S-PLUS: Photometric Recalibration with the Stellar Color Regression Method and an Improved Gaia XP Synthetic Photometry Method

We present a comprehensive recalibration of narrowband/medium-band and broadband photometry from the Southern Photometric Local Universe Survey (S-PLUS) by leveraging two approaches: an improved Gaia XP synthetic photometry (XPSP) method with corrected Gaia XP spectra, and the stellar color regression (SCR) method with corrected Gaia Early Data Release 3 photometric data and spectroscopic data from LAMOST Data Release 7. Through the use of millions of stars as standards per band, we demonstrate the existence of position-dependent systematic errors, up to 23 mmag for the main survey region, in the S-PLUS iDR4 photometric data. A comparison between the XPSP and SCR methods reveals minor differences in zero-point offsets, typically within the range of 1–6 mmag, indicating the accuracy of the recalibration, and a twofold to threefold improvement in the zero-point precision. During this process, we also verify and correct for systematic errors related to CCD position. The corrected S-PLUS iDR4 photometric data will provide a solid data foundation for conducting scientific research that relies on high-precision calibration. Our results underscore the power of the XPSP method in combination with the SCR method, showcasing their effectiveness in enhancing calibration precision for wide-field surveys when combined with Gaia photometry and XP spectra, to be applied for other S-PLUS subsurveys.

