Estimation of calibration lines for water ice content using 1.5-μm absorption band depth for lunar polar exploration

Abstract

Water ice trapped in permanently shadowed regions (PSRs) near the lunar poles is an important research target for understanding the distribution of light elements in the Moon. Herein, we examined the relationship between the near-infrared water absorption band depth and water ice content in mineral powders with a low ice content; these minerals serve as analogs to the regolith in the lunar PSRs. Four base minerals—olivine, plagioclase, clinopyroxene, and a mixture of these—were prepared at two grain size fractions. We constructed calibration lines based on the correlation between the water ice content ranging from 0.3 to 2.2 wt.% and 1.5-μm water absorption band depth. Results show that the calibration-line gradients, which are key parameters for determining the water ice content based on the absorption band depth, depend on the mineral species and grain size. The calibration-line gradients increased with increasing mineral grain size and correlated with variations in 1.5-μm reflectance among the dry mineral samples. The calibration-line gradients and reflectance values decrease in the order of clinopyroxene, plagioclase, mixture, and olivine. Combining the effects of mineral reflectance and grain size, we establish a predictive relationship for estimating the water ice content based on the observed 1.5-μm absorption bands. The proposed relationship provides a practical method to determine the ice content for future in situ landing explorations of the lunar PSRs, even when the exact regolith composition is unknown.

Introduction

Several studies have indicated the presence of water ice on the Moon (e.g., Arnold, 1979). Because the Moon’s spin axis is inclined by 1.6° from the ecliptic, permanently shadowed regions (PSRs) exist in its crater bottoms at the lunar poles (Arnold, 1979; Noda et al., 2008). Several mechanisms have been proposed to deliver water to the lunar surface, including impacts by water-bearing meteoroids or comets (Weissman, 1989), volcanic degassing from the lunar interior (Needham and Kring, 2017), and solar-wind reduction by the use of iron in regolith (Zeller et al., 1966; Nakauchi et al., 2021). Remote-sensing observations suggest that the water ice content in PSRs can reach several weight percentages (wt.%). For example, the water ice content has been estimated as 1–3 wt.% based on the Lunar Prospector Neutron Spectrometer data (Feldman et al., 2000), 0.5–4.0 wt.% using the Lunar Exploration Neutron Detector data (Mitrofanov et al., 2010, 2012), and 5.6 ± 2.9 wt.% through the Lunar Crater Observation and Sensing Satellite (LCROSS) analyses of Cabeus Crater ejecta (Colaprete et al., 2010). Based on observed spectral features in the near-infrared region, Li et al. (2018) suggested that water ice covers ~3.5% of the area in PSRs. However, no landing observations have been conducted in the lunar PSRs.

To compensate for the absence of in-situ observational data in lunar PSRs, the Japan Aerospace Exploration Agency (JAXA) and Indian Space Research Organization (ISRO) are planning a joint lunar polar exploration mission for directly detecting water ice in PSRs (Hoshino et al., 2020). An Advanced Lunar Imaging Spectrometer (ALIS)—a visible–near-infrared (VIS–NIR) imaging spectrometer developed by one of the present authors (Kazuto Saiki), which operates in the 750–1650 nm range—was selected to be onboard the rover for water ice detection in June 2020.

Near-infrared observations are effective for detecting water ice based on its characteristic absorption bands, with band wavelengths identifying compositions and band depths indicating abundance. However, the spectral properties and reflectance values of regolith depend on the grain size and mineral species, complicating the precise evaluation of trace components. Water ice exhibits prominent absorption bands at 1.5 and 2.0 μm in the VIS–NIR range. This study focuses on the 1.5-μm absorption band because of two main reasons. First, as the 2.0-μm band can be used for water ice detection as demonstrated in several studies, it overlaps the absorption features of pyroxenes (Cloutis, 2002), its absorption strength can vary by the mineral composition rather than by the water ice content alone. Second, the 1.5-μm band falls within the operational range of InGaAs imaging sensors. These sensors can operate at moderate detector temperatures (typically around 15–25°C), eliminating the need for extensive cooling. This characteristic simplifies observation systems and improves their practicality. ALIS is one such system that utilizes these sensors.

Previous studies (Clark and Lucey, 1984; Yoldi et al., 2021; Ogishima and Saiki, 2021) have investigated the effects of different factors, such as water ice content, base regolith composition, and regolith size, on the 1.5-μm water absorption band of mineral–ice mixtures. Clark and Lucey (1984) conducted NIR spectroscopic observations of mixtures predominantly containing water ice (70–100 wt.%), with minor amounts of Mauna Kea soil (0–30 wt.%). They reported that the 1.5-μm absorption band depth depends on the ice content. Yoldi et al. (2021) performed the NIR hyperspectral imaging of ice-mixing regolith simulants and suggested that the 1.5-μm water absorption band depth varied with the regolith grain size and linearly increased with the water ice content. However, they primarily simulated the Martian polar cap soils with much higher ice contents (>10 wt.%) than expected in the lunar PSRs. Therefore, it is uncertain whether calibration lines derived from experiments with tens of weight percentages of water ice can be appropriately extrapolated to estimate much lower ice contents, such as those expected in the lunar PSRs.

Ogishima and Saiki (2021) developed an apparatus that produced minute quantities of ice (approximately <2 wt.%) on mineral powders that were analogs of the lunar regolith and measured their NIR spectra, observing that the water absorption band depth varied between plagioclase and olivine. However, as their experiment only used a single grain size range (125–250 μm), they could not reveal the effect of the grain size on the water absorption band depth. After sieving the Apollo regolith samples, Graf (1993) revealed their grain size distributions, with the peaks typically ranging from 50 to 200 μm.

These studies demonstrate that the mineral species and grain size affect the 1.5-μm absorption band depth. As the precise composition and optical properties of the actual PSR regolith remain unknown, a model that accounts for these variables is invaluable for estimating the water ice content from the observed 1.5-μm absorption band depth for future landing missions. Therefore, a comprehensive understanding of the influences of these variables on water absorption features is necessary for the reliable in situ estimation of lunar ice.

Materials and Methods

Sample materials

The influence of mineral type on the depth of the water ice absorption band was examined on four types of mineral powders: olivine, plagioclase, clinopyroxene, and an equal-mass mixture of these minerals. Plagioclase is the primary mineral in lunar highland anorthosites, while olivine and pyroxene are the principal mafic minerals in lunar mare basalts and mantle rocks. The olivine (Fo90) used in this study consisted of gem-grade crystals sourced from San Carlos (California, USA). Single-crystal plagioclase (An60) was obtained from Casas Grandes (Chihuahua, Mexico), and large single crystals of clinopyroxene (Wo46En53Fs1, identified as diopside) were acquired from Dog Lake (Ontario, Canada). All samples were purchased from Nichika Inc. Their chemical compositions were approximated using energy-dispersive X-ray spectroscopy (Noran System7, Thermo Fisher Scientific, Massachusetts, USA) attached to a scanning electron microscope (SEM) (JSM-6010A, JEOL Ltd., Tokyo, Japan). The analysis revealed a small number of automorphic phlogopite inclusions in the clinopyroxene crystals. To avoid the influence of these weathered clay minerals, the clinopyroxene crystals were cracked and the phlogopite inclusions were removed by hand.

