# Partial Least Squares Modeling of Lunar Surface FeO Content with Clementine Ultraviolet-Visible Images

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## Abstract

To accurately predict the iron abundance of the Moon has long been the goal for lunar remote sensing studies. In this paper, we present a new iron model based on partial least squares regression (PLS) method and apply this model to map the global lunar iron distribution using Clementine ultraviolet-visible (UVVIS) dataset. Our iron model has taken into account of more calibration sites other than Apollo and Luna sample-return sites and stations (i.e., the six additional highland or immature sites) in combination with more spectral bands (5 bands and 2 band ratios), in order to derive reliable FeO content and improve the robustness of the PLS model. By comparing the PLS-derived iron map with Lucey’s band-ratio FeO map and Lawrence’s Lunar Prospector (LP) FeO map, the differences are mostly within 1 wt% in FeO content. Moreover, PLS-derived FeO is more consistent with LP’s result which was derived by direct measurement of Fe gamma-ray line (7.6 MeV) rather than the Lucey’s experiential algorithm applying only two bands (750, 950 nm) of Clementine UVVIS dataset. With a global mode of 5.1 wt%, PLS-derived iron map is also validated by FeO abundances of lunar feldspathic meteorites and in support of the lunar magma ocean hypothesis.

## Keywords

Lunar iron content Partial least squares regression (PLS) Spectroscopy Clementine UVVIS## 1.1 Introduction

As one of the major rock-forming elements, iron is closely related to lunar mafic mineral assemblages and rock types; thus the accurate estimation of iron abundance would provide important information of lunar geochemistry, petrogenesis, as well as the crustal evolution. Iron is often expressed as FeO in astrochemistry. The absorption properties in the ultraviolet-visible (UVVIS) and near-infrared (NIR) spectral regions of iron-bearing minerals (e.g., pyroxene, olivine, and ilmenite) are dominated by Fe^{2+} or Ti^{4+}/Ti^{3+} (Lucey et al. 1998). The absorption features from lunar sample or remotely sensed spectra would mix up influences from the exposures of lunar soils to the space environment, i.e., the Moon has been suffering from bombardments by micrometeorites, solar wind ions, cosmic rays, and solar flare particles (Fischer and Pieters 1994, 1996). Sustained bombardments will cause the lunar surface material change in petrography and chemistry. These changes include reduction of mean grain size, the production of nanophase iron (npFe^{0}) and complex glass-welded aggregates of lithic and mineral fragments (agglutinates), and so on (Fischer and Pieters 1994, 1996; Mckay et al. 1974). This so-called process “space weathering” will bring about the maturation of lunar regolith, i.e., the mature regolith usually has suffered from a longer time of space weathering compared to immature regolith. Space weathering will cause an overall reduction in the reflectance, and reduce the absorption band strengths, creating and steepening a red-sloped continuum (Fischer and Pieters 1994, 1996).

_{2}abundance has an effect on the relationship between Fe content and Fe parameter, and they optimized this method by adding TiO

_{2}-sensitive regression parameters into the regression of iron content. Wilcox et al. (2005) developed a new algorithm to determine the iron content in lunar mare regions based on the findings that the maturity trends in lunar mare area are more parallel than radial. They collected more than 9,000 craters from mare regions and make a 950/750 nm vs. 750 nm reflectance plot with these data and found the radial trends were disobeyed. While iron abundance was still orthogonal to maturity trends, the maturity trends were parallel to each other, suggesting new trends of iron distribution in lunar mare. Their new iron model has absolute uncertainty similar to Lucey 2000’s model (1.5 wt%), while it allows better compensation for the maturity-induced iron uncertainties (<0.5 wt%).

Except for NIR/VIS ratio methods mentioned above, many other approaches like utilizing infrared continuum slope of the spectrum in order to suppress the effect of topography (Le Mouelic et al. 2002) and iron absorption band depth (Fischer and Pieters 1994) have been proposed in the iron modeling. These methods are limited by the data calibration and quality of Clementine NIR dataset. Statistical relationships between spectral and chemical abundance of lunar soils have also been evaluated by Pieters et al. 2002 for their applications of remotely compositional analysis. She firstly applied principle component analysis (PCA) regression method with lunar mare soil spectra produced by Lunar Soil Characterization Consortium (LSCC) to define and evaluate the correlations between chemical abundance and spectral parameters (Pieters et al. 2002). Then she also derived three statistical relations between spectral and mineral parameters using LSCC data and applied them to Clementine UVVIS data (Pieters et al. 2006).

