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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 25 — Dec. 5, 2011
  • pp: 24986–25005
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Inherent optical property estimation in deep waters

Eric Rehm and Norman J. McCormick  »View Author Affiliations


Optics Express, Vol. 19, Issue 25, pp. 24986-25005 (2011)
http://dx.doi.org/10.1364/OE.19.024986


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Abstract

We develop two algorithms for determining two inherent optical properties (IOPs) from radiometric measurements in vertically homogeneous waters. The first algorithm is for estimation of the ratio of the backscattering to absorption coefficients from measurements of only the vertically upward radiance and the downward planar irradiance at depths where the light field is in the asymptotic regime. The second algorithm enables estimation of the absorption coefficient from measurement of the diffuse attenuation coefficient in the asymptotic regime after use of the first algorithm. Multiplication of the two estimates leads to an estimate for the backscattering coefficient. The algorithms, based upon the use of a simplified phase function and the asymptotic eigenmode, are shown to potentially provide good starting conditions for iteratively determining the absorption and backscattering coefficients of a wide variety of waters. The uncertainty in the estimates defines a subspace for IOPs that may reduce ambiguity in such iterative solutions. Because of the ease of estimating the backscattering to absorption ratio from in-water measurements, this IOP deserves further investigation as a proxy for biogeochemical quantities in the open ocean.

© 2011 OSA

1. Introduction

The estimation of ocean inverse optical properties (IOPs) from radiometric measurements is a long-standing and important problem in ocean optics [1

1. H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).

], forming the basis of remote sensing of ocean water constituents and biogeochemical properties. The solution for one or more IOPs from apparent optical properties (AOPs) is an example of such a problem. Using just the knowledge of the in-water light field, the inverse problem for IOPs can be solved by closed-form (explicit) methods or iterative (implicit) methods [2

2. N. J. McCormick, “Inverse radiative transfer problems: a review,” Nucl. Sci. Eng. 112, 185–198 (1992).

, 3

3. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

].

Explicit methods do this by manipulation of the radiative transfer equation (RTE) or analytic solutions to derive formulas that estimate IOPs from AOPs. The explicit method is illustrated by the early work of Gordon et al. [4

4. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975). [CrossRef] [PubMed]

] who showed, assuming quasi-single scattering [5

5. H. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973). [CrossRef] [PubMed]

] and employing Monte Carlo simulations, that remote sensing reflectance was strongly correlated with the ratio of the backscattering to absorption coefficients, bb/a. Explicit algorithms also can be derived from solutions to the RTE expressed as a linear combination of eigenfunctions [6

6. N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996). [CrossRef]

], or by carrying out a large number of radiative transfer simulations in order to parameterize relationships between IOPs and AOPs [7

7. H. Loisel and D. Stramski, “Estimation of inherent optical properties of natural waters from the irradiance attenuation coefficient and reflectance in the presence of Raman scattering,” Appl. Opt. 39, 3001–3011 (2000). [CrossRef]

, 8

8. 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, 628–641 (2000). [CrossRef]

].

Implicit methods estimate IOPs by repeatedly solving the RTE, starting with an initial estimate or guess of the IOPs. At each iteration, radiometric measurements (or AOPs derived therein) are compared to the RTE solution and then an objective function is computed that expresses the difference between the RTE solution and the measured values. The IOPs are then adjusted to reduce the objective function. Iteration is complete when the objective function is reduced to an acceptably small value. Implicit approaches to estimating absorption and backscattering in homogeneous waters have been developed that estimate IOPs from surface illumination and depth profiles of easily-measured radiometric quantities, while allowing for general multiple scattering as well as inelastic scattering [9

9. H. R. Gordon and G. C. Boynton, “A radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997). [CrossRef] [PubMed]

11

11. E. Rehm, “Inverting light with constraints,” Poster session presented at Ocean Optics XIX, Barga, Italy (2008).

].

Two inversion algorithms are developed here, one for estimating the bb/a ratio and the other for the absorption coefficient a. For estimating bb/a, we use a hybrid approach where we iteratively solve a system of equations derived from an explicit eigenfunction formulation of the RTE. Our algorithm for bb/a requires one fitted parameter F that corrects for an oversimplification of our assumed phase function. That phase function enables us to use a simplified solution of the radiative transfer equation which depends only on a and bb. We also assume that the optical measurements are made in the asymptotic regime. (The depths at which the light reaches an asymptotic distribution depends on the IOPs and the surface illumination and occurs at shallower depths when the incident light is more diffuse [3

3. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

, 12

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

, 13

13. B. D. Piening and N. J. McCormick, “Asymptotic optical depths in source-free ocean waters,” Appl. Opt. 42, 5382–5387 (2003). [CrossRef] [PubMed]

].) With the values of F and bb/a, our second algorithm can be used to estimate a from measurement of the in-water downward diffuse attenuation coefficient. Combination of the two estimates directly leads to an estimate of the backscattering coefficient bb. Alternatively, a can be estimated from bb/a, measurements of bb at one or more wavelengths, and a model for the spectral behavior of bb(λ).

The resulting estimates of a and bb can be used, for example, as the initial starting estimate, instead of a guess, for any implicit method, such as one using a nonlinear global optimization technique [11

11. E. Rehm, “Inverting light with constraints,” Poster session presented at Ocean Optics XIX, Barga, Italy (2008).

]. Furthermore, uncertainty in estimates from our algorithms can be used as inversion constraints [11

11. E. Rehm, “Inverting light with constraints,” Poster session presented at Ocean Optics XIX, Barga, Italy (2008).

, 14

14. S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global scale applications,” Appl. Opt. 41, 2705–2714 (2002). [CrossRef] [PubMed]

, 15

15. IOCCG, “Remote sensing of inherent optical properties: fundamentals, tests of algorithms, and applications,” Reports of the International Ocean-Colour Coordinating Group, 5, Z.-P. Lee, ed. (IOCCG, Dartmouth, Canada, 2006).

].