Accurate and uniform photometric calibration presents a challenging task, yet is crucial for wide-field surveys due to rapid fluctuations in Earth's atmospheric opacity on time scales of seconds to minutes, instrumental effects (e.g., flat-field corrections), and electronics instability (e.g., variation in detector gain over time).Traditional optical photometric calibration relies on networks of standard stars with well-determined photometry, such as Landolt (1992Landolt ( , 2009Landolt ( , 2013) ) and Stetson (2000).However, the limited number of standard stars hinder traditional methods from meeting the calibration accuracy expectations of modern wide-field photometric surveys.Over the past two decades, significant advancements have been made in achieving high-precision calibration using various methods, broadly categorized into "hardwaredriven" and "software-driven" approaches, as discussed by Huang & Yuan (2022).Hardware-driven methods include the Ubercalibration method (Padmanabhan et al. 2008), the Hypercalibration method (Finkbeiner et al. 2016), and the Forward Global Calibration Method (Burke et al. 2018).The software-driven approaches involve techniques such as the Stellar Locus Regression method (High et al. 2009), the Stellar Color Regression method (SCR; Yuan et al. 2015a), and the Stellar Locus method (López-Sanjuan et al. 2019).
The central idea of the SCR method is to predict the intrinsic colors of stars by utilizing stellar-atmospheric parameters, which has proven to be particularly effective in photometric re-calibration of wide-field surveys.For instance, when applied to the Sloan Digital Sky Survey (SDSS; York et al. 2000) Stripe 82 (Ivezić et al. 2007), it achieved a precision of 2-5 mmag in the SDSS colors.Additionally, it has been employed for data from Gaia Data Release 2 and Early Data Release 3 (EDR3) to correct for magnitude/color-dependent systematic errors in the Gaia photometry (Niu et al. 2021a,b), yielding an unprecedented precision of 1 mmag.Huang et al. (2021) utilized the SCR approach to recalibrate the second data release (DR2) of the SkyMapper Southern Survey (SMSS; Wolf et al. 2018), revealing large zero-point offsets in the uv-bands.Huang & Yuan (2022) applied the method to SDSS Stripe 82 standardstar catalogs (Ivezić et al. 2007;Thanjavur et al. 2021), achieving a precision of 5 mmag in the SDSS u-band and 2 mmag in the griz-bands (Yuan et al. 2015a)).In addition, Xiao & Yuan (2022) and Xiao et al. (2023b) applied the SCR method to the Pan-STARRS1 (PS1; Tonry et al. 2012) data, effectively correcting for significant large-scale and small-scale spatial variations in the magnitude offsets and magnitude-dependent systematic errors.Other applications include Xiao et al. (2023, in prep), who use the SCR method to perform re-calibration on the J-PLUS DR3 photometric data, accurately measuring and correcting for the PS1 systematic errors and the metallicity-dependent systematic errors present in the J-PLUS DR3 photometric data.Xiao et al. (2023a) also performed the photometric calibration of Nanshan one-meter wide-field telescope gri-band imaging Data of the Stellar Abundance and Galactic Evolution Survey (SAGES; Zheng et al. 2018Zheng et al. , 2019) ) using the SCR method, achieving 1-2 mmag precision in the zero-points.
More recently, comprehensive corrections to the Gaia XP spectra have been provided by Huang et al. (2023), utilizing spectra from CALSPEC (Bohlin et al. 2014;Bohlin & Lockwood 2022) and Hubble's Next Generation Spectral Library (NGSL; Koleva & Vazdekis 2012).In this process, the spectroscopy-based SCR method (Yuan et al. 2015a) was employed as well.Based on the corrected Gaia XP spectra, Xiao et al. (2023, in prep) further develop the XP spectra-based photometric synthesis (XPSP, hereafter) method, and applied it to the photometric calibration of J-PLUS DR3 data.The consistency between the J-PLUS zero-points predicted by the XPSP method after XP spectra correction and the SCR method is better than 5 mmag, which represents a twofold improvement compared to the consistency between the J-PLUS zero-points predicted by the XPSP method with uncorrected XP spectra and the SCR method.
Located at the Cerro Tololo Interamerican Observatory, the Southern Photometric Local Universe Survey (S-PLUS 15 ; Mendes de Oliveira et al. 2019) employs a 83 cm telescope to obtain images on a single CCD.The photometric calibration of S-PLUS DR4 is carried out using photometric data from GALEX, SDSS, Pan-Starrs, Skymapper, and so on, along with the spectral energy distribution (SED) information for calibration sources (Almeida-Fernandes et al. 2022).However, this method i) relies on reference catalogs that do not have uniform calibration precision across the S-PLUS footprint; ii) this approach relies on synthetic stellar models, and will inherit any systematic errors present in these (for instance, Almeida-Fernandes et al. 2022 observe zero-point offsets as high as 50 mmag for J0395 just by changing the synthetic spectral library); and iii) it relies on Schlegel et al. (1998) extinction maps, and thus fails at low Galactic latitudes and exhibits spatially-dependent systematic errors, up to 0.02 mag (Sun et al. 2022); iv) and aperture corrections for the determination of aperture magnitudes.Improvement of the photometric calibration of S-PLUS is crucial, given the importance of high-precision investigations, in particular those that seek accurate determinations of stellar parameters and elemental abundances.
In this study, we utilize both an improved XPSP method and the SCR method to conduct photometric re-calibration of the S-PLUS DR4 data (Herpich et al., in prep.), aiming to achieve uniform photometry with accuracy better than 1%.The structure of this paper is as follows.We present the data used in this work in Section 2. The predictions of S-PLUS magnitudes with the XPSP method and the SCR method are presented in Section 3, followed by a description of the systematic errors presented in S-PLUS DR4 data in Section 4. A discussion is carried out in Section 5. Finally, we provide brief conclusions in Section 6.

