In Search of the Edge: A Bayesian Exploration of the Detectability of Red Edges in Exoplanet Reflection Spectra

2.1.1. Atmospheric Model

We generate an atmospheric model resembling the modern Earth using Exo-Prime2 (see, e.g., Kasting & Ackerman 1986; Pavlov et al. 2000; Pavlov & Kasting 2002; Segura et al. 2005, 2007; Kaltenegger et al. 2010; Madden & Kaltenegger 2020)—a 1D radiative–convective terrestrial atmosphere code. Exo-Prime2 couples 1D climate and photochemistry models to compute the vertical temperature structure and profiles of atmospheric mixing ratio for a planet, assuming an incident stellar spectrum and planetary outgassing rates. Exo-Prime2 also includes feedback from wavelength-dependent surface albedos and clouds. For Earth-like clouds, we use the MODIS 20 μm cloud albedo model (King et al. 1997; Rossow & Schiffer 1999), which provides a reasonable average for many clouds of different droplet size. The application of Exo-Prime2 to model Earth-like planets around different host stars and through geological time has been extensively described in the literature (e.g., Kaltenegger et al. 2010; Rugheimer et al. 2013; Rugheimer & Kaltenegger 2018; Madden & Kaltenegger 2020; Kaltenegger & Lin 2021; Lin et al. 2022). The resulting pressure–temperature (P–T) and mixing ratio profiles computed by Exo-Prime2 are shown in Figure 1.

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2.1.2. Surface Model

We model a representative Earth-like surface using wavelength-dependent albedos from the USGS and ASTER spectral libraries (Baldridge et al. 2009; Clark et al. 2007; Kokaly et al. 2017). We create an average present-day Earth surface albedo from eight raw albedos of snow, water, coast, sand, trees, grass, basalt, and granite (after Kaltenegger et al. 2007). We assume an Earth-like surface consisting of 70% ocean, 28% land, and 2% coast. The land surface consists of 30% grass, 30% trees, 9% granite, 9% basalt, 15% snow, and 7% sand. We use the surface-fraction-weighted albedo (see Figure 2, bottom panel) in our 1D radiative transfer models.

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2.1.3. Computation of Reflection Spectra

A distant observer directly imaging an exoplanet measures the wavelength-dependent planet–star flux ratio. At wavelengths where reflected light dominates over thermal emission, the flux ratio can be expressed as

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{p}(\alpha ,\lambda )}{{F}_{s}(\lambda )}={A}_{g}(\lambda )\,{\rm{\Phi }}(\alpha ,\lambda ){\left(\displaystyle \frac{{R}_{p}}{d}\right)}^{2}\end{eqnarray} \tag{ 1 }$$

where A_g is the planet's geometric albedo spectrum, Φ is the phase function, α is the orbital phase, R_p is the planetary radius, and d is the planet–star orbital distance. The geometric albedo is traditionally defined as the ratio of the observed flux from the planet at full phase to that from a perfectly reflecting Lambert disk. The phase function encodes the dilution of the planetary brightness for phase angles without full illumination (Φ = 1 when α = 0). While the geometric albedo encodes information about an atmosphere's composition, temperature, cloud properties, and surface reflection, the phase function is controlled by both the stellar illumination and atmospheric scattering.

We generate model reflection spectra for an Earth-like planet around a Sun-like star using the open-source radiative transfer code PICASO (Batalha et al. 2019). To compute F_p /F_s from the geometric albedo we provide PICASO with R_p and d (fixed to 1 au) and assume observations at full phase (Φ = 1) unless otherwise noted. We note that observations will more typically occur at nonzero phase, which would dampen the resultant reflection spectra. However, in this proof-of-concept study, we choose to focus on full phase to reduce the complexity of the radiative transfer calculations required within the retrieval framework. We set the planetary reference radius such that r(P = 1 bar) = R_⊕, the surface at 1 bar, and the surface gravity to 9.81 m s⁻². For the stellar spectrum, we used PICASO to interpolate the Castelli & Kurucz (2003) grid for a Sun-analog star with T_eff = 5780 K, $\mathrm{log}\,g$ = 4.437, and [Fe/H] = 0.0122.

For the radiative transfer calculation, we provide PICASO with the P–T profile, mixing ratio profiles, and the wavelength-dependent surface albedo from our Earth-like Exo-Prime2 model. Our computations of reflection spectra span the near-UV to near-IR, ranging from 0.3 to 2.5 μm. We consider molecular line opacity for H₂O, O₂, O₃, CH₄, CO₂, and N₂O (see Batalha et al. 2019, for details on the opacity database), alongside Rayleigh scattering from N₂ and O₂. For computational efficiency, we downsampled PICASO's molecular cross sections by 10× (from R = 10,000). We tested different resampling factors and find that 10× downsampling provides a reliable balance between speed and accuracy.

