Photometric Re-calibration of VPHAS+ $u$-band Photometry with the Stellar Colour Regression Method and Gaia DR3

In recent years, numerous wide-field photometric surveys have been conducted (York et al., 2000; The Dark Energy Survey Collaboration, 2005; Skrutskie et al., 2006; Wright et al., 2010; Chambers et al., 2016; Gaia Collaboration et al., 2016; Zou et al., 2017; Onken et al., 2019; Željko Ivezić et al., 2019), producing revolutionary impacts on the advancement of astronomy. Uniform and high-precision flux calibration, which ensures a high degree of consistency in the flux measurements between targets that are widely separated in the sky and guarantees that these measurements do not vary with changes in the telescope’s working environment, detector position, or time of observation, is one of the key factors determining the success of wide-field surveys (Kent et al., 2009; Huang et al., 2022). The precision of flux calibration sets the lower limit on the error in stellar brightness measurements, which is significant for the determination of stellar photometric parameters, the selection of special astronomical objects, and the study of the structure of the Milky Way. Therefore, for the successful execution of wide-field photometric surveys, the undertaking of high-precision flux calibration is of utmost importance.

In traditional astronomical observations, absolute flux calibration is commonly performed using standard stars that are known to have a good internal consistency of brightness (Landolt, 1992; Yan et al., 2000; Zhou et al., 2001). However, due to a shortage of standard stars, the Forbes effect, the colour-dependent atmospheric extinction effect, and flat-field precision, the classical standard star method has difficulties to meet the milli-magnitude level calibration precision required by contemporary wide-field photometric surveys (Huang et al., 2022). In recent years, astronomers have developed a series of new flux calibration methods for wide-field photometric surveys. These methods typically focus only on the relative flux calibration process, with the absolute zero point still being provided by observing standard stars or calibrated to specific standard photometric systems. These include the Uber-calibration method (Padmanabhan et al., 2008), the Hyper-calibration method (Finkbeiner et al., 2016), the Forward Global Calibration Method (Burke et al., 2017), and the XP Synthetic Photometry method (Xiao et al., 2023b, 2024); as well as the Stellar Locus Regression method (High et al., 2009), Stellar Locus method (López-Sanjuan et al., 2019), and Stellar Colour Regression (SCR) method (Yuan et al., 2015; Huang & Yuan, 2022), among others.

Through the application of these methods, we have now achieved typical flux calibration precision of a few milli-magnitude in ground-based optical bands (Huang et al., 2022). However, the calibration precision in the near-ultraviolet $u$ band, especially for the $u$ band photometry in the Galactic disk, is still significantly larger than that in the optical bands. This is due to factors such as more significant atmospheric extinction effects in the $u$ band, more sensitive to stellar atmospheric parameters, lower detector response, fainter stellar magnitudes, insufficient numbers of standard stars, and more serious interstellar extinction effects in the Galactic disk. Nevertheless, the $u$ band is a very important part of stellar parameter determination and the identification of special astronomical objects. Previous work has attempted to accurately calibrate the $u$ band photometry for stars at high Galactic latitudes using stellar atmospheric parameters. Yuan et al. (2015) and Huang & Yuan (2022) have refined the $u$ band photometric calibration for the SDSS Stripe 82, achieving a precision of approximately 5 mmag in $(u-g)$ colour and $u$ magnitude, respectively. In a similar vein, Huang et al. (2021) managed to re-calibrate the $u$ band data for the SkyMapper Southern Survey at high Galactic latitudes with a precision of up to 1 per cent. Furthermore, the $u$ band photometry for the J-PLUS and S-PLUS surveys have been re-calibrated by Xiao et al. (2023c) and Xiao et al. (2023a), respectively, attaining calibration precision around of 5 mmag.

The VST Photometric H$\alpha$ Survey (VPHAS+; Drew et al. 2014) uses VST/OmegaCAM to survey the Southern Galactic Plane and Bulge in $u,g,r,i$, H$\alpha$ down to magnitudes greater than 20. The $u$ band photometric data from VPHAS+ are essential for astronomical research, as they allow for the precise determination of stellar parameters near the Galactic centre and the study of the metallicity of the inner Galactic disk and the bulge. They are also crucial for the selection of OB stars in the inner Galactic disk. Chen et al. (2019b), using the VPHAS+ DR2 $u$ band data, identified nearly 27,000 candidates for O-type and early B-type stars, but due to the poor photometric precision, most of these sources were false positives. After validation with Gaia Data Release (DR2) data (Gaia Collaboration et al., 2018), of the nearly 27,000 candidates, only 6,858 were confirmed as candidates for O-type and early B-type stars.

The VPHAS+ observations concluded in 2018, and VPHAS+ Data Release (DR4) released catalogues of objects based on all observational data, including aperture magnitudes in the $u$ band for stars. The main goal of this paper is to assess the $u$ band photometric precision of stars in this VPHAS+ final data release and attempt to further calibrate the photometry. The structure of this paper is as follows: in Section 2, we introduce the data used; in Section 3, we present the methods employed. Our results are showcased and discussed in Section 4, and concluded in Section 5.

