# The Sentinel 2 MSI Spectral Mixing Space

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

^{2}pixels =) 13.3 × 10

^{9}Sentinel 2 spectra we identify a second set of (12 categories × 10 examples =) 120 subtiles from specific land cover subcategories. Diverse sets of either 5 or 10 subtiles, each 10 × 10 km in area, are chosen from agricultural (10), evaporite (10), cryospheric (10), volcanic (5), urban (5), shallow marine (10), sand dune (10), closed (10) and open (10) canopy forest, scrub/shrub (5), tundra (5), wetland (10), and igneous/sedimentary/metamorphic rock + alluvium (10 + 10) landscapes for a total 120,000,000 subcategory-specific spectra. False color composites of both of these aggregate mosaics are shown in Figure 2. Tile IDs and subcategory subtile locations are given in Appendix B. All data were downloaded free-of-charge as Level 1C exoatmospheric reflectance from the USGS EarthExplorer data portal (https://earthexplorer.usgs.gov/, accessed on 1 October 2022).

#### 2.2. Methods

^{9}Sentinel 2 spectra from 110 spectral diversity hotspots provides a basis for statistical assessment of the information content of the MSI spectral bands. The parametric Pearson correlation coefficient and the non-parametric Mutual Information metric were computed for all band pairs. The Pearson correlation coefficient, r

_{xy}, is given as:

_{KL}is the Kullback–Leibler divergence, p

_{X,Y}is the joint distribution of X and Y, and p

_{X}and p

_{Y}are the marginal distributions of X and Y. The MI of a variable with itself is defined as its self-information. Both mutual and self-information are bounded by [0, +$\infty $], and MI $\le $ SI. Conceptually, both self-information and MI can be understood as measures of “surprise”—the less probable are more surprising than probable events, and events with 100% probability are totally “unsurprising” (information = 0). Computation of MI was performed in Python using scikit-learn (package

`sklearn.featureselection.mutual_info_regression`, with the implementation of [18,19]). As with the correlations, MI is computed for all 55 band pairs.

`n_components`,

`n_neighbors`, and

`min_dist`. Results were found to be relatively insensitive to all three hyperparameters, within at least 2 orders of magnitude of variability. All UMAP shown in this analysis used

`n_components`= 2,

`n_neighbors`= 5 or 500, and

`min_dist`= 0.1.

F_{S}E_{1} + F_{V}E_{1} + F_{D}E_{1} O_{1} |

. . . |

. . = . In matrix notation: O = FE + ε |

. . . |

F_{S}E_{11} + F_{V}E_{11} + F_{D}E_{11} O_{11} |

**E**is the 3 column matrix of 11-band endmember vectors, O is the observed spectral vector to be modeled, F

_{S|V|D}is the vector of endmember fractions to be estimated, and ε is the model misfit to be minimized by the inversion. In addition, a unit sum constraint equation is included. The least squares solution, F = (

**E**

^{T}

**E**)

^{−1}

**E**

^{T}O [22] for the S,V,D endmember fraction estimates gives fractions well-bounded [0, 1]. Model validity is assessed by the Root Mean Square (RMS) of the difference between observed and modeled spectra using the S, V, D estimates and endmember spectra in the forward model (L

_{2}norm).

## 3. Results

_{i}) is more physically plausible than an SVD model using an outer Substrate endmember (S

_{o}) composed of pure sand. However, the outer Substrate endmember could be used for modeling landscapes where bright sands are prominent. Similarly, an outer Vegetation endmember (V

_{o}) composed of a single pixel spectrum is less representative than an inner Vegetation endmember (V

_{i}) composed of an average of several individual spectra at the more densely occupied inner Vegetation apex of the mixing space. Comparisons of inner and outer S and V endmembers are shown in Figure 5. All 5 endmember spectra are given in Table 3.

_{i}and V

_{i}endmembers yields the SVD fraction space shown in Figure 5. As expected, S fractions for the high albedo sands outside the triangular model exceed 1.0 with Dark fractions <0, but all other fraction estimates are well-bounded [0, 1]. Relatively small percentages of the binary S-D and V-D mixtures have V and S (respectively) fractions are slightly negative, but almost all are within 5% of 0. As shown in Figure 6, the spectra with these slightly negative near-zero fractions are limited to a few spatially contiguous geographies (e.g., mangroves, dunes or volcanic ash deposits). The distribution of RMS misfit between the observed and modeled spectra for the 80 subcategory composite has <6% misfit for >99% of 80,000,000 Sentinel 2 spectra, with the upper tail of higher misfits also limited to a few specific land covers not represented in the SVD model (e.g., turbid water, evaporites and light snow).