Our PICASO model accounting for Earth-like clouds assumes optical properties consistent with water. Specifically, we use an asymmetry factor of 0.85 and a single-scattering albedo of 1.0 (after Feng et al. 2018). We place the cloud base at $\mathrm{log}\,{p}_{b}=-0.23$ (pressure in bars), set its vertical extent to $\mathrm{log}\,{dp}=$ −0.53, and set the cloud optical depth to $\mathrm{log}\tau =$ −1.0. We selected these values for the vertical extent of the cloud by calibrating the continuum flux from 0.4 to 1 μm of our 1D models to reproduce the reflection spectrum of Earth from Robinson et al. (2011). We note that assuming a constant cloud albedo increases the reflected flux beyond 1 μm compared to Robinson et al. (2011), but does not significantly influence our analysis.

2.1.4. Impact of the Red Edge on Reflection Spectra

We have developed a Bayesian retrieval wrapper around the PICASO radiative transfer code. We employ this retrieval framework in subsequent sections to demonstrate that information on the red edge can be reliably retrieved from reflection spectra of exo-Earths. Here, we describe the simulated data used in our retrievals and our retrieval configuration.

2.2.1. Simulated Data and Noise Model

Our aim is to investigate the retrievability of the red edge as a function of data quality, rather than for a specific future mission architecture. Consequently, we generated several synthetic data sets, for both the cloud-free and cloudy models described in Section 2.1.3, spanning signal-to-noise ratios of S/N = 5, 10, 15, and 20 (at a reference wavelength of 0.55 μm) and spectral resolutions of R = 70 and 140. We account for wavelength-dependent noise for the simulated observations using a scaling relation for a noise model with constant spectral resolution as done in Robinson et al. (2016) and Feng et al. (2018):

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}(\lambda )\propto q(\lambda )\,{ \mathcal T }(\lambda )\,{A}_{g}(\lambda )\,{\rm{\Phi }}(\alpha ,\lambda )\,B(\lambda )\,\lambda \end{eqnarray} \tag{ 2 }$$

where q is the detector quantum efficiency, ${ \mathcal T }$ is the throughput, and B is a blackbody representing the parent star (see Feng et al. 2018). We adopt functions for q and ${ \mathcal T }$ from the Python package coronagraph, ³ which is an open-source noise simulator for coronagraph-based observations of directly imaged exoplanets (e.g., Robinson et al. 2016; Lustig-Yaeger et al. 2019). For the blackbody, we use T_eff = 5780 K.

When generating each simulated data set, we do not randomize the placement of each data point by sampling from a Gaussian distribution. Rather, the data are centered on the (true) planet-to-star flux ratio—corresponding to the model after binning down to the data resolution—and assigned error bars according to our noise model at the desired S/N. We note that running retrievals on a data set with Gaussian noise can bias the retrieval results, especially for low spectral resolution and S/N (see Feng et al. 2018). However, running retrievals with Gaussian scatter still allows for spectral features to be recovered (see, e.g., Lin et al. 2021, Appendix A). To avoid biasing our retrieval results to a specific random noise draw, we run "scatter-free" retrievals, which produce posterior distributions equivalent to the ensemble average over many individual noise instances. We include an example retrieval with Gaussian scatter in Appendix A.

2.2.2. Retrieval Configuration

Bayesian retrieval codes repeatedly call a parameterized radiative transfer forward model to identify the range of bulk planetary, atmospheric, and surface properties consistent with a given data set. Our retrieval framework employs PICASO (Batalha et al. 2019) as the radiative transfer forward model and the MultiNest (Feroz & Hobson 2008; Feroz et al. 2009, 2019) wrapper PyMultiNest (Buchner et al. 2014) for the sampling algorithm used to explore the parameter space.

We parameterize the atmospheric and surface properties using a simplified prescription that captures the salient features shaping terrestrial exoplanet reflection spectra. We parameterize the P–T profile with an isotherm. We assume H₂O, O₂, O₃, CH₄, CO₂, and N₂O are the main spectrally active gases with sufficient abundances to shape the spectrum for Earth-like planets, with each gas ascribed a single free parameter for the uniform-in-altitude volume mixing ratio. We also assume the primary atmospheric gas is N₂, with its mixing ratio determined by the condition that mixing ratios must sum to one. We prescribe three further free parameters for the planetary radius and gravity (evaluated at 1 mbar) and the surface pressure. These choices are similar to those made by other reflected-light retrieval studies (e.g., Feng et al. 2018; Damiano & Hu 2022; Robinson & Salvador 2023; Wang et al. 2022).