Although many iron models have been put forward as discussed above, a quantitatively accurate iron model is still in need, especially for the exploration of the potentials of multispectral imaging data like Clementine UVVIS and other lunar hyperspectral datasets (e.g., data from Moon Mineralogy Mapper (M^{3}), Interference Imaging Spectrometer (IIM), etc.). In this paper, we choose to build iron abundance models with partial least squares (PLS) regression method. PLS is known as the second generation of regression method, which performs well in multivariable regression especially when multiple correlations exist among variables. Li (2006) made a comparison between PLS and PCA in deriving chemical and mineral abundances using data from LSCC. He found PLS models use less components and perform better than PCA in the estimation of lunar chemical and mineral abundances. However, Li didn’t apply his result to lunar remotely sensed images. Our PLS-derived iron model is developed with intent to explore the potential of the UVVIS imaging dataset. During the modeling, we find it is easy to reach a good regression relationship (high correlation efficient (R^{2}) value) between spectra parameters and iron contents, while maturity suppressing is more difficult to attain. We have tested many different variables in PLS modeling to find the most applicable one for Clementine UVVIS images and compare our results with previous studies to evaluate the robustness of the PLS model.

## 1.2 Data

The lunar remote sensing images used in this study are from Clementine UVVIS Digital Image Model (DIM) published by US Geological Survey (USGS) Astrogeology Team at Flagstaff, Arizona (NASA PDS Geosciences Node). The DIM has five bands with a nominal ground resolution at 100 m/pixel, and the center wavelengths (spectra resolutions) of the five filters are A, 415 nm (40 nm); B, 750 nm (10 nm); C, 900 nm (20 nm); D, 950 nm (30 nm); and E, 1,000 nm (30 nm) (Eliason et al. 1999). This dataset is archived in the NASA Planetary Data System, and each image has undergone radiometric and geometric correction, spectral registration, and photometric normalization by Integrated Software for Imagers and Spectrometers (ISIS) processing system.

Iron abundance and reflectance values of supplementary data from Clementine UVVIS

Clementine spectra | ||||||
---|---|---|---|---|---|---|

Sites | 415 nm | 750 nm | 900 nm | 950 nm | 1,000 nm | FeO (wt%) |

Farside-1 | 0.1196 | 0.2022 | 0.2178 | 0.2250 | 0.2302 | 2.3 |

Farside-2 | 0.1195 | 0.2032 | 0.2183 | 0.2271 | 0.2336 | 1.7 |

Farside-3 | 0.1330 | 0.2271 | 0.2372 | 0.2447 | 0.2546 | 3.8 |

Fresh-1 | 0.0794 | 0.1265 | 0.1140 | 0.1108 | 0.1111 | 17.2 |

Fresh-2 | 0.0767 | 0.1229 | 0.11155 | 0.1088 | 0.1103 | 17.4 |

Fresh-3 | 0.0811 | 0.1353 | 0.1202 | 0.1174 | 0.1203 | 16.8 |

## 1.3 Partial Least Squares Regression Method and Data Processing

PLS is a new kind of multivariate statistics regression method, which was developed by Herman Wold in 1966 (Li 2006). Comparing to other regression methods (like PCA regression), PLS has many advantages, especially in resolving mutual influence problems among variables. PLS has already been utilized for analyzing material compositions from laboratory and remote sensing spectra datasets. Li (2006, 2008) resampled LSCC bidirectional reflectance data into the airborne visible/infrared imaging spectrometer (AVIRIS) spectral resolution and derived several composition derivation models such as iron and TiO_{2} with PLS regression method (Li 2006, 2008, 2011). Li’s model was based on laboratory data and was not applied to remotely sensed data, making it difficult to evaluate the ability of the model in maturity suppressing.

