## Fast light calculations for ocean ecosystem and inverse models |

Optics Express, Vol. 19, Issue 20, pp. 18927-18944 (2011)

http://dx.doi.org/10.1364/OE.19.018927

Acrobat PDF (1016 KB)

### Abstract

Ocean physical-biological-optical ecosystem models can require light calculations at thousands of grid points and time steps. Implicit inverse models that recover ocean absorption and scattering properties from measured light variables can require thousands of solutions of the radiative transfer equation. An extremely fast radiative transfer code, EcoLight-S(ubroutine), has been developed to address these needs. EcoLight-S requires less than one second on a moderately fast computer to compute spectral irradiances over near-ultraviolet to near-infrared wavelengths with errors in the photosyntheically available radiation (PAR) of no more than ten percent throughout the euphotic zone. It is thus possible to replace simple and often inaccurate analytical PAR or spectral irradiance models with more accurate radiative transfer calculations, with very little computational penalty. EcoLight-S is applicable to Case 2 and optically shallow waters for which no analytical light models exist. EcoLight-S also computes upwelling and downwelling plane irradiances, nadir and zenith radiances, and the remote-sensing reflectance. These quantities allow ecosystem predictions to be validated with optical measurements obtained from in-water instruments or remotely sensed imagery.

© 2011 OSA

## 1. Introduction

1. C. D. Mobley, *Light and Water: Radiative Transfer in Natural Waters* (Academic, 1994), http://www.curtismobley.com/lightandwater.zip.

3. C. D. Mobley and L. K. Sundman, *HydroLight Technical Documentation* (Sequoia Scientific, Inc., 2008), http://www.hydrolight.info.

*L*(

*z*,

*θ*,

*ϕ*,

*λ*) as functions of depth

*z*, polar

*θ*and azimuthal

*ϕ*directions, and wavelength

*λ*for any plane-parallel water body. It therefore meets the need for generality in ecosystem and implicit inverse models. However, HydroLight is used primarily as a research tool and its emphasis is on accuracy of the RTE solution, with run time being of secondary importance. Moreover, it computes the full angular distribution of the radiance. It is therefore computationally much too expensive to be practicable for ecosystem models that require light calculations at thousands of grid points and time steps or for inverse models that require hundreds of RTE solutions to achieve convergence.

4. C. D. Mobley, *EcoLight-S 1.0 User’s Guide and Technical Documentation* (Sequoia Scientific, Inc., 2011), http://www.hydrolight.info. [PubMed]

## 2. RTE formulation

*c*is the beam attenuation coefficient (the sum of the absorption and scattering coefficients);

*β*is the volume scattering function (VSF), which describes how radiance is scattered from direction (

*θ*′,

*ϕ*′) into direction (

*θ*,

*ϕ*); and

*S*is a source function representing the contributions of inelastic scattering at wavelength

*λ*. Depth

*z*is measured positive downward from the mean sea surface,

*θ*= 0 corresponds to light heading straight down, and

*ϕ*= 0 can be chosen for convenience, e.g. in the sun’s azimuthal direction.

*L*(

*z*,

*θ*,

*ϕ*,

*λ*) obtained from Eq. (1) gives far more information than is needed by ecosystem models. The spectral total scalar irradiance is the quantity needed for calculations of photosynthesis and heating of the water. Indeed, most ecosystem biological models parameterize the light in terms of PAR, The

*λ*/

*hc*factor, where

*h*is Planck’s constant and

*c*is the speed of light, converts the energy units of

*E*

_{o}(W m

^{−2}nm

^{−1}) to quantum units (photons m

^{−2}s

^{−1}) as needed for photosynthesis. Water heating rates by short-wave radiation are usually computed by where

*T*is the temperature,

*t*is time,

*ρ*is the density of sea water,

*c*is the specific heat of sea water, and

_{p}*Ē*and

_{d}*Ē*are respectively the downwelling and upwelling plane irradiances integrated over visible and near-infrared wavelengths, typically 400 to 1000 nm. These irradiances are computed from the radiance by with a similar equation over the upward directions for

_{u}*Ē*.

