The Bayesian nonlinear inversion used for MAJIS temperature retrieval builds upon techniques previously validated across multiple planetary missions. This methodology was routinely used on data acquired by the Dawn/VIR imaging spectrometer, which achieved nearly global coverage at asteroid Vesta and dwarf planet Ceres, leveraging the nonlinear radiance-temperature relationship to infer thermophysical properties while accommodating rapid rotational cycles (Tosi et al., 2014, 2015, 2018; Capria et al., 2014). The methodology's adaptability was further demonstrated on infrared data acquired by Rosetta/VIRTIS at the asteroid Lutetia (Keihm et al., 2012) and especially at the nucleus of comet 67P/Churyumov–Gerasimenko, resolving self-heating phenomena within shadowed concavities and diurnal thermal gradients at 15 mperpixel resolution (Tosi et al., 2019). Most directly, a similar Bayesian workflow was applied to JIRAM data of the Moon acquired during the 2013 Earth flyby of Juno to derive temperatures validated against LRO/Diviner data (Adriani et al., 2016).

The details of the Bayesian approach to nonlinear inversion are explained in the Appendix A of Tosi et al. (2014). Briefly, this technique addresses the inherent challenge of solving for multiple unknowns – temperature and wavelength-dependent emissivity – from a limited set of spectral radiance measurements. For a retrieval involving N spectral channels, the state vector includes N emissivity values plus the surface temperature, for a total of N+1 free parameters.

The method follows the classical maximum-a-posteriori (MAP) formulation of Rodgers (2000), solved iteratively using a Gauss–Newton scheme. The radiative transfer model is linearized around an initial a priori state, and the optimal solution is obtained by minimizing a quadratic cost function combining the spectral misfit and the prior constraints. The associated posterior covariance matrix – derived from the inverse of the approximate Hessian of the cost function – provides the formal uncertainties of the retrieved parameters. In our implementation, the observation error covariance matrix Se is defined using the in-flight Noise Equivalent Spectral Radiance (NESR) of MAJIS. In the absence of a validated model for channel-to-channel noise correlations, Se is assumed diagonal, with wavelength-dependent variances given by the squared NESR. The a priori covariance matrix Sa applies soft constraints on temperature (± 30 K) and on emissivity, enforcing physically plausible values (ε≤1) and spectral smoothness. In the present implementation, no a priori cross-covariances are prescribed between temperature and emissivity, and Sa is taken as diagonal. Temperature and emissivity are therefore constrained independently in the prior, while their coupling arises through the forward model and is reflected in the posterior covariance. Emissivity correlations between neighbouring spectral channels are represented through off-diagonal terms in Sa, modelled with a Gaussian correlation function: for spectral channels i and j at wavelengths λi and λj, we set: 1cov(εi,εj)=σεi⋅σεj⋅e-(λi-λj)2l2i,j=1…N where σεi is the a priori standard deviation assigned to emissivity at channel i and l is the correlation length (chosen comparable to the spectral sampling, in our case 20 nm which is about three MAJIS-IR spectral channels). This regularization mitigates the intrinsic temperature–emissivity degeneracy, especially near the crossover region where reflected and emitted components are comparable, and stabilizes the emissivity solution while preserving physically plausible variability (Rodgers, 2000).

The physical basis of the approach lies in the radiance equation for airless bodies, where the measured spectrum combines reflected solar and thermally emitted radiation. At thermal equilibrium, Kirchhoff's law states that the spectral directional emissivity equals the corresponding spectral directional absorptivity. For an opaque surface (negligible transmittance), this can be written as εd(λ)=1-rhd(λ), where rhd is the hemispherical–directional spectral reflectance. In the Bayesian retrieval, emissivity is treated as a spectral (channel-by-channel) directional quantity; no bolometric Kirchhoff relation is assumed. However, the system remains underconstrained without additional prior information. The inversion therefore begins with an estimate of surface temperature obtained from the brightness temperature in the dominant thermal-emission region, assuming an initial constant emissivity. An important refinement is the dynamic determination of the crossover wavelength – the point at which reflected and emitted radiation are equal – based on this temperature estimate. The spectral domain used in the inversion is then defined as extending from 0.5 µm shortward of this crossover to the long-wavelength limit of MAJIS sensitivity, ensuring that the crossover region is always included for a stable emissivity solution.

