Bayesian data analysis for sky-averaged 21-cm experiments in the presence of ionospheric effects

List of different types of beams.

	Beam type
B	Original antenna directivity
\|$\hat{B}$\|	Antenna directivity including ionospheric effects
\|$\hat{B}_t$\|	Antenna directivity including time-dependent ionospheric effects

	Beam type
B	Original antenna directivity
\|$\hat{B}$\|	Antenna directivity including ionospheric effects
\|$\hat{B}_t$\|	Antenna directivity including time-dependent ionospheric effects

Table 1.

List of different types of beams.

	Beam type
B	Original antenna directivity
\|$\hat{B}$\|	Antenna directivity including ionospheric effects
\|$\hat{B}_t$\|	Antenna directivity including time-dependent ionospheric effects

	Beam type
B	Original antenna directivity
\|$\hat{B}$\|	Antenna directivity including ionospheric effects
\|$\hat{B}_t$\|	Antenna directivity including time-dependent ionospheric effects

2.2 Time-varying ionospheric effects

The real ionosphere varies with time, and the chromatic ionospheric effects on the foregrounds do not cancel out over time. Hence, we model the ionosphere layers based on actual observational data (Shen et al. 2021). The data are collected from Lowell GIRO Data Center at station Louisvale, South Africa (https://ulcar.uml.edu/DIDBase/). Data of the parameters used to characterize the D-layer are mostly absent during night time due to low electron density. Therefore, in our model, the D-layer parameters are scaled from the available data of the F-layer parameters based on the fiducial theoretical values. In this work, the ionospheric effects are applied to the beam, resulting in time-varying beams when generating antenna temperature data.

3 DATA ANALYSIS PIPELINE

The REACH data analysis pipeline (Anstey et al. 2021) uses POLYCHORD, a nested sampling algorithm based on Bayesian inference (Handley, Hobson & Lasenby 2015). It adopts a physically motivated method of modelling the foregrounds of 21-cm experiments in order to fit the systematic chromatic distortions as part of the foregrounds.

3.1 Bayesian inference

Bayesian inference is useful in performing parameter estimation and model comparison. Algorithms based on Markov–Chain Monte Carlo (MCMC) methods, such as the Metropolis Hastings (MH) algorithm (Chib & Greenberg 1995) and its more sophisticated variants (Foreman-Mackey et al. 2013), are used to achieve parameter estimation. Model comparison requires calculating evidence, which is the integration of likelihood over the prior density.

By Bayes’ theorem, one can derive the equation

$$\begin{eqnarray} \begin{aligned} \mathrm{P}(\theta _\mathcal {M}|\mathcal {D},\mathcal {M}) &= \frac{\mathrm{P}(\mathcal {D}|\theta _\mathcal {M},\mathcal {M})}{\mathrm{P}(\theta _\mathcal {D}|\mathcal {M})}, \quad \mathrm{or} \quad \mathcal {P} = \frac{\mathcal {L}\pi }{\mathcal {Z}}, \end{aligned} \end{eqnarray}$$

(4)

where |$\mathcal {P}$| is the posterior distribution. |$\mathcal {L}$| is the likelihood, which is the probability of the data given a model |$\mathcal {M}$| and the set of parameters |$\theta _\mathcal {M}$| describing the model. |$\mathcal {\pi }$| is the prior distribution of the parameters, and |$\mathcal {Z}$| is the Bayesian evidence or marginal likelihood:

$$\begin{eqnarray} \begin{aligned} \mathcal {Z} = \int \mathrm{P}(\mathcal {D}|\theta _\mathcal {M},\mathcal {M})\mathrm{P}(\theta _\mathcal {D}|\mathcal {M})\mathrm{ d}\theta _\mathcal {M} = \int \mathcal {L}\pi \mathrm{ d}\theta _\mathcal {M}. \end{aligned} \end{eqnarray}$$

(5)

In this paper, we calculate the value of likelihood |$\mathcal {L}$| to estimate the confidence of the signal detection in the data analysis pipeline.

3.2 Foreground modelling

A sky model T_sky(ν, θ, ϕ) is generated across the observing band by dividing the sky into N regions of similar spectral indices and scaling a base map assuming a distinct uniform spectral index in each region. This sky map is then convolved with a antenna beam model B(ν, θ, ϕ) to produce a model of foregrounds and chromaticity parametrized by the spectral indices of the regions. In this work, number of regions is 25. An approximate sky map and an antenna directivity pattern generated from physically motivated parameters for each observing frequency are convolved by the equation

$$\begin{eqnarray} \begin{aligned} T_{\mathrm{D}} = \frac{1}{4\pi } \int ^{4\pi } _0 B(\nu , \theta , \phi) \int ^{t_{\mathrm{end}}} _{t_{\mathrm{start}}} T_{\mathrm{sky}}(\nu , \theta , \phi) \mathrm{ d}t \mathrm{ d}\Omega + \hat{\sigma }, \end{aligned} \end{eqnarray}$$