_{u}*ϕ*, as seen in Eqs. (2) and (5). The azimuthal dependence of the radiance, obtained at considerable computational expense in HydroLight, is thus lost when computing irradiances. In most waters Raman scatter by water and fluorescence by chlorophyll and colored dissolved organic matter (CDOM) have little effect on PAR (see Table 2 below and the discussion thereof). Including these inelastic scatter processes requires additional inputs and significantly increases run times. It is therefore reasonable to solve an azimuthally integrated, source-free version of the RTE, where is the azimuthally integrated VSF. Equation (6) yields the azimuthally integrated radiance, which then can be integrated over polar angle

*θ*to obtain the irradiances. Equation (6) is the form of the RTE solved by EcoLight-S (and also by the standard version of EcoLight that is included with HydroLight, except that the standard EcoLight retains the source term for inelastic effects). The azimuthally integrated RTE yields the same nadir and zenith radiances as HydroLight because in both cases the radiances for

*θ*= 0 and

*π*have no azimuthal dependence. EcoLight-S thus computes the same nadir-viewing water-leaving radiance, remote-sensing reflectance, and in-water zenith and nadir radiances as HydroLight. These are the radiometric quantities most often used, along with the plane irradiances, in inverse models.

*θ*,

*ϕ*) directions is discretized into

*M*

*θ*bins and

*N*

*ϕ*bins. The run time is proportional to (

*MN*)

^{2}because radiance can be scattered from any one bin into any other. Removing the azimuthal dependence thus reduces the run time by a factor of

*N*

^{2}. Compared to the standard HydroLight angular resolution of 15 deg in

*ϕ*, i.e.

*N*= 24, this is a reduction of (24)

^{2}in run time. The mathematical details of directional discretization are given in [1

1. C. D. Mobley, *Light and Water: Radiative Transfer in Natural Waters* (Academic, 1994), http://www.curtismobley.com/lightandwater.zip.

4. C. D. Mobley, *EcoLight-S 1.0 User’s Guide and Technical Documentation* (Sequoia Scientific, Inc., 2011), http://www.hydrolight.info. [PubMed]

### 2.1. Depth specification

1. C. D. Mobley, *Light and Water: Radiative Transfer in Natural Waters* (Academic, 1994), http://www.curtismobley.com/lightandwater.zip.

### 2.2. IOP specification

5. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. Jaffe, ed., Proc. SPIE2258 (with corrections), 194–201 (1994). [CrossRef]

6. C. D. Mobley, L. K. Sundaman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. **41**(6), 1036–1050 (2002). [CrossRef]

6. C. D. Mobley, L. K. Sundaman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. **41**(6), 1036–1050 (2002). [CrossRef]

7. W. Freda and J. Piskozub, “Improved method of Fournier-Forand marine phase function parameterization,” Opt. Express **15**(20), 12763–21768 (2007). [CrossRef] [PubMed]

## 3. RTE solution options

*E*

_{o}(

*z*,

*λ*) has been computed to some depth

*z*

_{o}deep enough to be free of surface boundary effects, it is not necessary to continue solving the RTE to greater depths, which is computationally expensive. As shown below, in many cases of practical interest it is possible to extrapolate the accurately computed upper-water-column irradiances to greater depths and still obtain irradiances that are acceptably accurate for ecosystem predictions. Likewise, it may not be necessary to solve the RTE at every wavelength used by a particular ecosystem model in order to obtain acceptably accurate irradiances at the needed wavelength resolution. In models requiring high wavelength resolution (e.g., the EcoSim biological model [8

8. W. P. Bissett, J. J. Walsh, D. A. Dieterle, and K. L. Carder, “Carbon cycling in the upper waters of the Sargasso Sea: I. numerical simulation of differential carbon and nitrogen fluxes,” Deep-Sea Res. **46**, 205–269 (1999a). [CrossRef]

9. W. P. Bissett, K. L. Carder, J. J. Walsh, and D. A. Dieterle, “Carbon cycling in the upper waters of the Sargasso Sea: II. numerical simulation of apparent and inherent optical properties,” Deep-Sea Res. **46**, 271–317 (1999b). [CrossRef]

*E*

_{o}(

*z*,

*λ*) at 5 nm resolution from 400 to 700 nm.), omitting every other wavelength cuts the EcoLight-S run time by roughly one half. Models using only PAR can solve the RTE at relatively few wavelengths and still obtain PAR profiles accurate to within a few percent.

### 3.1. Dynamic depth solutions

*z*

_{o}to which the RTE will be solved at a particular wavelength, note that the scalar irradiance can be written as This equation defines

*K*

_{o}, the diffuse attenuation coefficient for scalar irradiance. Note that

*K*

_{o}is not known until after the RTE has been solved. However, except very near the sea surface where boundary effects are important,

*K*

_{o}is approximately equal to

*K*

_{d}, the diffuse attenuation coefficient for downwelling plane irradiance (

*K*

_{o}becomes exactly equal to

*K*

_{d}at great depths in homogeneous water). To first order (e.g., Ref [1

*Light and Water: Radiative Transfer in Natural Waters* (Academic, 1994), http://www.curtismobley.com/lightandwater.zip.

