Portable Self-Calibrating Absolute Radiation Source for Field Calibration of Ground-Based Lunar Observation System

Highlights

What are the main findings?

• Developed a self-calibrating spectral radiometric quantity transfer scheme traceable to the SI based on the electrical substitution radiometry principle and hyperspectral reconstruction method.
• Designed and implemented a portable field radiometric calibration system for on-site calibration of lunar observation systems, leveraging an integrated structural design concept.

What is the implication of the main finding?

• Provided SI-traceable field radiation references for ground-based lunar observation systems, significantly enhancing the data quality of long-term lunar observations and advancing the construction of high-precision lunar models and comprehensive lunar calibration.
• The radiation calibration scheme based on the electrical substitution principle offers reliable self-calibrating radiation references for field observations, representing a transformative advancement in improving the long-term reliability of field optical instruments.

Abstract

To enhance the field calibration capability of ground-based lunar observation instruments for long-term continuous monitoring and to optimize the stability and traceability of lunar observation data, this manuscript presents the development of a SI traceable Portable Self-calibrating Absolute Radiation Source (PSARS) based on an electrical substitute radiometer. A self-calibrating radiation transfer model has been established. The system features a “+” structure layout centered around an integrating sphere, which ensures uniformity of the light source while improving system integration. Preliminary performance testing results indicate that PSARS achieves excellent radiative planar uniformity and angular uniformity within the targeted area, both exceeding 99%. During the self-calibration cycle of PSARS, the detector demonstrates high measurement stability for the built-in light source. Ultimately, through comparative validation and uncertainty assessment, the self-calibration accuracy of spectral irradiance for PSARS in the 400–1000 nm wavelength range is better than 2%, meeting the demands for high-frequency, high-stability, and high-precision real-time on-site radiometric calibration under ground-based lunar observation field test conditions. This provides technical support for the construction of high-precision lunar models and the widespread application of lunar calibration technologies.

1. Introduction

The main technical bottleneck currently faced by space optical remote sensing calibration is the establishment of a unified space radiation standard, which is essential for improving and maintaining the accuracy and comparability of remote sensing data over extended observation periods. This is a common challenge in the international remote sensing community. Various countries are actively developing space radiation standard satellite projects; however, due to the significant technical challenges, none have yet been successfully deployed in orbit [1,2,3,4]. The Moon possesses a radiation stability far exceeding that of Earth’s climate change rates (10⁻⁸/year) [5], which gives it the potential to serve as a natural space radiation reference. It can not only act as an ideal observation target for the calibration transfer of space radiation reference satellites but also, with comprehensive lunar calibration, help achieve unified traceability for space remote sensing payloads. Consequently, lunar calibration technology has become a focal point of research in space optical remote sensing, offering tremendous prospects for enhancing the accuracy and comparability of remote sensing data and effectively connecting multi-task payload observation data. However, the complex variations in lunar spectral radiation make the establishment of an absolute lunar space radiation standard highly reliant on precise lunar spectral radiation models, highlighting the urgent need for a long-term continuous lunar spectral observation database [6,7,8]. Ground-based lunar observations can provide comprehensive, high-density, long-term data across all lunar phases. For example, the U.S. Geological Survey’s RObotic Lunar Observatory (ROLO) project obtained over 110,000 lunar images through continuous observation from 1995 to 2003, which were used to construct the ROLO lunar radiation model [9]. This model has been applied for on-orbit performance verification of various payloads, including MODIS and VIIRS, and is recognized internationally as the most accurate lunar radiation model [10,11,12]. The Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP) of the Chinese Academy of Sciences has proposed the Full-field Integrated Radiometric Observation System (FIROS) for ground-based lunar hyperspectral irradiance measurement. By integrating and dispersing the radiation of the entire lunar disk, FIROS achieves hyperspectral irradiance measurements of the Moon in the 400–1000 nm range [13].

The construction of a comprehensive lunar radiation model typically requires several years or even more than a decade of continuous lunar observation data [8]. Long-term continuous observations impose stringent demands on data consistency. Ground-based lunar observation instruments undergo thorough calibration in the laboratory, and are typically equipped with high-precision temperature control devices to ensure stable responses in field test environments. However, during the transport of the instruments to the field test site, they may be subjected to risks such as vibrations and contamination of optical components, leading to a breakdown in the traceability chain between the instrument’s response and the laboratory standards [14,15,16]. Furthermore, the degradation and aging of optical components during prolonged field observations can lead to response drift and trends in the observational data. This negatively impacts the stability and measurement precision of the instruments, resulting in poor consistency in observational data sequences and low absolute accuracy, making it challenging to meet the data quality requirements for constructing high-precision lunar radiation models. For complex field observation systems, frequently returning them to the laboratory for calibration and inspection is clearly impractical due to the significant human and material resources required. High-frequency transportation also carries the risk of damaging the instruments and disrupts the continuity of field experiments, leading to data loss. Therefore, equipping dedicated field radiation calibration methods for real-time response monitoring and correction of observational instruments during the observation period is crucial for enhancing data quality.

Currently, the field calibration capabilities of ground-based lunar observation instruments internationally are quite limited. For example, the ROLO observation system uses the star method for radiometric calibration, which is constrained by the absolute radiative accuracy of the stars, resulting in poor traceability of observational data and an absolute accuracy of the lunar model of only 5% to 10% [17,18]. This inadequacy significantly restricts the comprehensive application of lunar absolute calibration technology. The common radiometric calibration methods for field optical remote sensing instruments include the Langley method, radiance method, and reflectance method based on standard detectors [19,20,21,22]. The radiance method typically uses a standard irradiance lamp to illuminate a diffuser with good uniformity for calibration. However, due to the inherent uncertainties of the light source and the diffuser, achieving high calibration accuracy is challenging. Additionally, the radiation characteristics of the diffuser can degrade due to environmental contamination, leading to a breakdown in the calibration transfer chain with laboratory standards. The precision of the Langley method depends on atmospheric conditions; poor atmospheric conditions lead to degraded calibration accuracy. The reflectance method based on standard detectors performs calibration by having a reference detector and the sensor to be calibrated simultaneously observe a diffuser. However, the measurement stability of the reference detector in the field is also affected by environmental factors, resulting in a disruption of the traceability chain with laboratory standards. Existing field calibration methods struggle to meet the precision and stability requirements for long-term ground-based lunar observations. Therefore, establishing a long-term stable high-precision field radiometric calibration system is crucial for enhancing ground-based lunar observation capabilities and ensuring long-term stability.

