# Retrieval of Phytoplankton Pigments from Underway Spectrophotometry in the Fram Strait

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## Abstract

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## 1. Introduction

## 2. Data and Methods

#### 2.1. Data Collection

#### 2.2. Retrieval of Phytoplankton Pigments

#### 2.2.1. Gaussian Decomposition

#### 2.2.2. Matrix Inversion Technique

#### Singular Value Decomposition—Non-Negative Least Squares (SVD-NNLS)

#### Non-Negative Least Squares—Non-Negative Least Squares (NNLS-NNLS)

#### Sensitivity Analysis

#### 2.2.3. Normalization of ${a}_{ph}\left(\lambda \right)$ by Pigment Package Effect

#### 2.2.4. Statistics

## 3. Results

#### 3.1. Characteristics of the Pigment Retrieval Data Set

#### 3.2. Gaussian Decomposition

#### 3.3. Matrix Inversion Technique

#### 3.3.1. The Number of Pigment Types to Be Estimated

#### 3.3.2. SVD-NNLS

#### 3.3.3. Intercomparison between SVD-NNLS Applications

#### 3.3.4. Feasibility of SVD-NNLS-9 for Multispectral ${a}_{ph}\left(\lambda \right)$

#### 3.4. Gaussian Decomposition versus SVD-NNLS

## 4. Discussion

#### 4.1. Gaussian Decomposition

#### 4.2. Matrix Inversion Technique

#### 4.3. Applications

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Description |

${a}_{gaus}\left(\lambda \right)$ | Gaussian absorption coefficient (Equation (2)) |

${a}_{NAP}\left(\lambda \right)$ | spectral absorption coefficient by non-algal particles |

${a}_{p}\left(\lambda \right)$ | spectral particulate absorption coefficient |

${a}_{ph}\left(\lambda \right)$ | spectral phytoplankton absorption coefficient |

${\widehat{a}}_{ph}\left(\lambda \right)$ | pigment package effect normalized ${a}_{ph}\left(\lambda \right)$ (Equation (7)) |

A${}_{ph}$ | matrix of ${a}_{ph}\left(\lambda \right)$ (Equations (4) and (5)) |

${a}^{*}\left(\lambda \right)$ | real or measured pigment-specific absorption coefficient |

${\tilde{a}}^{*}\left(\lambda \right)$ | SVD or NNLS derived pigment-specific absorption coefficient (Equations (4) and (5)) |

${\tilde{a}}^{*+}\left(\lambda \right)$ | absolute values of ${\tilde{a}}^{*}\left(\lambda \right)$ (Equation (6)) |

$\left|\right|{\tilde{a}}^{*}\left(\lambda \right)\left|\right|$ | norm of ${\tilde{a}}^{*}\left(\lambda \right)$ (Equation (6)) |

$\tilde{\mathrm{A}}$ | matrix of ${\tilde{a}}^{*}\left(\lambda \right)$ (Equations (4) and (5)) |

A | regression coefficient of pigment concentration-${a}_{gaus}\left({\lambda}_{0}\right)$ power relationship (Equation (3)) |

B | regression coefficient (power) of pigment concentration-${a}_{gaus}\left({\lambda}_{0}\right)$ power relationship |

(Equation (3)) | |

c | HPLC derived pigment concentration |

C | matrix of c (Equation (4)) |

C${}^{+}$ | Moore–Penrose pseudoinverse of matrix C |

${c}_{TChl-a}$ | HPLC derived TChl-a concentration |

${\tilde{c}}^{*}$ | estimated pigment concentration |

$\tilde{C}$ | matrix of ${\tilde{c}}^{*}$ (Equation (5)) |

CD | quartile coefficient of dispersion |

m | number of pigment types |

MAE | mean absolute error (Equation (8)) |

MPE | median absolute percentage error (Equation (9)) |

n | number of samples |

${n}_{cond}$ | condition number of matrix C |

${R}^{2}$ | determination coefficient |

S | spectral exponent of ${a}_{NAP}\left(\lambda \right)$ (Equation (1)) |

$SI$ | similarity index between two ${\tilde{a}}^{*+}\left(\lambda \right)$ (Equation (6)) |

Spearman’s $\rho $ | Spearman${}^{\prime}$s rank correlation coefficient |

${Q}_{a}^{*}\left(\lambda \right)$ | pigment package effect index |

${\lambda}_{0}$ | peak wavelength of a Gaussian function |

$\sigma $ | width of a Gaussian function |

${\sigma}_{SD}\left(\lambda \right)$ | standard deviation of the 20-minute averaged matched AC-S ${a}_{p}\left(\lambda \right)$ spectra (Equation (2)) |

${\chi}^{2}$ | cost function of Gaussian decomposition (Equation (2)) |

## Appendix A. Cross-Validation Results of NNLS-NNLS

**(a)**NNLS-NNLS-6 (${a}_{\mathit{ph}}\left(\lambda \right)$ Based) and NNLS-NNLS-9 (${\widehat{a}}_{\mathit{ph}}\left(\lambda \right)$ Based)

Pigments | Perturb 1 ${}^{\mathbf{a}}$ | Perturb 2 ${}^{\mathbf{a}}$ | Perturb 3 ${}^{\mathbf{a}}$ | Perturb 1 ${}^{\mathbf{b}}$ | Perturb 2 ${}^{\mathbf{b}}$ | Perturb 3 ${}^{\mathbf{b}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | |

TChl-a | 0.22(0.16) | 24.2 | 0.22(0.17) | 21.5 | 0.22(0.16) | 23.6 | 0.05(0.04) | 4.2 | 0.07(0.05) | 6.6 | 0.08(0.05) | 6.6 |

TChl-b | - | - | - | - | - | - | 0.03(0.24) | 64.4 | 0.03(0.24) | 55.2 | 0.03(0.24) | 58.2 |

