Vicarious Calibration of Thermal Infrared Imagers Using Thermophysical Models of the Lunar Surface

Highlights

What are the main findings?

• Lunar TPMs enable calibration transfer of thermal infrared imagers with a transfer error bounded by 4% in the LWIR and 6% in the MWIR.
• MWIR calibration accuracy at wavelengths below 5 μm is limited by the radiometric scale of the reference mission (SLSTR), with a 5.2% inter-sensor disagreement between Sentinel-3A and Sentinel-3B at elevated brightness temperatures.

What is the implication of the main finding?

• The Moon provides a practical high-temperature calibration reference for thermal infrared missions lacking onboard blackbody calibration.

Abstract

Wildfire detection and characterization from space critically depend on accurate thermal infrared measurements across a wide dynamic range. OroraTech’s SAFIRE payloads feature mid-wave infrared (MWIR) and long-wave infrared (LWIR) bands optimized for this purpose, yet the volume and power constraints of many CubeSat platforms preclude the use of onboard calibration sources. This reflects a broader limitation of current Earth observation systems: the absence of robust calibration and validation methodologies at high brightness temperatures relevant to active fires. We evaluate the potential of using thermophysical models (TPMs) of the Moon as a vicarious calibration reference for calibration transfer between instruments. The Moon reaches surface temperatures of up to 400 K, offering a stable target in the thermal domain that is visible from many different orbits. Previous research has established the use of TPMs for Moon observations in the infrared; however, uncertainties in surface properties remain, particularly in the mid-wave infrared. We investigate these effects using SAFIRE observations of the lunar surface acquired over a wide range of lunar phase angles (from waxing $-81.5°$ to waning +122.2°). We constrain the TPMs using observations from the Sentinel-3 Sea and Land Surface Temperature Radiometer (SLSTR) fire channels and uncover a 5.2% inter-sensor disagreement in the MWIR between Sentinel-3A and Sentinel-3B at high brightness temperatures, corresponding to ∼2.2 K at 400 K. Applying the same methodology to observations of our SAFIRE payloads bounds the calibration transfer error at 5.8% in the MWIR and 4.2% in the LWIR for lunar phase angles within ±45°. These results establish lunar vicarious calibration as a viable approach in the thermal domain for high-temperature applications, providing a pathway toward improved fire radiative power retrievals and enhanced global wildfire monitoring.

1. Introduction

Accurate radiometric calibration at high brightness temperatures remains a fundamental challenge for thermal infrared Earth observation, particularly for small satellite platforms that lack onboard calibration sources. This limitation directly impacts applications such as wildfire detection, where signals often exceed the calibrated range of conventional systems. While vicarious calibration using natural targets offers a potential alternative, suitable references in the thermal domain are scarce. The Moon provides a unique opportunity in this context, combining geometric stability, global observability, and surface temperatures reaching up to 400 K.

The Moon has long served as a radiometric reference for spaceborne instruments because of its geometric stability, lack of atmosphere, long-term photometric consistency and well-known physical and thermal properties. In the visible and near-infrared, disk-integrated irradiance models such as the ROLO framework provide phase- and libration-dependent reflectance predictions suitable for calibration transfer [1]. In the mid-wave and long-wave infrared (MWIR/LWIR), however, the dominant signal is thermal emission rather than reflected sunlight. Consequently, radiometric prediction requires physically based thermophysical modeling of the lunar surface rather than purely photometric approaches such as the modeling of reflectance. State-of-the-art lunar thermophysical models (TPMs) solve the one-dimensional heat conduction equation in a regolith column subject to a time-varying surface energy balance. At the surface, absorbed solar irradiance, thermal reradiation, and conductive flux into the subsurface are balanced; at depth, the boundary is typically assumed thermally insulating or fixed at a geothermal gradient. Diviner observations have demonstrated that the upper lunar regolith is not homogeneous: bulk density and thermal conductivity increase significantly within the upper few centimeters, affecting nighttime cooling rates [2]. Global inversion of Diviner nighttime temperatures has further constrained near-surface thermal inertia and depth-dependent conductivity profiles, providing globally consistent thermophysical parameters for forward modeling [3].

A key source of uncertainty in lunar thermal emission modeling is small-scale surface roughness. Roughness modifies both absorbed solar flux (through facet orientation and shadowing) and emitted radiance (through anisothermality and mutual self-heating). Diviner-based studies have shown that lunar thermal infrared measurements are strongly influenced by RMS slope distributions on millimeter to centimeter scales, and that roughness effects can dominate directional brightness temperature variations [4]. Thermophysical treatments of rough surfaces typically employ statistical facet distributions or crater-based self-heating formalisms originally developed for airless bodies (e.g., beaming models [5]). These approaches are directly relevant for calibration scenarios where emission angle and phase angle differ from nadir daytime geometries. Topographic self-heating is especially critical in high-latitude or cratered terrain. Earlier crater-resolved thermophysical modeling demonstrated that multiple scattering of both sunlight and thermal infrared radiation within topography can substantially alter surface temperature distributions [6]. Such effects become important when modeling disk-integrated irradiance at large phase angles.

In addition to bulk thermophysical properties, spectral emissivity must be treated carefully in MWIR/LWIR simulation. Diviner-derived Christiansen feature mapping illustrates that emissivity varies systematically with composition and requires correction for temperature-dependent and photometric effects [7]. Laboratory and modeling studies further show that emissivity measured under ambient terrestrial conditions can differ from that under lunar vacuum and strong vertical thermal gradients. Radiative transfer modeling of near-surface gradients in particulate media demonstrates that thermal emission spectra are modified by the top few hundred microns of regolith, requiring physically consistent treatment in forward models [8]. Experimental validation under simulated lunar conditions confirms that emissivity spectra shift measurably when thermal gradients and vacuum are included [9,10].

Beyond disk-resolved thermophysical studies, several works have explicitly addressed disk-integrated lunar thermal emission for calibration applications. Clementine LWIR data provided early global brightness temperature mapping near 8–9 µm, establishing baseline thermal behavior for midday conditions [11]. More recently, full-disk thermophysical modeling has been validated directly against spaceborne infrared sounders. Müller et al. benchmarked a physically based TPM against High-resolution Infrared Radiation Sounder (HIRS) lunar intrusion measurements in 19 channels spanning 3.75–15 µm and a wide phase-angle range, achieving agreement at the 5% level (and better than 10% at $\lambda$ < 5 µm) when incorporating wavelength-dependent emissivity and realistic roughness assumptions [12]. This study explicitly positions the Moon as a calibration target for infrared instrumentation and demonstrates that asteroid-style TPM frameworks can be adapted successfully to lunar full-disk irradiance prediction. Complementing this disk-integrated validation, Wohlfarth et al. introduced a combined reflectance and thermal radiance model that treats unresolved roughness using rough fractal surfaces and includes self-scattering, self-heating, and disk-resolved bolometric albedo for whole planetary disks [13]. They validated the approach with disk-resolved lunar observations from Gaofen-4 in the 3.5–4.1 µm range (where reflected and thermal components overlap) and with Diviner thermal channels at 8.25 µm and 25–41 µm, reporting nearly exact agreement; they further demonstrated consistency with BepiColombo/MERTIS lunar flyby radiance profiles and discuss implications for emissivity calibration and thermal-excess correction workflows [13].

Together, these works establish a mature thermophysical modeling framework capable of predicting disk-integrated lunar irradiance in the MWIR and LWIR at calibration-relevant accuracy. In this work, we leverage the models of Müller et al. and Wohlfarth et al. to demonstrate radiometric calibration transfer between the SLSTR and SAFIRE instruments. This approach is particularly relevant for sensors lacking onboard calibration sources and for high-temperature regimes where conventional calibration is limited. We show that lunar thermophysical models enable calibration residuals on the order of a few percent, supporting their use for operational thermal infrared missions.

2. Reference Datasets

This work is based on observations from the Sea and Land Surface Temperature Radiometer (SLSTR) onboard Sentinel-3, as well as the proprietary SAFIRE instrument onboard OroraTech’s satellites. Table 1 summarizes the total count of observations used.

Table 1. Total count of lunar observations from different instruments in the used reference datasets.