Since our simulated observations incorporate an Earth-like wavelength-dependent surface albedo, we propose a new parametric treatment for wavelength-dependent surface albedos:

$$\begin{eqnarray}{A}_{s}(\lambda )=\left\{\begin{array}{ll}{A}_{s,1} & \lambda \lt {\lambda }_{1}\\ {A}_{s,2} & {\lambda }_{1}\leqslant \lambda \leqslant {\lambda }_{2}\\ {A}_{s,3} & \lambda \gt {\lambda }_{2}\end{array}\right.\end{eqnarray} \tag{ 3 }$$

where A_s,1, A_s,2, and A_s,3 define the surface albedo in three distinct wavelength regions, λ₁ marks the transition from A_s,1 → A_s,2, and λ₂ marks the transition from A_s,2 → A_s,3. This prescription for surface albedo thus has five free parameters. To avoid discontinuities at λ₁ and λ₂, we compute this function on a wavelength grid at R = 1000 and convolve it with a Gaussian with a standard deviation of 28 wavelength grid spaces (corresponding to 28 nm at 1 μm). Our albedo parameterization thus resembles a smoothed double-step function (similar to the function used by Taylor et al. 2021 to parameterize the single-scattering albedo of clouds in the nightside emission spectra of giant planets). We shall demonstrate in subsequent sections that the proposed parameterization is sufficiently flexible to capture both the strong wavelength dependence of the red edge and a possible secondary reflectance edge in the infrared (see Figure 2).

For retrievals including clouds, we add three further parameters: the cloud-base pressure, its vertical pressure extent, and optical depth. Following Feng et al. (2018), we assume water-like clouds with a fixed asymmetry parameter (0.85) and single-scattering albedo (1.0). In total, the most complex retrievals we consider thus have a total of 18 free parameters (summarized in Table 1). We validated our retrieval framework against simulated data from the Robinson et al. (2011) model (see Appendix B).

Table 1. Free Parameters Included in Our PICASO Retrievals

Parameter	Description	Reference Value	Prior Range
$\mathrm{log}$O2	Oxygen mixing ratio	−0.678	[−10, 0]
$\mathrm{log}$O3	Ozone mixing ratio	−6.25	[−10, −1]
$\mathrm{log}$H2O	Water vapor mixing ratio	−2.72	[−10, −1]
$\mathrm{log}$CO2	Carbon dioxide mixing ratio	−3.44	[−10, −1]
$\mathrm{log}$CH4	Methane mixing ratio	−5.77	[−10, −1]
$\mathrm{log}$N2O	Nitrous oxide mixing ratio	−6.55	[−10, −1]
logp0	Surface pressure	0.0	[−2, 2]
Rp	Planet radius at 1 mbar	1.007	[0.5, 2.0]
g	Gravity at 1 mbar	9.66	[1.0, 25]
T	Temperature	289	[100, 800]
λ1	Albedo transition point	0.72	[0.3, 2.5]
λ2	Albedo transition point	1.40	[0.3, 2.5]
As,1	Surface albedo	0.09	[0, 1]
As,2	Surface albedo	0.15	[0, 1]
As,3	Surface albedo	0.06	[0, 1]
$\mathrm{log}\ {p}_{{\rm{b}}}$	Cloud-base pressure	−0.23	[−2, 2]
$\mathrm{log}\ {dp}$	Cloud width	−0.53	[−2,2]
$\mathrm{log}\ \tau$	Cloud optical depth	−1.0	[−2, 2]

Note. The reference values for each parameter correspond to either "ground truth" values from the input model (e.g., planet radius and cloud properties; see Section 2.1) or representative average values (e.g., mixing ratios and albedo parameters; see Section 2.2.2). All priors are uniform distributions.

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Our retrieval analysis covers multiple model and data scenarios. First, in Section 3, we evaluate the retrievability of albedo changes for our cloud-free model, since this model has the strongest spectral red edge. We initially ran four retrievals on the simulated data at (S/N)_ref = 5, 10, 15, 20 and R = 70, where we parameterize the planet's wavelength-dependent surface albedo with Equation (3). Then, we ran a similar set of retrievals with a constant-in-wavelength surface albedo. Doing so enables us to perform Bayesian model comparisons between the wavelength-dependent surface and the constant-in-wavelength surface models (e.g., Benneke & Seager 2013; Trotta 2017). We also ran a retrieval with R = 140 at (S/N)_ref = 10 to investigate the impact of retrieving data at a higher spectral resolution. Lastly, we ran a retrieval at R = 70 and (S/N)_ref = 10 for a more realistic data set including a cloud deck to investigate how clouds impact the retrieval results. All our MultiNest retrievals use 2000 live points, which typically involve the computation of 10⁶ models.