## 4. Discussion

#### 4.1. Spectral Information Content

_{v}< ~0.5) trending toward the Dark endmember and toward the ternary mixing NPV + soil + shadow region near the S-D mixing continuum. Empirically, these mixing trends appear to correspond to closed and open canopy forest. Figure 5 also shows two distinct subparallel clusters on a single mixing trend on the V-D mixing continuum in the PC 3 vs. 2 space. These features are not seen in either the Landsat or MODIS mixing spaces [7,8].

#### 4.2. Spectral Dimensionality and Mixing Space Topology

#### 4.3. The SVD Model

#### 4.4. Manifold Topology and Spectral Resolution

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Correlation and Mutual Information

**Figure A1.**Correlation vs. Mutual Information estimates and band to band bivariate distributions for the 80 land cover specific subset mosaic. For all band to band pairs (

**top**) correlation and Mutual Information estimates show a correlation of 0.88 and MI of 1.195 (r on MI) and 1.16 (MI on r), with some degree of Log-linear scaling on the lower tail of the distribution and clear nonlinearity on the upper tail. The range of both metrics suggests that almost all non-adjacent, and some adjacent, band combinations provide some discriminative utility for at least some land cover subcategories. Bivariate distributions of MSI band 8 with all other bands show considerable deviations from linearity for all but band 7.