As an advanced statistical method, the principle of PLS analyzing can be expressed as: PLS = PCA + CCA + MLR (CCA, classical component analysis; MLR, multiple linear regression). The key to PLS modeling is to determine the number of latent variables (LVs), which are also called the components. Covariance between each corresponding component of independent variable and dependent variable should be kept maximum; this can be considered as a combination of LVs searching conditions of PCA and CCA.

*n*×

*m*matrix

*X*, and dependent variance is an

*n*×

*p*matrix

*Y*, we first standardize matrixes

*X*and

*Y*before modeling in order to reach a more stable result. Following PLS rules while regressing

*X*and

*Y*, finally, we can get the relations listed below (Eqs. 1.1 and 1.2). Both

*X*and

*Y*are decomposed into two parts: a matrix product term and a residual term. The matrix product term consists of a score matrix and a loading matrix, score matrixes are

*T*for

*X*and

*U*for

*Y*, and they are both

*n*×

*a*matrixes; loading matrixes are

*P*for

*X*and

*Q*for

*Y*, and they are both

*m*×

*a*matrixes.

*E*and

*F*are residual matrixes. The goal of regression is to find the correlative relation between

*X*and

*Y*(Eq. 1.3) while keeping residual matrixes

*E*and

*F*minimum:

*α*is absorbance. The derived absorbance

*α*is assumed to have a linear relationship with the abundance of composition (Li 2006; Yen et al. 1998; Whiting et al. 2004; Milliken and Mustard 2005):

## 1.4 PLS Modeling

### 1.4.1 Iron Modeling

*X*is a 53 × 5 matrix, and

*Y*is a 53 × 1 matrix. After transforming reflectance into absorbance, we standardized both

*X*and

*Y*in order to get a more stable model. While modeling, the most important thing is to derive reasonable iron content as well as suppress the space weathering effect at the same time. All of the five bands are included in the dependent variables to keep the maximum potential, and they are expressed by

*A*

_{1}−

*A*

_{5}, respectively. Band ratios are helpful especially when extracting chemical abundances, and they are also indications of maturity degree. Our model takes account of the typical NIR/VIS ratio (950 nm/750 nm), which is used in Lucey’s algorithm. Pieters et al. (2002) have tested the correlations between composition and spectral ratios, and experiments showed that the highest correlation for iron is 1,000/400 nm. Hence, we also bring it into our model, expressed by 1,000/415 nm. Finally, all the variables chosen to build model are listed in Eq. 1.5,

*c*

_{0}−

*c*

_{7}are regression coefficients, and

*A*

_{1}−

*A*

_{5}represent five absorption bands of Clementine data:

After inputting all the data into PLS toolbox, leave-one-out cross-validation is executed during modeling. The cross-validation means modeling with one variable left out until all the variables have been left out once; thus we would derive a regression model in each cross-validation and compute the root mean square error of cross-validation (RMSECV) for every leave-one-out model by Eq. 1.6; *k* is the number of variable that is left out. Usually, the one with minimum RMSECV will be chosen as the best LV number. After the number of components is determined, the total root mean square (RMSE) can be calculated by Eq. 1.7.

*R*

^{2}) of this model is 0.918, and RMSE is 1.44 wt%, indicating a good regression of iron abundance has been achieved:

### 1.4.2 Results and Analysis

PLS-derived iron map is shown in Fig. 1.4c in comparison with the result of Lucey’s work (Fig. 1.4b). Considering the maturity-suppressing ability can be indicated by the small fresh craters in the mare region, the difference between our model and Lucey’s is subtle, i.e., most of small fresh craters (bright spots in the 750 nm reflectance image (Fig. 1.4a)) are invisible in the PLS-derived FeO map in Fig. 1.4b, c. The distribution of FeO in the mare area is relatively homogenous, which indicates that the maturity-suppressing ability of our model is comparable to Lucey’s algorithm.