Instrument	# Sensors	# Obs. MWIR	# Obs. LWIR	Datetime Range	Lunar Phase Angle Range
SLSTR	2	20	20	31 August 2023–9 August 2025	$+5.2°$–$+6.9°$
SAFIRE	15	65	83	30 May 2024–8 October 2025	$-81.5°$–$+122.2°$

2.1. Lunar Observations of SLSTR

The SLSTR is a dual-view conical scanning radiometer aboard the Sentinel-3A (S3A) and Sentinel-3B (S3B) satellites, developed within the Copernicus program to provide high-accuracy measurements of sea and land surface temperature. The instrument builds upon the heritage of the (A)ATSR series and is specifically designed for climate-quality observations, with a radiometric stability of 0.1 K/decade and onboard calibration targets [14]. SLSTR features nine spectral bands spanning the visible, near-infrared, short-wave infrared, and thermal infrared regions, with spatial resolutions of 500 m (VIS/SWIR) and 1 km (TIR) at nadir and a swath width of approximately 1400 km. A key characteristic of SLSTR is its dual-view geometry, providing near-nadir and backward views to enable improved atmospheric correction and enhanced temperature retrieval accuracy. The instrument employs a rotating scan mirror, allowing near-simultaneous acquisition across all spectral channels with minimal temporal offset. Radiometric calibration is ensured through onboard blackbody references for the thermal channels and solar diffusers for the reflective bands. The thermal channels are calibrated against two onboard blackbodies: an unheated blackbody near the passive instrument thermal background (∼260 K) and a heated blackbody at approximately 302 K [15], supporting its role in long-term climate data records and operational products such as sea surface temperature, land surface temperature, fire radiative power, and aerosol properties. The Optical Mission Performance Cluster (Opt-MPC) of ESA has provided us with observations of the lunar surface from both SLSTR instruments. These lunar observations are done only in the fire bands, due to their extended dynamic range compared to the standard thermal channels. The F1 band provides accurate Level-1 quantification over 300–480 K [16], well above the saturation of the standard MIR channel S7 at ∼312 K; F2 likewise employs an extended dynamic range relative to channel S8 [14], though a specific saturation brightness temperature is not publicly documented. Figure 1 shows the spectral response function of the two fire bands F1 and F2.

Figure 1. Spectral response functions of both SLSTR instruments onboard Sentinel-3A and 3B [17].

Figure 2 shows exemplary lunar observations of SLSTR in the F1 and F2 bands. We use Planck’s law to convert the brightness temperature observations to spectral radiance. The retrieved spectral radiance is defined at the effective wavelength of its respective band. The line-scanner detector of SLSTR oversamples the lunar target in the along-track direction, due to its design for nadir operations. The oversampling in along- (ALT) and across-track (ACT) direction are taken into account in Equation (1) when aggregating the spectral radiance L of the lunar disk in each band. We estimate the oversampling in the ALT direction using the in situ pitch rate and integration time for each of the N pixels. The IFOV $\theta$ of each observed pixel is identical per observation given the design of the detector and can be factored out from the sum. All lunar observations of SLSTR have been acquired at lunar phase angles between $+5.2°$ and $+6.9°$.

$$E={\int }_{\mathsf{\Omega }}Ld\mathsf{\Omega }={\theta }_{\mathrm{alt}}{\theta }_{\mathrm{act}}{\displaystyle \sum _i^N}{L}_i$$

(1)

Figure 2. Exemplary observation of the lunar surface from SLSTR onboard Sentinel-3B in November, 2023. The two spectral channels can be acquired quasi-simultaneously given the scanning mode of the instrument. The observations were provided by the Opt-MPC.

We can describe the uncertainty of the disk-integrated irradiance $u\left(E\right)$ as shown in Equation (2), assuming near exact knowledge of the IFOV in ACT direction ${\theta }_{\mathrm{act}}$ given by the geometry of the sensor and the scene. The uncertainty of the IFOV in ALT direction $u\left({\theta }_{\mathrm{alt}}\right)$ is driven by the uncertainty of the pitch rate, which we obtain as statistical uncertainties (type A) from the Attitude Determination and Control System (ADCS) data at observation time. The uncertainty of the pixel-wise radiance $u\left(L_i\right)$ is obtained by propagating the pixel-wise uncertainties of the brightness temperatures $u\left(T_i\right)$, as given by the observation. The Algorithm and Theoretical Basis Document (ATBD) [18] for the uncertainties of SLSTR reports around 0.2 K uncertainty for brightness temperatures captured by the fire bands close to 400 K. In practice, the radiometric uncertainty of the radiance dominates over the geometric uncertainties driven by the pointing stability of the instrument.

$$u^2\left(E\right)={\left({\theta }_{\mathrm{act}}{\displaystyle \sum _{i=1}^N}{L}_i\right)}^2{u}^2\left({\theta }_{\mathrm{alt}}\right)+{\left({\theta }_{\mathrm{alt}}{\theta }_{\mathrm{act}}\right)}^2{\left({\displaystyle \sum _{i=1}^N}u\left(L_i\right)\right)}^2$$

(2)

2.2. Lunar Observations of SAFIRE

The SAFIRE instrument is the current generation of TIR imagers developed by OroraTech. Its design was centered around the task of wildfire detection, which put emphasis on long-wave and especially mid-wave thermal infrared bands. Additionally, the imagers were designed to maximize coverage and thus allow for a high revisit rate with fewer spacecraft. The current generation of imagers features three spectral bands at a 200 m GSD with around 420 km swath when looking nadir; when observing the Moon, the lunar disk spans approximately 25 pixels across its diameter, corresponding to a spatial sampling of ∼140 km per pixel projected onto the lunar surface. The imagers are deployed on a fleet of cubesats in LEO at approximately 550 km (height). Our calibration efforts focus on the mid-wave (M0) and second of the long-wave bands (L2), hence their applications in wildfire and land surface temperature products. The SAFIRE detectors saturate at approximately 650 K, though the exact value depends on the detector operating mode and other dynamic system properties. The spectral response function of each of the used instruments is shown in Figure 3. The MWIR bands show a small leakage in the LWIR region, which is due to design tradeoffs that we made during the selection of the filter material. The leakage will be removed in future instrument iterations by improved filter design. Unlike SLSTR, we can only observe the lunar surface in one specific spectral band at a given time.

Figure 3. Mean and standard deviation of the relative spectral response functions of all 15 SAFIRE imagers used in this study. The MWIR filters show a leak in the LWIR spectral range. The leakage has been removed in the latest sensor generation.

The incident radiance of the lunar surface is converted to an analogue voltage signal by the uncooled microbolometers on the instrument’s focal plane array. The microbolometer pixels respond linearly to incident radiance within their operating range and have a fixed noise floor that is mostly independent of the incoming signal [19]. This is because noise in uncooled microbolometer systems is governed by electrical and thermal fluctuations of the sensor, and not bound to the counting statistics of the individual photons. The raw measurement ${\mathrm{DN}}_{\mathrm{raw}}$ is corrected for electrical drifts of the readout amplifiers ${\mathrm{DN}}_{\mathrm{cds}}$, the self-emission of the uncooled instrument ${\mathrm{DN}}_{\mathrm{self}}$, as well as relative changes in the gains of the individual microbolometer pixels ${\alpha }_{\mathrm{flatfield}}$. The absolute radiometric gain ${\alpha }_{\mathrm{gain}}$ converting DN to radiance is then derived during the calibration described by this work. A simplified radiometric model of the SAFIRE imager is described in Equation (3). The thermal self-emission is inferred using the temperatures T given by sensors on the telescope structure, as well as temporal dynamics t acquired by taking shutter frames during the observation. Measurements of extended targets when pointing at Earth’s surface further require correction of diffuse straylight contributions, which are omitted in this model. We assume these straylight effects to be negligible as the Moon covers only a small fraction of the instrument’s field of view.

$$L=({\mathrm{DN}}_{\mathrm{raw}}-{\mathrm{DN}}_{\mathrm{cds}}\left(t\right)-{\mathrm{DN}}_{\mathrm{self}}(t,T))/{\alpha }_{\mathrm{flatfield}}/{\alpha }_{\mathrm{gain}}$$

(3)

Figure 4 shows an exemplary lunar observation of SAFIRE. We derive residual background, after applying the radiometric model, by averaging the deep space signal in an annular region surrounding the lunar crop. The per-pixel residual background is subtracted from all pixels before integration. The summation region is a fixed crop centered on the lunar centroid that encompasses the full disk, including the unilluminated portion; after background subtraction, unilluminated pixels contribute negligibly to the sum. The disk-integrated spectral irradiance E is then derived by summing all N pixels within the crop using the known instantaneous field of view (IFOV) ${\omega }_i$ of the respective imager and pixel i, as shown in Equation (4). A curve-of-growth analysis varying the summation radius around the centroid confirms that the disk-integrated irradiance plateaus with a scatter below $0.1\%$, demonstrating that pixel discretization and disk-extraction errors are negligible.

$$E={\int }_{\mathsf{\Omega }}Ld\mathsf{\Omega }={\displaystyle \sum _i^N}{\omega }_i·L_i$$

(4)

Figure 4. Exemplary observations of the lunar surface as seen from SAFIRE onboard FOREST-2 for two different dates in 2025. The samples were corrected by applying the radiometric gain derived in this work. The M0 sample was taken at $-33.0°$ and the L2 sample at $+3.2°$ lunar phase respectively.

