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Extracting the Epoch of Reionization Signal with 3D U-Net Neural Networks Using Data-driven Systematic Effect Model

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A 3D U-Net can pull the reionization 21-cm signal out of realistic SKA-Low mocks, with 4,380 hours of integration restoring the EoR window and 13,140 hours reaching nearly all scales.

desk verdict Useful SKA-Low EoR forecasting that may be inflated by rotation-augmented leakage; worth a serious referee but needs a disjoint test split. read the letter →

arxiv 2412.16853 v2 pith:FOXEQY4R submitted 2024-12-22 astro-ph.IM astro-ph.COgr-qchep-ph

Li-Yang Gao , Léon V. E. Koopmans , Florent G. Mertens , Satyapan Munshi , Yichao Li , Stefanie A. Brackenhoff , Emilio Ceccotti , J. Kariuki Chege
show 6 more authors
Anshuman Acharya Raghunath Ghara Sambit K. Giri Ilian T. Iliev Garrelt Mellema Xin Zhang
This is my paper · ORCID
classification astro-ph.IMastro-ph.COgr-qchep-ph
keywords 21-cmcosmologyEpochofReionizationpowerspectrumestimationdeeplearningU-NetGaussianprocessregressionsystematiceffectsSKA-Low
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

The paper asks whether a deep neural network can pull the faint 21-cm signal from the Epoch of Reionization out of radio data where foregrounds, thermal noise, and systematic errors are far brighter. It trains a 3D U-Net on mock SKA-Low observations whose noise and systematics are generated with Gaussian process regression from real LOFAR North Celestial Pole data, and claims that the 21-cm power spectrum can be recovered reliably once enough integration time accumulates: at 1752 hours with thermal noise alone, at 4380 hours within the EoR window when excess variance is included, and across nearly all scales at 13140 hours. The paper's key conclusion is that frequency-incoherent excess variance, not thermal noise, is the fundamental obstacle, because its structure cannot be learned or subtracted by the network. This matters because it tests whether machine-learning extraction can survive data-driven systematics as currently observed in an operating instrument.

What carries the argument

The machinery has three parts. The first is the 3D U-Net, a convolutional encoder-decoder network whose skip connections preserve small-scale structure, trained with a Log-Cosh loss on 64-by-64 frequency-channel slices to convert contaminated input cubes into cleaned EoR cubes. The second is a data-driven systematic model built with Gaussian process regression: Matérn kernels describe the foreground, thermal noise, and excess variance, and MCMC fits to real LOFAR NCP observations fix their hyperparameters (foreground coherence scales of 30 MHz and 8.1 MHz; excess-variance coherence length 0.26 MHz with amplitude 2.18 times the thermal-noise variance). The third is the evaluation metric, the 2D cross and coherence power spectra between target and predicted images, which determine which $(k_\perp, k_\parallel)$ regions are reliably recovered and which are not.

What would settle it

A direct falsifier would be to fit the same Gaussian process covariance model to SKA-Low commissioning or deep-field data at 134–146 MHz and compare the recovered excess-variance hyperparameters with the LOFAR-derived values of $l_{\rm ex}=0.26$ MHz and $\sigma^2_{\rm ex}=2.18\sigma^2_n$; if the coherence length or the amplitude-to-noise ratio is substantially different, the simulated milestones in this paper do not apply to the real instrument. A second, stricter test is to run the trained U-Net on an independent mock where the truth is known and check whether the recovered 2D power spectrum at 4380 hours stays within the reported coherence scatter inside the EoR window.

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Extended reading notes

Core claim

On its own terms, the paper demonstrates that a 3D U-Net can be trained to map contaminated SKA-Low-like data cubes onto the underlying EoR 21-cm brightness field, and that the fidelity of that mapping is set by the integration time and by which systematic components are present. With thermal noise corresponding to 1752 hours of observation, the network recovers the 21-cm 2D power spectrum reliably above the horizon delay line, and robustness tests show that signal recovered in the wedge is genuine rather than a network extrapolation. Adding the fixed LOFAR foreground residual leaves the region above the horizon intact but creates inconsistencies below the horizon line. When the LOFAR-derived excess variance is added, reliable power-spectrum estimates within the EoR window require 4380 hours, and estimates across nearly all scales (including below the horizon) require 13140 hours; the mean two-dimensional coherence between predicted and target images reaches 0.49, 0.68, and 0.85 at 1752, 4380, and 13140 hours respectively. The paper concludes that the frequency-incoherence of the excess variance is what ultimately limits deep-learning extraction.

