Multiband Spectropolarimetry of Lunar Maria, Pyroclastics, Fresh Craters, and Swirl Materials

Imaging polarimetry is a well-known method for examining the small-scale structure of the surface regolith of airless celestial bodies. In this study, we examine (for the first time) the wavelength-dependent polarization behavior of selected lunar areas, including maria, highlands, fresh craters, pyroclastic deposits, and the Reiner Gamma swirl, based on telescopic multiband UBVRI imaging polarimetry at phase angles within the range of the positive polarization branch. The terrain-dependent spectropolarimetric behavior is studied for the first time in this work. For each study area, we conduct a mapping of the relative regolith grain size, an analysis of the exponent of the Umov law, and the wavelength dependence of the degree of linear polarization. Furthermore, we perform area-specific principal component analyses of the degree of linear polarization, followed by unsupervised machine learning (clustering) to segment different terrain types. We find that fresh mare craters and high-titanium pyroclastic deposits have an increased regolith grain size, whereas crater ray material, low-titanium pyroclastic material, and the Reiner Gamma swirl are more finely grained than the average regolith. The degree of linear polarization decreases with increasing wavelength-dependent albedo according to a power law whose exponent is itself positively correlated with the albedo. For a constant albedo and grain size, the degree of linear polarization increases linearly with wavelength. The clustering step yields a library of terrain-dependent prototype spectra of the degree and angle of linear polarization.


Introduction
Already for about a century, polarimetric measurements of the Moon have been used to reveal the structural properties of its surficial regolith (see, e.g., Shkuratov et al. 2015 for an overview).However, it took until about 30 yr ago for lunar imaging polarimetry to become popular, which allowed for the mapping of the polarization state of the lunar surface at a resolution of a few kilometers using telescopic data (Shkuratov & Opanasenko 1992;Dollfus 1998;Shkuratov et al. 2007).The outcomes were primarily based on the analysis of the degree of linear polarization (DoLP) at phase angles around 90°(the so-called positive polarization branch around the maximum DoLP) for estimating the regolith grain size and the microscopic surface roughness.In particular, Shkuratov & Opanasenko (1992) established the dependence of the DoLP on the albedo and interpreted local deviations from the fitted relation as variations in regolith grain size.Shkuratov et al. (2007) found the value of the phase angle at which the DoLP obtains its maximum (which is always close to 90°) to be correlated with the optical maturity of the lunar regolith.They also analyzed the dependence of the polarization behavior on wavelength by computing images of the ratios between the components parallel and perpendicular to the plane of linear polarization at two different wavelengths.A summary of these works can be found in the overview article of Shkuratov et al. (2011).The contrast in DoLP between different lunar terrains was decomposed by Dollfus (1998Dollfus ( , 1999) ) into different contributions ascribed to variations in albedo, regolith grain size, microscopic surface roughness, and multiple scattering.Variations in roughness and grain size were observed across mare surfaces, and the rays of the Messier crater pair were found to be more coarsely grained than the surrounding mare.More recently, Jeong et al. (2015) constructed grain-size maps of the lunar nearside, based on imaging polarimetry, and examined the terrain dependence of the ratios between the components parallel and perpendicular to the plane of polarization in different spectral bands.They found the grain size in most nearside maria to be larger by about 15% than in the nearside highlands, and a general trend that the median grain size increases with latitude.Bhatt et al. (2023) studied the polarimetric and photometric characteristics of the Reiner Gamma swirl and, for comparison, the nearby craters Aristarchus and Kepler.They found conspicuous variations in grain size within Reiner Gamma, along with a significantly decreased opposition effect strength at its central oval, but only marginal differences in surface roughness relative to its surroundings.The first orbital imaging polarimetric studies of the lunar surface are currently being conducted by the polarimetric camera PolCam on board the Korea Pathfinder Lunar Orbiter/Danuri spacecraft (Jeong et al. 2020;Sim et al. 2020).The PolCam observations will increase the spatial resolution of lunar imaging polarimetry by up to 2 orders of magnitude in comparison to existing telescopic data sets.
This study aims at an identification of the wavelength dependence of the polarization properties of the light scattered by the lunar regolith, where the DoLP will be analyzed in the UBVRI spectral bands, ranging from the near-ultraviolet throughout the visible into the near-infrared domain.We will focus our analysis on: (i) the mare area of western Oceanus Procellarum (WOP), comprising the large rayed craters Aristarchus and Kepler, the Marius Hills volcanic complex, and the well-known swirl structure Reiner Gamma; (ii) the mixed mare/highland area between the crater Copernicus and Mare Vaporum (MV), exhibiting three regional pyroclastic deposits; and (iii) the floor-fractured crater Atlas (AT), exhibiting two localized pyroclastic deposits on its floor (see the first row of Figure 1 for intensity images).In particular, we will derive the relative regolith grain size in the study areas, examine the Umov exponent describing the relation between the DoLP and albedo within and across the individual spectral bands, and perform a principal component analysis (PCA) of our five-band DoLP data, to use it as the basis of an unsupervised classification (clustering) of our DoLP data.Extracting the cluster prototype spectra will then enable us to create a library of representative spectropolarimetric characteristics of different lunar terrain types.