(6)

where |$\hat{\sigma }$| is uncorrelated Gaussian noise, to produce a parametrized model that describes both the foregrounds and the chromatic distortion of the antenna. Then we use the Bayesian nested sampling algorithm PolyChord to fit this foreground model to the data with a parametrized 21-cm signal model (Anstey et al. 2021). In our work, the foregrounds are modelled by beams including ionospheric effects |$\hat{B}$| or time-dependent ionospheric effects |$\hat{B}_t$|⁠. For cases modelled by the same beam including ionospheric effects |$\hat{B}$|⁠, equation (6) becomes

$$\begin{eqnarray} \begin{aligned} T_{\mathrm{D}} = \frac{1}{4\pi } \int ^{4\pi } _0 \hat{B}(\nu , \theta , \phi) \int ^{t_{\mathrm{end}}} _{t_{\mathrm{start}}} T_{\mathrm{sky}}(\nu , \theta , \phi) \mathrm{ d}t \mathrm{ d}\Omega + \hat{\sigma }, \end{aligned} \end{eqnarray}$$

(7)

and for cases modelled by beams including time-dependent ionospheric effects |$\hat{B}_t$|⁠,

$$\begin{eqnarray} \begin{aligned} T_{\mathrm{D}} = \frac{1}{4\pi } \int ^{4\pi } _0 \int ^{t_{\mathrm{end}}} _{t_{\mathrm{start}}} \hat{B}_t(\nu , \theta , \phi ,t) T_{\mathrm{sky}}(\nu , \theta , \phi) \mathrm{ d}t \mathrm{ d}\Omega + \hat{\sigma }. \end{aligned} \end{eqnarray}$$

(8)

3.3 Likelihood of the local sidereal times (LST) model

The model adopted in this work is the local sidereal times (LST) model, in which the data are not integrated over the observing time (Anstey et al. in preparation). The general form of likelihood for the LST model is:

$$\begin{eqnarray} \log {\mathcal {L}} &=& \sum _i\sum _j\left[-\frac{1}{2}\log (2\pi \theta _{\sigma }) \right. \\ && \left. - \ \frac{1}{2}\left(\frac{T_{\mathrm{D}i,j}-T_{\mathrm{M}i,j}(\theta _\mathrm{F},\theta _\mathrm{S})}{\theta _{\sigma }}\right)^2\right], \end{eqnarray}$$

(9)

where i indicates frequency bins and j indicates time bins. T_D is the observational data and T_M is the parametrized model. In this equation, θ_F, θ_S, θ_σ are foreground, signal, and noise parameters, respectively. T_{Mi, j}(θ_F, θ_S) can be split into foreground T_F and signal T_S components. The global 21-cm signal has no time dependencies, meaning equation (9) can be written as

$$\begin{eqnarray} \log {\mathcal {L}} &=& -\frac{1}{2}\sum _i\sum _j\log (2\pi \theta _{\sigma }) \\ &&- \ \frac{1}{2}\sum _i\sum _j\left(\frac{T_{\mathrm{D}i,j}-T_{\mathrm{F}i,j}(\theta _\mathrm{F})-T_{\mathrm{S}i}(\theta _\mathrm{S})}{\theta _{\sigma }}\right)^2. \end{eqnarray}$$

(10)

A detection that is considered significant should show a high difference between the evidences yielded by the foreground model including a Gaussian signal model, |$\log \mathcal {Z}_{\mathrm{s}}$|⁠, and the one without, |$\log \mathcal {Z}_{\mathrm{ns}}$|⁠: |$\Delta \log \mathcal {Z} = \log \mathcal {Z}_{\mathrm{s}}- \log \mathcal {Z}_{\mathrm{ns}}$|⁠, where the subscripts s and ns indicate that the foreground model includes and excludes a signal, respectively. Root-mean-square error (RMSE) of the detected signal with respect to the injected signal is also measured to check that the fitted signal is not actually fitting the foregrounds instead of the injected signal. As the values of likelihood and log evidence are calculated by linearly summing over all data points, more time-steps, meaning more data points, could result in bigger values.

4 RESULTS

In practice, data are collected chronologically at points with a fixed interval between each during observation time and analysed. In our work, however, there are gaps between the selected time-steps in most of the cases. It is because the ionosphere can suffer significant change even in a short time frame. One of the aims of this work is to analyse how much error we could afford using the REACH data analysis pipeline, and therefore we start with the more ideal cases. With that in mind, we choose time-steps that yield better evidences or signal detection as a group, and do the error analyses from those. This means the time-steps are not necessarily temporally close to each other, and the intervals between the selected time-steps are meaningless in this work. The injected Gaussian signal adopted in the paper has an amplitude of 0.53 K and a standard deviation of 18.7 MHz, centring at 78.1 MHz, which is compatible with the EDGES (Bowman et al. 2018) signal in amplitude and centre frequency.

We fit the foreground model using one constant beam for all time-steps or more beams that correspond to the different time-steps, distorted by refraction and absorption due to the ionosphere based on the ionospheric model, to estimate how frequently the beam should be redetermined to achieve a meaningful detection. Fixed errors are introduced to the ionospheric parameters when generating the antenna temperature data, and the data are fitted using the original beam (the beam without applied errors) in the data analysis pipeline to see how much deviation in the parameters the foreground model is able to tolerate.