2. Radiative transfer equation

For our investigation the radiance L(z,μ,φ) for z ≥ 0 is taken to be a function of the direction cosine μ, measured with respect to the depth z in the water, and ϕ, the azimuthal angle measured in the plane perpendicular to the z-axis. The water is taken to be spatially uniform and deep enough that bottom effects can be neglected. For scattering given by the phase function β̃(μ0) = β̃(μ′,φ′,μ,φ) for μ0 the cosine of the angle between {μ′,φ′} and {μ,φ}, the radiative transfer equation for the semi-infinite medium is [3

3. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

]
μzL(z,μ,φ)+cL(z,μ,φ)=b02π11β˜(μ,φ,μ,φ)L(z,μ,φ)dμdφ,z0.
(1)
Here c = a + b is the sum of the absorption and scattering coefficients. Notice that this formulation and subsequent algorithm development ignores inelastic scattering. As a result, the algorithm developed here will not be applied to spectral regions in which inelastic scattering can make a significant contribution to the upward radiance [10

10. G. C. Boynton and H. R. Gordon, “Irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: Raman-scattering effects,” Appl. Opt. 39, 3012–3022 (2000). [CrossRef]

, 16

16. T. G. Peacock, K. L. Carder, C. O. Davis, and R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE 1302, 303–319 (1990).

, 17

17. H. R. Gordon, “Contribution of Raman scattering to water-leaving radiance: a reexamination,” Appl. Opt. 38, 3166–3174 (1999). [CrossRef]

].

Because the objective here is to develop algorithms for measurements done with a planar irradiance detector and a radiance detector aimed in the vertically upward direction for which μ = −1, there is no azimuthal dependence of the data so the analysis will be done for the azimuthally averaged radiance
L(z,μ)=(2π)102πL(z,μ,φ)dφ
(2)
with
μzL(z,μ)+cL(z,μ)=b11β˜(μ,μ)L(z,μ)dμ,z0,
(3)
and
β˜(μ,μ)=02πβ˜(μ,φ,μ,φ=0)dφ.
(4)

There are two assumptions needed to develop the algorithms. The first is that water beneath measurement depth zm is sufficiently spatially uniform and deep enough that zm is approaching the so-called “asymptotic regime” where the angular distribution of the radiance will not perceptibly change with increasing depth. Another interpretation of this first assumption is that zm must be deep enough that the directional nature of the incident surface illumination does not affect the directional nature of the underwater light field. The second assumption is that the angular distribution of the volume scattering function is so strongly forward that an “isotropic plus delta forward” synthetic scattering phase function model can be used,
β˜(μ,μ)=12[b˜bF+2(1b˜bF)δ(μμ)]
(7)
where b = bb/b is the backscattering ratio and F is the adjustment factor to be determined from numerical simulations or experimental data. Our phase function satisfies the proper normalization condition
11β˜(μ,μ)dμ=1.
(8)
A primary advantage of this phase function is that it leads directly to a radiative transfer equation that depends only on a and bb, but not b, which has been shown to be a significant advantage for remote sensing applications [4

4. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975). [CrossRef] [PubMed]

, 18

18. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977). [CrossRef]

].

The phase function of Eq. (7) is a slight extension of the simplest form of the classic “delta-Eddington” scattering model [19

19. J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976). [CrossRef]

] to which it degenerates if F = 1. Although this elementary model for the scattering phase function does not account for the directional dependence of backscattering, such a dependence has been shown to give errors in underwater radiances and planar irradiances on the order of 10% or less [20

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

].

Substitution of the phase function of Eq. (7) into Eq. (3), followed by changes in the variables to obtain a dimensionless optical distance,
τ=(cb+bb˜b/F)z=az[1+bb/aF],
(9)
results in the radiative transfer equation
μτL(τ,μ)+L(τ,μ)=ϖ211L(τ,μ)dμ,τ0
(10)
where
ϖ=bb/abb/a+F.
(11)
We also note that
ϖ1ϖ=bb/aF.
(12)

It is the parameter F that first must be obtained from either Hydrolight [21

21. C. D. Mobley and L. K. Sundman, Hydrolight 5.0, Ecolight 5.0 Technical Documentation (Sequoia Scientific, Inc., 2008).

] calculations with known values of bb/a or from experimental data in which both rrs(zm) and bb/a have been measured. Once obtained, the resulting F subsequently can be used in the algorithm for determining bb/a for waters where no or limited IOP measurements are available. Notice that with this model the objective is to avoid the dependence on b by having the scattering depend instead on bb, which is consistent with the quasi-single-scattering approximation [3

3. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

5

5. H. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973). [CrossRef] [PubMed]

].

3. Algorithm for backscattering-to-absorption ratio

The initial step in the derivation of the algorithm for bb/a follows by combining Eq. (10) written for μ and for −μ in a way that the derivative with respect to τ can be eliminated [22

22. N. J. McCormick, “Analytic inverse radiative transfer equations for atmospheric and hydrologic optics,” J. Opt. Soc. Am. A 21, 1009–1017 (2004). [CrossRef]

]. This is done by first multiplying the equation for μ by μL(τ,−μ) and the one written for −μ by μL(τ,μ), integrating both equations over −1 ≤ μ ≤ 1, and subtracting the results to obtain
ddτ11μ2L(τ,μ)L(τ,μ)dμ=ϖ11μL(τ,μ)dμ11L(τ,μ)dμ.
(13)
To simplify this equation, multiply Eq. (10) by μ, integrate over −1 ≤ μ ≤ 1, and rearrange the result to obtain the so-called diffusion equation,
11L(τ,μ)dμ=(1ϖ)1ddτ11μL(τ,μ)dμ.
(14)
Substitution of this result into Eq. (13) gives
dG(τ)dτ=ddτ[401μ2L(τ,μ)L(τ,μ)dμϖ1ϖ(11μL(τ,μ)dμ)2]=0.
(15)
Integration of dG(τ)/dτ over τ from an arbitrary measurement depth τm or zm to infinity, where the light field vanishes, allows us to write
ϖ1ϖ[11μL(zm,μ)dμ]2=401μ2L(zm,μ)L(zm,μ)dμ.
(16)
After division by [Ed(zm)/2π]2 and with Eqs. (5) and (12), Eq. (16) can be written as
bb/aF[12π01μL(zm,μ)Ed(zm)dμ]2=16π201μL(zm,μ)Ed(zm)μL(zm,μ)Ed(zm)dμ.
(17)