This manuscript designs and develops a Portable Self-calibrating Absolute Radiation Source (PSARS) for field calibration of ground-based lunar observation systems, enabling real-time monitoring and correction of the response of ground-based lunar observation instruments. The remainder of this paper is organized as follows: the second section explains the radiation transfer concepts and working principles of PSARS; the third section details the design scheme and simulation results of PSARS; the fourth section presents the test results and performance evaluations of PSARS; the fifth section discusses the uncertainty assessment results of PSARS’s self-calibration; and the sixth and seventh sections provide discussions and conclusions, respectively.

2. Design Concept

2.1. Technical Constraints Analysis

To improve the quality of lunar observation data, observation sites are typically located in remote areas away from cities to avoid the effects of urban light pollution on measurements. As a result, the testing environment is often quite complex. Based on the usage scenarios for PSARS and the data quality requirements needed for constructing lunar models, the design requirements for the PSARS system can be clearly defined:

• SI Traceability: For ground-based lunar spectral radiation measurements, ensuring data comparability requires that the final observational data possess a unified traceability. Laboratory calibration can achieve SI traceability of the instrument’s response. In response to the needs for constructing lunar radiation models, PSARS is designed to support lunar hyperspectral irradiance observation instruments in the 400–1000 nm wavelength range. To address the full-spectrum response variations in the instruments during long-term field observations, PSARS must have the capability for self-calibration of hyperspectral irradiance in the 400–1000 nm range, with the self-calibration results traceable to the International System of Units (SI).
• Portability: Currently, the most mature laboratory calibration solutions are detector-based methods. The SIRCUS (Spectral Irradiance and Radiance Responsivity Calibrations using Uniform Sources) calibration system developed by NIST can achieve calibration uncertainties better than 0.3% [23]. However, due to the system’s complexity and size, it is unsuitable for field calibration of remote sensing instruments. While the portable version, T-SIRCUS, has been simplified, the inherent complexity of the tunable laser still makes it challenging to achieve frequent field calibrations [24]. A portable calibration system can provide high-frequency, low-cost, real-time calibration for field observation systems, significantly enhancing the monitoring capability of the system’s status.
• Radiative Stability: There are significant technical barriers to taking laboratory radiation standards into the field and maintaining them over long periods, primarily due to the stability of the radiation reference [25,26,27,28,29,30,31]. Internationally, research on in-field calibration systems for observation equipment has been widely conducted, including systems like TSARS (Transfer Standard Absolute Radiance Source) for the Global Ozone Monitoring Experiment (GOME 2)–FM3 and SQM (SeaWiFS Quality Monitor) for monitoring the SeaWiFS ocean cruising experiment. Most of these systems utilize monitoring detectors for stability monitoring. However, the response of these monitoring detectors also degrades over time. For example, the SQM experienced a 0.6% decrease in response over a single 36-day cruise experiment, leading to the inability of the field calibration system to maintain long-term stability [23]. Therefore, ensuring the radiative stability of field calibration systems is essential for accurately monitoring the response of observational instruments and maintaining the consistency of long-term field measurement data.

Based on the on-site calibration requirements of the ground-based lunar observation system and the technical requirements mentioned above, the design specifications for the PSARS are proposed as shown in

Table 1. PSARS design specifications.

Design Parameter	Specification Requirement
Spectral Range	400–1000 nm
Radiation Uniformity	>98%
Total Mass	<20 kg
Irradiance Self-Calibration Uncertainty	<2% (Traceable to SI)

. The spectral range of the PSARS output light needs to cover 400–1000 nm, with a radiation uniformity better than 98%. To ensure portability, the total mass of the system must be less than 20 kg. Additionally, to maintain the accuracy of the ground-based lunar observation system over the long term, the PSARS must possess self-calibration capability for spectral irradiance with an uncertainty better than 2%.

2.2. Radiation Transfer Principle of PSARS

The main bottleneck of current field calibration technology lies in the lack of a long-term stable radiation standard. The principle of electrical substitution measurement has been around for over 100 years and is the primary method for achieving absolute optical radiation measurements, with results directly traceable to the International System of Units (SI). Thanks to the unique principles of electrical substitution radiometry, the Electrical Substitution Radiometer (ESR) exhibits radiation stability that is virtually unaffected by time and environmental changes. It is currently recognized as the most stable laboratory radiation standard internationally, offering traceability and high stability [32,33,34,35,36]. Currently, ESR are gradually being developed in a planar and lightweight direction. Based on the design requirements outlined in the previous section, a custom planar ESR has been chosen as the absolute radiation standard for PSARS. With the foundation of the ESR, PSARS can achieve radiation stability that shows minimal decay over time, and when combined with an appropriate self-calibration scheme, it can facilitate hyperspectral irradiance self-calibration in the field.

As shown in Figure 1, the main components of PSARS and their functions are outlined. Based on the need for hyperspectral irradiance self-calibration, PSARS primarily consists of five parts: the absolute radiation reference (ESR), a secondary radiation reference (FR), an integrating sphere for light homogenization, a multispectral monochromatic light source, and a full-spectrum polychromatic light source. The ESR is a thermoelectric type radiometer that lacks spectral resolution and can only provide radiation references at characteristic wavelengths. Additionally, the ESR has a relatively long measurement time, making it unsuitable for high-frequency measurements. To complete the hyperspectral irradiance self-calibration of PSARS, a faster channel filtered radiometer (FR) is introduced as the secondary radiation standard to perform the self-calibration process in a shorter time frame. Figure 2 illustrates the calibration principle of PSARS in the field. Since tunable lasers that provide continuous wavelength scanning monochromatic light are too complex, PSARS adopts a “multispectral calibration + spectral reconstruction” approach for hyperspectral irradiance self-calibration. The feasibility of this approach (as shown in Figure 2) has been confirmed in the benchmark transfer chain (BTC) of Chinese radiometric benchmark satellite LIBRA [37,38]. The red arrows represent the multispectral self-calibration process of PSARS. In the system, a multispectral monochromatic light source illuminates the integrating sphere to serve as a medium, with the ESR used as a radiative invariant to periodically calibrate the response of the FR, ensuring the long-term stability of the system. The green arrows in Figure 2 indicate the process of using PSARS for hyperspectral irradiance calibration of the lunar observation instruments. To achieve full-spectrum hyperspectral radiation calibration for the lunar observation system, a polychromatic light source that covers the observation wavelength range is introduced. The calibrated FR measures the multispectral irradiance of the integrating sphere illuminated by the polychromatic light source. Using hyperspectral reconstruction algorithms, the hyperspectral irradiance of the integrating sphere source is obtained, completing the hyperspectral irradiance self-calibration of PSARS, and enabling the field calibration of the hyperspectral irradiance response of the lunar observation instruments.