To simulate the lunar regolith, the minerals were first crushed in a ball mill and then wet-sieved into two grain size fractions: 75–125 μm and 180–250 μm. Any very fine particles adhering to the surfaces were removed by ultrasonic cleaning in ion-exchanged water for 5 min. Subsequently, the processed mineral samples were heated at 120°C for over 2 h with an electric heater to remove residual moisture. The sieved and cleaned base mineral samples were observed under a stereo microscope (Fig. 1). All samples exhibited similar particle shapes.

Fig. 1.

Enlarged images of the prepared base mineral simulants used in experiments captured using the Keyence VR-3000 wide-area 3D measurement system: (a) olivine, (b) plagioclase, and (c) clinopyroxene with 180–250-μm grains; (d) olivine, (e) plagioclase, and (f) clinopyroxene with 75–125-μm grains.

Experimental micro-ice attachment system

The regolith of lunar PSRs containing small amounts of water ice was simulated using the micro-ice attachment apparatus of Ogishima and Saiki (2021). This setup replicates the process by which water molecules traveling on ballistic trajectories outside PSRs are cold-trapped by regolith. The resulting fine grains of water ice condense and adhere to the sample minerals. The ice-attached samples were fabricated as follows. The workflow in this experiment procedure is shown in Fig. 2.

Fig. 2.

The workflow schematic of this experimental procedure. The background colors in the figure indicate the temperatures at which the samples were handled during each step; the same color represents the same temperature environment. Photographs of the glove box (B) and the cooling stage (C) are shown in Fig. 3a and b, respectively. A photograph of the micro-ice attachment apparatus (A) can be seen in Ogishima and Saiki (2021).

First, the mineral samples were immersed in liquid nitrogen to cool them below its boiling point of 77 K. Once sufficiently cooled, these samples were removed from liquid nitrogen kept in a box filled with dry nitrogen gas using a spatula (#1 in Fig. 2). The samples were then dropped through a 2-m-high vertical cylinder maintained at room temperature and filled with ambient air, allowing water vapor from the surrounding environment to condense onto the surfaces of mineral powders as they fell (#2). The micro-ice-attached samples were gathered in a collecting box filled with liquid nitrogen at the bottom of the cylinder (#3). To obtain samples with different ice amounts, this falling process was repeated 1–6 times (#4). After each drop, the samples were collected and cooled to 77 K by immersing in liquid nitrogen. In some cases, steam from boiling water was injected into the cylinder to further adjust the ice content. Except when falling through the cylinder, all samples were kept in the collecting box filled with liquid nitrogen, preserving the adhered water ice grains.

Following collection, the samples were placed on a copper sample holder cooled to ~77 K with liquid nitrogen (#5) and weighed (#6) in a nitrogen-purged glovebox (Fig. 3a). Although the sample holder was not exposed to liquid nitrogen, the sample was kept well below freezing during the weighing process. To minimize spectral interference during later measurements, the holder was coated with black heat-resistant paint (Asahipen Co., Osaka, Japan), which exhibits a flat spectral response with reflectances between 7% and 10% across the NIR wavelength range. During the NIR measurements, the weighed samples were transferred to a cooling stage (Fig. 3b) maintained at 218–228 K and purged with nitrogen gas (#7 and #8).

Fig. 3.

(a) Photograph of the glove box used in the experiment. The ice-attached samples dropped from the cylinder were kept cool in a nitrogen-filled collecting box (1) and transferred to a cooled sample holder inside the nitrogen-purged glove box, weighed on an electric balance (2), and then sealed in the cooling stage (3). (b) Schematic of the cooling stage. The cooling stage was purged with dry nitrogen and sealed. Its copper component was maintained at the stage temperature (218–228 K). Spectroscopic observations of the sample were conducted through the silica glass window.

Once the spectral measurements were completed, the samples were removed from the nitrogen-purged environment and heated at 120°C for over 2 h using an electric heater to evaporate the ice (#9). After heating, the 1.5-μm absorption band was effectively eliminated (Fig. 4). The dried samples were reweighed (#10) and sealed in a nitrogen-purged environment within the cooling stage and measured under the same environmental conditions as the ice-attached samples (#11).

Fig. 4.

Representative spectra of dry samples after heating to remove ice (dashed lines) and ice-attached samples (solid lines): olivine with (a) 180–250-μm and (b) 75–125-μm grains; plagioclase with (c) 180–250-μm and (d) 75–125-μm grains; clinopyroxene with (e) 180–250 μm and (f) 75–125-μm grains; mixed samples with (g) 180–250-μm and (h) 75–125-μm grains.

The amount of ice adhering to the mineral grains was determined as the mass difference between the ice-attached and dry samples. The uncertainty in the ice mass measurements was determined by the precision of the electronic balance (0.001 g). The calculation details are provided in the Appendix.

Optical microscopy observations of ice-attached samples

The morphologies and distributions of ice adhering to the mineral powders were observed under an optical microscope. A portion of the ice-attached samples produced in the micro-ice attachment experiments was placed in a pre-cooled plastic box purged with nitrogen gas and maintained at liquid nitrogen temperature. To record the size distribution of the ice particles on the mineral surfaces, some samples were photographed under the stereo microscope at a temperature below the melting point of ice. The photographed samples were left at ambient temperature for approximately 30 min to remove the water ice from the grains. The dried samples were re-photographed over the same area.

Near-infrared spectroscopic observations

An imaging spectrometer, designed and built by Saiki et al. (2021), integrating a grism with a Xenics Bobcat 640-GigE image sensor, was used for NIR spectral measurements over a wavelength range of 900–1640 nm. For all measurements, an illumination geometry with an incidence angle of 30° and an emission angle of 0° was employed, consistent with previous spectroscopic studies in planetary sciences (Mustard and Pieters, 1989; Yamada et al., 1999; Izawa et al., 2019; Ogishima and Saiki, 2021).

A schematic of the optical system is shown in Fig. 5. Light reflected from the sample was imaged and then passed through a slit (labeled a in Fig. 5), capturing a field of view of 123 mm × 610 μm. The transmitted light was subsequently dispersed in the wavelength direction through a grism and imaged onto a sensor (b in Fig. 5) with resolutions of 512 and 640 pixels along the spatial axis (y in Fig. 5) and wavelength axis (x in Fig. 5), respectively. Along the spatial axis, a length of 123 mm (y-axis) was resolved into 512 pixels (240 μm per pixel), while the entire 123 mm × 610 μm region was dispersed along the wavelength direction (x-axis). Consequently, the instantaneous field of view for each pixel on the VIS–NIR image sensor was approximately 610 μm × 240 μm.

