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Realization of a multifrequency celestial reference frame through a combination of normal equation systems

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Astronomers stack three radio frequency bands into one celestial reference frame that stays aligned with ICRF3 to within 3 microarcseconds and shows no detectable deformation.

desk verdict Useful and genuinely novel as a first full-covariance multifrequency VLBI frame combination, but the headline deformation-free claim is partly enforced by a tuned XKa weight, so the validation is weaker than the abstract suggests. read the letter →

arxiv 1908.11697 v1 pith:RRGH74M4 submitted 2019-08-30 astro-ph.IM astro-ph.EP

Maria Karbon , Axel Nothnagel This is my paper · ORCID
classification astro-ph.IMastro-ph.EP
keywords celestialreferenceframeICRF3VLBInormalequationsystemsHelmertblockingcoreshiftmultifrequencyastrometrycovariancetransfer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a celestial reference frame built by stacking independent VLBI catalogs at 8, 24, and 32 GHz, rather than by aligning finished position lists, carries the full covariance information across all sources and frequencies, and that the resulting ComboKXKa frame is free of detectable deformation. A sympathetic reader would care because the current ICRF3 treats its three frequency catalogs as standalone products aligned through common sources, whereas this method jointly re-determines common sources from all observations. The claimed product contains 4,617 compact radio sources, aligns with ICRF3 within 3 microarcseconds at the defining sources, and shows an average positional uncertainty of 0.1 mas in right ascension and declination. The paper also argues that core shifts are not an obstacle at current VLBI precision because their offsets have random orientation and add only white noise.

What carries the argument

The load-bearing object is the normal equation system (NEQ) from each VLBI global solution, with all non-source parameters already eliminated, carrying the variance-covariance relationships of the estimated source positions. The combination itself is Helmert stacking: the NEQ matrices and right-hand side vectors of common parameters are summed, a no-net-rotation datum is imposed with the ICRF2 defining sources, and the stacked system is solved. Because the XKa input arrived as a covariance matrix rather than NEQs, it is first converted back to datum-free normal equations using the Grafarend-Sanso identity; the XKa system is then scaled down empirically so its network-geometry rotations no longer dominate. This machinery makes the full covariance matrix of all 4,617 sources, across frequency bands, a direct output of the combination rather than an a posteriori construct.

What would settle it

Measure the angular separation between radio cores of the same compact sources at 8, 24, and 32 GHz with a VLBI campaign reaching per-source precision well below 0.1 mas; if the offsets are not isotropically distributed but systematically aligned with jet directions or grow with frequency separation for many sources, the white-noise assumption fails.

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Extended reading notes

Core claim

The paper's central claim is that ComboKXKa is a rigorous, multifrequency realization of the ICRS: 4,617 compact radio sources positioned jointly from SX (8 GHz), K (24 GHz), and XKa (32 GHz) observations, with all variance-covariance information carried through the combination. The frame is aligned with ICRF3 to within 3 microarcseconds and has an average positional uncertainty of 0.1 mas in both coordinates; rotation and deformation parameters from vector-spherical-harmonic analysis show no significant deformations once the XKa solution is down-weighted. Adding the higher-frequency catalogs also extends the frame southward, since 16 of the 31 XKa-only sources lie below -30 degrees declination. Comparisons with Gaia-CRF2 are called inconclusive because the transformation parameters depend strongly on source selection, yet the paper states that significant differences between all frames are attested.

Load-bearing premise

The load-bearing premise is that frequency-dependent source position offsets, or core shifts, are random in orientation and too small to be detected at current VLBI precision, so combining 8, 24, and 32 GHz positions adds only white noise.

Editorial extensions

If this is right

  • The same pipeline can be applied to future ICRF realizations directly from normal equations of all analysis centers, replacing the monolithic-solution-plus-alignment scheme.
  • Common sources present in more than one catalog are effectively re-determined from the union of observations, which improves formal errors for some sources and mitigates network deficiencies.
  • The southern extension from XKa-only sources improves sky coverage in the deep south, a region where SX catalogs remain sparsely populated.
  • The full covariance matrix enables proper statistical interpretation of the frame, including frame ties and comparisons with Gaia, without requiring ad hoc error inflation.
  • The absence of detectable deformation suggests that the weak network geometry of the XKa solution can be prevented from propagating into a stacked product by appropriate weighting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A dedicated VLBI campaign measuring the same compact sources in all three bands with matched networks could convert the random core-shift assumption into a measured correction; if core-shift vectors align with jet position angles, the stacked frame would need per-source frequency offsets.
  • The empirical down-weighting of XKa (a variance factor of 2 plus an additional 0.05 mas-squared inflation) indicates the combination is only as good as the weakest geometry; future frames could formalize such weights from VSH residuals instead of tuning them.
  • If Gaia DR3 confirms the radio-optical differences seen here, the combined multifrequency radio frame could serve as a clean intermediate frame for disentangling optical structure effects from true source position offsets.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper presents a method for combining multifrequency VLBI celestial reference frame catalogs (SX, K, XKa) via stacking of normal equation systems, thereby carrying full variance-covariance information into the combined solution. The resulting catalog, ComboKXKa, contains 4617 sources and is compared with ICRF2, ICRF3, and Gaia-CRF2 using vector spherical harmonic decomposition. The central claims are that the combination is rigorous, that the frame is aligned with ICRF3 within 3 microarcseconds, that no significant deformations are present, and that core-shift effects are negligible at current precision.