Figure 3 illustrates the schematic of PSARS field calibration. In lunar observation mode, the lunar observation instrument performs continuous lunar tracking measurements each night using a tracking turntable. In calibration mode, the lunar observation instrument is oriented towards PSARS via the tracking turntable, conducting observations 1–2 times each night. By combining these observations with the hyperspectral irradiance values obtained from PSARS at a distance $l$ from its output port, the response of the lunar observation system can be monitored and calibrated. Here, $l$ represents the distance between the aperture of the integrating sphere and the entrance pupil of the lunar observation system, calculated using the aperture diameter of the integrating sphere $D_{IS}$ and the field of view angle ${\theta }_{M\mathrm{oon}}$ of the Moon as seen from Earth, as shown in Equation (1). This calculation is used to simulate the field of view angle for lunar observations, thereby reducing calibration errors introduced by field of view mismatches.

$$l={\displaystyle \frac{D_{IS}}{2\cdot \tan ({\theta }_{M\mathrm{oon}}/2)}}$$

(1)

2.3. PSARS Self-Calibration Model

Based on the PSARS radiation transfer scheme, a hyperspectral irradiance self-calibration model has been established. The FR is a crucial component of PSARS; however, the integrated multi-channel filters and photodetectors within the FR are susceptible to environmental influences, leading to drift in its response in the field. The response of the ESR remains nearly unchanged in field conditions, enabling real-time calibration of the FR’s response. This stability is key to PSARS maintaining long-term self-calibration. The ESR measurement provides radiative power, while the target measurement for the FR is irradiance. To achieve accurate transfer of radiative values and self-calibration of PSARS’s spectral irradiance, it is necessary to establish the relationship between the monochromatic light power measured by the ESR and the monochromatic light irradiance measured by the FR in the laboratory. This requires calibrating the relationship between the FR measurements and the irradiance output from PSARS using a standard irradiance instrument in the lab. Figure 4 illustrates the calibration principle for the FR in the laboratory. A SVC HR-1024i spectral radiometer with an irradiance probe is used as the irradiance standard, and its irradiance response is calibrated using a reference lamp traceable to the National Metrology Institute of China. Through laboratory calibration, the irradiance response of the FR can be traced to SI. When the full-spectrum light source built into PSARS is activated, the SVC HR-1024i measures the multispectral irradiance at a distance $l$ from the output port of the integrating sphere and transmits this information to the FR.

The irradiance response of the FR at the initial time $t_0$ can be expressed as:

$$S{E}_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{t0}={\displaystyle \frac{I_{FR}{\left({\lambda }_n\right)}_{t0}-I_{FR}0}{E{\left({\lambda }_n\right)}_{t0}}}$$

(2)

where $S{E}_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{t0}$ is the irradiance response of the FR at time $t_0$ when the full-spectrum light source is activated and switched to the spectral channel with a central wavelength of ${\lambda }_n$, $I_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{t0}$ is the measurement value from the FR, $I_{FR}0$ is the laboratory dark background measurement of the FR, and $E{\left({\lambda }_{\mathrm{n}}\right)}_{t0}$ is the irradiance at wavelength ${\lambda }_n$ measured by PSARS at distance $l$ using the SVC spectral radiometer.

Assuming that the irradiance response of FR does not change in the laboratory, that is, $S{E}_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{t0}=S{E}_{FR}\left({\lambda }_{\mathrm{n}}\right)$, then the built-in laser can help establish the relative coefficient ${\sigma }_{ESR-FR}\left({\lambda }_{\mathrm{n}}\right)$ between the monochromatic light power measured by ESR and the monochromatic light irradiance measured by FR. With the full spectrum light source turned off and the internal laser turned on, the relative coefficient ${\sigma }_{ESR-FR}\left({\lambda }_{\mathrm{n}}\right)$ can be expressed as:

$${\sigma }_{ESR-FR}\left({\lambda }_{\mathrm{n}}\right)={\displaystyle \frac{{\varphi }_{ESR}\left({\lambda }_n\right)}{E\left({\lambda }_n\right)}}={\displaystyle \frac{{\varphi }_{ESR}\left({\lambda }_n\right)}{\left(I_{FR}\left({\lambda }_n\right)-I_{FR}0\right)/S{E}_{FR}\left({\lambda }_n\right)}}$$

(3)

where ${\varphi }_{ESR}\left({\lambda }_{\mathrm{n}}\right)$ is the monochromatic light radiative power measured by the ESR at wavelength ${\lambda }_n$, $E_{FR}\left({\lambda }_{\mathrm{n}}\right)$ is the monochromatic light irradiance measured by the FR, $I_{FR}\left({\lambda }_n\right)$ is the measurement response value of the FR at the monochromatic light wavelength ${\lambda }_n$, and $I_{FR}0$ is the laboratory dark background measurement of the FR.

