Optical inversions of the water column based on glider measurements

Catherine Mitchell; Howard R. Gordon; Bruce Bowler; Dave Drapeau; William M. Balch

doi:10.1364/OE.26.032824

Optical inversions of the water column based on glider measurements

Catherine Mitchell, Howard R. Gordon, Bruce Bowler, Dave Drapeau, and William M. Balch

Optics Express
Vol. 26,
Issue 25,
pp. 32824-32838
(2018)
•https://doi.org/10.1364/OE.26.032824

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Catherine Mitchell, Howard R. Gordon, Bruce Bowler, Dave Drapeau, and William M. Balch, "Optical inversions of the water column based on glider measurements," Opt. Express 26, 32824-32838 (2018)

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Related Topics
Table of Contents Category
- Atmospheric and Oceanic Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: August 3, 2018
- Revised Manuscript: September 28, 2018
- Manuscript Accepted: September 29, 2018
- Published: November 30, 2018

Accessible

Open Access

Abstract

We demonstrate a method for estimating absorption and backscattering coefficients by inverting glider-measured profiles of the downwelling irradiance and upwelling radiance. The inversion method was validated against approximately 1,300 profiles of data from 22 glider missions within the Gulf of Maine over a 10 year period. The backscattering coefficient at 532 nm was estimated with a mean absolute error of 21% and bias of 0.01% compared to measured values. We could only quantitatively evaluate the absorption coefficient against the fluorometry data, but found that profiles of fluorescence and absorption were in quantitative agreement. With absorption and backscattering coefficients acting as a basis for studying the biogeochemical parameters of the constituents in the water column, these results show the potential of bio-optical gliders for studying marine ecosystems under varying sky conditions.

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References

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[Crossref]
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[Crossref]
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Y. Fan, W. Li, V. S. Calzado, C. Trees, S. Stamnes, G. Fournier, D. McKee, and K. Stamnes, “Inferring inherent optical properties and water constituent profiles from apparent optical properties,” Opt. Express 23(15), A987–A1009 (2015).
[Crossref] [PubMed]
K. S. Johnson, W. M. Berelson, E. S. Boss, Z. Chase, H. Claustre, S. R. Emerson, N. Gruber, A. Kortzinger, M. J. Perry, and S. C. Riser, “Observing Biogeochemical Cycles At Global Scales With Profiling Floats and Gliders: Prospects for a global array,” Oceanography (Wash. D.C.) 22(3), 216–225 (2009).
[Crossref]
K. Niewiadomska, H. Claustre, L. Prieur, and F. d’Ortenzio, “Submesoscale physical-biogeochemical coupling across the Ligurian current (northwestern Mediterranean) using a bio-optical glider,” Limnol. Oceanogr. 53(5part2), 2210–2225 (2008).
[Crossref]
O. Schofield, R. Chant, B. Cahill, R. Castelao, D. Gong, A. Kahl, J. Kohut, M. Montes-Hugo, R. Ramadurai, P. Ramey, X. Yi, and S. Glenn, “The Decadal View of the Mid-Atlantic Bight from the COOLroom: Is Our Coastal System Changing?” Oceanography (Wash. D.C.) 21(4), 108–117 (2008).
[Crossref]
J. Bouffard, A. Pascual, S. Ruiz, Y. Faugère, and J. Tintoré, “Coastal and mesoscale dynamics characterization using altimetry and gliders: A case study in the Balearic Sea,” J. Geophys. Res. 115(C10), C10029 (2010).
[Crossref]
L. Merckelbach, D. Smeed, and G. Griffiths, “Vertical water velocities from underwater gliders,” J. Atmos. Ocean. Technol. 27(3), 547–563 (2010).
[Crossref]
F. Bourrin, G. Many, X. Durrieu de Madron, J. Martín, P. Puig, L. Houpert, P. Testor, S. Kunesch, K. Mahiouz, and L. Béguery, “Glider monitoring of shelf suspended particle dynamics and transport during storm and flooding conditions,” Cont. Shelf Res. 109, 135–149 (2015).
[Crossref]
V. S. Hemsley, T. J. Smyth, A. P. Martin, E. Frajka-Williams, A. F. Thompson, G. Damerell, and S. C. Painter, “Estimating Oceanic Primary Production Using Vertical Irradiance and Chlorophyll Profiles from Ocean Gliders in the North Atlantic,” Environ. Sci. Technol. 49(19), 11612–11621 (2015).
[Crossref] [PubMed]
N. Briggs, M. J. Perry, I. Cetinić, C. Lee, E. D’Asaro, A. M. Gray, and E. Rehm, “High-resolution observations of aggregate flux during a sub-polar North Atlantic spring bloom,” Deep. Res. Part I Oceanogr. Res. Pap. 58, 1031–1039 (2011).
C. A. Brown, Y. Huot, M. J. Purcell, J. J. Cullen, and M. R. Lewis, “Mapping coastal optical and biogeochemical variability using an autonomous underwater vehicle and a new bio-optical inversion algorithm,” Limnol. Oceanogr. Methods 2(8), 262–281 (2004).
[Crossref]
M. A. Moline, I. Robbins, B. Zelenke, W. S. Pegau, and H. Wijesekera, “Evaluation of bio-optical inversion of spectral irradiance measured from an autonomous underwater vehicle,” J. Geophys. Res. Oceans 117, 1–12 (2012).
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[Crossref] [PubMed]
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[Crossref]
W. M. Balch, D. T. Drapeau, B. C. Bowler, E. S. Booth, J. I. Goes, A. Ashe, and J. M. Frye, “A multi-year record of hydrographic and bio-optical properties in the Gulf of Maine: I. Spatial and temporal variability,” Prog. Oceanogr. 63(1-2), 57–98 (2004).
[Crossref]
E. Organelli, H. Claustre, A. Bricaud, C. Schmechtig, A. Poteau, X. Xing, L. Prieur, F. D’Ortenzio, G. Dall’Olmo, and V. Vellucci, “A novel near-real-time quality-control procedure for radiometric profiles measured by bio-argo floats: Protocols and performances,” J. Atmos. Ocean. Technol. 33(5), 937–951 (2016).
[Crossref]
B. N. Seegers, R. P. Stumpf, B. A. Schaeffer, K. A. Loftin, and P. J. Werdell, “Performance metrics for the assessment of satellite data products: An ocean color case study,” Opt. Express 26(6), 7404–7422 (2018).
[Crossref] [PubMed]
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[Crossref]
W. M. Balch, D. T. Drapeau, B. C. Bowler, and T. Huntington, “Step-changes in the physical, chemical and biological characteristics of the Gulf of Maine, as documented by the GNATS time series,” Mar. Ecol. Prog. Ser. 450, 11–35 (2012).
[Crossref]
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2018 (3)