We summarize the prior range for each retrieval free parameter in Table 1. We generally allow generous prior ranges, encompassing a wide range of physically plausible values, with all priors being uniform distributions. As in Feng et al. (2018), we allow for oxygen-rich atmospheres by extending its prior range to 100% (but rejecting any parameter combinations where the sum of the non-N₂ mixing ratios exceeds unity). Our prior ranges for the planet radius and gravity terminate at 2.0 R_⊕ and 25.0 m s⁻², respectively, since this study focuses on Earth-like planets. We note that PICASO requires that the pressure corresponding to R_p must be less than the highest pressure in the atmospheric pressure grid (i.e., the surface pressure). We circumvent this issue by defining the planet radius and gravity parameters at 1 mbar, such that the range of the surface pressure prior (10⁻²–10² bar) is always deeper than 1 mbar.

We also include "reference values" for each parameter in Table 1, which correspond closest to the original input Exo-Prime2 model (see Section 2.1), for comparison to the retrieval results. Since the input gas mixing ratios depend on height while the retrievals assume uniform mixing ratios, we set the reference values as the average of the true mixing ratio profile from the surface to 25 km altitude. Our reference temperature is the planet's surface temperature. For the albedo parameters, our reference values are determined by averaging the surface albedo over 0.3–0.72 μm, 0.72–1.4 μm, and 1.4–2.5 μm. The reference planet radius and gravity correspond to the true values from the Exo-Prime2 model (scaled to 1 mbar). Similarly, the reference values for surface pressure and cloud parameters correspond exactly to the original model inputs (see Section 2.1.3).

We now present the results of our retrievals including wavelength-dependent surface albedos.

We first assess whether a retrieval model assuming a constant-in-wavelength surface albedo can adequately fit the reflection spectrum of an exoplanet with a realistic Earth-like surface. We have already seen in Section 2.1.4 and Figure 2 that the Earth's red edge induces a sharp change in F_p /F_* around 750 nm, so here we quantify whether such a spectral signature is detectable and its impact on atmospheric retrievals.

In Figure 3, we demonstrate that a uniform albedo model often struggles to capture the spectral morphology of an Earth-like exoplanet. Our "ground truth" model is the cloud-free scenario described in Section 2.1, which produces the strongest red edge, while the simulated data (here, (S/N)_ref = 10) and retrieval configuration are detailed in Section 2.2. We see that the retrieved spectrum for the uniform albedo model begins to deviate from the simulated data for optical wavelengths longer than 0.65 μm. In the optical and near IR, where the S/N is highest, the uniform albedo model is often discrepant with the data to 2σ. The root cause of this model–data mismatch is that the uniform albedo model has a roughly constant continuum F_p /F_* from 0.6 to 0.9 μm (outside O₂ absorption features), which cannot reproduce the sharp change in spectral continuum associated with the vegetation red edge near 0.72 μm.

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In contrast, our wavelength-dependent parameterization of surface albedo well matches the reflection spectrum of our Earth-like exoplanet. Equation (3) allows our retrieval code to reproduce both the large increase in F_p /F_* at 0.72 μm caused by the vegetation red edge and the general morphology of the spectrum in the visible and near IR. Statistically, our retrieval including a wavelength-dependent albedo is favored over the uniform model with a Bayes factor of $\mathrm{ln}{ \mathcal B }=30.4$ (equivalent to 8.1σ using the relations in Benneke & Seager 2012), which would be considered a conclusive detection on the Jeffrey's scale of Bayesian model comparison (e.g., Trotta 2017).

We illustrate why the uniform albedo model struggles to fit our data by comparing the retrieved surface albedos in the bottom panel of Figure 3. While the uniform albedo model correctly captures the surface albedo in the near-IR wavelengths beyond 1.4 μm, it significantly underestimates the true surface albedo from 0.75 to 1.35 μm. However, our proposed parameterization demonstrates that one can retrieve wavelength-dependent surface properties—in particular the location of the vegetation red edge—at even a moderate signal-to-noise ratio ((S/N)_ref = 10). The retrieved surface albedo profile also correctly implies a decrease in the albedo for near-IR wavelengths beyond 1.3 μm. While our wavelength-dependent surface retrieval somewhat overestimates the magnitude of the albedo (likely due to other complexities not captured in the model, such as variable atmospheric abundances with height), it correctly captures the general shape of the wavelength-dependent albedo profile and lies within 2σ of the true Earth-like surface albedo.