Load-bearing premise

All of the quantitative integration-time milestones rest on the assumption that the excess variance measured in LOFAR's North Celestial Pole data—a coherence length of 0.26 MHz and an amplitude 2.18 times the thermal-noise variance—transfers to SKA-Low with the same ratio to thermal noise; the paper notes SKA-Low will likely have less excess variance, which would change the numbers.

Editorial extensions

If this is right

  • With 1752 hours of thermal noise alone, the U-Net gives reliable 2D power-spectrum predictions above the SKA-Low horizon line, and its recovered wedge-region signal is genuine recovery rather than a learned extrapolation.
  • Adding the fixed foreground residual does not hurt recovery above the horizon, but produces the same inconsistency below the horizon delay line that other methods exhibit.
  • Including excess variance, the mean 2D coherence between target and predicted images is 0.49 at 1752 hours, 0.68 at 4380 hours, and 0.85 at 13140 hours, with reliable estimation inside the EoR window at 4380 hours and across nearly all scales at 13140 hours.
  • The full, most realistic data case (foreground residual plus thermal noise plus excess variance) performs as well as the no-foreground case above the horizon; foreground power only degrades the region below the horizon line.
  • Because the excess variance is largely incoherent in frequency, deep learning cannot remove it, so the practical route to shorter integration times is reducing the excess variance through better calibration and foreground subtraction.

Reading between the lines

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

  • If SKA-Low's excess variance is weaker than LOFAR's, as the paper itself suspects from better beam control, the 4380-hour and 13140-hour milestones would be upper bounds; the ladder of required integration times would shift downward.
  • The same training recipe could be transferred to other 21-cm arrays by refitting the Gaussian process kernels to each instrument's own residual cubes, making the method instrument-agnostic in principle.
  • A testable extension is to inject partially frequency-coherent excess variance into the mocks: if the U-Net then recovers the signal at shorter integration times, the paper's claim that incoherence is the hard limit would be directly confirmed.
  • The coherence metric tracks only power-spectrum agreement; a next step would be questioning whether the network's recovered maps also preserve phase information, which matters for tomographic and bispectrum analyses.
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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

3 major / 6 minor

Summary. The paper trains a 3D U-Net to recover the Epoch of Reionization 21-cm brightness-temperature signal from simulated SKA-Low observations. The simulations combine EoR cubes from 21cmFAST with thermal noise, foreground residuals obtained from real LOFAR NCP observations, and Gaussian-process-regression-based excess variance, all restricted to LOFAR-like uv-coverage. The network is trained on 536 cubes (augmented from 134 underlying realizations by 90/180/270-degree rotations) and tested on 8 held-out cubes. The main reported results are: with 1752 hours of integration and thermal noise only, the U-Net recovers the 2D EoR power spectrum with coherence close to unity; adding fixed foreground residuals leaves recovery good above the horizon line but inconsistent below it; including excess variance, 4380 hours gives reliable recovery within the EoR window and 13140 hours gives reliable recovery across nearly all scales, with the frequency-incoherent excess variance identified as the limiting factor. A wedge-filter robustness test is presented as evidence that the network does not hallucinate wedge modes.

Significance. If the central claim holds, the paper would provide a useful quantitative forecast of how deep-learning-based signal separation could perform on SKA-Low EoR data, and it would identify frequency-incoherent excess variance as the key systematic that limits such methods. The work has genuine strengths: the systematic-effect model is data-driven, using foreground residuals and GPR-derived excess variance from real LOFAR observations; the wedge-filter experiment in Section 5.1.1 is a real self-check that the network does not predict filtered modes; and the LOFAR results in Appendices B and C act as a negative control showing degraded performance at higher noise. The main obstacle is that the train/test split may not be disjoint at the level of the underlying 21cmFAST realizations, which would directly undermine the reported recovery quality. The excess-variance transfer from LOFAR to SKA is also a significant model assumption that the quantitative milestones depend on. With a realization-disjoint evaluation and a sensitivity analysis of the excess-variance parameters, the paper would be a solid contribution; in its current form the central claim is not yet fully supported.