*K*

_{d}≈

*a*/

μ ¯

_{d}, where

*a*is the absorption coefficient and

μ ¯

_{d}is the mean cosine of the downwelling radiance distribution. In typical waters,

μ ¯

_{d}≈ 3/4. Thus, in Eq. (7)

*K*

_{o}can be roughly approximated by the absorption coefficient

*a*(

*z*,

*λ*). Equation (7) can then be rewritten as Here

*F*

_{o}is the fraction of the surface scalar irradiance that penetrates to depth

*z*

_{o}at the given wavelength.

*F*

_{o}, 0 ≤

*F*

_{o}≤ 1, as one of the inputs to EcoLight-S. Equation (8) is then solved for the depth

*z*

_{o}corresponding to the chosen

*F*

_{o}and the known absorption coefficient. In practice, because of the use of homogeneous layers, this amounts to recursively computing

*f*

_{o}= 1 and then

*f*=

_{k}*f*

_{k−1}exp[−

*a*(

*k*,

*λ*) Δ

*z*],

_{k}*k*= 1, 2, 3,... until

*f*<

_{k}*F*

_{o}. Here

*a*(

*k*,

*λ*) is the absorption coefficient for depth layer

*k*, which has thickness Δ

*z*. The next deeper layer mid-point or boundary depth is then taken to be

_{k}*z*

_{o}. Thus the RTE is always solved to a grid output depth greater than or equal to the actual depth corresponding to

*F*

_{o}and the IOPs. Moreover, this estimate of

*z*

_{o}will always be greater than the actual

*z*

_{o}value because the mean cosine factor is omitted. Omitting the mean cosine is equivalent to having too little absorption; hence the solution will go too deep. After the RTE is solved at the first wavelength, the actual value of

*z*

_{o}can be obtained from the computed irradiance. This value is then used to adjust the estimated

*z*

_{o}at the next wavelength, and so on for subsequent wavelenths. After the first wavelength, which always goes too deep, this algorithm results in

*z*

_{o}estimates that are very close to the actual

*F*

_{o}depths (Fig. 1 below).

*F*

_{o}= 0.1 would result in solving the RTE to the 10% irradiance level at each wavelength, i.e., to the depth where the irradiance has decreased to 0.1 or 10% of its value at the sea surface. There is also an option in the code to let the user-selected value of

*F*

_{o}be the spectral scalar irradiance

*E*

_{o}(

*z*

_{o},

*λ*) in W m

^{−2}nm

^{−1}. With that option, using

*F*

_{o}= 0.1 would result in solving the RTE to a depth where the irradiance has decreased to 0.1 W m

^{−2}nm

^{−1}, regardless of the surface value. That option is more reasonable for calculations of photosynthesis because it is irradiance magnitudes, not percentages, that determine photosynthesis. Regardless of which option is used to estimate depth

*z*

_{o}, the bottom boundary condition is then applied at the next layer midpoint or boundary depth

*z*below

_{k}*z*

_{o}, and the RTE is solved only between the surface and

*z*. The resulting radiances and derived quantities are then accurate down to depth

_{k}*z*.

_{k}### 3.2. Extrapolation below the solution depth

*z*at a particular wavelength, the computed values of

_{k}*E*

_{o}(

*z*) and

_{k}*E*

_{d}(

*z*) are extrapolated to greater depths as follows. The extrapolation is based on Eq. (7), except that the mean cosine factors can now be included in the approximations for

_{k}*K*

_{d}≈

*a*/

μ ¯

_{d}and

*K*

_{o}≈

*a*/

*. In addition to*μ ¯

*E*

_{o}(

*z*) and

_{k}*E*

_{d}(

*z*), the solution of the RTE gives all irradiances, in particular the downwelling scalar irradiance

_{k}*E*

_{od}(

*z*) and the upwelling plane irradiance

_{k}*E*

_{u}(

*z*). These irradiances are used to compute the mean cosines at the last solved depth. These values of

_{k}μ ¯

_{d}(

*z*) and

_{k}*(*μ ¯

*z*) are then used at all lower depths. Thus Eq. (7) becomes A similar equation using

_{k}μ ¯

_{d}(

*z*) holds for

_{k}*E*

_{d}. Equation (10) is then applied (in layer summation form) to the homogeneous layers, beginning at the last computed depth

*z*and extending to the maximum depth. The upwelling quantities

_{k}*E*

_{u}and

*L*

_{u}are not extrapolated below the maximum solution depth because they can be strongly influenced by bottom reflectance in shallow waters. In simulations with optically shallow bottoms the maximum solution depth determined by the absorption coefficient is usually deeper at visible wavelengths than the actual bottom depth, in which case all quantities are accurately computed down to the physical bottom.

