SimBAL: Bayesian Modeling for BAL Quasar Spectra

• SimBAL is a forward-modeling framework for broad absorption line quasar spectroscopy that uses Cloudy-generated ionic column grids and Bayesian MCMC analysis.
• It integrates photoionization calculations with synthetic spectral fitting to overcome challenges of blended, saturated, and partially covered absorption features.
• The approach employs velocity-resolved tophat and power-law partial-covering models to constrain ionization, density, kinematics, and outflow energetics.

Searching arXiv for papers on SimBAL and its later applications. arxiv_search(query="SimBAL quasar Leighly broad absorption line", max_results=10, sort_by="relevance") SimBAL is a spectral-synthesis and forward-modeling framework for broad absorption line (BAL) quasar spectroscopy that infers physical outflow properties by comparing synthetic spectra, generated from photoionization calculations, directly to observed data. It was introduced to address regimes in which conventional apparent-optical-depth or line-by-line analyses become unreliable, especially when BAL troughs are broad, blended, saturated, and affected by partial covering (Leighly et al., 2018). In SimBAL, grids of ionic column densities computed with Cloudy are mapped into synthetic absorption spectra and fit in a Bayesian framework, typically with MCMC, yielding constraints on ionization parameter, density, column density, kinematics, and inhomogeneous covering structure as functions of velocity (Leighly et al., 2018). Subsequent work has extended the method to multiwavelength covering-fraction tests in LoBAL quasars (Leighly et al., 2018), highly blended FeLoBAL and overlapping-trough systems (Choi et al., 2020, Choi et al., 2022), multi-epoch variability analyses (Green et al., 2024), high-redshift feedback studies (Bischetti et al., 2024), and extremely high-velocity outflows (Hidalgo et al., 19 Aug 2025).

1. Conceptual basis and motivation

SimBAL was introduced as a reversal of the standard BAL-analysis workflow. Rather than first measuring apparent optical depths or fitting individual troughs and only then comparing those measurements to photoionization models, it starts from photoionization models, constructs synthetic spectra directly, and compares those spectra with the observed data (Leighly et al., 2018). This design targets a longstanding problem in BAL spectroscopy: the most diagnostic transitions are often blended, partially saturated, and subject to partial covering, so a decomposition into isolated ionic measurements can be ambiguous or impossible.

The original motivation was empirical as well as methodological. Earlier BAL analyses, including approaches based on measured line complexes followed by Cloudy comparison, could yield physically plausible ionic columns yet fail to reproduce the full synthetic spectrum (Leighly et al., 2018). SimBAL was designed to avoid that inconsistency by fitting the spectrum itself. This also permits use of constraints from nondetections: the absence of specific transitions enters the likelihood naturally, rather than being discarded as noninformation (Leighly et al., 2018).

This suggests that SimBAL is best understood not merely as a line-fitting code but as an inference engine that couples photoionization physics, radiative transfer parameterization, and Bayesian calibration. In the published applications, this architecture is repeatedly used where blending is so severe that simpler diagnostics either become lower limits or fail structurally, as in FeLoBALs (Choi et al., 2020, Choi et al., 2022) and extremely broad EHVO troughs (Hidalgo et al., 19 Aug 2025).

2. Core model architecture

At its core, SimBAL uses large grids of ionic column densities generated by Cloudy as a function of the physical state of the gas. In the original implementation, the Cloudy C13.03 grids span $-4.0 < \log U < 2$, $2.8 < \log n < 9$, and $21.5 < \log N_H - \log U < 23.7$, with ionic columns extracted for 179 ground and excited states and combined with a line list of 6267 transitions (Leighly et al., 2018). Later FeLoBAL work using Cloudy C17 expanded this to 76488 transitions, 281 ions, and 997 ionic states when excited states are counted, together with a flexible oversampling scheme near the hydrogen ionization front and 619,721 grid points (Choi et al., 2020).