Chl-c1/2 | - | - | - | - | - | - | 0.11(0.32) | 59.7 | 0.12(0.34) | 66.6 | 0.12(0.33) | 69.6 |

But | - | - | - | - | - | - | 0.03(0.22) | 82.2 | 0.03(0.22) | 76.0 | 0.03(0.23) | 79.6 |

Diadino | 0.14(0.38) | 267.8 | 0.45(0.43) | 2496 | 0.15(0.37) | 210.4 | 0.33(0.60) | 238.2 | 0.21(0.48) | 161.2 | 0.23(0.48) | 222.3 |

Fuco | 0.10(0.27) | 46.5 | 0.10(0.27) | 46.0 | 0.10(0.27) | 46.4 | 0.13(0.22) | 61.8 | 0.11(0.20) | 50.1 | 0.13(0.21) | 60.1 |

Hex | 0.10(0.23) | 57.4 | 0.13(0.24) | 61.0 | 0.14(0.25) | 65.3 | 0.09(0.20) | 45.4 | 0.08(0.21) | 42.1 | 0.09(0.21) | 49.1 |

Peri | 0.04(0.25) | 185.5 | 0.04(0.25) | 164.7 | 0.04(0.24) | 162.2 | 0.02(0.13) | 67.4 | 0.02(0.20) | 104.0 | 0.02(0.19) | 91.9 |

Pheo-a | 0.01(0.02) | 88.5 | 0.02(0.02) | 94.3 | 0.02(0.02) | 97.4 | 0.07(0.05) | 322.9 | 0.05(0.05) | 231.7 | 0.05(0.05) | 234.1 |

Pigments | Perturb 1 ${}^{\mathbf{a}}$ | Perturb 2 ${}^{\mathbf{a}}$ | Perturb 3 ${}^{\mathbf{a}}$ | |||
---|---|---|---|---|---|---|

MAE | MPE | MAE | MPE | MAE | MPE | |

TChl-a | 0.21(0.14) | 22.0 | 0.18(0.12) | 18.4 | 0.21(0.07) | 21.5 |

TChl-b | 0.04(0.31) | 71.9 | 0.04(0.29) | 62.5 | 0.04(0.27) | 64.2 |

Chl-c1/2 | 0.06(0.22) | 34.1 | 0.06(0.21) | 32.4 | 0.06(0.30) | 34.4 |

PPC | 0.18(0.27) | 96.1 | 0.18(0.28) | 85.0 | 0.17(0.22) | 81.5 |

PSC | 0.21(0.20) | 46.0 | 0.20(0.20) | 41.0 | 0.22(0.22) | 45.2 |

Pigments | Perturb 1 ${}^{\mathbf{a}}$ | Perturb 2 ${}^{\mathbf{a}}$ | Perturb 3 ${}^{\mathbf{a}}$ | Perturb 1 ${}^{\mathbf{b}}$ | Perturb 2 ${}^{\mathbf{b}}$ | Perturb 3 ${}^{\mathbf{b}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | |

TChl-a | 0.25(0.15) | 26.6 | 0.24(0.15) | 26.0 | 0.25(0.15) | 26.9 | 0.08(0.05) | 5.54 | 0.10(0.06) | 8.1 | 0.10(0.06) | 8.1 |

Chl-c1/2 | 0.05(0.20) | 35.0 | 0.05(0.20) | 35.1 | 0.06(0.21) | 35.7 | 0.06(0.22) | 38.6 | 0.06(0.23) | 41.5 | 0.06(0.23) | 43.1 |

Diadino | 0.14(0.44) | 189.3 | 0.14(0.43) | 188.8 | 0.14(0.43) | 186.0 | 0.10(0.31) | 118.1 | 0.10(0.31) | 117.3 | 0.10(0.32) | 117.4 |

Hex | 0.17(0.35) | 46.2 | 0.17(0.35) | 47.3 | 0.17(0.35) | 48.7 | 0.23(0.37) | 123.1 | 0.24(0.37) | 114.3 | 0.24(0.37) | 110.8 |

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**Figure 1.**Cruise tracks for PS93.2 (July–August 2015), PS99.2 (June–July 2016) and PS107 (July–August 2017). Symbols denote locations where both AC-S and HPLC data were collected. Bathymetric grid data are extracted from the International Bathymetric Chart of the Arctic Ocean Version 3.0 [62]. Lambert azimuthal equal-area projection was used for mapping.

**Figure 2.**Schematic overview of the steps of applying Gaussian decomposition for phytoplankton pigment retrieval.

**Figure 3.**Schematic overview of the steps of applying the matrix inversion technique for phytoplankton pigment retrieval.

**Figure 4.**(

**a**) Variations of the AC-S derived ${a}_{p}\left(440\right)$ as a power function of TChl-a concentration; (

**b**) the Spearman${}^{\prime}$s rank correlation coefficients between the concentrations of phytoplankton pigments in our data set (linear color bar scale).

**Figure 5.**Concentrations of phytoplankton pigments measured by HPLC versus the magnitudes of the corresponding Gaussian functions obtained from the Gaussian decomposition of both ${a}_{ph}\left(\lambda \right)$ (

**a**,

**c**,

**e**,

**g**,

**i**) and ${\widehat{a}}_{ph}\left(\lambda \right)$ (

**b**,

**d**,

**f**,

**h**,

**j**,

**l**). The results of Chase et al. [22] are based on ${a}_{ph}\left(\lambda \right)$.

**Figure 6.**Variations in the minimum values of the condition number (${n}_{cond}$) of matrix C in Equation (4) with different pigment combination (m pigment types to be estimated): (

**a**) pigment data unperturbed; (

**b**) pigment data perturbed.

**Figure 7.**Pigment-specific absorption spectra obtained from SVD-NNLS-9 (

**a**,

**b**), SVD-NNLS-5${}^{\prime}$ (

**c**,

**d**) and Bricaud-SVD-NNLS-4 (

**e**,

**f**) without data perturbations, respectively. Cases with and without package effect normalization were compared.

**Figure 8.**Scatter plots of SVD-NNLS-9 estimated pigment concentrations versus measured pigment concentrations for unperturbed training data set. The symbols of green circles, red crosses and blue stars represent the data from the cruises PS93.2, PS99.2 and PS107, respectively. Dash lines denote the 50% error lines, and solid lines are one-to-one lines.