In this work, we adopt an updated SCR (Yuan et al., 2015) method. Unlike the initial implementation of SCR, which predicted intrinsic stellar colours based on stellar atmospheric parameters such as effective temperature $T_{\rm eff}$ and metallicity [Fe/H] from spectroscopic surveys, our method predicts intrinsic stellar colours from intrinsic Gaia XP spectra. The detailed methodology employed is as follows.

1. 1.

   We begin by establishing the relationship between a star’s intrinsic XP spectrum and its intrinsic colour $(u-G_{\rm BP})_{0}$ using the selected Control Sample. When obtaining the intrinsic XP spectra, we have applied a fixed extinction coefficient at each wavelength, sourced from Zhang et al. (2023b). This approach is justified given that the wavelength range corresponding to each flux measurement is only 10 nm. However, to obtain the intrinsic colour $(u-G_{\rm BP})_{0}$, considering the broad $G_{\rm BP}$ band and the $u$ band that covers the near-ultraviolet portion of the stellar SED where variations are prominent, the extinction coefficient $R=E(u-G_{\rm BP})/E0$ cannot be assumed as a constant (e.g., Zhang & Yuan 2023; Sun et al. 2023). Here, $E0$ is the extinction from Zhang et al. (2023b), and $R$ must at least vary with the effective temperature (or intrinsic colour) of the star. Initially, we use the extinction coefficient $R$ values provided by Zhang et al. (2023b) as a starting point for calculating the intrinsic colour $(u-G_{\rm BP})_{0}$.

   After establishing the intrinsic colour $(u-G_{\rm BP})_{0}$, we employ a machine learning method, random forest regression (RFR), to train the relationship between the intrinsic XP spectra and $(u-G_{\rm BP})_{0}$. We randomly divide the Control Sample, using 75% for training and 25% for testing. The RFR model’s input parameters are the standardized intrinsic XP spectra ($F^{\prime}=(F-\mu)/\sigma$, where $F$ and $F^{\prime}$ are the original and standardized fluxes, respectively, and $\mu$ and $\sigma$ are the mean and standard deviation). The output parameter is $(u-G_{\rm BP})_{0}$. This study uses the Python SCIKIT-LEARN package (Pedregosa et al., 2011) to train the RFR model. After testing a range of RFR model parameters, we select the best-performing set for training.

2. 2.

   The RFR model obtained from the previous step will be applied to the intrinsic XP spectra of stars in the Calibration Sample to obtain the predicted colour $(u-G_{\rm BP})^{\rm mod}_{0}$. Thus, we can determine the reddening value for each star in the Calibration Sample as $E(u-G_{\rm BP})=(u-G_{\rm BP})-(u-G_{\rm BP})^{\rm mod}_{0}$. By combining this with the extinction values $E0$ provided by Zhang et al. (2023b), we can fit the extinction coefficient $R$ and the zero-point offset $\Delta M$. We use the following formula to relate $E(u-G_{\rm BP})$ to $E0$:

   $$E(u-G_{\rm BP})=(a_{0}C_{0}^{2}+a_{1}C_{0}+a_{2})E0+\delta M,$$

   (1)

   where $C_{0}$ is the intrinsic colour $(u-G_{\rm BP})^{\rm mod}_{0}$, $a_{0}$, $a_{1}$, and $a_{2}$ are coefficients describing how the extinction coefficient $R$ varies with intrinsic colour $C_{0}$, and $\delta M$ represents the zero-point differences between the control and calibration samples. We iterate this step with the newly obtained extinction coefficients fed back into step (i), limiting the iterations to no more than five, as results generally stabilize after two or three iterations.

3. 3.

   With the relationship between intrinsic stellar spectra and colours obtained in step (i), and the extinction coefficient $R$ obtained in step (ii), we can predict the model magnitudes for each star in the Calibration Sample:

   $$u_{\rm mod}=G_{\rm BP}+(u-G_{\rm BP})^{\rm mod}_{0}+R\times E0.$$

   (2)

   The magnitude offsets $\Delta u$ between the model and observed magnitudes can then be determined using:

   $$\Delta u=u_{\rm mod}-u_{\rm obs}-\delta M.$$

   (3)

   After obtaining the initial magnitude offsets $\Delta u$, we have performed a preliminary photometric calibration on the Control Sample to achieve a more uniform photometry across the reference stars. This preliminary calibration entails a straightforward adjustment of the zero-point differences in the magnitudes observed during each visit. We have computed the average $\Delta u$ for the Calibration Sample stars across each observational visit and applied the correction to derive the calibrated $u$-band magnitudes: $u_{1}=u+\Delta u(\text{visit})$, where $u_{1}$ and $u$ are the post- and pre-calibration $u$-band magnitudes respectively, and $\Delta u(\text{visit})$ represents the average magnitude offset for each observational visit. The corrected magnitudes are then fed back into step (i) to iterate the calibration process.