Data Acquisition
For the acquisition of the lunar polarization imaging data examined in this study, a monochrome industrial camera DZK 33GX250 manufactured by the company The Imaging Source was used.To the individual pixels of the camera sensor, polarization filters rotated by 0°, 45°, 90°, and 135°are attached (The Imaging Source 2019).In this way, it is possible to infer from each 2 × 2 subset of pixels the unpolarized intensity, the DoLP, and the angle of linear polarization (AoLP) relative to the direction of the pixel rows.This camera was mounted on a Newton reflector telescope of 200 mm diameter with a primary focal length of 1.2 m.The telescope facility is located near Dortmund, Germany.The focal length was extended to 3.0 m using a Barlow lens, resulting in an image scale of about 0.9 km per pixel of the polarization images.The acquisition details of our spectropolarimetric images are given in Table 1.
To construct a polarization data set, we employed a "lucky imaging" technique by acquiring 3500 individual video frames in the Johnson-Cousins U, B, V, R, and I bands, respectively, ranging from the near-ultraviolet through the visible to the near-infrared spectral domain (Table 2).From each video frame, the subframes corresponding to the four polarization directions were extracted.For each polarization direction, the sharpest 10% of the subframes were selected and stacked, respectively, with the software Autostakkert3 (Kraaikamp 2023).This image-stacking procedure resulted in one image of 16 bits pixel depth per polarization direction.The 45°, 90°, and 135°images were then coregistered at subpixel accuracy with respect to the 0°image, where the image transformations were computed by minimizing the sum of squared differences between the high-frequency components of each image pair (Bhatt et al. 2023).A sinusoidal function was then fitted to the pixel intensities at each position of the resulting quadruple of coregistered images, yielding images of the intensity F, the DoLP P, and the AoLP W, as shown in Figure 1.The measurement error of the DoLP was found to be 0.01, by analyzing artificial quadruples of polarization images with the same noise level as the observed ones.Eventually, the U-, B-, V-, and R-band data of each region were all coregistered with respect to the I-band data, the latter being least affected by atmospheric seeing and sensor noise.The local variations of the AoLP are likely to be spurious for the reasons explained in Section 3.1.
By measuring a fully linearly polarized light source (a light bulb combined with linear polarizer), we found that the polarization sensitivity of our setup corresponds to 1.0 for the U, B, V, and R bands and to 0.89 for the I band.That is, the measured DoLP values in the UBVR range are not affected by systematic errors, whereas our measurements systematically underestimate the DoLP in the I band by a fraction of 11%.

Georeferencing of the Telescopic Data
Georeferencing is an important step in processing telescopebased lunar observations, in order to obtain the corresponding reflectance for deriving the physical parameters discussed in this work.For the georeferencing of the telescopic data, we constructed an artificial image of the Moon from the Lunar Reconnaissance Orbiter Wide Angle Camera mosaic (Wagner et al. 2015) for each observation, using a perspective projection, assuming the same lunar distance and subobserver longitude and latitude as given at the time of data acquisition.The intensity channel in the I band of each observation was then coregistered manually to the artificial image.Since the selenographic coordinates of the pixel grid of the artificial image are known, we inverted the transformation and converted the telescopic data to georeferenced maps in simple cylindrical projection (Snyder 1987).

