Epoch Of Reionization Signal Research Articles

21-cm signal from the Epoch of Reionization: a machine learning upgrade to foreground removal with Gaussian process regression

ABSTRACT In recent years, a Gaussian process regression (GPR)-based framework has been developed for foreground mitigation from data collected by the LOw-Frequency ARray (LOFAR), to measure the 21-cm signal power spectrum from the Epoch of Reionization (EoR) and cosmic dawn. However, it has been noted that through this method there can be a significant amount of signal loss if the EoR signal covariance is misestimated. To obtain better covariance models, we propose to use a kernel trained on the grizzly simulations using a Variational Auto-Encoder (VAE)-based algorithm. In this work, we explore the abilities of this machine learning-based kernel (VAE kernel) used with GPR, by testing it on mock signals from a variety of simulations, exploring noise levels corresponding to ≈10 nights (≈141 h) and ≈100 nights (≈1410 h) of observations with LOFAR. Our work suggests the possibility of successful extraction of the 21-cm signal within 2σ uncertainty in most cases using the VAE kernel, with better recovery of both shape and power than with previously used covariance models. We also explore the role of the excess noise component identified in past applications of GPR and additionally analyse the possibility of redshift dependence on the performance of the VAE kernel. The latter allows us to prepare for future LOFAR observations at a range of redshifts, as well as compare with results from other telescopes.

Search for the Epoch of Reionization with HERA: upper limits on the closure phase delay power spectrum

ABSTRACT Radio interferometers aiming to measure the power spectrum of the redshifted 21 cm line during the Epoch of Reionization (EoR) need to achieve an unprecedented dynamic range to separate the weak signal from overwhelming foreground emissions. Calibration inaccuracies can compromise the sensitivity of these measurements to the effect that a detection of the EoR is precluded. An alternative to standard analysis techniques makes use of the closure phase, which allows one to bypass antenna-based direction-independent calibration. Similarly to standard approaches, we use a delay spectrum technique to search for the EoR signal. Using 94 nights of data observed with Phase I of the Hydrogen Epoch of Reionization Array (HERA), we place approximate constraints on the 21 cm power spectrum at z = 7.7. We find at 95 per cent confidence that the 21 cm EoR brightness temperature is ≤(372)2 ‘pseudo’ mK2 at 1.14 ‘pseudo’ h Mpc−1, where the ‘pseudo’ emphasizes that these limits are to be interpreted as approximations to the actual distance scales and brightness temperatures. Using a fiducial EoR model, we demonstrate the feasibility of detecting the EoR with the full array. Compared to standard methods, the closure phase processing is relatively simple, thereby providing an important independent check on results derived using visibility intensities, or related.

The Impact of Beam Variations on Power Spectrum Estimation for 21 cm Cosmology. I. Simulations of Foreground Contamination for HERA

Detecting cosmological signals from the Epoch of Reionization (EoR) requires high-precision calibration to isolate the cosmological signals from foreground emission. In radio interferometry, the perturbed primary beams of antenna elements can disrupt the precise calibration, which results in the contamination of the foreground-free region, or the EoR window, in the cylindrically averaged power spectrum. For the Hydrogen Epoch of Reionization Array (HERA), we simulate and characterize the perturbed primary beams that are induced by feed motions, such as axial, lateral, and tilting motions, above the 14 m dish. To understand the effect of the perturbed beams, visibility measurements are modeled with two different foreground components, point sources and diffuse sources, and we find that different feed motions present a different reaction to each type of sky source. HERA’s redundant baseline calibration in the presence of nonredundant antenna beams due to feed motions introduces chromatic errors in the gain solutions, producing foreground power leakage into the EoR window. The observed leakage from the vertical feed motions comes predominantly from point sources around the zenith. Furthermore, the observed leakage from the horizontal and tilting feed motions comes predominantly from the diffuse components near the horizon. Mitigation of the chromatic gain errors will be necessary for robust detections of the EoR signals with minimal foreground bias, and this will be discussed in a subsequent paper.