The location of the antenna used to simulate the data is set in the Karoo radio reserve in South Africa (latitude: 30.7°S, longitude: 21.4°E, height: 25 m). The data for the ionospheric parameters are collected from Lowell GIRO Data Center at station Louisvale, South Africa (https://ulcar.uml.edu/DIDBase/). The time-steps taken are at the time specified after 00:00:00 UTC on 2019 Jan 1.

4.1 Foregrounds modelled with a constant beam

4.1.1 Two time-steps

We select antenna temperature data at four different time-steps [t₁, t₂, t₃, t₄] that yield reasonably good |$\Delta \log \mathcal {Z}$| and relatively low RMSE (<0.1K) with that of time-step t₀ when analysed using the LST version of the REACH data analysis pipeline, and the foreground signal is modelled using the same beam (⁠|$B_{t_0}$|⁠) for two different time-steps [t₀, t_i], the log evidences |$\log \mathcal {Z}$| of which are shown in Table 2. All analyses shown in this work are based on the time-steps listed in Table 2.

Table 2.

RMSE and Log evidence |$\log \mathcal {Z}$|⁠: the foreground signal is modelled using one single beam (⁠|$B_{t_0}$|⁠). Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₁	t₀, t₁	0.066	231.5 ± 0.3	62.8
C₂	t₀, t₂	0.035	520.2 ± 0.3	180.4
C₃	t₀, t₃	0.056	93.3 ± 0.4	161.8
C₄	t₀, t₄	0.024	519.0 ± 0.3	253.5

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₁	t₀, t₁	0.066	231.5 ± 0.3	62.8
C₂	t₀, t₂	0.035	520.2 ± 0.3	180.4
C₃	t₀, t₃	0.056	93.3 ± 0.4	161.8
C₄	t₀, t₄	0.024	519.0 ± 0.3	253.5

Note. [t₀, t₁, t₂, t₃, t₄] = [21^○00′, 21^○15′, 45^○15′, 46^○15′, 47^○45′]

Table 2.

RMSE and Log evidence |$\log \mathcal {Z}$|⁠: the foreground signal is modelled using one single beam (⁠|$B_{t_0}$|⁠). Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₁	t₀, t₁	0.066	231.5 ± 0.3	62.8
C₂	t₀, t₂	0.035	520.2 ± 0.3	180.4
C₃	t₀, t₃	0.056	93.3 ± 0.4	161.8
C₄	t₀, t₄	0.024	519.0 ± 0.3	253.5

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₁	t₀, t₁	0.066	231.5 ± 0.3	62.8
C₂	t₀, t₂	0.035	520.2 ± 0.3	180.4
C₃	t₀, t₃	0.056	93.3 ± 0.4	161.8
C₄	t₀, t₄	0.024	519.0 ± 0.3	253.5

Note. [t₀, t₁, t₂, t₃, t₄] = [21^○00′, 21^○15′, 45^○15′, 46^○15′, 47^○45′]

The settings used to model the foregrounds in the four cases with two time-steps are identical, so the value of the log evidence |$\log (\mathcal {Z}_{s})$| should be comparable, as demonstrated by the negative correlation between |$\log (\mathcal {Z}_{s})$| and RMSE; a higher |$\log (\mathcal {Z}_{s})$| could suggest better signal detection. The fitted signal and residual plots of case C₁[t₀, t₁] are shown in Fig. 1. As one can see, in these cases, the signal, shown in red, are fitted at significant detection.

$A two time-step case, the foregrounds fitted using one single beam $B_{t_0}$ (case C1[t0, t1] of Table 2, the RMSE of the reconstructed signal is 0.066 K and $\Delta \log \mathcal {Z} = 62$). The reconstructed global 21-cm signal is shown in blue, and the data with the fitted foregrounds removed is shown in red. The grey dashed line is the injected signal. The detected signal (reconstructed signal) is significant and the residuals are low for both time-steps.$

Figure 1.

A two time-step case, the foregrounds fitted using one single beam |$B_{t_0}$| (case C₁[t₀, t₁] of Table 2, the RMSE of the reconstructed signal is 0.066 K and |$\Delta \log \mathcal {Z} = 62$|⁠). The reconstructed global 21-cm signal is shown in blue, and the data with the fitted foregrounds removed is shown in red. The grey dashed line is the injected signal. The detected signal (reconstructed signal) is significant and the residuals are low for both time-steps.