## Appendix B

c1 | c2 |

S2A_MSIL1C_20160723T143750_T19KEQ | S2A_MSIL1C_20170205T210921_N0204_R057_T04QHH |

S2A_MSIL1C_20160723T143750__T19KER | S2B_MSIL1C_20180311T185149_N0206_R113_T10SFJ |

S2A_MSIL1C_20170118T081241_N0204_R078_T35MRV | S2A_MSIL1C_20170315T101021_N0204_R022_T32TPP |

S2A_MSIL1C_20170119T074231_N0204_R092_T36JTT | S2A_MSIL1C_20170412T074611_N0204_R135_T37PDQ |

S2A_MSIL1C_20170119T074231_N0204_R092_T36JUT | S2A_MSIL1C_20170427T021921_N0205_R060_T50HLH |

S2A_MSIL1C_20170119T143731_N0204_R096_T20NNM | S2A_MSIL1C_20170427T153621_N0205_R068_T18NTP |

S2A_MSIL1C_20170124T051101_N0204_R019_T43PGL | S2A_MSIL1C_20170428T215921_N0205_R086_T01KFS |

S2A_MSIL1C_20170124T051101_N0204_R019_T44RQV | S2A_MSIL1C_20170506T054641_N0205_R048_T42QXM |

S2A_MSIL1C_20170124T165551_N0204_R026_T14QQE | S2A_MSIL1C_20170508T012701_N0205_R074_T54STE |

S2A_MSIL1C_20170125T202521_N0204_R042_T58CDU | S2A_MSIL1C_20170604T043701_N0205_R033_T45RYL |

c3 | c4 |

S2A_MSIL1C_20170613T182921_N0205_R027_T11SMB | S2A_MSIL1C_20170723T064631_N0205_R020_T41TKG |

S2A_MSIL1C_20170620T181921_N0205_R127_T12TTK | S2A_MSIL1C_20170723T182921_N0205_R027_T11UQQ |

S2A_MSIL1C_20170621T074941_N0205_R135_T37RGL | S2A_MSIL1C_20170724T145731_N0205_R039_T18LZL |

S2A_MSIL1C_20170627T180911_N0205_R084_T12SUF | S2A_MSIL1C_20170830T125301_N0205_R138_T27WXM |

S2A_MSIL1C_20170627T180911_N0205_R084_T12SUG | S2A_MSIL1C_20170830T131241_N0205_R138_T23KLP |

S2A_MSIL1C_20170628T173901_N0205_R098_T13SCS | S2A_MSIL1C_20170908T063621_N0205_R120_T40QFK |

S2A_MSIL1C_20170704T013711_N0205_R031_T52MHD | S2A_MSIL1C_20170914T065621_N0205_R063_T40TFQ |

S2A_MSIL1C_20170705T022551_N0205_R046_T50NMN | S2A_MSIL1C_20170915T213531_N0205_R086_T06WVS |

S2A_MSIL1C_20170718T210021_N0205_R100_T08WNB | S2A_MSIL1C_20170916T055631_N0205_R091_T42RUN |

S2A_MSIL1C_20170719T084601_N0205_R107_T41XNE | S2A_MSIL1C_20170917T190351_N0205_R113_T10SFG |

c5 | c6 |

S2A_MSIL1C_20170919T142931_N0205_R139_T23VMH | S2A_MSIL1C_20171117T064141_N0206_R120_T40RFU |

S2B_MSIL1C_20180328T183949_N0206_R070_T11SKA | S2A_MSIL1C_20171129T142031_N0206_R010_T18FXJ |

S2A_MSIL1C_20170923T074231_N0205_R049_T37PHN | S2A_MSIL1C_20171201T150711_N0206_R039_T18LZH |

S2A_MSIL1C_20171002T150621_N0205_R039_T19LBE | S2A_MSIL1C_20171203T034121_N0206_R061_T48QUM |

S2A_MSIL1C_20171003T143321_N0205_R053_T20MQC | S2A_MSIL1C_20171207T082321_N0206_R121_T34HCH |

S2A_MSIL1C_20171013T080931_N0205_R049_T25CEM | S2A_MSIL1C_20171208T111441_N0206_R137_T29QKD |

S2A_MSIL1C_20171016T073911_N0205_R092_T36MZC | S2A_MSIL1C_20171209T072301_N0206_R006_T38QND |

S2A_MSIL1C_20171017T103021_N0205_R108_T32TLQ | S2A_MSIL1C_20171210T065251_N0206_R020_T40QCJ |

S2A_MSIL1C_20171107T070231_N0206_R120_T39LUC | S2A_MSIL1C_20160615T183312_N0204_R127_T11SPS |

S2A_MSIL1C_20171117T064141_N0206_R120_T40RFU | S2A_OPER_PRD_MSIL1C_PDMC_20150813T101657 |

c7 | c8 |

S2A_MSIL1C_20150813T101026_N0204_R022_T32UPU | S2A_OPER_MSI_L1C_TL_EPA__20161012T193400_A006777_T55KCB |

S2A_MSIL1C_20151022T184002_N0204_R027_T11SMA | S2B_MSIL1C_20170713T023549_N0205_R089_T51RTN |

S2A_OPER_PRD_MSIL1C_PDMC_20151206T145051 | S2B_MSIL1C_20170723T124309_N0205_R095_T28WDT |

S2A_OPER_PRD_MSIL1C_PDMC_20160318T145513_01 | S2B_MSIL1C_20170727T053639_N0205_R005_T43SFV |

S2A_OPER_MSI_L1C_TL_SGS__20161011T162433_A006812_T32WPT | S2B_MSIL1C_20170730T040549_N0205_R047_T47SND |

S2A_OPER_MSI_L1C_TL_SGS__20161013T032322_A006834_T56LKR | S2B_MSIL1C_20170816T005709_N0205_R002_T53JQJ |