External calibrator in global signal experiment for detection of the epoch of reionization

We present a conceptual design study of external calibrators in the 21 cm experiment towards detecting the globally averaged radiation of the epoch of reionization (EoR). Employment of external calibrator instead of internal calibrator commonly used in current EoR experiments allows removing instrumental effects such as beam pattern, receiver gain and instability of the system if the conventional three-position switch measurements are implemented in a short time interval. Furthermore, in the new design the antenna system is placed in an underground anechoic chamber with an open/closing ceiling to maximally reduce the environmental effect such as RFI and ground radiation/reflection. It appears that three of the four external calibrators proposed in this paper, including two indoor artificial transmitters and one outdoor celestial radiation (the Galactic polarization), fail to meet our purpose. Diurnal motion of the Galactic diffuse emission turns out to be the most probable source as an external calibrator, for which we have discussed the observational strategy and the algorithm of extracting the EoR signal.

Recovering the wedge modes lost to 21-cm foregrounds

ABSTRACT One of the critical challenges facing imaging studies of the 21-cm signal at the Epoch of Reionization (EoR) is the separation of astrophysical foreground contamination. These foregrounds are known to lie in a wedge-shaped region of (k⊥, k∥) Fourier space. Removing these Fourier modes excises the foregrounds at grave expense to image fidelity, since the cosmological information at these modes is also removed by the wedge filter. However, the 21-cm EoR signal is non-Gaussian, meaning that the lost wedge modes are correlated to the surviving modes by some covariance matrix. We have developed a machine learning-based method that exploits this information to identify ionized regions within a wedge-filtered image. Our method reliably identifies the largest ionized regions and can reconstruct their shape, size, and location within an image. We further demonstrate that our method remains viable when instrumental effects are accounted for, using the Hydrogen EoR Array and the Square Kilometre Array as fiducial instruments. The ability to recover spatial information from wedge-filtered images unlocks the potential for imaging studies using current- and next-generation instruments without relying on detailed models of the astrophysical foregrounds themselves.

Extracting the Optical Depth to Reionization τ from 21 cm Data Using Machine Learning Techniques

Upcoming measurements of the high-redshift 21 cm signal from the Epoch of Reionization (EoR) are a promising probe of the astrophysics of the first galaxies and of cosmological parameters. In particular, the optical depth τ to the last scattering surface of the cosmic microwave background (CMB) should be tightly constrained by direct measurements of the neutral hydrogen state at high redshift. A robust measurement of τ from 21 cm data would help eliminate it as a nuisance parameter from CMB estimates of cosmological parameters. Previous proposals for extracting τ from future 21 cm data sets have typically used the 21 cm power spectra generated by semi-numerical models to reconstruct the reionization history. We present here a different approach which uses convolution neural networks (CNNs) trained on mock images of the 21 cm EoR signal to extract τ. We construct a CNN that improves upon on previously proposed architectures, and perform an automated hyperparameter optimization. We show that well-trained CNNs are able to accurately predict τ, even when removing Fourier modes that are expected to be corrupted by bright foreground contamination of the 21 cm signal. Typical random errors for an optimized network are less than 3.06%, with biases factors of several smaller. While preliminary, this approach could yield constraints on τ that improve upon sample-variance limited measurements of the low-ℓ EE observations of the CMB, making this approach a valuable complement to more traditional methods of inferring τ.

Gaussian process foreground subtraction and power spectrum estimation for 21 cm cosmology

ABSTRACT One of the primary challenges in enabling the scientific potential of 21 cm intensity mapping at the epoch of reionization (EoR) is the separation of astrophysical foreground contamination. Recent works have claimed that Gaussian process regression (GPR) can robustly perform this separation, particularly at low Fourier k wavenumbers where the EoR signal reaches its peak signal-to-noise ratio. We revisit this topic by casting GPR foreground subtraction (GPR-FS) into the quadratic estimator formalism, thereby putting its statistical properties on stronger theoretical footing. We find that GPR-FS can distort the window functions at these low k modes, which, without proper decorrelation, make it difficult to probe the EoR power spectrum. Incidentally, we also show that GPR-FS is in fact closely related to the widely studied inverse covariance weighting of the optimal quadratic estimator. As a case study, we look at recent power spectrum upper limits from the Low-Frequency Array (LOFAR) that utilized GPR-FS. We pay close attention to their normalization scheme, showing that it is particularly sensitive to signal loss when the EoR covariance is misestimated. This has possible ramifications for recent astrophysical interpretations of the LOFAR limits, because many of the EoR models ruled out do not fall within the bounds of the covariance models explored by LOFAR. Being more robust to this bias, we conclude that the quadratic estimator is a more natural framework for implementing GPR-FS and computing the 21 cm power spectrum.

