This paper is devoted to the derivation of a fast and accurate model of scalar irradiance for stratified Case 2 waters. Five strategies are formulated and employed in the new model, including (1) reallocating the sky radiance, (2) approximating the influence of the air-water interface, (3) constructing a look-up table of average cosine based on the single-scattering albedo and the backscatter fraction, (4) calculating the phase function of surrogate particles in Case 2 waters, and (5) using the average cosine as an index to cope with stratified waters. A comprehensive model-to-model comparison shows that the new model runs more than 1,400 times faster than the commercially-available Hydrolight model, while it limits the percentage error to 2.03% and the maximum error to less than 6.81%. This new model can be used interactively in models of the oceanic system, such as biogeochemical models or the heat budget part of global circulation models.
©2006 Optical Society of America
The quantitative study of the interactions of light with the aquatic environment on earth, such as oceans, estuaries, lakes, rivers and other water bodies, plays an essential role in hydrologic optics . In particular, an accurate simulation of the vertical profile of underwater scalar irradiance E 0(z) is a prerequisite for modeling hydrologic systems, such as the photosynthesis process in biogeochemical models or the heat budget part of ocean circulation models . After being scattered and absorbed by the atmosphere, the photons penetrate through the wavy boundary of air-water interface from all possible directions and interact with water bodies. A bipartite classification is generally used to categorize the water bodies as Case 1 and Case 2 waters [3, 4]. By definition, phytoplankton with their accompanying and covarying retinue of material of biological origin is the principal component responsible for variations in optical properties of Case 1 waters, while inorganic particles in suspension and yellow substances all influence the optical properties of Case 2 waters. There has been a lot of research interest in investigating Case 2 waters since over 60% of the human population, and 90% of the world’s fish catch are directly linked to the coastal zones of Case 2 waters . However, the study of radiative transfer in Case 2 waters is quite challenging because of the complex constitution of their optical components [6, 7].
This paper aims at deriving a fast yet accurate approximation for calculating the underwater scalar irradiance for stratified Case 2 waters. In essence this is the extension of earlier work on the case of homogeneous Case 1 waters . A comprehensive model-to-model comparison shows that the new model runs more than 1,400 times faster than the full Hydrolight code, while it limits the percentage error to 2.03% and the maximum error to less than 6.81%. This new model can be used interactively in models of the oceanic system, such as biogeochemical models or the heat budget part of global circulation models.
2. Radiative transfer in aquatic environments
For a body of water with a constant index of refraction, the radiative transfer equation (RTE) can be derived from more fundamental equations , such as Maxwell’s electromagnetic equation [8, 9], quantum mechanics  and the Boltzmann equation . RTE states that the rate of change in radiance L through a distance r along the direction at a spatial position is the linear combination of loss due to attenuation -cL and gain due to scattering L * and internal emission :
where c is the beam attenuation coefficient. The radiative energy transfers within the water body at a constant speed v that equals the speed of light divided by the index of refraction. Therefore, the total rate of change along r is
where ≡ /v is the unit vector along the path r. Note that the time scale for reaching the steady state in studies of hydrologic optics is generally much shorter than the sampling time, therefore, the time-independent assumption is valid and the term of partial derivative with t is dropped. L * can be further divided into inelastic scattering and elastic scattering . For the elastic scattering, the volume scattering phase function β describes the probability of a photon scattered to the direction from all directions ′ in a unit sphere Ξ. For the inelastic scattering, is usually considered as an extra source that elevates the energy level at this wavelength. The common practice is to combine and the emission term into an effective source function S. Liu et al.  tested the sensitivity of various factors in simulating the vertical profile of underwater scalar irradiance. They concluded that S plays a negligible role. Therefore, the general form of the monochromatic RTE is
Plass and Kattawar  first applied the Monte-Carlo (MC) method to solve the general form of RTE in hydrologic optics. The basic principle of the MC method is to simulate a beam of light by using a very large number of photons. Following the path of each photon, a series of random numbers can be used to determine the photon’s life history according to different probabilities for different phenomena. The final light field is the cumulative contribution of all the photons. Based on the recent work of Liu and Woods , a 3D backward MC model was developed to simulate the light field in coastal waters with uneven bottom geometry . Their model is able to provide solutions to the most general form of RTE. However, if the information required is not merely the surface reflectance but the detailed light field under the sea surface, the MC approach would require quite considerable computer resources that would therefore limit its practical applications .