*E*

_{o}(

*z*,

_{k}*λ*) and the other irradiances incorporate all of the effects of the surface boundary and of the water IOPs above the maximum depth

*z*to which the RTE was solved. The extrapolations based on Eqs. (9) and (10) will be reasonably accurate if the variability in the mean cosines is not great below depth

_{k}*z*and if the IOPs covary with the absorption coefficient. This is often a good approximation, but might not be the case, for example, if there were a layer of highly scattering but non-absorbing particles below depth

_{k}*z*. In such a case, it would be necessary to solve the RTE to a depth deeper than the scattering layer if high accuracy is required for the computed irradiances below the scattering layer.

_{k}### 3.3. Wavelength skipping

8. W. P. Bissett, J. J. Walsh, D. A. Dieterle, and K. L. Carder, “Carbon cycling in the upper waters of the Sargasso Sea: I. numerical simulation of differential carbon and nitrogen fluxes,” Deep-Sea Res. **46**, 205–269 (1999a). [CrossRef]

9. W. P. Bissett, K. L. Carder, J. J. Walsh, and D. A. Dieterle, “Carbon cycling in the upper waters of the Sargasso Sea: II. numerical simulation of apparent and inherent optical properties,” Deep-Sea Res. **46**, 271–317 (1999b). [CrossRef]

*nwskip*, between RTE solutions. The RTE is always solved at the first and last user wavelengths. Thus if the user’s input wavelengths (where the IOPs and other inputs are defined) are 400 to 700 nm by 10 nm, and

*nwskip*= 0, the RTE will be solved for 10 nm wide bands centered at 405, 415, ...685, 695 nm. If

*nwskip*= 1, the RTE will be solved at 405, 425, ..., 665, 685, and 695 nm. Values at 415, 435, ..., 675 nm will be computed by linear interpolation at each depth between the computed values. If

*nwskip*= 2, the RTE will be solved at 405, 435, ..., 665, and 695 nm, with values at 415 and 425 being obtained by interpolation between the computed 405 and 435 nm values, and so on.

### 3.4. Other optimizations

*F*

_{o}or what wavelength resolution are adequate because different applications have different accuracy requirements, and the acceptable optimizations for a given accuracy may depend on water IOPs, bottom conditions, and the like. Some experimentation may be necessary to determine the acceptable amount of depth and wavelength optimization for a given application, before production runs are begun.

*K*functions and reflectances using the IOPs at the deepest input depth. However, those computations add to the run time and can be omitted if the asymptotic values are not of interest. Likewise, the computation of PAR can be omitted if the user’s application does not require PAR.

## 4. Example simulations

*Chl*= 0.5 mg m

^{−3}plus a gaussian profile with its maximum value of 2.0 mg m

^{−3}at 15 m depth. This continuous

*Chl*(

*z*) profile is shown by the red line in the left panel of Fig. 1. The EcoLight-S water column was modeled as 5 m thick homogeneous layers, which is typical of the depth resolution used in ocean ecosystem models. The layer chlorophyll values are the average of the continuous profile within each layer. The layer chlorophyll values were converted to layer absorption, scatter, and backscatter coefficients using a bio-optical model [10

10. C. D. Mobley, “A new IOP model for Case 1 water,” in Ocean Optics Web Book, http://www.oceanopticsbook.info/view/optical_constituents_of_the_ocean/__level_2/a_new_iop_model_for_case_1_water.

^{−1}. The bottom boundary condition was for infinitely deep water below 50 m. The dynamic depth option was selected with

*F*

_{o}= 0.1, interpreted as a percent of the surface irradiance at each wavelength. The greatest depth requested for output was 50 m. The solution wavelengths were 5 nm bands from 400 to 700 nm; no wavelength skipping was done.

*F*

_{o}= 0.1 or 10% irradiance level at each wavelength. Note that for the first wavelength band (400-405 nm, plotted at 402.5 nm) the RTE was solved considerably deeper than necessary. The estimated solution depths at subsequent wavelengths were only slightly deeper than the actual 10% irradiance depths as determined after the RTE was solved. The five-meter layer thicknesses make it easy to see (from the red dots) that the RTE was always solved to either a layer boundary or layer midpoint depth (2.5 m increments), where the output was saved. The main computational savings in this run come at wavelengths greater than 600 nm, where the optical depth of the bottom at 50 m increases from 30 (at 600 nm) to 48 (at 700 nm), but the solution optical depths decrease from 8 to 4. The corresponding run without dynamic depths, for which the RTE was solved to 50 m at each wavelength, took 1.42 s, but the run with dynamic depths and

*F*

_{o}= 0.1 took only 0.53 s. The two PAR profiles agreed to within 0.4% at all depths down to 50 m.