Understanding the HERA Phase I receiver system with simulations and its impact on the detectability of the EoR delay power spectrum

ABSTRACT The detection of the Epoch of Reionization (EoR) delay power spectrum using a ‘foreground avoidance method’ highly depends on the instrument chromaticity. The systematic effects induced by the radio telescope spread the foreground signal in the delay domain, which contaminates the EoR window theoretically observable. Applied to the Hydrogen Epoch of Reionization Array (HERA), this paper combines detailed electromagnetic and electrical simulations in order to model the chromatic effects of the instrument, and quantify its frequency and time responses. In particular, the effects of the analogue receiver, transmission cables, and mutual coupling are included. These simulations are able to accurately predict the intensity of the reflections occurring in the 150-m cable which links the antenna to the backend. They also show that electromagnetic waves can propagate from one dish to another one through large sections of the array due to mutual coupling. The simulated system time response is attenuated by a factor 104 after a characteristic delay which depends on the size of the array and on the antenna position. Ultimately, the system response is attenuated by a factor 105 after 1400 ns because of the reflections in the cable, which corresponds to characterizable k∥-modes above 0.7 $h\,\,\rm {Mpc}^{-1}$ at 150 MHz. Thus, this new study shows that the detection of the EoR signal with HERA Phase I will be more challenging than expected. On the other hand, it improves our understanding of the telescope, which is essential to mitigate the instrument chromaticity.

Quantifying excess power from radio frequency interference in Epoch of Reionization measurements

ABSTRACT We quantify the effect of radio frequency interference (RFI) on measurements of the 21-cm power spectrum during the Epoch of Reionization (EoR). Specifically, we investigate how the frequency structure of RFI source emission generates contamination in higher order wave modes, which is much more problematic than smooth-spectrum foreground sources. Using a relatively optimistic EoR model, we find that even a single relatively dim RFI source can overwhelm the EoR power spectrum signal of $\sim 10\, {\rm mK}^2$ for modes $0.1 \ \lt k \lt 2 \, h\, {\rm Mpc}^{-1}$. If the total apparent RFI flux density in the final power spectrum integration is kept below 1 mJy, an EoR signal resembling this optimistic model should be detectable for modes $k \lt 0.9\, h\, {\rm Mpc}^{-1}$, given no other systematic contaminants and an error tolerance as high as 10 per cent. More pessimistic models will be more restrictive. These results emphasize the need for highly effective RFI mitigation strategies for telescopes used to search for the EoR.

A Monte Carlo Implementation of Galactic Free–Free Emission for the EoR Foreground Models

Abstract The overwhelming foreground causes severe contamination on the detection of 21 cm signal during the Epoch of Reionization (EoR). Among various foreground components, the Galactic free–free emission is less studied, so that its impact on the EoR observation remains unclear. To better constrain this emission, we perform Monte Carlo simulation of Hα emission, which comprises direct and scattered Hα radiation from H ii regions and warm ionized medium (WIM). The positions and radii of H ii regions are quoted from the Wide-Field Infrared Survey Explorer H ii catalog, and the WIM is described by an axisymmetric model. The scattering is off dust and free electrons that are realized by applying an exponential fitting to the HI4PI H i map and an exponential disk model, respectively. The simulated Hα intensity, the Simfast21 software, and the latest SKA1-Low layout configuration are employed to simulate the SKA “observed” images of Galactic free–free emission and the EoR signal. By analyzing the one-dimensional power spectra, we find that the Galactic free–free emission can be about 105.4–102.1, 105.0–101.7, and 104.3–101.1 times more luminous than the EoR signal on scales of in the 116–124, 146–154, and 186–194 MHz frequency bands, respectively. We further calculate the two-dimensional power spectra inside the EoR window and show that the power leaked by Galactic free–free emission can still be significant, as the power ratios can reach about 110%–8000%, 30%–2400%, and 10%–250% on scales of in three frequency bands. Therefore, we indicate that the Galactic free–free emission should be carefully treated in future EoR detections.