major comments (3)
  1. [§2.1 and §4] All 536 data cubes are generated by rotating only 134 underlying 21cmFAST realizations (Section 2.1), and Section 4 splits them into 512 training, 16 validation, and 8 test cubes without stating that the test cubes use realizations absent from training. Because a 90/180/270-degree rotation leaves the ionization morphology exactly recognizable and does not change the 21-cm power spectrum, if any test cube shares its underlying realization with a training cube the network can memorize the target morphology instead of recovering the signal from the noisy input. The reported coherence near unity in Fig. 10 and the quantitative milestones in Section 5.3 could therefore be inflated by this leakage. I ask the authors to split by underlying realization (for example, hold out several complete realizations at the cube-generation stage) or, failing that, to report the exact realization IDs in the test set and demonstrate that no rotated sibling appears in the training set; the wedge robustness test in Section 5.1.1 does not remove this concern because the EoR_rev image is derived from the same EoR cube.
  2. [§3.2–3.3 and §5.3] The quantitative conclusions—for example that 1752 hours gives reliable recovery above the wedge, 4380 hours within the EoR window, and 13140 hours below the horizon—are conditional on the LOFAR-derived excess-variance model (l_ex = 0.26 MHz, sigma2_ex = 2.18 sigma2_n) and on the assumption that the ratio sigma2_ex/sigma2_n and the coherence scale are unchanged for SKA-Low. The paper itself concedes in Section 5.3 that SKA-Low likely has less excess variance, which would change the required integration times and the k_perp = 0.113 transition. Since these milestones are the main quantitative output of the paper, please add a sensitivity analysis that varies sigma2_ex/sigma2_n and l_ex over a reasonable range (for instance 0.5–2.18 and 0.1–1.0 MHz) and shows how the coherence maps and the quoted transition scales change, or alternatively recast the conclusions explicitly as predictions of the LOFAR-transfer model rather than of SKA-Low itself.
  3. [§5.3 and Figs. 16–17] The paper states mean values of the coherence power spectrum (0.49, 0.68, and 0.85 for 1752, 4380, and 13140 hours) but gives no dispersion over the 8 test cubes and no per-cube results. With a test set this small, the claim that recovery is 'reliable' needs at least the range or standard deviation across test cubes, and the paper should state whether the 8 test cubes come from 8 distinct underlying realizations. Please add per-cube coherence statistics or a scatter band to the reported figures.
minor comments (6)
  1. [Eq. (7)] The word 'corss' in 'we define the 2D corss power spectrum' should be 'cross'.
  2. [Acknowledgements] The Acknowledgements section contains a duplicated sentence: the ERC 'CoDEX' grant and the SERB-DST Ramanujan Fellowship are each listed twice; remove the duplicate.
  3. [§3.2] The statement 'Since LOFAR and SKA have similar antenna placement strategies, we believe that they have statistically similar behavior in their observations' is a model assumption rather than an established fact; it should be explicitly labeled as an assumption and cross-referenced with the caveat in Section 5.3.
  4. [Fig. 12 caption] The caption's phrase 'we find that these two spectra are different in scale before and after filtering' is ambiguous; clarify whether 'scale' refers to amplitude normalization or to spatial/angular scale.
  5. [Eqs. (7)–(8)] The ensemble average in the definitions of the cross and coherence power spectra is not specified; in practice it appears to be an average over k-space annuli or over test cubes, and this should be stated explicitly.
  6. [General] A brief code and data availability statement would help reproducibility, particularly for the ps_eor and 21cmFAST versions used and for any trained network weights or seed values.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: held-out test cubes and external LOFAR-derived inputs support the U-Net recovery claims.