4.1.2 Three or more time-steps

Table 3 shows the evidence of the cases consisting three or more time-steps. It shows that when the foregrounds are modelled using only one beam, all data sets analysed should have good evidence between themselves, such as [t₀, t₂] and [t₀, t₄] (see Table 2), to still yield a good evidence in the cases with more time-steps. For the other cases, one observes that the evidences drop to negative values, but their respective |$\Delta \log \mathcal {Z}$| and RMSE still imply significant detection; Fig. 2 shows the plot of the fitted signals as well as the residuals of case C₅[t₀, t₁, t₂] at each time-step, and the reconstructed signals in both time-steps are not too different from the injected signal, as implied by the RMSE. The plots of case C₆[t₀, t₂, t₄] are shown in Fig. 3. Despite the negative, low evidences for multiple time-step cases, their RMSE’s and reconstructed signals still show significant detection, even in the five time-step case. The plots shown in Fig. 4. One should note that the time-steps selected for the analyses are the more ideal cases, which might not be the case in actual observations.

$A three time-step case, the foregrounds fitted using one single beam $B_{t_0}$ (Case C5[t0, t1, t2] of Table 3, the RMSE of the reconstructed signal is 0.082 K and $\Delta \log \mathcal {Z}=190$). The reconstructed global 21-cm signal is shown in blue, and the data with the fitted foreground removed is shown in red. The grey dashed line is the injected signal. The detected signal (reconstructed signal) is still significant despite its low evidence $\log \mathcal {Z}$ and high residuals in the last two time-steps.$

Figure 2.

A three time-step case, the foregrounds fitted using one single beam |$B_{t_0}$| (Case C₅[t₀, t₁, t₂] of Table 3, the RMSE of the reconstructed signal is 0.082 K and |$\Delta \log \mathcal {Z}=190$|⁠). The reconstructed global 21-cm signal is shown in blue, and the data with the fitted foreground removed is shown in red. The grey dashed line is the injected signal. The detected signal (reconstructed signal) is still significant despite its low evidence |$\log \mathcal {Z}$| and high residuals in the last two time-steps.

$A three time-step case, the foregrounds fitted using one single beam $B_{t_0}$ (Case C6[t0, t2, t4] of Table 3, the RMSE of the reconstructed signal is 0.034 K and $\Delta \log \mathcal {Z} = 317$). The reconstructed global 21-cm signal is shown in blue, and the data with the fitted foreground removed is shown in red. The grey dashed line is the injected signal. The detected signal (reconstructed signal) is significant and the residuals are low for all time-steps.$

Figure 3.

A three time-step case, the foregrounds fitted using one single beam |$B_{t_0}$| (Case C₆[t₀, t₂, t₄] of Table 3, the RMSE of the reconstructed signal is 0.034 K and |$\Delta \log \mathcal {Z} = 317$|⁠). The reconstructed global 21-cm signal is shown in blue, and the data with the fitted foreground removed is shown in red. The grey dashed line is the injected signal. The detected signal (reconstructed signal) is significant and the residuals are low for all time-steps.

$A five time-step case, the foregrounds fitted using one single beam $B_{t_0}$ (Case C9[t0, t1, t2, t3, t4] of Table 3, the RMSE of the reconstructed signal is 0.063 K and $\Delta \log \mathcal {Z} = 103$). The reconstructed global 21-cm signal is shown in blue, and the data with the fitted foreground removed is shown in red. The grey dashed line is the injected signal. The detected signal (reconstructed signal) is still significant despite its low evidence $\log \mathcal {Z}$ and high residuals.$

Figure 4.

A five time-step case, the foregrounds fitted using one single beam |$B_{t_0}$| (Case C₉[t₀, t₁, t₂, t₃, t₄] of Table 3, the RMSE of the reconstructed signal is 0.063 K and |$\Delta \log \mathcal {Z} = 103$|⁠). The reconstructed global 21-cm signal is shown in blue, and the data with the fitted foreground removed is shown in red. The grey dashed line is the injected signal. The detected signal (reconstructed signal) is still significant despite its low evidence |$\log \mathcal {Z}$| and high residuals.

Table 3.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: the foreground signal is modelled using one single beam (⁠|$B_{t_0}$|⁠). Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₅	t₀, t₁, t₂	0.082	−4720.2 ± 0.4	194.4
C₆	t₀, t₂, t₄	0.034	752.6 ± 0.4	316.1
C₇	t₀, t₁, t₂, t₄	0.055	−4665.1 ± 0.4	196.5
C₈	t₀, t₂, t₃, t₄	0.083	−7308.7 ± 0.4	53.0
C₉	t₀, t₁, t₂, t₃, t₄	0.063	−9387.3 ± 0.4	103.0

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₅	t₀, t₁, t₂	0.082	−4720.2 ± 0.4	194.4
C₆	t₀, t₂, t₄	0.034	752.6 ± 0.4	316.1
C₇	t₀, t₁, t₂, t₄	0.055	−4665.1 ± 0.4	196.5
C₈	t₀, t₂, t₃, t₄	0.083	−7308.7 ± 0.4	53.0
C₉	t₀, t₁, t₂, t₃, t₄	0.063	−9387.3 ± 0.4	103.0

Table 3.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: the foreground signal is modelled using one single beam (⁠|$B_{t_0}$|⁠). Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₅	t₀, t₁, t₂	0.082	−4720.2 ± 0.4	194.4
C₆	t₀, t₂, t₄	0.034	752.6 ± 0.4	316.1
C₇	t₀, t₁, t₂, t₄	0.055	−4665.1 ± 0.4	196.5
C₈	t₀, t₂, t₃, t₄	0.083	−7308.7 ± 0.4	53.0
C₉	t₀, t₁, t₂, t₃, t₄	0.063	−9387.3 ± 0.4	103.0