The fitted physical parameterization varies somewhat by application, but the common parameter set includes ionization parameter $\log U$, gas density $\log n$, the column-density parameter $\log N_H - \log U$, kinematic parameters, and a covering-fraction parameter $\log a$ (Leighly et al., 2018, Hidalgo et al., 19 Aug 2025, Choi et al., 2022). In some early work, the total column density $N_H$ is also discussed directly as a derived quantity (Leighly et al., 2018). The synthetic spectrum is generated in velocity space, either with Gaussian opacity profiles or, more commonly in later work, with adjacent rectangular “tophat” bins that flexibly represent variation across the trough (Leighly et al., 2018, Hidalgo et al., 19 Aug 2025, Bischetti et al., 2024).

The comparison to data is Bayesian. SimBAL generates spectra in real time and compares them with observed spectra via a $\chi^2$-based likelihood explored with emcee MCMC (Leighly et al., 2018). The original paper describes flat priors on the gas parameters to confine sampling to the Cloudy grid and Gaussian priors on line offsets and widths based on inspection (Leighly et al., 2018). Posterior chains provide both parameter estimates and derived physical quantities such as radius, mass outflow rate, momentum flux, and kinetic luminosity.

A concise representation of the original logic is:

$$\text{Cloudy} \rightarrow \text{ionic columns} \rightarrow \text{synthetic spectrum} \rightarrow \text{likelihood} \rightarrow \text{posterior}$$

as explicitly described in the foundational study (Leighly et al., 2018).

3. Treatment of partial covering and velocity structure

A defining feature of SimBAL is its use of an inhomogeneous, power-law partial-covering formalism. The optical depth is parameterized as

$2.8 < \log n < 9$0

where $2.8 < \log n < 9$1 is a fractional surface-area coordinate and $2.8 < \log n < 9$2 is the covering-fraction index (Leighly et al., 2018, Leighly et al., 2018, Green et al., 2024). The parameter is fit as $2.8 < \log n < 9$3, and a larger $2.8 < \log n < 9$4 corresponds to a smaller effective covering fraction (Leighly et al., 2018, Hidalgo et al., 19 Aug 2025). This prescription was adopted because it is mathematically commutative and because it captures the empirical behavior that stronger lines often appear to cover more of the source than weaker lines (Leighly et al., 2018).

The formalism is central to several SimBAL results. In the SDSS J0850+4451 studies, the model showed that covering fraction decreases strongly with velocity and is often the most strongly varying parameter controlling apparent trough morphology (Leighly et al., 2018). In the multiwavelength follow-up, extrapolation of the UV solution to optical and near-infrared data predicted $2.8 < \log n < 9$5, $2.8 < \log n < 9$6, and $2.8 < \log n < 9$7 troughs that were too deep, leading to the conclusion that the covering fraction is wavelength dependent and that the UV covering fraction is about 2.5 times higher than in the optical/NIR (Leighly et al., 2018). The long-wavelength fit required an upward shift of $2.8 < \log n < 9$8, interpreted geometrically as evidence for a clumpy, structured absorber rather than a smooth screen (Leighly et al., 2018).

The method also uses velocity-resolved parameterizations to recover internal outflow structure. The original “accordion model” represents the trough as a sequence of adjacent velocity bins; these bins are not interpreted as distinct physical clouds but as a flexible representation of changing outflow conditions with velocity (Leighly et al., 2018). In SDSS J0850+4451, the best-performing tophat accordion models improved markedly over single-Gaussian fits and revealed that $2.8 < \log n < 9$9 and column density increase with velocity while covering fraction decreases (Leighly et al., 2018). In the z$21.5 < \log N_H - \log U < 23.7$0 quasar J0923+0402, a 25-bin velocity-resolved tophat accordion fit from $21.5 < \log N_H - \log U < 23.7$1 to $21.5 < \log N_H - \log U < 23.7$2 showed that the observed LoBAL/high-ionization morphology could be explained without requiring strong velocity-dependent ionization changes; instead, column density and partial covering dominate the profile structure (Bischetti et al., 2024).