The integrals in Eq. (17) involve radiance data that are not available because L(zm, μ) is not measured, so we introduce two new radiometric functions
Λ(zm)=2π01μL(zm,μ)Lu(zm)dμ=Eu(zm)Lu(zm),
(18)
Ω(zm)=16π201μ2L(zm,μ)Ed(zm)L(zm,μ)Lu(zm)dμ,
(19)
where the upward planar irradiance is
Eu(zm)=2π01μL(zm,μ)dμ.
(20)
With the help of Eqs. (5), (6), (18) and (19), Eq. (17) then can be written in terms of the measured rrs(zm) value as
bb/aF=Ω(zm)rrs(zm)[1Λ(zm)rrs(zm)]2.
(21)

Equations (18) and (19) can be evaluated by invoking the second assumption for the algorithm that the measurements are made at any asymptotic depth zas. With the approximation of L(zas, μ) to evaluate Ω(zm) and Λ(zm), from Eq. (21) the final form for the bb/a algorithm is
bb/aF=Ω(zas)rrs(zm)[1Λ(zas)rrs(zm)]2
(22)
where
Λ(zas)=2π01μL(zas,μ)Lu(zas)dμ=Eu(zas)Lu(zas)
(23)
Ω(zas)=16π201μ2L(zas,μ)L(zas,μ)dμEd(zas)Lu(zas).
(24)

To evaluate L(zas, μ) in Eqs. (23) and (24), an analytic representation can be used for the radiance at depths where it is asymptotic. The eigenmode expansion method [23

23. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

,24

24. N. J. McCormick and I. Kuščer, “Singular eigenfunction expansions in neutron transport theory,” in Advances in Nuclear Science and Technology7, E. J. Henley and J. Lewins, eds. (Academic, 1973), pp. 181–282.

] shows that the asymptotic radiance far from the surface or bottom consists of only the dominant eigenmode [25

25. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992). [CrossRef]

] corresponding to the dominant eigenvalue ν0,
L(τas,μ)=A(ν0)ϕ(ν0,μ)exp(τas/ν0),1μ1,
(25)
or, from Eq. (9),
L(zas,μ)=A(ν0)ϕ(ν0,μ)exp{[1+bb/(aF)]azas/ν0},1μ1.
(26)
(The A(ν0) is a constant that depends on the surface illumination.) Substitution of Eq. (25) into Eq. (10) gives
(1μ/ν0)ϕ(ν0,μ)=(ϖ/2)11ϕ(ν0,μ)dμ.
(27)
Because this equation is homogeneous and the dominant eigenvalue satisfies 1 < ν0 ≤ ∞, the eigenfunction ϕ(ν0, μ) can be normalized with
11ϕ(ν0,μ)dμ=1
(28)
so from Eqs. (27) and (28),
ϕ(ν0,μ)=ϖν0/2ν0μ,1μ1.
(29)
Substitution of Eq. (29) into Eq. (28) gives the eigenvalue ν0 as the positive root of the transcendental equation
1ϖν02ln(ν0+1ν01)=0.
(30)
From Eq. (11) it follows that ϖ depends on F, so ν0 determined from Eq. (30) also does. Another very useful equation from the eigenfunction method is obtained by integrating Eq. (27) over −1 ≤ μ ≤ 1 to obtain, after use of Eqs. (11) and (28),
11μϕ(ν0,μ)dμ=ν0(1ϖ)=ν0[1+bb/(aF)]1.
(31)
With L(zas, μ) from Eqs. (26) and (29) it follows from Eq. (18) that
Λ(zas)=2π(ν0+1)01μ(ν0+μ)1dμ=2π(ν0+1)[1ν0ln(ν0+1ν0)].
(32)
In a similar way it follows from Eq. (19) that
Ω(zas)=16π2(ν0+1)01μ2(ν02μ2)1dμ01μ(ν0μ)1dμ=8π2(ν0+1)[1+ν0ln(ν0ν01)]111μ(ν0μ)1dμ=Fbb/a16π2(ν0+1)[1+ν0ln(ν0ν01)]1,
(33)
where Eqs. (12) and (31) have been used.

To summarize, for a specified value of bb/a, F must be determined consistent with Eqs. (11), (22), (30), (32), and (33).

4. Algorithm for absorption and backscattering coefficients

The algorithm for determining the absorption coefficient a ultimately follows from the Gershun equation
a=KE(zm)μ¯(zm),
(34)
where the diffuse attenuation coefficient and mean cosine of the radiance are
KE(zm)=d[ln11μL(zm,μ)dμ]/dzm=d{ln[Ed(zm)Eu(zm)]}/dzm
(35)
μ¯(zm)=[11μL(zm,μ)dμ]/[11L(zm,μ)dμ].
(36)
Because Kd(zm) is measured instead of KE (zm), it is again necessary to invoke the assumption that measurements are made at depths zmzas where KE (zas) = Kd (zas). Thus an algorithm for determining a can be obtained from
a=K^d(zas)μ¯(zas),
(37)
where d(zas), an estimate of the diffuse attenuation coefficient, follows from Eq. (35) as
K^d(zas)Δ[lnEd(zas)]Δzas.
(38)
Alternatively, from values of Kd(zn), n = 1 to 3, where the depths zn are such that the radiance is approaching the asymptotic regime, an estimate of d(zas) can be obtained from [6

6. N. J. McCormick, “Analytical transport theory applications in optical oceanography,” Ann. Nucl. Energy 23, 381–395 (1996). [CrossRef]

, 26

26. J. R. V. Zaneveld, “An asymptotic closure theory of irradiance in the sea and its inversion to obtain the vertical structure of inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989). [CrossRef]

]
K^d(zas)=Kd(z1)Kd(z3)[Kd(z2)]2Kd(z1)+Kd(z3)2Kd(z2)
(39)
provided z2 = 2−1(z1 + z3).

