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Gaia Early Data Release 3: Modelling and calibration of Gaia's point and line spread functions

N. Rowell, M. Davidson, L. Lindegren, F. van Leeuwen, J. Castañeda, C. Fabricius, U. Bastian, N. C. Hambly, J. Hernández, A. Bombrun, D. W. Evans, F. De Angeli, M. Riello, D. Busonero, C. Crowley, A. Mora, U. Lammers, G. Gracia, J. Portell, M. Biermann, A. G. A. Brown

arXiv:2012.02069v1astro-ph.IM

TL;DR

Gaia’s astrometric precision depends on accurately reconstructing source images for PSF fitting. This paper develops and calibrates time- and colour-dependent PSF and LSF models for EDR3, finding improved calibration and reduced systematic errors while identifying remaining limitations and future improvements.

  • Problem

    Accurate PSF reconstruction is needed to estimate Gaia source locations and fluxes, but Gaia’s unusual, undersampled observations and varying PSF dependences require dedicated modelling.

  • Method

    The paper models one-dimensional LSF and two-dimensional PSF profiles with basis components, fits selected primary sources, and merges independent time calibrations with a square root information filter.

  • Results

    The calibration reproduces strong time and colour dependences, shows stable goodness of fit across the instrument, and supports EDR3 improvements beyond those expected from increased observations alone.

  • Takeaways & Limitations

    The EDR3 PLSF calibration is a major processing advance that contributes to reduced systematic errors in core astrometric and photometric products.

  • Takeaways & Limitations

    Using a single effective-wavenumber colour parameter limits modelling for sources with non-stellar spectral energy distributions.

Abstract

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Context: The unprecedented astrometric precision of the Gaia mission relies on accurate estimates of the locations of sources in the Gaia data stream. This is ultimately performed by point spread function (PSF) fitting, which in turn requires an accurate reconstruction of the PSF. Gaia Early Data Release 3 (EDR3) will, for the first time, use a PSF calibration that models several of the strongest dependences, leading to signficantly reduced systematic errors. Aims: We describe the PSF model and calibration pipeline implemented for Gaia EDR3, including an analysis of the calibration results over the 34 months of data. We include a discussion of the limitations of the current pipeline and directions for future releases. This will be of use both to users of Gaia data and as a reference for other precision astrometry missions. Methods: We develop models of the 1D line spread function (LSF) and 2D PSF profiles based on a linear combination of basis components. We fit the models to selected primary sources in independent time ranges, using simple parameterisations for the colour and other dependences. Variation in time is smoothed by merging the independent calibrations in a square root information filter, with resets at certain mission events that induce a discontinuous change in the PSF. Results: The PSF calibration shows strong time and colour dependences that accurately reproduce the varying state of the Gaia astrometric instrument. Analysis of the residuals reveals both the performance and the limitations of the current models and calibration pipeline, and indicates the directions for future development. Conclusions: The PSF modelling and calibration carried out for Gaia EDR3 represents a major step forwards in the data processing and will lead to reduced systematic errors in the core mission data products. Further significant improvements are expected in the future data releases.

1. Introduction

Gaia EDR3 improves source-location and flux measurements through dedicated PSF and LSF calibration within a complete reprocessing of the first 34 months of mission data.

  • EDR3 reprocesses the first 34 months of Gaia observations, enabling improved calibration and instrument models beyond gains from additional observations alone.
  • PSF and LSF calibration is central to improving the accuracy and precision of single-observation source locations and G-band fluxes.The LSF is the one-dimensional image obtained by marginalising a two-dimensional point-source image, and PSF and LSF are calibrated independently.
  • PLSF shape varies with time, colour, focal-plane position, and other Gaia-specific dependences, making accurate calibration necessary to reduce astrometric and photometric systematic errors.
  • The paper presents the PSF and LSF models and calibrations adopted in cyclic processing to produce EDR3 data products.The same calibrations apply to full DR3 because they are not updated and the relevant EDR3 astrometry and integrated photometry are not recomputed.

2. Description of the instrument and observations

Gaia’s shared focal plane uses Sky Mapper and Astrometric Field CCDs to detect, window, bin, and measure sources as they transit the instrument.

  • Gaia’s focal plane contains 106 CCDs arranged in seven rows, including 14 Sky Mapper and 62 Astrometric Field CCDs used for unfiltered G-band observations.
  • Stars enter the focal plane from the left and drift rightward across the CCD arrangement for roughly one minute.
  • Sky Mapper and Astrometric Field 1 CCDs detect sources, estimate magnitudes, and predict motion to support windowing that reduces telemetry and readout noise.The reduced data volume introduces processing complications, and most windows use on-chip binning to marginalise the across-scan dimension.
  • A stellar image crosses an individual CCD in about 4.42 seconds, while TDI gates shorten integration for bright stars expected to saturate the detector.
  • PLSF calibration is split into independent units defined by field of view, CCD, TDI gate, and window class.There are 248 one-dimensional LSF calibration units covering AF CCDs, two window classes, and two fields of view.