Regolith Grain Size
For estimating the regolith grain size, we have generally followed the method proposed by Shkuratov & Opanasenko (1992), based on the relation between the surface albedo A and the DoLP P near the phase angle of strongest polarization.As a general trend, P is known to decrease with increasing value of A, a behavior that is known as the Umov law (Shkuratov & Opanasenko 1992).Accordingly, the first step of the method corresponds to a fit of the power law to all pixels of band k of the region under study, yielding estimated values of the parameters a k and b k .In this work, we used the spectral reflectance R(λ k ) at the center wavelength λ k of the filter band k as a proxy for the albedo A(λ k ).The spectral reflectance was derived from hyperspectral image data acquired by the Moon Mineralogy Mapper (M 3 ) instrument on board the Indian lunar orbiter Chandrayaan-1 (Green et al. 2011).The spectral reflectance was resampled to a scale of 20 pixels per degree, corrected for topography, and normalized photometrically to the standard configuration of 30°incidence angle and 0°emission angle (Wöhler et al. 2014;Wohlfarth et al. 2023).
The applied photometric normalization removes the lunar limb brightening and the opposition effect.A similar replacement of the albedo A by a phase-angle-specific spectral reflectance has been suggested by Shkuratov & Opanasenko (1992).They established an absolute, laboratory-based calibration of the mean grain size d based on the pixel-specific value of b k computed according to Equation (1).As the absolute scaling of the spectral reflectance used as a proxy for A depends, e.g., on the assumed phase angle, we modified the technique of Shkuratov & Opanasenko (1992) in the way that we computed the grain size d rel relative to the average grain size of the area under study, resulting in Here, P pix (λ k ) and R pix (λ k ) denote the corresponding values of the DoLP and spectral reflectance for a specific image pixel, and we set c = 2.9 according to the laboratory-based calibration of Shkuratov & Opanasenko (1992).Correspondingly, individual pixels for which b k pix is larger than the fitted value of b k , thus showing an "excess" of P pix (λ k ), are associated with grain sizes larger than the average value, whereas pixels with a "deficiency" of P pix (λ k ) have smaller grain sizes.In principle, a separate grain-size map can be computed for each band k.

Pixel-wise Umov Exponent
The albedo and thus also the spectral reflectance of the lunar surface commonly increase with increasing wavelength in the range spanned by the UBVRI bands (e.g., Smrekar & Pieters 1985).This behavior explains why the DoLP typically Note.The B filter had to be exchanged in 2023 January.
shows the tendency to decrease with increasing wavelength.We found that for an individual pixel, the dependence follows a power law of the form where the "across-band Umov exponent" ò is usually negative but strongly depends on the phase angle and, for a constant phase angle, on the terrain type.As in Section 2.3, we used the M 3 spectral reflectance as a proxy for the albedo and performed a pixel-wise estimation of the Umov exponent ò.The wavelength of the U band is not covered by the M 3 spectral range, and for phase angles around 90°, where the DoLP obtains its maximum, the I-band DoLP generally deviates from Equation (3), because it increases relative to the R band (see Section 3.7 for details).Thus, we restricted the computation of ò to the B, V, and R bands.The value of ò is positively correlated with the spectral reflectance, but deviations from this correlation occur for specific terrains (see Section 3).Additionally, we computed for each study area and each wavelength λ k the "within-band Umov exponent" δ k = − 1/a k , with a k fitted according to Equation (1).

Dependence of DoLP on Wavelength
To examine the dependence of the DoLP on the wavelength of the light scattered by the lunar regolith, we used the estimated band-specific values of a k and b k , as defined by Equation (1), and computed the corresponding values of ˜( ) k for a fixed spectral reflectance R.This procedure removed the effect of variable regolith grain size, because for given spectral reflectance and parameters a k and b k , Equation (1) yields the DoLP for a constant grain size corresponding to the average grain size of the study area.As a result, we obtained ˜( ) l P curves unaffected by grain-size variations for a set of constant spectral reflectance values, as shown in Figure 5.

PCA and Clustering
In order to visualize the acquired five-band DoLP data, we applied a PCA (e.g., Marsland 2015) to each set of measurements.We only retained the three most significant principal components (out of five) of the DoLP, because they contained more than 95% of the information for all acquired data sets.The resulting PCA features are visualized as RGB color composites in Figure 6.The color differences in these RGB images correspond to differences in the wavelength dependencies of the DoLP, respectively.
In order to extract a small set of typical DoLP spectra, we applied a clustering technique known as the self-organizing feature map (SOM; see, e.g., Marsland 2015 for a summary of the algorithm) to the set of six-dimensional PCA feature vectors derived from the DoLP data.The SOM was configured in the way that it consists of 3 × 3 neurons, such that it subdivides the data into nine clusters.This procedure results in a cluster index image, in which each pixel is assigned to one of the nine clusters.The cluster indices are strongly correlated with different types of terrain present in the area under study.Furthermore, all DoLP spectra belonging to the same cluster were averaged, thus yielding a set of nine DoLP prototype spectra that are representative of the area under study.The cluster index maps in combination with the prototype spectra for the first time allow for an intuitive understanding of the typical polarization behavior of the occurring terrain types.In principle, this approach can be further extended to other airless bodies for analyzing unknown terrain types, relying on the understanding developed based on the lunar terrains.