We further show the posteriors of our retrieved albedo parameters in Figure 4. All five parameters are well constrained by the data, demonstrating that reflection spectra are highly sensitive to the wavelength dependence of the surface albedo. In particular, the parameter encoding the wavelength location of the red edge, λ₁, is retrieved to a remarkable precision of 8 nm. Similarly, the retrieval identifies a secondary albedo change near 1.4 μm, encoded by λ₂, with a precision of ≈200 nm (though the long posterior tail to higher values indicates this is harder to constrain than λ₁). The three albedo parameters are slightly overestimated, as noted above, but are consistent within 2σ with the reference values (see Table 1). Overall, Figure 4 shows that Equation (3) offers a parameterization capable of capturing the key wavelength-dependent features of a realistic Earth-like surface.

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The assumption of a uniform surface albedo can bias inferred properties of an exoplanet. Since a retrieval code employs every available means to minimize model–data residuals, it can attempt to compensate for the non-inclusion of a wavelength-dependent surface by modifying the retrieved abundances of chemical species in the atmosphere (since their cross sections are also wavelength-dependent). In Figure 5, we demonstrate that one consequence from assuming a uniform surface albedo is biased abundance inferences for several key molecules in our model. Specifically, we find that the volume mixing ratios of O₃ and H₂O are overestimated by an order of magnitude and the bulk atmospheric gas would be identified as O₂ rather than N₂. This finding underscores an important point: accurate abundance inferences for atmospheric gases can depend on the inclusion of a wavelength-dependent surface albedo in retrieval frameworks based on reflected light. Since Figure 5 corresponds to the moderate case of R = 70 and (S/N)_ref = 10, wavelength-dependent surface spectral properties will be an important consideration for future direct-imaging missions for exoplanets, especially for missions focused on Earth-like exoplanets.

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Figure 6 shows our retrieved spectra and surface albedo profiles for (S/N)_ref = 5, 10, 15, and 20. We see that even at (S/N)_ref = 5, the retrieval correctly identifies a sharp rise in the surface albedo near 0.7 μm—consistent with the wavelength of the vegetation red edge on modern Earth. The uncertainty in the retrieved wavelength of this feature is remarkably small (λ₁ determined to ≈15 nm). This suggests that sudden changes in surface albedo are an effect of first-order importance even for observations with low signal-to-noise ratio. With a doubling to (S/N)_ref = 10, we see further improvements in the retrieved surface albedo profile: (i) the uncertainty in the location of the sharp rise in surface albedo is halved (λ₁ determined to ≈8 nm); (ii) a hint emerges of a secondary albedo change near 1.4 μm (λ₂ determined to ≈200 nm); and (iii) the true surface albedo profile is correctly captured throughout most of the wavelength range to within 2σ. For (S/N)_ref = 15, the retrieval becomes more confident about the existence of a secondary albedo edge (λ₂ determined to ≈70 nm). Finally, at (S/N)_ref = 20 the retrieved model attains even better overall agreement with the true albedo model. We also find that the tendency to overestimate the retrieved albedo (see Section 3.1) becomes less prevalent for higher S/N.

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Table 2 quantifies the preference for our wavelength-dependent albedo model (Equation (3)) over a uniform albedo model. For (S/N)_ref = 5, our Bayesian model comparison finds moderate evidence for a nonuniform surface albedo (2.7σ). A slight increase to (S/N)_ref = 10 suffices to conclusively detect at least one discontinuity (8.1σ). Further increases in S/N can help to detect a nonuniform surface albedo also for cloudy atmospheres, where the effect is smaller because clouds block part of the light from the underlying surface from view (see Section 4.3). Since our retrievals thus far have only considered data at R = 70, we next explore variable spectral resolution for a fixed signal-to-noise ratio.

Figure 7 shows how the retrieved surface albedo profile changes with spectral resolution. Specifically, we illustrate the expected improvement from doubling the spectral resolution from R = 70 to R = 140 (hence doubling the number of data points from 0.3–2.5 μm). We see that the retrieved albedo from the higher-resolution data is in better agreement with the true albedo, especially at longer wavelengths where the errors bars are largest. Further, the uncertainty on the retrieved albedo parameters decreases (A_s,1 and A_s,2 improve by ≈25%; A_s,3 improves by ≈50%). We further note that at R = 140 the retrieval of the second, smaller edge around 1.4 μm (λ₂) improves (by ≈68%), resulting in a retrieved albedo shape more consistent with the true Earth-like surface.