3. Description of the PLSF models

Gaia requires a dedicated PLSF model because its extreme centroiding accuracy, unusual PSF dependences, and undersampled, windowed observations exceed the assumptions of standard PSF software.

  • Gaia’s extreme centroiding requirements and unusual, undersampled observations motivate a dedicated PLSF model and calibration pipeline.The observations are highly windowed and marginalised, creating needs specific to Gaia.

3.1. Basics

The calibrated PLSF represents an effective, pixel-aware image model subject to continuity and flux-preservation constraints, with distinct treatment of one- and two-dimensional observations.

  • Basics: The observed effective PSF includes pixelation, charge diffusion, and TDI-related smearing rather than directly revealing the intrinsic instrumental PSF.
  • Basics: PLSF models must be continuous in value and first derivative and have a shift-invariant sampled sum to preserve source flux across sub-pixel positions.The shift-invariant sum expresses the constraint that received photoelectron counts are approximately independent of sub-pixel source location.
  • Basics: LSF and PSF models are normalised over one and two dimensions respectively, while finite across-scan windows leave AC flux loss to photometric calibration.
  • Basics: The PLSF models use window geometries specified by magnitude and CCD strip, with AL/AC sampling, binning, and LSF-versus-PSF outputs.On-chip and numerical binning are combined to optimise onboard performance and telemetry.
  • Basics: Positivity is physically required for the true PSF but is not enforced in the fitted model, so noisy calibration results can contain negative values where the true PSF is likely small.

3.2. Formulation of the 1D LSF model

The 1D LSF is represented as a mean profile plus weighted basis components derived from optical simulations, with calibrated weights capturing the observed asymmetric profile. The model uses AL displacements and is designed for both telescopes.

  • 3.2. Formulation of the 1D LSF model: The LSF is constructed as a mean profile H0 plus a weighted sum of N basis components Hn.The basis weights are the model parameters calibrated for observations.
  • 3.2. Formulation of the 1D LSF model: For EDR3, the LSF origin shift u0 is fixed to zero, leaving the basis weights hn as the free parameters requiring calibration.The weights are represented by multidimensional spline functions of observation parameters.
  • 3.2. Formulation of the 1D LSF model: The AL coordinate u measures displacements in the along-scan direction rather than an absolute detector location.Gaia’s TDI operation removes a direct correspondence between AL coordinate and detector position.
  • 3.2. Formulation of the 1D LSF model: The model’s basis functions are derived from simulations of Gaia’s optical system using random wavefront-error realisations and diffraction calculations.The simulated bases are intended to span LSFs associated with the instrument’s optical configuration space.
  • 3.2. Formulation of the 1D LSF model: The simulated basis components can have odd or even symmetry, but the calibrated LSF itself is asymmetric.The asymmetry arises through the fitted weighted combination of basis components.

3.3. Formulation of the 2D PSF model

The 2D PSF formulation progresses from an AL×AC product of 1D LSFs to pseudo-shapelets and then compound shapelets. The latter reduces model dimensionality while allowing asymmetric Gaia PSF structure to be represented.

  • 3.3. Formulation of the 2D PSF model: The original AL×AC model formed the 2D PSF as the outer product of independently calibrated AL and AC 1D LSF models.This factorisation was motivated by the separability of Fraunhofer diffraction from a rectangular pupil.
  • 3.3. Formulation of the 2D PSF model: Wavefront errors introduce significant asymmetric PSF features that the AL×AC factorisation cannot represent, motivating a new EDR3 formulation.The AL×AC model had been used for DR1 and DR2, but a replacement was required for EDR3.
  • 3.3.1. The pseudo-shapelets model: The pseudo-shapelets model builds 2D basis components as outer products of the 1D LSF functions, replacing Hermite functions with Gaia’s LSF basis functions.The model represents the PSF as a weighted sum of these 2D components.
  • 3.3.1. The pseudo-shapelets model: The pseudo-shapelets generalisation uses independent coefficients h_nm rather than products h_n h_m, providing greater freedom to reproduce asymmetric features.Low-order components include products such as H1(u)H0(v), H2(u)H0(v), H1(u)H2(v), and H2(u)H2(v).
  • 3.3.1. The pseudo-shapelets model: The pseudo-shapelets model requires 440 2D bases for 20 1D components plus the mean, creating a major calibration and sampling burden.Many bases have low importance, and the model provides no rigorous ranking for truncation.
  • 3.3.1. The pseudo-shapelets model: Observed Gaia PSFs have significant asymmetric structure and lower dimensionality than the 440-component pseudo-shapelets space.These observations motivated the compound shapelets model.
  • 3.3.2. The compound shapelets model: Compound shapelets calibrate fixed linear combinations of pseudo-shapelets designed to model the principal components of observed Gaia PSFs.Each combination becomes one full 2D basis component, with only its amplitude calibrated afterward.
  • 3.3.2. The compound shapelets model: EDR3 used 30 compound-shapelet basis components, with independent sets generated for FOV1 and FOV2.The constant matrix β defines each compound basis, while the free parameters are the weights gm.