Therefore, when performing field radiation calibration, the irradiance response of the FR $S_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}}$ is given by:

$$S{E}_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}}={\displaystyle \frac{I_{FR}{\left({\lambda }_n\right)}_{\mathrm{t}}-I_{FR}0}{E{\left({\lambda }_n\right)}_{\mathrm{t}}}}={\displaystyle \frac{I_{FR}{\left({\lambda }_n\right)}_{\mathrm{t}}-I_{FR}0}{{\varphi }_{ESR}{\left({\lambda }_n\right)}_t/{\sigma }_{ESR-FR}\left({\lambda }_n\right)}}$$

(4)

where $I_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_t$ is the measured original value of FR when the built-in laser wavelength is ${\lambda }_n$ at any time t, $E{\left({\lambda }_{\mathrm{n}}\right)}_t$ is the irradiance of the light source, and ${\varphi }_{ESR}{\left({\lambda }_{\mathrm{n}}\right)}_t$ is the radiated power measured by ESR. Therefore, when the full spectrum light source is turned on, the multi-spectral irradiance of the PSARS $E_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t$ can be calibrated by FR.

Where $I_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_t$ is the measurement value of the FR at any time t in the field test environment when the built-in monochromatic light wavelength is ${\lambda }_n$; $E{\left({\lambda }_{\mathrm{n}}\right)}_t$ is the irradiance of the light source at the monochromatic wavelength ${\lambda }_n$, derived from the radiative power measured by the ESR and the laboratory-calibrated power-irradiance conversion factor ${\varphi }_{ESR}{\left({\lambda }_{\mathrm{n}}\right)}_t$; ${\varphi }_{ESR}{\left({\lambda }_n\right)}_t$ is the radiative power measured by the ESR at the monochromatic wavelength ${\lambda }_n$; $I_{FR}{\left({\lambda }_n\right)}_{\mathrm{t}}$ is the measurement response value of the FR at time t for the monochromatic light wavelength ${\lambda }_n$; and $I_{FR}0$ is the dark background measurement of the FR.

Therefore, when the full-spectrum halogen lamp light source is activated at field observation time t, the multispectral irradiance $E_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t$ of PSARS can be calibrated using the FR that has been corrected by the ESR,

$$E_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t={\displaystyle \frac{I_{full-spectral}{\left({\lambda }_n\right)}_t-I_{FR}0}{SE{\left({\lambda }_n\right)}_t}}$$

(5)

That is, hyperspectral reconstruction can be carried out through spectral fitting and spectral interpolation to obtain the hyperspectral irradiance of the PSARS:

By performing spectral fitting and spectral interpolation for hyperspectral reconstruction, the hyperspectral irradiance $E_{full-spectral}{\left(\lambda \right)}_t$ of the full-spectrum light source can be obtained, thereby achieving the hyperspectral irradiance self-calibration of PSARS,

$$E_{full-spectral}{\left(\lambda \right)}_t=f\left(E_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t\right)$$

(6)

where $f\left(E_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t\right)$ stands for the hyperspectral reconstruction algorithm.

3. Design and Simulation

3.1. Overall Design of PSARS

a.

Main body structure

PSARS not only needs to monitor the radiation response of lunar observation instruments but also must provide a uniform target light source. As shown in Figure 3, PSARS integrates a light source system composed of a multispectral monochromatic light source and a full-spectrum light source, along with a detector system consisting of an ESR and an FR. To meet the portability requirements of the calibration system, a highly integrated structural design is necessary. The integrating sphere is a commonly used optical component in spectral radiation measurement, primarily functioning to spatially integrate the radiation flux. Light entering the integrating sphere undergoes numerous reflections, creating an approximately ideal Lambertian source at the output port. Additionally, the integrating sphere offers strong structural extensibility, allowing for high levels of integration. PSARS employs a design scheme that uses the integrating sphere as the main structural element, integrating all components into a single unit. This not only ensures the portability of PSARS but also maintains the stability and enclosure of the system structure. As shown in Equation (7), the radiation equation for the integrating sphere predicts the luminance $L_i$ as a function of the input radiation flux ${\mathsf{\Phi }}_i$, the internal surface area $A_s$, the reflectivity $\rho$, and the port fraction $f$. It can be observed that as the sphere’s diameter and opening ratio increase, the radiation level decreases. Considering the incident flux of the light source and the matching degree between PSARS and lunar radiation, the internal diameter of the integrating sphere is designed to be 250 mm, with an output port diameter of 80 mm. The inner wall is coated with a high-reflectivity polytetrafluoroethylene (PTFE) layer, which has a reflectivity greater than 97% in the observation wavelength range.

$$L_i={\displaystyle \frac{{\mathsf{\Phi }}_i}{\pi A_s}}\ast {\displaystyle \frac{\rho }{\rho \left(1-f\right)}}$$

(7)

b.

Light source system

The light source system primarily consists of a multispectral monochromatic light source and a full-spectrum polychromatic light source. The spectrum of the halogen lamp closely resembles that of the moon and features a smooth spectral curve, making it suitable for hyperspectral reconstruction. Therefore, two identical 35 W halogen lamps are used as the full-spectrum light source (with one serving as a backup), and the lamp bases are connected to heat sinks for cooling. The power of the halogen lamps needs to be selected so that, after passing through the integrating sphere, the radiation level of the light source matches that of the moon. For the monochromatic light system, a multi-channel laser array is used to inject light into the integrating sphere via optical fibers, with the wavelengths of the lasers corresponding one-to-one with the center wavelengths of the spectral channels of the FR (see Table 1). Each wavelength laser in the multi-channel array outputs more than 25 mW of power, with a wavelength stability better than 0.2 nm and a spectral linewidth of less than 0.5 nm. DBR or DFB lasers with a single-chip integrated grating are employed to achieve a narrow linewidth. The lasers are packaged in a 14-pin butterfly package, allowing simultaneous monitoring and control of the laser chip’s operating temperature and current. Each wavelength laser is equipped with a TEC (thermoelectric cooler) to ensure that the laser’s performance remains stable in the field environment, preventing wavelength drift and power fluctuations due to temperature changes. Since the output light from the optical fibers has a certain divergence angle, it creates a larger light spot on the walls of the integrating sphere, causing some of the first-reflected light to enter the detector. To address this, a fiber collimator array is introduced when the light enters the integrating sphere to narrow the beam.

c.

