Spectrophotometric Analysis of the Ryugu Rock Seen by MASCOT: Searching for a Carbonaceous Chondrite Analog

The analysis in this paper concerns night-time images that show the surface of Ryugu in reflected LED light. mascot spent two nights on Ryugu. Images from the first night show only the night sky, but images from the second night show a rock in clear detail. Table 1 lists all image sets acquired by MASCam during these two nights. Five image sets (#11–15) were acquired in the second night. The first set (11) was taken at dusk and shows rocks in the background still illuminated by the Sun. The last set (15) was taken at early dawn and some images show evidence of daylight in the top right corner, a potential source of stray light. The three remaining sets (12, 13, and 14) are suitable for spectrophotometric analysis. These sets consist of six images: one bias image, one dark image, and one image for each LED color (Table 2). A bias image has the smallest available exposure time of 0.2138 ms. Set 14 is of especially good quality because it was acquired at subzero temperatures, for which detector dark current was negligible (Jaumann et al. 2017).

The images of the empty sky from the first night were expected to be mostly devoid of signal, but instead show LED stray light that may be internal to the optics. It was not noticed before launch but was subsequently identified in images acquired on ground. We used the images from the first night to construct stray light patterns for the purpose of calibration. First, we calibrated the images from sets 7 and 8 by subtracting bias and dark current and applying the nonlinearity correction (Jaumann et al. 2017). The bulk dark current was very low, but the presence of many relatively hot pixels made a correction necessary. We then constructed a stray light pattern for each LED color as the average of the two calibrated images (one from either set). The final stray light patterns shown in Figure 1 have been convolved by a Gaussian kernel 9 × 9 pixels in size with a FWHM of 3 pixels to reduce noise. The patterns show that stray light in the image center is strongest for the blue LED and that the infrared LEDs cause strong stray light in the upper right corner.

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The LED images of sets 13, 14, and 15 (all numbered #751–754) were calibrated as described in Jaumann et al. (2017) with a few modifications. The first step is to produce a clean image by subtracting the bias image (#755) and applying the nonlinearity correction (Jaumann et al. 2017). While a dark image was acquired (#750), we omitted the step of correcting for dark current because the low temperature made dark current negligible. We verified that the dark current image contained only noise; subtracting it in an effort to correct for dark current merely increased the noise in the calibrated image. The resulting clean images ${{\boldsymbol{C}}}_{i}$ (images are denoted in bold) were calibrated to radiance in W m⁻² sr⁻¹ as

$$\begin{eqnarray}&&{{\boldsymbol{L}}}_{i}=\displaystyle \frac{{{\boldsymbol{C}}}_{i}/{t}_{\exp }-{{\boldsymbol{S}}}_{i}}{{R}_{i}{{\boldsymbol{V}}}_{i}},\end{eqnarray} \tag{ 1 }$$

with the exposure time ${t}_{\exp }$ in ms, responsivity factors R_i in (m² sr mJ⁻¹), and color ratio frames ${{\boldsymbol{V}}}_{i}$ (see below). Index i indicates the LED color (Table 3). In this paper, we refer to $({{\boldsymbol{L}}}_{1},{{\boldsymbol{L}}}_{2},{{\boldsymbol{L}}}_{3},{{\boldsymbol{L}}}_{4})$ as $({\boldsymbol{B}},{\boldsymbol{G}},{\boldsymbol{R}},{\boldsymbol{I}})$, corresponding to the blue, green, red, and infrared radiance images. Ideally, correcting for differences in irradiance between the LEDs of different colors is done when converting radiance to reflectance. But as the LED irradiance critically depends on topography (which is not accurately known over the entire field of view), dividing by the ratio frames in Equation (1) ensures that color composites constructed from radiance images are correctly balanced. The stray light images ${{\boldsymbol{S}}}_{i}$, described in the previous section have the unit DN ms⁻¹. Figure 2 shows that the stray light correction is substantial for the Ryugu images. Because we also identified stray light in older images that were used to derive the responsivity factors R_i in Jaumann et al. (2017), we re-derived the factors from those images with stray light subtracted (Table 3). The correction is minor, only 2% at most. The LEDs were regularly monitored during the cruise phase and were found to be stable. The revised factors are listed in Table 3. The exposure time of the LED images in sets 13, 14, and 15 was rather long, at almost 3 seconds, as the auto-exposure algorithm (Jaumann et al. 2017) successfully compensated for the relatively large distance to the dark rock. Only a handful of pixels, inside the brightest inclusions, were overexposed. Overexposure is defined as an image pixel having a signal larger than 12.5 kDN after bias substraction. Because the nonlinearity correction fails for such high values (Jaumann et al. 2017), overexposed pixels were excluded from the analysis. For set 14, the number of overexposed pixels is 2 in the blue image, 5 in the green image, and 37 in the red image. No pixels were overexposed in the infrared image because of the comparatively low LED flux at this wavelength.