Many efforts have been made to find a fast and accurate solution of the RTE without running the full MC simulation, and one way it can be done is by further simplifying the RTE. For the body of water with a flat bottom and where the horizontal variance of inherent optical properties (IOPs) is much less than their vertical variance, the assumption of plane-parallel waters can be made to reduce the spatial variable (vector) to one scalar variable, depth z. The general form of RTE, therefore, can be simplified as an integro-differential equation
By introducing the assumption of quad-averaging, the integro-differential equation can be further simplified to a set (typically hundreds) of ordinary differential equations. The invariant imbedding technique can be employed to imbed the air-sea and bottom boundary conditions to the set of ODEs, and the detailed distribution of L(z;θ;ϕ;λ) can then be solved simultaneously. The associated computer code was initially developed in 1979 and has since gone through a series of modifications [15–17]. Version 3.0 was funded by the US Office of Naval Research and renamed as Hydrolight in year 1995 . The latest version of Hydrolight, 4.3, was released in March 2003 as commercial software. However, it is not always practical to use Hydrolight to calculate global underwater light fields, as the computational time required for running a full Hydrolight simulation interactively in biogeochemical models or global circulation models is often too long . Because of this, the general approach still relies on the empirical algorithm of K to calculate the underwater light field, resulting in considerable errors . The progress of modeling the oceanic system is impeded somewhat in that while we know that error exists, and have the tool to correct the error, we simply cannot afford to use that tool. As a result, we all tolerate and live with error in simpler but faster models, despite that Hydrolight has been available for several years.
3. Theory of approximation
Liu et al.  employed four strategies to accelerate the simulation of the scalar irradiance E 0(z) without losing accuracy compared with Hydrolight. Their model runs more than fourteen thousand times faster than the full Hydrolight code, while limiting the percentage error to 2.20% and the maximum error to less than 4.78%. However, their model was limited to (1) vertically homogeneous water columns, (2) Case 1 waters or Case 2 waters that happen to be gelbstoff rich, (3) spectral ranges from 400 to 700 nm, and (4) chlorophyll concentration (Chl) from 0 to 10.0 mg∙m-3. This paper modifies one strategy described in Liu et al.  and proposes three new strategies to remove those limitations, resulting in a new model of E 0(z) that can be applied to the stratified Case 2 waters. Details of these strategies are given as follows.
3.1 Reallocation of sky radiance
In the realm of hydrologic optics, the incident radiative energy originates from the Sun. After being scattered and absorbed by the atmosphere, the photons penetrate through the wavy boundary of the air-water interface from all possible directions (θ;ϕ) and interact with the bodies of water. Because the quantity of concern, E 0(z), is computed from integrals of the radiance over the entire sphere Ξ,
The incident photons which arrive at the sea surface (z = 0+) from all possible direction (θ;ϕ) can be reallocated to the plane of the Sun (θ;ϕ 0) and the same integrated result of E 0(z) still can be obtained. In other words, the planar irradiance incident on the surface can be expressed as
where the cumulative radiance in the azimuthal direction is
Applying the superposition principle at each depth z, the quantity of concern E 0(z) can be quickly calculated by summing the contribution from each light source in the plane of the Sun (θ;ϕ 0) by
where E̅0(z;θ;ϕ 0) is the profile of E 0 obtained by placing in a black sky a unit radiance in (θ;ϕ 0). As long as the distribution of incident radiance L(0+;θ;ϕ) is given and E 0(z;θ;ϕ 0) is calculated and stored in advance, solving E 0(z) is simply a procedure of multiplication and summation that can be done with very high efficiency . Note that no approximation needs to be made at this stage.