S2A_OPER_MSI_L1C_TL_EPA__20161012T193400_A006777_T55LCC | S2B_MSIL1C_20170817T114639_N0205_R023_T33XWF |

S2A_OPER_MSI_L1C_TL_MTI__20161014T211238_A006858_T15MXV | S2B_MSIL1C_20170824T145909_N0205_R125_T22WEV |

S2A_OPER_MSI_L1C_TL_SGS__20161017T100159_A006894_T45QYG | S2B_MSIL1C_20170826T155519_N0205_R011_T17NMJ |

S2A_OPER_MSI_L1C_TL_MTI__20161018T111609_A006910_T38RPV | S2B_MSIL1C_20170905T085549_N0205_R007_T35TMF |

c9 | c10 |

S2B_MSIL1C_20170906T002659_N0205_R016_T55KCA | S2B_MSIL1C_20171013T081959_N0205_R121_T36SYF |

S2B_MSIL1C_20170912T084549_N0205_R107_T36TUL | S2B_MSIL1C_20171019T083959_N0205_R064_T36STF |

S2B_MSIL1C_20170912T170949_N0205_R112_T14RLP | S2B_MSIL1C_20171101T004649_N0206_R102_T54JTL |

S2B_MSIL1C_20170916T215519_N0205_R029_T06WVB | S2B_MSIL1C_20171103T061009_N0206_R134_T42SWC |

S2B_MSIL1C_20170918T054629_N0205_R048_T43SDT | S2B_MSIL1C_20171103T061009_N0206_R134_T42SWD |

S2B_MSIL1C_20170918T205119_N0205_R057_T07VEG | S2B_MSIL1C_20171116T132219_N0206_R038_T23KKP |

S2B_MSIL1C_20170919T140039_N0205_R067_T21KVA | S2B_MSIL1C_20171123T043059_N0206_R133_T45QYE |

S2B_MSIL1C_20170929T222959_N0205_R072_T60KWF | S2B_MSIL1C_20171130T160619_N0206_R097_T17RMH |

S2B_MSIL1C_20171008T105009_N0205_R051_T30TYN | S2B_MSIL1C_20171202T064229_N0206_R120_T40RGU |

S2B_MSIL1C_20171009T003649_N0205_R059_T55MDP | S2B_MSIL1C_20171207T105419_N0206_R051_T30RVT |

c11 | |

S2B_MSIL1C_20171208T052209_N0206_R062_T44SMD | |

S2B_MSIL1C_20180729T141049_N0206_R110_T21LTC | |

S2B_MSIL1C_20171208T084329_N0206_R064_T33JWN | |

S2B_MSIL1C_20171212T064249_N0206_R120_T40QEL | |

S2B_MSIL1C_20171212T064249_N0206_R120_T40QFH | |

S2B_MSIL1C_20171212T100359_N0206_R122_T32RLQ | |

S2B_MSIL1C_20180622T085559_N0206_R007_T34RGS | |

S2B_MSIL1C_20171214T155519_N0206_R011_T18RUN | |

S2B_MSIL1C_20171215T152629_N0206_R025_T18NUF | |

S2B_MSIL1C_20171227T160459_N0206_R054_T17QME |

Agriculture | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170205T210921_N0204_R057_T04QHH | 4N | 868610 | 2223190 |

S2A_MSIL1C_20170315T101021_N0204_R022_T32TPP | 32N | 623950 | 4864330 |

S2A_MSIL1C_20170508T012701_N0205_R074_T54STE | 54N | 269220 | 3988590 |

S2A_MSIL1C_20170723T064631_N0205_R020_T41TKG | 41N | 266210 | 4645260 |

S2A_MSIL1C_20170917T190351_N0205_R113_T10SFG | 10N | 688930 | 4167330 |

S2A_OPER_PRD_MSIL1C_PDMC_20161017T044357 | 45N | 723470 | 2625060 |

S2B_MSIL1C_20170730T040549_N0205_R047_T47SND | 47N | 554190 | 4363690 |

S2B_MSIL1C_20170918T054629_N0205_R048_T43SDT | 43N | 459570 | 3800040 |

S2B_MSIL1C_20171008T105009_N0205_R051_T30TYN | 30N | 702100 | 4787760 |

S2B_MSIL1C_20171013T081959_N0205_R121_T36SYF | 36N | 778000 | 4095680 |

Sand | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170628T173901_N0205_R098_T13SCS | 13N | 372290 | 3654900 |

S2A_MSIL1C_20170908T063621_N0205_R120_T40QFK | 40N | 653400 | 2447190 |

S2A_MSIL1C_20171119T040041_N0206_R004_T48TUK | 48N | 305540 | 4438710 |

S2A_MSIL1C_20171208T111441_N0206_R137_T29QKD | 29N | 291550 | 2399280 |

S2A_MSIL1C_20171209T072301_N0206_R006_T38QND | 38N | 527910 | 1890720 |

S2B_MSIL1C_20171207T105419_N0206_R051_T30RVT | 30N | 481880 | 3290910 |

S2B_MSIL1C_20171208T084329_N0206_R064_T33JWN | 33S | 541880 | 7265640 |

S2B_MSIL1C_20171212T100359_N0206_R122_T32RLQ | 32N | 339750 | 2966720 |

S2B_MSIL1C_20171212T100359_N0206_R122_T32RLR | 32N | 331950 | 3100020 |

Lava & Ash | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170205T210921_N0204_R057_T04QHH | 4N | 861160 | 2206290 |