Detector system

The ESR (Electrical Substitution Radiometer) is an extremely important part of the PSARS system. To enhance the system’s portability and integration, a flat-type ESR has been designed as the absolute radiation standard. Its structural design profile and detector layout are shown in Figure 5. The customized flat ESR uses a lightweight ceramic material with a high thermal conductivity (36 W * m⁻¹ * K⁻¹) as the base for the planar optical absorption detector. This design reduces the overall mass and lowers the effective heat capacity, thereby improving the detector’s response speed. The design features a multi-layer thermocouple structure that efficiently utilizes the planar space of the detector. Using MENS (Micro-Electro-Mechanical Systems) fabrication technology, hundreds of pairs of thermocouples are integrated on the surface of the flat detector base to enhance sensitivity. Additionally, electrical heating wires are fabricated on the underside of the detector using MENS technology. This design mimics the measurement principle of cavity detectors and allows for periodic electrical substitution self-calibration of the detector. This approach ensures high self-calibration accuracy while being insensitive to variations in the external environmental temperature over a relatively wide measurement temperature range.

The ESR is also equipped with a shutter to control the reception of light signals. When the shutter is open, the detector converts the absorbed incident light power into an increase in temperature at the detector hotspot using black paint with an emissivity better than 0.99. When the shutter is closed, an electric voltage is applied to the heating wire to generate the same temperature increase at the hotspot. By precisely measuring the applied electrical power, the incident light power can be obtained. This compensatory mechanism helps eliminate the impact of environmental temperature fluctuations, enhancing measurement accuracy. Additionally, a water cooling system is integrated into the ESR to ensure operational stability. The electrical substitution principle used in the ESR is currently the most stable method for absolute radiation measurement, allowing it to disregard environmental temperature variations while maintaining a certain level of measurement accuracy across a wide temperature range. Furthermore, the measured incident light power can be traced back to the SI units. The instability of the ESR response is primarily influenced by the emissivity of its black paint. Since the ESR is mainly used to measure lasers in the visible spectrum, its emissivity stability—when not exposed to ultraviolet radiation—is significantly better than 1% per year. This is precisely why the ESR is adopted as the radiation reference for PSARS self-calibration.

The structural profile of the FR is shown in Figure 6. It uses the Hamamatsu S2281 silicon photodiode as the detector, with the output signal read by a Keysight 34401A six-and-a-half-digit voltmeter. The front end of the detector is equipped with two precision apertures that limit the field of view to ±2.86°, ensuring that the observation area on the inner wall of the integrating sphere matches the projected area of the aperture on the inner wall, thereby reducing radiation transfer measurement errors. Additionally, a stray light aperture is employed to effectively suppress stray light. The FR integrates a filter wheel for spectroscopic measurement, consisting of filters, a filter wheel, a motor, an encoder, and other components. The wheel contains six custom filters covering the spectral range of 400–1000 nm, with specific parameters listed in

Table 2. Filter parameters.

Channel Number	Filter Diameter/mm	Central Wavelength /nm	Spectral Bandwidth /nm
1	25.4	404.1	10.6
2	25.4	532.2	10.6
3	25.4	632.8	9.5
4	25.4	780.4	9.8
5	25.4	851.8	9.9
6	25.4	940.4	10.2

. The motor drives the filter wheel to enable spectral channel switching and dark background measurement, with a positioning accuracy of better than 0.1° to minimize the effects of misalignment on transmittance at different positions. The selected filters exhibit excellent out-of-band suppression, with a cutoff depth better than OD5 within the operational wavelength range of the FR. The front and rear apertures are made of stainless steel to minimize the impact of thermal expansion and contraction on the aperture diameter while providing strong corrosion resistance. All other parts of the FR are constructed from aluminum to reduce the overall weight of the instrument.

d.

Structural design

Since the integrating sphere uses multiple reflections on its inner wall to achieve uniform illumination, integrating multiple detectors and light sources on the limited surface area of the sphere’s wall is critical. The positioning and distribution of the detectors and light sources significantly impact the uniformity of the outgoing light from the sphere and the measurement stability of the detectors. Therefore, PSARS employs an integrated structural design where all detectors and light sources are located within the hemisphere of the integrating sphere’s exit aperture, ensuring that the rear hemisphere has no openings. This configuration guarantees the uniformity of the outgoing light from the sphere. Additionally, the detector measurements must avoid interference from direct light from the source and first-reflection light from the sphere’s wall. After extensive simulation analysis, a “+” cross-distribution positioning scheme was chosen. Figure 7 shows the overall structure of PSARS, with the x-y plane serving as the light source plane for arranging the laser entry ports and full-spectrum light source installation locations, and the y-z plane serving as the detector plane for positioning the ESR and FR. With this design, the outgoing light from the integrating sphere achieves excellent uniformity, and the measurement accuracy of the detectors remains largely unaffected.

3.2. Simulation and Analysis

During field calibration, there may be angular offsets or shifts between the lunar observation system and PSARS, and the non-uniformity of the light source can introduce certain errors into the calibration. Therefore, the uniformity of the outgoing light from the calibration source will directly affect the accuracy of the field calibration. We aim for PSARS to achieve a uniformity of over 99%, or even higher. To validate the feasibility of the proposed optical structure design, we imported the model into optical simulation software and conducted a simulation analysis of the outgoing light distribution from PSARS using the Monte Carlo ray-tracing method. The grid distribution of the radiance at the exit aperture of the integrating sphere is shown in Figure 8. The simulation results indicate that the integrating sphere achieves excellent radiation uniformity in over 85% of the exit aperture area, confirming the viability of the “+” cross-structure design scheme.

4. PSARS Performance Evaluation

4.1. Radiance Uniformity

Figure 9 shows the physical image of PSARS, which achieves a very high level of integration and portability. The figure also displays details of the filter wheel and the laser fiber coupling optics. The multi-channel lasers are connected to the main body via flexible fiber optics, enhancing the freedom of movement for PSARS. The system employs a main control computer to facilitate the simultaneous switching of the FR spectral channels and laser wavelengths, as well as the automatic data reading and storage for both ESR and FR. All data collection and self-calibration operations are completed through pre-set programs.