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₅	t₀, t₁, t₂	0.082	−4720.2 ± 0.4	194.4
C₆	t₀, t₂, t₄	0.034	752.6 ± 0.4	316.1
C₇	t₀, t₁, t₂, t₄	0.055	−4665.1 ± 0.4	196.5
C₈	t₀, t₂, t₃, t₄	0.083	−7308.7 ± 0.4	53.0
C₉	t₀, t₁, t₂, t₃, t₄	0.063	−9387.3 ± 0.4	103.0

4.2 Foregrounds modelled using time-varying beams

The results shown in Section 4.1 where the foreground signal is modelled using only one beam for multiple time-steps demonstrate that this approach can be problematic in the presence of the chromatic effects introduced by a changing ionosphere. Therefore, in this section we test time-varying beams to see how much the analyses can be improved by introducing more corresponding beams when fitting the foregrounds for cases with multiple time-steps.

Table 4 shows two cases that include three time-steps where the foreground signal is modelled using two corresponding beams. We explore how the signal detection of the two time-step cases with foregrounds modelled using one constant beam would differ when introducing an extra time-step with its foregrounds modelled using its corresponding beam. This extra time-step is t₂, shown in Table 4 in red. One observes that although [t₀, t₁] as a pair does yield reasonably good evidence (Table 2), the evidence when analysed together with another time-step t₂ (case C₁₀), modelled using its corresponding beam, drops to negative values. The other case C₁₁, on the other hand, as [t₀, t₄] yields good evidence as a pair, yields high evidence as well as |$\Delta \log \mathcal {Z}$| when analysed with the same time-step (t₂) as in case C₁₀. The RMSE also decreases which suggests better signal detection. This again shows that signal detection could be improved by including more time-steps as long as they, in smaller groups, yield better evidence and signal detection.

Table 4.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: foregrounds modelled using two beams; the foregrounds of the time-step coloured in red are modelled using its corresponding beam, and the rest using |$B_{t_0}$|⁠. The cases coloured in grey are the references from Table 3. Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₁₀	t₀, t₁, t₂	0.056	−177.2 ± 0.3	9478.2
C₅	t₀, t₁, t₂	0.082	−4720.2 ± 0.4	194.4
C₁₁	t₀, t₂, t₄	0.022	796.9 ± 0.4	385.5
C₆	t₀, t₂, t₄	0.034	752.6 ± 0.4	316.1

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₁₀	t₀, t₁, t₂	0.056	−177.2 ± 0.3	9478.2
C₅	t₀, t₁, t₂	0.082	−4720.2 ± 0.4	194.4
C₁₁	t₀, t₂, t₄	0.022	796.9 ± 0.4	385.5
C₆	t₀, t₂, t₄	0.034	752.6 ± 0.4	316.1

Table 4.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: foregrounds modelled using two beams; the foregrounds of the time-step coloured in red are modelled using its corresponding beam, and the rest using |$B_{t_0}$|⁠. The cases coloured in grey are the references from Table 3. Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₁₀	t₀, t₁, t₂	0.056	−177.2 ± 0.3	9478.2
C₅	t₀, t₁, t₂	0.082	−4720.2 ± 0.4	194.4
C₁₁	t₀, t₂, t₄	0.022	796.9 ± 0.4	385.5
C₆	t₀, t₂, t₄	0.034	752.6 ± 0.4	316.1

Case	[t]	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
C₁₀	t₀, t₁, t₂	0.056	−177.2 ± 0.3	9478.2
C₅	t₀, t₁, t₂	0.082	−4720.2 ± 0.4	194.4
C₁₁	t₀, t₂, t₄	0.022	796.9 ± 0.4	385.5
C₆	t₀, t₂, t₄	0.034	752.6 ± 0.4	316.1

4.3 Deviation in modelled ionospheric parameters

In this section, a fixed amount of error [2 per cent, 5 per cent, 10 per cent, 15 per cent] is applied to the ionospheric parameters (a fixed percentage of increment in electron density N_e, collision frequency ν_c, and layer thickness), when generating the antenna temperature data to determine how much error would lead to unsuccessful detection. A steady decrease in antenna temperature T_A, due to the applied error increasing the chromatic ionospheric absorption, is shown in Fig. 5. The data analysis pipeline then fit the foregrounds to the antenna temperature data using the beam distorted by the ionosphere, but with no error applied to the parameters. Fig. 6 shows how much the beam gain changes at 50, 125, and 200 MHz when a 10 per cent deviation (10 per cent increment) is applied to the ionospheric parameters with respect to when no deviation is applied.

Figure 5.

The change in integrated antenna temperature ΔT = T′ − T and the percentage of change ΔT/T with respect to frequency when a fixed amount of error [2 per cent, 5 per cent, 10 per cent, 15 per cent] is applied to the ionospheric parameters, electron density N_e, collision frequency ν_c, and layer thickness, when modelling the antenna beam subjected to chromatic ionospheric effects.