Preliminary analysis indicates a radiometric uncertainty of approximately 1–2% prior to the application of the gain factor ${\alpha }_{\mathrm{gain}}$. The full propagation of uncertainties through the SAFIRE radiometric model depends on the uncertainty of ${\alpha }_{\mathrm{gain}}$, which is itself derived within this work. To avoid circular dependence, the uncertainty associated with ${\alpha }_{\mathrm{gain}}$ is not iteratively propagated within the current analysis. The lunar observations of SAFIRE have been acquired at lunar phase angles between $-81.5°$ and $+122.2°$.

3. Modeling

We test and compare two independently developed TPMs within this work. Both models simulate the reflection and emission of thermal radiance from the lunar surface in the mid- to long-wave infrared regime. They share a common physical foundation based on radiative energy balance and subsurface heat conduction, but differ in their treatment of surface roughness, spatial resolution, and radiative transfer. Both models account for the illumination and observing geometries at the time of simulation.

• TPM Müller (2021) [12] models the Moon as a single effective thermal emitter using a disk-integrated formulation and a wavelength-dependent emissivity with a common roughness parameter.
• TPM Wohlfarth (2023) [13,20] models the Moon as a spatially resolved, rough, radiatively interacting surface, accounting for surface geometry and coupling between reflected and emitted radiation.

3.1. Shared Physics of Modeling the Lunar Surface

At the core of thermophysical planetary models is the fundamental energy balance equation. At any given point, the surface temperature T is governed by the equilibrium between incident radiation, subsurface conduction, and thermal emission

$$(1-A_{\mathrm{dh}})(S_{0}cos\left(i\right)+F_{\mathrm{sca}})+(1-A_{\mathrm{dh},\mathrm{th}})F_{\mathrm{rad}}+k{\left.{\displaystyle \frac{\partial T}{\partial z}}\right|}_{z=0}=ϵ\sigma T^4$$

(5)

where $A_{\mathrm{dh}}$ is the bolometric albedo, $S_0$ is the solar constant at 1 AU, $\sigma$ is the Stefan–Boltzmann constant, $ϵ$ is the bolometric emissivity, and k is the thermal conductivity. This formulation accounts for secondary radiative contributions from adjacent topography via self-scattering $F_{\mathrm{sca}}$ and self-heating $F_{\mathrm{rad}}$, similarly to [21]. Figure 5 visualizes the interaction of surface geometry and the different contributions of radiative transfer in the TPM. Note that the bolometric albedo for thermally emitted radiation, $A_{\mathrm{dh},\mathrm{th}}$, does not necessarily equal the solar bolometric albedo, $A_{\mathrm{dh}}$. This is because the two parameters are integrated over distinct spectral energy distributions. Specifically, $A_{\mathrm{dh}}$ is weighted by the solar irradiance spectrum $E_0\left(\lambda \right)$, while $A_{\mathrm{dh},\mathrm{th}}$ is weighted by the thermal emission spectrum $X\left(\lambda \right)$ of the body itself. Because the directional-hemispherical reflectance $r_{\mathrm{dh}}(i,\lambda )$ is wavelength-dependent, these integrals yield different values; for the thermal infrared regime, we typically assume $A_{\mathrm{dh},\mathrm{th}}\sim 0.05$ (see also Equations (14) and (15) in [13]). The last term represents conductive heat flux into the subsurface. The temporal evolution of the temperature profile $T(z,t)$ is described by the one-dimensional heat conduction equation

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{\partial }{\partial z}}\left(k{\displaystyle \frac{\partial T}{\partial z}}\right)$$

(6)

where $\rho$ is the bulk density, and c is the specific heat capacity. Together, they define the thermal inertia $\mathrm{\Gamma }=\sqrt{k\rho c}$, which controls the amplitude and phase of the diurnal temperature cycle. In reality, planetary surfaces are rough, i.e., they exhibit topographic structure down to scales of millimeters and below. These small-scale structures cannot be resolved by infrared detectors. Consequently, the measured radiance at the detector is the superposition of the thermal radiance of multiple facets with individual temperatures and viewing geometries. The thermally emitted spectral radiance $X_{\mathrm{n}}\left(\lambda \right)$ thus becomes

$$X_{\mathrm{n}}\left(\lambda \right)={\displaystyle \frac{{\sum }_{\mathrm{j}}P(\lambda ,T_{\mathrm{j}})b_{\mathrm{j}}{v}_{\mathrm{j}}cos\left(e_{\mathrm{j}}\right)}{{\sum }_{\mathrm{j}}{b}_{\mathrm{j}}{v}_{\mathrm{j}}cos\left(e_{\mathrm{j}}\right)}},$$

(7)

where P denotes the Planck function, $b_j$ is the area of a small surface facet, and $v_j$ denotes whether the facet j can be seen by the observer. The total radiance I that emerges from an individual facet is expressed as the sum of reflected solar and thermally emitted components, governed by illumination, viewing geometry, and surface properties.

$$I(i,e,\psi ,\lambda )=r(i,e,\psi ,\lambda )·E_0\left(\lambda \right)+{ϵ}_{\mathrm{d}}(\lambda ,e)·X(i,e,\psi ,\lambda ,T)$$

(8)

Figure 5. Geometric conventions for a thermal roughness model of TPM Wohlfarth. Blue arrows indicate the mapping from planetary facets to the unresolved rough surface and observing geometry; yellow dashed arrows denote incident solar radiation, red arrows denote thermal radiation, and black arrows denote facet-to-facet radiative exchange. (Left): The planetary model is divided into N facets indexed with n. (Middle): Each facet is associated with a fractal surface with M = 200 × 200 elements with edge width 1 mm. (Right): Relationship between physical quantities. Adopted from [13,21].

Here, $E_0\left(\lambda \right)$ denotes the incident solar spectral irradiance, $r(i,e,\psi ,\lambda )$ is the bidirectional reflectance, and ${ϵ}_{\mathrm{d}}\left(\lambda \right)$ is the spectral directional emissivity. The variables i, e, and $\psi$ represent the angles of incidence, emission, and azimuth, respectively, and X is the thermally emitted radiance of a perfect emitter with unresolved surface roughness at temperature T. All in all, rough surfaces lead to deviations from ideal single-temperature blackbody behavior, particularly at large phase angles and at shorter wavelengths at the Wien side of the Planck function. The relative contributions of reflected and emitted radiation are strongly wavelength-dependent. Reflected radiation dominates in the visible and near-infrared (NIR), while thermal emission dominates in the thermal infrared. In the transition region between approximately $2.5$ and 7 μm, both components contribute significantly and must be treated jointly [22].

3.2. Distinct Features of TPMs

Despite their shared physical basis, the two TPMs differ in their implementation. The model of Müller et al. (2021) was originally developed for asteroid thermophysical modeling [12,23,24,25,26] and has been successfully applied to study a variety of Near-Earth Asteroids, Main-Belt Asteroids, and Trans Neptunian Objects. For studying the Moon, the model parameters can simply be adjusted to lunar conditions. Lunar radiance is computed by integrating an oblate spheroidal shape model with its true spin pole, and assuming global photometric parameters. Surface roughness is modeled with hemispherical segments and parameterized via the rms-slope, including shadowing and mutual heating effects. The model employs an empirically adjusted, wavelength-dependent hemispherical emissivity to reproduce observed disk-integrated radiances. This approach prioritizes computational efficiency and robustness, making it well suited for calibration applications involving spatially unresolved observations.

The model of Wohlfarth (2023) has been primarily designed for disk-resolved simulations of the lunar surface [13]. It spatially resolves the photometric behavior of the lunar surface and models surface roughness with fractal descriptions of subpixel slopes. Each subpixel surface element is treated individually, accounting for its local illumination and viewing geometry. The model includes shadowing, mutual heating, and self-scattering effects, enabling a physically consistent treatment of anisothermality and directional emission. In addition, it explicitly separates reflected and emitted radiance, which is particularly important in the MWIR (<5 μm) where both contributions are significant.