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Figure 8 shows how the inclusion of clouds affects the retrieved spectrum and surface albedo profile. Our cloud properties for this demonstration were chosen to resemble the continuum flux from the model in Robinson et al. (2011) (see Section 2.1.1). We see that while the data are well fit, the retrieved albedo profile is generally overestimated when clouds are included (see also Wang et al. 2022 for a discussion on cloud–surface degeneracies). Nevertheless, the wavelength of the red edge is still reliably retrieved and well constrained even in the presence of clouds (λ₁ determined to 17 nm). The retrieved albedo also shows a slight decrease at longer wavelengths, but the secondary albedo change near 1.4 μm is not well constrained. Overall, the presence of a cloud deck can lead one to infer an artificially brighter surface outside the 1σ uncertainty region of the retrieved albedo profile.

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Clouds increase the minimum signal-to-noise ratio required to detect a wavelength-dependent surface feature. For example, Table 2 demonstrates that the detection significance for a nonuniform surface albedo at (S/N)_ref = 10 drops from 8.1σ (cloud-free) to 2.9σ (including clouds). Such lower significances arise from cloud–surface degeneracies broadening albedo uncertainties (see Figure 8). However, we still find a detection of a wavelength-dependent surface albedo for (S/N)_ref = 15 (5.6σ). These results show that, while clouds can complicate the inference of wavelength-dependent surface features, it is still possible to identify nonuniform surface albedos for Earth-like cloud coverage.

Figure 9 (top left) shows that the reliable inference of bulk planetary properties requires moderate signal-to-noise ratios for cloud-free models. At (S/N)_ref = 5, the 1 mbar radius is underestimated and the surface pressure slightly overestimated. The temperature is correctly retrieved, albeit with broad uncertainty (≈100 K). We find that (S/N)_ref = 10 is the minimum to reliably retrieve these parameters. In particular, for the temperature a well-defined peak appears around 300 K and the uncertainty shrinks to ≈70 K. The surface pressure and planetary radius are less biased for (S/N)_ref ≥ 10, with the reference values correctly retrieved within 2σ. For the gravity at 1 mbar, we find only a lower limit for all our signal-to-noise ratios.

Figure 9 (top right) shows that the abundances of gases with strong absorption features in the optical and near infrared (O₃, O₂, and H₂O) are generally well constrained. The O₃ abundance can always be constrained to better than a factor of 2 (0.3 dex), even for (S/N)_ref = 5. The ease of constraining O₃ is driven by its strong absorption at optical and near-UV wavelengths. The O₂ posterior is the broadest due to the smaller number of data points spanning its narrow absorption features. Nevertheless, O₂ can be constrained to 0.3 dex for (S/N)_ref = 20. We note that the O₃, O₂, and H₂O abundances are slightly underestimated for (S/N)_ref = 5, but are reliably retrieved for (S/N)_ref ≥ 10.

Our retrievals are unable to detect gases with only weak absorption features in the modeled wavelength range. We can place an upper limit on the CH₄ abundance for (S/N)_ref = 5, but CO₂ and N₂O require (S/N)_ref = 15 for upper limits. We find a tentative hint of CO₂ at (S/N)_ref = 20 centered on the reference value, but the posterior tail to lower abundances indicates a nondetection of CO₂ absorption. Constraints on gases such as CO₂ and CH₄ for a modern Earth-like atmosphere at low resolution and low signal-to-noise ratio would benefit from observations of thermal emission in the mid-infrared (e.g., Des Marais et al. 2002; Kaltenegger et al. 2007; Konrad et al. 2022).

Figure 9 (bottom) highlights trends in the retrieved surface albedo parameters. As discussed in Sections 3 and 4, our main results are: (i) including wavelength-dependent surface albedo in retrievals can improve the accuracy of atmospheric inferences; and (ii) even for low to moderate signal-to-noise ratios one can constrain the wavelength-dependent surface albedo for an Earth-analog planet. We highlight here that the λ₁ posterior demonstrates that a sharp change occurs in the surface albedo around 0.72 μm, even for (S/N)_ref = 5, which is consistent with the modern Earth's red edge. This albedo transition is remarkably well constrained, as indicated by the narrow 1σ intervals in Table 2. Similarly, the posteriors for λ₂ at (S/N)_ref ≥ 10 indicate that there is another sharp feature in the wavelength-dependent surface albedo around 1.4 μm. Compared with the posteriors for the bulk planetary properties and molecular abundances, these results suggest that changes in surface albedo are some of the most reliable features to detect in reflection spectra of Earth-like planets.