S-PLUS Data Release 4
The S-PLUS DR4 encompasses 1629 pointings, covering approximately 3000 deg 2 of the Southern sky, including the Main Survey with PStotal and PSF photometry, the Magellanic Clouds (MCs) with PStotal and PSF photometry, and the Disk Survey with PSF photometry (Herpich et al., in prep).The PStotal photometry was the one used for the calibration, and is the best representation for the total magnitude of a point source in the S-PLUS catalogs (for the aperture photometry).The S-PLUS data were obtained using the T80-South telescope 16 .The panoramic camera features a single chargecoupled device (CCD) with a resolution of 9.2k × 9.2k pixels, a field of view (FoV) measuring 1.4 • × 1.4 • , and a pixel scale of 0.55 ′′ pix −1 (Marin-Franch et al. 2015).It employs 5 broad-band filters (uJAVA, gSDSS, rSDSS, iSDSS, and zSDSS) and 7 medium-band filters (J0378, J0395, J0410, J0430, J0515, J0660, and J0861) within the optical range.It is essential to note that the S-PLUS DR4 magnitudes mentioned in this paper refer to the magnitudes calibrated following Almeida-Fernandes et al. (2022).2.2.Gaia Early Data Release 3 The Gaia EDR3 (Gaia Collaboration et al. 2021a,b) provides the most precise photometric data available to date for approximately 1.8 billion stars.The magnitudes in the G, G BP , and G RP bands have been uniformly calibrated with accuracy at the mmag level (e.g., Abbott et al. 2021;Niu et al. 2021c).To address magnitudedependent systematic errors, which are estimated to be around 1% in these bands for Gaia EDR3, Yang et al. (2021) utilized approximately 10,000 Landolt standard stars from Clem & Landolt (2013).In our study, we adopt the magnitudes of G, G BP , and G RP as corrected by Yang et al. (2021) by default.
2.3.Gaia Data Release 3 Gaia DR3 (Carrasco et al. 2021;Gaia Collaboration et al. 2022), based on 34 months of observations, provides very low-resolution (λ/∆λ ∼ 50) XP spectra for approximately 220 million sources, with the majority having magnitudes G < 17.65.The XP spectra cover a wavelength range from 336 to 1020 nm, and have undergone precise internal calibrations (Carrasco et al. 2021;De Angeli et al. 2022) as well as external calibrations (Montegriffo et al. 2022).However, it is crucial to note that Gaia XP spectra are subject to systematic errors that depend on magnitude, color, and extinction, especially at wavelengths below 400 nm (see Montegriffo et al. 2022;Huang et al. 2023).A comprehensive set of corrections, based on reference spectra from CALSPEC and NGSL, have been provided by Huang et al. (2023).In this paper, the term "corrected Gaia XP spectra" refers to the Gaia XP spectra as rectified by Huang et al. (2023).

LAMOST Data Release 7
The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST; Cui et al. 2012;Deng et al. 2012;Zhao et al. 2012;Liu et al. 2014) is a quasi-meridian reflecting Schmidt telescope equipped with 4000 fibers and a field-of-view spanning 20 deg 2 .LAMOST's Data Release 7 (DR7; Luo et al. 2015) presents a comprehensive data set comprising 10,640,255 low-resolution spectra, over the full optical wavelength range from 369 to 910 nm, with a spectral resolution of R ≈ 1800.To derive fundamental stellar parameters, including effective temperature (T eff ), surface gravity (log g), and metallicity ([Fe/H]), the LAMOST Stellar Parameter Pipeline (LASP; Wu et al. 2011) has been employed.The internal precision typically attained for these parameters is approximately 110 K for T eff , 0.2 dex for log g, and 0.1 dex for [Fe/H] ≳ −2.5 (Luo et al. 2015).

PREDICTIONS OF S-PLUS MAGNITUDES
In this section, we describe how to obtain the predicted magnitudes for the 12 photometric bands of S-PLUS using the improved XPSP method and SCR method.