These differences result in complementary strengths. The Müller model provides accurate and efficient predictions of disk-integrated radiance and is therefore well suited for radiometric calibration and cross-instrument comparison. The Wohlfarth model, on the other hand, captures the physical effects of surface roughness and radiative interaction in greater detail, enabling improved interpretation of phase-angle dependencies and spatially resolved observations. In this work, both models are evaluated in the context of lunar vicarious calibration. Their differing levels of physical complexity allow us to assess the sensitivity of calibration results to model assumptions, particularly with respect to emissivity, roughness, and the treatment of mixed reflected and emitted radiance in the mid-wave infrared.

3.3. Previous Validation of TPMs

The TPM Müller [12] model was validated using spaceborne observations of the Moon obtained from infrared sounders, in particular, lunar intrusion measurements from the High-resolution Infrared Radiation Sounder (HIRS) instrument series. These observations provide disk-integrated radiances of the Moon across multiple spectral channels in the mid- to long-wave infrared, covering a wavelength range from approximately 3.75 to 15 μm and a wide range of phase angles of ±73°. The model was evaluated by comparing simulated disk-integrated radiances with measured brightness temperatures across 19 HIRS channels for multiple observation geometries. Particular emphasis was placed on the ability of the model to reproduce the spectral shape and phase-angle dependence of the observed lunar signal. The model employs a wavelength-dependent hemispherical emissivity, which is adjusted to match measured radiances over the full spectral range. Validation results demonstrate that the TPM is capable of reproducing observed lunar radiances with an accuracy on the order of ∼5% in most thermal infrared channels, with somewhat larger deviations (up to ∼10%) in the shorter wavelength range below 5 μm. The model was shown to capture the overall phase-angle dependence and spectral behavior of the lunar emission, supporting its applicability for radiometric absolute calibration of spaceborne infrared instruments using disk-integrated observations.

Further, the TPM Wohlfarth [13,20] model was previously validated using independent disk-resolved spaceborne observations of the Moon. Lunar observations from the Gaofen-4 geostationary satellite were used for model fitting and validation in the mid-infrared. The dataset consists of repeated disk-resolved observations acquired in 2018 [22] in a single MWIR channel spanning 3.50–4.10 μm (centered at 3.77 μm) at approximately 4 km spatial resolution projected onto the lunar surface. The measurements span moderate phase angles ($-30.09°$ to $-26.92°$) and sample the full range of incidence and emergence angles, from 0° at the sub-solar or sub-spacecraft point to 90° at the illuminated or visible limb, respectively. Measurements from the Diviner Lunar Radiometer [27] were used to validate the model in the thermal infrared. Diviner observes reflected solar radiation and lunar thermal emission across nine spectral channels spanning 0.3–400 μm, including channels sensitive to the Christiansen feature at daytime temperatures and broadband channels capturing long-wavelength emission. Model performance was assessed using nadir observations and Emission Phase Function (EPF) observations, which sample a wide range of emission and azimuth angles under nearly constant solar incidence, providing strong constraints on surface roughness effects. EPF sequences comprise a four-minute pattern of off-nadir pointings, with emission angles varying in nine steps between ∼0° and 75° and azimuth angles divided into two groups depending on the subsolar position. These multi-angular measurements are particularly suited to test thermal models under varying combinations of incidence, emission, and azimuth angles [4]. For validation, Refs [13,20] used the same EPF datasets as in [4].

3.4. Evaluation of Models

Both TPMs are evaluated for a given tuple of observer position, datetime and spectral band. The models operate at a fixed wavelength grid spacing of 0.1 µm over the interval of 3 µm to 15 µm. The calculated spectral flux $E_{\lambda }$ is then normalized to the effective wavelength of the respective band by integration over its SRF to obtain an effective spectral flux $E_{\mathrm{TPM}}$ as shown in Equation (9).

$$E_{\mathrm{TPM}}={\displaystyle \frac{\int \mathrm{SRF}\left(\lambda \right)·E_{\lambda }d\lambda }{\int \mathrm{SRF}\left(\lambda \right)d\lambda }}$$

(9)

Figure 6 shows the experimental setup by denoting the relationship between the different data sources. The TPMs are calibrated on independent reference observations. TPM Müller uses data from HIRS to constrain the spectral emissivity over the whole lunar surface as described in [12]. In contrast, TPM Wohlfarth employs the SLSTR observations in this study to fine-tune its model parametrization. We discuss the results of this in Section 4.

Figure 6. Relationship between data sources in this study. TPM Wohlfarth is used to transfer radiometric calibration from SLSTR onto the SAFIRE platform. TPM Müller is employed as an independent source for validation. The phase angle range $\varphi$ is indicated for each dataset [12,13].

4. Results

In this section, we discuss the results of calibration and validation by applying the reference datasets discussed in Section 2 using the TPMs described in Section 3. The two models are handled in distinct ways. We fine-tune the parametrization of TPM Wohlfarth using the observations of SLSTR and then use it to calibrate the radiometric gain ${\alpha }_{\mathrm{gain}}$ of the SAFIRE instruments (see Equation (3)). In contrast, TPM Müller is evaluated independently without any change in the parametrization discussed in [12].

4.1. Calibration of TPM Wohlfarth Using SLSTR

The directional–hemispherical albedo, $A_{\mathrm{dh}}$, and surface roughness, ${\theta }_{\mathrm{rms}}$, are intrinsic parameters that jointly control thermal behavior across wavelengths in thermophysical models. In practice, however, both are sensitive to model formulation and data quality: estimates of $A_{\mathrm{dh}}$ vary across studies [2,28,29,30], and surface roughness also differ slightly in the literature [4,12,13,20,31,32]. We therefore treat $A_{\mathrm{dh}}$ and ${\theta }_{\mathrm{rms}}$ as effective parameters, tuning them to absorb model–data discrepancies. This follows the strategy adopted in TPM Wohlfarth, where the same parameters are adjusted to reproduce observations from Diviner Lunar Radiometer Experiment, GF-4, and MERTIS [20]. We opt for fine-tuning the model parametrization of TPM Wohlfarth, as it was originally obtained for disk-resolved observations and shows elevated residuals in the few-percent order against the disk-integrated SLSTR observations prior to modification.

Here, disk-integrated irradiances from SLSTR are used to constrain these parameters by minimizing the root-mean-square error (RMSE) between modeled and observed irradiances (Figure 7). Due to the computational cost of the model, we employ a grid search over the parameter space. The resulting best-fit parameters are summarized in Table 2. Figure 8 shows the resulting scaled bolometric albedo with an average magnitude of 0.13. The retrieved roughness values should not be interpreted as revised geophysical estimates of the global lunar surface roughness. Recent analyses based on Diviner observations indicate globally representative RMS slopes of approximately 31–32° [32]. In the present study, ${\theta }_{\mathrm{rms}}$ acts as an effective calibration parameter within the TPM Wohlfarth formulation, absorbing residual uncertainties related to emissivity, directional reflectance, unresolved roughness effects, and other model assumptions. The lower MWIR value of ${\theta }_{\mathrm{rms}}=22°$ therefore reflects the parameterization that best reproduces the reference observations within the adopted modeling framework, rather than evidence for a physically smoother lunar surface at these wavelengths. Residual errors are below 3% in the MWIR band and below 1% in the LWIR band, when controlling for outliers in the F2 band. These outliers deviate abruptly from the smooth annual cycle reproduced by both TPMs, with no corresponding anomaly in the simultaneous F1 observations; they are excluded from the parameter fit (see Appendix A, Figure A1).

Figure 7. Observed vs. modeled disk-integrated irradiance for SLSTR/F1 (3.74 μm, (left)) and SLSTR/F2 (10.85 μm, (right)) after calibration of TPM Wohlfarth against SLSTR. The relative root-mean-square error (rRMSE; normalized to the mean of all observations) and absolute RMSE are annotated in each panel. Six outliers in the LWIR range (faint markers) are excluded from the calibration, as they are likely caused by operational anomalies during acquisition. Error bars represent the disk-integrated flux uncertainty, which is well below 1% in both bands and does not account for systematic differences between S3A and S3B.

Table 2. Results of parameter grid search for the calibration of TPM Wohlfarth against SLSTR. The effective emissivity in the MWIR range is inferred by the TPM during runtime based on the mapped bidirectional reflectance estimate obtained from the HAPKE model [13] (see Figure 8 for exemplary values). In contrast, reflectance is neglected in the LWIR part of the TPM. It is therefore run with a scalar emissivity value for the entire lunar disk that is subject to calibration against SLSTR. The lower bound of the roughness search (20°) is the lower limit of the model’s tabulated roughness range, not an author choice.