Fig. 5.

Schematic illustrating the relationship between the experimentally obtained spectral data and observed sample surfaces. One slit defines a field of view of 610 μm (width) ×123 mm (length), within which the sample is placed. The VIS–NIR sensor covers a length of 123 mm in 512 pixels along the spatial dimension (~240 μm/pixel) and 640 pixels along the spectral axis. Because the slit has a finite width of 610 μm, the signal is spatially averaged across this width during acquisition. As a result, each pixel represents a sample area of 240 μm × 610 μm. During each observation, spectral data were acquired from the central 30 pixels of the sample (red dashed line).

As the range of mineral grain size in this study was 75–250 μm, a single pixel along the y-axis of the image sensor could not provide a well-averaged bulk spectrum of the powder surface. Because only a few mineral grains were present within this field of view, specular reflections and shadows from individual grain shapes became more pronounced, causing random variations in the observed reflectance. To eliminate this influence of individual random grains, the spectra were averaged over a sufficiently large area of the powder surface. First, 30 contiguous pixels along the y-axis of the image sensor, located in the central area of the sample where stray light from the sample holder walls was absent, were selected (c in Fig. 5). Additional reflectance spectra were measured at 10 locations on the sample surface by randomly rotating the sample holder (d in Fig. 5) relative to the slit. At each location, 30 contiguous central pixels were acquired, and the spectra were averaged over all 300 pixels. Averaging spatial information from these 300 pixels reduced the reflectance offset caused by individual grain effects, allowing reflectance variations to converge to within 1% in the vicinity of the 1.5-μm water absorption band.

Absorption spectra analysis

Band depth parameter

To quantify the relative strengths of absorption bands, many researchers adopt the band depth (Clark and Roush, 1984; Bell and Crisp, 1993; Milliken and Mustard, 2005; Ogishima and Saiki, 2021), which is a function of the reflectance $R_b(\lambda )$ at the wavelength of maximum absorption and the continuum line $R_c(\lambda )$ at the same wavelength (Clark and Roush, 1984). In this study, the continuum slope was determined by fitting a straight line through the two shoulders of the absorption depression. The band depth was then computed as${(R}_c-R_b)/R_c$.

To determine the two contact points of the continuum, the observed spectra were first smoothed with smoothing spline fitting. Starting with two arbitrary points around the depression of the spectral curve, the right-hand point was moved until the tangent slope reached a maximum. The right-hand point was then fixed and the left-hand point was moved until the tangent slope was minimized. The right-hand point was then adjusted to again maximize the tangent slope. This process was iterated until the contact points converged across the wavelength resolution range. The resulting straight line represented the continuum connecting the two shoulders of the absorption band. The band depth error was calculated as described in the Appendix.

Smooth spline fitting

To correctly determine the continuum line, which is tangent to the absorption band, we obtained a mathematically smooth spectral curve. Specifically, we smoothed the reflectance spectra using the smooth spline function of the R language (Fig. 6). Mathematically, the smoothing spline seeks the function $g$ that minimizes the following expression:

Fig. 6.

Observed reflectance and spline fitted spectrum of ice-attached plagioclase with180–250-μm grains and an ice content of 1.22 wt.% (#14 in Table 1) at approximately 1.5-μm absorption band, presented as an example.

  

$$\sum _{i=1}^n{(y_i-g\left(x_i\right))}^2+\lambda {\int }_{}^{}{g}^{″}{(x)}^{2}dt,$$

where $y_i$ denotes the measurement data with errors, $g$ is the fitting function, $\lambda$ is a nonnegative smoothing parameter, and $g^{″}$ is the second derivative of $g$ (James et al., 2013). The first term ensures that $g$ closely follows the data, while the second term penalizes excessive curvature. When $\lambda =0$, the method reduces to spline interpolation that exactly connects all data points with a curve. As $\lambda$ increases, the curve becomes smoother and loses its fine details. Here, we set $\lambda =0.4$, which optimally reduces the reflectance variation while preserving the characteristic shape of the 1.5-μm water absorption band.

Results

Micro-ice attachment experiments were conducted on eight mineral sample types: two grain size ranges (75–125 and 180–250 μm) of four base mineral species (olivine, plagioclase, clinopyroxene, and a three-mineral mixture). Multiple samples of each mineral type were prepared with varying water ice contents of approximately 0.3–2.2 wt.%. Figure 7 presents the representative microscopic images of ice-attached olivine grains under two selected conditions of different base mineral grain sizes and water ice contents as well as their dry samples. During experiments, ice particles with approximate diameters of 10 μm adhered to the surfaces of these four base minerals. Occasionally, a few large water-ice grains were attached not to the base mineral but to other ice grains, forming aggregated grains (e.g., the lower left arrow in Fig. 7c). Under our experimental conditions, the size of each ice grain remained approximately constant (~10 μm) and independent of the size of base mineral grains or total amount of ice. Increasing the ice content increased only the number of ice grains adhered to the mineral surface, not the size of ice grains.

Fig. 7.

Optical microscope images showing olivine samples before and after drying, for two different ice mixing ratios, taken from the same areas. (a) frozen olivine samples with 75–125-μm grains containing 1.73 wt.% ice mixing, and (b) dry olivine samples within the area of (a), (c) Frozen olivine samples with 180–250-μm grains containing 1.39 wt.% ice content, (d) dry olivine samples within the area of (c). Images (a) and (b) were taken with a wide-area 3D measurement system (Keyence VR-3000) and (c) and (d) were taken under a Stereozoom S9i microscope (Leica Microsystems Ltd.). Ice grains (pointed by orange arrows) attached to mineral surfaces in (a) and (c) are absent in (b) and (d).

Water ice absorption was detected at around 1.5 μm in all recorded spectra. All spectra are provided in Fig. S1, which is available through the Osaka University Knowledge Archive (https://doi.org/10.60574/87068). Figure 4 presents representative spectra from eight different sample types. To quantify the ice content, the band depth parameters were calculated (Table 1). To determine the relationship between the band depth and ice content, these parameters are plotted for each mineral type in Fig. 8 and for each grain size in Fig. 9. The calibration lines were fitted through the origin, and their slopes varied depending on the mineral species and mineral grain size.

Table 1.