XPSP Method with Corrected Gaia XP Spectra
The synthetic photometry method involves projecting the Spectral Energy Distribution (SED) at the top of atmosphere of a source onto the transmission curve of the photometric system.Following by Xiao et al. (in prep.), we compute the synthetic magnitude in the AB system (Oke & Gunn 1983;Fukugita et al. 1996) for each S-PLUS band.
To account for the uJAVA-band's wavelength range (322 to 382 nm), slightly bluer than that of the Gaia XP spectra (336 to 1020 nm), we perform numerical extrapolation to extend the Gaia XP spectra.For each source, we obtain a linear function for the Gaia XP spectra flux density with wavelength through fitting of the Gaia XP spectral data over the range of 336 nm to 382 nm for individual stars.This approach has been proposed and validated in the process of re-calibration of J-PLUS photometry, after evaluating multiple extrapolation methods (Xiao et al. in prep.).
3.2.SCR Method with Gaia Photometry and LAMOST Spectra To obtain a cross-validation of the calibration results obtained by the XPSP method, we also employ the independent SCR method to re-calibrate the Main Survey sample.
The SCR method comprises two key techniques: intrinsic color prediction and reddening correction.The former can be performed based on either spectroscopic or photometric data, while the latter necessitates precise measurement of the reddening coefficients relative to extinction values.The SCR method typically involves defining the relationship between the intrinsic colors and the physical quantities using a sample of low-extinction stars, which is then applied to the entire sample to obtain predicted magnitudes.A detailed description of the SCR method is as follows.
For each color, we fit the intrinsic color as a function of T eff and [Fe/H] using a two-dimensional polynomial.Specifically, we use second-order polynomials for the G RP − i, G RP − J0861, and G RP − z colors, and third-order polynomials for the other colors.The intrinsic colors (C 0 ) can then be estimated using observed colors C minus the product of reddening coefficients and extinction E(G BP − G RP ).
The fitting results of the intrinsic colors as a function of T eff , [Fe/H], and extinction of E(G BP − G RP ) are shown in Figure 2, and the corresponding fitting parameters are listed in Table 1.The intrinsic-color fitting residuals are, respectively, 49, 60, 60, 26, 26, 25, 28, 22, 26, 20, 19, and 18 and G RP − J0861 colors, suggesting that S-PLUS magnitudes can be predicted for individual stars with a precision of 20 to 60 mmag using the Gaia and LAMOST data.Furthermore, the fitting residuals exhibit no dependence on T eff , [Fe/H], and E(G BP − G RP ).
Having obtained the intrinsic-color fitting functions, we apply them to the calibration stars to obtain the derived magnitudes m SCR for each image using Equation 1: (1)

SYSTEMATIC ERRORS IN S-PLUS DR4
In this section, we present the process of accurately measuring systematic errors in the S-PLUS photometric data, as well as the their tracing and correction.
4.1.Dependence on G and G BP − G RP Figure 2 illustrates the relationship between the magnitude offsets predicted by the SCR method and the S-PLUS magnitudes, considering the G magnitude and intrinsic color (G BP − G RP ) 0 of the calibration samples.We observe no discernible dependence with respect to either G magnitudes or (G BP −G RP ) 0 color.This indicates that the detector possesses excellent linearity.
We also investigated the differences between the XPSP method magnitude predictions and S-PLUS magnitudes as functions of the G magnitude and G BP − G RP color.There is a slightly dependence on G BP − G RP color, especially in the bluer and redder range, in the uJAVA and g bands, as shown in Figure 3.There is no dependence on G magnitude for all the filters.We attribute method can be found in Xiao et al. (in prep.).
For the calibration of the uJAVA and gSDSS bands, we selectively choose stars from specific G BP − G RP ranges of (0.5, 1.2) and (0.5, 1.8), respectively.Moreover, the fraction of stars falling outside the prescribed color range is only 2-3 per cent.

Spatial Variations
We plotted the spatial distribution of the difference between zero-points for the XPSP method and the S-PLUS magnitudes, as shown in Figures 4, A1, A2, A3,  and A4.The difference in the zero-points between the XPSP method and the S-PLUS magnitudes is computed as the median value of the difference between the XPSP predicted magnitudes and the S-PLUS magnitudes on each image.We observe strong spatial variations in the difference of the zero-points, caused by calibration errors in S-PLUS, which are more pronounced in the blue filters.Simultaneously, we noticed spatial correlations in the differences in the zero-points between the different S-PLUS bands.The reasons for this are discussed in detail in Section 4.3.
To quantitatively estimate calibration errors in the S-PLUS photometry, we consider histograms of the difference in zero-points between the XPSP method and the S-PLUS magnitudes, as shown in Figure 5.By fitting a Gaussian distribution, we estimated the standard deviations for each band.Here, to better illustrate the effect, we forcibly set the overall zero-point difference to zero.During the re-calibration process, we calibrate the zero-point of the S-PLUS magnitudes to the XPSP method.These values indicate the internal precision of S-PLUS DR4, as also mentioned in Almeida-Fernandes et al. (2022), and listed in Table 2.

