Simulation-based inference for joint 21 cm gain calibration and imaging with SKA-Low

Joy Sanghavi1, Sambit Giri2, Daan Meerburg3, Christoph Weniger1

1GRAPPA & Institute of Physics, University of Amsterdam

2Stockholm University

3Van Swinderen Institute, University of Groningen

EuCAIFCon 2026 · Heidelberg built on FALCON j.k.sanghavi@uva.nl
GRAPPA University of Amsterdam

The problem

Vab = ga Sab gb*

Every station-pair/baseline measures one Fourier mode of the sky, scaled by a complex gain that bundles amplifier response, cable delay and phase of both stations. Incorrect calibration gives noisy estimates for the 21 cm power-spectrum.

Global phase is unobservable, so gauge-fix g0 = 1 and gains for telescope a is written as:

ga = (1 + Re(z)a) + i Im(z)a
64stations,
2016 baselines
126real gain parameters
= 2(Na − 1)
𝒰(−1, 1)prior on each —
far broader than reality
SKA uv tracks
The uv tracks the baselines sweep out as the sky rotates — nB is how many times each cell is visited. Coverage is wildly uneven.

Estimator

qφ(θ | x) = 𝒩(θ; μφ(x), Σ)

A data-dependent mean — an MLP between two EMA whitening transforms — and one global covariance, estimated online from the prediction residuals r = θ − μφ(x) and eigendecomposed each step.

Why not just a flow? At D = 126 on a buffer stored on memory, flows train unreliably. Meanwhile the gain block is locally gaussian and unimodal.1

1 Ewall-Wice, Dillon, Liu & Hewitt, MNRAS 470, 1849 (2017)

Model design

calibrator leg — gains only, sky known science leg — sky × gains z 126 gains · root gains g₀ = 1 gauge-fixed Sref known calibrator · fixed V_ref = G S G† observed qφ( z | Vref ) gaussian full-cov · 126-D pw 4 params · root δ → δT_b GRF sky at z = 9 Fourier Image FFT of the sky shared UV tracks 2016 baselines obsx image domain P(k)_obs observed qψ( pw | P(k)obs , Vref ) masked autoregressive flow · 4-D evidence steer next simulations
▸
The whole model

The posterior factorises exactly:

p(z, π | Vref, Pobs) ∝
  p(z | Vref) · p(π | Pobs, Vref)
1
z — the gains

126 real perturbations, each 𝒰(−1, 1).

a batch of 128 buffer draws of z three individual buffer draws of z

A buffer batch, and three single draws across all 126 components.

2
gains
ga = (1 + zRa) + i zIa
buffer draws of the gains on the complex plane

Three buffer draws on the complex plane.

3
Sref — the calibrator
the calibrator: one bright Gaussian source

A bright known source. Fixed across every simulation.

4
Vref — observed
the calibrator as the array observes it
5
Gain estimator
qφ(z|Vref) = 𝒩(μφ, Σ)

Full-covariance Gaussian over all 126 gains at once. Mean is a 3×256 MLP; Σ is one global matrix updated by EMA and eigendecomposed every step.

6
pw — the sky
P(k) = A (k/kb)n₁ or n₂
buffer draws of the broken power law

Buffer draws of P(k): amplitude, early draws are anywhere in the prior box.

7
δ → δTb
density field drawn from buffer samples

Top: Buffer draws during training of the temperature field. Bottom: True temperature field

brightness temperature field
8
Fourier image
Fourier transform of the sky

The sky in Fourier space.

9
UV tracks

Gain-weighted uv tracks interact with the sky.

Due to this interaction, the power spectrum posteriors have a joint distribution with the calibrator.
10
obsx → P(k)obs
image-domain reconstruction, two uv subsets

The image the array reconstructs.

11
Power spectrum estimator

A MAF flow on 4 parameters, conditioned on the 'dirty' power spectrum and on Vref.

Embedding: MLP over log P(k) ⊕ extracting features from certain Vref baselines as log-magnitude, cos φ, sin φ.

12
Steer, then repeat

Falcon tempers each posterior into a proposal and simulates again.

Posterior vs prior

marginal posteriors for the first sixteen gain parameters, on the full prior range
Sixteen of the 126 gain parameters, prior bounds.
convergence of the four power-spectrum parameters during training
The running [min, max] envelope of every buffer draw, for all four spectral parameters.

Power spectrum results

recovered power spectrum against the truth
Posterior-predictive band from all 1500 draws, against the input spectrum.
SBI posterior against the Fisher forecast, on prior-bound axes
SBI posterior (red) against an independent Fisher forecast (blue), on prior-bound axes.

Next steps

  • Noise and diffuse foregrounds switched on
  • Time-changing/direction-dependent gains
  • Complex parametric power spectrum
  • Check out FALCON for more complex Bayesian hierarchical modeling

Thank you. Questions?

Joy Sanghavi · j.k.sanghavi@uva.nl

GRAPPA, University of Amsterdam · paper in preparation · Poster No. 43

GRAPPA University of Amsterdam
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