*F*

_{o}values and no wavelength skipping. The inset labels in the second panel from the left show the values of

*F*

_{o}, the wavelength resolution (5 nm in all cases, corresponding to

*nwskip*= 0), and the run times in seconds. The label colors correspond to the curve colors in all panels. The third panel shows the relative error in percent, of the optimized PAR profiles compared to the unoptimized profile, and the last panel shows the

*actual error*=

*PAR*(optimized) –

*PAR*(unoptimized). The colored dots on the PAR profiles in the second panel show the maximum depth to which the RTE was solved for the various

*F*

_{o}values, after the initial solution. These runs show for example that, for this chlorophyll profile, the RTE can be solved to only the 20% irradiance depth at each wavelength (green curves) with resulting errors in PAR of less than 3% or 1

*μ*mol photons m

^{−2}s

^{−1}down to 50 m, which is roughly the depth at which PAR has decreased to 1% of its surface value. Solving the RTE to just the 50% irradiance depth (red curves), which is above the depth of the cholorphyll maximum, still gives PAR values within 9% or 7

*μ*mol photons m

^{−2}s

^{−1}. The optimized run times are all less than 1 second.

*μ*mol photons m

^{−2}s

^{−1}for microplankton. The common rule of thumb for the depth of the euphotic zone is the depth where PAR has decreased to 1% of its mid-day surface value, which is about 50 m for the water properties of Fig. 2. It is reasonable to assume that errors in PAR below the depth where PAR is roughly 10

*μ*mol photons m

^{−2}s

^{−1}will not greatly affect primary production calculations in ecosystem models. In the simulation of Fig. 2 that depth is about 60 m.

*λ*= 10 (

*nwskip*= 1) means that the RTE was solved at every other one of the 5 nm bands;

*λ*= 50 (

*nwskip*= 9) corresponds to RTE solutions only at every tenth band, centered at 402.5, 452.5, ..., 697.5 nm. Even for just 50 nm resolution, PAR is computed within 3% of the unoptimized 5 nm values. The largest magnitude errors are now near the sea surface, with the largest error of 33

*μ*mol photons m

^{−2}s

^{−1}(2885 vs. 2852

*μ*mol photons m

^{−2}s

^{−1}) being at the surface for the simulation at 20 nm resolution.

*nwskip*= 3 solves the RTE at 20 nm resolution. Strictly speaking this is not the same radiative transfer problem as using a 10 nm resolution and skipping one wavelength between solutions, or using a 20 nm resolution with no wavelength skipping. However, the results and run times for these cases are very similar. For these three cases and the current chlorophyll profile, the PAR profiles are the same to within ±2% and the run times are the same to within a few hundredths of a second.

*F*

_{o}and wavelength resolution for the same conditions as Fig. 1. Solving the RTE to only the 50% irradiance level at 25 nm resolution still gives PAR errors of less than 10% down to 50 m with a run time of only 0.05 s. The less-optimized runs all have PAR errors of less than 4% and run times of 0.2 s or less.

*E*

_{o}(

*z*,

*λ*) for the unoptimized solution of Fig. 4, and for the optimized solution with

*F*

_{o}= 0.2 and 25 nm resolution (

*nwskip*= 4), which is the orange curve of that figure. The black squares in the upper right panel show the wavelengths and

*F*

_{o}depths to which the RTE was solved in the optimized run. Visual comparison of the contour plots of the upper panels of Fig. 5 show that the unoptimized and optimized irradiances are similar, although the effects of interpolation between the solved wavelengths are apparent. In particular, the scalloped appearance of the contours in the upper right panel at red wavelengths and large depths occurs because the linear interpolation between the solution wavelengths at a given depth does not accurately describe the non-linear change with wavelength in

*E*

_{o}(

*z*,

*λ*), which results from the rapid increase in water absorption with wavelength. The lower left panel shows the relative errors in percent computed as in Eq. (11). Most of the water column between 400 and 600 nm has errors of less than ±10% (blue colors), but the relative errors are large for the unsolved wavelengths in the red spectral region at large depths (red colors). These large relative errors occur only where the irradiances are several order of magnitude less than the maximum irradiances at the same depth. The small irradiances with the large relative errors make little contribution to the PAR value. The lower right panel shows that the errors in the irradiance magnitudes themselves are usually less than ±0.02, and almost always less than ±0.05 W m

^{−2}nm

^{−1}.