P. J. Werdell, L. I. W. Mckinna, E. Boss, S. G. Ackleson, S. E. Craig, W. W. Gregg, Z. Lee, S. Maritorena, C. S. Roesler, C. S. Rousseaux, D. Stramski, J. M. Sullivan, M. S. Twardowski, M. Tzortziou, and X. Zhang, “An overview of approaches and challenges for retrieving marine inherent optical properties from ocean color remote sensing,” Prog. Oceanogr. 160, 186–212 (2018).
[Crossref]

M. Ramírez-Pérez, M. Twardowski, C. Trees, J. Piera, and D. McKee, “Inversion of In Situ Light Absorption and Attenuation Measurements to Estimate Constituent Concentrations in Optically Complex Shelf Seas,” J. Geophys. Res. Oceans 123(1), 720–737 (2018).
[Crossref]

B. N. Seegers, R. P. Stumpf, B. A. Schaeffer, K. A. Loftin, and P. J. Werdell, “Performance metrics for the assessment of satellite data products: An ocean color case study,” Opt. Express 26(6), 7404–7422 (2018).
[Crossref] [PubMed]

2016 (3)

E. Organelli, H. Claustre, A. Bricaud, C. Schmechtig, A. Poteau, X. Xing, L. Prieur, F. D’Ortenzio, G. Dall’Olmo, and V. Vellucci, “A novel near-real-time quality-control procedure for radiometric profiles measured by bio-argo floats: Protocols and performances,” J. Atmos. Ocean. Technol. 33(5), 937–951 (2016).
[Crossref]

W. M. Balch, T. Huntington, G. Aiken, D. T. Drapeau, B. C. Bowler, L. C. Lubelczyk, and K. Butler, “Toward a quantitative and empirical dissolved organic carbon budget for the Gulf of Maine, a semienclosed shelf sea,” Global Biogeochem. Cycles 30(2), 268–292 (2016).
[Crossref]