Results
The focus of this study is on three distinct areas of the Moon covered by our UBVRI spectropolarimetric data.
1. WOP: western Oceanus Procellarum with the rayed craters Aristarchus and Kepler, the Marius Hills volcanic complex, and the swirl Reiner Gamma, surrounded by mare basalts of a variety of chemical compositions (e.g., Hiesinger et al. 2003).2. MV: the large pyroclastic deposits of Mare Vaporum, Sinus Aestuum, and Rima Bode (e.g., Gaddis et al. 2003).3. AT: the region around the floor-fractured crater Atlas, exhibiting two small pyroclastic deposits (Gaddis et al. 2012), and the lava-flooded crater Hercules, surrounded by typical highland surface to its west, north, and east, an unnamed bright rayed highland crater to its east, and smoother, mare-like material to its south.
In this section, we will discuss in some detail the WOP data of 2022 December 16, the MV data of 2023 March 2, and the AT data of 2023 February 28.The dependencies of the terrainspecific prototype spectra and the within-band Umov exponent on the phase angle will be illustrated based on all five observations of the AT area.Our telescopic data set is provided as online Supplementary Material.

Intensity and DoLP
The I-band intensity, DoLP, and AoLP of the WOP, MV, and AT areas are shown as georeferenced maps in simple cylindrical projection in Figure 1.As predicted by the Umov law, the DoLP is inversely correlated with the intensity in all three study areas.The smallest DoLP values are found at bright and fresh craters, in particular at Aristarchus in the WOP area and at an unnamed bright highland crater to the east of Atlas.In Figure 1, the AoLP is strongly correlated with the DoLP.This behavior suggests that the apparent spatial variations of the AoLP with an amplitude of up to several degrees are likely to be spurious.Possibly, weak crosstalk between the polarization channels (which are all acquired simultaneously by the same image sensor) caused by a small amount of stray light (whose amount would depend on the surface albedo) distributed across several neighboring sensor pixels may easily induce spurious local, albedo-dependent AoLP variations.This might explain why spatial AoLP variations of a similar order as those visible in Figure 1 have not been described in the pertinent literature (see Shkuratov et al. 2011 for a summary), where a more traditional setup relying on consecutive acquisitions of the polarization channels was used.Alternatively, subpixel inaccuracies in the mutual coregistration of the polarization channels, e.g., caused by subtly different distortions in each channel, would cause AoLP anomalies in particular in strongly structured areas with large intensity gradients.For these reasons, we will not further analyze the AoLP variations in this study, but will look more thoroughly into this matter by checking for consistency with future additional data acquired with different telescope/sensor combinations.

Relative Grain Size
The relative grain-size maps obtained for the different spectral bands are well consistent with each other for the examined study regions.The relative grain-size maps obtained in the I band are shown in Figure 2. The vertical stripes visible in these maps (and the maps of other parameters whose computation involves M 3 data) are due to calibration artifacts in the M 3 spectral radiance (Green et al. 2011).
In the WOP area, the grain size is larger by about 10% for the older mare units around Reiner Gamma, which appears darker than the younger mare units around and in between Kepler and Aristarchus (age units as per Hiesinger et al. 2003).The pyroclastic deposit northwest of the crater Aristarchus is at least in part more finely grained than the surrounding mare, which has also been observed by Shkuratov & Opanasenko (1992).The interiors of the relatively young craters Aristarchus and Kepler stand out due to their anomalously large grain size.The structure of Reiner Gamma appears to be more finely grained than the surrounding mare, even more finely than the small ray system of the crater Lichtenberg and the extended rays crossing the surface of Oceanus Procellarum between the craters Seleucus and Aristarchus.Small craters scattered all over Oceanus Procellarum appear as coarsely grained anomalies.The contrast in grain size of the ray systems of Aristarchus and Kepler relative to the surrounding mare only amounts to about 3%-5%.
Parts of the highland surfaces of the MV area appear to have a slightly larger grain size than the maria, but these possible differences are obscured by the stripe artifacts, making it difficult to tell whether they are real.The strongest grain-size anomaly corresponds to the Rima Bode pyroclastic deposit, but also the Sinus Aestuum and MV deposits stand out due to their grain size, which is increased by about 20% compared to the surrounding surface.
In the AT area, the lava-flooded floor of the crater Hercules and the northern pyroclastic deposit on the floor of Atlas exhibit a grain-size increase by about 15% relative to the surrounding surface.Due to its high DoLP, the southern pyroclastic deposit on the floor of Atlas stands out as a localized anomaly with a grain size larger by up to 20%-40% than for the surrounding surface.The small bright rayed crater east of Atlas exhibits a similarly pronounced grain-size anomaly.

