Bibliography

This page documents the key literature on systematic effects mitigation in photometric galaxy surveys. For each reference, the main method(s), their mathematical formulation, pros/cons, and implementation status in sys_mapping are described.

Ross et al. 2011 

Ross, N. P., et al. (2011). Ameliorating Systematic Uncertainties in the Angular Clustering of Galaxies: A Study using the SDSS-III BOSS.
MNRAS 424(1): 564–586.
ADS · arXiv:1105.2320

Methods: Masking of stellar contamination, dust extinction corrections, sky background removal, and jack-knife covariance estimation for systematic null tests. Systematic corrections are applied by assigning per-pixel weights based on observational condition maps (seeing, airmass, star density, Galactic extinction).

Key equation — Landy-Szalay two-point estimator after systematic weighting:

\[w(\theta) = \frac{DD - 2\,DR + RR}{RR}\]

Null test: cross-correlation of the weight map with each template should be consistent with zero for a well-corrected field.

Jack-knife covariance:

\[\sigma_{\mathrm{jack}}^2(\theta) = \frac{N-1}{N} \sum_{i=1}^{N} \bigl[w(\theta) - w_i(\theta)\bigr]^2\]

where \(w_i\) is the estimator with patch \(i\) removed.

Pros and cons

Pros: Intuitive per-pixel weighting. Jack-knife uncertainties are conservative and capture spatial correlations. Multiple independent null tests can be run simultaneously.

Cons: Masking reduces effective area. Jack-knife requires many patches for the prefactor to be accurate. Correlations between systematics are not jointly modelled.

Implementation status in sys_mapping

Landy-Szalay estimator: implemented (utils.measure_two_point_function).
Jack-knife covariance: implemented (bootstrap.jackknife_covariance).
Null test cross-correlations: implemented (diagnostics.null_test_cross_correlations).

Ho et al. 2012 

Ho, S., et al. (2012). Clustering of Sloan Digital Sky Survey III photometric luminous galaxies: the measurement, systematics and cosmological implications.
ApJ 761(1): 14.
ADS · arXiv:1201.2137

Methods: Optimal quadratic estimator (QMV) for the galaxy angular power spectrum with systematic control. The covariance-weighted estimator minimises variance while accounting for mode-coupling from the survey mask. Systematic templates are handled via mode projection in harmonic space.

Key equations — optimal quadratic estimator and Fisher matrix:

\[\hat{C}_\ell^{ss} = \sum_i N_i^{-1}\, d^T\, E_i\, d, \qquad E_i = C^{-1}\,\frac{\partial C}{\partial C_i}\,C^{-1}\]

\[N_{\ell\ell'} = \frac{1}{2}\,\mathrm{tr}\!\left[ C^{-1}\frac{\partial C}{\partial C_\ell} C^{-1}\frac{\partial C}{\partial C_{\ell'}}\right]\]

Pros and cons

Pros: Statistically optimal (minimum variance). Rigorous treatment of data covariance including noise and mask coupling.

Cons: Requires explicit covariance matrix inversion — scales poorly to large pixel numbers. Systematic templates must be characterised a priori.

Implementation status in sys_mapping

Harmonic-space pseudo-\(C_\ell\) estimator (simpler, mask-based): implemented (power_spectrum.measure_pseudo_cl).
Full QMV optimal estimator: not planned (high computational cost; NaMaster/pymaster covers this use case).

Elsner, Leistedt & Peiris 2016 

Elsner, F., Leistedt, B., Peiris, H. V. (2016). Unbiased methods for removing systematics from galaxy clustering measurements.
MNRAS 456(2): 2095–2104.
ADS · arXiv:1509.08933

Methods: Three complementary harmonic-space approaches with closed-form bias expressions:

Template subtraction (TS): subtract \(\hat\alpha_i\,\tilde C_\ell^{t_i}\) from the measured pseudo-\(C_\ell\).
Basic mode projection (BMP): marginalise over template modes via a modified pixel covariance.
Extended mode projection (EMP): threshold-based exclusion of modes dominated by templates, with an analytical bias correction.

Key equations:

Cleaned power spectrum (TS):

\[\tilde C_\ell^{\rm TS} = \hat C_\ell^{d\times d} - \sum_i \hat\alpha_i\,\hat C_\ell^{t_i\times t_i}\]

Additive bias from template subtraction (\(n\) templates):

\[b_\ell = -\frac{n}{2\ell+1}\]

EMP bias (Eq. 27 of Elsner+16):

\[b_\ell = -\sqrt{\frac{2}{\pi}}\, \frac{k^2}{2(2\ell+1)}\, e^{k^2/(2C_\ell^{ss})}\, \mathrm{erfc}\!\left(\frac{k}{\sqrt{2C_\ell^{ss}}}\right)\]

Pros and cons

Pros: Closed-form bias corrections; analytically tractable. Template subtraction is fast. Mode projection does not require fitting contamination amplitudes.