S2A_MSIL1C_20171016T073911_N0205_R092_T36MZC | 36S | 819250 | 9703580 |

S2A_MSIL1C_20171016T073911_N0205_R092_T36MZC | 36S | 834220 | 9768640 |

S2A_OPER_PRD_MSIL1C_PDMC_20161014T163303 | 15S | 652170 | 9967520 |

S2B_MSIL1C_20170723T124309_N0205_R095_T28WDT | 28N | 399960 | 7200220 |

Urban | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170508T012701_N0205_R074_T54STE | 54N | 269890 | 3950620 |

S2A_MSIL1C_20170830T131241_N0205_R138_T23KLP | 23S | 328970 | 7398470 |

S2A_MSIL1C_20170916T055631_N0205_R091_T42RUN | 42N | 300000 | 2758120 |

S2A_MSIL1C_20171017T103021_N0205_R108_T32TLQ | 32N | 390060 | 4999690 |

S2B_MSIL1C_20170912T170949_N0205_R112_T14RLP | 14N | 364980 | 2848280 |

Forest—1 | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170118T081241_N0204_R078_T35MRV | 35S | 831290 | 9963030 |

S2A_MSIL1C_20170119T074231_N0204_R092_T36JTT | 36S | 284150 | 7247210 |

S2A_MSIL1C_20170205T210921_N0204_R057_T04QHH | 4N | 847400 | 2230620 |

S2A_MSIL1C_20170427T021921_N0205_R060_T50HLH | 50S | 355240 | 6230970 |

S2A_MSIL1C_20170508T012701_N0205_R074_T54STE | 54N | 257880 | 3907290 |

S2A_MSIL1C_20170604T043701_N0205_R033_T45RYL | 45N | 794940 | 3088140 |

S2A_MSIL1C_20170705T022551_N0205_R046_T50NMN | 50N | 450950 | 704020 |

S2A_MSIL1C_20170724T145731_N0205_R039_T18LZL | 18S | 875170 | 8546360 |

S2A_MSIL1C_20170724T145731_N0205_R039_T19LBF | 19S | 215640 | 8582190 |

S2A_MSIL1C_20170830T131241_N0205_R138_T23KLP | 23S | 321220 | 7348390 |

Forest—2 | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170917T190351_N0205_R113_T10SFG | 10N | 607440 | 4106660 |

S2A_OPER_PRD_MSIL1C_PDMC_20151206T145051 | 20N | 469370 | 431170 |

S2B_MSIL1C_20170713T023549_N0205_R089_T51RTN | 51N | 231700 | 3257530 |

S2B_MSIL1C_20170718T101029_N0205_R022_T32TQS | 32N | 773730 | 5121020 |

S2B_MSIL1C_20170906T002659_N0205_R016_T55KCA | 55S | 353630 | 8006280 |

S2B_MSIL1C_20170912T084549_N0205_R107_T36TUL | 36N | 335150 | 4512660 |

S2B_MSIL1C_20171009T003649_N0205_R059_T55MDP | 55S | 469610 | 9317570 |

S2B_MSIL1C_20171013T081959_N0205_R121_T36SYF | 36N | 791100 | 4092030 |

S2B_MSIL1C_20171116T132219_N0206_R038_T23KKP | 23S | 215910 | 7344400 |

S2B_MSIL1C_20171215T152629_N0206_R025_T18NUF | 18N | 381240 | 26200 |

Senescent Vegetation | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170119T074231_N0204_R092_T36JUT | 36S | 387540 | 7237130 |

S2A_MSIL1C_20170119T074231_N0204_R092_T36JUT | 36S | 381920 | 7259800 |

S2A_MSIL1C_20170119T074231_N0204_R092_T36JUT | 36S | 375110 | 7261040 |

S2A_MSIL1C_20170119T074231_N0204_R092_T36JUT | 36S | 379990 | 7209420 |

S2A_MSIL1C_20170516T154911_N0205_R054_T18TWQ | 18N | 563770 | 4938390 |

Tundra & Wetlands | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170718T210021_N0205_R100_T08WNB | 8N | 508380 | 7654750 |