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Table 3. Revised Responsivity Factors R_i for the LED Colors

i	LED	λeff	R
		(nm)	(m2 sr mJ−1)
1	blue	${471}_{-16}^{+6}$	109.7 ± 1.2
2	green	${532}_{-24}^{+12}$	130.7 ± 1.3
3	red	${630}_{-7}^{+10}$	127.3 ± 1.4
4	infrared	${809}_{-17}^{+18}$	95.4 ± 1.3

Note. The range for the effective wavelength ${\lambda }_{\mathrm{eff}}$ is the FWHM.

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The Ryugu rock color composite shown in Jaumann et al. (2019) features an obvious calibration artifact in the form of a green bar at the bottom. The images in this composite had been calibrated using the color ratio frames ${{\boldsymbol{V}}}_{i}$ as constructed from images of a barium sulfate plate in the nominal landing configuration, i.e., parallel to the bottom side of both mascot and the camera. In reality, mascot did not come to rest on a flat surface. Fortunately, we had also acquired images of the plate tilted at various angles prior to launch in anticipation of such an occurrence. Figure 3 shows the calibrated green image ${{\boldsymbol{G}}}_{{\rm{p}}}$ of the plate tilted forward at an angle of 30° with respect to the plane parallel to the bottom of the camera (plate images are indicated with the subscript "p"). This configuration better matches both the distance and orientation of the Ryugu rock with respect to MASCam. Calibrating the plate images included a correction for stray light. The figure also shows a map of the distance to the plate surface, which can be compared to the map of the distance to the Ryugu surface which we will introduce in the following section. We were able to accurately derive the distance to the plate from images of a chessboard pattern that was positioned at the same distance and tilt angle as the plate. From the tilted plate images we constructed the new ratio frames ${{\boldsymbol{V}}}_{i}$ in Figure 3, which are defined with respect to the green image as ${{\boldsymbol{V}}}_{1}={{\boldsymbol{B}}}_{{\rm{p}}}/{{\boldsymbol{G}}}_{{\rm{p}}}$, ${{\boldsymbol{V}}}_{2}=1$, ${{\boldsymbol{V}}}_{3}={{\boldsymbol{R}}}_{{\rm{p}}}/{{\boldsymbol{G}}}_{{\rm{p}}}$, and ${{\boldsymbol{V}}}_{4}={{\boldsymbol{I}}}_{{\rm{p}}}/{{\boldsymbol{G}}}_{{\rm{p}}}$. The ratio frames are normalized at a location near the image center for which the responsivities in Table 3 were derived. Division as in Equation (1) ensures that the barium sulfate plate would appear white in a color composite of radiance images, at least at the location of normalization. In addition, the irradiance   J in (W m⁻²) of the green LEDs at unit distance could be determined from ${{\boldsymbol{G}}}_{{\rm{p}}}$, where we assumed that the plate has Lambertian reflective properties. For this, we calculated the distance from the center of the LED array to the plate surface and the incidence angle of the light on the plate coming from the same direction, where we used the fact that the LED array is located 1.8 cm below the aperture. The irradiance image   J in Figure 3 appears homogeneous over most of the frame, as expected. The irradiance apparently increases toward the top corners, which is likely an artifact resulting from the LED array being extended rather than a point source and/or non-Lambertian reflective properties of the plate. We judge the irradiance image to be reliable in the bottom half of the frame. Knowing the irradiance allows us to estimate Ryugu's absolute reflectance (radiance factor, I/F) in image pixel $(x,y)$ as

$$\begin{eqnarray}&&{\left({r}_{{\rm{F}}}\right)}_{x,y}^{i}=\pi \displaystyle \frac{{d}_{x,y}^{2}{L}_{x,y}^{i}}{{d}_{\mathrm{ref}}^{2}{J}_{x,y}},\end{eqnarray} \tag{ 2 }$$

with d the distance from the center of the LED array to the Ryugu surface in centimeters and reference distance ${d}_{\mathrm{ref}}=1\,\mathrm{cm}$.