### 1.4.3 PLS Modeling of Highland Areas

As discussed above, although the two iron maps behave similarly in iron abundance and maturity suppressing, they still exhibit discrepancy in FeO modeling of highland regions. The iron abundance of our model for highland region is a little higher than Lucey’s, which is shown both in iron map (Fig. 1.4b, c) and the statistical results of iron map (Fig. 1.5). As is known to all, statistical methods strongly depend on the sampling data points, i.e., when the sampling data points lack of a specific range of FeO abundance, the result may tend to behave deviate from that range. During the modeling, although six supplementary data are added for lunar sampling stations, the highland data sources are only composed of Apollo 16 sampling stations and 3 added lunar farside sites. The limited proportion of highland spectra to the total modeling data may lead to the overestimation of iron abundance in highland areas during the PLS modeling. To testify this hypothesis, we derive another iron model using only Apollo 16 and Apollo 17 sampling sites, in order to increase the proportion of highland sampling sites. Data processing pipeline follows the first PLS model (Eq. 1.5).

## 1.5 Global Iron Mapping and Analysis

### 1.5.1 Global Iron Mapping

### 1.5.2 Comparison with Former Works

Comparison of FeO abundance between different algorithms

FeO model | Global mean (wt%) | Global mode (wt%) | Peak value in mare (wt%) |
---|---|---|---|

Lucey 2000 | 7.8 | 4.7 | 17.1 |

Lunar Prospector | 7.8 | 6.4 | 17 |

PLS model | 7.6 | 5.1 | 16.9 |

In order to show the global difference between PLS model and Lucey’s method, we apply Lucey’s algorithm to Clementine DIM and make a difference map (PLS FeO minus Lucey’s FeO), as shown in Fig. 1.10a. Most of the difference distributes within −0.9 to 1.0 wt% which is shown in green color in the difference map. PLS model gets an even higher iron abundance than Lucey’s result in lunar farside, which is consistent with the statistical result comparison (Fig. 1.9b, c). Another discovery from the difference map is that iron in fresh craters derived by PLS is often lower than Lucey’s result, represented by Tycho crater (located in south-southwest of the map, about 43.3°S, 11.2°W). This may be caused by different degrees of maturity suppressing between the two methods. Statistical result (Fig. 1.10b) shows a nearly Gaussian distribution of the difference, with an average value around −0.27 wt%, and root mean square error (RMS) is 1.13 wt%, indicating a relatively small difference of the global iron abundance between the two maps.

Considering the different spatial resolutions of the FeO maps from the spectral and gamma-ray datasets, in order to compare in detail with iron map derived by LP, we resample the Clementine iron map derived by PLS to the same spatial resolution as LP iron map (15 km/pixel) and make a global difference map (regions exceed 70°S–70°N are not included) (Fig. 1.11a). The difference distribution on the global map isn’t very homogenous, but when we take a look at the global iron distribution, we find the global difference is concentrated within −0.9 to 1.0 which is colored by green, and iron abundance detected by LP in mare areas and high-latitude regions is higher than PLS model. The difference in high latitudes tends to be greater; this effect may be the influence of topographic shading or illumination conditions.

Furthermore, our iron content of PLS model for the South Pole-Aitken (SPA) basin is higher than LP’s but lower than Lucey’s (Fig. 1.12c). As the largest impact crater on the Moon, the SPA impact event didn’t penetrate the materials from lunar mantle, which are expected to be more mafic and iron rich. The specific noritic mineralogy may account for this low FeO concentration (Lucey 2004). From the statistical result of the difference map, the global average of iron difference is within 1 wt% and the RMS is 2.3 wt%, suggesting a good consistency of PLS model and LP iron map.

In a word, we find the iron map from PLS model agrees with those from the Lucey’s and LP’s, though subtle difference appears for the global maps. This suggests our PLS model is a robust algorithm for the extraction of lunar iron content. Furthermore, the application of PLS method on the global lunar mosaic seems to be more consistent with LP results than those of Lucey 2000. Note that our PLS model explored the potential for all the five available Clementine UVVIS bands and some more spectral parameter than Lucey et al. (2000), who used only Clementine’s two bands (950 and 750 nm) to derive iron content. We believe our PLS model is a good test and validation for the lunar elemental mapping with available lunar spectral data and those from future lunar missions.