Cons: Template subtraction biases scale as \(n/(2\ell+1)\) — significant at low \(\ell\) with many templates. EMP bias depends on signal-to-noise and threshold choice. Mixing from the survey mask breaks spherical harmonic orthogonality.

Implementation status in sys_mapping

Template subtraction in harmonic space: implemented (power_spectrum.subtract_template_cl).
Harmonic bias formula: implemented (power_spectrum.harmonic_bias).
Basic and extended mode projection: implemented (power_spectrum.mode_projection_bias).
Full correction pipeline: implemented (correction.correct_power_spectrum_harmonic).

Leistedt & Peiris 2014 

Leistedt, B., Peiris, H. V. (2014). Exploiting the full potential of photometric quasar surveys: optimal power spectra through blind mitigation of systematics.
MNRAS 444(1): 2–19.
ADS · arXiv:1404.6530

Methods: Introduces extended mode projection (EMP) as a blind systematic mitigation strategy. Templates are identified from the data themselves by iteratively projecting out the modes most correlated with external maps, without prior knowledge of contamination amplitudes.

Key idea: project a set of templates \(\{f_i\}\) out of the data vector before power spectrum estimation:

\[d_{\rm proj} = d - \sum_i (d \cdot f_i)\, f_i / (f_i \cdot f_i)\]

This is iterated until the cross-power spectrum of the cleaned field with each template is consistent with zero.

Pros and cons

Pros: Blind — no prior knowledge of contamination amplitudes needed. Naturally handles correlated templates.

Cons: Each projected template costs one mode per \(\ell\) band — significant variance penalty for large template sets. Power loss must be corrected via a transfer function.

Implementation status in sys_mapping

EMP basis: implemented as part of power_spectrum.mode_projection_bias.
Blind iterative variant: planned (future extension of power_spectrum).

Elsner, Leistedt & Peiris 2017 

Elsner, F., Leistedt, B., Peiris, H. V. (2017). Unbiased pseudo-Cℓ power spectrum estimation with mode projection.
MNRAS 465(2): 1847–1855.
ADS · arXiv:1609.03577

Methods: Integrates mode projection directly into the pseudo-\(C_\ell\) estimator, deriving exact closed-form expressions for the deprojection bias within the MASTER/PCL framework. Extends the 2016 work to handle arbitrary mask geometry and pixel weights.

Key result: the pseudo-\(C_\ell\) after EMP is related to the true power spectrum by a modified coupling matrix \(M_{\ell\ell'}^{\rm proj}\) that is efficiently computable from the mask.

Pros and cons

Pros: Provides exact bias correction in the pseudo-\(C_\ell\) framework. Suitable for non-Gaussian fields and realistic survey footprints.

Cons: Coupling matrix computation scales as \(O(\ell_{\max}^3)\); large surveys require optimised implementations (e.g. NaMaster).

Implementation status in sys_mapping

Coupling matrix computation: partially implemented via power_spectrum.mode_projection_bias (analytical approximation).
Full coupling matrix: deferred (requires NaMaster as optional backend).

Weaverdyck & Huterer 2021 

Weaverdyck, N., Huterer, D. (2021). Mitigating contamination in LSS surveys: a comparison of methods.
MNRAS 503(4): 5061–5084.
ADS · arXiv:2007.14499

Methods: Systematic comparison of four decontamination methods applied to the same simulated and real data:

DES-Y1 iterative method: multiplicative weights \(w = (1 + \alpha T)^{-1}\) iterated to convergence.
Template subtraction: subtract the cross-correlation component.
Mode projection: harmonic-space marginalisation.
ElasticNet regression: L1+L2 penalised linear regression.

ElasticNet loss:

\[\mathcal{L} = \frac{1}{2N_{\rm pix}} \left\|\delta_g - \sum_i\alpha_i\,t_i\right\|_2^2 + \lambda_1\|\boldsymbol\alpha\|_1 + \lambda_2\|\boldsymbol\alpha\|_2^2\]

Per-pixel weight from regression:

\[w(p) = \frac{1}{1 + \hat{\boldsymbol\alpha}\cdot\mathbf{t}(p)}\]

Pros and cons

Pros: ElasticNet naturally prevents overfitting and finds sparse solutions. The comparison framework reveals method-dependent biases. No iterative convergence issue for ElasticNet.

Cons: Regularisation hyperparameters (\(\lambda_1, \lambda_2\)) require tuning. No single method dominates in all scenarios.