full rationale

The paper's derivation chain is not circular. The EoR signal cubes are generated independently with 21cmFAST; thermal noise and excess variance are simulated with the pseor/GPR pipeline using kernel hyperparameters (l_ex = 0.26 MHz, sigma2_ex = 2.18 sigma2_n) measured from real LOFAR observations in Mertens et al. (2020). That citation is external, observationally anchored data rather than a result defined by the present paper's claims, so overlapping authorship does not make the input self-referential. The U-Net is trained on 512 cubes, validated on 16, and tested on 8 cubes that are not used for weight fitting, so the reported 2D cross and coherence power spectra are genuine prediction metrics within the simulation. The robustness test with the 30-degree wedge is also internally meaningful: the network fails to recover the filtered wedge modes, which the authors use to argue that below-wedge recovery in later tests is not a learned interpolation. The assumptions transferring LOFAR excess variance to SKA-Low are explicitly acknowledged ('we believe that they have statistically similar behavior', 'assuming that the ratio between excess variance and thermal noise is invariant') and the authors concede SKA may have less excess variance; these are model assumptions that affect the quantitative milestones but do not reduce a prediction to an input by definition. One non-circularity concern is worth noting: the paper does not explicitly state whether the 536 augmented cubes are split into training/validation/test in a realization-disjoint manner; the counts (512=128x4, 16=4x4, 8=2x4) are consistent with a disjoint split, but if test cubes were rotations of training cubes, the high coherence could be inflated by leakage. That would be a data-splitting flaw, not a circular derivation, and the paper's own statements do not establish that the flaw occurred. Overall, the central claims are self-contained against the paper's simulations and external inputs.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new physical entities. Its quantitative conclusions rest on GPR kernel hyperparameters fitted to real LOFAR NCP data in Mertens et al. (2020) and transferred to SKA-Low SCP simulations, plus assumptions that excess variance scales with thermal noise, that the foreground residual is fixed, and that 21cmFAST provides realistic EoR ground truth. The free parameters are the transferred GPR hyperparameters, the chosen 21cmFAST ionizing efficiency, and the chosen wedge angle.

free parameters (6)
  • Excess variance coherence scale (l_ex) = 0.26 MHz
    MCMC fit to LOFAR NCP data in Mertens et al. (2020), Table 1; used as input for generating nsex cubes in Section 3.3. Central to the paper's main finding that frequency-incoherent excess variance limits U-Net recovery.
  • Excess variance amplitude ratio (sigma2_ex / sigma2_n) = 2.18
    Best-fit ratio from Mertens et al. (2020), Table 1; used in Section 3.3 to generate excess variance cubes scaled with thermal noise variance at each integration time.
  • Mode-mixing foreground kernel scale (l_mix) = 8.1 MHz
    Best fit from Mertens et al. (2020), Table 1; part of the foreground residual model K_mix that shapes fg_fix in Section 3.2.
  • Mode-mixing variance (sigma2_mix) = 50.4 sigma2_n
    Best fit from Mertens et al. (2020), Table 1; used in the fixed foreground residual model in Section 3.2.
  • 21cmFAST ionizing efficiency (zeta) = 30
    Chosen astrophysical parameter for EoR signal generation in Section 2.1; the EoR morphology and power spectrum used as ground truth depend on it.
  • Filter wedge angle = 30 degrees
    Chosen by hand in Section 5.1.1 for the robustness test that removes foreground-like modes from EoR images and checks whether the network re-creates them.
assumptions (7)
  • domain assumption Excess variance measured in LOFAR NCP observations is statistically representative of SKA-Low SCP observations.
    Section 3.2: 'Since LOFAR and SKA have similar antenna placement strategies, we believe that they have statistically similar behavior in their observations.' Load-bearing because the central conclusion about excess variance limiting U-Net recovery transfers to SKA only under this assumption.
  • domain assumption The ratio of excess variance to thermal noise is invariant with integration time.
    Section 3.3: 'assuming that the ratio between excess variance and thermal noise is invariant.' Used to rescale excess variance for the 4380 and 13140 hour simulations.
  • domain assumption The smooth foreground residual is fixed and coherent across observations.
    Section 3.2 and 5.2: the foreground residual fg_fix is kept fixed across all training samples, motivated by the foreground not changing during observations; the network is therefore never tested against varying foreground realizations.
  • domain assumption The GPR component decomposition in Eqs. (2) and (3) correctly separates foreground, excess variance, thermal noise, and EoR signal.
    Section 3.1: the covariance model K = K_fg + K_ex + K_th + K_EoR with Matern kernels assumes components are distinguishable by frequency coherence; if the excess variance is not cleanly separable, the mock cubes misrepresent real systematics.
  • domain assumption 21cmFAST produces realistic EoR 21-cm signal morphologies.
    Section 2.1: the EoR signal used as ground truth for training and evaluation comes from 21cmFAST; the realism of the extracted signal is conditional on this generator.
  • ad hoc to paper Excess variance can be modeled as a Gaussian random field with a Matern 5/2 kernel and no correlated phase structure.
    Section 5.3: 'the nsex we use here is a Gaussian random field, but in real observations, the nsex could have some correlated structures.' The authors note real correlated structures could change results.
  • standard math Flat-sky approximation for the horizon line relation in Eq. (5).
    Section 5: horizon delay line computed with the flat-sky approximation k_parallel = k_perp D_M H0 E / (c(1+z)).