Tracing and Correction
In order to trace the origin of the systematic errors in the S-PLUS photometry, we plot correlations between the zero-point offsets for each band pair in Figure 6, along with their corresponding correlation coefficients when the correlation coefficients are greater than 0.7.We find a strong correlation between photometric bands with similar central wavelengths (e.g., ∆iSDSS vs. ∆zSDSS); the data points are distributed closely along the one-toone line.This phenomenon is predominantly driven by systematic errors in the reference photometric data in the respective bands.For example, the systematic errors in the S-PLUS i-and z-bands are predominantly influenced by the systematic errors in the Pan-STARRS and SDSS photometric data (e.g., the color and photometric re-calibration of SDSS Stripe 82 can be observed in Yuan et al. (2015a); Huang & Yuan (2022) and while the photometric re-calibration of PS1 can be seen Xiao & Yuan (2022); Xiao et al. (2023b)).
To correct the above systematic errors, we perform a smoothed interpolation algorithm with a linear kernel for each image.The magnitude correction of a certain star in the field of view is obtained by taking the magnitude offsets of the adjacent 20 calibration stars.The corrected magnitude m corr can be computed as where m obs is the observed magnitude from S-PLUS DR4, and ∆m(R.A., decl.) is the position-dependent magnitude offset.The re-calibrated S-PLUS DR4 data is publicly available.

DISCUSSION
This section applies to the S-PLUS DR4 Main Survey data, using it as an illustration for discussion.

Final Accuracies
Figure 7 depicts a comparison of zero-points between the XPSP method and the SCR method for all twelve S-PLUS filters.The differences between these zero-points are computed as the median value of the difference between the XPSP and SCR predicted magnitudes for each image.From inspection, all the points are consistently distributed along the one-to-one line for each band.
To quantitatively estimate the final accuracies of the re-calibration in this work, we present the difference in the zero-points between the XPSP method and SCR method, as a function of star numbers, in Figure 8. Notably, the standard deviations start at higher values, then decrease and converge to stable values as the numbers of stars increase.The convergence value represents the recalibrated accuracy using the XPSP method, which is 1-6 mmag for each of the 12 bands.The final accuracy of the S-PLUS DR4 data in the 12 bands are similar in the Main Survey, MCs and Disk Survey, listed in last column of Table 2.

External Check by White Dwarf Loci
We provide an independent check of the re-calibration using a white dwarf (WD) locus, known for its stability and uniformity at different spatial locations.
To accomplish this, we cross-match the WD catalog constructed by Gaia Collaboration et al. ( 2022 XPSP method and SCR method for calibrating the photometric zero-points.2022) found spatial structures in the residuals of the flat-field correction in the Galactic (X, Y ) plane, and corrected them with numerical interpolation.In this study, we further investigate whether there are any systematic errors related in this plane before and after re-calibration of the S-PLUS photome-try.Specifically, we focus on the gSDSS, J0515, and J0861 bands as examples.We selected three images with ID of iDR4 3 HYDRA-0161, iDR4 3 HYDRA-0152, and iDR4 3 HYDRA-0145 for this investigation, because they contain the highest number of reference stars, approximately 10,000 to 20,000 stars.The images can be retrieved from the S-PLUS cloud.17 From Figure 10, we observe distinct spatial structures in the stellar flat-fields, with variations larger than 0.01 mag.Notably, the structures for each image dif-  shows a trend of smaller values on the left side and larger values on the right side.However, despite these variations, the structures are consistent for different wavelength observations of each image.These structures, which may occur as residual artifacts following sky flatfield correction, can be effectively corrected during the re-calibration process, as shown in Figure 11.