	MWIR Range		LWIR Range
Parameter	Optimal	Search Range	Optimal	Search Range
Albedo Scaling $a_{A_{\mathrm{dh}}}$	1.1	0.8–2.1	1.1	0.8–2.1
Roughness ${\theta }_{\mathrm{rms}}$	22°	20–36°	32°	20–36°
Emissivity $ϵ$	inferred	-	0.98	0.92–0.98

Figure 8. Exemplary model parametrization of TPM Wohlfarth after its calibration against SLSTR. (Left): The bolometric albedo $A_{\mathrm{dh}}$ after application of scaling factor $a_{A_{\mathrm{dh}}}$. The initial albedo estimate is derived based on observations of the M3 instrument as described in [20]. (Right): Exemplary directional emissivity in the MWIR range of a given sample. The emissivity will be slightly different for each model run, as it depends on the viewing geometry [13]. The albedo and emissivities are only computed for pixels visible by the observing instrument.

The retrieved parameters indicate a lower effective roughness in the MWIR (${\theta }_{\mathrm{rms}}=22°$) compared to the LWIR (${\theta }_{\mathrm{rms}}=32°$), which is consistent with the model validation results against GF4 and Diviner [20].

Despite the low residuals, we observe a systematic discrepancy in the calibration of the F1 band between the SLSTR instruments onboard Sentinel-3A (S3A) and Sentinel-3B (S3B). The observed RMSE against the calibrated TPM Wohlfarth of approximately 3% exceeds the reported uncertainty of the on-ground calibration for this band, as also shown by the error bars [18,33]. This discrepancy is likely related to increased uncertainty at elevated brightness temperatures, as the instrument calibration is limited to approximately 320 K. Yet, the observed calibration residual of SLSTR/F1 still complies with the original design requirement of the instrument, stating <3 K for brightness temperatures below 500 K for the fire bands of the instrument [34].

The TPM by Müller [12] follows a different calibration strategy. In that framework, surface roughness (${\theta }_{\mathrm{rms}}$∼32°) and albedo ($A_{\mathrm{dh}}$∼0.1) are treated as fixed global parameters, and discrepancies between model and observations are primarily compensated through the derivation of a wavelength-dependent spectral emissivity. This approach agrees well with the global surface roughness estimate of 31–32° for mares and highlands, respectively, as derived in [32]. While this approach is well suited for spectrally resolved datasets such as HIRS, it is not directly applicable in this study due to the limited spectral sampling of SLSTR, which provides only two thermal bands. Instead, we employ TPM Müller using its given parametrization as a way of validating our results through an independent model. Figure 9 shows the modeling results of both TPMs when compared against the observations from SLSTR. The predictions of TPM Müller show 3.9% RMSE against SLSTR/F1 (MWIR) and 5.6% RMSE against SLSTR/F2 (LWIR). Both models show a similar precision between epochs, given their standard deviation of relative flux against SLSTR. The lowered accuracy (elevated rRMSE) in the LWIR band between TPM Müller and SLSTR points towards a disagreement between SLSTR/F2 and HIRS, given the model’s origin of parametrization. In summary, the obtained results mostly agree with previous model validations stating an accuracy <10% for the MWIR and <5% for the LWIR region [12].

Figure 9. Comparison of the relative flux of two TPM models against SLSTR observations after calibration of TPM Wohlfarth against SLSTR. TPM Müller has not been modified and predicts the disk-integrated flux at the epochs of the SLSTR observations based on its native parametrization. The mean and standard deviation of relative flux are given for each model. The solid line marks a relative flux of 1, and the dashed lines mark the ±10% bounds. Outliers in the F2 band are denoted as grey markers.

4.2. Calibration of SAFIRE Using TPM Wohlfarth

Figure 10 shows the integrated lunar observations of all SAFIRE imagers across a phase angle range of $-81.5°$ to $+122.2°$. For calibration, we selected up to five observations within a phase angle range of ±45° for each imager and band to determine a separate scalar radiometric gain ${\alpha }_{\mathrm{gain}}$ per imager and band (see Equation (3)). The gain is specific to each sensor in the fleet and is not shared across instruments. This number was not chosen as an optimized hyperparameter, but reflects the largest common set of usable lunar observations available for most imager-band combinations in the current dataset. Since the calibration estimates only a single multiplicative gain factor, the problem is not underconstrained even for a small number of observations. Using multiple acquisitions nevertheless reduces sensitivity to individual observation anomalies, pointing uncertainties, and residual background effects, while applying the same selection rule across the fleet avoids introducing sensor-dependent calibration choices. The resulting gain was obtained from the ratio between observed and modeled disk-integrated irradiance over the selected samples. The ±45° restriction is supported by the sensitivity analysis in Section 4.3, which shows the phase-transfer systematic is bounded at ≤1.0% in M0 and ≤1.5% in L2 within this interval.

Figure 10. Comparison of spectral flux from all observations of the SAFIRE imagers against both TPM candidates. The SAFIRE imagers have been calibrated against TPM Wohlfarth. The relative RMSE (rRMSE) is normalized to the mean of all observations within $\left|\varphi \right|\le 45°$.

Figure 11 shows the calibration and validation split. In the LWIR band, calibration samples are concentrated near full Moon due to limited observation availability within the ±45° interval.

Figure 11. Spectral flux of all 15 SAFIRE imagers against TPM Wohlfarth (orange) and TPM Müller (green), separated into calibration samples (top row, $\left|\varphi \right|\le 45°$) and validation samples (bottom row). The rRMSE values annotated in each panel are normalized to the mean of the respective sample set.

Using this calibration strategy, SAFIRE achieves a calibration rRMSE of $1.8\%$ in M0 and $2.2\%$ in L2 within $\left|\varphi \right|\le 45°$ against TPM Wohlfarth. On the held-out validation set, the rRMSE is $3.2\%$ in M0 and $6.0\%$ in L2. The elevated validation rRMSE, particularly in L2, is consistent with the phase-angle-dependent model degradation beyond $\left|\varphi \right|=45°$ characterized in Section 4.3; these out-of-window samples are not used for the calibration and do not affect the transfer bound. The uncertainty of the calibration transfer is characterized in Section 4.4.

Validation against the independently parametrized TPM Müller results in larger residuals of approximately $6.6\%$ in M0 and $4\%$ in L2. These values should not be interpreted as an alternative estimate of the SAFIRE gain uncertainty, but rather as an external consistency check that includes uncertainties of the independent model itself. The larger discrepancy in the MWIR is expected because this spectral range is more sensitive to the treatment of reflected solar radiance, directional emissivity, and surface roughness, and because TPM Müller was not specifically optimized for separating reflected and emitted radiance contributions in this regime. The agreement nevertheless remains within the previously reported validation accuracy of lunar thermal models at wavelengths below 5 µm, supporting the consistency of the calibration transfer.

We observe nonlinear deviations between model and observations as a function of lunar phase angle, as shown in the upper row of Figure 12. The residuals exhibit two distinct features: a ∼5% deficit between ±50° and ±80° present in both bands, and an increasing L2 divergence beyond ±90° absent in M0. Both are investigated in Section 4.3 and discussed in Section 5.3.

Figure 12. (Top): Relative flux from all SAFIRE observations against both TPM candidates after calibration of SAFIRE against TPM Wohlfarth. The solid line marks a relative flux of 1, and the dashed lines mark the ±10% bounds. (Bottom): Relative flux from TPM Wohlfarth and TPM Müller for all SAFIRE observational epochs. Statistics (mean, standard deviation) are computed within $\left|\varphi \right|\le 45°$.

Similar phase-dependent behavior has been reported in disk-resolved lunar observations [35], which exhibit enhanced beaming at low incidence angles and a pronounced decrease at high incidence. Although disk-integrated observations over a full libration cycle are not directly comparable to disk-resolved varying incidence geometries, both highlight the sensitivity of model performance to the treatment of sub-pixel roughness. At the smallest scales, unresolved grain-scale cavities may enhance near-nadir beaming, further contributing to the residual non-linearity observed at low phase angles.

The bottom row in Figure 12 shows the direct comparison of both TPMs evaluated at the epochs of the SAFIRE dataset. The models exhibit stable agreement within a phase angle range of approximately ±45°. Toward larger phase angles, both the model-to-observation agreement and the inter-model agreement degrade nonlinearly, with growing sensitivity to differences in model assumptions under more extreme illumination geometries. The origin of these trends is addressed in Section 4.3.