To verify the uniformity of the outgoing light from PSARS, tests were conducted on both the radiative surface uniformity and angular uniformity of the PSARS integrating sphere exit port, as shown in Figure 10. The surface uniformity was assessed using a gridded scanning method. A small-field Gershun tube radiometer was fixed on a motorized two-dimensional track, with the motion direction of the track perpendicular to the normal of the integrating sphere exit port. This setup allowed the Gershun tube detector to scan point-by-point across the target plane to measure the radiance at each local position of the integrating sphere. The scanning area had a diameter of 80 mm, with a step size of 10 mm. For the angular uniformity measurement, the Gershun tube detector was fixed on a circular sliding rail with a radius of 500 mm, focusing on the center of the integrating sphere opening as the target. The radiance was measured at different angles, with a scanning range of ±30° and a step size of 2°. Measurements were conducted in the light source plane and the detector plane, as shown in Figure 8. The detector was connected to an amplifier box and then to a calibrated voltmeter, measuring three sets of data at each measurement point and taking the average as the final measurement value for that point.

The test results for the surface uniformity are shown in Figure 11, where X and Z represent the relative positions of the test points in two directions with respect to the center of the integrating sphere opening. Equation (8) is used to calculate the uniformity $\mathrm{u}$, where ${\mathrm{E}}_{\max }$ is the maximum value in the grid data and ${\mathrm{E}}_{\min }$ is the minimum value. The calculations reveal that the surface uniformity within 85% of the exit port area is outstanding, achieving a value of 99.4%.

$$\mathrm{u}=1-{\displaystyle \frac{{\mathrm{E}}_{\max }-{\mathrm{E}}_{\min }}{{\mathrm{E}}_{\max }+{\mathrm{E}}_{\min }}}$$

(8)

Figure 12 shows the normalized radiance as a function of measurement angle in two planes. It can be observed that there is excellent angular uniformity within ±30° on the detector plane. However, it is noteworthy that on the light source plane, when the measurement angle exceeds ±15°, the deviation significantly increases. This is due to the fact that only one halogen lamp was turned on during the measurement, causing the direct light from the lamp to affect the results. Nonetheless, in another dimension, the energy distribution of the lamp is uniform across both positive and negative angles.

Table 3. Radiance angular uniformity.

	±30°	±20°	±14°	±10°	±6°
Light Source Plane	0.9775	0.9918	0.9994	0.9994	0.9994
Detector Plane	0.9976	0.9976	0.9980	0.9984	0.9992

presents the angular uniformity of radiance over different ranges in the two planes.

4.2. Radiation Matching Degree

To minimize the calibration error introduced by the nonlinearity of the FR’s response, the radiation levels of the integrating sphere when both the laser and the halogen lamp are on should be as close as possible. To verify this, the calibrated SVC HR-1024i spectroradiometer was used to measure the irradiance of PSARS when the monochromatic and polychromatic light sources were activated. Figure 13 shows the irradiance ratio in both cases, indicating that the irradiance difference falls within the linear range of the FR. Therefore, the introduced error is negligible.

To reduce the calibration error introduced by measurement nonlinearity, PSARS should have a good match with the radiation levels of the lunar observation target. Figure 14a compares the spectral irradiance of PSARS in the 400–1000 nm range with the lunar spectral irradiance predicted by the MT2009 model at a phase angle of 30°. The spectral irradiance of PSARS was measured at the field calibration distance $l$ using the SVC HR-1024i spectroradiometer equipped with an irradiance probe. Figure 14b shows the spectral irradiance ratio of the two, indicating that although the peak spectral values differ, the matching of PSARS with the lunar spectral irradiance is within the same order of magnitude across the 400–1000 nm range. Based on the linearity assessment of the lunar observation system, the introduced error is within 0.2%.

4.3. Detector Measurement Stability

During the entire self-calibration process of PSARS, measurement stability is a major source of error affecting calibration accuracy. The measurement stability of both ESR and FR for the laser and full-spectrum light source was validated (notably, ESR does not need to measure the full-spectrum light source during the PSARS self-calibration process). Figure 15, Figure 16 and Figure 17 show the measured values over time. The measurement duration for the laser was set to 10 min, while the duration for the full-spectrum light source was approximately 1.5 min, based on the actual conditions of PSARS self-calibration.

Table 4. Measurement stability of ESR and FR for PSARS built-in light source.

Laser Measured by FR
Wavelength/nm	405	532	633	780	852	940
STD	1.147 × 10−9	2.175 × 10−9	1.046 × 10−9	1.496 × 10−9	1.970 × 10−9	2.025 × 10−9
AVE	6.543 × 10−7	1.814 × 10−6	6.524 × 10−7	1.110 × 10−6	1.261 × 10−6	1.401 × 10−6
RSTD	1.753 × 10−3	1.199 × 10−3	1.603 × 10−3	1.347 × 10−3	1.562 × 10−3	1.445 × 10−3
Laser measured by ESR
STD	1.422 × 10−6	1.781 × 10−6	2.578 × 10−6	1.317 × 10−6	9.887 × 10−7	1.909 × 10−6
AVE	3.304 × 10−4	3.042 × 10−4	3.300 × 10−4	3.059 × 10−4	3.086 × 10−4	3.038 × 10−4
RSTD	4.304 × 10−3	5.854 × 10−3	7.812 × 10−3	4.303 × 10−3	3.204 × 10−3	6.283 × 10−3
Full spectral source measured by FR
STD	3.534 × 10−4	3.526 × 10−5	2.371 × 10−5	5.557 × 10−5	4.067 × 10−5	5.575 × 10−5
AVE	3.907	1.059	1.148	3.189	4.054	4.866
RSTD	9.044 × 10−5	3.331 × 10−5	2.066 × 10−5	1.743 × 10−5	1.003 × 10−5	1.146 × 10−5

summarizes the standard deviation (STD), average value (AVE), and relative standard deviation (RSTD) representing measurement stability for the measurements at various characteristic wavelengths. It can be observed that the laser stability measured by FR is better than 0.2% across all channels, while the full-spectrum light source stability is better than 0.01%. ESR’s measurement stability for lasers at all wavelengths is also better than 0.8%. It is worth noting that the measurement results encompass both the signal-to-noise ratio of the detector and the stability of the light source.