**Figure 9.**SVD-NNLS-9 estimated Fuco and Hex concentrations from underway spectrophotometry during the cruise periods of PS93.2 (

**a**), PS99.2 (

**b**) and PS107 (

**c**).

**Table 1.**Abbreviations of phytoplankton pigments and pigment groups analyzed in this study, and the minimum, maximum, mean and standard deviation of the pigment concentrations (mg m${}^{-3}$).

Pigment/Pigment Group | Abbreviation | Minimum | Maximum | Mean | Standard Deviation |
---|---|---|---|---|---|

alloxanthin | Allo | 0.00 | 0.16 | 0.01 | 0.01 |

chlorophyll-c1/2 | Chl-c1/2 | 0.00 | 0.94 | 0.15 | 0.15 |

chlorophyll-c3 | Chl-c3 | 0.00 | 0.83 | 0.08 | 0.11 |

$\alpha $-carotene | $\alpha $-Caro | 0.00 | 0.04 | 0.00 | 0.01 |

$\beta $-carotene | $\beta $-Caro | 0.00 | 0.07 | 0.02 | 0.01 |

diadinoxanthin | Diadino | 0.00 | 0.49 | 0.10 | 0.08 |

diatoxanthin | Diato | 0.00 | 0.05 | 0.01 | 0.01 |

fucoxanthin | Fuco | 0.01 | 1.28 | 0.22 | 0.21 |

${19}^{\prime}$-hexanoyloxyfucoxanthin | Hex | 0.00 | 1.63 | 0.23 | 0.24 |

${19}^{\prime}$-butanoyloxyfucoxanthin | But | 0.00 | 0.51 | 0.04 | 0.05 |

neoxanthin | Neo | 0.00 | 0.02 | 0.00 | 0.00 |

lutein | Lut | 0.00 | 0.01 | 0.00 | 0.00 |

peridinin | Peri | 0.00 | 0.45 | 0.03 | 0.06 |

prasinoxanthin | Prasino | 0.00 | 0.05 | 0.00 | 0.01 |

pheophytin-a | Pheo-a | 0.00 | 1.31 | 0.02 | 0.10 |

pheophorbide-a | Phide-a | 0.00 | 0.17 | 0.01 | 0.02 |

violaxanthin | Viola | 0.00 | 0.03 | 0.01 | 0.01 |

zeaxanthin | Zea | 0.00 | 0.08 | 0.01 | 0.01 |

total chlorophyll-a | TChl-a | 0.06 | 3.87 | 0.86 | 0.66 |

total chlorophyll-b | TChl-b | 0.00 | 0.22 | 0.06 | 0.03 |

total chlorophyll-c | TChl-c | 0.00 | 1.62 | 0.23 | 0.24 |

photosynthetic carotenoids | PSC | 0.02 | 3.56 | 0.52 | 0.49 |

photoprotective carotenoids | PPC | 0.01 | 0.64 | 0.17 | 0.11 |

${\mathit{\lambda}}_{0}\phantom{\rule{4pt}{0ex}}\left[\mathbf{nm}\right]$ | $\mathit{\sigma}\phantom{\rule{4pt}{0ex}}\left[\mathbf{nm}\right]$ | Pigment | A | B | ${\mathit{R}}^{2}$ | MAE ${}^{\mathbf{b}}$ | MPE ${}^{\mathbf{b}}$ | N | MAE ${}^{\mathbf{c}}$ | MPE ${}^{\mathbf{c}}$ |
---|---|---|---|---|---|---|---|---|---|---|

406 | 16 | TChl-a | 17.60 ± 4.03 | 0.90 ± 0.08 | 0.75 | 0.28(0.18) | 26.2 | 274 | - | - |

434 | 12 | TChl-a | 41.61 ± 6.71 | 1.12 ± 0.05 | 0.87 | 0.21(0.13) | 20.7 | 297 | 0.22(0.13) | 20.8 |

453 | 12 | TChl-b & c ${}^{\mathrm{a}}$ | 1.18 ± 0.20 | 1.23 ± 0.05 | 0.92 | 0.00(0.14) | 21.1 | 297 | - | - |

470 | 13 | TChl-b | 0.38 ± 0.11 | 0.50 ± 0.08 | 0.52 | 0.02(0.17) | 29.5 | 296 | - | - |

492 | 16 | PPC | 1.23 ± 0.38 | 0.54 ± 0.09 | 0.50 | 0.06(0.18) | 30.6 | 298 | 0.06(0.18) | 31.5 |

523 | 14 | PSC | 25.25 ± 7.58 | 0.92 ± 0.08 | 0.76 | 0.20(0.21) | 33.4 | 298 | 0.20(0.21) | 34.0 |

550 | 14 | phycoerythrin | - | - | - | - | - | - | - | - |

584 | 16 | Chl-c1/2 | 12.18 ± 5.18 | 0.85 ± 0.09 | 0.68 | 0.07(0.26) | 44.9 | 297 | - | - |

617 | 13 | TChl-a | 21.00 ± 7.85 | 0.57 ± 0.07 | 0.66 | 0.33(0.20) | 36.8 | 295 | - | - |