Figure 13 shows the mean and standard deviation of the ratio of spectral flux retrieved from both models for the set of SAFIRE observations. The largest divergence between the models is observed in the MWIR region. This is consistent with the increased uncertainty in this spectral interval, arising from less well-characterized surface properties as well as the previously discussed differences between the two models.

Figure 13. Comparison of the mean and standard deviation of the ratio of spectral flux from both model candidates. The models were evaluated for the 64 epochs of the SAFIRE/M0 dataset. The spectral response functions of SAFIRE are indicated in blue.

4.3. Sensitivity Analysis and Phase Transferability

TPM Wohlfarth is calibrated against SLSTR observations concentrated near a fixed lunar phase of approximately 6°. Because $a_{A_{\mathrm{dh}}}$ and ${\theta }_{\mathrm{rms}}$ jointly control the phase-dependent shape of the emission model, parameter sets that agree at small phase angles may diverge when evaluated at wider phase angles. The left panels of Figure 14 show the rRMSE surfaces of the SLSTR grid search for the MWIR (F1, top) and LWIR (F2, bottom) bands. The two surfaces have different characters. The MWIR surface is a monotonic ramp descending toward the lower roughness boundary: rRMSE rises from ∼3% at the optimum to ≳10% at the literature roughness ${\theta }_{\mathrm{rms}}=32°$ [32], placing that value outside the admissible region and confirming that the physically motivated roughness is inadequate for the disk-integrated MWIR band. The optimum sits at ${\theta }_{\mathrm{rms}}=22°$, one grid step from 20°, which is the lower limit of the model’s tabulated roughness range (not an author choice); the surface gives no indication that the true minimum lies above this boundary. The MWIR surface spans rRMSE up to approximately 15%, consistent with the steep Planck sensitivity at 3.74 μm ($hc/\lambda kT\sim 9.6$ at 400 K). The LWIR surface, by contrast, shows an elongated degeneracy valley, making the effective-parameter interpretation of Section 4.1 geometrically visible; it spans a narrower range up to ∼4%.

Figure 14. (Left): rRMSE surface from the SLSTR parameter grid search over albedo scaling $a_{A_{\mathrm{dh}}}$ and roughness ${\theta }_{\mathrm{rms}}$ for the MWIR/F1 band (top) and LWIR/F2 band at $ϵ=0.98$ (bottom; emissivity is swept from $0.92$ to $0.98$ in the full grid search). For the MWIR (top), the surface is a monotonic ramp descending toward the 20° lower roughness boundary (the lower limit of the model’s tabulated range). For the LWIR (bottom), it shows an elongated degeneracy valley. The red dot marks the optimal parameter set. (Right): SAFIRE/TPM ratio versus lunar phase angle for the admissible candidate set (grey) and the nominal optimum (red). Annotated values give the standard deviation of the candidate spread at each phase bin. All candidates are level-matched near full Moon by construction; the fan width reflects pure model-shape divergence. Candidate selection thresholds: ≤3.5% rRMSE for M0; ≤1% rRMSE for L2.

To bound the systematic uncertainty from this degeneracy, we define an admissible set as parameter combinations whose rRMSE against SLSTR falls within the residual floor of each band: ≤3.5% for F1 and ≤1% for F2. These thresholds correspond to each band’s own residual floor, dominated by model error and the S3A/S3B systematic offset rather than SLSTR noise, so parameter sets agreeing within that floor cannot be distinguished by the available dataset. For the F2 band, the grid search is run for each emissivity value from $0.92$ to $0.98$ in steps of $0.02$, yielding a separate rRMSE surface per $ϵ$; the left panel of Figure 14 shows the $ϵ=0.98$ surface. Admissible candidates are drawn from all emissivity-specific surfaces. In the LWIR branch with reflectance neglected, $ϵ$ acts as a pure level-scaling degenerate with the radiometric gain and does not affect the phase-dependent model shape, so the candidate fan captures the full admissible parameter volume.

Each admissible parameter set is re-run over a representative subset of SAFIRE epochs spanning the full phase range of the dataset, with ${\alpha }_{\mathrm{gain}}$ held fixed at its nominal calibration value. Because all candidates were selected by agreement with SLSTR near full Moon, they are level-matched at small phase angles by construction. The resulting fan of SAFIRE/TPM ratio curves therefore isolates model-shape divergence as a function of phase angle; a scalar gain shift would translate all curves uniformly without altering the spread.

The right panels of Figure 14 show the candidate fan for M0 and L2, with the standard deviation of the candidate spread annotated at representative phase bins. Within $\left|\varphi \right|\le 45°$, the spread among all admissible candidates is ≤1.0% in M0 and ≤1.5% in L2. This bounds the phase-transfer systematic directly: no retuning within the SLSTR-admissible parameter volume shifts the derived gain by more than these amounts over the calibration interval.

At larger phase angles, the divergence grows substantially. In the L2 band, the candidate spread reaches 8.7% at +122°, comparable in magnitude to the observed divergence itself. Every admissible parameter set diverges similarly, demonstrating that the large-phase L2 discrepancy is not an artifact of inadequate roughness tuning within the available parameter space: no retuning can remove it. The rapid growth of the candidate spread beyond ±90° also provides an independent quantitative justification for the $\left|\varphi \right|\le 45°$ calibration window. The physical interpretation of the remaining phase-dependent residuals is discussed in Section 5.3.

4.4. Sources of Error in the Calibration Transfer

Table 3 identifies the individual contributions to the lunar calibration transfer. Items in grey are absorbed into a parent empirical residual and shown for completeness. All values are evaluated within $\left|\varphi \right|\le 45°$.

The three dominant systematic contributions are the SLSTR–TPM Wohlfarth residual (item 3), the phase-transfer sensitivity (item 4), and the TPM Wohlfarth–SAFIRE residual (item 8). Independence among these cannot be supported: the SLSTR–TPM Wohlfarth residual (item 3) and the TPM Wohlfarth–SAFIRE residual (item 8) share the same model (the SLSTR fit absorbs TPM Wohlfarth error at ∼6° phase into the fitted parameters, so any systematic model error propagates coherently into every SAFIRE epoch), and the S3A/S3B inter-sensor disagreement (item 2) is a bias rather than zero-mean scatter, so it cannot reduce under $\sqrt{N}$. These items are therefore combined as a linear sum, yielding a worst-case upper bound on the calibration transfer error of 5.8% in M0 and 4.2% in L2.

Table 3. Sources of error in the lunar calibration transfer, evaluated within $\left|\varphi \right|\le 45°$. Items in grey are absorbed into a parent empirical residual and listed for completeness. Because the terms are not independent (items 3 and 8 share the same model; item 2 is a systematic bias rather than zero-mean scatter), item 9 is a linear upper bound on the transfer error, not a propagated standard uncertainty. Item 4 quantifies the spread of predictions across the SLSTR-admissible parameter set (Section 4.3).

Item	Name	Type	M0	L2	Reference
Included in item 3
1	Radiometric Uncertainty of SLSTR	Random	∼0.1%	∼0.1%	Equation (2)
2	S3A/S3B Inter-Sensor Disagreement (F1)	Systematic	5.2%	—	Figure 7
3	SLSTR → TPM Wohlfarth Residual	Systematic	2.7%	0.5%	Figure 7
4	Sensitivity of TPM Parametrization to Phase Transfer ($\left|\varphi \right|<45°$)	Systematic	<1.0%	<1.5%	Section 4.3
Included in item 8
5	NEDT of SAFIRE at 300 K	Random	∼1 K	∼0.1 K	Section 2
6	Out-of-band Leakage, M0 only (SRF-integrated; see Section 5.4)	Systematic	modelled a	—	Equation (9)
7	Disk-Integration Error	Systematic	∼0.1%	∼0.1%	Section 2
8	TPM Wohlfarth → SAFIRE Residual	Systematic	2.1%	2.2%	Figure 10
9	Upper Bound on Calibration Transfer Error (3 + 4 + 8)	Bound	5.8%	4.2%	—
Excluded from bound (not characterized here)
—	SLSTR absolute calibration traceability at elevated brightness temperature	Excluded	—	—	—
—	Absolute accuracy of thermophysical models	Excluded	—	—	—

^a Leak contributes ∼22–26% of total in-band M0 signal at lunar (400 K) temperatures, falling to ∼3.3% at 600 K and below ∼1.5% at 800 K. Captured self-consistently in ${\alpha }_{\mathrm{gain}}$ via full-SRF integration (Equation (9)), so not an added term in item 9, but relevant to narrowband interpretation of M0 at moderately elevated temperatures (Section 5.4).