C. Mitchell, A. Cunningham, and D. McKee, “Derivation of the specific optical properties of suspended mineral particles and their contribution to the attenuation of solar irradiance in offshore waters by ocean color remote sensing,” J. Geophys. Res. Oceans 121(1), 104–117 (2016).
[Crossref]

2015 (4)

C. Mitchell and A. Cunningham, “Remote sensing of spatio-temporal relationships between the partitioned absorption coefficients of phytoplankton cells and mineral particles and euphotic zone depths in a partially mixed shelf sea,” Remote Sens. Environ. 160, 193–205 (2015).
[Crossref]

Y. Fan, W. Li, V. S. Calzado, C. Trees, S. Stamnes, G. Fournier, D. McKee, and K. Stamnes, “Inferring inherent optical properties and water constituent profiles from apparent optical properties,” Opt. Express 23(15), A987–A1009 (2015).
[Crossref] [PubMed]

F. Bourrin, G. Many, X. Durrieu de Madron, J. Martín, P. Puig, L. Houpert, P. Testor, S. Kunesch, K. Mahiouz, and L. Béguery, “Glider monitoring of shelf suspended particle dynamics and transport during storm and flooding conditions,” Cont. Shelf Res. 109, 135–149 (2015).
[Crossref]

V. S. Hemsley, T. J. Smyth, A. P. Martin, E. Frajka-Williams, A. F. Thompson, G. Damerell, and S. C. Painter, “Estimating Oceanic Primary Production Using Vertical Irradiance and Chlorophyll Profiles from Ocean Gliders in the North Atlantic,” Environ. Sci. Technol. 49(19), 11612–11621 (2015).
[Crossref] [PubMed]

2014 (1)

M. K. Barnes, G. H. Tilstone, T. J. Smyth, D. J. Suggett, R. Astoreca, C. Lancelot, and J. C. Kromkamp, “Absorption-based algorithm of primary production for total and size-fractionated phytoplankton in coastal waters,” Mar. Ecol. Prog. Ser. 504, 73–89 (2014).
[Crossref]

2013 (3)

A. Cunningham, L. Ramage, and D. McKee, “Relationships between the inherent optical properties and the depth of penetration of solar radiation in optically complex waters,” J. Geophys. Res. 118, 2310–2317 (2013).

G. Zheng and D. Stramski, “A model for partitioning the light absorption coefficient of suspended marine particles into phytoplankton and nonalgal components,” J. Geophys. Res. Oceans 118(6), 2977–2991 (2013).
[Crossref]

E. Rehm and C. D. Mobley, “Estimation of hyperspectral inherent optical properties from in-water radiometry: error analysis and application to in situ data,” Appl. Opt. 52(4), 795–817 (2013).
[Crossref] [PubMed]

2012 (2)

M. A. Moline, I. Robbins, B. Zelenke, W. S. Pegau, and H. Wijesekera, “Evaluation of bio-optical inversion of spectral irradiance measured from an autonomous underwater vehicle,” J. Geophys. Res. Oceans 117, 1–12 (2012).

W. M. Balch, D. T. Drapeau, B. C. Bowler, and T. Huntington, “Step-changes in the physical, chemical and biological characteristics of the Gulf of Maine, as documented by the GNATS time series,” Mar. Ecol. Prog. Ser. 450, 11–35 (2012).
[Crossref]

2011 (3)

C. D. Mobley, “Fast light calculations for ocean ecosystem and inverse models,” Opt. Express 19(20), 18927–18944 (2011).
[Crossref] [PubMed]

N. Briggs, M. J. Perry, I. Cetinić, C. Lee, E. D’Asaro, A. M. Gray, and E. Rehm, “High-resolution observations of aggregate flux during a sub-polar North Atlantic spring bloom,” Deep. Res. Part I Oceanogr. Res. Pap. 58, 1031–1039 (2011).

Z. Lee, V. P. Lance, S. Shang, R. Vaillancourt, S. Freeman, B. Lubac, B. R. Hargreaves, C. Del Castillo, R. Miller, M. Twardowski, and G. Wei, “An assessment of optical properties and primary production derived from remote sensing in the Southern Ocean (SO GasEx),” J. Geophys. Res. Oceans 116, 1–15 (2011).