S2A_MSIL1C_20170718T210021_N0205_R100_T08WNB | 8N | 540940 | 7608620 |

S2A_OPER_PRD_MSIL1C_PDMC_20160318T145513 | 19S | 495986 | 7997974 |

S2B_MSIL1C_20170916T215519_N0205_R029_T06WVB | 6N | 442210 | 7700040 |

S2B_MSIL1C_20170916T215519_N0205_R029_T06WVB | 6N | 458950 | 7676830 |

Mangroves | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170427T153621_N0205_R068_T18NTP | 18N | 258620 | 824760 |

S2A_MSIL1C_20170704T013711_N0205_R031_T52MHD | 52S | 814620 | 9839210 |

S2A_MSIL1C_20170705T022551_N0205_R046_T50NMN | 50N | 498390 | 752360 |

S2A_MSIL1C_20170705T022551_N0205_R046_T50NMN | 50N | 423780 | 704730 |

S2A_MSIL1C_20170916T055631_N0205_R091_T42RUN | 42N | 319520 | 2736030 |

S2A_OPER_PRD_MSIL1C_PDMC_20161018T073751 | 38N | 655730 | 3419140 |

S2B_MSIL1C_20170826T155519_N0205_R011_T17NMJ | 17N | 472220 | 875270 |

S2B_MSIL1C_20170919T140039_N0205_R067_T21KVA | 21S | 445610 | 8017250 |

S2B_MSIL1C_20171123T043059_N0206_R133_T45QYE | 45N | 756960 | 2481220 |

S2B_MSIL1C_20171123T043059_N0206_R133_T45QYE | 45N | 763390 | 2429410 |

Rock & Alluvium—1 | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20160723T143750_T19KER | 19S | 506000 | 7534310 |

S2A_MSIL1C_20170124T051101_N0204_R019_T44RQV | 44N | 781870 | 3417600 |

S2A_MSIL1C_20170412T074611_N0204_R135_T37PDQ | 37N | 467190 | 1496550 |

S2A_MSIL1C_20170412T074611_N0204_R135_T37PDQ | 37N | 415880 | 1480390 |

S2A_MSIL1C_20170613T182921_N0205_R027_T11SMB | 11N | 478340 | 4162580 |

S2A_MSIL1C_20170613T182921_N0205_R027_T11SMB | 11N | 441920 | 4110190 |

S2A_MSIL1C_20170613T182921_N0205_R027_T11SMB | 11N | 424630 | 4194020 |

S2A_MSIL1C_20170613T182921_N0205_R027_T11SMB | 11N | 429810 | 4180830 |

S2A_MSIL1C_20170627T180911_N0205_R084_T12SUF | 12N | 310360 | 4011400 |

S2A_MSIL1C_20170627T180911_N0205_R084_T12SUF | 12N | 304930 | 4096250 |

Rock & Alluvium—2 | |||

TileID | UTM Zone | Easting | Northing |

S2A_MSIL1C_20170627T180911_N0205_R084_T12SUG | 12N | 393280 | 4169500 |

S2A_MSIL1C_20170908T063621_N0205_R120_T40QFK | 40N | 664760 | 2494790 |

S2A_MSIL1C_20171201T150711_N0206_R039_T18LZH | 18S | 866060 | 8213050 |

S2A_MSIL1C_20171207T082321_N0206_R121_T34HCH | 34S | 395100 | 6286480 |

S2A_OPER_PRD_MSIL1C_PDMC_20151022T184002 | 11N | 516790 | 4027140 |

S2A_OPER_PRD_MSIL1C_PDMC_20160318T145513 | 19S | 486817 | 8008443 |

S2B_MSIL1C_20171103T061009_N0206_R134_T42SWC | 42N | 576560 | 3774420 |

S2B_MSIL1C_20171103T061009_N0206_R134_T42SWD | 42N | 544220 | 3856340 |

S2B_MSIL1C_20171202T064229_N0206_R120_T40RGU | 40N | 768340 | 3304040 |

S2B_MSIL1C_20171212T064249_N0206_R120_T40QEL | 40N | 520620 | 2570980 |

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**Figure 1.**Geographic and climatic distributions of 110 Sentinel 2 tiles from spectral diversity hotspots. Geographic distribution of sample sites is guided by climatic and geologic diversity as well as overall species biodiversity. Individual tile selection criteria favor spectral diversity arising from land cover diversity within and across biomes. Tile geographic coverage corresponds well to global land area distribution within the climatic parameter space (lower left) from [14]. All biomes are well represented. Biome classification (lower right) adapted from [15].