4.4. Spectral Irradiance Self-Calibration Accuracy

Experiments were conducted to validate the self-calibration capability of PSARS in the laboratory. After completing the calibration of FR, the spectral channel irradiance response values $S{E}_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{t0}$ were obtained. When the built-in tungsten lamp was turned on, the calibrated FR was used to measure the multispectral irradiance of PSARS. Simultaneously, the SVC HR-1024i spectroradiometer with an irradiance probe measured the irradiance at a distance $l$ from the PSARS output (which corresponds to the entrance pupil of the lunar observation system during field calibration). The multispectral irradiance measured by FR was fitted multiple times to achieve hyperspectral reconstruction, which was then compared with the hyperspectral irradiance measured by SVC. The hyperspectral reconstruction algorithm obtained through fitting is represented as follows:

$$\begin{array}{l}{E}_{full-spectral}{\left(\lambda \right)}_t=f\left(E_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t\right)=\\ \left\{\begin{array}{l}-1.1156\times {10}^{-13}{\lambda }^5+5.9591\times {10}^{-10}{\lambda }^4-1.2021\times {10}^{-6}{\lambda }^3+0.0011057{\lambda }^2\\ -0.4376\lambda +61.621(400\le \lambda <532)\\ -3.4672\times {10}^{-13}{\lambda }^5+4.3529\times {10}^{-10}{\lambda }^4-1.0864\times {10}^{-7}{\lambda }^3\\ (532\le \lambda \le 1000)\end{array}\right.\end{array}$$

(9)

where $E_{full-spectral}{\left(\lambda \right)}_t$ is the spectral irradiance of PSARS at the standard distance $l$ obtained through self-calibration when the tungsten lamp is turned on, and ${\lambda }_n$ is the wavelength.

The comparison results are shown in Figure 18a, where the red solid line represents the spectral irradiance measured by the SVC spectrometer at the standard distance $l$, the purple stars indicate the multispectral irradiance measured by the calibrated FR, and the black dashed line represents the hyperspectral irradiance obtained from the fitting of the multispectral data measured by the FR. Figure 18b displays the relative deviation between the SVC standard radiometer measurements and the PSARS self-calibrated spectral irradiance. As shown in the figure, across the 400–1000 nm range, the deviation is better than 0.4% for most wavelengths, except near 700 nm and beyond 900 nm. The non-smooth spectrum beyond 900 nm is mainly influenced by gas absorption bands. From Figure 18a, a slight dip is observed near 700 nm, which may be due to uneven reflectivity of the integrating sphere’s inner wall; this can be optimized by selecting materials with smoother reflectivity spectra. The comparison results fall within the measurement uncertainty of the SVC, confirming the feasibility of the PSARS self-calibration method.

5. Uncertainty Analysis

According to the radiation transfer model presented in Section 3, the spectral irradiance self-calibration process of PSARS is mainly divided into three steps: a. converting the radiation power reference of the ESR to the multispectral irradiance reference of the FR, b. calibrating the multispectral irradiance of the integrating sphere with the already calibrated FR full-spectrum light source, and c. performing hyperspectral reconstruction of the spectral irradiance from the full-spectrum light source. Based on the results from simulations and experimental tests, a detailed analysis of the uncertainties associated with the spectral irradiance self-calibration of PSARS has been conducted, with each uncertainty component listed in

Table 5. PSARS spectral radiance self-calibration uncertainty.

Parameter	Physical Significance	Uncertainty	Combined Uncertainty
$S_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}0}$	FR laboratory calibration		1.17%
	▪ SVC measurement uncertainty	1.1%
	▪ SVC measurement repeatability	0.2%
	▪ FR full-spectrum source measurement repeatability	0.01%
	▪ Stray light	0.3%
	▪ Alignment error	0.2%
${\sigma }_{ESR-FR}\left({\lambda }_{\mathrm{n}}\right)$	Relative coefficients between ESR and FR		1.43%
	▪ $S_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}0}$ uncertainty	1.17%
	▪ ESR laser measurement repeatability	0.8%
	▪ FR laser measurement repeatability	0.2%
	▪ ESR photoelectric non-equivalence	0.1%
$S_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}}$	Multi-spectral irradiance response of FR		1.65%
	▪ ${\sigma }_{ESR-FR}\left({\lambda }_{\mathrm{n}}\right)$ uncertainty	1.43%
	▪ ESR laser measurement repeatability	0.8%
	▪ FR laser measurement repeatability	0.2%
	▪ ESR photoelectric non-equivalence	0.1%
	▪ Response nonlinearity	0.05%
$L_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t$	Multi-spectral irradiance of full-spectrum light source		1.65%
	▪ $S_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}}$ uncertainty	1.65%
	▪ FR full-spectrum source measurement repeatability	0.01%
$L_{full-spectral}{\left(\lambda \right)}_t$	Hyper-spectral irradiance of PSARS full spectrum source		1.69%
	▪ $L_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t$ uncertainty	1.65%
	▪ Hyperspectral reconstruction	0.3%

:

a.

As shown in Equation (4), the uncertainty of the FR irradiance responsivity $S{E}_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}}$ during PSARS self-calibration primarily originates from the measurement repeatability of laser measurements by the ESR and FR, the uncertainty of the relative coefficient ${\sigma }_{ESR-FR}\left({\lambda }_{\mathrm{n}}\right)$, the photoelectric non-equivalence of the ESR, and the nonlinearity error of the FR. The FR utilizes an S2281 photodetector, whose response nonlinearity is better than 0.05%. The uncertainty of ${\sigma }_{ESR-FR}\left({\lambda }_{\mathrm{n}}\right)$ mainly stems from the uncertainty of $S_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}0}$, the measurement repeatability of laser measurements by the ESR and FR, and the photoelectric non-equivalence of the ESR. Based on the measurement results in Section 4.3, the measurement repeatability of laser measurements by the ESR and FR is better than 0.8% and 0.2%, respectively. The ESR employs electrical substitution radiometry, and its photoelectric non-equivalence is better than 0.1%. The uncertainty of $S_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}0}$ is primarily derived from the irradiance measurement uncertainty of the SVC (including standard lamp uncertainty of 1%, stray light of 0.3%, measurement repeatability of 0.2%, alignment error of 0.2%, nonlinearity error of 0.2%, etc.), the influence of stray light, the measurement repeatability of the FR for the full-spectrum light source, and alignment error. Among these, the standard lamp uncertainty is provided by the National Institute of Metrology of China, while other influencing factors of the SVC measurement uncertainty are obtained from laboratory measurement data analysis. During laboratory calibration, a laser rangefinder is used for auxiliary alignment, and analysis shows that the error introduced by alignment is better than 0.2%. According to the measurement results in Section 4.3, the measurement repeatability of the FR for the full-spectrum light source is better than 0.01%. Finally, by combining test and analysis data, the uncertainty of $S_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}0}$ is determined to be approximately 1.65%.

b.