638 | 11 | Chl-c1/2 | 49.89 ± 16.13 | 1.03 ± 0.06 | 0.81 | 0.05(0.20) | 33.4 | 297 | 0.06(0.20) | 33.5 |

660 | 11 | TChl-b | 0.66 ± 0.21 | 0.44 ± 0.06 | 0.57 | 0.02(0.16) | 29.1 | 293 | 0.02(0.16) | 29.3 |

675 | 10 | TChl-a | 19.70 ± 3.92 | 0.76 ± 0.05 | 0.82 | 0.24(0.14) | 24.9 | 298 | 0.25(0.14) | 25.3 |

${\mathit{\lambda}}_{0}\phantom{\rule{4pt}{0ex}}\left[\mathbf{nm}\right]$ | Pigment | A | B | ${\mathit{R}}^{2}$ | MAE ${}^{\mathbf{b}}$ | MPE ${}^{\mathbf{b}}$ | N | MAE ${}^{\mathbf{c}}$ | MPE ${}^{\mathbf{c}}$ |

406 | TChl-a | 12.50 ± 1.37 | 0.96 ± 0.048 | 0.88 | 0.22(0.15) | 19.5 | 274 | - | - |

434 | TChl-a | 19.23 ± 1.22 | 1.07 ± 0.026 | 0.96 | 0.13(0.08) | 11.9 | 297 | 0.13(0.08) | 12.2 |

453 | TChl-b & c ${}^{\mathrm{a}}$ | 0.39 ± 0.05 | 1.10 ± 0.05 | 0.92 | 0.00(0.15) | 25.8 | 297 | - | - |

470 | TChl-b | 0.30 ± 0.06 | 0.51 ± 0.07 | 0.60 | 0.02(0.15) | 26.3 | 296 | - | - |

492 | PPC | 1.89 ± 0.32 | 0.77 ± 0.06 | 0.77 | 0.05(0.14) | 21.4 | 298 | 0.05(0.14) | 21.8 |

523 | PSC | 44.04 ± 5.31 | 1.19 ± 0.04 | 0.92 | 0.11(0.14) | 20.4 | 298 | 0.11(0.14) | 20.5 |

550 | phycoerythrin | - | - | - | - | - | - | - | - |

584 | Chl-c1/2 | 16.73 ± 4.31 | 1.00 ± 0.06 | 0.82 | 0.06(0.22) | 36.2 | 297 | - | - |

617 | TChl-a | 64.19 ± 13.15 | 0.83 ± 0.04 | 0.84 | 0.22(0.15) | 24.1 | 295 | - | - |

638 | Chl-c1/2 | 34.11 ± 7.20 | 1.06 ± 0.05 | 0.91 | 0.04(0.17) | 27.2 | 297 | 0.04(0.18) | 27.2 |

660 | TChl-b | 0.47 ± 0.12 | 0.41 ± 0.05 | 0.62 | 0.02(0.15) | 27.9 | 293 | 0.02(0.15) | 27.4 |

675 | TChl-a | 33.57 ± 0.72 | 1.00 ± 0.01 | 1.00 | 0.05(0.03) | 3.6 | 298 | 0.05(0.03) | 3.6 |

Method | Pigments | ${\mathit{a}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ${\widehat{\mathit{a}}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ||||
---|---|---|---|---|---|---|---|

${\mathit{n}}_{\mathit{cond}}$ | Maximum $\mathit{SI}$ | m | ${\mathit{n}}_{\mathit{cond}}$ | Maximum $\mathit{SI}$ | m | ||

SVD-NNLS | Fram-20 | 54.9 | 0.86 | 9 | 54.9 | 0.85 | 9 |

NNLS-NNLS | Fram-20 | 30.7 | 0.79 | 6 | 54.9 | 0.86 | 9 |

SVD-NNLS-5${}^{\prime}$ | Gauss-5 | 47.8 | 0.81 | 5 | 47.8 | 0.84 | 5 |

NNLS-NNLS-5${}^{\prime}$ | Gauss-5 | 47.8 | 0.79 | 5 | 47.8 | - | 5 |

Bricaud-SVD-NNLS | Bricaud-12 | 45.1 | 0.72 | 4 | 45.1 | 0.76 | 4 |

Bricaud-NNLS-NNLS | Bricaud-12 | 45.1 | 0.83 | 4 | 45.1 | 0.76 | 4 |

Method | Pigments | ${\mathit{a}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ${\widehat{\mathit{a}}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ||||
---|---|---|---|---|---|---|---|

${\mathit{n}}_{\mathit{cond}}$ | Maximum $\mathit{SI}$ | m | ${\mathit{n}}_{\mathit{cond}}$ | Maximum $\mathit{SI}$ | m | ||

SVD-NNLS | Fram-20 | 59.0 | 0.86 | 9 | 59.0 | 0.81 | 9 |

NNLS-NNLS | Fram-20 | 30.7 | 0.75 | 6 | 59.0 | 0.85 | 9 |

SVD-NNLS-5${}^{\prime}$ | Gauss-5 | 47.8 | 0.74 | 5 | 47.8 | 0.82 | 5 |

NNLS-NNLS-5${}^{\prime}$ | Gauss-5 | 47.8 | 0.78 | 5 | 47.8 | - | 5 |

Bricaud-SVD-NNLS | Bricaud-12 | 45.8 | 0.72 | 4 | 45.8 | 0.76 | 4 |

Bricaud-NNLS-NNLS | Bricaud-12 | 45.8 | 0.83 | 4 | 45.8 | 0.76 | 4 |

Method | Pigments | ${\mathit{a}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ${\widehat{\mathit{a}}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ||||
---|---|---|---|---|---|---|---|