3.2 Approximation of the influence of the air-water interface
The air-water interface is usually roughened by wind, and the influence of this on simulating E 0(z) can be regarded as redistributing the radiance field after the beam of light penetrates the interface. To consider this effect of redistribution, the profile of Ē0(z;θ;ϕ 0) can be scaled by multiplying a corrective factor for wind speed CW that is defined as
where Ē0(0-;θ;ϕ 0)|Vwind and E 0(0-;θ;ϕ 0)|V'wind are the subsurface values of E 0 obtained by placing in a black sky a unit light source in (θ;ϕ 0) under the condition of surface wind speed Vwind and V́wind , respectively. Given CW(θ;ϕ 0)|Vwind→V'wind, the vertical profile of E 0(z) calculated at wind speed Vwind (Eq. 8) can be easily extended to E 0(z) at wind speed V′wind by
To be more precise, CW is influenced by water constituents as well. Liu et al.  demonstrated that CW is only a weak function of water constituents. It is therefore feasible to construct a look-up table (LUT) based on the situation of clear water for quick reference to the CW. Liu et al.  calculated CW at 31 fixed wavelength from 400 nm to 700 nm at steps of 10 nm. To have a flexibility on the wavelength, the main variables of the new LUT are selected as ω 0(0.01 – 0.99), V wind (0.0 – 15.0) and θ (0.0 – 90.0). A standard volume scattering phase function for water molecules is obtained from the analytic Fournier-Forand phase function  by giving the backscatter fraction (BF) a value of 0.5. A section of the LUT is denoted as LUTCW and given in Table 1. Note that the table is for pure water with an assumed absorption thus that ω 0 can be varied from 0.01 to 0.99.
3.3 Look-up table of average cosine based on ω0 and BF
The key to achieving a fast and accurate simulation of E 0(z) is the existence of a proper mathematical model that is able to approximate the vertical profile of the average cosine μ̄(z) with a set of only five parameters (B 0, B 1, P, B 2, Q)
yet which still retains a very high accuracy , where the optical depth ζ = cz and (z) is
Liu et al.  demonstrated that a LUT can then be constructed for quick reference to set of five parameters (B 0, B 1, P, B 2, Q) from the given IOPs. With this LUT, the net diffuse attenuation coefficient KNET (z) can be calculated from μ̄(z) by use of Gershun’s equation 
The definition of KNET (z) is
where Ed and Eu are the downwelling and upwelling planar irradiance, respectively. Therefore, KNET (z) gives us the profile of E(z) that can be converted to E 0(z) using the definition of μ̄(z) (Eq. 12).
The major limitation of the LUT approach is the parameterization of IOPs. The earlier work  was based on the biooptical model of Case 1 waters that uses the Chl, λ and the backscatter fraction (BF) as the main variables. As a result, the model of Liu et al.  cannot be applied to Case 2 waters. The main variable Chl is limited to the range from 0 to 10.0 mg∙m-3, and λ is limited to the range from 400 to 700 nm as well. Considering the fact that all IOPs can be derived from two principal IOPs: the absorption coefficient a and the phase scattering function β(ψ) , where ψ is the scattering angle, the two principal IOPs should be selected as the main variables in the LUT. However, β(ψ) describes the probability of scattering in the ψ direction, which varies from a symmetric distribution (molecular scale) to a heavily peak distribution in the forward direction (large particle). To consider a large range of the distribution of β (ψ), it is necessary to find an accurate and efficient way to parameterize β(ψ). Fournier and Forand (FF)  proposed an analytical phase function for ocean water, which parameterized β(ψ) as a function of the backscatter fraction (BF), defined as the backscattering coefficient bb divided by the scattering coefficient b. A large range in β(ψ) can be obtained by varying the value of BF. For example, BF=0.0183 provides a very good fit to the Petzold’s average particle phase scattering function, while BF=0.5 yields the pure water phase function. Mobley el al.  did a thorough sensitivity test of phase function effects on the oceanic light field. They suggested that the physically based FF phase function can be used to generate realistic phase functions having any desired BF. Therefore, BF is a good choice for parameterizing β(ψ).