Absorption BD and ice content for each experiment

Run number	Mineral type	Particle size	Band depth	Ice content (wt.%)
#1	Olivine	180–250 μm	0.033 ± 0.012	0.65 ± 0.08
#2	Olivine	180–250 μm	0.025 ± 0.011	0.63 ± 0.08
#3	Olivine	180–250 μm	0.043 ± 0.010	0.91 ± 0.09
#4	Olivine	180–250 μm	0.033 ± 0.011	0.86 ± 0.10
#5	Olivine	180–250 μm	0.113 ± 0.009	2.01 ± 0.09
#6	Olivine	75–125 μm	0.004 ± 0.007	0.31 ± 0.10
#7	Olivine	75–125 μm	0.005 ± 0.007	0.45 ± 0.11
#8	Olivine	75–125 μm	0.011 ± 0.008	0.65 ± 0.12
#9	Olivine	75–125 μm	0.033 ± 0.008	1.82 ± 0.13
#10	Olivine	75–125 μm	0.025 ± 0.007	1.14 ± 0.15
#11	Olivine	75–125 μm	0.003 ± 0.009	0.41 ± 0.12
#12	Plagioclase	180–250 μm	0.141 ± 0.008	1.56 ± 0.09
#13	Plagioclase	180–250 μm	0.115 ± 0.008	1.45 ± 0.12
#14	Plagioclase	180–250 μm	0.103 ± 0.009	1.22 ± 0.16
#15	Plagioclase	180–250 μm	0.061 ± 0.006	0.69 ± 0.11
#16	Plagioclase	180–250 μm	0.020 ± 0.007	0.33 ± 0.13
#17	Plagioclase	75–125 μm	0.023 ± 0.004	0.49 ± 0.12
#18	Plagioclase	75–125 μm	0.010 ± 0.004	0.59 ± 0.15
#19	Plagioclase	75–125 μm	0.013 ± 0.004	0.49 ± 0.12
#20	Plagioclase	75–125 μm	0.024 ± 0.004	0.73 ± 0.15
#21	Plagioclase	75–125 μm	0.137 ± 0.006	2.02 ± 0.19
#22	Plagioclase	75–125 μm	0.042 ± 0.005	0.73 ± 0.13
#23	CPX*	180–250 μm	0.105 ± 0.007	0.86 ± 0.10
#24	CPX	180–250 μm	0.042 ± 0.007	0.36 ± 0.07
#25	CPX	180–250 μm	0.066 ± 0.006	0.50 ± 0.08
#26	CPX	180–250 μm	0.024 ± 0.007	0.38 ± 0.08
#27	CPX	180–250 μm	0.126 ± 0.009	1.04 ± 0.11
#28	CPX	75–125 μm	0.104 ± 0.006	0.87 ± 0.12
#29	CPX	75–125 μm	0.033 ± 0.006	0.59 ± 0.13
#30	CPX	75–125 μm	0.003 ± 0.005	0.35 ± 0.17
#31	CPX	75–125 μm	0.075 ± 0.006	1.77 ± 0.13
#32	CPX	75–125 μm	0.036 ± 0.004	0.65 ± 0.16
#33	CPX	75–125 μm	0.039 ± 0.007	0.85 ± 0.19
#34	CPX	75–125 μm	0.015 ± 0.005	0.42 ± 0.14
#35	CPX	75–125 μm	0.107 ± 0.011	1.49 ± 0.21
#36	Mixture	180–250 μm	0.095 ± 0.011	1.76 ± 0.11
#37	Mixture	180–250 μm	0.008 ± 0.008	0.74 ± 0.08
#38	Mixture	180–250 μm	0.070 ± 0.013	1.23 ± 0.11
#39	Mixture	180–250 μm	0.116 ± 0.012	1.40 ± 0.12
#40	Mixture	180–250 μm	0.020 ± 0.010	0.73 ± 0.13
#41	Mixture	75–125 μm	0.007 ± 0.005	0.68 ± 0.11
#42	Mixture	75–125 μm	0.004 ± 0.004	0.52 ± 0.10
#43	Mixture	75–125 μm	0.068 ± 0.007	2.15 ± 0.13
#44	Mixture	75–125 μm	0.004 ± 0.006	0.83 ± 0.24
#45	Mixture	75–125 μm	0.009 ± 0.005	0.81 ± 0.12
#46	Mixture	75–125 μm	0.080 ± 0.013	1.67 ± 0.22

*clinopyroxene

Fig. 8.

Plots of the ~1.5-μm water absorption BD versus ice content (wt.%) for (a) olivine, (b) plagioclase, (c) clinopyroxene, and (d) a mixture of the three minerals, where the fitting lines pass through the origin. Results are shown for the 180–250-μm grains (open circles and dashed lines) and the 75–125-μm grains (closed circles and solid lines). Error bars represent a 1σ range, with the horizontal axis error being determined from Eq. (S1) and vertical axis error being calculated from Eq. (S2).

Fig. 9.

Plots of the ~1.5-μm water absorption BD versus ice content (wt.%) on the four studies minerals, along with their fitting lines. Panels (a) and (b) show the results of the 75–125-μm and 180–250-μm grains, respectively. Note that the gradient lines of plagioclase and clinopyroxene nearly coincide in (a). Error bars represent a 1σ range obtained as described in the caption of Fig. 6.

Comparing the plots of different mineral grain sizes, the calibration lines for the coarse powders of all mineral types exhibited steeper slopes than those for fine grains. Meanwhile, comparing the plots of different mineral species, the slopes of the calibration lines for each grain size decreased in the order of clinopyroxene, plagioclase, three-mineral mixture, and olivine.

Discussion

Spectral distributions of samples in three-dimensional space

The calibration line we acquired can determine the ice content for a given band depth. Owing to the graph’s configuration—where the x-axis represents the ice content and the y-axis represents the band depth—a greater calibration-line gradient indicates that a smaller ice content is inferred from the same band depth. The calibration-line gradients differed considerably depending on the base mineral grain size and mineral species. For example, the calibration-line gradient for 75–125-μm olivine was 1.83, which is approximately 6.5 times lower than that for 180–250-μm clinopyroxene (11.87), as shown in Fig. 8. This indicates that the estimated ice contents can differ by a factor of 6.5 at the same band depth. Examining factors controlling this variation, we found that differences among mineral species could be represented by a simple parameter: the 1.5-μm reflectance of the dry base mineral samples. When the median mineral grain size (in μm), 1.5-μm reflectance of dry base minerals, and calibration-line gradient are plotted along the x, y, and z axes, respectively, in a three-dimensional (3D) space, they appear to align on a single plane (Fig. 10). Therefore, their relationship could be described by a plane equation of the form z = ax + by + c. Two additional data points from Ogishima and Saiki (2021) obtained under the same experimental system were distributed near the same plane, further validating this relationship. Based on the eight data points of this study and additional data points from Ogishima and Saiki (2021), we derived the following equation:

Fig. 10.

(a) Spectral features of samples in the 3D space spanned by the grain size, base sample reflectance at the water absorption wavelength of 1.5 μm, and calibration-line gradient. Two triangular symbols represent the results of Ogishima and Saiki (2021). Error ranges along the y and z axes are the 1σ errors in the reflectance of the base samples and calibration-line gradients, respectively. Reflectance error was determined from variations in the dry-sample spectra during each experiment. Because Ogishima and Saiki (2021) determined dry reflectance from one spectrum per mineral, the error range of their reflectance data could not be calculated. (b) A plot of the measured value from a different viewing direction. Clearly, all sample points are nearly distributed on a single plane.