A plausible implication is that SimBAL’s velocity binning is less a decomposition into discrete absorbers than a regularized coordinate system for mapping opacity, ionization, and covering structure across the outflow.

4. Derived quantities and physical interpretation

SimBAL applications routinely convert fitted parameters into physical radii and energetics using the standard ionization-parameter relation

$21.5 < \log N_H - \log U < 23.7$3

with $21.5 < \log N_H - \log U < 23.7$4 the hydrogen-ionizing photon rate, $21.5 < \log N_H - \log U < 23.7$5 the absorber distance, $21.5 < \log N_H - \log U < 23.7$6 the gas density, and $21.5 < \log N_H - \log U < 23.7$7 the speed of light (Leighly et al., 2018, Green et al., 2024, Hidalgo et al., 19 Aug 2025, Bischetti et al., 2024, Choi et al., 2022). Once $21.5 < \log N_H - \log U < 23.7$8, $21.5 < \log N_H - \log U < 23.7$9, and $\log U$0 are specified or constrained, the absorber distance follows. This is how SimBAL moves from spectral fitting to geometry.

A recurrent correction concerns the conversion from the column-density parameter to a covering-fraction-weighted total column density:

$\log U$1

a relation used explicitly in the FeLoBAL survey and the EHVO analysis (Choi et al., 2022, Hidalgo et al., 19 Aug 2025). This correction matters because modifying the covering-fraction structure changes the inferred total column and hence the energetics. In the J0850+4451 multiwavelength covering-fraction study, using the long-wavelength covering fraction lowered the best estimate of the total hydrogen column density by a factor of about 1.6 relative to the UV-only solution (Leighly et al., 2018).

Mass outflow rate and kinetic luminosity are then computed with the standard thin-shell style relations

$\log U$2

with $\log U$3 and typically $\log U$4 assumed as the global covering fraction (Leighly et al., 2018, Choi et al., 2022, Green et al., 2024, Hidalgo et al., 19 Aug 2025, Choi et al., 2020). Several studies emphasize the strong velocity leverage, effectively $\log U$5 under the adopted scaling (Hidalgo et al., 19 Aug 2025, Choi et al., 2022).

The physical interpretations vary by object class. In the original SDSS J0850+4451 analysis, SimBAL placed the outflow at 1–3 pc, found total $\log U$6 for solar metallicity and 22.4 for $\log U$7, and derived a mass outflow rate of 17–28 $\log U$8 with kinetic power 0.8–0.9% of $\log U$9 (Leighly et al., 2018). In the overlapping-trough FeLoBAL SDSS J1352+4239, SimBAL inferred $\log n$0, radius $\log n$1 pc, $\log n$2, and $\log n$3, with $\log n$4 (Choi et al., 2020). In the high-redshift quasar J0923+0402, SimBAL delivered $\log n$5, $\log n$6, and a BAL radius bracketed as $\log n$7 pc (Bischetti et al., 2024).

5. Observational domains and representative applications

SimBAL has been applied across a range of BAL phenomenology. The table summarizes representative studies documented in the literature.

Object / study	SimBAL role	Principal result
SDSS J085053.12+445122.5 I (Leighly et al., 2018)	Foundational HST/COS UV spectral synthesis	Velocity-resolved $\log n$8, $\log n$9, density, and covering-fraction structure; outflow at 1–3 pc
SDSS J085053.12+445122.5 II (Leighly et al., 2018)	UV solution extrapolated to optical/NIR	UV covering fraction about 2.5 times higher than optical/NIR; clumpy structured absorber
SDSS J135246.37+423923.5 (Choi et al., 2020)	Overlapping-trough FeLoBAL forward modeling	Extremely fast FeLoBAL outflow; scattered light and radiation filtering required
50 low-z FeLoBAL quasars (Choi et al., 2022)	First systematic sample analysis	Wide range of $\log N_H - \log U$0, density, and radius; 18% with feedback-level outflows
WPVS 007 (Green et al., 2024)	First simultaneous multi-epoch SimBAL analysis	Variability primarily driven by changes in covering fraction
J0923+0402 at $\log N_H - \log U$1 (Bischetti et al., 2024)	BAL fit in multi-phase feedback study	Powerful compact LoBAL wind on $\log N_H - \log U$2 pc scale
J164653.72+243942.2 (Hidalgo et al., 19 Aug 2025)	EHVO best-estimate forward modeling with multi-epoch tying	Variability best explained by changing $\log N_H - \log U$3; pc-scale extremely high-velocity outflow