${\mathit{n}}_{\mathit{cond}}$ | Maximum $\mathit{SI}$ | m | ${\mathit{n}}_{\mathit{cond}}$ | Maximum $\mathit{SI}$ | m | ||

SVD-NNLS | Fram-20 | 54.9 | 0.84 | 9 | 54.9 | 0.81 | 9 |

NNLS-NNLS | Fram-20 | 30.7 | 0.81 | 6 | 54.9 | 0.86 | 9 |

SVD-NNLS-5${}^{\prime}$ | Gauss-5 | 47.8 | 0.79 | 5 | 47.8 | 0.82 | 5 |

NNLS-NNLS-5${}^{\prime}$ | Gauss-5 | 47.8 | 0.80 | 5 | 47.8 | - | 5 |

Bricaud-SVD-NNLS | Bricaud-12 | 45.1 | 0.72 | 4 | 45.1 | 0.76 | 4 |

Bricaud-NNLS-NNLS | Bricaud-12 | 45.1 | 0.83 | 4 | 45.1 | 0.76 | 4 |

Pigments | ${\mathit{a}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ${\widehat{\mathit{a}}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Slope | Intercept | ${\mathit{R}}^{2}$ | MAE | MPE | N | Slope | Intercept | ${\mathit{R}}^{2}$ | MAE | MPE | N | |

TChl-a | 0.97 $\pm \phantom{\rule{4pt}{0ex}}0.04$ | 0.00 $\pm \phantom{\rule{4pt}{0ex}}0.01$ | 0.92 | 0.15(0.10) | 12.8 | 295 | 1.00 $\pm \phantom{\rule{4pt}{0ex}}0.00$ | −0.00 $\pm \phantom{\rule{4pt}{0ex}}0.00$ | 1.00 | 0.01(0.01) | 0.74 | 298 |

TChl-b | 0.31 $\pm \phantom{\rule{4pt}{0ex}}0.15$ | −0.81 $\pm \phantom{\rule{4pt}{0ex}}0.20$ | 0.31 | 0.04(0.32) | 58.3 | 268 | 0.41 $\pm \phantom{\rule{4pt}{0ex}}0.15$ | −0.70 $\pm \phantom{\rule{4pt}{0ex}}0.20$ | 0.25 | 0.04(0.30) | 52.6 | 269 |

Chl-c1/2 | 0.65 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | −0.21 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | 0.72 | 0.06(0.25) | 42.6 | 286 | 0.60 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | −0.27 $\pm \phantom{\rule{4pt}{0ex}}0.07$ | 0.63 | 0.06(0.26) | 37.1 | 275 |

But | 0.43 $\pm \phantom{\rule{4pt}{0ex}}0.10$ | −0.53 $\pm \phantom{\rule{4pt}{0ex}}0.17$ | 0.38 | 0.04(0.43) | 100.0 | 209 | 0.54 $\pm \phantom{\rule{4pt}{0ex}}0.08$ | −0.45 $\pm \phantom{\rule{4pt}{0ex}}0.14$ | 0.62 | 0.03(0.35) | 62.6 | 206 |

Diadino | 0.39 $\pm \phantom{\rule{4pt}{0ex}}0.07$ | −0.45 $\pm \phantom{\rule{4pt}{0ex}}0.08$ | 0.40 | 0.06(0.30) | 57.8 | 283 | 0.38 $\pm \phantom{\rule{4pt}{0ex}}0.08$ | −0.52 $\pm \phantom{\rule{4pt}{0ex}}0.09$ | 0.46 | 0.06(0.30) | 49.2 | 284 |

Fuco | 0.84 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | −0.03 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | 0.82 | 0.08(0.23) | 33.7 | 276 | 0.85 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | −0.07 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | 0.79 | 0.07(0.22) | 31.6 | 286 |

Hex | 0.62 $\pm \phantom{\rule{4pt}{0ex}}0.05$ | −0.16 $\pm \phantom{\rule{4pt}{0ex}}0.05$ | 0.69 | 0.09(0.24) | 37.3 | 266 | 0.72 $\pm \phantom{\rule{4pt}{0ex}}0.04$ | −0.13 $\pm \phantom{\rule{4pt}{0ex}}0.04$ | 0.83 | 0.07(0.20) | 32.0 | 270 |

Peri | 0.74 $\pm \phantom{\rule{4pt}{0ex}}0.15$ | −0.30 $\pm \phantom{\rule{4pt}{0ex}}0.23$ | 0.54 | 0.04(0.36) | 57.5 | 134 | 0.67 $\pm \phantom{\rule{4pt}{0ex}}0.17$ | −0.37 $\pm \phantom{\rule{4pt}{0ex}}0.26$ | 0.43 | 0.05(0.40) | 66.6 | 128 |

Pheo-a | 0.06 $\pm \phantom{\rule{4pt}{0ex}}0.87$ | −0.31 $\pm \phantom{\rule{4pt}{0ex}}0.65$ | -0.16 | 0.59(0.53) | 166.0 | 22 | −0.04 $\pm \phantom{\rule{4pt}{0ex}}1.02$ | −0.39 $\pm \phantom{\rule{4pt}{0ex}}0.77$ | −0.34 | 0.64(0.58) | 132.0 | 22 |

Pigments | ${\mathit{a}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ${\widehat{\mathit{a}}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Slope | Intercept | ${\mathit{R}}^{2}$ | MAE | MPE | N | Slope | Intercept | ${\mathit{R}}^{2}$ | MAE | MPE | N | |

TChl-a | 1.04 $\pm \phantom{\rule{4pt}{0ex}}0.04$ | −0.00 $\pm \phantom{\rule{4pt}{0ex}}0.02$ | 0.91 | 0.16(0.12) | 16.2 | 295 | 1.00 $\pm \phantom{\rule{4pt}{0ex}}0.00$ | −0.00 $\pm \phantom{\rule{4pt}{0ex}}0.00$ | 1.00 | 0.02(0.01) | 1.8 | 298 |