3.4. Modelling of the major PLSF dependences

The PLSF model represents colour, focal-plane position, across-scan rate, gate length, and time dependences through weighted basis components and spline functions. Time is handled separately by merging half-revolution calibrations to follow smooth evolution and discontinuities.

  • Colour dependence: Colour dependence is parameterised by the effective wavenumber νeff, the photon-weighted inverse wavelength.Chromatic shifts in the PLSF centroid are expected to be linear in νeff.
  • Across-scan rate: Across-scan image motion broadens the PSF, with the effect controlled by integration time and disabled for gates 10 and shorter.For the longest integration time, the broadening amplitude is around 4.5 pixels; gates 10 and shorter correspond to integration times ≤1 second.
  • Across-scan rate: EDR3 models native AC rate by using the signed ˙µ rather than |˙µ|, capturing asymmetry caused by small CCD rotational misalignments.The broadening affects PSF observations only; the LSF has no AC-rate dependence.
  • Across-scan position: Residual focal-plane variation is modelled as a continuous function of AC position µ within independently calibrated devices.The AC coordinate runs continuously from 13.5 to 1979.5 across the CCD image area.
  • Time dependence: Time variation is calibrated independently in 0.5-revolution steps and merged with a square root information filter to track smooth changes and discontinuities.Resets accommodate mission events such as decontaminations and refocuses.

4. Calibration pipeline

The calibration pipeline selects and prepares suitable science observations, balancing them across time and observation-parameter space before fitting the PLSF models. This process addresses non-uniform source distributions, instrumental defects, and uncertainty in predicted source locations.

  • Pipeline overview: The pipeline self-calibrates from regular science observations rather than requiring special calibration data.Selected observations are processed through multiple eligibility, selection, and preparation stages.
  • Eligibility of observations: Eligible transits require valid νeff, a valid astrometric solution, and astrometric excess noise below 0.5 mas.The astrometric solution also supplies predicted source locations and AC rates used to align observations.
  • Selection of observations: The first selection stage throttles eligible observations uniformly in time, while the second selects uniformly across parameter space, especially νeff and ˙µ.This balances sparse regions, including extreme colours, so the model remains constrained across rare object types.
  • Selection of observations: 1,604,769 observations passed the first selection stage, but correlations between νeff and AC rate vary over time and constrain short-range calibration.The AC rate follows a one-revolution sinusoid with an amplitude around 1 pixel per second and differs in phase between telescopes.
  • Observation preparation: Selected samples are converted to electrons, corrected for bias, background, and dark signal, then masked for saturation, non-linearity, cosmic rays, and CCD defects.Unmasked samples are normalised for flux outside the observed window and assigned uncertainties from noise, calibration errors, and predicted-location errors.

4.2. Partial solution

Partial PLSF solutions are computed independently for each calibration unit and half-revolution interval by fitting spline-weighted basis-component amplitudes to preprocessed sample residuals. A numerically stable linear-algebra procedure produces each interval’s solution.

  • Solution construction: 10,529,472 partial solutions were computed for 1,268 calibration units over 4,152 revolutions.Many are only partial because their observations do not fully constrain the model parameters.
  • Solution construction: The fitted parameters are spline coefficients that interpolate the amplitudes of the PLSF basis components over observation parameters.The parameter vector contains the spline parameters for each basis component, solved jointly using a shared spline configuration.
  • Linear fit: The observation vector contains preprocessed samples after subtraction of the fixed mean PLSF, so the fit solves only the basis-component amplitudes.Sample uncertainties weight the observation vector and design matrix.
  • Linear fit: Householder orthogonal transformations reduce the design matrix to an upper-triangular form to obtain each partial solution.The same approach extends from the example LSF model to the PSF model.

4.3. Running solution

The running solution combines noisy, independently calibrated half-revolution results to improve parameter constraints while following the PLSF’s time evolution.

  • Running solution: Independent half-revolution solutions are merged into an updated solution at each time step, weighted according to time difference.This reduces noise and compensates for incomplete parameter coverage caused by sparse observations.