Significance. If the claims hold, the paper provides a useful methodological advance for future ICRF realizations: it is the first application of full covariance transfer through normal equation stacking to a multifrequency VLBI frame, and the resulting catalog with cross-frequency covariance information would be a valuable community product. The authors also ship the catalog and covariance matrix publicly, which is a concrete strength. However, the central validation is weakened by the empirical tuning of the XKa weight, the self-referential comparison with ICRF3, and the admitted low power of the core-shift test, so the 'deformation-free' claim is not established as an independent outcome.

major comments (4)
  1. [Sec. 4, step 3 and Sec. 5.2] The XKa down-weighting is circular with respect to the deformation claim. Section 4 step 3 states that the weighting parameter of 0.05 mas^2 'was determined empirically in such a manner that the originally dominant rotations are just no longer discernible in the residuals,' and Section 5.2 reports that for ComboKXKa 'the rotation in R2 is very small' with the explicit note that without the scaling it would be -30 microas. The absence of a significant R2 rotation is therefore enforced by construction rather than demonstrated, yet the abstract and conclusions cite the small rotations as evidence that 'No significant deformations can be identified.' This circularity also affects the stated axis stability of 3 microas, since the stability is estimated from the same weighted solution. The authors should present the weighting as a deliberate regularization choice, quantify the resulting systematic uncertainty (e.g., by repeating the analysis with a range of weights), and avoid citing the suppressed R2 as an independent validation.
  2. [Abstract and Sec. 6] The alignment claim in the abstract ('aligned with ICRF3 within 3 microas') is not supported by the reported transformation parameters. Table 3 gives |R| = 13 +/- 1 microas for ComboKXKa with respect to ICRF3 and |R| = 31 +/- 7 microas with respect to ICRF2. The 3 microas value that appears in Sec. 5.2 refers to the scatter of rotation parameters across different source subsets (axis stability), not to the absolute alignment. The abstract and Sec. 6 should be reworded to distinguish 'axis stability of 3 microas' from 'rotation magnitude of about 13 microas with respect to ICRF3,' and the wording in Sec. 6 ('aligned with ICRF2 within +/-3 microas') should be corrected; as written it is inconsistent with Table 3.
  3. [Sec. 2.1.1 and Sec. 5.2] The comparison of the combined product with ICRF3 is partly self-referential. As the paper notes, the GSF SX solution 'is identical in content to that used for the determination of the SX catalog of ICRF3,' so the small deformation parameters between ComboKXKa and ICRF3 largely reflect the propagation of the same SX input. Because the K and XKa catalogs are down-weighted and the XKa weight is empirically tuned, the agreement with ICRF3 does not provide an independent external check of the frame. The authors should state this limitation explicitly when interpreting the ICRF3 comparison, and should place more weight on the Gaia-CRF2 comparison (which they admit is inconclusive) or on comparisons against realizations not built from the same input.
  4. [Sec. 3] The core-shift analysis does not substantiate the claim that core shifts 'only add white noise.' The angular-separation test in Fig. 3 has low power, as the authors themselves conclude: 'the failure of any statistical testing of core shifts of a large number of sources, as the individual source position is too inaccurate at the current state and the standard deviations are too optimistic.' Absence of detection is not evidence of random orientation, and the two K-band outliers (3C119, 2018+295) are dismissed as analysis artifacts without a quantitative argument beyond their large separations. The conclusion in Sec. 1 that 'we demonstrate that the effect is of a random nature for catalog combinations' therefore overstates what the data show. The paper should present the randomness of core shifts as an assumption required by current precision, not as a demonstrated property, and should discuss the potential impact of a systematic frequency-dependent component on the combined positions.
minor comments (4)
  1. [Table 3] The column header 'CommboKXKa' contains a typo; it should be 'ComboKXKa'.
  2. [Abstract and throughout] The abstract uses 'Ghz' but the standard unit symbol is 'GHz'; please correct for consistency.
  3. [Sec. 5.4] In the text, 'aM2,0 which describes a sharing of the two hemispheres' should read 'shearing,' not 'sharing,' to match the earlier definition in Sec. 2.4.
  4. [Tables 1 and 4] The column labels 'wmean' and 'σwmean' are ambiguous; consider relabeling as 'weighted mean' and 'weighted standard deviation' or adding a footnote explaining the notation.