Priors constrain the physically meaningful variability of the unknowns (± 30 K for temperature, and wavelength-dependent bounds on emissivity to ensure ε≤1 and avoid nonphysical solutions). In practice, these values represent soft constraints: they define the width of the a priori covariance matrix but do not impose hard limits. The Gauss–Newton iterations may therefore converge outside these ranges if supported by the radiance data. Only two hard constraints are applied. First, emissivity values exceeding unity are clipped to ε=1. Second, spectral segments are discarded when the retrieved emissivity is inconsistent with Kirchhoff's law by more than 4 %, i.e. when r(λ)=1-ε(λ) indicates non-physical behaviour near the crossover region. These steps prevent non-physical behaviour while preserving the flexibility inherent in the Bayesian formulation. Formal uncertainties of temperature and emissivity are computed from the posterior covariance and incorporate the in-flight NESR, which increases toward longer wavelengths due to spectrometer background.

A Bayesian formulation offers significant advantages over a simple least-squares fit when estimating temperatures from infrared spectra, particularly when the available wavelengths lie in the Wien tail of the Planck function. In this regime, radiance depends exponentially on temperature, creating a strong degeneracy between temperature and emissivity and making least-squares solutions highly sensitive to noise and to the choice of initial emissivity. In practice, this ambiguity becomes stronger when the Jacobian sensitivities of the radiance to temperature and emissivity are similar, especially in the reflected/thermal crossover regime. In the present framework, this degeneracy is regularized through the covariance structure imposed on the emissivity spectrum, which enforces smoothness across adjacent wavelengths while allowing physically plausible variability. Although the Gauss–Newton implementation retrieves only the local posterior maximum rather than the full posterior distribution, the posterior covariance naturally provides formal uncertainties that reflect the structure of the inverse problem. It is therefore used here to diagnose the uncertainties and correlations of the retrieved state, rather than to impose a prior relationship between temperature and emissivity. By suppressing unrealistic high-frequency oscillations – especially near the reflected/thermal crossover – the covariance-based regularization stabilizes the solution without imposing mineralogical assumptions.

Compared to approaches that fit a Planck function with a single emissivity value, the Bayesian retrieval therefore uses the covariance-based prior to regularize the temperature–emissivity degeneracy in a physically motivated way. Tighter spectral correlations suppress noise more efficiently but increase the risk of over-regularization, whereas looser correlations allow greater variability at the expense of stability.

Subsequent iterations refine temperature and emissivity simultaneously. For each pixel, the algorithm iterates up to 50 times, recalculating the Jacobian matrix (radiance sensitivity to parameter changes) at each step to update the solution and its uncertainties. Post-retrieval validation removes emissivity data affected by saturation or inconsistent with the Bayesian solution, ensuring robustness. The method achieves convergence even with suboptimal initial guesses, yielding reliable results for temperatures > 170 K (the exact limit depends on sensitivity range and instrument thermal conditions) for dayside measurements. Typical formal uncertainties are 1–3 K for temperature in well lit areas and 0.01–0.20 for emissivity, increasing for low-temperature and/or high incidence angle scenarios.

For MAJIS, the heritage methodology was systematically adapted to lunar conditions. First, we consider data acquired only in the infrared (IR) channel, which covers the spectral range 2.28–5.56 µm. Raw data underwent reinsertion of the background signal (which is automatically subtracted in the calibration pipeline) to identify saturation artifacts, which appear as plateaus in the spectral profile, sometimes with oscillations. Spectral pixels (“spectels”) exceeding 10 400 Digital Numbers (DN) – a threshold empirically derived from C4 data, which experience the highest solar illumination – were flagged and excluded. Synthetic radiance spectra were generated by combining thermal emission with reflected solar components modelled using the MODTRAN Cebula + Kurucz extraterrestrial solar spectrum (https://www.nlr.gov/grid/solar-resource/spectra, last access: 18 June 2026) scaled for the heliocentric distance of the Moon at the time of the observation (1.014 au). Several hypotheses on the initial emissivity (0.70–0.95) were evaluated, and the initial temperature was set equal to the brightness temperature in the 5.27–5.56 µm range, allowing both parameters to vary within noise constraints until convergence. Two spectral ranges were tested: the computationally efficient 4.5–5.56 µm window (Table A1) and the broader 3.0–5.56 µm interval (Table A2), the latter reducing saturation biases in high-temperature regions at the cost of increased processing load. We treat this as a stability–information trade-off: the wider range increases leverage but also amplifies calibration/reflectance sensitivities in the crossover domain. The Lommel–Seeliger disk function (Hapke, 1981) – a photometric law optimized for the Moon – was also tested to account for illumination and viewing geometry (Table A3).