Fig. 2 illustrates the spatial distribution of the $\Delta u$ for the Control Sample both before and after the preliminary flux calibration following the first iteration. Prior to the correction, a clear spatial variation in the magnitude offsets can be observed, with larger offsets in the upper-left region of the Diagram (Dec $>0$ deg) compared to the lower-right region (Dec $<-1$ deg). The post-correction panel shows a more uniform distribution of magnitude offsets across different visits.

We first present some intermediate results obtained during the process of deriving the $u$-band model magnitudes. For training, a subset consisting of 25 per cent of the Control Sample is randomly selected as the test sample to gauge the performance of our RFR model. Fig. 3 displays a comparison between the intrinsic colours $(u-G_{\rm BP})_{0}$ predicted by our RFR model for the test sample stars and the observed values. The prediction closely matches the observations with a consistency parameter, the $r^{2}$ score, reaching 0.98. The dispersion between predicted and observed values is 0.064 mag. This outcome aligns with our expectations since the intrinsic XP spectra of stars contain essential information about their atmospheric parameters and intrinsic colours. In modelling the intrinsic colours, we utilized spectral fluxes directly, avoiding possible systematic errors inherent in measuring atmospheric parameters (Shen et al., 2022; Yang et al., 2024).

Fig. 4 shows the relationship between the reddening values $E(u-G_{\rm BP})$ for the Calibration Sample stars and the $E0$ values from Zhang et al. (2023b). There are clear differences in the $R$ values for stars of different intrinsic colours (or effective temperatures). Our fitted results, based on Equation (1), correspond well with the data. The fitted relation for the extinction coefficient is as follows:

The magnitude difference $\delta M$ between the Control Sample and the Calibration Sample is determined to be 0.009 mag.

Based on the corrected XP spectra (Huang et al., 2024) and the $G_{\rm BP}$ photometric data from Gaia DR3, along with the individual stellar extinction values provided by Zhang et al. (2023b), we have applied the novel SCR method to valid and enhance the photometric calibration of the $u$-band magnitudes for the VPHAS+ DR4. In this work, we improve the previously established SCR approach by employing a machine learning technique, specifically RFR, to train the relationship between the extinction-corrected intrinsic XP spectra and the intrinsic stellar colours. Our method leverages the vast number of XP spectra available in Gaia DR3, effectively overcoming the limitations of traditional SCR methods, which rely heavily on spectroscopic survey data. As a result, we obtain a high-quality Calibration Sample of nearly 3 million stars, even for the surveys near the Galactic centre of VPHAS+. The RFR model, trained to predict the intrinsic colours of stars based on their XP spectra, shows remarkable consistency with the actual observations, achieving an $r^{2}$ score of 0.98. We also explore the relationship between colour excess $E(u-G{\rm BP})$ and extinction values $E0$ from Zhang et al. (2023b), observing how the extinction coefficients vary with intrinsic colour.

In this study, we have determined the model $u$-band magnitudes for stars in our Calibration Sample. Through our analysis, we have unveiled spatial and temporal variations in the $u$-band magnitude offsets within the VPHAS+ dataset, potentially linked to differing observational conditions. For each observational visit, the computed average magnitude offset, $\Delta u_{1}({\text{visit}})$, exhibits a mean value of $-0.036$ mag and a standard deviation of $0.063$ mag. Moreover, we have discerned variations in the magnitude offsets among different CCD chips within a single visit, which also fluctuate over time. These differences, denoted as $\Delta u_{2}(\text{CCD, visit})$, yield a mean value of $0.0096$ mag and a standard deviation of $0.022$ mag. Additionally, we report the detection of consistent magnitude differences at the pixel level within each CCD, which appear to be stable over time, with a mean offset $\Delta u_{3}{\text{(pixel)}}$ of $0.0003$ mag and a standard deviation of $0.009$ mag.

Upon correcting for these varying magnitude offsets across observational visits ($\Delta u_{1}({\text{visit}})$), CCD chips ($\Delta u_{2}(\text{CCD, visit})$), and pixel bins ($\Delta u_{3}{\text{(pixel)}}$), we have derived the revised $u$-band magnitudes. These corrections led to a reduction in the standard deviation between the corrected magnitudes and the model magnitudes, decreasing from $0.088$ mag to $0.065$ mag. Notably, the corrected magnitudes exhibit no residual dependence on the stellar magnitude, colour, or extinction values. This improvement underscores the significance of our corrections and the enhanced reliability of the $u$-band magnitude determinations as a result of this study.

This work is partially supported by the National Key R&D Program of China No. 2019YFA0405500, National Natural Science Foundation of China 12173034, 12322304, 12222301 and 12173007, and Yunnan University grant No. C619300A034. We acknowledge the science research grants from the China Manned Space Project with NO. CMS-CSST-2021-A09, CMS-CSST-2021-A08 and CMS-CSST-2021-B03.

This work is based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 177.D-3023, as part of the VST Photometric H$\alpha$ Survey of the Southern Galactic Plane and Bulge (VPHAS+, www.vphas.eu).

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

The data underlying this article is available in the manuscript.