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Cite this review

Pith. "Pith review of Extracting the Epoch of Reionization Signal with 3D U-Net Neural Networks Using Data-driven Systematic Effect Model." pith.science (2026). https://pith.science/paper/FOXEQY4R

@misc{pith2026241216853,
  author       = {Pith},
  title        = {Pith review of: Extracting the Epoch of Reionization Signal with 3D U-Net Neural Networks Using Data-driven Systematic Effect Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FOXEQY4R}},
  note         = {Machine review of arXiv:2412.16853}
}
read the original abstract

Neutral hydrogen (HI) serves as a crucial probe for the Cosmic Dawn and the Epoch of Reionization (EoR). Actual observations of the 21-cm signal often encounter challenges such as thermal noise and various systematic effects. To overcome these challenges, we simulate SKA-Low-depth images in South Celestial Pole (SCP) field and process them with a deep learning method. We utilized foreground residuals acquired by LOFAR during actual North Celestial Pole (NCP) field observations, thermal and excess variances calculated via Gaussian process regression (GPR), and 21-cm signals generated with 21cmFAST for signal extraction tests. Our approach to overcome these foreground, thermal noise, and excess variance components employs a 3D U-Net neural network architecture for image analysis. When considering thermal noise corresponding to 1752 hours of integration time, U-Net provides reliable 2D power spectrum predictions, and robustness tests ensure that we get realistic EoR signals. Adding foreground residuals, however, causes inconsistencies below the horizon delay-line. Lastly, evaluating both thermal noise and excess variances with observations up to 4380 hours and 13140 hours ensures reliable power spectrum estimations within the EoR window and across nearly all scales, respectively. The incoherence of excess variances in the frequency direction can greatly affect deep learning to extract 21-cm signals.

Figures

Figures reproduced from arXiv: 2412.16853 by the authors.