Geometric precision was ensured through SPICE kernel navigation data (Acton, 1996; Acton et al., 2018) and instrument kernels, defining ellipsoid-level planetocentric coordinates as well as solar incidence and emission angles. The LEGA observations enable an update of the MAJIS instrument kernel, with details provided in a companion paper dedicated to the geometric calibration (Seignovert et al., 2026). Topographic effects could in principle be mitigated using a digital elevation/shape model to compute local illumination and emission angles; however, no DEM-based topographic correction is applied in this work. While such corrections are critical for irregular bodies (e.g. asteroids and cometary nuclei), approximating the Moon as a smooth ellipsoid is adequate for the large-scale interpretation of MAJIS data at a spatial resolution of a few hundred metres per pixel. On the other hand, this approximation does not capture unresolved local slopes and shadowing, which may contribute to pixel-scale discrepancies. Emissivity outputs were restricted to the spectral range used for temperature retrieval (4.5–5.56 and 3.0–5.56 µm).

Using the range 4.5–5.56 µm, the temperature values obtained through the Bayesian inversion across the four MAJIS lunar observations (C1 to C4) reveal systematic dependencies on both the level of solar illumination and the choice of prior emissivity. As expected, the mean surface temperature increases from C1 through C4, reflecting progressively stronger solar heating (Figs. 1 and 2). C1, acquired under grazing illumination, shows the lowest mean temperatures (∼ 176–184 K), while C4, obtained under near-equatorial midday conditions, yields mean temperatures approaching 356–343 K, depending on the emissivity prior (Fig. 3 and Table A1). To quantify the dependence of the retrieval on the emissivity prior, we evaluate the spread of the retrieved mean temperatures obtained using ε0 = 0.70, 0.80, 0.90, and 0.95 (Table A1). For the 4.5–5.56 µm retrieval window, the mean temperature varies by ΔT_mean ≈ 8.6 K (C1), 2.2 K (C2), 9.2 K (C3), and 12.9 K (C4) across the tested priors. This confirms that the sensitivity to ε0 is generally limited for intermediate illumination conditions (C2), whereas it becomes more significant under extreme thermal regimes, where the radiance–temperature relationship is strongly nonlinear and temperature–emissivity degeneracy is enhanced.

Representative signal-to-noise conditions for two example pixels (optimal and sub-optimal) are illustrated in Fig. A1, where the measured radiance ± NESR is shown on the same radiance scale for reference, and the corresponding median signal-to-noise ratio (SNR) and 5 %–95 % SNR range are reported explicitly.

As expected, the choice of initial emissivity (ε0) influences the retrieved temperatures. Lower assumed emissivity values yield higher temperature estimates, due to the inverse relationship between emissivity and brightness temperature in the Planck function. In high-SNR observations such as C4, this leads to differences in mean temperature as large as 13 K across the range of tested priors. In contrast, datasets acquired at higher incidence angles (e.g., C1 and C2) exhibit smaller temperature shifts across priors, likely because shadowed regions dominate the radiance signal, flattening the temperature distribution and dampening the sensitivity to ε0.

The uncertainties associated with the retrieved temperatures show a more pronounced dependence on both solar illumination and the width of the emissivity prior. Formal uncertainties and 1σ dispersions are significantly higher in C1 and C2, particularly when using broad priors (e.g., ε0 = 0.70 ± 0.15), reflecting the limited thermal contrast and stronger model degeneracy under low radiance conditions. In contrast, observations with higher insolation – namely C3 and C4 – yield markedly lower formal errors and temperature dispersion. This reduction in spread is not only a function of radiometric quality but also of scene illumination: in C4, the entire observed area is well lit, and the few remaining shadows are due solely to local topography. This homogeneity in thermal forcing minimizes temperature variability across the scene and enhances retrieval precision, reinforcing the robustness of the Bayesian approach under favourable conditions.

Results show that extending the lower bound of the spectral window from 4.5 to 3.0 µm produces a significant change in the character of the inversion (Fig. 4 and Table A2). When using the full 3.0–5.56 µm range, the retrieval yields temperature values that are nearly invariant with respect to the emissivity prior, especially in the well illuminated datasets C3 and C4. In these cases, the mean surface temperatures converge to a fixed value regardless of the prior, and, while the associated 1σdispersion is substantially similar to the 4.5–5.56 µm case, the formal uncertainties collapse to zero, indicating that the inversion algorithm has converged to a narrow solution space (Fig. 5). Although such tight convergence might suggest improved precision, it more likely reflects numerical over-constraint. This behaviour is consistent with an over-dominant contribution from shorter wavelengths – between 3.0 and 4.2 µm – where thermal emission is still weak and the radiance spectrum is more sensitive to the shape of the assumed reflectance curve and potential calibration uncertainties.