Circularity Check

2 steps flagged · score 6.0 of 10

XKa weighting is tuned to hide R2 rotations, which are then reported as absent; ICRF3 alignment is partly built-in via the identical GSF input.

  1. fitted input called prediction [Section 4, step 3 and Section 5.2]
    "Because initial results have shown that the XKa solution introduces significant rotations around R2, we chose to down-weight this catalog. The weighting parameter of 0.05 mas 2 was determined empirically in such a manner that the originally dominant rotations are just no longer discernible in the residuals. [...] This is the axis mostly affected by XKa, or rather its scaling. Without the scaling this parameter amounts to -30 µas and clearly dominates the rotations."

    The XKa down-weighting is not estimated from the data's covariance but is adjusted until the XKa-induced R2 rotation disappears from the residuals. Section 5.2 then reports R2 = -2 ± 1 µas and axis stability to within 3 µas, and the abstract concludes that no significant deformations can be identified. The absence of the R2 rotation is therefore not an independent outcome of the combination; it is the fitting target used to choose 0.05 mas^2. The claim that the frame is deformation-free is, for this degree of freedom, enforced by construction rather than demonstrated by the combination.

  2. self definitional [Sections 2.1.1, 2.4, and 5.2]
    "We note that ICRF3 at SX frequency and GSF are the same catalog simply represented in different ways. Only the formal errors of GSF were inflated a posteriori according to Eq. 7 for the reported ICRF3 uncertainties. [...] This was expected, as the combination contains the ICRF3 equivalent GSF."

    The reported alignment with ICRF3 within 3 µas is not an external validation of the combination: the GSF SX input is identical to ICRF3's SX catalog, and the same ICRF2 defining sources set the datum. The combined frame therefore inherits the orientation and source positions of ICRF3's SX component almost by construction, so the small rotations and deformations with respect to ICRF3 are largely built-in. The authors disclose this, but the abstract's alignment-with-ICRF3 statement still presents an input identity as a result.

full rationale

The derivation core is standard and self-contained: Helmert stacking of normal equation systems (Eqs. 12-22) is described explicitly, and the combination itself is not posited in the inputs. I do not count the core-shift random-orientation assumption as circular; it is a weakly supported empirical assumption, not a construction. The load-bearing circularity is in the validation narrative: the XKa variance floor is hand-adjusted until its dominant R2 rotation is no longer discernible, and the same absence is then reported as evidence that the frame is stable and deformation-free. This is a fitted input presented as a prediction. A second, milder self-referentiality affects the 'aligned with ICRF3 within 3 µas' claim: the GSF input is identical to ICRF3's SX catalog and the same defining sources set the datum, so small rotations with respect to ICRF3 are partly guaranteed before the combination is performed. The method is real and the catalog is a useful product, but the specific conclusion that the multifrequency frame is deformation-free is partially enforced by the tuning parameter rather than independently established.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the random-orientation core shift assumption, the validity of reconstructing XKa NEQs from its covariance, and the choice of the GSF SX solution as representative. The empirical XKa weighting is the main free parameter that shapes the validation results.

free parameters (3)
  • XKa variance factor = 2.0
    No a posteriori variance factor was available for the XKa solution, so the authors set it to 2 (Sec. 4, Step 3), inflating XKa uncertainties.
  • XKa weight = 0.05 mas^2
    The XKa catalog was down-weighted by this empirically determined scaling so that initially dominant R2 rotations would no longer be discernible (Sec. 4, Step 3). This is a parameter fitted to the validation target.
  • Core shift test thresholds = rho=10 mas, X=4.1
    Thresholds in the angular separation test for core shift (Sec. 3, Fig. 3). The authors call them 'a bit arbitrary' but state that relaxing them does not change the picture.
assumptions (3)
  • domain assumption Core shift offsets between frequencies have random orientation and no global systematic component
    Sec. 3 assumes jets show no preferred orientation so core shifts add only white noise. The paper's own test cannot detect core shifts at current precision, which is not the same as proving they are absent.
  • standard math The XKa covariance matrix can be transformed into a datum-free NEQ via C_xx = (N_free + B^T B)^-1 - B^T(B B^T B B^T)^-1 B
    Sec. 4, Step 1 uses the Grafarend and Sanso formulation. This is mathematically standard, but relies on the provided covariance being a pseudo-inverse of the free NEQ in the same datum, which is not independently verified in the paper.
  • domain assumption The GSF SX solution can stand in for the three SX analysis center solutions
    Sec. 2.3 discards GFZ and VIE solutions, arguing GSF shows smallest deviations from ICRF2 because the same analysis center produced both. This makes the combined frame heavily dependent on a single analysis chain.