  

$${\text{gradient}}_{\text{experiment}}=0.0517x+26.0y-17.4,$$(1)

where x is the grain size of the base mineral (in μm) and y is the reflectance of the dry mineral powder at 1.5 μm.

To assess the accuracy of this plane model, we calculated the relative error between the experimentally observed calibration-line gradient and that predicted by Eq. (1) using the median mineral grain sizes and reflectances of the dry mineral powders from Table 2. The resulting relative errors ranged from a small percent to ~35% (Table 3).

Table 2.

Summary of median grain sizes of base minerals, reflectances of dry minerals at 1.5 μm, and calibration-line gradients in the experimental and simulated data

Sample types		Median grain size	Reflectance†	Calibration‡ gradient
Run #1–#5	Olivine (C)*	215	0.460 ± 0.019	5.18 ± 0.38
Run #6–#11	Olivine (F)#	100	0.542 ± 0.025	1.83 ± 0.16
Run #12–#16	Plagioclase (C)	215	0.565 ± 0.029	8.48 ± 0.32
Run #17–#22	Plagioclase (F)	100	0.679 ± 0.020	5.83 ± 0.80
Run #23–#27	CPX (C)	215	0.610 ± 0.018	11.87 ± 0.87
Run #28–#35	CPX (F)	100	0.665 ± 0.047	5.91 ± 0.98
Run #36–#40	Mixture (C)	215	0.509 ± 0.034	5.71 ± 1.14
Run #41–#46	Mixture (F)	100	0.634 ± 0.038	3.11 ± 0.71
experiment (Ogishima and Saiki, 2021)	Olivine	187.5	0.420	3.76 ± 0.69
	Plagioclase	187.5	0.675	8.02 ± 0.71

*Coarse-grained (180–250 μm) samples

^#Fine-grained (75–125 μm) samples

^†Reflectance errors were measured from the reflectances obtained in each spectroscopic observation of the dried samples. The error range of the reflectances in Ogishima and Saiki (2021) was not calculated.

^‡The calibration gradients in Ogishima and Saiki (2021) differ from those in the original paper because we recalculated them for a calibration line passing through the origin, aligning them with our data. Gradient errors were calculated as the standard error of the slope derived from ordinary least squares regression through the origin.

Table 3.

Comparison of the observed calibration-line gradients and those predicted by Eq. (1) using the x and y values in Table 2

Sample types		Observed gradient	Predicted gradient	Relative error (%)
Run #1–#5	Olivine (C)*	5.18	5.62	8.49
Run #6–#11	Olivine (F)#	1.83	1.81	1.09
Run #12–#16	Plagioclase (C)	8.48	8.34	1.65
Run #17–#22	Plagioclase (F)	5.83	5.38	7.72
Run #23–#27	CPX (C)	11.87	9.50	19.97
Run #28–#35	CPX (F)	5.91	5.00	15.40
Run #36–#40	Mixture (C)	5.71	6.89	20.67
Run #41–#46	Mixture (F)	3.11	4.21	35.37
experiment (Ogishima and Saiki, 2021)	Olivine	3.76	3.16	15.96
	Plagioclase	8.02	9.80	22.19

*Coarse-grained (180–250 μm) samples

^#Fine-grained (75–125 μm) samples

The distribution of the calibration-line gradients on the plane resulted from two obvious experimental observations. First, the water ice absorption band depth was affected by the grain size of the base minerals: for the same mineral species, samples with larger base mineral grain sizes exhibited higher calibration-line gradients. Second, the calibration-line gradients were influenced by mineral species. Differences in the mineral species could be represented by a simple parameter—the 1.5-μm reflectance of dry mineral powders. For the same grain size, the calibration-line gradient linearly increased with the reflectance values at 1.5 μm. The alignment of the sample points on the plane in a 3D space was not coincidental and could be understood based on optical scattering principles.

According to Hapke’s (1993) particle scattering model, the bidirectional reflectance of a surface comprising grains—such as regolith or our present samples—results from a combination of single scattering by individual grains and multiple scattering between grains. Generally, as the particle size increased, the optical path length within each grain increased, increasing incident light absorption. Meanwhile, finer grains demonstrated a higher surface-area-to-volume ratio, enhancing the contribution of surface scattering to the reflectance relative to internal transmission and thus increasing overall reflectance (Hapke, 1993). Thus, even for the same dry base mineral, smaller grain sizes yielded higher reflectance values at 1.5 μm (Table 3). In our experiments, ice grains were attached to the surfaces of other grains (Fig. 7). Because smaller mineral grains have higher surface-to-volume ratios, the area density of ice grains on the mineral surfaces was lower for a given ice content. Consequently, the possibility of surface-scattered light to pass through ice grains decreased, weakening the 1.5-μm band (Fig. 11a). Meanwhile, the lower reflectance of the base minerals reduced the ratio of internally scattered to surface-scattered light. Thus, the 1.5-μm band weakened with decreasing contribution of multiple scatterings by water ice (Fig. 11b).

Fig. 11.

Schematic illustrating how the size and reflectance of dry mineral grains affect the absorption BD of water ice: effects of varying (a) the grain size and (b) the reflectance of the base mineral.

Based on the above discussion, the observed plane is not a mere coincidence; rather, it holds as long as the grain size and 1.5-μm reflectance of the actual dry lunar PSR regolith fall within the ranges covered by our experimental conditions. Because the detailed characteristics and composition of the natural PSR regolith remain uncertain, this approach allows us to determine calibration lines based on two simple parameters obtainable from in situ observations: regolith grain size and reflectance. As remote-sensing observations cannot directly measure the regolith grain size in the lunar PSRs, our two-parameter model is particularly valuable for in situ lunar landing missions. High-resolution imaging and direct sampling during future missions will improve the reliability of regolith grain size and reflectance estimations, enabling the development of a more accurate ice quantification model. Therefore, Eq. (1) provides a practical method for determining the water ice content for future lunar PSR explorations without requiring additional micro-ice attachment experiments.

Suitability of our experimental samples for lunar PSR regolith

This section assesses the validity of our laboratory samples as lunar regolith analogs and discusses the limitations of our experimental approach.

In our plane model, the calibration-line gradient representing variations in the 1.5-μm water ice band depth linearly depends on the mineral grain size and the 1.5-μm reflectance of regolith. The mineral species dominantly affects the 1.5-μm reflectance, which is directly related to the absorbance in mineral grains. Higher internal transmission means more light from the mineral grain interior influences the water ice grains adhered to its surfaces, thereby strengthening the water ice absorption band.