*relative to the unoptimized values*are determined mostly by the solution depth and wavelength optimizations. What is not easily seen in the scale of the

*PAR*(

*z*) profiles of those figures is that the PAR profiles depend on the depth resolution of the IOPs in regions where the water is inhomogeneous. Table 1 shows the values of PAR at 15 m, the depth of the chlorophyll maximum, for the unoptimized and optimized red curves, which have

*F*

_{o}= 0.5 and

*λ*= 25 (

*nwskip*= 4), of Figs. 4 and 6. These values illustrate that for either depth resolution the relative and actual errors are almost the same, but that the PAR values are different for the different layer thicknesses. For this example, the PAR values are about 8% less for the 5 m layer thickness than for the 1 m layers, and these differences are almost the same for the unoptimized and optimized solutions.

*z*and run time. These runs all gave the same PAR profiles to within 1% near the surface and below 25 m. However, near the chlorophyll maximum the difference between the high-resolution (Δ

*z*= 0.5 m) and the low-resolution (5 m) cases was as much as 11% or 65

*μ*mol photons m

^{−2}s

^{−1}. The unoptimized run times increased from 1.42 s for the 5 m layers to 2.09 s for the 0.5 m layers. The output files for 0.5 m layers are ten times as large as the files for 5 m layers because output is returned at all layer boundaries and mid-points. Corresponding runs optimized with

*F*

_{o}= 0.2 and 25 nm wavelength resolution (as shown in Table 1) are the same as the unoptimized runs to within 2% down to 30 m, and differ by less than 4% at 50 m. The optimized run times were 0.09 s (5 m layers) to 0.16 s (0.5 m layers). It should be remembered that different depth resolutions of the continuous profile are different radiative transfer problems even though the column-integrated chlorophyll values are the same. Thus the differences seen in Table 1 and Fig. 7 are not errors in the same sense as those due to solution depth and wavelength optimizations, they simply show that different IOP profiles have different PAR profiles.

^{−3}, CDOM absorption of

*a*

_{CDOM}(440) = 0.2 m

^{−1}, and “brown earth” minerals with a concentration of 1 gm m

^{−3}. These values were converted to IOPs using HydroLight’s generic bio-geo-optical model for Case 2 water. The backscatter fraction by phytoplankton was taken to be 0.01 at all wavelengths. The backscatter fraction of the minerals was wavelength dependent and given by 0.03(550/

*λ*)

^{0.5}. These concentrations and spectral functions give roughly equal contributions to absorption by the phytoplankton and minerals, with CDOM absorption being greater than either at blue wavelengths. Scattering by phytoplankton and minerals is roughly equal, but the backscatter coefficient for minerals is 2-3 times that of the phytoplankton. CDOM was assumed to be nonscattering. The sun was placed at the zenith in a clear sky; the resulting above-surface PAR value was 2882

*μ*mol photons m

^{−2}s

^{−1}. The sea surface was level. The HydroLight and EcoLight runs were made from 350–700 nm with 10 nm bandwidths so that inelastic scatter effects (Raman scatter by the water and chlorophyll and CDOM fluorescence) from below 400 nm would be accounted for in the PAR wavelengths of 400-700 nm. The EcoLight-S unoptimized runs were from 400 to 700 nm by 10 nm. The pure water runs were made to a depth of 400 m, and the Case 2 runs to a depth of 20 m.

*F*

_{o}from 0.1 to 0.2 (solving the RTE to a shallower depth) for a given wavelength resolution has little effect on the PAR difference at 400 m, but decreasing the wavelength resolution from 10 to 20 nm for a given

*F*

_{o}significantly increases the error. However, the oppose occurs with the Case 2 water: changing the wavelength resolution from 10 to 20 nm has little effect, but changing

*F*

_{o}from 0.1 to 0.2 triples the error at 20 m. However, regardless of which EcoLight-S optimization is used, these runs show that even in extreme cases of pure water or very turbid water, it is possible to compute PAR values to within roughly 10% in a few tenths of a second of computer time. The optimized EcoLight-S runs are usually more than 1,000 times faster than the HydroLight run with inelastic effects.

*R*

_{rs}. Figure 8 shows the

*R*

_{rs}spectra for the pure and Case 2 water simulations of Table 2. The solid lines are the spectra computed by HydroLight 5.1 including inelastic effects. The red open circles are the unoptimized EcoLight-S values, and the black dots are the EcoLight-S values with

*F*

_{o}= 0.2 and 10 nm resolution. The EcoLight-S values are always less than the HydroLight values because of the omission of inelastic effects, but the difference is noticeable only in extreme cases such as the pure water simulation, for which Raman scatter makes its maximum contribution to the water-leaving radiance. The contributions of Raman scatter and chlorophyll and CDOM fluorescence to

*R*

_{rs}are insignificant in the Case 2 water case except in the chlorophyll fluorescence band at 685 nm where there is a small fluorescence peak (the HydroLight runs assumed a 2% quantum efficiency for chlorophyll fluorescence). The optimized and unoptimized EcoLight-S runs are indistinguishable in this figure because

*R*

_{rs}is determined by the near-surface light field, which is being accurately computed in all cases, regardless of the effect of the

*F*

_{o}value on the light field at depth.