Comparing corresponding rows across Tables A1 and A2 shows systematic differences in mean temperatures. For example, in C4 under ε0 = 0.90 ± 0.10, the mean temperature increases from 345.2 K (4.5–5.56 µm) to 348.6 K (3.0–5.56 µm), while in C1 it decreases from 184.3 K (4.5–5.56 µm) to 180.2 K (3.0–5.56 µm). These small shifts reflect the different spectral leverage of the two wavelength ranges, as shorter wavelengths provide enhanced sensitivity to the thermal continuum at higher temperatures. In warm, well illuminated conditions, the additional thermal signal from the 3.0–4.5 µm interval strengthens the fit to the observed radiance, improving the stability of the retrieval where the signal in the 4.5–5.56 µm range alone would begin to saturate.

Despite these potential advantages, the retrieval from the broader 3.0–5.56 µm range may occasionally yield physically implausible outliers. For instance, in C3, a peak temperature of 430.9 K is reported under ε0 = 0.80 – a value well above what is expected for lunar surface conditions at that incidence angle. This suggests that residual artifacts or noise in the shortwave portion of the spectrum may corrupt the inversion under some conditions. These effects are not observed in the results from the 4.5–5.56 µm range, where temperature statistics behave smoothly and remain consistent with physical expectations.

Overall, these findings indicate that while the inclusion of wavelengths between 3.0 and 4.5 µm can, in principle, enhance retrievals in low-radiance scenes by increasing the spectral baseline, this benefit is offset by increased sensitivity to calibration uncertainties and reflectance modelling. In high-SNR, thermally bright observations such as C3 and C4, the narrower 4.5–5.56 µm range appears to offer more stable and physically reliable results. It minimizes sensitivity to the crossover region and avoids the numerical dominance of channels with mixed solar and thermal contributions.

These considerations support a context-dependent approach to spectral selection: the full 3.0–5.56 µm range may prove useful in cold or poorly illuminated scenes, whereas the more conservative 4.5–5.56 µm window is preferable when strong thermal emission is present. Future improvements in the treatment of solar reflectance and absolute calibration in the shortwave infrared may enable more confident use of the full range, but under current processing conditions, restricting the inversion to wavelengths dominated by thermal emission yields more robust and interpretable results.

A final test was performed by applying a Lommel–Seeliger photometric correction to the MAJIS spectra in the range 3.0–5.56 µm before retrieval, with the aim of reducing possible biases linked to viewing and illumination geometry. The Lommel–Seeliger law, originally formulated in the late 19th century to describe single scattering from dark planetary surfaces (Lommel, 1887; Seeliger, 1888), has long been applied to lunar observations as a simple and effective photometric correction. Its role was later formalized within the more general radiative transfer framework of Hapke (1981), where it is recognized as a limiting case particularly suited to the Moon's porous regolith under moderate illumination conditions. The Lommel–Seeliger correction was implemented pixel-by-pixel through the factor μ/(μ+μ0), where μ=cos(i) and μ0 = cos(e) being i and e the solar incidence and emission angle, respectively, and normalizing the observed radiance to a canonical geometry (i = 60°, e = 30°) reproducing the high phase angle (∼ 90°) of the JUICE lunar flyby. However, it is precisely this extreme phase angle that makes the correction sub-optimal, which explains why the retrieved temperatures (Fig. 6 and Table A3) are virtually indistinguishable from those obtained without photometric adjustment. This outcome indicates that, within the spectral interval used for retrieval and at the spatial resolution of MAJIS, photometric effects have little influence on the thermal inversion. The Moon's global shape can be approximated by a smooth ellipsoid at the few-hundred-meter pixel scale of MAJIS, so the primary driver of variability in the thermal signal is the distribution of insolation rather than geometric scattering. The Lommel–Seeliger law, while traditionally adopted for the diffuse scattering properties of the lunar regolith and effective at moderate phase angles where single scattering dominates, becomes less appropriate at the extreme phase angles encountered during the JUICE flyby. In this geometry, multiple scattering, surface roughness, and shadowing effects grow increasingly important, and a single-parameter correction cannot capture such complexity. The absence of significant differences between corrected and uncorrected retrievals is therefore unsurprising and further supports the robustness of the Bayesian thermal inversion under the observing conditions of the JUICE flyby.

For the subsequent intercomparison with the other methods (Sect. 2.5), we adopt the Bayesian retrieval parameters that provide the most physically plausible and stable results: the 4.5–5.56 µm spectral range, an emissivity prior of ε0 = 0.70 ± 0.15, and no photometric correction.