Umov Exponent
For all three study regions, the across-band Umov exponent ò exhibits a strong positive correlation (with correlation coefficients between 0.6 and 0.9) with the logarithm of the R-band reflectance.We fitted a second-order polynomial to the relation ò versus R log and then mapped the residual  res of ò with respect to the established relation (Figure 3).
In the  res map of the WOP area, the crater Aristarchus, the Kepler rays, and Reiner Gamma stand out as positive anomalies, whereas the ray near Seleucus, the rims of large craters near the very western border of Oceanus Procellarum, the Marius Hills area, the Aristarchus pyroclastic deposit, and the small craters scattered across the mare surface appear as negative anomalies.The  res map of the MV area is more difficult to interpret because it appears to be affected by oblique illumination.The Rima Bode and MV pyroclastic deposits and the surroundings of the well-preserved crater Bode appear as negative anomalies, and the rays of the crater Copernicus on the smooth surface of Sinus Aestuum as positive anomalies.The  res map of the AT area is largely inconspicuous, but shows a small positive anomaly for a part of the southern pyroclastic deposit on the floor of Atlas and a more extended but less pronounced negative anomaly at the northern deposit.The bright impact crater east of Atlas stands out as a positive  res anomaly.
Table 3 shows that the within-band Umov exponent δ k describing the area-specific dependence of the DoLP P(λ k ) on the spectral reflectance R(λ k ) is always negative, because in a given wavelength band the DoLP always decreases with increasing albedo.For all study areas, δ k increases with increasing wavelength λ k at least across the BVR range (Table 3), i.e., P(λ k ) decreases less strongly with increasing R(λ k ) at long wavelengths than at short wavelengths.For the AT area, Figure 4 displays the variations of δ k with wavelength and phase angle.The data obtained at the smallest observed phase angle of 38°give no meaningful result due to a near-zero correlation between DoLP and the albedo.To illustrate this behavior in more detail, Figure 9 shows DoLP maps of the AT area for all five observed phase angles.These maps reveal that P does not only weaken in general with decreasing phase angle, but also becomes more and more uniform, in particular resulting in a decreasing contrast between mare and highland surfaces.

Dependence of Normalized DoLP on Wavelength
Not surprisingly, the DoLP P normalized with respect to the albedo and grain size increases the complete UBVRI wavelength range when the spectral reflectance decreases.In addition to that, it turns out to be a general, region-independent behavior that P increases with increasing wavelength for constant values of the spectral reflectance and grain size (Figure 5).Furthermore, the slope ˜l dP d is largely independent of the spectral reflectance.
The values of P in Figure 5 are lower in the AT area than in the MV area, despite the higher phase angle of the AT data (74°f or AT versus 52°for MV).This somewhat unexpected behavior can be explained by the much higher mean albedo of the AT area, which is about twice as high as the mean albedo of the MV area.As a consequence, for the AT area, the effect of the Umov law prevails above the effect of the higher phase angle.

PCA
The scores PC1, PC2, and PC3 on the three most significant principal components of P over the spectral bands cannot be attributed to a specific physical meaning, but differences in their configurations within one map correspond to differences in the polarization behavior of the surface (see also Wöhler et al. 2023).We thus visualize the PCA results as false-color RGB composite images for P for each study area, respectively (Figures 6 and 11), where the first, second, and third principal components were assigned to the red, green, and blue channels of the color composite, respectively.In all three study areas, the value of the P-derived PC1 is correlated with the albedo.More subtle differences between various terrain types are revealed by PC2 and PC3, as shown in Figure 11.
Figure 6 shows that two mare units of WOP can be differentiated using the principal components of P. Furthermore, the P-derived PCA images emphasize the pyroclastic deposits in the MV and AT regions.According to Figure 11, two mare units differ strongly by their PC2 values in the P-derived PCA of the WOP area.The inner ejecta blanket and the rays of Kepler stand out as negative anomalies, whereas the intense rays crossing the surface of westernmost Oceanus Procellarum show up due to their high values.Reiner Gamma does not show a strong contrast in PC2 with the surrounding surface.In the PC3 map, different mare units all across Oceanus Procellarum can be distinguished, especially the Marius Hills region.The main oval of Reiner Gamma appears bright, similar to Aristarchus and Kepler, but the tail shows up as a low-value lineament (Figure 11).
In the P-derived PC2 map of the MV area, the Rima Bode and Sinus Aestuum pyroclastic deposits stand out as positive anomalies, whereas in the PC3 map, the Rima Bode deposit is a positive anomaly and the Sinus Aestuum and MV deposits are negative anomalies.The P-derived PC2 and PC3 maps of the AT area are without structure except for topography-related artifacts, with the exception of the southern pyroclastic deposit appearing as a positive anomaly in PC2 and a negative anomaly in PC3.