## 1.6 Indication of Lunar Magma Ocean Hypothesis

After lunar sample returned to earth, a lot of laboratory experimental analyses have been done to extract information of the lunar mineralogy and petrogenesis, which are very helpful in understanding lunar origin and evolution progress in a global or local scale. Samples from lunar highland regions contain higher plagioclase abundance and hence are rich in Al and poor in Fe compared to those from mare regions (Lucey et al. 1995). These rocks are interpreted as forming from a global circling magma ocean, and plagioclase floated in it (Wood et al. 1970; Warren and Haskin 1991). The magma ocean hypothesis was developed following the first sample return from the Moon. The crystallization of the magma ocean would result in FeO poor anorthosite rocks concentrating in the crust.

As was mentioned by Lucey et al., the key test of the magma ocean hypothesis is the abundance of anorthosite (Lucey et al. 1995). Usually, anorthosite assembles in lunar highlands, so iron abundances in these regions could represent the global anorthosite concentration. In remote detection, the global mode of FeO concentration represents iron abundance of lunar highland regions. The global mode of iron abundance derived by Lucey et al. in 1995 is about 3 wt% (in Fe, and 3.96 wt% in FeO), and he improved the algorithm in year 1998, and the new global mode of FeO is 4.8 wt% (Lucey et al. 1998). These results are consistent with our result (5.1 wt%). As far as lunar meteorites study are concerned, Korotev et al. studied eight best characterized feldspathic lunar meteorites and showed the average concentration of FeO is 4.4 ± 0.5 wt%, and the FeO range is 3–6 wt% (Korotev et al. 1996; 2003; Korotev 2005). The global mode of PLS-derived FeO map is 5.1 wt%, which also agrees well with meteorite studies and remote sensing results (Lucey et al. 1995, 1998, 2000; Lawrence et al. 2002) and thus could also support the magma ocean hypothesis of lunar crust.

## 1.7 Conclusions

We derived a new iron model with PLS method, which has been verified to be able to derive robust iron abundances for the Moon. We apply this model to Clementine DIM and obtain global distribution of iron. Our results show that peak distribution of iron abundance in highlands and mare regions are 5.1 wt% and 16.9 wt%, respectively. Comparing our iron map to Lucey’s algorithm as well as that detected by Lunar Prospector gamma-ray spectrometer, we find the three results agree well in mare regions, while PLS model and LP iron maps show higher iron content in highlands. Local comparisons (e.g., Tycho crater and SPA basin) also suggest our PLS model is reliable and more consistent with the LP results. Besides, the PLS model-derived iron abundance peak of lunar farside is 5.1 wt%, which agrees well with the lunar meteorites that are assumed from lunar highland. Our global FeO distributions are also consistent with the lunar magma ocean hypothesis as has been presented by previous work (Lucey et al. 1995, 1998; Wood et al. 1970; Warren and Haskin 1991).

Although our PLS algorithms have already shown its potential for extraction of lunar iron abundance, it should be kept in mind that there are limitations, i.e., the exact physical significance of PLS is not as evident as experience algorithms, and PLS regression highly depends on the type of the modeling data inputs. More lunar samples and precise geographic location of them would definitely contribute to the improvement of PLS modeling for iron. Interference Imaging Spectrometer (IIM) onboard Chang’E-1 has achieved the abundance of some key elements of the Moon (Ling et al. 2011; Wu et al. 2012; Jin et al. 2013). As is known, China’s new lunar lander and rover mission, “Chang’E-3” lander and “Yutu” rover, respectively, have launched in December 2013, and the rover will be released to detect mineral distribution of the lunar surface, especially in Sinus Iridum (Liu et al. 2013), which has never been set foot on by any lander or rover before. Spectral data from VIS-NIR Imaging Spectrometer (VNIS) onboard Yutu rover may provide good opportunities and more constraints for lunar compositional studies and as PLS modeling of lunar iron abundance as well.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (11003012, U1231103), the Natural Science Foundation of Shandong Province (ZR2011AQ001), Independent Innovation Foundation of Shandong University (2013ZRQP004), and Graduate Innovation Foundation of Shandong University at WeiHai, GIFSDUWH (yjs13026).

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