TChl-b | 0.54 $\pm \phantom{\rule{4pt}{0ex}}0.21$ | −0.54 $\pm \phantom{\rule{4pt}{0ex}}0.29$ | 0.12 | 0.05(0.38) | 72.6 | 250 | 0.55 $\pm \phantom{\rule{4pt}{0ex}}0.19$ | −0.54 $\pm \phantom{\rule{4pt}{0ex}}0.26$ | 0.18 | 0.05(0.34) | 62.0 | 247 |

Chl-c1/2 | 0.55 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | −0.25 $\pm \phantom{\rule{4pt}{0ex}}0.07$ | 0.67 | 0.09(0.33) | 62.4 | 268 | 0.52 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | −0.31 $\pm \phantom{\rule{4pt}{0ex}}0.07$ | 0.61 | 0.08(0.30) | 59.1 | 269 |

PPC | 0.50 $\pm \phantom{\rule{4pt}{0ex}}0.10$ | −0.26 $\pm \phantom{\rule{4pt}{0ex}}0.09$ | 0.50 | 0.10(0.27) | 55.2 | 278 | 0.51 $\pm \phantom{\rule{4pt}{0ex}}0.08$ | −0.25 $\pm \phantom{\rule{4pt}{0ex}}0.08$ | 0.58 | 0.10(0.27) | 55.9 | 288 |

PSC | 0.66 $\pm \phantom{\rule{4pt}{0ex}}0.05$ | −0.02 $\pm \phantom{\rule{4pt}{0ex}}0.03$ | 0.75 | 0.20(0.22) | 37.8 | 288 | 0.70 $\pm \phantom{\rule{4pt}{0ex}}0.04$ | −0.02 $\pm \phantom{\rule{4pt}{0ex}}0.02$ | 0.83 | 0.15(0.19) | 32.1 | 292 |

Pigments | ${\mathit{a}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ${\widehat{\mathit{a}}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ Based | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Slope | Intercept | ${\mathit{R}}^{2}$ | MAE | MPE | N | Slope | Intercept | ${\mathit{R}}^{2}$ | MAE | MPE | N | |

TChl-a | 0.85 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | 0.01 $\pm \phantom{\rule{4pt}{0ex}}0.02$ | 0.75 | 0.26(0.15) | 25.5 | 298 | 0.98 $\pm \phantom{\rule{4pt}{0ex}}0.01$ | 0.01 $\pm \phantom{\rule{4pt}{0ex}}0.00$ | 0.98 | 0.08(0.05) | 5.3 | 298 |

Chl-c1/2 | 0.69 $\pm \phantom{\rule{4pt}{0ex}}0.04$ | −0.20 $\pm \phantom{\rule{4pt}{0ex}}0.05$ | 0.80 | 0.05(0.19) | 31.2 | 294 | 0.69 $\pm \phantom{\rule{4pt}{0ex}}0.06$ | −0.24 $\pm \phantom{\rule{4pt}{0ex}}0.07$ | 0.65 | 0.06(0.22) | 36.3 | 278 |

Diadino | 0.60 $\pm \phantom{\rule{4pt}{0ex}}0.08$ | −0.22 $\pm \phantom{\rule{4pt}{0ex}}0.10$ | 0.71 | 0.08(0.33) | 75.6 | 245 | 0.68 $\pm \phantom{\rule{4pt}{0ex}}0.16$ | −0.04 $\pm \phantom{\rule{4pt}{0ex}}0.18$ | 0.37 | 0.14(0.44) | 111.8 | 206 |

Hex | 0.10 $\pm \phantom{\rule{4pt}{0ex}}0.12$ | −0.66 $\pm \phantom{\rule{4pt}{0ex}}0.12$ | −0.01 | 0.32(0.60) | 83.2 | 227 | 0.15 $\pm \phantom{\rule{4pt}{0ex}}0.13$ | −0.41 $\pm \phantom{\rule{4pt}{0ex}}0.14$ | 0.10 | 0.40(0.62) | 126.2 | 175 |

Pigments | Perturb 1 ${}^{\mathbf{a}}$ | Perturb 2 ${}^{\mathbf{a}}$ | Perturb 3 ${}^{\mathbf{a}}$ | Perturb 1 ${}^{\mathbf{b}}$ | Perturb 2 ${}^{\mathbf{b}}$ | Perturb 3 ${}^{\mathbf{b}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | |

TChl-a | 0.22(0.15) | 22.3 | 0.17(0.12) | 16.5 | 0.21(0.15) | 21.0 | 0.01(0.01) | 1.1 | 0.05(0.03) | 4.3 | 0.05(0.03) | 4.3 |

TChl-b | 0.04(0.29) | 60.2 | 0.03(0.27) | 53.7 | 0.03(0.27) | 53.6 | 0.04(0.28) | 61.4 | 0.03(0.27) | 52.9 | 0.04(0.27) | 56.3 |

Chl-c1/2 | 0.07(0.26) | 45.3 | 0.07(0.25) | 41.1 | 0.08(0.26) | 44.1 | 0.09(0.30) | 52.9 | 0.08(0.27) | 44.6 | 0.09(0.29) | 50.4 |

But | 0.03(0.32) | 104.8 | 0.03(0.31) | 80.7 | 0.03(0.31) | 81.8 | 0.02(0.25) | 68.7 | 0.02(0.26) | 67.2 | 0.03(0.27) | 69.8 |

Diadino | 0.07(0.32) | 64.4 | 0.08(0.31) | 61.9 | 0.08(0.31) | 65.2 | 0.07(0.32) | 66.0 | 0.07(0.31) | 59.7 | 0.08(0.32) | 64.7 |