2010 (2)

J. Bouffard, A. Pascual, S. Ruiz, Y. Faugère, and J. Tintoré, “Coastal and mesoscale dynamics characterization using altimetry and gliders: A case study in the Balearic Sea,” J. Geophys. Res. 115(C10), C10029 (2010).
[Crossref]

L. Merckelbach, D. Smeed, and G. Griffiths, “Vertical water velocities from underwater gliders,” J. Atmos. Ocean. Technol. 27(3), 547–563 (2010).
[Crossref]

2009 (2)

H. R. Gordon, M. R. Lewis, S. D. McLean, M. S. Twardowski, S. A. Freeman, K. J. Voss, and G. C. Boynton, “Spectra of particulate backscattering in natural waters,” Opt. Express 17(18), 16192–16208 (2009).
[Crossref] [PubMed]

K. S. Johnson, W. M. Berelson, E. S. Boss, Z. Chase, H. Claustre, S. R. Emerson, N. Gruber, A. Kortzinger, M. J. Perry, and S. C. Riser, “Observing Biogeochemical Cycles At Global Scales With Profiling Floats and Gliders: Prospects for a global array,” Oceanography (Wash. D.C.) 22(3), 216–225 (2009).
[Crossref]

2008 (2)

K. Niewiadomska, H. Claustre, L. Prieur, and F. d’Ortenzio, “Submesoscale physical-biogeochemical coupling across the Ligurian current (northwestern Mediterranean) using a bio-optical glider,” Limnol. Oceanogr. 53(5part2), 2210–2225 (2008).
[Crossref]

O. Schofield, R. Chant, B. Cahill, R. Castelao, D. Gong, A. Kahl, J. Kohut, M. Montes-Hugo, R. Ramadurai, P. Ramey, X. Yi, and S. Glenn, “The Decadal View of the Mid-Atlantic Bight from the COOLroom: Is Our Coastal System Changing?” Oceanography (Wash. D.C.) 21(4), 108–117 (2008).
[Crossref]

2007 (1)

M. Defoin-Platel and M. Chami, “How ambiguous is the inverse problem of ocean color in coastal waters?” J. Geophys. Res. 112(C3), C03004 (2007).
[Crossref]

2004 (3)

O. Schofield, T. Bergmann, M. J. Oliver, A. Irwin, G. Kirkpatrick, W. P. Bissett, M. A. Moline, and C. Orrico, “Inversion of spectral absorption in the optically complex coastal waters of the Mid-Atlantic Bight,” J. Geophys. Res. 109(C12), C12S04 (2004).
[Crossref]

C. A. Brown, Y. Huot, M. J. Purcell, J. J. Cullen, and M. R. Lewis, “Mapping coastal optical and biogeochemical variability using an autonomous underwater vehicle and a new bio-optical inversion algorithm,” Limnol. Oceanogr. Methods 2(8), 262–281 (2004).
[Crossref]

W. M. Balch, D. T. Drapeau, B. C. Bowler, E. S. Booth, J. I. Goes, A. Ashe, and J. M. Frye, “A multi-year record of hydrographic and bio-optical properties in the Gulf of Maine: I. Spatial and temporal variability,” Prog. Oceanogr. 63(1-2), 57–98 (2004).
[Crossref]

2003 (1)

D. McKee, A. Cunningham, and S. Craig, “Estimation of absorption and backscattering coefficients from in situ radiometric measurements: theory and validation in case II waters,” Appl. Opt. 42(15), 2804–2810 (2003).
[Crossref] [PubMed]

2002 (1)

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41(6), 1035–1050 (2002).
[Crossref] [PubMed]

2000 (1)

M. Stramska, D. Stramski, B. G. Mitchell, and C. D. Mobley, “Estimation of the Absorption and Backscattering Coefficients from in-Water Radiometric Measurements,” Limnol. Oceanogr. 45(3), 628–641 (2000).
[Crossref]

1999 (2)

R. A. Leathers, C. S. Roesler, and N. J. McCormick, “Ocean inherent optical property determination from in-water light field measurements,” Appl. Opt. 38(24), 5096–5103 (1999).
[Crossref] [PubMed]

D. Stramski, R. A. Reynolds, M. Kahru, and B. G. Mitchell, “Estimation of particulate organic carbon in the ocean from satellite remote sensing,” Science 285(5425), 239–242 (1999).
[Crossref] [PubMed]

1998 (2)

H. R. Gordon and G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: vertically stratified water bodies,” Appl. Opt. 37(18), 3886–3896 (1998).
[Crossref] [PubMed]

A. Bricaud, A. Morel, M. Babin, K. Allali, and H. Claustre, “Variations of light absorption by suspended particles with chlorophyll a concentration in oceanic (case 1) waters: Analysis and implications for bio-optical models,” J. Geophys. Res. Oceans 103(C13), 31033–31044 (1998).
[Crossref]

1997 (1)

R. A. Leathers and N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36(33), 8685–8698 (1997).
[Crossref] [PubMed]

1993 (1)

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32(36), 7484–7504 (1993).
[Crossref] [PubMed]

1991 (1)

H. R. Gordon, “Absorption and Scattering Estimates from Irradiance Measurements: Monte Carlo Simulations,” Limnol. Oceanogr. 36(4), 769–777 (1991).
[Crossref]

Ackleson, S. G.