To estimate μ̄(zas) in order to obtain the absorption coefficient from Eq. (37), it follows from Eqs. (26), (28), and (31) that
μ¯(zas)=[11μϕ(ν0,μ)dμ]/[11ϕ(ν0,μ)dμ]=ν0[1+(bb/(aF)]1.
(40)
Thus, the algorithm for determining the absorption coefficient from Eq. (37) then becomes
a=K^d(zas)ν0[1+(bb/(aF)]1,
(41)
where bb/a has been obtained from Eq. (22) and used, along with the previously determined F, to compute the ν0.

A second approach for determining the absorption coefficient from knowledge of bb/a and F is to take the ratio of measurements in the asymptotic regime at two locations z1 and z2 and to use
L(zas,1)=Lu(zas)=A(ν0)ϕ(ν0,1)exp[azas[1+bb/(aF)]/ν0].
(42)
For z1 and z2 in the asymptotic radiance regime this leads to the approximation
a=ν0(z2z1)[1+bb/(aF)]ln[Lu(z1)Lu(z2)].
(43)
Because of the small magnitude of Lu(zas), however, this algorithm is expected to be prone to numerical difficulties. Another possibility, of course, is if bb is measured or can be estimated at the wavelengths of interest, then a can be directly estimated from bb/a as
a=bb,measbb/a.
(44)

Finally, the estimated values of bb/a from Eq. (22) and a from Eq. (41) or (43) can be combined to give an estimated value of bb from
bb=(bb/a)a.
(45)

5. Data and methods

5.1. Synthetic data sets

We assessed the ability of our extended delta-Eddington model to estimate the scattering phase function using synthetic data. Radiometric quantities Lu and Ed were simulated at 490 nm using a simple Case 1 chlorophyll-based bio-optical model embedded in Hydrolight [21

21. C. D. Mobley and L. K. Sundman, Hydrolight 5.0, Ecolight 5.0 Technical Documentation (Sequoia Scientific, Inc., 2008).

] to generate inherent optical properties. Simulations were performed using chlorophyll concentrations Chl (0.01, 0.03, 0.1, 0.3, 1, 3, 10 mg m−3) and three zenith angles (15°,30°,60°), yielding 7x3 = 21 total simulations. While simulation output was requested every 5 m from 0 m to 50 m, radiometric data for three output depths (5, 10, 30 m) from each simulation was included in this synthetic data set, resulting in 63 simulated data points.

One synthetic data set was created allowing Hydrolight to select a Fournier-Forand scattering phase function β̃FF [27

27. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. S. Jaffe (ed), Proc. SPIE 2258, 194–201 (1994).

] based on a specified particle backscattering ratio bp = bbp/bp, which resulted in a distinct scattering phase function selection for each chlorophyll concentration. To assess the ability of our algorithm to retrieve bb/a in the absence of measurement noise and natural variability, the entire first synthetic data set was used both as a training set to determine the fitted parameter F and also as a validation set to compare values of bb/a generated by the Case 1 bio-optical model to those estimated from corresponding simulated values of rrs = Lu/Ed.

A second, somewhat independent, synthetic validation set was created using the same cross product of chlorophyll concentrations, depths, and zenith angles, but using a single Petzold scattering phase function β̃P [28

28. T. J. Petzold, “Volume scattering functions for selected ocean waters,” Tech. Rep. SIO 72–78 (Scripps Institution of Oceanography, La Jolla, Calif., 1972), pp. 1–78.

] for all simulations of radiometric quantities. This second synthetic data set was used solely as a validation set to test the sensitivity of our algorithm’s estimates of bb/a to any assumed phase function effects in the parameter F fitted using the first training set.

5.2. NAB08 data set

We also tested our algorithm’s performance using depth profiles of Lu(z), Ed(z), a(z), and bb(z) from a calibration campaign carried out as part of the 2008 North Atlantic Bloom Experiment (NAB08) in support of long term (51 d) autonomous physical, optical, and radiometric measurements from a Lagrangian float [29

29. E. A. D’Asaro, C. Lee, M. Perry, K. Fennel, E. Rehm, A. Gray, N. Briggs, and K. Gudmundsson, “The 2008 North Atlantic bloom experiment I: overview and strategy,” Eos, Transactions, American Geophysical Union89(53), Fall Meeting Supplement, Abstract OS24A-08 (2008), http://adsabs.harvard.edu/abs/2008AGUFMOS24A..08D.

]. The data were obtained at eight process cruise stations during the spring bloom of May 2008, representing two distinct phytoplankton bloom communities: a large diatom-dominated community (Chl ∼ 2 – 5 mg m−3) during the bloom and mixed ciliate/picoeukaryote community (Chl < 1 mg m−3) following the bloom peak.

5.2.1. NAB08 data collection and processing

Temperature, salinity, pressure (CTD), and bio-optical profiles to approximately 80 m depth were performed during a cruise on the R/V Knorr from 1–22 May 2008. A Satlantic free-falling optical profiler was used to measure downward spectral irradiance, Ed (z,λ), and upward spectral radiance, Lu(z,λ) at 3.3 nm increments from 350 to 800 nm with a spectral accuracy of 0.3 nm and a spectral resolution of 10 nm. Profiles were taken within 1.5 h of local noon. Radiometric data was processed using 1 m bins with ProSoft 8-RC5 software (Satlantic, Inc.) to determine of Ed (z,λ), Lu(z,λ), and Kd (z,λ).