Based on the work of Liu et al. , we reconstructed a new LUT with two non-dimensional variables BF (0.0001 – 0.5) and ω 0 (0.01 – 0.99) for quick reference to a set of parameters (B 0, B 1, P, B 2, Q) used by the McCormick five-parameter model  of μ̄(ζ). The single-scattering albedo ω 0 is defined as b divided by c. A section of the LUT is denoted as LUTμ and given in Table 2, where the Pearson correlation coefficient PCC shows how good the McCormick five-parameter model fits the vertical profiles of the average cosine. Throughout the entire LUT, the average value of PCC is 0.999577 and the lowest value is 0.980118.
Implementation of the new LUT is explained as follows. Given the IOPs of Case 2 waters, ω 0 and BFP can be calculated. The bulk property of BF is given by
where BFw =0.5. Referencing to the new LUT, ω 0 and BFP give us a profile of μ̄p(ζ), while ω 0 and BF give us another profile of μ̄′(ζ) that can be regarded as the value for a pure medium with a backscatter fraction BF. However, neither μ̄p(ζ) nor μ̄′(ζ) represents the actual profile of μ̄(ζ), because both particles and water molecules contribute to the effect of scattering. The earlier work  suggested that a constant offset Δ exists between μ̄p(ζ) and μ̄(ζ). Figure 1 illustrates that the assumption of a constant offset only held at depths where μ̄(ζ) has reached its asymptotic solution. To give a better approximation of μ̄(ζ) from μ̄p(ζ) and μ̄′(ζ), this research proposes a new approach to describe the vertical profile of offset Δ(ζ). The offset near the surface is given by
while the offset at infinite depth is
μ̄(ζ) can be estimated from μ̄p(ζ) by adding the offset Δ(ζ), i.e.,
Where the vertical profile of offset Δ(ζ) is given by
To illustrate the validity of Eq. (18), a total of six cases with various particle sizes and concentrations were selected by setting Chl (1.0, and 10.0 mg∙m-3) and BFP (0.00915, 0.0183 and 0.0366), respectively. Other conditions of computation were set as typical Case 1 waters and are listed in the caption of Fig. 1. Both Hydrolight and our approach are employed to simulate the profiles of μ̄(ζ). The corresponding profiles of μ̄′(ζ) and μ̄p(ζ) are also plotted for comparison. The deviation between μ̄′(ζ) and μ̄p(ζ) is apparent as the size of the particles and/or the concentration of particles is reduced, such as the case of Fig. 1(a). For all six cases, Eq. (18) gives a sound estimation of μ̄(ζ) that corresponds closely to the result from Hydrolight.
3.4 Phase function of surrogate particles in Case 2 waters
Based on their optical characteristics, the Case 2 waters are generally categorized into four components with different parameterizations of IOPs, including pure water, large particles (assumed to absorb light like Chl and scatter light like large particles), CDOM (assumed to absorb light but not scatter it) and small particles. Note the component of small particles represents minerals in waters that can both absorb and scatter light. The amount of each component does not need to be controlled by Chl and the backscattering fraction for large particles BFl and small particles BFs do not need to be constant values either. To extend our approach to the general situation of Case 2 waters, it is equivalent to find a surrogate particle with backscattering fraction BFp that is able to exhibit the same optical properties as those two types of particles presented. The normalized phase scattering function β̃ for the four-component Case 2 waters is
where β̃l (ψ,λ) and β̃s(ψ,λ) can be obtained from the analytic Fournier-Forand phase function  by giving the values of BFl and BFs , respectively. Assuming a surrogate particle implies that
Therefore, BFp is the value that the resulting phase function gives the best fit to β̃p(ψ,λ).