Table 3. Sources of error in the lunar calibration transfer, evaluated within $\left|\varphi \right|\le 45°$. Items in grey are absorbed into a parent empirical residual and listed for completeness. Because the terms are not independent (items 3 and 8 share the same model; item 2 is a systematic bias rather than zero-mean scatter), item 9 is a linear upper bound on the transfer error, not a propagated standard uncertainty. Item 4 quantifies the spread of predictions across the SLSTR-admissible parameter set (Section 4.3).

Item	Name	Type	M0	L2	Reference
Included in item 3
1	Radiometric Uncertainty of SLSTR	Random	∼0.1%	∼0.1%	Equation (2)
2	S3A/S3B Inter-Sensor Disagreement (F1)	Systematic	5.2%	—	Figure 7
3	SLSTR → TPM Wohlfarth Residual	Systematic	2.7%	0.5%	Figure 7
4	Sensitivity of TPM Parametrization to Phase Transfer ($\left|\varphi \right|<45°$)	Systematic	<1.0%	<1.5%	Section 4.3
Included in item 8
5	NEDT of SAFIRE at 300 K	Random	∼1 K	∼0.1 K	Section 2
6	Out-of-band Leakage, M0 only (SRF-integrated; see Section 5.4)	Systematic	modelled a	—	Equation (9)
7	Disk-Integration Error	Systematic	∼0.1%	∼0.1%	Section 2
8	TPM Wohlfarth → SAFIRE Residual	Systematic	2.1%	2.2%	Figure 10
9	Upper Bound on Calibration Transfer Error (3 + 4 + 8)	Bound	5.8%	4.2%	—
Excluded from bound (not characterized here)
—	SLSTR absolute calibration traceability at elevated brightness temperature	Excluded	—	—	—
—	Absolute accuracy of thermophysical models	Excluded	—	—	—

^a Leak contributes ∼22–26% of total in-band M0 signal at lunar (400 K) temperatures, falling to ∼3.3% at 600 K and below ∼1.5% at 800 K. Captured self-consistently in ${\alpha }_{\mathrm{gain}}$ via full-SRF integration (Equation (9)), so not an added term in item 9, but relevant to narrowband interpretation of M0 at moderately elevated temperatures (Section 5.4).

In the MWIR, the SLSTR–TPM Wohlfarth residual (item 3, 2.7%) is dominated by the inter-sensor disagreement between S3A and S3B. The mean SLSTR/TPM Wohlfarth ratio is $0.9753\pm 0.0093$ for S3A and $1.0269\pm 0.0061$ for S3B over 20 epochs each. The inter-sensor disagreement is $\Delta =5.2$%, with each sensor offset by ∼2.6% from the joint TPM fit. At 3.74 μm and 400 K (Planck sensitivity $x=hc/\lambda kT$∼9.6), this corresponds to ∼2.2 K, within the <3 K design requirement [34] but consuming most of the allowance. The per-sensor scatter (0.93% and 0.61%) is consistent with the ∼0.2 K uncertainty reported in the calibration ATBD [18]: each instrument is individually precise, and the disagreement is an absolute-scale offset between two well-calibrated radiometers. The MWIR transfer bound is therefore set by the reference mission, not by SAFIRE or the TPM. Note that S3B observes at systematically higher flux levels than S3A in the F1 band (Figure 7); if the two sensor flux ranges do not overlap, the inter-sensor offset and a shared flux-dependent non-linearity are degenerate on this dataset. Either interpretation leaves the bound unchanged. In the LWIR, the SLSTR–TPM Wohlfarth residual (item 3, 0.5%) is small, and the bound is dominated by the TPM Wohlfarth–SAFIRE residual (item 8, 2.2%).

The SLSTR radiometric uncertainty (item 1, Equation (2), ∼0.1%) is more than an order of magnitude below the S3A/S3B inter-sensor disagreement (item 2). The SAFIRE noise floor (NEDT, item 5, ∼1 K in M0 at 400 K) and the disk-integration error (item 7, ∼0.1%) are likewise negligible relative to their parent residuals. These terms enter the SLSTR–TPM Wohlfarth residual (item 3) and TPM Wohlfarth–SAFIRE residual (item 8) by construction and do not independently constrain the bound.

The quoted bound characterizes the transfer of the SLSTR radiometric scale onto SAFIRE within $\left|\varphi \right|\le 45°$. It does not include the absolute calibration traceability of SLSTR at elevated brightness temperature, nor the absolute accuracy of the thermophysical models; both are listed at the foot of Table 3 and excluded from item 9. The figure is the transfer precision relative to the SLSTR reference, not a full radiometric uncertainty for SAFIRE.

The external consistency check against TPM Müller (6.6% in M0, 4.0% in L2) should not be read as confirming the bound. The SAFIRE/Müller residual is dominated by the inter-model difference rather than instrument error; that is precisely the absolute-accuracy term the bound excludes by construction.

5. Discussion

Unlike reflective solar bands, the thermal infrared lacks a widely accepted set of natural calibration references at elevated brightness temperatures. Conventional vicarious calibration approaches typically rely on terrestrial targets such as deserts, oceans, or instrumented ground sites. While these targets are valuable for validating radiometric performance under nominal Earth observation conditions, they rarely reach the brightness temperatures encountered in active wildfire observations and are subject to atmospheric effects, seasonal variability, and uncertainties in surface properties. The Moon provides a complementary reference target that is free from atmospheric interference, exhibits long-term physical stability, and regularly reaches surface temperatures approaching 400 K during the lunar day. In addition, its visibility from a wide range of orbital configurations enables the use of a common radiometric reference across multiple missions and sensor architectures. These characteristics make the lunar surface particularly attractive for calibration activities targeting the high-temperature regime relevant to wildfire detection and fire radiative power retrieval.

5.1. Implications for Lunar Vicarious Calibration

The results indicate that TPMs of the lunar surface can support calibration of radiometric gains in the thermal infrared with a transfer error bounded by 5.8% in M0 (MWIR) and 4.2% in L2 (LWIR) for $\left|\varphi \right|\le 45°$ (Table 3). This level of agreement is comparable to typical calibration requirements for wildfire monitoring applications and suggests that the Moon can serve as a viable in-flight calibration reference for thermal infrared imagers.

An important implication is that lunar calibration can complement onboard calibration systems. Instruments such as SLSTR employ well-characterized preflight calibration and in-flight blackbody references, yet we observe a 5.2% inter-sensor disagreement in the MWIR band (F1) between Sentinel-3A and Sentinel-3B, with each sensor offset by ∼2.6% from the joint TPM fit; at 400 K, this corresponds to ∼2.2 K (Figure 7). This is within the instrument’s <3 K design requirement but points to increasing calibration uncertainty at elevated brightness temperatures (>320 K), which are particularly relevant for applications such as wildfire detection (Table 3).

For instruments without onboard calibration targets, such as SAFIRE, the results suggest that TPM-based calibration transfer bounds the transfer error at 4.2% in the LWIR and 5.8% in the MWIR (Table 3). This indicates that lunar vicarious calibration may serve not only as a fallback solution, but as a complementary approach in regimes where traditional calibration sources are limited.

A further advantage is the potential for cross-instrument calibration. Because the Moon is observable from a wide range of orbital configurations, it provides a common reference that can support consistency across different missions and sensor designs. This is particularly relevant for constellations such as SAFIRE, where inter-sensor consistency is required for generating uniform data products.

We restrict the application of TPMs to absolute lunar phase angles below ±45°, given the decreased observation time to model towards larger phase angles (see Figure 12). This constrains the time interval for calibration to around 8 days per month.

5.2. Interpretation of Model Differences

The comparison between the TPMs of Wohlfarth and Müller reveals systematic differences in model performance, particularly in the MWIR (see Figure 13). The Wohlfarth model, which accounts for surface roughness, radiative coupling, and the separation of reflected and emitted components, shows closer agreement with SAFIRE observations. This is expected given that the SAFIRE observations are calibrated against this model. The larger MWIR divergence between the models likely reflects differences in the treatment of directional emissivity, roughness-induced thermal beaming, and the separation of reflected and emitted radiance. In contrast, validation of the calibrated observation against the independently parametrized TPM Müller model exhibits slightly elevated residuals, especially in the MWIR band. Still, the residuals of both models against both datasets are within the validation performance established by [12], citing <5% accuracy in the LWIR and <10% accuracy for shorter wavelengths in the MWIR.