Fig. 1 An IOP and a radiometric profile from the NAB08 dataset. (A) Typical IOPs were nearly homogeneous in the lit portion of the water column. (B) Radiometric profiles of Lu and Ed at 488 nm show little curvature until 40 m, another indication of vertically homogeneous water.

For each profile, radiometric and IOP data were sampled at eight depths (5, 10, 15, 20, 25, 30, 35, 40 m) at AC-9 wavelengths (412, 440, 488, 510, 532 and 555 nm). For wavelengths ≥ 555 nm we expect our algorithms to work poorly because our formulation of the radiative transfer equation (Eq. 1) does not include inelastic scattering [10

10. G. C. Boynton and H. R. Gordon, “Irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: Raman-scattering effects,” Appl. Opt. 39, 3012–3022 (2000). [CrossRef]

, 17

17. H. R. Gordon, “Contribution of Raman scattering to water-leaving radiance: a reexamination,” Appl. Opt. 38, 3166–3174 (1999). [CrossRef]

]. The 40 m depth was omitted for two profiles where radiometric measurements were below the noise level of the radiometer. Example contemporaneous radiometric and IOP profiles at 488 nm for a single station are shown in Fig. 1; bb(488) is estimated from bb(470) and bb(700) as described above and bb/a is computed from a(488) and the estimated bb(488).

When processing the measured data, no attempt was made to incorporate possible instrumentation inaccuracies. Radiometric measurements were discarded for Lu(z) < 10−4 μW cm−2 sr−1 or Ed(z) < 10−2 μW cm−2.

5.2.2. Cross-validation approach

We used the NAB08 data set to both determine the fitted parameter F as well as test the algorithm. We were also interested in answering the question: “How many calibration profiles are needed to determine F for a specified accuracy?” To answer this question as fairly as possible, we employed a cross-validation approach [32

32. B. Efron and G. Gong, “A leisurely look at the bootstrap, the jackknife, and cross-validation,” Am. Stat. 37, 36–48 (1983). [CrossRef]

] to estimate error in our predictions of IOPs from F, rrs and Kd(zas). Cross-validation is a resampling technique used to estimate how accurately a predictive model will perform in practice. Our cross-validation scheme iteratively partitions the n = 8 profiles into a training set used to determine the fitted parameter F and a validation set used to estimate the IOPs. A training set consists of a k-element subset of profiles, k = 1 ... 7, chosen from the set of eight profiles, resulting in Mk=(nk) possible training set combinations for a given k. A validation set of profiles is simply the complementary set of profiles for any training set combination, assuring that training data is never used as validation data. Examples for profile numbers included for k = 1,2 are shown in Table 1.

Table 1. Example training and validation sets of NAB08 calibration profiles for k = 1,2 used for cross validation.

table-icon
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Traditional cross-validation yields a nearly unbiased estimate of prediction error by removing one data point at a time from the data set used for prediction. A weakness of our approach is that the number of members of each set and the number of sets vary with each iteration k.

5.3. Determination of fitted parameter F

For each training set, F was found using the contemporaneous values of rrs = Lu/Ed, bb, and a available from Hydrolight simulation or field measurement. The problem was solved as a system of three equations GF = 0 with three unknowns XF, where
XF=[ν0bb/aF].
(46)
A nonlinear least squares equation solver (function fsolve in MATLAB [33

33. MATLAB® Optimization Toolbox™ 5 User’s Guide, (The MathWorks Inc., 2010). [PubMed]

]) provides a solution XF that minimizes the cost function gF = GFTGF, where the first two elements of GF correspond to Eqs. (22) and (30), and the final element provides the necessary third constraint by matching the estimated and measured values of bb/a:
GF=[1ϖν02ln(ν0+1ν01)bb/aFΩ(zas)rrs(zm)[1Λ(zas)rrs(zm)]2(bb/a)(bb/a)measured].
(47)
The required initial point for the solver, XF0, was chosen as
XF0=[1.00050.20.2].
(48)
Finally, using the results for all the depths of given profile and wavelength, a simple linear least squares fit was made to estimate F as a linear function of bb/a,
F(bb/a)=p1(bb/a)+p2.
(49)

5.4. Determination of bb/a

The validation sets of profiles were used to test the algorithm. The linear equation Eq. (49) for the fitted parameter F(bb/a) determined from the training set and rrs = Lu/Ed computed from simulated or field radiometric data were used to estimate bb/a. The contemporaneous measured or simulated values of bb and a were reserved for subsequent error analysis (see section 5.5). Similar to the approach used to determine fitted parameter F in section 5.3, this problem was solved as a system of two equations G = 0 and two unknowns X, where
X=[ν0bb/a].
(50)
The same nonlinear least squares equation solver was used to minimize the cost function g = GTG, where the first two elements of G again correspond to Eqs. (22) and (30):
G=[1ϖν02ln(ν0+1ν01)bb/aFΩ(zas)rrs(zm)[1Λ(zas)rrs(zm)]2].
(51)
The required initial point for the solver, X0, was chosen as
X0=[1.00050.2].
(52)

5.5. Error analysis

Assessment of systematic biases and typical uncertainty between estimated IOPs and measured IOPs is made using relative percent difference ψ and its absolute value |ψ|, respectively. In the formulation of the relative percent differences, measured IOP data are considered the reference values, so for a paired estimate and measurement the relative error is
ψi(zj)=100IOPi,est(zj)IOPi,meas(zj)IOPi,meas(zj),
(53)
where an IOP may be bb/a, a, or bb, IOPi,est (zj) is an estimated value, and IOPi,meas(zj) is the corresponding measured value for a given validation set i = 1 ...Mk at depth zj. Similarly, the average absolute percent difference is computed with |ψi(zj)|. Outliers were excluded to prevent biased estimates by removing single ψi(zj) values exceeding the average plus or minus three times the standard deviation σ of the total number of values within each average.