An example of calculating BFp based on the technique of optimization is illustrated in Fig. 2. A set of β̃p(ψ,λ) for ocean water is obtained by varying BFp from 0.0001 to 0.5 (dashed lines). The thick solid line gives the corresponding β̃p(ψ,λ) of a four-component Case 2 waters, with which BFl =0.001, BFs = 0.08 and ratiol = 0.31. Note that
Based on the result of optimization, setting BFp at a value of 0.053316 gives the best fit (thin solid line).
The process of searching for the optimized solution of BFp consumes considerable computer time. To avoid repetitive calculation, the concept of constructing a LUT is again adopted. Three variables, ratiol , BFl and BFs are selected, with ratiol ranging from 0 to 1, BFl ranging from 0.0001 to 0.02 to represent large particles, and BFs ranging from 0.018 to 0.3 to represent small particles. A section of the LUT is denoted as LUTBF and given in Table 3. Throughout the entire LUTBF, the average value of PCC is 0.989577 and the lowest value is 0.930118. Note that the inelastic scattering effects were not included in the Hydrolight runs used to generate the three LUTs needed for the model of this paper.
3.5 Use average cosine as an index to cope with stratified waters
The stratified water column can be divided into many thin layers, each with its own constitution of IOPs. Since a highly efficient solution method for homogeneous waters has been developed, it is advantageous overall to solve the RTE within each layer, and then to combine these “layer solutions” to obtain the solution of the RTE for an inhomogeneous body of water . However, to determine the attenuation of radiative energy between each layer requires information about the radiance distribution at that particular depth. Recall that μ̄(ζ) is a crude but useful one-parameter measure of the directional structures of the light field in spatially uniform waters. It can be used as an index to provide rough information about the radiance distribution at that depth, and hence the attenuation of light within that layer. This idea is implemented and examined in Fig. 3.
Lewis et al.  suggested that a typical profile of chlorophyll concentration for Case 1 waters can be approximated as a background value plus a Gaussian
Similar parameterization of vertical profile was employed by Albert and Mobley  as well. Figure 3(a) shows a profile of chlorophyll with one meter intervals, which was obtained by setting in Eq. (23) the parameters Co , h, s, z max values of 1, 20, 1.5, 7, respectively. This profile can also be treated as a combination of stratified layers with a total of eight concentrations of chlorophyll. Considering eight cases of homogeneous water bodies, each has one of those chlorophyll concentrations under the same conditions of incident light (as listed in caption). A set of μ̄i(z) is calculated and illustrated in Fig. 3(b), where the subscript i is the index of the layer. These “layer solutions” of μ̄i(z) can be combined to approximate the full solution of μ̄(z) for the corresponding case of inhomogeneous bodies of water under the same conditions of incident light (Fig. 3c). Note that the deviation of μ̄(z) between the approximation and the full Hydrolight run of the inhomogeneous case is small, and that the deviation of E 0(z) is even lower (Fig. 3d). This verifies that the average cosine is a good index to extend the results of earlier work on homogeneous waters to the case of stratified waters.
A comprehensive model-to-model comparison was made to examine the accuracy, flexibility and applicability of the new model that includes all the aforementioned strategies. A total of 20 cases were compared by randomly specifying values to parameters, including the solar zenith angle θS , cloudiness, surface wind speed Vwind , the backscattering fraction for large particles BFl , and the backscattering fraction for mineral particles BFS (Table 4). The stratified layers were composed by randomly specifying values to four parameters of Eq. (23) to describe the vertical profiles of chlorophyll concentration Chl(z) and the mineral particle concentration M(z), respectively (Table 5). The CDOM absorption at a reference wavelength a g,440(z) was set to be varied linearly from the surface value to the bottom value , as listed in Table 5. Note that the spectral values of CDOM absorption are approximated as an exponential decay function
where the normalized chlorophyll-specific absorption coefficient (λ) is given in the paper of Morel . The spectral absorption aw (λ) and scattering coefficient bw (λ) for pure water are given in the papers of Pope and Fry  and Smith and Baker , respectively. The mineral absorption coefficient is given by
and the mineral scattering coefficient is
Where the mass-specific mineral absorption coefficient (λ) and scattering coefficient (λ) are taken from published data . With the set of bio-optical models described from Eq. (24) to Eq. (28), the vertical profiles of Chl(z), a g,440(z) and M(z) can be converted to the vertical profiles of total absorption coefficient a(z,λ) and total scattering coefficient b(z,λ).