The foundational application to SDSS J0850+4451 established the method’s capacity to recover velocity-resolved physical conditions from HST/COS far-UV data and showed that simple single-Gaussian opacity profiles were inadequate (Leighly et al., 2018). The follow-up study extended the analysis into rest-frame optical and near-IR bands, especially He I* $\log N_H - \log U$4, to exploit the fact that the continuum-emitting region is much larger at longer wavelengths (Leighly et al., 2018). Because the 1 $\log N_H - \log U$5m emission region was estimated to be about a factor of 12 larger in radius and about 140 in area than the 1100 Å region, the weaker long-wavelength absorption implied that absorber uniformity depends on source scale (Leighly et al., 2018).

In FeLoBAL work, SimBAL has been particularly important because near-UV spectra can contain thousands of overlapping Fe II transitions. The single-object analysis of SDSS J1352+4239 showed that the code could simultaneously fit continuum, emission, absorption, reddening, and a scattered-light component in a heavily absorbed overlapping-trough quasar (Choi et al., 2020). The broader survey of 50 low-redshift FeLoBAL quasars then used SimBAL uniformly to derive ranges of $\log N_H - \log U$6, $\log N_H - \log U$7, and $\log N_H - \log U$8 pc, finding no evidence for disk winds at $\log N_H - \log U$9 pc in that sample (Choi et al., 2022).

Time-domain applications represent another expansion. In WPVS 007, the first simultaneous multi-epoch SimBAL analysis fit pairs of epochs while allowing only one parameter class to vary and concluded that covering-fraction changes dominate the variability (Green et al., 2024). The EHVO quasar J1646 used a similar multi-epoch tying strategy across DR5 and DR9, and again the best model was the one in which only $\log a$0 changed, with $\log a$1 and $\log a$2 held fixed (Hidalgo et al., 19 Aug 2025).

6. Scientific inferences enabled by SimBAL

Several scientific themes recur across SimBAL studies. The first is that covering fraction is often the dominant control on observed trough shape. In SDSS J0850+4451, the covering fraction decreases strongly with velocity (Leighly et al., 2018). In WPVS 007, BAL variability is determined primarily by changes in covering fraction rather than ionization (Green et al., 2024). In J1646, the extreme two-epoch variability is likewise best explained by changes in $\log a$3, not by changes in ionization or total column density (Hidalgo et al., 19 Aug 2025). These convergent results suggest that absorber substructure and projected source coverage are central observables rather than nuisance parameters.

A second theme is that multi-line forward modeling yields systematically stronger physical constraints than AOD baselines. The J1646 paper explicitly contrasts SimBAL with AOD: AOD fits only continuum-normalized absorption, does not model emission-line fill-in, relies mostly on C IV, and returns conservative lower limits, whereas SimBAL fits continuum, emission lines, and absorption together, uses multiple ions, and recovers hidden opacity beneath emission peaks (Hidalgo et al., 19 Aug 2025). In that case, AOD gives $\log a$4 and $\log a$5 in the two epochs, while SimBAL returns $\log a$6 and $\log a$7 (Hidalgo et al., 19 Aug 2025).

A third theme is that SimBAL can expose nontrivial outflow geometry. The multiwavelength J0850+4451 result rejects the “giant screen” picture because the absorber does not fully cover even the compact UV continuum source (Leighly et al., 2018). The favored interpretation is a distribution of small clumps, either internally structured or clustered, that more effectively cover the smaller UV source than the larger optical/NIR source (Leighly et al., 2018). In J0923+0402, the balance of high- and low-ionization BAL features is explained without invoking strong ionization stratification, instead pointing to velocity-dependent opacity and covering structure (Bischetti et al., 2024).