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Pith. "Pith review of Realization of a multifrequency celestial reference frame through a combination of normal equation systems." pith.science (2026). https://pith.science/paper/RRGH74M4

@misc{pith2026190811697,
  author       = {Pith},
  title        = {Pith review of: Realization of a multifrequency celestial reference frame through a combination of normal equation systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RRGH74M4}},
  note         = {Machine review of arXiv:1908.11697}
}
abstract

Context. We present a celestial reference frame (CRF) based on the combination of independent, multifrequency radio source position catalogs using nearly 40 years of Very Long Baseline Interferometry observations at the standard geodetic frequencies at SX band and about 15 years of observations at higher frequencies (K and XKa). The final catalog contains 4617 sources. Aims. We produce a multifrequency catalog of radio source positions with full variance-covariance information across all radio source positions of all input catalogs. Methods. We combined three catalogs, one observed at 8 GHz (X band), one at 24 GHz (K band) and one at 32 GHz (Ka band). Rather than only using the radio source positions, we developed a new, rigorous combination approach by carrying over the full covariance information through the process of adding normal equation systems. Special validation routines were used to characterize the random and systematic errors between the input reference frames and the combined catalog. Results. The resulting CRF contains precise positions of 4617 compact radio astronomical objects, 4536 measured at 8 Ghz, 824 sources also observed at 24 GHz, and 674 at 32 GHz. The frame is aligned with ICRF3 within 3 $\mu$as and shows an average positional uncertainty of 0.1 mas in right ascension and declination. No significant deformations can be identified. Comparisons with Gaia-CRF remain inconclusive, nonetheless significant differences between all frames can be attested.

Figures

Figures reproduced from arXiv: 1908.11697 by the authors.

Figure 1
Figure 1. Sources unique to one catalog. Light blue: GFZ, purple: K band, and black: XKa. 2.3. Quick look data characterization Before a combination is performed, the input data needs to be characterized in terms of their quality and possible deviations. We, therefore, first determined differences with respect to ICRF2 (Fey et al. 2015). The results are summarized in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Residuals w.r.t. ICRF2 (left) and standard deviations (right) vs. declination. The top row shows GSF (magenta diamonds), VIE (dark blue circles), and GFZ (light blue squares); the bottom row shows K band (purple triangles) and XKa (black triangles). duced scale of the plots. Nevertheless, the overall patterns re￾main the same, hence, the majority of sources fall in the cen￾tral bin (> 80% for K band, > 90% for XKa) … view at source ↗
Figure 3
Figure 3. Angular separation ρ against the normalized separation Xnorm for (from left to right) VIE (dark blue), GFZ (light blue), K band (purple), and XKa (black) with respect to GSF. The vertical line is located at Xnorm=4.1, the horizontal at ρ=10 mas. sources, as the individual source position is too inaccurate at the current state and the standard deviations are too optimistic. Fur￾ther, in geodetic VLBI the source posit… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Flowchart of the combination strategy. The relation of the a priori CRF and that estimated can be de￾scribed by a rotation matrix B containing three rotation an￾gles r. Alternatively, a parameter dz can also be introduced, which accounts for a global translation of the…
Figure 5
Figure 5. Figure 5: Residuals of the defining sources plotted over declination with respect to ICRF3. Left: gray diamonds for ICRF2, purple circles for K band and black squares for XKa; right: purple triangles for ComboKX and black triangles for ComboKXKa [PITH_FULL_IMAGE:figures/full_fi…
Figure 6
Figure 6. Figure 6: Residuals with respect to ICRF3, shown in purple ComboKX and in black ComboKXKa. Top: ICRF3 defining sources only. Bottom: All sources. Sources with residuals >2 mas or σ >2 mas were excluded. The solid line denotes the ecliptic, the dashed line the galactic equator. c…
Figure 7
Figure 7. Figure 7: Distribution of the standard deviations of the estimated source positions, in pink for GSF, green for ICRF3, gray for ICRF2, and black for ComboKXKa; on the left for right ascensions, on the right for declination. The ComboKX values in purple are subdued under the Comb…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.