Clustering
For each study area, the SOM-based clustering stage yields a set of nine prototype P spectra that represent the corresponding area, along with a cluster index map (Figure 7).The cluster index map of the WOP area distinguishes between the three main basaltic units, the fresh material of the craters Aristarchus and Kepler, the inner and mutually overlapping outer ray systems of these craters, and Reiner Gamma.Interestingly, the main oval of the swirl is assigned the same cluster index as the inner continuous ejecta blanket of Kepler, whereas the swirl's tail is assigned the same index as parts of the outer Kepler rays and the surrounding mare.This clearly suggests differences in the regolith properties between the main oval and the tail of Reiner Gamma.The prototype P spectra exhibit a continuous transition from strongly wavelength-dependent mare spectra to nearly "flat" fresh crater spectra.
In the MV area, the P prototype spectra again form a regular sequence, where the dark pyroclastic deposits exhibit the highest DoLP, which decreases slightly toward long wavelengths, whereas the bright highlands and the ejecta blanket of Manilius have a lower DoLP, which increases slightly toward long wavelengths.The two "outlier" prototypes (in the dark blue color) correspond to shadowed parts of the surface.The   generally less strong wavelength dependence of the DoLP in the MV area when compared to the WOP area is presumably due to the fact that we imaged MV at a smaller phase angle (52°.0)than WOP (87°.5).The three pyroclastic deposits are assigned to the same cluster and differ from the surrounding mare material.
Also in the AT area, the cluster analysis distinguishes between the mare-like and highland-like terrains of the region.The two pyroclastic deposits on the floor of Atlas correspond to a unique cluster (indices 5 and 7; green and yellow), which can also be found in the northern floor of the neighboring crater Hercules that is flooded by dark lava.The smallest DoLP is found at the unnamed bright and fresh highland crater located to the east of Atlas.The P prototype spectra show a welldetectable decrease of the DoLP from the UB toward the VRI wavelength range, where the wavelength dependence is stronger than in MV and weaker than in WOP, which is presumably due to the intermediate phase angle at data acquisition of 73°.6.

Phase-angle-dependent P Prototypes
We acquired multiple sets of observations of the AT area at different phase angles (Table 1).We used these data to examine the phase-angle-dependent behavior of the P prototype spectra.Figure 8 shows the phase-angle dependence of the P prototype spectra of: (i) the pyroclastic deposits on the floor of the crater Atlas; (ii) the highland north of Atlas; (iii) the fresh highland crater east of Atlas; (iv) the mare south of Atlas; and (v) the mare-flooded floor of the crater Hercules.
For all terrains, the amplitude of the variation of P decreases with increasing wavelength, and there is a transition from a positive to a negative slope of the P spectrum between phase angles of 52°and 74°.The highest values of P are observed at a phase angle of 108°at short wavelengths for the pyroclastic deposits, the mare south of Atlas, and the mare-flooded floor of Hercules.At longer wavelengths, in particular in the I band, the maximum of P is observed at a phase angle of 85°, i.e., apparently the phase angle at which P obtains its maximum decreases with increasing wavelength (Figure 10).Although at phase angles of 108°and 74°, the value of P is lower by about 30% in the highland terrains than in the pyroclastic and mare terrains, this difference between the terrains decreases for lower phase angles and nearly vanishes at 38°.