As shown in Equation (5), the uncertainty of the multi-spectral irradiance $E_{full-spectral}{\left({\lambda }_{\mathrm{n}}\right)}_t$ of the full-spectrum light source during PSARS self-calibration primarily arises from the uncertainty of $S{E}_{FR}{\left({\lambda }_{\mathrm{n}}\right)}_{\mathrm{t}}$, the measurement repeatability of the FR for the full-spectrum light source. According to the measurement results in Section 4.3, the measurement repeatability of the FR for the full-spectrum light source is better than 0.01%. The resulting uncertainty of the multispectral irradiance for the full-spectrum light source is 1.65%.

c.

As shown in Equation (6), the uncertainty of the spectral irradiance $E_{full-spectral}{\left(\lambda \right)}_t$ of the emitted light from the PSARS full-spectrum light source is primarily composed of the uncertainty of $E_{full-spectral}{\left(\lambda \right)}_t$ and the uncertainty from hyperspectral reconstruction. The accuracy of hyperspectral reconstruction is better than 0.3% within the 400–1000 nm band. Through detailed analysis of the PSARS self-calibration process and various uncertainty components, the combined uncertainty is better than 1.69%, with the individual uncertainty components listed in Table 4.

6. Discussion

For ground-based lunar observations, maintaining data consistency and traceability throughout the entire lifecycle is crucial. However, the numerous uncertainties experienced by instruments in field test environments result in unpredictable changes in performance, making it difficult to accurately track and monitor these variations. This significantly limits the credibility of observational data and the potential value that should be realized, greatly affecting our observational capabilities regarding the Moon and the widespread application of lunar calibration technologies. This paper presents a highly integrated field calibration system—PSARS—whose portability allows it to be transported to any field observation site for long-term use. PSARS integrates both monochromatic and full-spectrum light sources, combining the stability of detector-based calibration methods like SIRCUS with the efficiency of standard light source calibration systems. It replaces the continuously tunable laser with a multispectral laser calibration approach and employs a full-spectrum light source to cover the calibration spectrum, utilizing a compact planar ESR instead of a low-temperature absolute radiometer to minimize the radiation transfer path. The feasibility of the PSARS design has been validated through simulations and experimental testing. From the final uncertainty analysis results, it is evident that the measurement stability of the ESR is one of the main sources of uncertainty in the spectral irradiance self-calibration of PSARS. This is related to the measurement principles of the ESR, which often struggle to balance small size and high precision. Consequently, a choice must be made. However, for the application scenarios of PSARS, the current achievable absolute calibration capability meets the needs of ground-based lunar observations. Currently, PSARS can only cover the 400–1000 nm wavelength range, but it has significant expansion potential to extend the spectral range to encompass the entire solar reflection spectrum. While PSARS sacrifices some absolute accuracy, it greatly enhances the system’s portability and versatility while ensuring calibration system stability, which is highly valuable. Moreover, by altering the FR calibration scheme in the laboratory, PSARS can also serve as a spectral radiance standard light source. Its radiance can match the observation targets of most field radiometers, offering very high universality and providing the possibility for establishing a global calibration network. It is worth mentioning that the PSARS multispectral laser light source is equipped with precision temperature control and constant current power supply modules, ensuring extremely high spectral stability even under field conditions. Figure 19 shows a schematic diagram of the spectral calibration of CIOMP’s FIROS lunar observation system using PSARS. By establishing the relationship between wavelength and the instrument’s response spectral channels, field spectral calibration can be achieved.

So far, PSARS has undergone preliminary verification of its optical performance in the laboratory. The next step will be to validate the stability and calibration accuracy of PSARS under different environmental conditions, which is crucial for its practical application. At the same time, we will continue to optimize the measurement stability of the planar ESR, as this is essential for enhancing the absolute calibration capability of PSARS and the traceability of remote sensing data.

7. Conclusions

This paper presents the design of a portable field radiation calibration light source system for ground-based lunar observations—PSARS. Based on the principle of electrical substitution radiometry and hyperspectral reconstruction algorithms, a new field radiation transfer scheme and hyperspectral irradiance self-calibration method are proposed. Detailed design work was conducted on PSARS, integrating the detector system, light source system, and the integrating sphere for uniform light distribution. The final optical structure was determined through iterative optimization and validated via optical simulation analysis. Preliminary laboratory performance tests of PSARS were conducted, and the results show that PSARS achieves excellent output light radiation uniformity while ensuring high integration. The surface uniformity in 85% of the output area is 99.4%, with angular uniformity better than 99.7% within ±30° around the output normal in the y-z plane, and better than 99% within ±20° in the x-y plane. The detector demonstrates good measurement stability for the built-in light source, meeting the self-calibration measurement requirements of PSARS. Finally, the self-calibration uncertainty of PSARS was validated through testing and detailed analysis, revealing a hyperspectral irradiance self-calibration accuracy better than 2% in the 400–1000 nm range.

Thanks to the characteristic of the ESR that its radiation response does not decay over time or with environmental changes, PSARS can maintain a high-precision, high-stability self-calibration capability for an extended period. It provides a real-time, traceable SI field irradiance standard that can be used for the absolute radiation calibration of ground-based lunar observation instruments. This has profound significance for ensuring the uniformity and continuity of observation data throughout the lifecycle of remote sensing instruments and for enhancing the accuracy of lunar radiation models.