Contribution of Galactic free–free emission to the foreground for EoR signal in SKA experiments

ABSTRACT The overwhelming foreground contamination hinders the accurate detection of the 21-cm signal of neutral hydrogen during the Epoch of Reionization (EoR). Among various foreground components, the Galactic free–free emission is less studied, so that its impact on the EoR observations remains unclear. In this work, we employ the observed H α intensity map with the correction of dust absorption and scattering, the simfast21 software, and the latest SKA1-Low layout configuration to simulate the SKA ‘observed’ images of Galactic free–free emission and the EoR signal. By calculating the 1D power spectra from the simulated image cubes, we find that the Galactic free–free emission is about 103.5–102.0, 103.0–101.3, and 102.5–101.0 times more luminous than the EoR signal on scales of 0.1 Mpc−1 < k < 2 Mpc−1 in the 116–124, 146–154, and 186–194 MHz frequency bands. We further analyse the 2D power spectra inside the properly defined EoR window and find that the leaked Galactic free–free emission can still cause non-negligible contamination, as the ratios of its power (amplitude squared) to the EoR signal power can reach about 200, 60, and 15 per cent on scales of k ∼ 1.2 Mpc−1 in three frequency bands, respectively. Therefore, we conclude that the Galactic free–free emission, as a severe contaminating foreground component, needs to be carefully treated in the forthcoming deep EoR observations.

Quantifying EoR delay spectrum contamination from diffuse radio emission

ABSTRACT The 21 cm hyperfine transition of neutral hydrogen offers a promising probe of the large-scale structure of the universe before and during the Epoch of Reionization (EoR), when the first ionizing sources formed. Bright radio emission from foreground sources remains the biggest obstacle to detecting the faint 21 cm signal. However, the expected smoothness of foreground power leaves a clean window in Fourier space where the EoR signal can potentially be seen over thermal noise. Though the boundary of this window is well defined in principle, spectral structure in foreground sources, instrumental chromaticity, and choice of spectral weighting in analysis all affect how much foreground power spills over into the EoR window. In this paper, we run a suite of numerical simulations of wide-field visibility measurements, with a variety of diffuse foreground models and instrument configurations, and measure the extent of contaminated Fourier modes in the EoR window using a delay-transform approach to estimate power spectra. We also test these effects with a model of the Hydrogen Epoch of Reionization Array (HERA) antenna beam generated from electromagnetic simulations, to take into account further chromatic effects in the real instrument. We find that foreground power spillover is dominated by the so-called pitchfork effect, in which diffuse foreground power is brightened near the horizon due to the shortening of baselines. As a result, the extent of contaminated modes in the EoR window is largely constant over time, except when the Galaxy is near the pointing centre.

The Impact of Realistic Foreground and Instrument Models on 21 cm Epoch of Reionization Experiments

Abstract Predictions for the ability of 21 cm interferometric experiments to discriminate Epoch of Reionization (EoR) signal models are typically limited by the simplicity of data models, whereby foreground signals and characteristics of the instrument are often simplified or neglected. To move toward more realistic scenarios, we explore the effects of applying more realistic foreground and instrument models to the 21 cm signal, as well as the ability to estimate astrophysical parameters with these additional complexities. We use a highly optimized version of 21cmFAST, integrated into 21cmMC, to generate lightcones of the brightness temperature fluctuation for Bayesian parameter estimation. We include a statistical point-source foreground model and an instrument model based on the Murchison Widefield Array scaled in observation time to have an effective sensitivity similar to the future Square Kilometre Array. We also extend the current likelihood prescription to account for the presence of beam convolution and foregrounds, the two-dimensional power spectrum (PS), and the correlation of PS modes. We use frequency bands between 150 and 180 MHz to constrain the ionizing efficiency (ζ), the minimum virial temperature of halos ( ), the soft X-ray emissivity per unit star formation rate (SFR) (L X /SFR), and the X-ray energy threshold (E 0). We find that the inclusion of realistic foregrounds and instrumental components biases the parameter constraints, due to unaccounted for cross-power between the EoR signal, foregrounds, and thermal noise. This causes estimates of ζ to be biased by up to 5σ, but the estimates of T vir, L X /SFR and E 0 remain unaffected and are all within 1σ.