While Figure 9 corresponded to cloud-free models, clouds can also increase the uncertainty on the retrieved atmospheric composition. Figure 10 shows that a cloud deck increases the abundance uncertainties for detectable species. The cloud parameters broaden the 1σ constraint for the oxygen, ozone, and water vapor abundances (from 0.55 to 0.61 dex for O₂; from 0.18 to 0.30 dex for O₃, and from 0.33 to 0.50 dex for H₂O). These effects can be attributed to the degeneracy that emerges between the location of the cloud base and the gas mixing ratios (see Appendix C). Despite the broader distributions, the O₂, O₃, and H₂O abundances are still retrieved to within 1σ of their reference values when clouds are included in our model.

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Finally, Figure 11 shows the cloud parameter constraints for R = 70 and (S/N)_ref = 10 (corresponding to the retrieved spectrum in Figure 8). The retrieved optical depth ($\mathrm{log}\tau$), pressure extent ($\mathrm{log}{dp}$), and base pressure ($\mathrm{log}{p}_{c}$) of the cloud deck are all correctly retrieved within 1σ. The posteriors for $\mathrm{log}\tau$ and $\mathrm{log}{dp}$ are broad due to the degenerate nature of these parameters. However, the bounded constraint on $\mathrm{log}{p}_{c}$ demonstrates that our retrieval technique correctly identifies the presence of a cloud deck as a necessary model component distinct from the wavelength-dependent surface albedo.

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Wavelength-dependent surface albedos significantly impact the reflection spectra of directly imaged terrestrial exoplanets (e.g., Figure 2). Our retrieval analysis demonstrates that future direct-imaging missions could find evidence of wavelength-dependent surface features and constrain the shape of surface albedo profiles. Our results complement and expand on the recent study by Wang et al. (2022), by offering a novel parameterization to retrieve changes in the surface albedo at a priori unknown wavelengths. In particular, our demonstration that sharp features like the modern Earth's vegetation red edge can be reliably retrieved (to 150 nm for cloud-free models at S/N = 5) is very promising for a proposed future large IR/optical/UV space-based telescope (e.g., Decadal Survey on Astronomy & Astrophysics 2020). While we find that clouds can result in overestimated surface albedos (in agreement with results from Feng et al. 2018, Robinson & Salvador 2023, and Damiano & Hu 2022), the red edge's wavelength location can nonetheless be correctly retrieved for modern Earth-like planets even for Earth-like cloud coverage.

More generally, detecting wavelength-dependent surface albedos from reflection spectra offers the opportunity to constrain the surface composition of rocky exoplanets. Other materials such as sand, basalt, and granite, which cover substantial regions of Earth's surface, have unique albedo profiles (see Figure 12) and shape the reflection spectra of Earth-like exoplanets (see, e.g., Madden & Kaltenegger 2020). These profiles could potentially be extracted from spectra of rocky worlds whose surfaces are dominated by these materials (Pham & Kaltenegger 2021, 2022). However, for retrieval purposes the flexibility of our surface albedo parameterization (Equation (3)) would need to be tested for these surface compositions. Our parameterization was inspired by the modern Earth's surface albedo, hence it is able to locate sharp albedo changes such as the red edge. Future work should investigate the flexibility of our parameterization for other surfaces, such as oceans or deserts, to determine whether a generalized parameterization is necessary.

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An additional caveat is that our retrievals have focused on zero orbital phase (i.e., full illumination), while future direct-imaging observations will be constrained to higher phase angle. The left panel of Figure 13 shows the impact of higher phase angles through PICASO calculations at orbital phases of 0°, 30°, 60°, and 90° for the model in Figure 2. We see that orbital phase acts to scale down F_p /F_*, while preserving the relative amplitude of the red edge relative to the surrounding continuum. The right panel of Figure 13 shows the impact of a nonzero orbital phase on a retrieval of our cloud-free Earth-like exoplanet model. Due to the computational requirements of nonzero orbital phase retrievals, for this demonstration we consider only an orbital phase of 60° and observations with (S/N)_ref = 20 and R = 70. We see that the location of the vegetation red edge can still be constrained for nonzero phases, but the uncertainty becomes about 50% larger due to the lower flux ratio. Future retrieval studies could investigate the impact of a wide range of nonzero orbital phases (e.g., Nayak et al. 2017), or multiphase observations (Damiano et al. 2020; Carrión-González et al. 2021), on wavelength-dependent surface albedo constraints. Nevertheless, our results show great promise for the detectability and characterization of rocky exoplanet surfaces from reflection spectra.