A fourth theme is feedback energetics. In the original J0850+4451 study, the momentum flux was consistent with $\log a$8, and the kinetic luminosity was 0.8–0.9% of $\log a$9 (Leighly et al., 2018). The FeLoBAL survey adopted $N_H$0 as a feedback threshold and found that 8 of 55 outflowing BAL components meet it, corresponding to roughly 18% of the full sample (Choi et al., 2022). The high-redshift J0923+0402 analysis concluded that the BAL wind carries $N_H$1, consistent with efficient AGN feedback (Bischetti et al., 2024). J1646 showed that even when column density is lower than typical BAL values, very high velocity can dominate the kinetic budget, yielding $N_H$2–47.2 for the EHVO component (Hidalgo et al., 19 Aug 2025).

7. Limitations, caveats, and related terminology

SimBAL’s published applications also identify recurrent limitations. Density is frequently weakly constrained unless a density-sensitive transition is present. In the original SDSS J0850+4451 analysis, $N_H$3 provided the density constraint needed for the radius estimate (Leighly et al., 2018). In J1646 and J0923+0402, the spectra lack such diagnostics, so the density is fixed at $N_H$4 as a modeling assumption rather than being measured directly (Hidalgo et al., 19 Aug 2025, Bischetti et al., 2024). Radius and energetics in those cases are therefore degenerate with the adopted density.

Global covering fraction $N_H$5 is another systematic. Many studies adopt $N_H$6, motivated by BAL fractions, but explicitly note that mass outflow rates and kinetic luminosities scale with this assumption (Leighly et al., 2018, Choi et al., 2022, Bischetti et al., 2024, Hidalgo et al., 19 Aug 2025). Metallicity and SED choices also matter. In SDSS J0850+4451, a hard SED required substantially larger columns and gave a poorer fit than a soft quasar-like SED, while enhanced metallicity $N_H$7 improved the fit and lowered the inferred total column (Leighly et al., 2018). In J0923+0402, the authors note that higher metallicity would lower the inferred column density and weaken the outflow energetics (Bischetti et al., 2024).

Model flexibility can itself be a source of ambiguity. The FeLoBAL survey introduced spectral PCA for emission-line modeling, priors on PCA coefficients to prevent the continuum model from spuriously fitting away absorption, and modified partial-covering schemes for especially opaque overlapping-trough objects (Choi et al., 2022). In SDSS J1352+4239, SimBAL had to include anomalous reddening and a scattered-light component with a best-fit scattering fraction of about $N_H$8 to reproduce the residual flux in the troughs (Choi et al., 2020). Those extensions show both the adaptability of the framework and the dependence of inference on the adopted composite source model.

A separate terminological caveat concerns names. In quasar spectroscopy, “SimBAL” refers to the spectral-synthesis BAL code introduced by Leighly and collaborators (Leighly et al., 2018). This should be distinguished from “SimBaL” in low-$N_H$9 CMB analysis, where the term denotes a simulation-based likelihood for Planck temperature-polarization spectra rather than a BAL-quasar spectral-synthesis code (Belsunce et al., 2021). The shared spelling reflects nomenclature overlap, not methodological continuity.

Overall, SimBAL occupies a specific niche in quasar-outflow research: it is a forward-modeling BAL inference framework optimized for spectra where line blending, partial covering, and non-black saturation render simpler techniques incomplete. The literature to date shows its utility in deriving velocity-resolved physical conditions, testing absorber geometry across wavelength and time, and quantifying outflow energetics from pc to sub-kpc scales across LoBAL, FeLoBAL, and EHVO phenomenology (Leighly et al., 2018, Leighly et al., 2018, Choi et al., 2020, Choi et al., 2022, Green et al., 2024, Bischetti et al., 2024, Hidalgo et al., 19 Aug 2025).