Implementation status in sys_mapping

ElasticNet regression: implemented (regression.elasticnet_contamination_fit).
Iterative OLS: implemented (regression.iterative_systematics_decontamination).
Method comparison framework: implemented (regression.method_comparison).

Rezaie et al. 2020 

Rezaie, M., et al. (2020). Improving Galaxy Clustering Measurements with Deep Learning: analysis of the DECaLS DR7 data.
ApJS 253(2): 32.
arXiv:1907.11355

Methods: Artificial neural networks (ANN) trained to predict the galaxy density field from observational systematic maps (Galactic extinction, seeing, stellar density). The ratio of the predicted to mean density defines per-pixel weights. The method is model-free and captures non-linear systematic dependencies.

Weight definition:

\[w(p) = \frac{\bar n_g}{n_g^{\rm ANN}(p)}\]

where \(n_g^{\rm ANN}\) is the ANN-predicted galaxy count from systematic templates.

Pros and cons

Pros: Captures non-linear and cross-template systematic dependencies. No functional form assumed for contamination.

Cons: Prone to overfitting when templates are correlated with large-scale structure. Requires careful validation (e.g. cross-validation on disjoint sky regions). Less interpretable than linear methods.

Implementation status in sys_mapping

ANN-based systematic mapping: not planned (out of scope for the current Bayesian likelihood framework; recommended as external pre-processing step).

Alonso et al. 2019 — NaMaster 

Alonso, D., Sanchez, J., Slosar, A. (2019). A unified pseudo-Cℓ framework.
MNRAS 484(3): 4127–4151.
ADS · arXiv:1809.09603

Methods: NaMaster (pymaster) provides a unified pseudo-\(C_\ell\) estimator with: - Exact mask mode-coupling matrix computation. - E/B-mode purification for spin-2 fields. - Template deprojection in harmonic space. - Support for arbitrary pixelisation schemes (HEALPix and flat-sky).

MASTER equation:

\[\langle\tilde C_\ell\rangle = \sum_{\ell'} M_{\ell\ell'}\,C_{\ell'}\]

where \(M_{\ell\ell'}\) is the mode-coupling matrix computed from the mask power spectrum.

Pros and cons

Pros: Highly optimised coupling matrix computation. Standard reference implementation used by DES, DESI, Euclid.

Cons: External dependency; adds build complexity. Coupling matrix computation can be expensive for large \(\ell_{\max}\).

Implementation status in sys_mapping

measure_pseudo_cl uses healpy.anafast (no NaMaster required).
NaMaster: optional backend — planned as an optional dependency for exact coupling matrix computation in power_spectrum.mode_projection_bias.

Berlfein, Mandelbaum & Schafer 2024 

Berlfein, F., Mandelbaum, R., Dodelson, S., Schafer, C. (2024). Joint inference of multiplicative and additive systematics in galaxy density fluctuations and clustering measurements.
MNRAS 531: 4954–4974.
ADS · arXiv:2401.12293

Methods: The primary reference for sys_mapping. Defines three nested contamination models and a joint Bayesian MCMC framework for inferring their parameters.

Contamination model (combined):

\[\hat\delta_g(p) = \delta_g(p)\!\left(1 + \sum_i b_i\,\delta_{t,i}(p)\right) + \sum_i a_i\,\delta_{t,i}(p)\]

Gaussian log-likelihood (Eq. 17):

\[\ln\mathcal{L} = -\frac{N_{\rm pix}}{2}\ln(2\pi\sigma^2) - \frac{1}{2\sigma^2}\sum_p \delta_g(p)^2 + \sum_p \ln|J(p)|\]

Two-point function correction (Eq. 15–16):

\[\hat w_{\rm corr}(\theta) = \frac{\hat w(\theta) - \sum_i\tilde a_i^2\,\xi_i(\theta)} {1 + \sum_i\tilde b_i^2\,\xi_i(\theta)}\]

Noise debiasing (Eq. 21):

\[\tilde a_i^2 = \max\!\left(\hat a_i^2 - \mathrm{Var}[\hat a_i],\, 0\right)\]

Pros and cons

Pros: Joint treatment of additive and multiplicative systematics avoids double-counting. PCA template rotation decorrelates parameters. Noise debiasing removes variance inflation. Skew-normal likelihood handles lognormal galaxy fields.

Cons: MCMC is slower than regression approaches. Assumes Gaussian/skew-normal pixel overdensities. Linear contamination model may miss non-linear systematics.

