuracy.

This study unequivocally shows that explicit treatment of radiative transfer plays a crucial role in improving calibration accuracy. The spectral calibration method proposed in this study solves the technical problem of accurate calibration of SO₂ UV cameras and has been well applied in remote monitoring of volcanic eruptions. Hopefully, this calibration method will also show great application prospects in the remote sensing monitoring of industrial chimney and ship exhaust.