Calibration and 21-cm power spectrum estimation in the presence of antenna beam variations

ABSTRACT Detecting a signal from the Epoch of Reionization (EoR) requires an exquisite understanding of Galactic and extragalactic foregrounds, low-frequency radio instruments, instrumental calibration, and data analysis pipelines. In this work, we build upon existing work that aims to understand the impact of calibration errors on 21-cm power spectrum (PS) measurements. It is well established that calibration errors have the potential to inhibit EoR detections by introducing additional spectral features that mimic the structure of EoR signals. We present a straightforward way to estimate the impact of a wide variety of modelling residuals in EoR PS estimation. We apply this framework to the specific case of broken dipoles in Murchison Widefield Array (MWA) to understand its effect and estimate its impact on PS estimation. Combining an estimate of the percentage of MWA tiles that have at least one broken dipole (15–40 per cent) with an analytic description of beam errors induced by such dipoles, we compute the residuals of the foregrounds after calibration and source subtraction. We find that that incorrect beam modelling introduces bias in the 2D-PS on the order of $\sim 10^3\, \mathrm{mK}^2 \, h^{-3}\, \mathrm{Mpc}^{3}$. Although this is three orders of magnitude lower than current lowest limits, it is two orders of magnitude higher than the expected signal. Determining the accuracy of both current beam models and direction-dependent calibration pipelines is therefore crucial in our search for an EoR signal.

Joint estimation of the Epoch of Reionization power spectrum and foregrounds

ABSTRACTThe power spectrum of redshifted 21 cm emission brightness temperature fluctuations is a powerful probe of the Epoch of Reionization (EoR). However, bright foreground emission presents a significant impediment to its unbiased recovery from interferometric data. We estimate the power spectrum within a Bayesian framework and demonstrate that incorporating a priori knowledge of the spectral structure of foregrounds in the large spectral scale component of the data model enables significantly improved modelling of the foregrounds without increasing the model complexity. We explore two astrophysically motivated parametrizations of the large spectral scale model: (i) a constant plus power-law model of the form $q_{0}+q_{1}(\nu /\nu _{0})^{b_{1}}$ for two values of b1: b1 = 〈β〉GDSE and b1 = 〈β〉EGS, the mean spectral indices of the Galactic diffuse synchrotron emission and extragalactic source foreground emission, respectively; and (ii) a constant plus double power-law model of the form $q_{0}+q_{1}(\nu /\nu _{0})^{b_{1}}+q_{2}(\nu /\nu _{0})^{b_{2}}$ with b1 = 〈β〉GDSE and b2 = 〈β〉EGS. We estimate the EoR power spectrum from simulated interferometric data consisting of an EoR signal, Galactic diffuse synchrotron emission, extragalactic sources, and diffuse free–free emission from the Galaxy. We show that, by jointly estimating a model of the EoR signal with the constant plus double power-law parametrization of the large spectral scale model, unbiased estimates of the EoR power spectrum are recoverable on all spatial scales accessible in the data set, including on the large spatial scales that were found to be contaminated in earlier work.

Contribution of Radio Halos to the Foreground for SKA EoR Experiments

The overwhelming foreground contamination is one of the primary impediments to probing the Epoch of Reionization (EoR) through measuring the redshifted 21 cm signal. Among various foreground components, radio halos are less studied and their impacts on the EoR observations are still poorly understood. In this work, we employ the Press-Schechter formalism, merger-induced turbulent re-acceleration model, and the latest SKA1-Low layout configuration to simulate the SKA "observed" images of radio halos. We calculate the one-dimensional power spectra from simulated images and find that radio halos can be about $10^4$, $10^3$ and $10^{2.5}$ times more luminous than the EoR signal on scales of $0.1\,\text{Mpc}^{-1} < k < 2\,\text{Mpc}^{-1}$ in the 120-128, 154-162, and 192-200 MHz bands, respectively. By examining the two-dimensional power spectra inside properly defined EoR windows, we find that the power leaked by radio halos can still be significant, as the power ratios of radio halos to the EoR signal on scales of $0.5\,\text{Mpc}^{-1} \lesssim k \lesssim 1\,\text{Mpc}^{-1}$ can be up to about 230-800%, 18-95%, and 7-40% in the three bands, when the 68% uncertainties caused by the variation of the number density of bright radio halos are considered. Furthermore, we find that radio halos located inside the far side-lobes of the station beam can also impose strong contamination within the EoR window. In conclusion, we argue that radio halos are severe foreground sources and need serious treatments in future EoR experiments.