Additionally, we observe increased residuals between TPM Müller and SLSTR in the F2 band (LWIR) as shown in Figure 9. The RMSE of 5.6% exceeds previous modeling results against the long-wave HIRS channels 8–12 (6.5–12.5 µm), which validated to 1–3% [12]. This could indicate a disagreement in the calibration between SLSTR/F2 and HIRS.

5.3. Phase-Angle Dependence and Unresolved Effects

The sensitivity analysis of Section 4.3 allows the phase-dependent residuals to be decomposed into two components with distinct physical characters.

The first component, a ∼5% deficit in both M0 and L2 between ±50° and ±80° together with the relative enhancement near opposition, is present in both bands and insensitive to the choice of admissible parameters. Its parameter-independence identifies it as a physical effect not fully captured by either model, most plausibly a combination of thermal beaming, shadow-hiding, and enhanced mutual heating between surface facets at low phase angles [35]. This is consistent with disk-resolved observations that exhibit enhanced beaming at low incidence angles and a pronounced decrease at high incidence.

The second component is the increasing L2 divergence beyond ±90°, which does not appear in M0 over the overlapping phase range (M0 data extend to +105°). The sensitivity analysis shows this cannot be attributed to inadequate roughness tuning: every admissible parameter set diverges, and no retuning within the SLSTR-admissible volume removes it. The effect is also spectrally selective in a way that eliminates instrumental stray light. Solar spectral irradiance at 3.8 μm exceeds that at 11.5 μm by a factor of approximately 20, so any scattered-sunlight path would manifest first and far more strongly in M0. Furthermore, the geometry is unfavorable for straylight to be negligible at large phase: solar elongation is approximately $180°-\alpha$, so the Sun is closest to the boresight precisely in this regime. Its absence in M0 under these conditions constitutes a spectral elimination, not an assumption.

The cause of the L2 large-phase divergence beyond ±90° remains unresolved and is a subject for future work. Straylight is eliminated on spectral grounds (Section 4.3), but no confirmed physical mechanism has been identified within the scope of this study. Because all L2 observations beyond ±90° are validation samples and the $\left|\varphi \right|\le 45°$ calibration window is unaffected, this does not impact operational calibration accuracy.

A further consideration is the difference in spatial sampling between the two instruments: SLSTR resolves the lunar disk at ∼1 km GSD while SAFIRE resolves it at 200 m GSD. Because the lunar disk exhibits strong temperature gradients, particularly near the terminator, instruments with different GSDs integrate this non-uniform temperature field differently when computing disk-integrated irradiance, which could in principle introduce a small bias in the calibration transfer. However, the SLSTR reference observations used to calibrate TPM Wohlfarth are confined to a narrow phase-angle range ($+5.2°$ to $+6.9°$), well away from the terminator, limiting the magnitude of this effect for the calibration transfer performed here. The scale effect is expected to become more relevant if the methodology is extended to larger phase angles, where temperature gradients across the disk are steeper; we flag this as a topic for future work.

5.4. Fundamental Limitations in MWIR Calibration

A central limitation highlighted by this study is the uncertainty in absolute thermal emission of the Moon in the MWIR. Even for well-calibrated reference instruments such as SLSTR, discrepancies between observations and TPM predictions remain at the few-percent level, indicating that the absolute radiometric scale in this spectral range is not yet fully constrained.

This has two implications. First, calibration uncertainties in the MWIR are not solely driven by instrument performance, but also by incomplete knowledge of the reference target itself. Second, it underscores the importance of physically consistent thermophysical modeling, as empirical calibration approaches alone may not capture all relevant effects.

A further consideration specific to the M0 band is the out-of-band spectral leakage visible in its SRF (Figure 3, Section 2). Because the leak sits in the 10.5–13 μm range, its fractional contribution to the total in-band signal is strongly temperature dependent: for a blackbody at 400 K, representative of lunar calibration conditions, the leak accounts for approximately 22–26% of the total integrated M0 signal, whereas for blackbodies at 600 K and 800 K, representative of elevated fire brightness temperatures, this fraction drops to approximately 3.3% and below 1.5%, respectively. A representative lunar TPM spectrum yields a comparable value (∼26%) to the 400 K blackbody case. This follows directly from the Planck function: at 400 K, the 11.5 μm region is intrinsically brighter than the 3.8 μm region, whereas at fire-relevant temperatures, the Wien-side growth of MWIR emission increasingly dominates and the leak’s contribution continues to shrink.

This does not affect the validity of the derived gain ${\alpha }_{\mathrm{gain}}$ itself. Both the TPM prediction (Equation (9)) and the SAFIRE observation are integrated over the same measured SRF, including the leak, so the calibration transfer is self-consistent for any source whose spectral shape is used consistently on both sides. The leak therefore does not introduce a bias into the calibration transfer bound reported in Table 3.

It does, however, imply that “M0 radiance”, as calibrated here, is a band-integrated quantity tied to the full SRF, not a narrowband quantity centered at 3.8 μm. Any downstream use that treats M0 as a narrowband channel, for instance, inverting a single-wavelength Planck function to retrieve brightness temperature, implicitly assumes a fixed relationship between the SRF-integrated signal and a narrowband equivalent. Because the leak fraction falls from roughly one quarter of the signal at lunar calibration temperatures to a few percent or less at fire-relevant temperatures, this assumption does not hold uniformly across that range, though the effect becomes progressively less significant at the highest brightness temperatures most relevant to active fire detection. Retrieval algorithms relying on M0 as a narrowband proxy at moderately elevated temperatures (of order 600 K) should characterize this effect independently. We flag this as a limitation for future work, consistent with Section 5.5 on extrapolating ${\alpha }_{\mathrm{gain}}$ beyond the lunar calibration regime. This leakage is specific to the current SAFIRE filter generation analyzed in this study; newer instrument iterations already in operation do not exhibit this out-of-band leak.

5.5. Operational Considerations and Future Work

From an operational perspective, lunar calibration offers several advantages, including the absence of additional hardware requirements, global observability, and the ability to perform cross-calibration between missions. However, its applicability is currently limited by phase-angle-dependent effects and model uncertainties at extreme geometries.

Future work should focus on improving the physical modeling of the lunar surface, particularly with respect to:

• Phase-angle-dependent and thermal beaming effects;
• Joint modeling of reflected and emitted radiation in the MWIR;
• Improved constraints on spectral emissivity under realistic lunar conditions.

Addressing these challenges will be important for extending lunar calibration to a broader range of observation geometries and improving absolute accuracy in the thermal infrared. Both thermophysical models used in this study, TPM Wohlfarth [13,20] and TPM Müller [12], are not publicly distributed as software; their underlying concepts and parametrizations are published in the cited references, but access to the implementations or pre-computed outputs requires contact with the respective authors. The calibration approach presented here addresses the radiometric gain component of the instrument model. Lunar observations of the Moon reach brightness temperatures of up to ∼400 K, which is well within the manufacturer’s specified operational range of the SAFIRE detectors, in which the detector response is considered linear. The transfer of the derived gain to fire-relevant brightness temperatures above this range will require dedicated validation and is the subject of future work. Well-calibrated radiometric products in the TIR domain are essential for precise detection and quantification of thermal anomalies on the ground such as wildfires. Additionally derived products include fire radiative power (FRP), land surface temperature (LST) and sea surface temperature (SST). Independent end-to-end validation against Earth observation targets and cross-sensor comparisons is performed as part of ongoing system-level calibration efforts, but is beyond the scope of this work.

6. Conclusions

We show that thermophysical models of the lunar surface can be used for vicarious calibration of thermal infrared imagers, bounding the calibration transfer error at 5.8% in the MWIR and 4.2% in the LWIR within $\left|\varphi \right|\le 45°$. Using SLSTR observations as a reference, a fleet of SAFIRE instruments is calibrated consistently without reliance on onboard blackbody targets.

The results further indicate that conventional calibration approaches exhibit limitations at elevated brightness temperatures, most directly evidenced by a 5.2% inter-sensor disagreement between Sentinel-3A and Sentinel-3B in the MWIR fire band. This disagreement, corresponding to ∼2.2 K at 400 K, is the dominant contributor to the MWIR transfer bound and shows that the reference mission, rather than SAFIRE or the thermophysical model, sets the current MWIR calibration floor.

At the same time, the absolute accuracy of the thermophysical models and the absolute radiometric traceability of the reference mission at elevated brightness temperature are excluded from the quoted bound and remain areas for future improvement.

Overall, lunar vicarious calibration emerges as a practical approach for thermal infrared missions, with particular relevance for small satellite constellations and high-temperature applications such as wildfire monitoring.