Fuco | 0.09(0.22) | 44.5 | 0.09(0.21) | 36.9 | 0.09(0.22) | 38.4 | 0.12(0.27) | 53.1 | 0.10(0.23) | 40.0 | 0.11(0.25) | 44.5 |

Hex | 0.09(0.24) | 43.1 | 0.10(0.26) | 42.6 | 0.11(0.26) | 44.9 | 0.09(0.23) | 42.4 | 0.08(0.22) | 36.2 | 0.10(0.24) | 42.1 |

Peri | 0.02(0.17) | 66.8 | 0.03(0.23) | 90.4 | 0.03(0.23) | 90.4 | 0.02(0.17) | 68.3 | 0.02(0.21) | 74.8 | 0.03(0.21) | 76.3 |

Pheo-a | 0.04(0.04) | 123.7 | 0.02(0.03) | 88.7 | 0.02(0.03) | 90.0 | 0.04(0.04) | 107.0 | 0.02(0.03) | 95.2 | 0.02(0.03) | 91.7 |

Pigments | Perturb 1 ${}^{\mathbf{a}}$ | Perturb 2 ${}^{\mathbf{a}}$ | Perturb 3 ${}^{\mathbf{a}}$ | Perturb 1 ${}^{\mathbf{b}}$ | Perturb 2 ${}^{\mathbf{b}}$ | Perturb 3 ${}^{\mathbf{b}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | |

TChl-a | 0.22(0.17) | 23.3 | 0.18( 0.13) | 17.3 | 0.22(0.16) | 21.7 | 0.02(0.02) | 2.2 | 0.05(0.03) | 4.7 | 0.05(0.04) | 4.9 |

TChl-b | 0.04(0.32) | 73.1 | 0.04(0.30) | 58.5 | 0.04(0.30) | 59.5 | 0.04(0.29) | 66.2 | 0.04(0.28) | 58.3 | 0.04(0.29) | 61.2 |

Chl-c1/2 | 0.09(0.30) | 68.5 | 0.07(0.26) | 49.2 | 0.08(0.27) | 50.4 | 0.09(0.31) | 67.1 | 0.07(0.26) | 47.9 | 0.09(0.29) | 54.3 |

PPC | 0.11(0.28) | 62.1 | 0.09(0.24) | 47.9 | 0.10(0.25) | 51.1 | 0.11(0.28) | 66.2 | 0.09(0.25) | 53.2 | 0.10(0.26) | 57.6 |

PSC | 0.21(0.23) | 42.8 | 0.20(0.22) | 35.0 | 0.21(0.23) | 38.0 | 0.20(0.24) | 42.8 | 0.16(0.19) | 28.7 | 0.20(0.23) | 37.9 |

Pigments | Perturb 1 ${}^{\mathbf{a}}$ | Perturb 2 ${}^{\mathbf{a}}$ | Perturb 3 ${}^{\mathbf{a}}$ | Perturb 1 ${}^{\mathbf{b}}$ | Perturb 2 ${}^{\mathbf{b}}$ | Perturb 3 ${}^{\mathbf{b}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | |

TChl-a | 0.26(0.16) | 26.3 | 0.26(0.15) | 25.3 | 0.27(0.16) | 26.6 | 0.08(0.05) | 5.7 | 0.10(0.06) | 8.3 | 0.10(0.06) | 8.4 |

Chl-c1/2 | 0.05(0.19) | 33.2 | 0.05(0.20) | 32.6 | 0.05(0.20) | 33.5 | 0.06(0.22) | 39.0 | 0.06(0.23) | 41.6 | 0.06(0.23) | 43.1 |

Diadino | 0.06(0.28) | 77.2 | 0.07(0.28) | 78.2 | 0.07(0.28) | 80.1 | 0.10(0.31) | 118.1 | 0.10(0.31) | 117.6 | 0.10(0.32) | 118.3 |

Hex | 0.24(0.46) | 83.6 | 0.24(0.44) | 82.1 | 0.24(0.45) | 82.7 | 0.23(0.37) | 123.2 | 0.24(0.37) | 112.2 | 0.24(0.38) | 109.8 |

^{a}${a}_{\mathit{ph}}\left(\lambda \right)$ based;

^{b}${\widehat{a}}_{\mathit{ph}}\left(\lambda \right)$ based.

**Table 6.**Statistics of phytoplankton pigment retrieval using SVD-NNLS-9 with ${a}_{ph}\left(\lambda \right)$ at ten MODIS bands based on leave-one-out cross-validation. MAE is in mg m${}^{-3}$ (values outside the parentheses were calculated with linear-scale values, while inside the parentheses with log10-scale values) and MPE in %. “Perturb 1, 2 and 3” represent the input data with perturbations of pigment concentrations solely, ${a}_{ph}\left(\lambda \right)$ solely and both, respectively.

Pigments | Perturb 1 ${}^{\mathbf{a}}$ | Perturb 2 ${}^{\mathbf{a}}$ | Perturb 3 ${}^{\mathbf{a}}$ | Perturb 1 ${}^{\mathbf{b}}$ | Perturb 2 ${}^{\mathbf{b}}$ | Perturb 3 ${}^{\mathbf{b}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | MAE | MPE | |

TChl-a | 0.25(0.18) | 28.8 | 0.21(0.14) | 19.8 | 0.28(0.18) | 29.9 | 0.01(0.01) | 0.3 | 0.03(0.02) | 2.0 | 0.03(0.02) | 1.9 |

TChl-b | 0.04(0.28) | 60.7 | 0.04(0.24) | 68.2 | 0.03(0.23) | 67.0 | 0.06(0.31) | 74.7 | 0.04(0.24) | 67.2 | 0.04(0.25) | 73.3 |