Discussion
In this section, we discuss the observed polarization effects and suggest possible reasons for their occurrence in a qualitative way.
The absence of a general contrast in grain size between the maria and highlands in the MV and AT study areas suggests that the grain size is mainly influenced by external factors, in particular micrometeoroid bombardment, and not by chemical composition (but, admittedly, the highland areas studied in this work are not extensive and are all adjacent to maria).In comparison, the results of Jeong et al. (2015) are ambiguous, as a clear difference in grain size between maria and highlands is visible in their Figure 7, but not in their Figure 18.The increased relative grain size of fresh craters, such as the large craters Aristarchus and Kepler, and the multitude of small craters scattered across the surface of Oceanus Procellarum presumably reflects the young ages of these structures, because time has so far been insufficient to reduce the grain size to the level of the surrounding surface by micrometeoroid impacts.Extended crater rays are generally more finely grained than the surrounding mare surface, which hints at a separation between coarse and fine grains on the rays.This mechanism is presumably due to a fast movement of small ejecta particles nearly parallel to the surface just after the crater-forming impact, so that they could exert a horizontal force on the regolith grains, whose effect possibly depends on their size.The observation that Reiner Gamma is more finely grained than the undisturbed surrounding mare surface hints at a separation between coarse and fine grains on the the swirl surface.This would support the interpretation of Hess et al. (2020Hess et al. ( , 2023) ) and Bhatt et al. (2023) that the Reiner Gamma swirl underwent an interaction between the regolith grains and the gas of a passing cometary coma as the source of a horizontal force on the regolith grains (where a force component due to the same interaction but with a direction perpendicular to the surface led to the soil compaction described by Hess et al. 2020).An alternative sorting mechanism (which cannot account for soil compaction, though) was proposed by Garrick-Bethell et al. (2011) and assumes the electrostatic lofting of small regolith particles at swirls associated with localized magnetic fields.In that model, the sources of the electrostatic forces are the gradients of the electric potential resulting from the separation between positively and negatively charged solar wind particles at localized magnetic anomalies, because protons penetrate more deeply into the magnetic field than electrons due to their higher mass-to-charge ratio.The increased grain size of the three regional pyroclastic deposits in the MV area and the two localized pyroclastic deposits on the floor of Atlas suggests that glassy pyroclastic material erupted by fire-fountaining is more resistant against destruction by micrometeoroid impacts than the surrounding, effusively erupted crystalline basaltic material.The same may be true for the Marius Hills region, which shows a slight positive contrast in relative grain size against the surrounding mare.In this region, the presence of a large number of cinder cones suggests the occurrence of explosive volcanism (e.g., Besse et al. 2011).All these volcanic structures consist of high-Ti basalts (Sato et al. 2017).In contrast, the regional pyroclastic deposit northwest of Aristarchus at least in part exhibits a smaller grain size than the surrounding mare (see also Shkuratov & Opanasenko 1992).Here, either the glassy particles ejected by fire-fountaining were smaller than typical mare regolith grains or the ejected particles were originally large but were more susceptible to destruction by micrometeoroids than the mare material.The latter explanation would require differences in material strength between the MV and AT pyroclastic deposits, on the one hand, and the Aristarchus pyroclastic deposit, on the other hand, which might be related to the peculiarly low, near-zero Ti content of the Aristarchus deposit in contrast to the high Ti content of the MV and AT deposits (Sato et al. 2017).
The within-band Umov exponent δ k is governed by the spectral reflectance R(λ k ) only, due to the constant wavelength.All values of δ k measured in this study are negative, i.e., in a specific spectral band, the DoLP always decreases with increasing spectral reflectance R(λ k ).Deviations from the within-band Umov law are commonly interpreted as variations in grain size (Shkuratov & Opanasenko 1992).The across-band Umov exponent ò differs from δ k and exceeds zero for bright surface structures like Aristarchus, i.e., in such cases, the DoLP increases with increasing reflectance.The reason for the observed strong correlation of ò with R is unknown so far, as is the mechanism behind the terrain-specific deviations  res from the fitted ò versus R relation.Modeling based on, e.g., light-scattering theory and/or radiative transfer is needed for a physical understanding of this empirically found polarization behavior.
The clusters determined by SOM analysis of the scores of the principal components derived from the P spectra represent a variety of lunar terrain types, as described in Section 3. In particular, they clearly extract the peculiar structure of the Reiner Gamma swirl and separate between pyroclastic deposits and maria in the MV and AT areas.The inferred cluster prototypes depict the corresponding wavelength dependencies of the DoLP.The DoLP spectra can partially be explained by the variable regolith grain size and the general increase of the spectral reflectance with increasing wavelength.However, a residual dependence on wavelength remains, even for constant spectral reflectance and uniform grain size.The reason for this behavior is unknown so far and needs to be examined by physical modeling.For Reiner Gamma, a polarization dichotomy between the main oval and the tail is indicated by the fact that our clustering analysis does not assign these parts of the swirl to the same cluster.Rather, the main oval belongs to the same cluster as the inner continuous ejecta blanket of Kepler, whereas the tail is assigned to the same cluster as Kepler's outer crater rays.The corresponding difference in DoLP behavior between the main oval and the tail may be explained by the main oval being bright due to low surface maturity preserved by magnetic shielding (Glotch et al. 2015), whereas the tail was found to be more mature, presumably due to the weaker magnetic field than at the main oval, where its brightness is at least partially due to soil compaction (Hess et al. 2020).