**Figure 2.**Sentinel 2 composites for 110 spectral diversity hotspot tiles (110 × 110 km) and specific land cover subcategories (10 × 10 km) selected from individual hotspot tiles. Identical 1% linear stretch applied to both mosaics.

**Figure 3.**Spectral dimensionality from variance partition. Five 20 tile subsets (

**left**) each have very similar variance partition to the full 110 tile aggregate (black) with 99% variance in the first 3 dimensions. The aggregate of 120 land cover subcategory subsets (

**right**-black) also has very similar variance partition with 98% in the first 3 dimensions. The individual land cover subcategories vary somewhat, with sand and ice + snow having lower dimensionality than more heterogeneous categories. All have <1% variance in all dimensions >4. Both mosaics can be considered 3D in the sense that the 3 low order dimensions represent >98% of total variance.

**Figure 4.**Spectral mixing spaces for Sentinel 2 mosaics. Orthogonal projections show the 3D topology of the PC spaces as continuous with clearly defined apexes corresponding to physically distinct spectral endmembers. Vegetation and Sand + Substrate endmembers show strongly linear mixing with the Dark (shadow or water) endmember. Both mosaics have very similar topology and endmembers in the 3D PC space, indicating that the 120 land cover subcategories capture the salient features of the 110 spectral diversity hotspots.

**Figure 5.**Sentinel 2 SVD spectral mixing space, spectral endmembers, and the corresponding SVD fraction space. An eight column (80,000,000 spectra) subset of the Land Cover Subcategory mosaic encompassing the SVD-bounded plane of the full mixing space is effectively 2D with PC dimensions 1 (81%) and 2 (14%) accounting for 95% of total variance, compared to PC 3 (2%). Maximum amplitude (Outer) and lower amplitude mean (Inner) endmember spectra for Substrate and Vegetation define bases for maximal and minimal SVD models (

**left**). Inversion of the minimal model provides liberal estimates of SVD fractions (

**right**), but excludes pure sand landscapes. Because sands lie outside the minimal SVD model, their Substrate fractions exceed 1.0 with Dark fractions < 0. The resulting planar SVD fraction distribution can be projected onto a 2D ternary diagram (

**lower right**) with no loss of information.

**Figure 6.**SVD model misfit and negative fraction distributions. RMS misfit map (linear stretch [0, 10%]) shows largest misfits associated with snow, evaporites and turbid water. RMS distribution shows 99% of spectra with <5% misfit (92% < 3%). Negative S (red) and V (green) fractions are well within 0.1 of zero.

**Figure 7.**Minimal model SVD fraction mosaic for the primary land cover subset. Linear stretch [0, 1] for all fractions. Sands are saturated red because they are outside the minimal model with S

_{I}fractions > 1.

**Figure 8.**UMAP manifolds with distinct clusters labeled in mixing space then back-projected into geographic space (on RMS misfit map). Distinct clusters of spectra in the mixing space correspond to geographically distinct and spatially contiguous land cover.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8a | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0.62 | 0.57 | 0.5 | 0.47 | 0.31 | 0.21 | 0.21 | 0.18 | 0.39 | 0.42 |