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The impact of the color correction (division by the ratio frames ${{\boldsymbol{V}}}_{i}$ in Equation (1)) on the calibration to radiance is shown in Figure 4. Here, we compare color composites of uncorrected radiance images, images corrected with the earlier set of ratio frames, and images corrected with the revised frames in Figure 3. The revised frames successfully prevent the green bar at the bottom from appearing on the surface close to the camera. The origin of the green bar lies in the obscuration of the bottom row of LEDs on the array by the physical structure surrounding the array. The bottom row has only red and blue LEDs (Figure 5), and therefore the bar at the bottom of the image for terrain that is close to the camera is green (and IR). Where the terrain is more distant (the apparent hollow at the bottom right), the green bar is still faintly visible. This illustrates the difficulty in calibrating images of a surface that is illuminated by this LED device, which has an illumination pattern that is different for each color and, moreover, depends on the distance to the surface. Other artifacts apparently remain in the images after calibration. One way to identify calibration artifacts is to construct ratios of the LED color images. Figure 6 shows three such ratios: ${\boldsymbol{G}}/{\boldsymbol{B}}$, ${\boldsymbol{R}}/{\boldsymbol{G}}$, and ${\boldsymbol{I}}/{\boldsymbol{R}}$. Both the ${\boldsymbol{G}}/{\boldsymbol{B}}$ and ${\boldsymbol{R}}/{\boldsymbol{G}}$ ratios shows traces of the green bar artifact at the bottom of the frame, but only for the more distant terrain. The ${\boldsymbol{G}}/{\boldsymbol{B}}$ ratio shows a darkening in the center of the frame that may indicate a slight overcorrection of stray light. The ${\boldsymbol{R}}/{\boldsymbol{G}}$ ratio shows bright vertical stripes at the left of the frame that represent an excess of red signal. This pattern, which had not been seen before, is probably stray light. The stripes are also faintly visible in the ${\boldsymbol{G}}/{\boldsymbol{B}}$ ratio. The ${\boldsymbol{I}}/{\boldsymbol{R}}$ ratio frame is relatively dark in the center. We suspect that this is not related to the reflective properties of the rock but to the illumination pattern of the infrared LEDs being different from that expected (i.e., an incorrect ${{\boldsymbol{V}}}_{4}$). Contours of rock topography are clearly visible in all ratio frames, either in black or white. This artifact also results from the arrangement of colors over the LED array (Figure 5). In this case, the terrain was illuminated by the top row of the array (green and IR), while from the perspective of the second row from the top (red and blue) it was in the shadow. This is consistent with the contours showing as white in the ${\boldsymbol{G}}/{\boldsymbol{B}}$ and ${\boldsymbol{I}}/{\boldsymbol{R}}$ ratio frames.