Implementation status in sys_mapping

All core methods from Berlfein+2024 are fully implemented:

Forward/inverse contamination model: contamination
JAX JIT-compiled likelihoods: likelihood
emcee MCMC inference: inference
PCA template rotation + noise debiasing: correction
Two-point function correction: contamination.compute_two_point_correction
Likelihood ratio model selection: model_selection

Rodríguez-Monroy et al. 2025 

Rodríguez-Monroy, M., et al. (2025). Dark Energy Survey Year 6 Results: improved mitigation of spatially varying observational systematics with masking.
arXiv:2509.07943.
ADS · arXiv:2509.07943

Methods: Two complementary strategies applied to DES Y6 galaxy clustering:

Iterative Systematics Decontamination (ISD): OLS regression on polynomial expansions of templates up to 3rd order, iterated to convergence.
Footprint masking: conservative masking of survey regions with high systematic sensitivity, enabling simpler and more robust mitigation.

ISD cleansed overdensity:

\[\delta_g^{\rm clean}(p) = \frac{\delta_g(p) - f_{\rm add}(p)}{1 + f_{\rm mult}(p)}\]

where \(f_{\rm add}\) and \(f_{\rm mult}\) are the additive and multiplicative systematic components estimated from polynomial template regression.

Pros and cons

Pros: ISD captures non-linear systematic dependencies via polynomial template expansion. Footprint masking provides conservative, robust improvement.

Cons: Polynomial expansion grows combinatorially with template number and polynomial order. Masking reduces effective survey area.

Implementation status in sys_mapping

ISD (polynomial OLS iteration): implemented (regression.iterative_systematics_decontamination).
Footprint masking diagnostics: implemented (diagnostics.footprint_mask_diagnostics).

Cornish et al. 2026 

Cornish, T., Alonso, D., Leistedt, B., Wolz, K. (2026). Systematics mitigation for catalogue-based angular power spectra.
MNRAS 547.
ADS · arXiv:2510.19912

Methods: Extends template deprojection to work directly on discrete source catalogues (not pixelised maps), avoiding pixelisation-induced power leakage. Introduces a transfer function calibration loop to correct for power loss from deprojection.

Catalogue-based deprojection:

\[a_i^{\rm deproj} = a_i - \sum_p A_p\, f_i^p\]

Transfer function calibration:

\[T_f(\ell) = \frac{\langle C_\ell^{\rm cont}\rangle_{\rm sim}} {\langle C_\ell^{\rm before}\rangle_{\rm sim}}\]

Pros and cons

Pros: Avoids pixelisation bias. Transfer function corrects for deprojection power loss. Exact treatment for discrete catalogues.

Cons: Transfer function calibration requires simulations (expensive). Mode-coupling increases in complexity for catalogue-based fields.

Implementation status in sys_mapping

Catalogue-based deprojection: deferred (planned as deprojection module once power_spectrum is validated).
Transfer function calibration: deferred.

Tanidis et al. 2026 

Tanidis, K., Alonso, D., Miller, L., Harnois-Déraps, J. (2026). Reconstructing spatially-varying multiplicative bias for Stage IV weak lensing galaxy surveys with a quadratic estimator.
MNRAS 547.
ADS · arXiv:2512.13679

Methods: Quadratic estimator exploiting E/B mode coupling to reconstruct spatially-varying multiplicative bias \(m(\hat n)\) in weak lensing surveys. Three signal-to-noise ratio definitions are introduced for template detection and ranking.

Quadratic estimator:

\[\hat m(\mathbf{L}) = N(\mathbf{L})^{-1} \int\frac{d^2\ell}{(2\pi)^2}\, E^{\rm obs}(\boldsymbol\ell)\, B^{\rm obs}(\boldsymbol\ell - \mathbf{L})\, W(\boldsymbol\ell, \boldsymbol\ell-\mathbf{L})\]

Three SNR definitions:

\[\mathrm{SNR}_{\rm template} = \frac{|\hat\alpha_i|}{\sigma_{\hat\alpha_i}}, \quad \mathrm{SNR}_{\rm data} = |\mathrm{Corr}(\delta_g, t_i)|, \quad \mathrm{SNR}_{\rm peak} = \frac{\max_\ell \hat C_\ell^{\delta_g t_i}}{\sigma_\ell}\]

Pros and cons

Pros: Detects percent-level multiplicative bias at high significance for Stage IV surveys. Mode-coupling signal is insensitive to additive (c-bias) contamination.

Cons: Primarily developed for weak lensing shear fields; adaptation to galaxy density requires modification. Requires knowledge of the cosmic shear power spectrum.

Implementation status in sys_mapping

SNR-based template ranking (all three definitions adapted for galaxy clustering): implemented (diagnostics.snr_template_ranking).
Full quadratic estimator for shear E/B coupling: not planned (specific to weak lensing; outside the scope of galaxy density maps).