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The potential habitability of Earth-like exoplanets orbiting Sun-like stars could be assessed by constraining wavelength-dependent surface properties. Future direct-imaging missions will focus on characterizing these atmospheres to search for biosignature gas pairs such as O₃ combined with CH₄. Our results suggest that such missions can also retrieve surface spectral features, and that not accounting for wavelength-dependent surfaces can bias the retrieved abundances of biosignature gas pairs. We stress that including wavelength-dependent surface albedos in retrievals is an opportunity, since it provides a science case for an expanded mission including searches for surface biosignatures. The vegetation red edge is one candidate, but its universality remains uncertain. Exoplanets could have reflectance edges at different wavelengths (see, e.g., Kiang et al. 2007) or photosynthetic organisms that do not show red edge features (see, e.g., Cockell et al. 2009). Some minerals also exhibit sharp spectral features near optical wavelengths (e.g., Seager et al. 2005; O'Malley-James & Kaltenegger 2018a). Thus, any detection of reflectance edges would need to be carefully placed in context with other signatures of habitability before attributing a biological origin.

The promising detectability of the modern Earth's red edge also suggests that other surface biosignatures would benefit from retrieval studies. One such alternative surface biosignature is biofluorescence. On Earth, coral and other organisms absorb harmful shortwave radiation and re-emit it at longer wavelengths as a protection mechanism. Like the vegetation red edge, biofluorescence can dramatically increase a planet's brightness at specific wavelengths (O'Malley-James & Kaltenegger 2018b, 2019b). Biofluorescence could therefore manifest as a time-dependent spectral edge that may be retrievable from reflection spectra of Earth-like exoplanets.

One of the primary science goals of future space-based observatories with direct-imaging capabilities will be to characterize the atmospheres of Earth-like exoplanets orbiting Sun-like stars. Our results indicate that a large space-based observatory with the capability to achieve F_p /F_* ∼ 10⁻¹⁰ could precisely constrain the abundances of several biosignature pair gases on modern Earth analogs. Assuming a cloud-free atmosphere, we showed that the O₂, O₃, and H₂O abundances can be constrained within 0.6 dex for S/N = 10 and R = 70. Crucially, the O₃ abundance can be constrained to within a factor of 2 (0.3 dex) even for S/N = 5. However, it will be more challenging for these missions to detect biosignature gases with weak absorption features. For instance, we could only place an upper limit on the CH₄ mixing ratio for a modern Earth-like analog. Furthermore, clouds can also impact our ability to constrain and detect some biosignatures. Clouds broaden the uncertainties in the retrieved molecular abundances of O₂, O₃, and H₂O, which makes it more challenging to constrain their abundances.

Our retrieval results suggest that meaningful information can be extracted from reflected light spectra of a modern Earth analog, even with a lower S/N than indicated by previous work, if the available data cover an expanded wavelength range. Feng et al. (2018) found that S/N = 15 is generally a prerequisite to constrain the abundances of O₂, O₃, and H₂O. In comparison, we find that these gases can be precisely constrained for S/N ≥ 10. These differences are mainly attributable to the wavelength range of the simulated data—we use 0.3–2.5 μm, while Feng et al. (2018) considered 0.4–1.0 μm. The longer wavelength coverage in our retrievals decreased the minimum S/N necessary to constrain H₂O because of the three additional water features at 1.1, 1.4, and 1.9 μm. Similarly, the short wavelength coverage of O₃ absorption lowered the S/N necessary for constraining the O₃ abundance. However, the difference in the minimum signal-to-noise ratios for retrieving the O₂ abundance is (i) due to our differing noise models (we assumed an agnostic detector efficiency while Feng et al. (2018) based their model on Roman Space Telescope-like detectors), and (ii) due to the wavelength-dependent surface features we added to the retrieval process.

Ultimately, a key driver of exoplanet science is the characterization of potentially habitable planets around Sun-like stars. The Astro 2020 Decadal Survey (Decadal Survey on Astronomy & Astrophysics 2020) specifically highlights the goal of searching for atmospheric biosignatures on Earth-like exoplanets orbiting Sun-like stars. Our findings indicate that a future direct-imaging mission observing reflected light could also detect surface biosignatures—including the vegetation red edge—via the retrieval of a wavelength-dependent surface albedo. Our odds of detecting life in the solar neighborhood can only be enhanced by considering all the ways life shapes its host planet. For a spectral edge encoded in light from a distant star may, one day, illuminate the surface of a world not too dissimilar to our own.