Li and Milliken (2016) developed an empirical thermal correction model tailored for spectra acquired by the Moon Mineralogy Mapper (M3), which operated over the spectral range 0.43–3.0 µm (Pieters et al., 2009). They identified that at wavelengths beyond ∼ 2 µm, thermal emission from the lunar surface contributes significantly to the measured radiance, particularly on the dayside. This thermal component can obscure or distort diagnostic absorption features associated with minerals and hydration, rendering accurate compositional analysis challenging.

To address this, they constructed a model grounded in laboratory measurements of lunar soils and glasses, leveraging the strong empirical correlation between reflectance at 1.55 µm (unaffected by thermal emission) and at 2.54 µm (where thermal emission may be substantial, depending on the surface temperature). This relationship follows a power law, enabling the prediction of the “true” reflectance at 2.54 µm based on the value at 1.55 µm. Any excess radiance at 2.54 µm is attributed to thermal emission, which can then be quantified and removed. Crucially, this approach allows for simultaneous estimation of the surface temperature, without requiring independent thermal measurements from instruments like Diviner, provided the observed materials match common lunar regolith compositions.

This method is particularly useful when applied to MAJIS data acquired in its visible and near-infrared (VISNIR) channel covering the spectral range 0.49–2.36 µm. While this does not extend as far into the thermal regime as M3 (and the MAJIS-IR channel itself), thermal contributions can still become non-negligible near the upper end of this spectral window – especially for observations near local noon or in equatorial regions. Applying the Li and Milliken (2016) model to MAJIS data enables the detection and removal of this thermal component, improving the fidelity of the retrieved reflectance spectra. In doing so, it not only enhances mineralogical interpretations in the near-IR but also yields estimates of surface temperature derived directly from the observed radiance, assuming that a portion of the MAJIS spectrum captures the onset of thermal emission. This makes the approach particularly attractive for MAJIS lunar observations where Diviner-like thermal constraints may be unavailable or spatially mismatched, while allowing a direct comparison with the Bayesian method.

By grounding the correction in the inherent spectral behaviour of lunar soils and leveraging the statistical predictability of thermally unaffected wavelengths, this model provides a robust and computationally efficient tool for disentangling thermal and reflective components in hyperspectral datasets such as those from MAJIS.

Formally, the method proposed by Li and Milliken (2016) relies on two main equations. The first is Kirchhoff's law of radiative equilibrium: 2I=1/πFSunR+IBB(T)(1-R) where I is the total radiance emitted by the surface, FSun the solar irradiance received by the surface, IBB the blackbody radiance at temperature T associated with the surface thermal emission, and R=1-ε the surface reflectance, which is related to the emissivity ε under radiative equilibrium. The second equation is an empirical power-law relationship between the reflectances at 1.55 and 2.54 µm, measured in the laboratory on lunar soil and glass samples under the geometry i = 30°, e = 0°: 3R2.54=1.124×R1.550.8793

Assuming that thermal emission is negligible at 1.55 µm, i.e., I1.55≈1/πFSunR1.55, the thermal emission of the surface – and therefore its temperature – can be directly derived from the radiances measured by MAJIS at 1.55 and 2.54 µm, together with the incoming solar flux, without requiring additional assumptions about the surface emissivity. This makes the method computationally very efficient.

Nevertheless, over the range of lunar surface temperatures, the radiance excess at 2.54 µm can be small or even undetectable due to the radiometric uncertainty of MAJIS (see Langevin et al., 2026), which tends to scatter data points around the power-law curve (Fig. 7). Consequently, all data points lying below the power-law relation correspond to surface temperatures that cannot be reliably estimated – or are highly uncertain – using this empirical approach. This limitation is most pronounced near the terminator (e.g., in C2), where thermal emission is weak and radiometric uncertainties dominate.

To assess the accuracy of the surface temperature estimates, in Fig. 7 we compare our results with those predicted by the two-layer thermal equilibrium model proposed by Vasavada et al. (1999), with updated parameters based on LRO/Diviner observations (Vasavada et al., 2012). Our surface temperature estimates show good agreement with the model for solar incidence angles below 55° but diverge significantly at higher incidence angles. This discrepancy likely arises from unresolved sub-pixel topographic effects (e.g. mixed slopes and shadowing) that are not represented in the ellipsoid-based geometry and can locally bias the measured radiance. Sub-pixel slopes and shadowing can dominate the measured radiance and produce local deviations from the expected thermal behaviour. In the MAJIS observations, this incidence range corresponds to the highlands observed at the end of C3 and the beginning of C4. Part of the observed differences may also reflect the intrinsic mismatch between the MAJIS physical-temperature retrieval and the Diviner brightness-temperature products used for comparison. In terms of accuracy, assuming a 10 % uncertainty in the radiometric flux measured by MAJIS, the corresponding uncertainties in surface temperature are estimated at ± 15 K for C2, ± 7 K for C3, and ± 5 K for C4.