4.4. Data gaps

The pipeline remains operational across observation gaps by propagating the last running calibration, while automated checks identify poorly constrained or invalid solutions.

  • Gaps arise from sparse calibration-unit observations, satellite anomalies, downlink problems, or transient auxiliary-calibration issues.
  • During gaps, the pipeline produces an identity partial solution that carries no weight, so the running calibration is propagated unchanged.
  • Automated qualification checks parameter ranges, undefined values, input gaps, and fit statistics to monitor pipeline integrity.
  • Some EDR3 calibration units remain inadequately calibrated, requiring failed solutions to be replaced with qualified alternatives.
  • Each half-revolution PLSF library is roughly 4.9MB, or ~570MB with covariance information, producing ~4.7TB of total pipeline products.

4.6. Implementation and execution

The PLSF calibration is a six-module Java pipeline integrated with Gaia’s CALIPD system and executed through batch processing on shared MareNostrum resources. Its outputs support image-parameter determination for source locations and fluxes, under a single-point-source assumption.

  • The PLSF calibration is implemented in Java within Gaia DPAC’s CALIPD system and divided into six sequential modules.
  • Execution at Barcelona uses MareNostrum’s hierarchical file system and shared job queue because database use is discouraged by the machine design.
  • Batch processing and partitioning allow loosely coupled, high-volume tasks to run concurrently and reduce elapsed processing time.
  • PLSF models are used during raw-data preprocessing to determine each window’s source location and flux image parameters.
  • The processing assumes each window contains a single point source, while separate subsystems handle extended objects and non-single stars outside EDR3.

5. Results

The EDR3 calibration captures strong temporal, colour, focal-plane-position, and scan-rate dependences, while residual analyses expose model incompleteness and time-dependent uncertainty. Image sharpness generally stabilizes after later decontaminations, but rapid post-decontamination changes remain difficult to track.

  • The second of two PLSF and AGIS calibration iterations supplied the inputs for the final EDR3 data products.
  • 5.1. Time evolution: After revolution 4124, image-sharpness degradation became modest, with later quality maintained through refocusing rather than further decontaminations.
  • 5.1. Time evolution: Post-decontamination cooling changes the PLSF rapidly, causing a short-lived systematic difference because the running solution merges information over a wide time range.
  • 5.2. Colour, AC position, and AC rate dependence: Longer effective wavelengths produce broader profiles with stronger diffraction features, and the two fields of view have different profile shapes.
  • 5.2. Colour, AC position, and AC rate dependence: Across the focal plane, image sharpness degrades toward the corners in both fields of view despite independently calibrated devices.
  • 5.2. Colour, AC position, and AC rate dependence: Non-zero AC rates broaden PSFs in the AC direction, with the strongest dependence in gate 0 and slight high-rate bimodality identified as a calibration artefact.
  • 5.3. Invalid solutions: SM calibration is destabilized by saturation, undersampling, higher read noise, AF-tailored basis components, gate-12 modelling problems, and noisier auxiliary locations.
  • 5.4. Fit statistics and calibration residuals: The reduced-chi-square statistic is generally stable, but is worse for AF1 and rises shortly after decontaminations when rapid PLSF evolution is poorly tracked.

6. Discussion

EDR3 introduced major PSF and LSF calibration advances, but residual systematic errors expose important limitations in modelling rate, colour, time, basis, and profile extent.

  • EDR3 activated time and colour dependences, introduced a full 2D PSF model, and closed the iterative loop with the astrometric solution.
  • Calibration uncertainty varies over time, increasing near running-solution resets and decreasing during Galactic-plane scans with denser, broader-colour observations.
  • The dominant PSF defect is a long-gate systematic error correlated with AC rate, producing bimodality and residual spatial structure because AL-rate dependence is omitted.
  • AL and AC rate mismatch shears the PSF, while an AC-only model cannot reproduce the generally non-monotonic relation between the two rates.
  • Other limitations include weighting-related systematic errors, less successful PSF colour calibration, post-decontamination non-convergence, and effective-wavenumber limits for non-stellar spectra.
  • Future work targets analytic rate modelling, revised optical basis components, improved SM calibration, and an extended PSF model for bright-star profiles.

7. Conclusions

The paper documents Gaia EDR3’s PLSF models, calibration pipeline, and products, which improve the processing and support reduced systematic errors in core mission data products.

  • EDR3 PLSF modelling and calibration represent a major processing advance and contribute to reduced systematic errors in core mission data products.
  • Astrometric and photometric EDR3 solutions improve relative to DR2 beyond gains expected from the increased observation count alone.
  • The paper provides a detailed description of the models, pipeline, and calibration products needed to understand EDR3 contents and survey properties.
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