## 5. Discussion

11. C. D. Mobley, L. K. Sundman, W. P. Bissett, and B. Cahill, “Fast and accurate irradiance calculations for ecosystem models,” Biogeosci. Discuss. **6**, 10625–10662 (2009), http://www.biogeosciences-discuss.net/6/10625/2009/bgd-6-10625-2009.pdf. [CrossRef]

12. A. F. Shchepetkin and J. C. McWilliams, “The regional ocean modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model,” Ocean Model. **9**, 347–404 (2005), https://www.myroms.org/wiki/index.php/Documentation_Portal. [CrossRef]

8. W. P. Bissett, J. J. Walsh, D. A. Dieterle, and K. L. Carder, “Carbon cycling in the upper waters of the Sargasso Sea: I. numerical simulation of differential carbon and nitrogen fluxes,” Deep-Sea Res. **46**, 205–269 (1999a). [CrossRef]

9. W. P. Bissett, K. L. Carder, J. J. Walsh, and D. A. Dieterle, “Carbon cycling in the upper waters of the Sargasso Sea: II. numerical simulation of apparent and inherent optical properties,” Deep-Sea Res. **46**, 271–317 (1999b). [CrossRef]

*E*

_{o}(

*z*,

*λ*) at the 5 nm resolution needed by EcoSim with less than a 30% increase in the total simulation time. EcoLight-S is currently being used as the optical component of a coupled ROMS-CoSiNE (Carbon Silicon Nitrogen Ecosystem [13, 14

14. M. Fujii, E. Boss, and F. Chai, “The value of adding optics to ecosystem models: a case study,” Biogeosciences **4**, 817–835 (2007), http://www.biogeosciences.net/4/817/2007/. [CrossRef]

16. E. Rehm, C. D. Mobley, and J. Smart, “Inverting light with constraints,” presented at Ocean Optics XIX, Barga, Italy, 6–10 Oct. 2008, http://staff.washington.edu/erehm/OOXIX-Rehm-final.pdf.

*E*

_{d}and

*L*

_{u}obtained from a glider. The effect of errors in the computed

*E*

_{d}and

*L*

_{u}, i.e., the acceptable level of EcoLight-S optimization, on the convergence of implicit inverse models and on the accuracy of their retrievals must be evaluated by the user for each particular inversion algorithm. Most users of EcoLight-S in an inversion model would probably choose to solve the RTE at the wavelengths and depths corresponding to the observational data used by the inversion algorithm, in which case the errors in the computed light quantities would be minimal.

## Acknowledgments

## References and links

1. | C. D. Mobley, |

2. | C. D. Mobley and L. K. Sundman, |

3. | C. D. Mobley and L. K. Sundman, |

4. | C. D. Mobley, |

5. | G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. Jaffe, ed., Proc. SPIE2258 (with corrections), 194–201 (1994). [CrossRef] |

6. | C. D. Mobley, L. K. Sundaman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. |

7. | W. Freda and J. Piskozub, “Improved method of Fournier-Forand marine phase function parameterization,” Opt. Express |

8. | W. P. Bissett, J. J. Walsh, D. A. Dieterle, and K. L. Carder, “Carbon cycling in the upper waters of the Sargasso Sea: I. numerical simulation of differential carbon and nitrogen fluxes,” Deep-Sea Res. |

9. | W. P. Bissett, K. L. Carder, J. J. Walsh, and D. A. Dieterle, “Carbon cycling in the upper waters of the Sargasso Sea: II. numerical simulation of apparent and inherent optical properties,” Deep-Sea Res. |

10. | C. D. Mobley, “A new IOP model for Case 1 water,” in Ocean Optics Web Book, http://www.oceanopticsbook.info/view/optical_constituents_of_the_ocean/__level_2/a_new_iop_model_for_case_1_water. |

11. | C. D. Mobley, L. K. Sundman, W. P. Bissett, and B. Cahill, “Fast and accurate irradiance calculations for ecosystem models,” Biogeosci. Discuss. |

12. | A. F. Shchepetkin and J. C. McWilliams, “The regional ocean modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model,” Ocean Model. |