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Inclusions were mapped in the radiance $({\boldsymbol{R}},{\boldsymbol{G}},{\boldsymbol{B}})$ color composite of set 14 at its full brightness range using the ArcGIS ⁵ mapping tools. One of us (H.S.) did the mapping to achieve consistent results. Inclusions were identified as areas of different brightness or color compared to their immediate surroundings ("matrix"). They were outlined by drawing a polygon between the pixels of highest color and brightness contrast at a constant zoom factor. While its vertices were mapped in a continuous coordinate system, the polygon was subsequently converted into a collection of (discrete) image pixels. We did not map inclusions inside inclusions. Identification of inclusions was limited to the well-illuminated terrain in the foreground of the scene, an area we also outlined with a polygon. Inside this area, we defined matrix pixels as all those not associated with inclusions, applying a threshold to the radiance to exclude shadows. The area of each inclusion was calculated from the number of image pixels it covered, the instantaneous field of view (IFOV) of those pixels, the distance from the camera aperture to the surface, and the angle between the surface normal and the line connecting the aperture and surface (emission angle). As such, the area is corrected to first order for projection effects, as inclusions seen from the side on a sloping surface are larger than they appear. The distance to the rock surface and local emission angle were estimated from the DTM in Scholten et al. (2019). The color-coded distance in Figure 7 shows that most of the rock in the foreground is roughly at 25 cm. The distance to the rock in the background is not known accurately because of a lack of stereo coverage, but is certainly larger than 40 cm. Mapping was restricted to inclusions on terrain closer than this distance. The surface recedes in an apparent hollow at the bottom right of the scene and is therefore out of focus. The uncertainty of the DTM is about 0.5 cm in the foreground at a distance of 20 cm and about 1.5 cm at a distance of 30–40 cm (Scholten et al. 2019). Figure 7 also shows the phase angle of illumination by the LEDs. The phase angle is generally low, around 5° for the well-illuminated surface in the foreground. The low phase angle ensures that we can actually distinguish the inclusions, unobscured by the strong shadows that characterize the day images of this rough surface. The phase angle decreases with distance from the camera, so it is smaller for the poorly illuminated terrain in the back and the hollow at bottom right. The phase angle was calculated on the assumption of illumination by a point source at the center of the LED array. In reality, any point of the surface received light with a variety of phase angles due to the extended size of the array ($4.2\,\times 0.9$ cm²). For example, the edges of the array illuminated the terrain in the foreground with a phase angle that was different by about 5° from the point-source estimate.

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Mapping ambiguities arose because of unclear boundaries and spectral variations within apparent inclusions. To illustrate these challenges, we enlarge the area in the foreground at the bottom of the frame in Figure 8 in which several features are highlighted. Feature 1 appears to be an inclusion with clear boundaries. However, it harbors pixels that appear either red or blue in the color composite, and its average color may therefore appear neutral. Feature 2 is a reddish inclusion whose boundaries are clearly defined in the color composite in (a) but it appears to be larger in the stretched composite in (b). Feature 3 was mapped as a collection of three inclusions, one of which is clearly redder than the others. But one could arguably draw an outline around the entire group to count it as a single inclusion. Feature 4 is marked as an inclusion on the basis of its slightly brighter appearance, but its boundaries are so indistinct that this feature might as well be an expression of local topography. Finally, the matrix in (b) (i.e., pixels that are not marked red) contains abundant small areas that are brighter than their surroundings. In fact, they are so abundant that one suspects that what we have defined as the "matrix" is not a homogenous substance. These examples make it clear that the "inclusions" that we refer to in this report are not uniquely defined.

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We derived the absolute reflectance (I/F) of the Ryugu rock by photometrically correcting the   G image through Equation (2). Figure 9 shows both the uncorrected (radiance) image and the corrected (reflectance) image. Because the DTM is increasingly uncertain beyond 35 cm, we restrict the correction to that distance. The brightness in the reflectance image is much more evenly distributed over the frame than in the radiance image, indicating a successful photometric correction (assuming that the rock reflectance is uniform). For example, the hollow in the foreground is no longer recognizable as such. The reflectance increases toward the terrain in the background, consistent with the lower average phase angle there (see Figure 7). We calculate the average reflectance for the terrain inside the circle in Figure 9 as ${\bar{r}}_{{\rm{F}}}=0.034\pm 0.006$ at an average phase angle of ${4.5}^{\circ }\pm {0.1}^{\circ }$. The circle was chosen such that it enclosed terrain that is comparatively free of image artifacts (see Figure 6) and is large yet small enough such that the phase angle does not vary by too much. While the standard deviation of the reflectance of the pixels inside the circle is 0.006, the true uncertainty of this estimate of the rock reflectance is smaller. It is derived from the uncertainty of the green responsivity factor (1%, Table 3), variations in the irradiance image   J (5%, Figure 3), and uncertainty in the distance to the rock in the DTM. The distance to the area in the circle is about 25 cm (Figure 7), for which the uncertainty is about 1 cm, or 4% (Scholten et al. 2019). Adding up these percentages, we arrive at a total uncertainty of 10% and estimate the reflectance of the Ryugu rock in the green channel as r_F = 0.034 ± 0.003 at a 4.5° phase angle. We can predict the reflectance of the average Ryugu surface for a phase angle of 4.5° using the photometric model parameters from Sugita et al. (2019), which agree with those of Ishiguro et al. (2014) and are valid for the Optical Navigation Camera (ONC) v-band at 549 ± 14 nm (Kameda et al. 2017). The predicted I/F values for the $(\iota ,\epsilon )=({4.5}^{\circ },{0}^{\circ })$, (2.2°, 2.3°), and (0°, 4.5°) geometries are all equal to 0.036°, which agrees very well with our reconstructed I/F of 0.034 ± 0.003 at ${532}_{-24}^{+12}$ nm. Thus, from a photometric point of view, the rock in front of mascot is typical for Ryugu.