Multi-component Decomposition of Cosmic Infrared Background Fluctuations

Abstract The near-infrared background between 0.5 and 2 μm contains a wealth of information related to radiative processes in the universe. Infrared background anisotropies encode the redshift-weighted total emission over cosmic history, including any spatially diffuse and extended contributions. The anisotropy power spectrum is dominated by undetected galaxies at small angular scales and a diffuse background of Galactic emission at large angular scales. In addition to these known sources, the infrared background also arises from intrahalo light (IHL) at z &lt; 3 associated with tidally stripped stars during galaxy mergers. Moreover, it contains information on the very first galaxies from the epoch of reionization (EoR). The EoR signal has a spectral energy distribution (SED) that goes to zero near optical wavelengths due to Lyman absorption, while other signals have spectra that vary smoothly with frequency. Due to differences in SEDs and spatial clustering, these components may be separated in a multi-wavelength-fluctuation experiment. To study the extent to which EoR fluctuations can be separated in the presence of IHL, and extragalactic and Galactic foregrounds, we develop a maximum likelihood technique that incorporates a full covariance matrix among all the frequencies at different angular scales. We apply this technique to simulated deep imaging data over a 2 × 45 deg2 sky area from 0.75 to 5 μm in 9 bands and find that such a “frequency tomography” can successfully reconstruct both the amplitude and spectral shape for representative EoR, IHL, and the foreground signals.

Fundamental Limitations on the Calibration of Redundant 21 cm Cosmology Instruments and Implications for HERA and the SKA

Abstract Precise instrument calibration is critical to the success of 21 cm cosmology experiments. Unmitigated errors in calibration contaminate the Epoch of Reionization (EoR) signal, precluding a detection. Barry et al. characterized one class of inherent errors that emerge from calibrating to an incomplete sky model; however, it has been unclear if errors in the sky model affect the calibration of redundant arrays. In this paper, we show that redundant calibration is vulnerable to errors from sky model incompleteness even in the limit of perfect antenna positioning and identical beams. These errors are at a level that can overwhelm the EoR signal and prevent a detection in crucial power spectrum modes. Finally, we suggest error mitigation strategies with implications for the Hydrogen Epoch of Reionization Array and the Square Kilometre Array.

Separating the EoR signal with a convolutional denoising autoencoder: a deep-learning-based method

When applying the foreground removal methods to uncover the faint cosmological signal from the epoch of reionization (EoR), the foreground spectra are assumed to be smooth. However, this assumption can be seriously violated in practice since the unresolved or mis-subtracted foreground sources, which are further complicated by the frequency-dependent beam effects of interferometers, will generate significant fluctuations along the frequency dimension. To address this issue, we propose a novel deep-learning-based method that uses a 9-layer convolutional denoising autoencoder (CDAE) to separate the EoR signal. After being trained on the SKA images simulated with realistic beam effects, the CDAE achieves excellent performance as the mean correlation coefficient ($\bar{\rho}$) between the reconstructed and input EoR signals reaches $0.929 \pm 0.045$. In comparison, the two representative traditional methods, namely the polynomial fitting method and the continuous wavelet transform method, both have difficulties in modelling and removing the foreground emission complicated with the beam effects, yielding only $\bar{\rho}_{\text{poly}} = 0.296 \pm 0.121$ and $\bar{\rho}_{\text{cwt}} = 0.198 \pm 0.160$, respectively. We conclude that, by hierarchically learning sophisticated features through multiple convolutional layers, the CDAE is a powerful tool that can be used to overcome the complicated beam effects and accurately separate the EoR signal. Our results also exhibit the great potential of deep-learning-based methods in future EoR experiments.

Bayesian power spectrum estimation at the Epoch of Reionization

We introduce a new method for performing robust Bayesian estimation of the three-dimensional spatial power spectrum at the Epoch of Reionization (EoR), from interferometric observations. The versatility of this technique allows us to present two approaches. First, when the observations span only a small number of independent spatial frequencies ($k$-modes) we sample directly from the spherical power spectrum coefficients that describe the EoR signal realisation. Second, when the number of $k$-modes to be included in the model becomes large, we sample from the joint probability density of the spherical power spectrum and the signal coefficients, using Hamiltonian Monte Carlo methods to explore this high dimensional ($\sim$ 20000) space efficiently. This approach has been successfully applied to simulated observations that include astrophysically realistic foregrounds in a companion publication (Sims et al. 2016). Here we focus on explaining the methodology in detail, and use simple foreground models to both demonstrate its efficacy, and highlight salient features. In particular, we show that including an arbitrary flat spectrum continuum foreground that is $10^8$ times greater in power than the EoR signal has no detectable impact on our parameter estimates of the EoR power spectrum recovered from the data.