Figure 6.

Difference in beam gain of the case modelled with 10 per cent deviation in ionospheric parameters with respect to the case where no deviation is applied at 50 MHz, 125 MHz, and 200 MHz, from top to bottom. The unit of the colour code is dB.

4.3.1 Two time-steps, foregrounds modelled using one beam and two corresponding beams

Table 5 shows the log evidences of the cases with different fixed deviations applied to the ionospheric parameters when the foregrounds of all the time-steps is modelled using only one beam |$B_{t_0}$|⁠. The top-first panel of Fig. 7 shows the RMSE and |$\Delta \log {\mathcal {Z}}$| with respect to the deviation applied; one observes that the RMSE increases with the growing deviation, as expected. |$\Delta \log {\mathcal {Z}}$| decreases except for the 15 per cent case, which might not seem reasonable; as the deviation reaches 10 per cent, the RMSE becomes really high and |$\log {\mathcal {Z}_s}$| turns negative, so the high |$\Delta \log {\mathcal {Z}}$| could mean the model is fitting other features in the foregrounds instead of the injected global 21-cm signal, and hence its high value most likely does not suggest better signal detection. As the configuration of the analysis pipeline is identical for all the cases with deviations, |$\log {\mathcal {Z}_s}$| should be comparable; it drops consistently as the deviation in ionospheric parameters increases, and turns negative as the deviation reaches 10 per cent, also suggesting unreliable signal detection when the deviation goes beyond 10 per cent.

$$\Delta \log {\mathcal {Z}}$ (blue), $\log {\mathcal {Z}}$ (red), and RMSE (olive) of the cases where a fixed amount of error [2 per cent, 5 per cent, 10 per cent, 15 per cent] is applied to the ionospheric parameters, electron density Ne, collision frequency νc, and layer thickness, when modelling the antenna beam subjected to chromatic ionospheric effects. The foregrounds in the data analysis pipeline are modelled using one single beam ($B_{t_0}$). The red dashed line is the 0 reference for $\log {\mathcal {Z}}$ and the olive dashed line is the 0.10 K reference for RMSE. Top left-hand panel: two time-step (N = 2) case [t0, t1], foregrounds modelled with one beam (Table 5). Top right-hand panel: two time-step (N = 2) case [t0, t1] (Table 6), foregrounds modelled with two beams. Bottom left-hand panel: three time-step (N = 3) case [t0, t1, t2], foregrounds modelled with one beam (Table 7). Bottom right-hand panel: three (N = 3) case [t0, t1, t2], foregrounds modelled with three beams (Table 8). All plots show increasing RMSE and decreasing $\log {\mathcal {Z}}$ with deviation, which is expected. $\Delta \log {\mathcal {Z}}$ does not show a consistent trend, and sometimes even increase abruptly in the cases with higher deviation; the value of $\Delta \log {\mathcal {Z}}$ is not reliable when the RMSE is high, as it implies the model is fitting the foregrounds instead of the injected signal. Overall, it suggests that the detection is not reliable as the deviation goes beyond 5 per cent.$

Figure 7.

|$\Delta \log {\mathcal {Z}}$| (blue), |$\log {\mathcal {Z}}$| (red), and RMSE (olive) of the cases where a fixed amount of error [2 per cent, 5 per cent, 10 per cent, 15 per cent] is applied to the ionospheric parameters, electron density N_e, collision frequency ν_c, and layer thickness, when modelling the antenna beam subjected to chromatic ionospheric effects. The foregrounds in the data analysis pipeline are modelled using one single beam (⁠|$B_{t_0}$|⁠). The red dashed line is the 0 reference for |$\log {\mathcal {Z}}$| and the olive dashed line is the 0.10 K reference for RMSE. Top left-hand panel: two time-step (N = 2) case [t₀, t₁], foregrounds modelled with one beam (Table 5). Top right-hand panel: two time-step (N = 2) case [t₀, t₁] (Table 6), foregrounds modelled with two beams. Bottom left-hand panel: three time-step (N = 3) case [t₀, t₁, t₂], foregrounds modelled with one beam (Table 7). Bottom right-hand panel: three (N = 3) case [t₀, t₁, t₂], foregrounds modelled with three beams (Table 8). All plots show increasing RMSE and decreasing |$\log {\mathcal {Z}}$| with deviation, which is expected. |$\Delta \log {\mathcal {Z}}$| does not show a consistent trend, and sometimes even increase abruptly in the cases with higher deviation; the value of |$\Delta \log {\mathcal {Z}}$| is not reliable when the RMSE is high, as it implies the model is fitting the foregrounds instead of the injected signal. Overall, it suggests that the detection is not reliable as the deviation goes beyond 5 per cent.