13. | F. Chai, R. C. Dugdale, T.-H. Peng, F. P. Wilkerson, and R. T. Barber, “One dimensional ecosystem model of the equatorial Pacific upwelling system, part I: model development and silicon and nitrogen cycle,” Deep-Sea Res. , Part II |

14. | M. Fujii, E. Boss, and F. Chai, “The value of adding optics to ecosystem models: a case study,” Biogeosciences |

15. | F. Chai, “Incorporating optics into physical and ecosystem modeling,” presented at the Gordon Research Conference on Coastal Ocean Modeling, South Hadley, MA, 26 June–1 July, 2011. |

16. | E. Rehm, C. D. Mobley, and J. Smart, “Inverting light with constraints,” presented at Ocean Optics XIX, Barga, Italy, 6–10 Oct. 2008, http://staff.washington.edu/erehm/OOXIX-Rehm-final.pdf. |

**OCIS Codes**

(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(010.5620) Atmospheric and oceanic optics : Radiative transfer

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: June 29, 2011

Revised Manuscript: August 11, 2011

Manuscript Accepted: August 23, 2011

Published: September 14, 2011

**Virtual Issues**

Vol. 6, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Curtis D. Mobley, "Fast light calculations for ocean ecosystem and inverse models," Opt. Express **19**, 18927-18944 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-18927

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### References

- C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994), http://www.curtismobley.com/lightandwater.zip .
- C. D. Mobley and L. K. Sundman, HydroLight User’s Guide (Sequoia Scientific, Inc., 2008), http://www.hydrolight.info .
- C. D. Mobley and L. K. Sundman, HydroLight Technical Documentation (Sequoia Scientific, Inc., 2008), http://www.hydrolight.info .
- C. D. Mobley, EcoLight-S 1.0 User’s Guide and Technical Documentation (Sequoia Scientific, Inc., 2011), http://www.hydrolight.info . [PubMed]
- G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” in Ocean Optics XII, J. Jaffe, ed., Proc. SPIE2258 (with corrections), 194–201 (1994). [CrossRef]
- C. D. Mobley, L. K. Sundaman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt.41(6), 1036–1050 (2002). [CrossRef]
- W. Freda and J. Piskozub, “Improved method of Fournier-Forand marine phase function parameterization,” Opt. Express15(20), 12763–21768 (2007). [CrossRef] [PubMed]
- W. P. Bissett, J. J. Walsh, D. A. Dieterle, and K. L. Carder, “Carbon cycling in the upper waters of the Sargasso Sea: I. numerical simulation of differential carbon and nitrogen fluxes,” Deep-Sea Res.46, 205–269 (1999a). [CrossRef]
- W. P. Bissett, K. L. Carder, J. J. Walsh, and D. A. Dieterle, “Carbon cycling in the upper waters of the Sargasso Sea: II. numerical simulation of apparent and inherent optical properties,” Deep-Sea Res.46, 271–317 (1999b). [CrossRef]
- C. D. Mobley, “A new IOP model for Case 1 water,” in Ocean Optics Web Book, http://www.oceanopticsbook.info/view/optical_constituents_of_the_ocean/__level_2/a_new_iop_model_for_case_1_water .
- C. D. Mobley, L. K. Sundman, W. P. Bissett, and B. Cahill, “Fast and accurate irradiance calculations for ecosystem models,” Biogeosci. Discuss.6, 10625–10662 (2009), http://www.biogeosciences-discuss.net/6/10625/2009/bgd-6-10625-2009.pdf . [CrossRef]
- A. F. Shchepetkin and J. C. McWilliams, “The regional ocean modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model,” Ocean Model.9, 347–404 (2005), https://www.myroms.org/wiki/index.php/Documentation_Portal . [CrossRef]
- F. Chai, R. C. Dugdale, T.-H. Peng, F. P. Wilkerson, and R. T. Barber, “One dimensional ecosystem model of the equatorial Pacific upwelling system, part I: model development and silicon and nitrogen cycle,” Deep-Sea Res., Part II 49, (13–14), 2713–2745 (2002).
- M. Fujii, E. Boss, and F. Chai, “The value of adding optics to ecosystem models: a case study,” Biogeosciences4, 817–835 (2007), http://www.biogeosciences.net/4/817/2007/ . [CrossRef]
- F. Chai, “Incorporating optics into physical and ecosystem modeling,” presented at the Gordon Research Conference on Coastal Ocean Modeling, South Hadley, MA, 26 June–1 July, 2011.
- E. Rehm, C. D. Mobley, and J. Smart, “Inverting light with constraints,” presented at Ocean Optics XIX, Barga, Italy, 6–10 Oct. 2008, http://staff.washington.edu/erehm/OOXIX-Rehm-final.pdf .

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