The error analysis was carried out using the NAB08 radiometric and IOP profile data for the Mk validation sets complementing the Mk training sets used to determine fitting factor F. We summarize estimation of the errors in three ways for an implicitly denoted k:
  1. Average absolute error |ψ(zj)|¯ of all Mk validation set combinations at each depth zj
    |ψ(zj)|¯=1Mki=1Mk|ψi(zj)|
    (54)
    in order to explore our assumption of asymptotic radiance.
  2. Depth-averaged absolute error |ψi|¯ and relative error ψi¯ for each validation set i
    |ψi|¯=18j=18|ψi(zj)|,
    (55a)
    ψi¯=18j=18ψi(zj)
    (55b)
    with which to explore the variability of the Mk different validation set combinations, assuming a vertically homogeneous regime.
  3. Average absolute error |ψ| and average relative error ψ
    |ψ|¯=18Mki=1Mkj=18|ψi(zj)|,
    (56a)
    ψ¯=18Mki=1Mkj=18ψi(zj)
    (56b)
    is used to explore the algorithm’s average performance across all depths and validation set combinations.

For the two synthetic data sets, ψ̄ and |ψ|¯ were computed as averages of individual errors ψi(zj) and |ψi(zj)| in IOPs estimated from the 63 simulated data points.

6. Results

6.1. Synthetic data sets

The results for the first synthetic data set constructed using β̃FF are shown in Table 2. For each value of chlorophyll concentration Chl, IOPs at 490 nm were generated. The parameter F was fitted for values of bb/a using all the simulated radiometric and IOP data and was essentially constant, varying from 0.074 to 0.076 while bb/a varied from 0.037 to 0.11. Using the simulated radiometric data to test our bb/a algorithm, the average relative and absolute errors were found to be small, demonstrating that once the fitted parameter F was determined, recovery of the original data is possible to within ∼4%. This implies that the variation of β̃FF due to changes in the backscattering ratio b̃bp was insignificant with respect to the phase function we assumed.

Table 2. Average relative error ψ̄ and absolute error |ψ|¯ (in percent) for estimates of bb/a from radiative transfer simulations at 490 nm. Chl in mg m−3. Parenthetical values indicate the range of |ψ|¯ over all depths.

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When we replaced the validation set by repeating the simulation except for a change from β̃FF to β̃P for all values of Chl, there was a greater error in the estimates of bb/a (right half of Table 2). As one might expect, the smallest errors corresponded to Chl = 0.10 to 1.0 mg m−3 where b̃bp was closest in both data sets. This is because F has been “tuned” to the set of Fournier-Forand scattering phase functions used in the first data set. The Petzold data set represents a different bio-optical regime determined by a different scattering phase function; we should not expect F tuned to one bio-optical regime to do as well in another.

6.2. NAB08 data set

For the field data at 488 nm, Fig. 2A shows the evolution of the fitted parameter F as the number of training profiles k increases. The slope and intercept of F(488) show convergence towards stable mean values, as one would expect as more of the data set is used to determine F; more of the natural variability of bb/a is captured as additional profiles cover the experimental area and duration of the bloom. A linear model of F(bb/a) as in Eq. (49), using data for all k = 6 training set combinations at each wavelength, demonstrates the wavelength dependence of F beyond 488 nm (Fig. 2B, Table 3).

Fig. 2 Estimates of F using rrs = Lu/Ed and IOP measurements from the NAB08 dataset. A) Values of fitted parameter F at 488 nm for all training set combinations (black x’s) and the linear fit for each value of k. Color indicates number of profiles k used in the training set. (B) fitted parameter F at selected wavelengths (color), created by linear fit for all k = 6 training set combinations.

Table 3. Slope, intercept and R2 for fitted parameter F = p1(bb/a) + p2 for AC-9 wavelengths ≤ 532 nm with a linear least squares fit using all k = 6 training set combinations.

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Figure 3 shows the evolution of the estimates of bb/a and average retrieval errors at 488 nm for k = 1 ...7. As the number of training profiles increases, estimated values of bb/a converge toward their measured values (Fig. 3A, Table 4) with a corresponding decrease in average absolute error |ψ|¯ (Fig. 3B, Table 4). For bb/a, the average retrieval error |ψ|¯ for k ≥ 4 is approximately ±20%, with some retrieval errors |ψi|¯ as high as 100%. The range of relative error ψi¯ for each validation set combination i remains large across all values of k, but the relative errors show little bias when averaged as ψ̄ (Fig. 3C, Table 5). As expected, we can conclude that taking multiple measurements within a vertically homogeneous water column will reduce the average absolute error in the IOP estimates.

Fig. 3 (A) Matchups of estimated and measured bb/a values show convergence towards 1:1 line (black dashed line) as k increases; each colored dot represents a single matchup. (B) Depth-averaged absolute error for each validation set ( |ψi|¯, colored dots) remains variable irrespective of the number of profiles k used to determine F. The overall average absolute error |ψ|¯ for the algorithm (black dashed line) decreases most from k = 1 to 4. (C) Similarly, relative error depth-averaged for each validation set ( ψi¯, colored dots) shows variability but little average bias ψ̄ (black dashed line). (D) Absolute error averaged across all validation set combinations at each depth |ψ(zj)|¯ decreases markedly in deep water.

Table 4. Relative and absolute error (in percent) for estimates of bb/a using different numbers of profiles k in the training set for fitted parameter F

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Table 5. Relative and absolute error (in percent) for estimates of a using Eqs. (41) and (44) using different numbers of profiles k in the training set for fitted parameter F

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The average absolute error at each depth |ψ(zj)|¯ is shown in Fig. 3D and decreases markedly after 25 m; this depth represents 3–5 optical depths for most profiles. This indicates that the radiance from 0–25 m is not necessarily in an asymptotic regime and a solution to the radiative transfer equation governed by the dominant eigenvalue ν0 (Eq. 25) is therefore approximate.