Table 5.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: two time-steps [t₀, t₁] with deviation in ionospheric parameters, the foreground signal is modelled using one single beam (⁠|$B_{t_0}$|⁠). Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.066	231.5 ± 0.3	62.8
2 per cent	0.076	189.1 ± 0.3	42.1
5 per cent	0.091	81.0 ± 0.3	16.6
10 per cent	0.260	−377.1 ± 0.4	0.2
15 per cent	0.283	−1822.5 ± 0.4	2.0E + 10

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.066	231.5 ± 0.3	62.8
2 per cent	0.076	189.1 ± 0.3	42.1
5 per cent	0.091	81.0 ± 0.3	16.6
10 per cent	0.260	−377.1 ± 0.4	0.2
15 per cent	0.283	−1822.5 ± 0.4	2.0E + 10

Table 5.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: two time-steps [t₀, t₁] with deviation in ionospheric parameters, the foreground signal is modelled using one single beam (⁠|$B_{t_0}$|⁠). Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.066	231.5 ± 0.3	62.8
2 per cent	0.076	189.1 ± 0.3	42.1
5 per cent	0.091	81.0 ± 0.3	16.6
10 per cent	0.260	−377.1 ± 0.4	0.2
15 per cent	0.283	−1822.5 ± 0.4	2.0E + 10

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.066	231.5 ± 0.3	62.8
2 per cent	0.076	189.1 ± 0.3	42.1
5 per cent	0.091	81.0 ± 0.3	16.6
10 per cent	0.260	−377.1 ± 0.4	0.2
15 per cent	0.283	−1822.5 ± 0.4	2.0E + 10

Table 6 shows the log evidences of the cases with deviations applied to the ionospheric parameters when the foregrounds of the time-steps is modelled using two corresponding beams. Signal detection of all the cases improves comparing with the cases in which the foregrounds are modelled using only one beam, as expected.

Table 6.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: two time-steps [t₀, t₁] with deviation in ionospheric parameters, the foreground signal is modelled using two corresponding beams. Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.017	580.2 ± 0.4	208.7
2 per cent	0.071	455.7 ± 0.4	98.1
5 per cent	0.169	164.4 ± 0.3	2.4
10 per cent	0.208	−740.2 ± 0.4	34.7
15 per cent	0.208	−3113.0 ± 0.4	99.6

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.017	580.2 ± 0.4	208.7
2 per cent	0.071	455.7 ± 0.4	98.1
5 per cent	0.169	164.4 ± 0.3	2.4
10 per cent	0.208	−740.2 ± 0.4	34.7
15 per cent	0.208	−3113.0 ± 0.4	99.6

Table 6.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: two time-steps [t₀, t₁] with deviation in ionospheric parameters, the foreground signal is modelled using two corresponding beams. Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.017	580.2 ± 0.4	208.7
2 per cent	0.071	455.7 ± 0.4	98.1
5 per cent	0.169	164.4 ± 0.3	2.4
10 per cent	0.208	−740.2 ± 0.4	34.7
15 per cent	0.208	−3113.0 ± 0.4	99.6

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.017	580.2 ± 0.4	208.7
2 per cent	0.071	455.7 ± 0.4	98.1
5 per cent	0.169	164.4 ± 0.3	2.4
10 per cent	0.208	−740.2 ± 0.4	34.7
15 per cent	0.208	−3113.0 ± 0.4	99.6

4.3.2 Three time-steps, foregrounds modelled using one beam and three corresponding beams

Table 8 shows the log evidences of the three time-step cases with different fixed deviations in ionospheric parameters when the foreground signal is modelled using three corresponding beams. The evidences of the lower deviation cases improve significantly with respect to the cases in which the foreground signal is modelled using only one beam.

Table 7.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: three time-steps [t₀, t₁, t₂] with deviation in ionospheric parameters, the foreground signal is modelled using one single beam (⁠|$B_{t_0}$|⁠). Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.082	−4723.2 ± 0.4	191.4
2 per cent	0.092	−5426.3 ± 0.4	131.5
5 per cent	0.099	−6803.2 ± 0.4	44.3
10 per cent	0.205	−9987.9 ± 0.4	15.4
15 per cent	0.211	−15659.0 ± 0.4	121.9

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.082	−4723.2 ± 0.4	191.4
2 per cent	0.092	−5426.3 ± 0.4	131.5
5 per cent	0.099	−6803.2 ± 0.4	44.3
10 per cent	0.205	−9987.9 ± 0.4	15.4
15 per cent	0.211	−15659.0 ± 0.4	121.9

Table 7.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: three time-steps [t₀, t₁, t₂] with deviation in ionospheric parameters, the foreground signal is modelled using one single beam (⁠|$B_{t_0}$|⁠). Case (C_i) gives the explored cases. [t] lists all the time-steps analysed in each case. RMSE is the root-mean-square error of the detected signal with respect to the injected signal. |$\log \mathcal {Z}_{\mathrm{s}}$| is the evidence yielded by the foreground model including a Gaussian signal model, and |$\Delta \log \mathcal {Z}$| is the difference between the evidence yielded by the foreground model including a Gaussian signal model and the one without.