Although our experimental samples represent major minerals of the lunar regolith, their solid solution compositions might not represent the typical mineral compositions in lunar PSRs. Specific measurement analyses of lunar PSR regolith are lacking but analyses of the Apollo samples reveal diverse compositions in lunar regolith minerals (Papike et al., 1991). The reflectance at 1.5 μm should be less affected by differences in mineral composition than by differences among the mineral species. For example, pyroxene exhibits two absorption bands (one near 1 μm and the other near 2 μm), and the band peaks of orthopyroxene shift to shorter wavelengths from those of clinopyroxene (Cloutis, 2002). The width of the primary absorption band of olivine at around 1 μm increases with increasing iron content (Sunshine and Pieters, 1998), and plagioclase features a shallow, broad band with a peak between 1.10 and 1.29 μm (Adams and Goullaud, 1978). The absorption bands of any of these mineral species that shift with variations in the solid solution composition are primarily centered around 1 μm and/or 2 μm, and they exert no substantial effect on the reflectance and spectral shape at 1.5 μm.

In summary, when the 1.5-μm reflectance values of regolith do not significantly differ from those of our experiments, the calibration-line gradients can be predicted using Eq. (1). Our approach is applicable to unknown mineral compositions in lunar PSRs.

The reflectances of our mineral powder samples range from ~0.4 to 0.7 at ~1.5 μm, exceeding those of typical lunar regolith (~0.2–0.3) (McCord and Johnson, 1970). However, the LOLA observations at 1.064 μm revealed that the reflectance of PSRs is systematically higher than that of adjacent non-PSRs (Lucey et al., 2014). Based on the reflectance distributions shown in Lucey et al. (2014), we estimated that regions within the PSRs exhibiting reflectance values greater than 0.4 are observed at a frequency of approximately 12% in the north pole PSRs and 1.1% in the south pole PSRs. The higher reflectance observed in PSRs is possibly because they comprise relatively immature regolith due to a weakened space weathering process and/or because of the presence of water ice deposits (Lucey et al., 2014). There is no evidence that all high-reflectance regions in PSRs are caused by water ice, and they are generally considered to result from the presence of immature regolith (Fisher et al., 2017). At least in the case of the Shackleton Crater PSR, where reflectances of up to 0.5 were observed at 1.05 μm using the Kaguya Multiband Imager and similarly by LOLA, the high reflectance is attributed to immature anorthosite regolith rather than water ice (Haruyama et al., 2013). Although the reflectance of lunar PSRs has not been directly observed at 1.5 μm, it is reasonable to expect higher reflectance at this wavelength because lunar regolith typically contains mafic minerals such as pyroxene and olivine, which cause absorption near 1 μm and consequently lower the reflectance at shorter wavelengths like 1.05 or 1.064 μm. Accordingly, the fraction of PSR regions to which our model is applicable may be greater than the ~12% (north pole) and ~1.1% (south pole), since those estimates are based on reflectance at 1.064 μm, which tends to be lower than at 1.5 μm.

Validity and limitations of our model: The ice morphology, grain-size effects, and calibration line estimation

This subsection discusses how the morphology and grain size of water ice affect the spectral behavior of the water ice–mineral mixtures and the resulting calibration lines. We further discuss the validity of the calibration line approximation in low ice contents.

The form of ice in lunar PSRs remains largely unknown. Our experiment simulated the process by which water molecules originating from outside lunar PSRs were cold-trapped on the mineral surface of PSR regolith. Ice grains are relatively uniformly attached to the surfaces of other grains. Several mixing modes have been proposed for water ice and regolith in PSRs, such as continuous ice coating on regolith surfaces and the presence of large ice-breccia mixtures (Cannon et al., 2021). If the actual form of water ice in lunar PSRs differs notably from our experimental conditions, our plane model cannot be used. Over long timescales, the continuous impact gardening process is thought to more strongly drive the sublimation of pure ice layers, rather than stabilizing them as a subsurface ice layer (Costello et al., 2021). This leads to repeated cycles of sublimation and cold trapping. Some sublimated water molecules may be cold-trapped again on the regolith surfaces of PSRs, forming ice grains, as produced in our experiment. Therefore, ice in PSRs can exist as fine ice grains mixed with regolith as observed in this study.

Further, ice grains in our experiments are uniformly sized (~10 μm), but water ice in PSRs may have varying size. Coarse ice grains with a size of 67 μm, as observed in intimately mixed samples by Yoldi et al. (2021), exhibit a higher water absorption band at 1.5 μm than fine ice grains with a size of 4.5 μm. To verify the effect of the ice grain size on calibration-line gradients, we conducted spectral simulations of the ice-attached samples using the Hapke mixing model (see Appendix for calculation details). Following Ogishima and Saiki (2021), we simulated the spectra and determined the calibration lines of a 215-μm grain mixture of three minerals with different water ice contents of 0.5, 1.0, 1.5, and 2.0 wt.% (Fig. 12). For ice grain sizes of 5, 10, and 20 μm, calibration-line gradients are calculated as 7.25, 8.55, and 9.33, respectively (Fig. 12). The Hapke model predicts higher calibration-line gradients for larger ice grain sizes, consistent with the Yoldi et al.’s (2021) experimental observations at a high ice content. The relative error caused by the change in the calibration-line gradient is approximately 15% when the ice grain size decreases from 10 to 5 μm and 9.1% when the ice grain size increases from 10 to 20 μm. In our experiments, the ice grain size is ~10 μm, and even if the grain size varies by several tens of percent, its impact on our results will be relatively small.

Fig. 12.

Plots of the ~1.5-μm water absorption BD versus ice content (wt.%) for the three-mineral mixture. Black data points and their calibration line are experimental results obtained for 180–250-μm mineral grains. Orange, green, and blue points and their corresponding calibration lines are predictions of the Hapke mixing model for ice grain sizes of 5, 10, and 20, respectively, on 215-μm mineral grains.

It may seem somewhat unrealistic to assume that the ice grains in lunar PSR regolith are significantly larger or smaller—by more than an order of magnitude—than those observed in our experiments, for the following reasons. Suppose the ice content is 1 wt.% and the ice grain size is ten times larger than in our experiment, comparable to that of the base mineral grains. Given that the mineral density is approximately three times higher than that of ice, this scenario would correspond to roughly one ice grain per ~33 regolith grains. Under such conditions, each water molecule arriving at the PSR would be ~33 times more likely to be cold-trapped on a mineral grain surface than on an ice grain surface. Therefore, before large ice grains could form, smaller grains would likely nucleate on mineral surfaces instead. Conversely, the formation of significantly finer ice grains is also implausible. PSR surfaces are continually exposed to radiation and micrometeorite impacts, which enhance sublimation. Water molecules tend to be more stable when located in interior regions than at the surface. Fine-grained ice is more susceptible to sublimation, whereas larger grains with lower curvature (i.e., lower surface energy) are more stable and tend to survive longer. Therefore, it is unlikely that extremely fine ice grains exist in large quantities as a uniform mixture within the regolith. As long as the ice content of lunar PSR regolith remains on the order of a few wt.% or less, it is reasonable to assume that the ice grains are of a similar order of magnitude in size to those observed in our experiments.