Given the values bm (z,λ) and bc (z,λ), the vertical profile of ratiol (z,λ) can be calculated from Eq. (20). Following the procedure described earlier, the backscattering fraction BFp (z,λ) of surrogate particles can be quickly referred to in the LUT from the given values of ratiol (z,λ), BFl and BFS . For illustration, Fig. 4(d), 4(e) and 4(f) show the vertical profiles of a(z), b(z) and BFp (z) respectively at wavelength 440 nm.
For each of the 20 cases described above, the vertical profiles of wavelength-integrated scalar irradiance in the Photosynthetical Active Radiation (PAR) range E 0,PAR(z) were simulated by use of the commercially available 4.3 version of the Hydrolight model and the new model developed in this work, respectively. A total of 35 wave bands were simulated, which ranged from 350 to 700 nm in steps of 10 nm. Figure 5 shows the comparison of E 0,PAR(z) from 0 to 50 m with one meter intervals. To quantitate the comparison, the correlation coefficient r, the maximum relative error ε max, the average relative error ε average and the percentage error ε% in the euphotic zone were calculated and given in Fig. 5. The definition of ε% is given by
where the root-mean-square-error in log 10 scale RMSElog10 is defined by
This kind of analysis puts the same emphasis on underestimates and overestimates. Comparing the results of our model to the results of the Hydrolight run, a very high correlation (r = 0.999924) as well as a large computational speed ratio (CSR) of 1402.8 is obtained for our model. The percentage error ε% is 2.03% and the maximum relative error ε max is not more than 6.81%. The average relative error ε average is as low as 1.62%. Figure 5 demonstrates that the new model can indeed provide an accurate and fast simulation of the vertical profile of E 0,PAR(z) for stratified Case 2 waters with a variety of compositions of the inherent optical properties.
This paper is devoted to the derivation of a fast and accurate model of scalar irradiance for stratified Case 2 waters. Over 60% of the human population and 90% of the world’s fish catch are directly linked to the coastal zone of Case 2 waters. The study of the radiative transfer in Case 2 waters is yet more challenging because of the complex constitution of their optical components. By definition, phytoplankton with their accompanying and covarying retinue of material of biological origin, as well as inorganic particles in suspension and yellow substances all influence the optical properties of Case 2 waters. Although the three-dimension MC optical model based on the backward ray-tracing technique is able to provide the solution to the most general form of radiative transfer equation, the computational resources required mean it is impractical to implement. A much faster calculation can be achieved by the commercially-available Hydrolight model that is based on the technique of invariant imbedding and the assumption of plane-parallel waters. However, it is not always practical to apply these numerical models to calculate global underwater light fields; the computational time required for running a full simulation interactively in biogeochemical models or global circulation models is often too long.
Extending earlier approaches for the case of homogeneous Case 1 waters, this paper describes a new model that runs more than 1,400 times faster than the full Hydrolight code, while it limits the percentage error to 2.03% and the maximum error to less than 6.81%. Having removed all limitations of earlier works, this new model is now fully comparable to the Hydrolight model in simulating the underwater scalar irradiance in Case 2 waters, yet still retains a very high computational speed. This new model can be used interactively in models of the oceanic system, such as biogeochemical models or the heat budget part of global circulation models.
This research was supported by National Science Council of Taiwan through grants NSC 94-2611-M-006-002. Thanks to Dr. Richard Miller for helps in Hydrolight simulation while the author held a National Research Council Research Associateship at NASA John C. Stennis Space Center.
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