Aiken, G.

Allali, K.

Ashe, A.

Astoreca, R.

Babin, M.

Balch, W. M.

Barnes, M. K.

Béguery, L.

Berelson, W. M.

Bergmann, T.

Bissett, W. P.

Booth, E. S.

Boss, E.

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41(6), 1035–1050 (2002).
[Crossref] [PubMed]

Boss, E. S.

Bouffard, J.

Bourrin, F.

Bowler, B. C.

Boynton, G. C.

Bricaud, A.

Briggs, N.

Brown, C. A.

Brown, O. B.

Butler, K.

Cahill, B.

Calzado, V. S.

Castelao, R.

Cetinic, I.

Chami, M.

M. Defoin-Platel and M. Chami, “How ambiguous is the inverse problem of ocean color in coastal waters?” J. Geophys. Res. 112(C3), C03004 (2007).
[Crossref]

Chant, R.

Chase, Z.

Claustre, H.

Craig, S.

Craig, S. E.

Cullen, J. J.

Cunningham, A.

D’Asaro, E.

D’Ortenzio, F.

Dall’Olmo, G.

Damerell, G.

Defoin-Platel, M.

M. Defoin-Platel and M. Chami, “How ambiguous is the inverse problem of ocean color in coastal waters?” J. Geophys. Res. 112(C3), C03004 (2007).
[Crossref]

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Fig. 1 Relative error (%) of the estimated values of (a)

a

and (b)

b_{b}

for profile aub2 and of (c)

a

and (d)

b_{b}

for profile a2b2. Note the details of differences between aub2 and a2b2 are described in the methods and Table 1.

Download Full Size | PPT Slide | PDF

Fig. 2 Inversion results for all glider missions (Henry in orange, Grampus in green), shown as

b_{b}

regressions on (a) linear scale, and (b) log scale, with the dashed line indicating the 1:1 line, the solid line the least squares regression and histograms on the top and right axes indicating the distribution of the data points; and (c) the frequency distribution of the ratio of

b_{b}

inverted to

b_{b}

measured, with the dashed lines indicating

\pm 25

Download Full Size | PPT Slide | PDF

Fig. 3 (a) the measured and estimated backscattering profiles for one yo, with a depth offset visible between the two profiles, (b) cross correlation between

b_{b}^{e s t}

and

b_{b}^{m e a}

(see text for details).

Download Full Size | PPT Slide | PDF

Fig. 4 Performance of the inversion method for estimating

b_{b}

for individual glider missions. On the x-axis, each glider mission is labeled by a letter (g = Grampus, h = Henry) and a mission number, and “all” refers to all the missions together. (a) slope of a least squares regression between

b_{b}^{e s t}

and

b_{b}^{m e a}

, (b)

R^{2}

for the least squares regression, (c) bias between

b_{b}^{e s t}

and

b_{b}^{m e a}

, and (d) MAE between

b_{b}^{e s t}

and

b_{b}^{m e a}

. All performance metrics were calculated on log-transformed data. See Section 2.4 for details of performance metrics and 2.3.1 for details of classification flags and types.

Download Full Size | PPT Slide | PDF

Fig. 5 Relationship between

a (443)

, the measured Chl fluorescence and the corresponding estimated

a_{p h} (443)

. The 1:1 line between

a (443)

and

a_{p h} (443)

is shown as reference.