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.082	−4723.2 ± 0.4	191.4
2 per cent	0.092	−5426.3 ± 0.4	131.5
5 per cent	0.099	−6803.2 ± 0.4	44.3
10 per cent	0.205	−9987.9 ± 0.4	15.4
15 per cent	0.211	−15659.0 ± 0.4	121.9

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.082	−4723.2 ± 0.4	191.4
2 per cent	0.092	−5426.3 ± 0.4	131.5
5 per cent	0.099	−6803.2 ± 0.4	44.3
10 per cent	0.205	−9987.9 ± 0.4	15.4
15 per cent	0.211	−15659.0 ± 0.4	121.9

Table 8.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: three time-steps [t₀, t₁, t₂] with deviation in ionospheric parameters, foregrounds modelled using three corresponding beams.

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.021	904.4 ± 0.4	1991.5
2 per cent	0.071	609.2 ± 0.4	2038.0
5 per cent	0.188	101.9 ± 0.4	2300.2
10 per cent	0.208	−1919.3 ± 0.4	2669.7
15 per cent	0.209	−6977.1 ± 0.4	210.4

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.021	904.4 ± 0.4	1991.5
2 per cent	0.071	609.2 ± 0.4	2038.0
5 per cent	0.188	101.9 ± 0.4	2300.2
10 per cent	0.208	−1919.3 ± 0.4	2669.7
15 per cent	0.209	−6977.1 ± 0.4	210.4

Table 8.

RMSE and log evidence |$\log \mathcal {Z}$|⁠: three time-steps [t₀, t₁, t₂] with deviation in ionospheric parameters, foregrounds modelled using three corresponding beams.

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.021	904.4 ± 0.4	1991.5
2 per cent	0.071	609.2 ± 0.4	2038.0
5 per cent	0.188	101.9 ± 0.4	2300.2
10 per cent	0.208	−1919.3 ± 0.4	2669.7
15 per cent	0.209	−6977.1 ± 0.4	210.4

Deviation	RMSE (K)	\|$\log \mathcal {Z}_{\mathrm{s}}$\|	\|$\Delta \log \mathcal {Z}$\|
0 per cent	0.021	904.4 ± 0.4	1991.5
2 per cent	0.071	609.2 ± 0.4	2038.0
5 per cent	0.188	101.9 ± 0.4	2300.2
10 per cent	0.208	−1919.3 ± 0.4	2669.7
15 per cent	0.209	−6977.1 ± 0.4	210.4

Like Section 4.3.1, where foregrounds are modelled using either one beam or three corresponding beams, both show reliable detection until the deviation reaches 10 per cent. The RMSE and |$\Delta \log {\mathcal {Z}}$| with respect to the deviation applied is shown in lower panels of Fig. 7.

5 CONCLUSIONS

In this paper, we study how the time-varying chromatic ionospheric effects could affect the detection of the global 21-cm line using the existing data analysis pipeline. We explore cases including multiple time-steps during observation time to explore time variability of the ionosphere to study how the chromatic ionospheric effects on the foregrounds affects the signal detection using the REACH data analysis pipeline. We analyse and compare the different cases in terms of Bayesian log evidence and RMSE of the detected signal. We assume that the antenna beam pattern is distorted by the chromatic ionospheric effects and thus it changes with time depending on the ionospheric condition. In actual observation, however, the antenna does not change. To determine how to effectively achieve signal detection, we model the foregrounds using different number of beams to see if the pipeline can still achieve significant detection using fewer beams. We also study to what degree of error the pipeline can still manage a detection for different precision in the ionospheric parameters we use. The results can be summarized by the following points:

Signal detection does not necessarily improve when more time-steps are analysed together using the foreground model fitted with only one beam.
In the case where the number of corresponding beams used to fit the foregrounds is lower than the number of time-steps, signal detection could improve with lower RMSE and higher evidences, when all its two time-step subgroups yield good enough signal detection in the first place. Nevertheless, for cases whose two time-step subgroups yield less significant detection, it is still possible to yield poorer yet statistically significant detection.
Using more corresponding beams at different time-steps when modelling the foregrounds greatly improves signal detection.
Less than 5 per cent error in our knowledge of the ionospheric parameters can lead to unsuccessful detection of the injected global 21-cm signal.

We find that the global signal can be robustly detected if the uncertainty in ionospheric modelling is below 5 per cent. In multiple time-step cases where the ionosphere does not differ too much at different time-steps, and when our knowledge of the ionospheric condition is good, signal detection can be significant. However, if the ionospheric condition differs drastically between the selected time-steps, the current data analysis pipeline might not be able to achieve statistically meaningful detection without changing the beam configuration with time.

ACKNOWLEDGEMENTS

ES thanks Cambridge Trust and Taiwan Ministry of Education for their support. DA is supported by STFC and EdLA is supported by STFC Ernest Rutherford Fellowship. AF is supported by the Royal Society University Research Fellowship.

DATA AVAILABILITY

Two sets of simulation data (integrated antenna temperature) for reference are available at doi:10.5281/zenodo.4456385. The ionospheric data collected from Lowell GIRO Data Center are available at https://ulcar.uml.edu/DIDBase/.

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