The Li and Milliken (2016) method also allows for an a posteriori determination of emissivity at 5.5 µm by inverting Kirchhoff's law, under the assumption that reflected sunlight at this wavelength is negligible, i.e., I5.5≈ε5.5IBB(T). This capability will be further exploited in Sect. 2.5.2, where emissivity retrievals obtained with different approaches are compared in detail.

An additional approach builds upon the roughness-informed model introduced by Wohlfarth et al. (2023), originally developed to improve our understanding of the lunar diurnal water cycle based on M3 observations. We adopt the treatment of both the reflectance and thermal-emission components as described by Wohlfarth et al. (2023). In this framework, the reflectance contribution is modelled according to the Hapke formalism (Hapke, 2005), adopting the set of Hapke parameters reported by Li and Li (2011), whereas the thermal-emission component is implemented following the approach detailed in Wohlfarth et al. (2023). We note, however, that under the high phase-angle conditions of the JUICE lunar flyby (∼ 90°), no photometric formulation can be regarded as strictly optimal, including Hapke-based corrections; in the present implementation, the Hapke term should therefore be viewed as a physically motivated approximation of the reflectance contribution rather than as a unique or fully constrained solution.

Unlike purely empirical approaches, the model represents sub-pixel roughness in terms of mm–cm scale RMS slope using synthetic fractal surfaces (Rozitis et al., 2020). These surfaces statistically reproduce unresolved topographic variability and allow the simulation of associated effects such as self-heating, sub-pixel shadowing, and multiple scattering. In this fractal representation, the surface is depicted as a series of discrete square plates, called “facets.” The model solves the energy balance for each facet m, where the total flux Fm is the sum of three components: direct solar irradiation, scattered sunlight, and thermal self-heating from other facets. In our case, we divide each MAJIS pixel into a 65 × 65 sub-pixel grid, where the thermal balance is computed individually. The resolution of the sub-pixel facets provides a good compromise between capturing sub-pixel temperatures and computational tractability.

Once a sub-pixel synthetic fractal roughness surface is generated, the thermal contribution of the mth facet is given by the energy balance equation: 4Fm=Fmsun+Fmsca+Fmrad where Fmsun is the flux of directly absorbed solar radiation, Fmsca is the contribution of scattered sunlight from neighbouring sub-pixel facets and Fmrad is the additional flux due to thermal self-heating. The thermal emission related to direct solar irradiation of the mth sub-pixel facet is modelled as: 5Fmsun=(1-Adh)Jm where Adh is the directional–hemispherical albedo for the reflected sunlight (Shkuratov et al., 2011), here computed for each pixel following the workflow of Wöhler et al. (2017) and intrinsically connected to the single-scattering albedo, which is a free parameter of the model, while Jm is the incident solar irradiance on the mth facet. The terms for scattering from neighbouring sub-pixel facets (Fmsca) and thermal self-heating (Fmrad) are, respectively: 6Fmsca=(1-Adh)Adh∑j≠m65×65fm,jJj+(1-Adh)×Adh2∑j≠m65×65fm,j∑k≠j65×65fj,kJk+…7Fmrad=(1-Adh,th)∑j≠m65×65fm,jFj+(1-Adh,th)×Adh,th∑j≠m65×65fm,j∑k≠j65×65fj,kFk+… where fi,j are the geometric view factors (Wohlfarth et al., 2023) and Adh,th is the directional–hemispherical albedo for self-heating (Wohlfarth et al., 2023), which is linked to the effective emissivity ε by Kirchhoff's law (Spencer, 1990). The reported equations are resolved in a matrix-vector notation to account for higher order terms for Fmsca and Fmrad, following the pipeline described by Wohlfarth et al. (2023). Subsequently, the Stefan–Boltzmann law relates the radiant flux of the mth sub-pixel facet Fm to its equilibrium temperature, which can be retrieved jointly with the effective emissivity ε: 8Tm=Fmεσ14

Once the temperatures of the sub-pixel facets are estimated, the thermal emission component linked to the nth pixel is defined as: 9Xn(λ)=∑m65×65Pm(λ)νmcosem∑m65×65νmcosem where Pm(λ) denotes the Planckian thermal emission of the mth sub-pixel facet, computed from its estimated equilibrium temperature; νm is a line-of-sight visibility factor to the sensor and em is the emission angle of the mth facet.