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The reconstructed reflectance spectrum of the Ryugu rock is shown in Figure 10. The mean reflectance is that of the pixels inside the circle in Figure 9, with the blue vertical error bars indicating the standard deviation. The sizes of the error bars reflect the brightness variations over the terrain, which includes shadows, but not the uncertainty of the spectral shape. The latter is the uncertainty of the spectral calibration, which derives from the uncertainty of the responsivity factors (only 1%) and that of the LED irradiance in the different color channels. The (spectral) uncertainty in the LED irradiance dominates the calibration uncertainty and concerns the color correction, i.e., division by the color ratio frames in Figure 3. As these frames are scene-dependent, their uncertainty is difficult to quantify. For example, frame ${{\boldsymbol{V}}}_{3}$ was constructed from images of a flat plate and has large-scale variations of around 15%. The uncertainty of the color correction is probably smaller than that. We adopted an uncertainty of 10% for the black vertical error bars on the rock reflectance spectrum in Figure 10. Within this uncertainty, our spectrum is consistent with average reflectance spectra of Ryugu from ONC images (Sugita et al. 2019). The slight excess of the MASCam reflectance in the red channel is not necessarily real as it is within the calibration uncertainty.

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3.2.1. Spatial Distribution

We mapped a total of 1443 inclusions inside an area that was well illuminated due to its proximity to the camera. The totality of this area minus inclusions is defined as the matrix, where we applied a lower threshold of 0.008 W m⁻² sr⁻¹ to the radiance (for all LED colors) to exclude areas in shadow. A map of the inclusions and matrix pixels is shown in Figure 11. The inclusions appear to be more or less uniformly distributed over the mapping area. We tried to estimate the total area covered by the inclusions and the matrix to estimate their respective volume abundances, which are thought to be diagnostic quantities. However, we found the total matrix area to be dominated by pixels with an emission angle close to 90°. The area derived for such pixels is probably very inaccurate. The resolution of the Scholten et al. (2019) DTM appears to be much lower than the actual scale of the surface roughness so the emission angle calculated for image pixels is generally only a first-order approximation. We therefore selected only matrix pixels with an emission angle smaller than 80° (excludes 1.4%). The total area of the matrix pixels is then 172 cm². For the same reason, we selected only inclusions with an average emission angle smaller than 80° (excludes 6). The total area of the mapped inclusions is then 13.6 cm². The areal abundance of inclusions is 100% × 13.6/(13.6 + 172) = 7.3%, which corresponds to a matrix areal abundance of 92.4%. As a test, we also estimated the pixel areas without the correction for emission angle and found inclusion and matrix abundances of 7.2% and 92.8%, respectively. These values are virtually identical to the earlier ones, implying that the retrieved abundances are robust. However, the terrain we mapped as the "matrix" may include unresolved inclusions, so our inclusion and matrix abundances are lower and upper limits, respectively. Finally, we performed a simple numerical simulation of small, randomly distributed spheres throughout a volume and verified that the volume abundance (vol%) can be estimated as the observed areal abundance on a planar surface. Thus, to first order, the volume abundance is equal to the retrieved areal abundance.