Figure 1
Figure 1. Number densities of LOFAR baselines (first line) and SKA baselines (second line) with 12-hour observation (first column) and 24-hour observation (second column) per day. SKA are significantly better than those of LOFAR. SKA’s 12-hour observations provide perfect coverage of the entire observation area, whereas LOFAR’s 12-hour observations re￾sult in some gaps in uv-coverage. Although a longer obser￾vation time leads… view at source ↗
Figure 2
Figure 2. Simulated slices of images (first line) and gridded visibil￾ities (second line) from SKA before applying LOFAR uv-coverage of EoR and nsth, where nsth is obtained based on LOFAR’s imag￾ing capabilities and sensitivity, and EoR is obtained from full uv￾coverage using 21cmFAST code. The units of images and gridded visibilities are both millikelvin (mK). is primarily shaped by the sizes and spatial arrangement of the i… view at source ↗
Figure 3
Figure 3. Simulated slices of images (first line) and gridded visibil￾ities (second line) from SKA before applying LOFAR uv-coverage of fgfix and nsex, where fgfix is the smooth foreground residual ob￾tained via GPR based on observations of the real NCP sky field, nsex is obtained based on LOFAR’s imaging capabilities, sensitiv￾ity, and nsth. The units of images and gridded visibilities are both mK [PITH_FULL_IMAGE:figures/f… view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Matern covariance functions for ´ Kmix and Kex. is kept fixed. Hence, random realizations of the residuals, during training of the network, arise only from the excess variance, thermal noise, and 21-cm signals, which will be elaborated on later. Since LOFAR and SKA hav…
Figure 6
Figure 6. Figure 6: Simulated slices of images (first line) and gridded visibilities (second line) from SKA after applying LOFAR uv-coverage of fgfix, nsth, nsex, and EoR for the slices in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Slices of the middle 64 × 64 pixel images (first line) of the four components fgfix, nsth, nsex, and EoR in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: Training process of CNNs with U-Net architecture. Each color represents a structure in the U-Net network, where yellow cubes represent the convolutional layers and ReLU sections, red cubes represent pooling layers in down-sampling, blue cubes represent the transposed c…
Figure 10
Figure 10. Figure 10: 2D cross power spectrum and 2D coherence power spec￾trum between the target EoR and U-Net predictive image when considering only the effects of nsth. The black dotted lines are hori￾zon lines for SKA-Low. age and the U-Net output are illustrated in [PITH_FULL_IMAGE:f…
Figure 9
Figure 9. Figure 9: Target EoR image and predictive images given by U-Net when considering only the effects of nsth. These images are in units of mK. of training epochs to maintain stability and prevent overfit￾ting, as will be demonstrated next. In order to quantitatively evaluate the pe…
Figure 11
Figure 11. Figure 11 [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: Original EoRori 2D power spectrum, EoRrev 2D power spectrum after removing the 30◦ filter wedge, and their 2D coherence power spectrum between them. We find that these two spectra are different in scale before and after filtering. Due to windowing effects based on the…
Figure 14
Figure 14. Figure 14: 2D cross power spectrum and 2D coherence power spec￾trum between the target EoR and U-Net predictive image when considering the effects of fgfix and nsth. The black dotted lines are horizon lines for SKA-Low. ined a 1752-hour observation for consistency with the pre￾v…
Figure 15
Figure 15. Figure 15: Images of the nsall at observation times 1752 hours, 4380 hours, and 13140 hours, respectively. These images are in units of mK. influences the required number of epochs to achieve stability in U-Net training. For the observation of 1752 hours, increas￾ing the number …
Figure 16
Figure 16. Figure 16: The 2D cross power spectra (first row) and their corresponding 2D coherence power spectra (second row) of predicted and target images derived from U-Net processing of simulated data (nsex + nsth + EoR), which were observed for durations of 1752 hours, 4380 hours, and …
Figure 17
Figure 17. Figure 17: The 2D cross power spectra (first row) and their corresponding 2D coherence power spectra (second row) of predicted and target images derived from U-Net processing of simulated data (fgfix + nsex + nsth + EoR), which were observed for durations of 1752 hours, 4380 hou…
Figure 18
Figure 18. Figure 18: 2D power spectrum of the target EoR image and the 2D power spectrum of the predicted image given by U-Net after 850 epochs for LOFAR scenario when only consider nsth. The black dotted lines are horizon lines for LOFAR. 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 k [h cMpc…
Figure 19
Figure 19. Figure 19: 2D cross power spectrum and 2D coherence power spectrum of the target and predicted images for LOFAR scenario when only consider nsth. The black dotted lines are horizon lines for LOFAR [PITH_FULL_IMAGE:figures/full_fig_p019_19.png]
Figure 20
Figure 20. Figure 20: 2D power spectrum of the target EoR image and the 2D power spectrum of the predicted image given by U-Net after 1500 epochs for LOFAR scenario when considering nsth and nsex. The black dotted lines are horizon lines for LOFAR. 0.050 0.075 0.100 0.125 0.150 0.175 0.200…
Figure 21
Figure 21. Figure 21: 2D cross power spectrum and 2D coherence power spectrum of the target and predicted images for LOFAR scenario when consid￾ering nsth and nsex. The black dotted lines are horizon lines for LOFAR [PITH_FULL_IMAGE:figures/full_fig_p020_21.png]

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  1. Constraints on the state of the IGM at $z\sim 8-10$ using redshifted 21-cm observations with LOFAR

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