The estimates of absorption at 488 nm and average retrieval errors are shown in Fig. 4 for two different algorithms: (1) absorption aKd(488) estimated from F, bb/a, ν0, and Kd(zas) using Eq. (41) (Fig. 4A–C) and (2) absorption aest (488) estimated from bb/a and measurements of bb(470) and bb(700) using Eq. (44) and the bio-optical model to estimate bb(488) (Fig. 4D–E). As we saw with estimates of bb/a, both absorption estimates improve as k increases, based on improved F and bb/a estimates. The absorption estimate aKd(488) (Fig. 4A–C) requires the estimation of the asymptotic quantity Kd(zas) from Ed data; estimates of Kd (zas) from radiometric profile samples below 25 m assured that we were close to the asymptotic regime, but in some cases IOPs changed below 30 m that challenge this assumption. Applying standard error propagation techniques, we can expect a 20% uncertainty in bb/a and similar 20% uncertainty in Kd (zas) to result in ∼30% uncertainty in aKd; this is close to the average absolute error |ψ|¯ shown in Fig. 4B for k ≥ 4. The use of measured values of backscattering improve the estimation of a(488) to a lower average absolute error |ψ|¯ of 15% for k ≥ 4. The improvements are most evident at lower values of absorption.

Fig. 4 Values of absorption a (A) and average error (B–C) at 488 nm, estimated from F, bb/a, ν0, and Kd (zas), using Eq. (41), and (D–F) estimated from bb/a and measurements of bb(470) and bb(700) using Eq. (44). Colors indicate the number of profiles k used to determine the value of F; remaining profiles are used to estimate bb/a, Kd (zas) and a.

Finally, estimates of backscattering at 488 nm are shown in Fig. 5. While the average absolute error is ∼20% (Fig. 5B), large individual errors can be seen in groups of matchups (Fig. 5A). Large overestimates in bb(488) are the result of overestimates in both bb/a and aKd; these cases are found in profiles taken near the bloom peak where both higher values of absorption and backscattering were common.

Fig. 5 Values of backscattering bb (A) and average error (B) and (C) at 488 nm, estimated from bb/a, F, and Kd (zas). Colors indicate the number of profiles k used to determine the value of F; remaining profiles are used to estimate bb/a, Kd (zas) and a.

The estimates of average error for any given validation set |ψi|¯ are roughly twice that of the average for all validation sets |ψ|¯. For example, the estimate of absorption a(488) has an average uncertainty of |ψ|¯=1516% for k ≥ 4 (Table 5), with uncertainty for individual validation sets |ψi|¯ exhibiting as much as 36% error (Fig. 4B). This information can be used to create a subspace of absorption values ranging from 0.64ā to 1.36ā, where ā represents the water column average absorption estimate. This water column average could be used as a starting point and the subspace values as bound constraints for a more sophisticated multi-parameter nonlinear inversion algorithm [11

11. E. Rehm, “Inverting light with constraints,” Poster session presented at Ocean Optics XIX, Barga, Italy (2008).

].

The validation results for other wavelengths below 555 nm are consistent with those at 488 nm (Tables 4, 5, 6): (1) little or no average bias ψ̄ is evident in relative errors; (2) the average absolute error |ψ|¯ in IOP estimates decreases as k increases, i.e., as more training profiles are used to estimate the fitted parameter F; (3) while parameterization of F as a function of bb/a (Eq. 49) varies with wavelength (Fig. 2B), there is no discernible wavelength trend in IOP errors for wavelengths less than 555 nm; and (4) retrievals of absorption a nearly always improve when measurements of bb are available, even when a bio-optical model is used to estimate bb(λ) at wavelengths other than the measurement wavelengths (470 and 700 nm) (Table 5).

Table 6. Relative and absolute error (in percent) for estimates of bb using different numbers of profiles k in the training set for fitted parameter F

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The 555 nm retrieval of bb/a is poor even when most of the profiles are used as training data (Table 4). This is due to Raman scattering which our algorithm does not incorporate. The slope of fitted parameter F (Eq. 49) does vary with wavelength (Fig. 2, Table 3). At wavelengths below 555 nm, we suspect F can help compensate for some Raman scattering at these wavelengths, while at 555 nm our assumption of no inelastic scattering in the formulation of the radiative transfer equation (Eq. 1) breaks down.

7. Discussion

We have developed a simple analytical algorithm that, when tuned using contemporaneous IOP and radiometric measurements, produces an IOP subspace defined by a mean IOP estimate and its associated uncertainty. Following the suggestion of Defoin-Platel and Chami [34

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

], the resulting IOP subspace can be used as a divide and conquer strategy to reduce or eliminate ambiguities of IOPs estimated from a wider set of radiometric measurements in the same waters where IOP measurements are limited or not available. For example, the predicted IOP subspace can be used to constrain a more sophisticated inversion algorithm that incorporates, for example, models of Raman scattering and chlorophyll fluorescence and/or more accurate scattering phase functions as well as computationally efficient radiative transfer models (e.g., Ecolight-S [35

35. C. D. Mobley, “Fast light calculations for ocean ecosystem and inverse models,” Opt. Express, submitted (2011). [CrossRef] [PubMed]

]) that do not require the assumptions of an asymptotic light regime employed in the algorithms presented here.

Our single-parameter bb/a algorithm converges quickly and reliably; no Monte Carlo simulations [36

36. H. R. Gordon and O. B. Brown, “irradiance reflectivity of a flat ocean as a function of its optical properties,” Appl. Opt. 12, 1549–1551 (1973). [CrossRef] [PubMed]

] or iterative solutions with forward problem solvers [3

3. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

,37

37. G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University Press, 1999). [CrossRef]

] are required. The algorithm works best when light field measurements are made in the asymptotic regime and the parameter F is tuned to the local environment (Fig. 3). With simulations, we have shown that the bb/a algorithm’s performance degrades as the scattering phase function of the radiometric measurements deviates from that used to generate the data (Table 2). Consistent with these results, we found that applying the parameterization of F developed for our synthetic data produced very poor estimates when applied to the NAB08 field data set. For the NAB08 calibration campaign, numerous radiometric and IOP profiles were available that adequately spanned the variability of waters. Estimating F directly from this calibration campaign data (Fig. 2) effectively tunes our simple scattering phase function β̃ (Eq. 7) to the North Atlantic waters under study. While we did not develop an algorithm to directly estimate the number of profiles k required for such a calibration campaign, the results presented here using the bio-optical data set (eight calibration profiles) from the NAB08 process cruise can be used as a guide.