Our experiments yield lower calibration-line gradients than the values predicted by the Hapke model. This difference may be attributed to the Hapke model assuming an ideal mixture in which ice and mineral grains are randomly distributed in space. In our samples, ice grains adhere to the mineral grain surfaces, potentially causing differences in the optical behavior compared with the ideal random mixture.

In some samples—particularly the mixture of three minerals with 75–125 μm grains—the data points with ice contents below 0.85 wt.% appear to lie below the calibration line (Fig. 8). This deviation was also observed in the Hapke model simulations. Similar to Fig. 12, we simulated the spectra for a mixture of three minerals with 215 μm and 100 μm grains and 10 μm water ice grains at 0.5 wt.% intervals, and determined the calibration lines for the 1.5 μm band depth versus ice content. The results suggest that data points at low ice contents (e.g., 0.5 wt.%) with 75–125-μm base mineral grains tend to fall below the calibration line (Fig. 13). The reason why band depth and ice content do not exhibit a linear relationship in the Hapke model can be explained: the bulk single-scattering albedo of a two-component powder (i.e., the base mineral and water ice) mixture varies non-linearly with changes in the mixing ratio (Eq. S9), and the relationship between single-scattering albedo and reflectance is also non-linear (Eq. S3). A similar deviation might also be present in our experimental results. We confirm that the influence of such potential deviations from the calibration line on its gradient is small. Even when the calibration gradients were derived using only data points with ice contents exceeding 0.85 wt.%, the resulting differences were approximately within the 1σ error ranges shown in Table 2. In other words, any possible deviation from a linear trend is likely incorporated within the error bounds of our planar model. Furthermore, if such deviations tend to occur below the fitted calibration line, as suggested in some low ice-content cases, then the ice content estimated from our model may be slightly underestimated in these regions.

Fig. 13.

Plots of the ~1.5-μm water absorption BD versus ice content (wt.%) for the three-mineral mixture with 100-μm and 215-μm grains simulated by Hapke’s model. The calibration lines are fitting lines passing through the origin in the same manner as in Figs. 8 and 9.

Comparison with previous studies

The relationship between the water ice content in regolith simulants and the water absorption band in NIR spectra was previously investigated by Yoldi et al. (2021), who presented the VIS–NIR spectra of two Mars regolith simulants containing 10–100 wt.% water ice. They created three types of ice–regolith mixtures: frost condensed on the sample surface, intimate mixtures of water ice particles and regolith, and water-saturated frozen soils. An intimate mixture describes a granular-scale mixture of the grains of the end-members. Water frost is a fresh ice resulting from both the direct condensation of atmospheric water on a cold surface and the deposition of ice grains, which are condensed in the atmosphere above the surface, onto a surface. Frozen soil is created by pouring water onto the powders until their pores saturate and a layer of water forms on top of the dust.

The state of the ice-attached samples in our study is similar to those of intimate mixtures described by Yoldi et al. (2021), with ice grains being distributed between mineral grains, rather than forming a continuous layer on the surface of the mineral grains (Fig. 7). Because the reflectances of water ice and dust intimate mixtures do not increase linearly with the water ice content (Yoldi et al., 2015, 2021), the calibration lines derived from experiments with tens of weight percentages of water ice cannot be extrapolated to estimate much lower ice contents, such as those expected in lunar PSRs. Furthermore, the experiments by Yoldi et al. (2021) were intended to simulate Martian regolith using regolith simulants that contained clay minerals exhibiting a 1.5-μm absorption band before ice mixing, and a dark basaltic powder. In this study, to align more closely with the expected ice contents in lunar PSRs, the ice contents in our experiments are an order of magnitude lower than those in Yoldi et al. (2021). The regolith simulants used in this study consisted of plagioclase, pyroxene, olivine, and their mixture, which are representative of primary minerals on the Moon. Our experimental conditions cover a mineral grain size range of 75–250 μm, water ice content of approximately 0.3–2.2 wt.%, and dry base mineral reflectance of about 0.4–0.7. Within this range, the calibration gradient can be predicted using Eq. (1), enabling water ice content estimation from the absorption band depth observed in ice mixed regolith.

To estimate the ice content based on the 1.5-μm water absorption band depth, Eq. (1) requires only two key parameters—the grain size and 1.5-μm reflectance of dry base regolith—while reasonably assuming that the regolith grain size and mineral composition remain consistent across local areas. Based on the 1.5-μm water absorption band depth, our plane model determines the calibration-line gradient and ice content of regolith whose detailed optical and mineral properties remain unknown.

In landing missions, even if the local PSR surface contains several weight percentages of water ice, the adjacent sunlit regions can be considered sufficiently dry for directly assessing differences between the dry surface and ice-mixing layer. The future manned or unmanned explorations of PSRs are expected to involve operations that traverse from nearby sunlit areas—where battery recharging and communication with Earth are possible—into PSRs. This operation facilitates comparisons between the PSR ice-mixing layers and dry surfaces of sunlit regions with minimal operational complexity. Leveraging regolith grain imaging and dry reflectance measurements in sunlit areas, combined with water absorption peaks observed in the NIR spectra of ice-mixing regolith in shadowed regions, future landing and in situ observations will enable a more precise determination of the water ice content using our model.

Conclusion

To obtain information relevant to future lunar polar exploration, we measured the NIR spectra of ice-attached powders prepared from four types of mineral powders (olivine, plagioclase, clinopyroxene, and their mixture) in two distinct grain size ranges (75–125 μm and 180–250 μm) with water contents varying from approximately 0.3 to 2.2 wt.%. By plotting the relationship between the 1.5-μm band depth and the water ice content for each of the eight samples, we quantified the 1.5-μm water absorption band depth and established calibration lines. Unlike previous studies, in which the water ice content of the samples exceeded 30 wt.%, our experiments were performed on samples containing low water ice content.

The water absorption band depth in our study was influenced by water ice content, mineral grain size, and mineral species. In particular, the calibration-line gradients were positively correlated with both mineral grain size and 1.5-μm dry reflectance. Although the dry reflectance is inherently dependent on mineral species, it can be directly measured; therefore, our model does not require prior knowledge of mineral composition. Notably, when the spectral features were plotted in the 3D space defined by the mineral grain size, dry reflectance at the water band wavelength, and calibration-line gradient, the sample points approximately occupied a single plane. This finding implies that the calibration-line gradient of a lunar PSR regolith, even of uncertain composition, can be predicted from its grain size and dry 1.5-μm reflectance provided that the state of regolith is similar to that of our experimental conditions, in which fine ice grains are mixed with the base minerals. As the regolith grain size and dry 1.5-μm reflectance are readily measurable in situ, our model provides a practical approach for estimating the water ice content in future lunar PSR landing missions without requiring detailed knowledge of the mineral composition, mineral mixing ratios, and optical behaviors of regolith.