The retrieval is performed in the 5.0–5.5 µm spectral range, which is dominated by thermal emission and largely unaffected by detector saturation. At this stage, single scattering albedo is optimized by minimizing the root mean square error (RMSE) between the modelled reflected radiance plus thermal emission Xn(λ) and the observed MAJIS spectra.

The synthetic fractal surfaces are generated by using as free parameter the mm–cm scale roughness. However, applying a full coupled reflectance–thermal inversion based on synthetic fractal surfaces on a pixel-by-pixel basis over the entire hyperspectral cubes would be computationally demanding and, more importantly, not warranted by the information content of all datasets. We therefore retrieve representative average mm–cm scale roughness values only for selected regions of interest (ROIs), chosen to sample the main terrain units.

In particular, we use the C4 radiance dataset for the mm–cm scale roughness retrieval because this hyperspectral cube provides the most favorable illumination geometry and the strongest thermal contribution. As a result, the modeled radiance exhibits the highest sensitivity to mm–cm scale roughness. In contrast, for cubes C1 and C2 the thermal signal is comparatively weak and the measured radiance is largely dominated by noise, which prevents a robust constraint on surface roughness and introduces a strong degeneracy with the single-scattering albedo. For this reason, retrieving mm–cm scale roughness independently for C1 and C2 would lead to poorly constrained solutions with limited physical interpretability. Instead, we adopt the terrain-representative mm–cm scale roughness values derived from the C4 cube for the analysis of the other datasets.

Nevertheless, degeneracies intrinsic to thermophysical models may still affect the retrieval even for C4. To assess and mitigate this effect, we performed 20 independent simulations for each ROI, varying the initial guesses for both roughness and single-scattering albedo. These simulations generally converge to slightly different solutions without any evident outlier, which can be used to estimate the uncertainties of the retrieved parameters and to evaluate the impact of model degeneracy. Within this framework, the mm–cm scale roughness is found to be well constrained, whereas the single-scattering albedo shows a maximum variability of about 20 % within both ROIs.

The ROIs selected to constrain the mm–cm scale roughness and single-scattering albedo at 5.5 µm in the C4 hyperspectral cube are listed in Table 2. The retrieved mm–cm scale roughness values are then compared with meter-scale roughness estimates derived from the LOLA Digital Roughness Map (LRO/LOLA LDRM_32 data; Smith et al., 2010, 2017). Both quantities are expressed as terrain RMS slope: the spatial distribution of the LOLA-derived meter-scale roughness is shown in Fig. 8, while the ROI-averaged values are reported in Table 2.

We assume that the average mm–cm scale roughness values reported in Table 2 are representative of the corresponding terrain units (mare-like and highland-like) across the C1–C4 hyperspectral cubes. These roughness values are expected to be comparable within the JUICE/MAJIS footprints, consistent with the behaviour observed across the lunar surface, as indicated by previous studies (e.g., Wohlfarth et al., 2023). With this approach, we account for realistic mm–cm scale average roughness values across different terrain types while avoiding a full pixel-by-pixel retrieval, which would be computationally expensive and strongly affected by parameter degeneracy in low-SNR hyperspectral cubes, as for C1 and C2. Results indicate that the average mm–cm scale roughness values are overall higher in terms of RMS slope than those inferred at the meter scale.

This outcome is broadly consistent with results from previous studies, which generally report roughness values of ∼ 20–35° depending on observation geometry and retrieval approach. Near-nadir observations typically yield mm–cm scale roughness estimates of ∼ 20–25° (e.g. Bandfield et al., 2015; Wohlfarth et al., 2023), whereas more oblique configurations commonly give higher values, generally around ∼ 30–35° (e.g. Rubanenko et al., 2020; Müller et al., 2021). The values retrieved in this study (Table 2) fall within the lower-to-intermediate part of this range. This is noteworthy because the roughness retrieval is based on C4, whose emission angles (Table 1) correspond to distinctly off-nadir viewing geometry rather than a near-nadir configuration. The comparatively moderate values obtained here may therefore reflect differences in instrument characteristics, phase angle (∼ 90°), spectral range, and modelling assumptions relative to previous studies. In addition, effective incidence angles are computed by combining the observing geometry with local terrain slopes from LOLA Digital Slope Map (LRO/LDSM_16 data) (Smith et al., 2010, 2017), thereby better reproducing the actual physical illumination conditions (Fig. 9).