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3.2.2. Spectral Properties

The color of the inclusions can be determined as the ratio of the color-corrected radiance images and thus does not depend on an accurate determination of the LED irradiance. We investigate the color of the inclusions as defined in Figure 11. We determined the average spectrum of each inclusion and subjected the body of spectra (n = 1441) to a principal component analysis (PCA) using the prcomp package in R. ⁶ The first three principal components (or eigenvectors) are shown in Figure 12, where we omitted the last component (PC4) as the highest components generally contain only noise. PC1 represents the average shape of the inclusion spectra, which is similar to the average spectrum of the rock itself (Figure 10). PC2 represents the dominant color variation, which in this case is a change in the spectral slope from blue to infrared, while PC3 expresses more subtle color variations that may exist in addition to the slope variation. The contribution of the three components to the variance is PC1: 97.9%, PC2: 1.4%, and PC3: 0.5%. The large contribution of PC1 is the consequence of the variable degree of illumination over the scene, where inclusions in the background are perceived as darker than those in the foreground. The dominant spectral variation (PC2) is therefore a change in spectral slope over the entire wavelength range of the LEDs. PC3 uncovers a possible variation in the red channel with respect to the other three, which may be related to red being slightly depressed in PC2. To better understand this variation, we evaluate the spectra of several individual inclusions. Many inclusions have a single pixel at their center that is much brighter than the others, indicating that they are unresolved, in which case the image merely represents the point-spread function (PSF) of the imaging system. We therefore focus only on that brightest pixel. To circumvent the uncertainties associated with unequal illumination patterns between the LEDs, we calculate the ratio of the radiance in the brightest pixel and the average radiance of the matrix, chosen as a (circular) area of uniform appearance in close proximity. Figure 13 shows the ratio spectra of 14 prominent inclusions, labeled a-n. Some of these are outside the terrain mapped earlier, as, in this case, the selection criterion is radiometric accuracy (lack of image artifacts) rather than DTM accuracy. We calculated ratio spectra for each of the image sets 13, 14, and 15 (Table 1) and the ratio spectra in the figure are averages over these three sets. A few inclusions (a, b, d) are very bright and very red. Inclusion a is the most extreme case, being more than 8 times brighter than the matrix in the infrared. Another inclusion (h) is also much brighter (by about a factor of 4) but blue instead. Other inclusions are typically twice as bright as the matrix, and either red, blue, or spectrally neutral. Variation in the red channel indeed exists. The radiance in the red channel appears to be reduced for inclusions c, e, and j, although the error bars are relatively large and may be slightly enhanced for inclusions i and m.

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We return to the inclusions as they are mapped in Figure 11. We express the dominant color variation with a single quantity: the relative spectral slope. In light of the lack of apparent artifacts in the mapping area in the ${\boldsymbol{G}}/{\boldsymbol{B}}$ ratio frame in Figure 6 and the "flat" appearance of the rock, we define the relative slope as $({\boldsymbol{G}}-{\boldsymbol{B}})/{\boldsymbol{B}}$. Figure 14 shows the relative slope as a function of inclusion area. The matrix pixels are neutral in color with a relative spectral slope of −0.003 ± 0.028. We now define "red" inclusions as those with a spectral slope larger than the average spectral slope of the matrix pixels plus one standard deviation and "blue" inclusions as those with a slope smaller than the average matrix slope minus one standard deviation. The figure shows that the vast majority of inclusions have spectral slopes that are in the range of those of the matrix pixels. This suggests that the matrix harbors unresolved or otherwise unrecognized inclusions. Also, some inclusions harbor both red and blue pixels (examples in Figure 8), which tends to reduce their average spectral slope to that of the matrix. The number of red inclusions is 1.2 times larger than the number of blue inclusions. This ratio is within the expected uncertainty assuming Poisson statistics, so the number of red and blue inclusions is not significantly different. However, the ratio of red over blue inclusions critically depends on the definition of spectral slope; had we defined the slope as $({\boldsymbol{R}}-{\boldsymbol{B}})/{\boldsymbol{B}}$ or $({\boldsymbol{I}}-{\boldsymbol{B}})/{\boldsymbol{B}}$, the ratio would be 1.4 and 4.5, respectively. Figure 14 also reveals that inclusion color is not a matter of size, although the largest inclusions in our sample (>3 mm²) are strictly neutral in color, perhaps as a result of averaging any spectral diversity inside. Finally, we counted nine very red inclusions (slope > 0.1), but none that are similarly blue (slope <− 0.1).