Appendix

Error propagation in ice content determination and detection limit of the ice content

Let $m_1$, $m_2$, and $m_3$ be the weights of the sample holder with the ice-attached mineral powder, the sample holder with the dry mineral powder, and the empty sample holder, respectively. The mass of the water ice is then determined as $\partial w=m_1-m_2$ and that of the base mineral sample is $m_s=m_2-m_3$. The ice content f is expressed as $f=\partial w/m_s$.

Denoting the measurement uncertainty of the electronic balance as ∂x (0.001 g), the uncertainty ∂f in the ice content can be evaluated using the following error propagation formula:

  

$$\partial f=\partial x\sqrt{{\left(\frac{1}{m_s}\right)}^2+{(\frac{1}{m_s}+\frac{\partial w}{m_s^2})}^2+{\left(\frac{\partial w}{m_s^2}\right)}^2}=\frac{\partial x}{m_s}\sqrt{2+2f+2f^2}.$$(S1)

Calculation of the band depth error

The band depth uncertainty was determined in terms of the standard deviation (SD) $\partial R$ of the unsmoothed spectrum relative to the smoothed spline fitting curve. To simplify the analysis, we assumed a uniform measurement uncertainty across the wavelength range near the 1.5-μm water absorption band. The band depth (BD) is defined as

  

$$BD=\frac{(R_c-R_b)}{R_c},$$

where $R_b$ is the reflectance at the absorption band peak (~1.5 μm) and $R_c$ is the continuum reflectance at the same wavelength. The measurement uncertainty in the reflectance, denoted as $\delta R$, is the average SD between the measured and smoothed spectra in the 1400–1640-nm range.

The uncertainty in the BD was determined using the error propagation formula. As $R_c$ and $R_b$ were independently measured with similar uncertainties, the uncertainty in their difference was calculated as

  

$$\partial (R_C-R_b)=\sqrt{2}\partial R.$$

Expressing the BD as the ratio of two independent quantities, A =$R_C-R_b$ and B =$R_C$, we applied the standard error propagation formula for division:

  

$$\delta BD=\sqrt{\frac{\delta A^2}{B^2}+\frac{A^2\delta B^2}{B^4}}.$$

Substituting $\delta A=\sqrt{2}\partial R,A=R_c-R_b,B=R_c,$ and $\delta B$ $=\delta R$, we obtain:

  

$$\partial BD=\frac{R_c-R_b}{R_c}\sqrt{{\left(\frac{\partial R}{R_c}\right)}^2+{\left(\frac{\sqrt{2}\partial R}{R_c-R_b}\right)}^2}.$$(S2)

This equation provides the uncertainty in the BD, accounting for the measurement uncertainty in the reflectance and its propagation through the BD calculation.

Hapke mixing model

As described in the main text, we compared our experimentally derived calibration lines with those simulated by the Hapke mixing spectral model. According to Hapke (1993), a reflectance spectrum R can be converted to a single-scattering albedo (SSA) using the following reflectance factor:

  

$$R(i,e,g)=\frac{w}{4}\frac{1}{{\mu }_0+\mu }[(1+B(g))p(g)+H\left({\mu }_0\right)H(\mu )-1],$$(S3)

where i, e, and g denote the incidence, viewing, and phase angles, respectively, ${\mu }_0=\cos i$, and $\mu =\cos e$. Both our measurements and models were performed in an observation geometry of i = 30°, e = 0°, and g = 30°. The opposition effect B(g) was assumed as 0 because the phase angle was sufficiently large. Following Mustard and Pieters (1989), the phase function $p(g)$ at g = 30° was defined as $p(g)=-0.75w+1.65$.

For isotropic scattering, we employed the Hapke approximation of the Chandrasekhar function

  

$$H(x)\sim \frac{1+2x}{1+2x\sqrt{1-w}}.$$

As indicated in Eq. (S3), under constant phase-angle conditions and negligible opposition effects, the reflectance can be approximated as a function of a single parameter, the SSA.

The SSA itself is a function of the optical constants (the real (n) and imaginary (k) components of the complex index of refraction) and the optical path length $⟨D⟩$, which can be related to the mineral grain size d. In terms of these parameters, the SSA becomes

  

$$w=S_e+(1-S_e)\frac{(1-S_i)\theta }{1-S_i\theta },$$(S4)

where the external scattering coefficient $S_e$ and internal scattering coefficient $S_i$ are respectively defined as

  

$$S_e=\frac{(n-1{)}^2+k^2}{(n+1{)}^2+k^2}+0.05,$$(S5)

  

$$S_i=1-\frac{4}{n(n+1{)}^2},$$(S6)

and the internal transmission factor $\theta$ is approximated as

  

$$\theta =\frac{r_i+\exp (-\sqrt{\alpha (\alpha +s)}⟨D⟩)}{1+r_i\exp (-\sqrt{\alpha (\alpha +s)}⟨D⟩)},$$(S7)

  

$$r_i=\frac{1-\sqrt{\frac{\alpha }{\alpha +s}}}{1+\sqrt{\frac{\alpha }{\alpha +s}}}.$$(S8)

Assuming that the particles are spherical grains, $⟨D⟩$ can be approximated as 2/3 of the grain size d. The volume absorption coefficient $\alpha$ is related to the imaginary index of refraction via the dispersion relation $\alpha =4\pi k/\lambda$. Further assuming minimal internal scattering, the near-surface internal scattering coefficient s is approximated as 0.

Hapke (1993) calculated the bulk SSA of an intimate mixture of powders as

  

$$w_{bulk}=\frac{w_2+f{w}_1}{1+f}$$(S9)

with

  

$$f=\frac{M_1}{M_2}\frac{{\rho }_2}{{\rho }_1}\frac{d_2}{d_1}.$$

Here, $M_j$ is the bulk mass, ${\rho }_j$ is the density, and $d_j$ is the grain size of the j-th component. In this study, the densities of clinopyroxene and water ice were assumed as 3.4 and 0.92 g/cm³, respectively.

We first determined the SSA of clinopyroxene by solving Eq. (S3) inversely using the Newton method. The complex refractive indices (n and k) of water ice were extracted from the data provided by Warren and Brandt (2008). Next, the SSAs of water ice with grain sizes of 5, 10, and 20 μm were calculated using Eqs. (S4)–(S8), and the SSA of the mixture was determined using Eq. (S9), along with the SSA of clinopyroxene. Finally, the spectra of the samples attached with ice grains of different sizes were obtained using Eq. (S3).

All raw data used in this study are accessible in the Osaka University Knowledge Archive (https://doi.org/10.60574/87068). This includes all spectral images (Fig. S1).

Acknowledgments

This study was supported by the Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for scientific research (grant no. 19H01953).

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