After constraining the mm–cm scale roughness in specific ROIs (Table 2), temperatures retrieved from each MAJIS hyperspectral cube are reported in Fig. 10. The retrieved temperatures exhibit a clear spatial correlation with the effective illumination conditions (Fig. 9). The highest temperature value (383 K) is observed in the northwestern sector of C4, consistent with the local incidence angles and in agreement with the location of maximum temperature identified by the other two methods. Furthermore, low-albedo terrains exhibit higher surface temperatures than brighter terrains at the same local solar time. This behaviour is evident in the transition region between Mare Tranquillitatis and the adjacent highland-like terrain in C4 (Fig. 10). Conversely, the lowest temperature values (< 150 K) occur in shadowed regions of C1 and C2, corresponding to pixels characterized by effective incidence angles exceeding 90° (Fig. 10).

To further assess the influence of roughness assumptions, we perform a similar analysis by assuming the mm–cm scale roughness to be equal to the meter-scale roughness measured by LRO/LOLA (Fig. 8). This test is intended to evaluate the similarities and differences in the resulting temperature estimates.

In our case, the mm–cm scale roughness is significantly higher than the meter-scale roughness (Table 2), which may lead to different contributions from self-heating, self-shadowing, and scattered sunlight. A direct comparison between the temperatures retrieved under the two assumptions for C3 and C4 is shown in Fig. 11.

The results shown in Fig. 11 highlight the differences between a full thermophysical inversion, in which the average sub-pixel roughness is retrieved, and the assumption that mm–cm scale roughness matches the meter-scale roughness derived from LRO/LOLA measurements (Fig. 9). Within the longitude range between 22.5 and 46.0°, the retrieved temperatures are partially comparable within the associated uncertainties – defined as the 1σ dispersion after binning the data with a longitudinal step of 0.1° – although temperatures derived using the mm–cm scale roughness are on average slightly higher. This result suggests that self-heating, self-shadowing, and scattering effects are mitigated – and therefore less sensitive to surface roughness – at low incidence angles, as expected from previous studies (Spencer, 1990; Wohlfarth et al., 2023).

Beyond 46° longitude, the two approaches begin to diverge (Fig. 11), with temperature differences exceeding the associated uncertainties and therefore no longer comparable within the 1σ error bars. In particular, the skin temperature retrieved using the mm–cm scale roughness values derived from the thermophysical modelling (Table 2) is systematically higher than that obtained when the meter-scale roughness derived from LRO/LOLA is adopted instead. This behaviour indicates that the terms dominating Eq. (4) become increasingly sensitive to micro-scale slopes at higher incidence angles. Higher roughness values correspond to steeper slopes at the mm–cm scale, which enhance the radiative interaction between facets and increase the contribution of the self-heating term, as expected (Spencer, 1990). The largest variation is observed for C3 at longitude of 56.2°, corresponding to a temperature difference of (15.3±11.8 K). The spatial distribution of the temperature differences between the two assumptions is reported in Fig. A2.

To place the MAJIS thermal observations into a broader lunar context, we compared our results with co-located datasets from the Diviner Lunar Radiometer Experiment aboard LRO (Williams et al., 2017). Diviner provides radiance and brightness temperature for channels 3–6 (7.8–25 µm), along with derived emissivity maps and the Christiansen Feature (CF) wavelength. For this work, we gathered Diviner radiances for channels 3–6 acquired under sub-solar longitudes between -26 and +18°. Following Williams et al. (2018), the field of view of each measurement was modelled using a Monte-Carlo distribution of 50 points to compute its effective footprint.

Radiances were then adjusted to the solar distance corresponding to the JUICE flyby (1.014 au) under the assumption of radiative equilibrium, and subsequently gridded at 128 pixel per degree. Because Diviner does not provide full spatial coverage at the exact solar geometry of the MAJIS observations (sub-solar longitude -4.1°, sub-solar latitude 1.0°), we estimated gap-free radiances by linearly interpolating the observations bracketing each pixel in sub-solar longitude. This procedure reproduces the large-scale thermal structure well, though orbit-to-orbit striping remains visible in places (Fig. 12). CF position and peak brightness temperature were determined by parabolic fitting of channels 3–5 (Greenhagen et al., 2010), and channel emissivities were evaluated relative to the CF peak temperature. We note that Diviner products provide brightness-temperature quantities (including the peak brightness temperature around the CF) rather than a direct estimate of physical thermodynamic surface temperature. In this work, the Diviner peak brightness temperature is used as a practical proxy for the surface thermal state, since it is derived near the CF where the effective emissivity is expected to be close to unity and the dependence on emissivity is reduced compared to single-channel brightness temperatures. This approach highlights compositional and textural contrasts between mare and highlands, although emissivity estimates at large incidence angles (i> 60°) are strongly affected by photometric roughness effects, particularly in channel 6.