Chl-c1/2 | 0.13(0.37) | 85.3 | 0.16(0.38) | 78.3 | 0.15(0.37) | 80.9 | 0.18(0.43) | 95.0 | 0.14(0.37) | 72.2 | 0.18(0.40) | 87.6 |

But | 0.06(0.39) | 185.5 | 0.08(0.41) | 265.0 | 0.08(0.40) | 266.3 | 0.03(0.30) | 87.1 | 0.05(0.33) | 162.1 | 0.05(0.34) | 165.8 |

Diadino | 0.15(0.49) | 151.7 | 0.24(0.48) | 188.5 | 0.23(0.45) | 196.0 | 0.18(0.49) | 168.5 | 0.20(0.45) | 165.4 | 0.23(0.47) | 193.4 |

Fuco | 0.24(0.38) | 114.7 | 0.23(0.34) | 76.9 | 0.21(0.32) | 76.0 | 0.35(0.47) | 148.4 | 0.23(0.36) | 78.8 | 0.31(0.39) | 97.1 |

Hex | 0.15(0.33) | 64.7 | 0.18(0.34) | 70.4 | 0.19(0.34) | 73.4 | 0.18(0.34) | 73.3 | 0.13(0.28) | 58.4 | 0.18(0.31) | 73.2 |

Peri | 0.02(0.14) | 89.7 | 0.04(0.26) | 196.6 | 0.05(0.26) | 211.8 | 0.02(0.12) | 82.2 | 0.03(0.21) | 141.0 | 0.04(0.22) | 148.6 |

Pheo-a | 0.07(0.05) | 312.4 | 0.05(0.04) | 296.1 | 0.06(0.04) | 281.0 | 0.08(0.05) | 341.4 | 0.05(0.04) | 309.0 | 0.05(0.04) | 285.9 |

**Table 7.**The range of values, median and quartile coefficient of dispersion (CD) for the Gaussian decomposition derived pigment-specific absorption coefficient at the corresponding wavelength ${a}_{i}^{*}\left({\lambda}_{0}\right)$ (in m${}^{2}$ mg${}^{-1}$).

${\mathit{\lambda}}_{0}$ (nm) | Pigment | Decomposition of ${\mathit{a}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ | Decomposition of ${\widehat{\mathit{a}}}_{\mathit{ph}}\left(\mathit{\lambda}\right)$ | Chase et al. (2013) [22] | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Range | Median | CD [%] | Range | Median | CD [%] | Range | Median | CD [%] | ||

434 | TChl-a | 0.006-0.153 | 0.036 | 21.2 | 0.020-0.179 | 0.064 | 12.9 | 0.015-0.165 | 0.065 | 35.5 |

675 | TChl-a | 0.007-0.060 | 0.017 | 28.8 | 0.024-0.046 | 0.030 | 5.9 | 0.007-0.065 | 0.019 | 22.6 |

660 | TChl-b | 0.003-0.346 | 0.060 | 38.7 | 0.006-0.540 | 0.100 | 44.5 | 0-0.408 | 0.072 | 43.0 |

638 | Chl-c1/2 | 0.004-0.163 | 0.024 | 40.0 | 0.010-0.333 | 0.039 | 31.6 | 0.010-0.247 | 0.051 | 41.2 |

492 | PPC | 0.059-0.827 | 0.142 | 31.9 | 0.077-1.012 | 0.253 | 21.5 | 0.049-0.797 | 0.097 | 40.7 |

523 | PSC | 0.011-0.192 | 0.029 | 41.5 | 0.024-0.483 | 0.048 | 28.3 | 0.010-0.243 | 0.035 | 47.2 |

**Table 8.**Statistics of phytoplankton pigments retrieval using Gaussian decomposition with package effect normalization based on leave-one-out cross-validation. Package effect normalization was performed with ${c}_{TChl-a}$ in Equation (7) calculated using cruise-specific ${a}_{ph}\left(675\right)$ (AC-S)-TChl-a (HPLC) relationships (see Section 3.1). MAE values outside the parentheses were calculated with linear-scale values, while inside the parentheses with log10-scale values.

${\mathit{\lambda}}_{0}$ [$\mathbf{nm}$] | Pigment | MAE [mg m${}^{-3}$] | MPE [%] |
---|---|---|---|

434 | TChl-a | 0.19(0.13) | 19.2 |

675 | TChl-a | 0.14(0.09) | 12.4 |

660 | TChl-b | 0.02(0.16) | 28.0 |

638 | Chl-c1/2 | 0.06(0.18) | 30.7 |

492 | PPC | 0.05(0.15) | 24.5 |

523 | PSC | 0.16(0.16) | 25.6 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Boss, E.; Chase, A.; Xi, H.; Zhang, X.; Röttgers, R.; Pan, Y.; Bracher, A.
Retrieval of Phytoplankton Pigments from Underway Spectrophotometry in the Fram Strait. *Remote Sens.* **2019**, *11*, 318.
https://doi.org/10.3390/rs11030318

**AMA Style**

Liu Y, Boss E, Chase A, Xi H, Zhang X, Röttgers R, Pan Y, Bracher A.
Retrieval of Phytoplankton Pigments from Underway Spectrophotometry in the Fram Strait. *Remote Sensing*. 2019; 11(3):318.
https://doi.org/10.3390/rs11030318

**Chicago/Turabian Style**

Liu, Yangyang, Emmanuel Boss, Alison Chase, Hongyan Xi, Xiaodong Zhang, Rüdiger Röttgers, Yanqun Pan, and Astrid Bracher.
2019. "Retrieval of Phytoplankton Pigments from Underway Spectrophotometry in the Fram Strait" *Remote Sensing* 11, no. 3: 318.
https://doi.org/10.3390/rs11030318