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3.2.3. Size Distribution

The smallest and largest inclusions in our sample have an area of 0.031 and 23 mm², respectively. The size distribution of the mapped inclusions is affected by three biases. Bias (1) is that the smallest inclusions can only be distinguished on terrain closest to the camera. Figure 15(a) shows the inclusion area as a function of distance to the camera aperture. It shows that inclusions were mapped on terrain at a distance between 19 and 34 cm. The dashed curve represents the area of a single pixel with an IFOV of 0.9 mrad seen face-on (zero emission angle) as a function of distance. The smallest inclusions cluster around this curve, indicating that their size is a single pixel. We cannot be sure that such inclusions are fully resolved, so their area is an upper limit. Another curve (dashed–dotted) represents inclusions that cover an area of two image pixels. The mostly empty space between the curves is a sampling gap. Similar gaps exist for higher pixel numbers. Bias (2) is introduced by the sloping terrain at larger distance; the higher emission angles there make it impossible to see many small inclusions. Bias (3) follows from the fact that most of the area we mapped is at a larger distance so there we find a relatively large number of large inclusions. The three kinds of bias make it difficult to evaluate whether inclusions are uniformly distributed over the rock, i.e., independent of their size. At least when we distinguish red and blue inclusions, again defining the relative slope as $({\boldsymbol{G}}-{\boldsymbol{B}})/{\boldsymbol{B}}$, we find little variation; both red and blue inclusions are found at any distance. However, if we define the relative spectral slope as $({\boldsymbol{R}}-{\boldsymbol{B}})/{\boldsymbol{B}}$, there appear to be more blue than red inclusions at a larger distance. The distribution is uniform again for $({\boldsymbol{I}}-{\boldsymbol{B}})/{\boldsymbol{B}}$, so the odd result for the red channel may reflect heterogeneity in the rock but could also be an artifact.

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The three biases make it necessary to restrict the size analysis to inclusions mapped on terrain relatively close to the camera. It appears that single-pixel-sized inclusions were only recognized up to a distance of 25.5 cm (dotted vertical line in Figure 15(a)), so we adopt this as a distance limit. Inclusion size is often reported in the literature as the diameter of a sphere. We therefore convert our inclusion area a into diameter $d=2\sqrt{a/\pi }$ of an equivalent disk. Figure 15(b) shows a histogram of the diameter of inclusions closer than 25.5 cm. The minimum and maximum diameter in the sample are 0.20 and 5.4 mm, respectively. The vast majority of inclusions is smaller than 1 mm in diameter and very few are larger than 2 mm. The histogram peaks around 0.5 mm diameter. However, the number in the smallest size bin (0.1–0.3 mm) is uncertain for two reasons. First, the area of a single pixel with an IFOV of 0.9 mrad seen face-on at a 20 cm distance corresponds to d = 0.20 mm. Thus, some inclusions assigned to this bin could actually be smaller than 0.1 mm, but so bright that they appeared bigger due to the detector's PSF. On the other hand, the rock displays many small, faint brightness features (see examples in Figure 8), which we assumed to be photometric variations resulting from topography, but might also be inclusions of the correct size that were too faint to be recognized. As there are competing biases for the number in the smallest size bin, it is probably not too far off and the peak in the size distribution around 0.5 mm is likely real.

3.2.4. Brightness Distribution

The successful photometric correction described in Section 3.1 allows us to reconstruct the absolute reflectance of the inclusions. Inclusions generally appear bright in the images and we did not unequivocally identify any that are darker. Figure 16 compares the reflectance of the inclusions with the average matrix reflectance, again distinguishing red and blue inclusions as defined in Figure 14. The figure confirms our impression that inclusions are generally brighter, by up to a factor of two, than the matrix at phase angles of around 5°. The brightness of inclusions seems to depend on neither color nor size.

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