Summary and Conclusion
In this study, we have described our multiband UBVRI imaging polarization measurements of selected lunar regions in WOP, MV, and around the crater Atlas.The terrains found in these regions include compositionally different maria, highlands, fresh craters, the Reiner Gamma swirl, and pyroclastic deposits.In particular, we conducted a region-specific mapping of the relative grain size, an analysis of the within-band and across-band Umov exponent, a study of the wavelength dependence of the DoLP after normalization to the albedo and grain size, and a PCA of the DoLP with a subsequent clustering stage, as well as established a library of DoLP spectra representative of different lunar terrain types.The main results of our study can be summarized as follows: 1.In the regional grain-size maps, negative anomalies are found for crater ray material and the Reiner Gamma swirl, whereas positive anomalies correspond to small fresh mare craters and pyroclastic deposits.The latter finding suggests that regolith grains enriched in glassy material (impact-induced or volcanic) are less susceptible to destruction by micrometeoroid bombardment than "normal," crystalline regolith grains.2. The across-band Umov exponent commonly has negative values and is positively correlated with the albedo.Deviations from that trend show localized anomalies that are either positive (crater Aristarchus and Reiner Gamma swirl) or negative (small fresh mare craters, crater ray material, and pyroclastic deposits).3. The DoLP P normalized with respect to the albedo and grain size increases with wavelength in the UBVRI spectral domain.The slope ˜( ) l l dP d is largely independent of the albedo.4. The PCAs of the DoLP and the subsequent clustering stage divide the surface into geologically meaningful units, including fresh crater material, continuous ejecta and ray material of the craters Aristarchus and Kepler, mare units of different composition, highland material, the Reiner Gamma main oval versus tail as at least two distinct clusters, and pyroclastic deposits.5.For all five examined terrain types in the AT area, the variations of the DoLP with phase angle decrease with increasing wavelength.Between phase angles of 52°and 74°, the slope of the P spectrum changes its sign.Furthermore, the phase angle at which the DoLP obtains its maximum value decreases with increasing wavelength.
For now, we have suggested qualitative explanations for some (but not all) spectral effects observed in our measured DoLP data.Future work will have to focus on the simulation-based study of the scattering of light by regolith grains of variable size, shape, and composition, along with observations at multiple phase angles of extended regions, in order to gain a more physically based, quantitative understanding of the terrain-specific, phase-angle-dependent spectropolarimetric behavior of the lunar regolith.

Appendix B Principal Component Maps
Georeferenced maps of the scores of the first three principal components of the WOP, MV, and AT areas are shown in Figure 11 for the P-derived PCA.These maps correspond to the individual color channels of the RGB composites shown in Figure 6.

Figure 1 .
Figure 1.Maps of the pixel intensity F (in DN), DoLP P, and AoLP W (in degrees) of the WOP, MV, and AT areas in the I band.

Figure 2
displays the maps of the d rel parameter obtained in the I band in logarithmic scale for the different terrains shown in Figure1.

Figure 2 .
Figure 2. Maps of the relative grain size d rel derived from the I-band data in the WOP, MV, and AT areas (base-10 logarithmic scaling).

Figure 3 .
Figure 3. Maps of the residual  res of the across-band Umov exponent ò vs. R-band reflectance relation for the WOP, MV, and AT areas.

Figure 5 .
Figure 5. Wavelength dependence of the DoLP P normalized to the albedo and average grain size d mean in the WOP, MV, and AT areas.

Figure 6 .
Figure6.RGB composite maps of the P-derived scores on the first three principal components for the WOP, MV, and AT areas.The first, second, and third principal components were assigned to the red, green, and blue channels of the color composite, respectively.

Figure 7 .
Figure 7. Cluster index maps and prototype P spectra of the study areas WOP, MV, and AT.The colors of the P spectra are the same as those of the corresponding regions in the cluster index map, respectively.

Figure 10 .
Figure 10.Dependence of P on phase angle for the five terrains examined in the AT area.The magenta, blue, green, red, and black curves correspond to the U, B, V, R, and I bands, respectively.

Table 1
Spectropolarimetric Observations Analyzed in This Study The acquisition times refer to the I-band data.The georeferenced data and derived products are available at doi:10.5281/zenodo.10692671.Center Wavelengths λ k (in Nanometers) of the Broadband Filters Used in This Study