0.62 | 1 | 0.96 | 0.85 | 0.8 | 0.53 | 0.35 | 0.37 | 0.3 | 0.67 | 0.71 |

0.57 | 0.96 | 1 | 0.95 | 0.93 | 0.7 | 0.52 | 0.55 | 0.48 | 0.82 | 0.85 |

0.5 | 0.85 | 0.95 | 1 | 0.99 | 0.76 | 0.57 | 0.6 | 0.53 | 0.92 | 0.95 |

0.47 | 0.8 | 0.93 | 0.99 | 1 | 0.84 | 0.65 | 0.69 | 0.63 | 0.95 | 0.96 |

0.31 | 0.53 | 0.7 | 0.76 | 0.84 | 1 | 0.9 | 0.96 | 0.94 | 0.87 | 0.8 |

0.21 | 0.35 | 0.52 | 0.57 | 0.65 | 0.9 | 1 | 0.91 | 0.92 | 0.71 | 0.62 |

0.21 | 0.37 | 0.55 | 0.6 | 0.69 | 0.96 | 0.91 | 1 | 0.98 | 0.76 | 0.66 |

0.18 | 0.3 | 0.48 | 0.53 | 0.63 | 0.94 | 0.92 | 0.98 | 1 | 0.71 | 0.6 |

0.39 | 0.67 | 0.82 | 0.92 | 0.95 | 0.87 | 0.71 | 0.76 | 0.71 | 1 | 0.98 |

0.42 | 0.71 | 0.85 | 0.95 | 0.96 | 0.8 | 0.62 | 0.66 | 0.6 | 0.98 | 1 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8a | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|

1.06 | 0.86 | 0.56 | 0.42 | 0.41 | 0.19 | 0.17 | 0.15 | 0.15 | 0.29 | 0.33 |

0.68 | 1.39 | 0.85 | 0.61 | 0.58 | 0.25 | 0.22 | 0.21 | 0.2 | 0.39 | 0.44 |

0.49 | 0.96 | 1.28 | 0.83 | 0.8 | 0.38 | 0.31 | 0.32 | 0.28 | 0.54 | 0.57 |

0.39 | 0.83 | 0.9 | 1.17 | 1.03 | 0.59 | 0.52 | 0.53 | 0.49 | 0.68 | 0.76 |

0.34 | 0.66 | 0.83 | 0.99 | 1.29 | 0.69 | 0.57 | 0.54 | 0.52 | 0.81 | 0.79 |

0.14 | 0.21 | 0.36 | 0.5 | 0.59 | 1.37 | 1.03 | 0.96 | 0.93 | 0.55 | 0.43 |

0.1 | 0.16 | 0.28 | 0.44 | 0.5 | 1.04 | 1.34 | 1.11 | 1.19 | 0.46 | 0.39 |

0.09 | 0.16 | 0.28 | 0.44 | 0.46 | 0.95 | 1.08 | 1.4 | 1.08 | 0.45 | 0.37 |

0.08 | 0.14 | 0.24 | 0.39 | 0.44 | 0.9 | 1.15 | 1.08 | 1.42 | 0.44 | 0.37 |

0.24 | 0.41 | 0.53 | 0.67 | 0.81 | 0.64 | 0.58 | 0.57 | 0.58 | 1.28 | 0.88 |

0.27 | 0.51 | 0.6 | 0.78 | 0.88 | 0.55 | 0.51 | 0.5 | 0.5 | 1.01 | 1.1 |

λ (nm) | S_{i} | V_{i} | D | S_{o} | V_{o} |
---|---|---|---|---|---|

443 | 1754 | 1084 | 1198 | 1536 | 1194 |

490 | 1799 | 827 | 946 | 1556 | 909 |

560 | 2154 | 892 | 739 | 2291 | 969 |

665 | 3028 | 410 | 280 | 5485 | 447 |

705 | 3303 | 1070 | 208 | 6236 | 1126 |

740 | 3472 | 4206 | 180 | 6889 | 4762 |

783 | 3656 | 5646 | 167 | 7323 | 6323 |

842 | 3566 | 5495 | 135 | 7176 | 6193 |

865 | 3686 | 6236 | 129 | 7530 | 6629 |

1610 | 5097 | 2101 | 26 | 10,252 | 1731 |

2190 | 4736 | 775 | 14 | 8745 | 712 |

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Small, C.; Sousa, D.
The Sentinel 2 MSI Spectral Mixing Space. *Remote Sens.* **2022**, *14*, 5748.
https://doi.org/10.3390/rs14225748

**AMA Style**

Small C, Sousa D.
The Sentinel 2 MSI Spectral Mixing Space. *Remote Sensing*. 2022; 14(22):5748.
https://doi.org/10.3390/rs14225748

**Chicago/Turabian Style**

Small, Christopher, and Daniel Sousa.
2022. "The Sentinel 2 MSI Spectral Mixing Space" *Remote Sensing* 14, no. 22: 5748.
https://doi.org/10.3390/rs14225748