CONCLUSIONS
In this paper, we present a re-calibration of S-PLUS photometry using millions of standards constructed by the XPSP method with corrected Gaia XP spectra.Ad-ditionally, we employ the SCR method with corrected Gaia EDR3 photometric data and spectroscopic data from LAMOST DR7 to construct a sample of about two hundred FGK dwarf standard stars per band, providing an independent validation.
Similarly, when comparing the zero-points between the XPSP and SCR methods, we find minor differences in zero-point offsets, approximately 3-6 mmag for the blue filters, 1-2 mmag for the SDSS-like filters, and 1-3 mmag for the redder filters.These results show that the re-calibration achieves an accuracy of approximately 1 to 6 mmag, when using the XPSP method in this work.
To validate our re-calibration results, we examine the color locus of white dwarfs, and as expected, the distribution of white dwarfs after re-calibration on the color-color diagram appears more compact than before calibration.Additionally, we discuss the minor systematic errors related to CCD position, and identify almost no remaining residuals in the flat-field correction of the S-PLUS photometry.The corrected S-PLUS DR4 photometric data will provide a solid data foundation for conducting scientific research that relies on high-calibration precision.
Overall, our results underscore the effectiveness of the XPSP method paired with the SCR method in improving calibration precision for wide-field surveys, when combined with Gaia photometry and XP spectra.The SCR method is not affected by the accuracy of the transmission curve, and can provide a more robust test and correction for magnitude-or color-dependent systematic errors presented in the photometry data.We propose that future releases of S-PLUS photometry should incorporate the XPSP method paired with the SCR method in their calibration process.

Figure 1 .
Figure 1.Histograms of the number of standard stars for the XPSP method in each image.The bands are labeled in each panel.

Figure 2 .
Figure 2. Two-dimensional polynomial fitting of intrinsic colors with respect to T eff and [Fe/H] for the calibration stars in the SCR method.The intrinsic colors include GBP − uJAVA, GBP − J0378, GBP − J0395, GBP − J0410, GBP − J0430, GBP − gSDSS, GBP − J0515, GRP − rSDSS, GRP − J0660, GRP − iSDSS, GRP − J0861, and GRP − zSDSS.From left to right, the fit results after 3σ clipping are shown in the first column, with the red and blue curves representing results for [Fe/H] = 0 and [Fe/H] = = −1, respectively.The fitting residuals are labeled in red.In the second to sixth columns, the residuals are plotted against T eff , [Fe/H], extinction of E(GBP − GRP), G magnitude, and (GBP − GRP)0 color, respectively.Zero residuals are denoted by black dotted lines.

Figure 3 .Figure 4 .Figure 5 .
Figure 3. Magnitude offsets between the XPSP predicted magnitudes and the S-PLUS magnitudes, as a function of GBP − GRP color, for all 12 bands.The colors represents the density of points, and the bands are indicated in each panel.Zero residuals and two color cuts are denoted by the black and green dotted lines, respectively.Color bars are plotted in the lower right corner for each panel.

Figure 6 .
Figure 6.Correlation plots between the zero-point offsets.The correlation coefficients are shown when they have values greater than 0.7.The colors in each panel indicates the number density of stars, and the dashed black line represents the one-to-one line.

Figure 9 .Figure 10 .
Figure 9.The WD loci G − uJAVA vs. GBP − GRP before (left panel) and after (right) re-calibration in the uJAVA-band.The red and black points represent the R.A. < 90 • and decl.< −40 • region and the R.A. > 180 • and decl.> −30 • region, respectively.The red-dotted curve and black curve correspond to the quadratic polynomial fitting results for the red dots and black dots, respectively.

5. 3 .
Residuals of the Flat-field Correction S-PLUS utilizes the sky flat-fielding technique (Mendes de Oliveira et al. 2019) for this correction.However, Almeida-Fernandes et al. (

Figure 11 .
Figure 11.Similar to Figure 10, but for the results after re-calibration.

Figure A1 .
Figure A1.Similar to Figure 4, but for the MCs with PStotal photometry.

Figure A2 .
Figure A2.Similar to Figure 4, but for the Main Survey with PSF photometry.

Figure A3 .
Figure A3.Similar to Figure 4, but for the MCs fields with PSF photometry.

Figure A4 .
Figure A4.Similar to Figure 4, but for the Disk Survey with PSF photometry.

Table 1
Coefficients used to Obtain Intrinsic Colors as Functions of T eff and [Fe/H] in the 12 bands.In the table, the symbol ei represents 10 where, x is T eff and y is [Fe/H].

Table 2
Internal Precision of the Photometric Calibration for the 12 S-PLUS Bands, in Units of mmag filters Main Ap Main PSF MCs Ap MCs PSF disk PSF Final Accuracy