If simple IOP measurements (e.g., bb(470) and bb(700)) can be made at the time of radiometric measurements, we can improve the mean absorption estimate and decrease its overall uncertainty, further reducing potential ambiguity and increasing the performance of more computationally intensive IOP estimation approaches that benefit from reduced bound constraints. This use of ancillary IOP data is an example of an “enrichment strategy” suggested by Defoin-Platel and Chami [34

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

].

The results presented here do not account for instrument error. If there is an expected uncertainty of ±17% in backscattering [38

38. G. DallOlmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6, 947–967 (2009). [CrossRef]

] and ±17% in absorption, as mentioned previously, this results in a relative instrumental error of 34% in bb/a, which is within the range of the average absolute errors shown in Table 4. Similarly, the resulting estimates for absorption and backscattering (Tables 5 and 6, respectively) are also comparable to the instrument error.

One might wonder, based on the assumptions introduced and the simplicity of the algorithm for bb/a, why it works as well as it does. One interpretation is that the algorithm is a “bilinear inverse method” that weights the measured data for rrs(zm) by the “importance” of the photons as given by the adjoint radiance L(zm, μ) ≡ L(zm, −μ) (e.g., Eq. 24) of the radiative transfer equation [39

39. J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

, 40

40. G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand-Reinhold, 1970).

]. Radiance, considered from the point of view of forward radiative transfer methods, conceptually traces every photon from the source to detector. Adjoint radiance can be visualized as tracing pseudo photons received by the detector backwards to the source [41

41. Q. Min and L. C. Harrison , “An adjoint formulation of the radiative transfer method,” J. Geophys. Res. 101, 1635–1640 (1996). [CrossRef]

]; these pseudo photons provide insight to the importance of “how much a given position-direction matters for a given radiometric observation” [42

42. A. B. Davis and Y. Knyazikhin, “A Primer in Three-Dimensional Radiative Transfer,” in Three-Dimensional Radiative Transfer in the Cloudy Atmosphere, A. Davis and A. Marshak, eds. (Springer-Verlag, 2005), pp. 153–242. [CrossRef]

].

As the ratio of two inherent optical properties (IOPs), bb/a is itself an inherent optical property. Particulate backscattering bbp has been shown to correlate to particulate organic carbon [38

38. G. DallOlmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6, 947–967 (2009). [CrossRef]

, 43

43. H. Loisel, J. Nicolas, P. Deschamps, and R. Frouin, “Seasonal and inter-annual variability of particulate organic matter in the global ocean,” Geophys. Res. Lett. 29(24), 2196 (2002). [CrossRef]

, 44

44. D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008). [CrossRef]

]; several bio-optical models relate chlorophyll concentration to particulate absorption [45

45. L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26(4), 671–689 (1981). [CrossRef]

, 46

46. A. Morel, “Light and marine photosynthesis: a spectral model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991). [CrossRef]

]. The ratio bb/a might, therefore, be close to a proxy for a carbon-to-chlorophyll ratio [47

47. M. J. Behrenfeld and E. Boss, “The beam attenuation to chlorophyll ratio: an optical index of phytoplankton physiology in the surface ocean?,” Deep-Sea Res. I 50, 1537–1549 (2003).

], and thus be a diagnostic IOP for plankton community structure. Given the ease of estimating bb/a from in-water measurements of Lu and Ed, this IOP deserves further investigation as a bio-optical proxy in the open ocean.

Acknowledgments

This work was supported by the National Science Foundation grants OCE0628379 and OCE0628107, and NASA grants NNX08AL92G and NNX-10AP29H. This work would not have been possible without the help of Emily Kallin, who assisted with the collection of all of the NAB08 bio-optical profile data used here and Mary Jane Perry and Emmanuel Boss who loaned their bio-optical and Satlantic radiometric profilers. We wish to thank Eric D’Asaro for statistical insight, Curtis Mobley for guidance with radiative transfer simulations, and the crew and technicians of the R.V. Knorr.

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H. Loisel, J. Nicolas, P. Deschamps, and R. Frouin, “Seasonal and inter-annual variability of particulate organic matter in the global ocean,” Geophys. Res. Lett. 29(24), 2196 (2002). [CrossRef]

44.

D. Stramski, R. A. Reynolds, M. Babin, S. Kaczmarek, M. R. Lewis, R. Röttgers, A. Sciandra, M. Stramska, M. S. Twardowski, B. A. Franz, and H. Claustre, “Relationships between the surface concentration of particulate organic carbon and optical properties in the eastern South Pacific and eastern Atlantic Oceans,” Biogeosciences 5, 171–201 (2008). [CrossRef]

45.

L. Prieur and S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific spectral absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26(4), 671–689 (1981). [CrossRef]

46.

A. Morel, “Light and marine photosynthesis: a spectral model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991). [CrossRef]

47.

M. J. Behrenfeld and E. Boss, “The beam attenuation to chlorophyll ratio: an optical index of phytoplankton physiology in the surface ocean?,” Deep-Sea Res. I 50, 1537–1549 (2003).

OCIS Codes
(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(010.7340) Atmospheric and oceanic optics : Water
(030.5620) Coherence and statistical optics : Radiative transfer
(100.3190) Image processing : Inverse problems
(160.4760) Materials : Optical properties
(290.4210) Scattering : Multiple scattering

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: July 27, 2011
Revised Manuscript: October 19, 2011
Manuscript Accepted: November 10, 2011
Published: November 22, 2011

Virtual Issues
Vol. 7, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Eric Rehm and Norman J. McCormick, "Inherent optical property estimation in deep waters," Opt. Express 19, 24986-25005 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-24986


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