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Tables (2)

Table 1 Modeled IOP profile parameters

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Table 2 Mean absolute relative error (%) for the retrieved $a$ and $b_{b}$ using modeled data

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Equations (8)

Equations on this page are rendered with MathJax. Learn more.

a^{(0)} (z) = μ_{0} K_{E}^{m} (z) = \cos (θ_{0 w}) [- \frac{d}{d z} \ln (E_{d}^{m} - E_{u}^{m})]

\begin{array}{l} b_{b}^{(0)} (z) & = & a^{(0)} (z) X^{m} (z) \\ = & 3 a^{(0)} (z) {R^{m} (z) - \frac{d R^{m} (z)}{d z} \int_{z}^{z_{m a x}} {[\frac{E_{d}^{m} (z^{'})}{E_{d}^{m} (z)}]}^{2} d z^{'}} \end{array}

a^{(1)} (z) = \bar{μ} (z) K_{E}^{m} (z) = \frac{E_{d}^{(1)} (z) - E_{u}^{(1)} (z)}{E_{0}^{(1)} (z)} [- \frac{d}{d z} \ln (E_{d}^{m} - E_{u}^{m})]

Δ X (z) = 3 {Δ R (z) - \frac{d Δ R (z)}{d z} \int_{z}^{z_{m a x}} {[\frac{E_{d}^{m} (z^{'})}{E_{d}^{m} (z)}]}^{2} d z^{'}}

δ^{n} = \frac{1}{2 N} \sum_{i = 1}^{N} | \ln (E_{d}^{(n)} (z_{i})) - \ln (E_{d}^{m} (z_{i})) | + \frac{1}{2 N} \sum_{i = 1}^{N} | \ln (E_{u}^{(n)} (z_{i})) - \ln (E_{u}^{m} (z_{i})) |

a (z) = a_{0} + a_{1} \exp [- \frac{{(z - z_{a})}^{2}}{2 σ_{a}^{2}}] b_{b} (z) = b_{0} + b_{1} \exp [- \frac{{(z - z_{b})}^{2}}{2 σ_{b}^{2}}]

b i a s = 10^(\frac{Σ_{i = 1}^{n} \log_{10} (y) - \log_{10} (x)}{n}), M A E = 10^(\frac{Σ_{i = 1}^{n} ∣ \log_{10} (y) - \log_{10} (x) ∣}{n})

a = μ_{d} K_{d} \frac{1 - R (K_{u} / K_{d})}{1 + R (K_{u} / K_{d})}

IOP Profiles	$a_{0}$ (m $^{- 1}$ )	$a_{1}$ (m $^{- 1}$ )	$b_{0}$ (m $^{- 1}$ )	$b_{1}$ (m $^{- 1}$ )	$σ_{a}$ (m)	$σ_{b}$ (m)	$z_{a}$ (m)	$z_{b}$ (m)
aub2	0.1	0	0.0072	0.0072	1	1	-	2
a2b2	0.1	0.1	0.0072	0.0072	1	1	3	1

IOP Profile	Solar Zenith Angle	Correct vs incorrect phase function	MARE $a$ (%) from this study		MARE $b_{b}$ (%) from this study	MARE $b_{b}$ (%) from [31]
aub2	0	Correct	0.62	0.68	3.48	2.81
		Incorrect	4.07	1.65	6.55	5.99
	60	Correct	0.37	1.20	3.55	3.16
		Incorrect	3.91	2.30	9.25	8.42
a2b2	0	Correct	0.80	1.72	1.65	2.32
		Incorrect	4.02	1.45	4.13	4.88
	60	Correct	0.65	1.92	1.10	1.79
		Incorrect	3.65	1.59	7.13	5.87

IOP Profiles	$a_{0}$ (m $^{- 1}$ )	$a_{1}$ (m $^{- 1}$ )	$b_{0}$ (m $^{- 1}$ )	$b_{1}$ (m $^{- 1}$ )	$σ_{a}$ (m)	$σ_{b}$ (m)	$z_{a}$ (m)	$z_{b}$ (m)
aub2	0.1	0	0.0072	0.0072	1	1	-	2
a2b2	0.1	0.1	0.0072	0.0072	1	1	3	1

IOP Profile	Solar Zenith Angle	Correct vs incorrect phase function	MARE $a$ (%) from this study		MARE $b_{b}$ (%) from this study	MARE $b_{b}$ (%) from [31]
aub2	0	Correct	0.62	0.68	3.48	2.81
		Incorrect	4.07	1.65	6.55	5.99
	60	Correct	0.37	1.20	3.55	3.16
		Incorrect	3.91	2.30	9.25	8.42
a2b2	0	Correct	0.80	1.72	1.65	2.32
		Incorrect	4.02	1.45	4.13	4.88
	60	Correct	0.65	1.92	1.10	1.79
		Incorrect	3.65	1.59	7.13	5.87

Abstract

References

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