A simple, surprisingly accurate, method for estimating the influence of Raman scattering on the upwelling light field in natural waters is developed. The method is based on the single (or quasi-single) scattering solution of the radiative transfer equation with the Raman source function. Given the light field at the excitation wavelength, accurate estimates (~1-10%) of the contribution of Raman scattering to the light field at the emission wavelength are obtained. The accuracy is only slightly degraded when typically measured aspects of the light field at the excitation are available.
© 2014 Optical Society of America
The backscattered solar radiance exiting natural waters contains information regarding the water’s constituents such as phytoplankton, other suspended material, dissolved organic material, etc . The goal of water color remote sensing is to use this radiance to estimate the concentration of such constituents. Raman scattering by water molecules is also an important contributor to the light field in natural waters [2–15]. In particular, as much as 25% of the solar reflected radiance in clear ocean water at wavelengths greater than 500 nm can be attributed to Raman scattering . The constituents influence the Raman contribution both through their effect on the excitation radiance as well as the emission radiance. It is important to be able to assess (1) the contribution of Raman scattering in predicting the influence of constituents on the light field through simulations, and (2) account for its contribution in experimental measurements of various aspects of the light field. In this paper we provide a simple procedure for such assessments. In particular, we show that determining the lowest order of Raman scattering – basically single Raman scattering – is sufficiently accurate to be useful in both of the above assessments.
Because it yields simple analytical expressions to complex radiative transfer problems, the single scattering approximation has played an important role, over and above pedagogy, in water color remote sensing. It was central to the original atmospheric correction algorithm for the Coastal Zone Color Scanner [16,17], and formed the basis for more accurate and complex approaches . The simple replacement of the beam attenuation coefficient by the sum of the absorption and backscattering coefficients – quasi-single scattering  – in the single scattering solution for the diffuse reflectance of natural waters enabled estimation of said reflectance with considerable accuracy for even highly multiple scattering media . Here, we apply it to the estimation of Raman-induced light fields.
We begin by providing basic definitions of important properties of the light field and the governing equation – the radiative transfer equation (RTE). Next, we solve the RTE for the contribution from Raman scattering in the lowest order. Finally, the resulting Raman contribution to various light field characteristics is compared with “exact” Monte Carlo simulations.
2. Characteristics and governing equations of the light field
Consider a horizontally homogeneous water body that is stratified in the z direction, which is into the water. Let L(z, u, ϕ, λ) be the radiance at a wavelength λ propagating in a direction specified by the polar and azimuth angles θ and ϕ, respectively, with u = cosθ, and θ measured from the + z axis. The downward (Ed), upward (Eu), and scalar (E0) irradiances are defined according to 
The radiance is governed by the radiative transfer equation Eq. (2)) can be written
3. The solution of the RTE for the inelastic component
The solution to Eq. (5) can be developed by considering the set of differential equations;Eq. (6) can be developed by introducing an integrating factor (exp[τ /u]) and performing integration from τa to τb:
For Raman scattering, βI = βR, the dependence of the scattering on direction is similar to that for Rayleigh scattering but with a different depolarization factor :4,11,12]
The quantities related to the light field that are most often measured experimentally are vertical profiles of zenith propagating radiance (Lup) and upward (Eu) and downward (Ed) propagating irradiance. Let us examine the Raman contribution to Lup first. In this case, u = −1, ϕ is irrelevant, and the expression for Q is easily found by replacing the integral over λE by simple multiplication by ΔλE:Eq. (1)). ThenEq. (15).
The lowest order approximation (Eq. (3)) to the desired upward radiance is then
To test its applicability in natural waters, we consider the contribution of Raman scattering in a particle-free water body beneath an aerosol-free atmosphere at 550 nm for solar zenith angles of 0 and 60 deg. In this case λE = 464 nm. Monte Carlo simulations of the light field at 464 nm show that in this case is essentially independent of depth and that E0(z, λE) decays exponentially with depth: When this is the case (and c is constant) the integration can be carried out yieldingEq. (2)) using Monte Carlo (MC) methods are provided in Table 1. Note that the difference between the Monte Carlo simulation of the radiance and that computed via Eq. (17) is less than 1%. This shows the accuracy that can be obtained by computing Lup(0)(0, λ) using Eq. (16) or (17) when the light field at the excitation wavelength is known. If we want the estimate Lup(0)(0, λ) in an experimental situation, usually the only properties of the excitation light field that would be measured are Ed(z, λE) and Lu(0)(0, λE) or Eu(z, λE). Thus, we need to estimate K0(z, λE), E0(0, λE) and . (Note however, that instrumentation capable of measuringthe entire radiance distribution exists [21,22], and has recently been miniaturized , so increasingly in the future these needed quantities at the excitation wavelength will be directly available.) The estimate of K0(z, λE) can be effected by approximating it with Kd(z, λE). The scalar irradiance is given by15]. If Lup(0, λE) is measured rather than Eu(0, λE), Eq. (18) with these approximations to μd and μu, can still be used with Eu(0, λE) replaced by πLup(0, λE). This assumes that Lu is independent of u. The final quantity required can be approximated through which is similar to the approximation The 4th column in Table 1 provides Lup(0) (0,λ) computed using these approximations and shows that again the agreement with the Monte Carlo simulations is excellent.
The upwelling irradiance in the lowest order is also easy to find in this approximation. Letting the upward radiance in the direction (μ,ϕ) isEq. (7) can be carried out analytically yieldingTable 1 compares the value of computed using Eq. (22) with the Monte Carlo simulations. Again, the results are excellent when the actual light field at the excitation wavelength is provided, and reasonably accurate when approximate values of the required quantities are used. In a like manner, the downwelling irradiance can be computed with similar accuracy; however, with the boundary conditions we used, the computation yields only the component directly generated from λE, to which must be added the upwelling Raman irradiance at λ reflected from the surface into the downwelling stream. The reflected Raman radiance just beneath the surface can be computed with reasonable accuracy; however, since this radiance is nearly diffuse its attenuation coefficient, which is critical in estimating its contribution at depth, is difficult to estimate. Considering these complications and the fact that the elastic component of Ed is much larger near the surface than the Raman component (by a factor of 100 or more) [3–9], computation of the Raman component is usually of little interest. In the light of these complications and facts, we omit the solution to Ed.
We now consider a more realistic example, e.g., a water body with particles, below an atmosphere with aerosols, Eq. (17) cannot be used because K0 depends strongly on depth, especially when the solar zenith angle is large. Using a bio-optical model similar to that in Ref , but tuned to better match experimental measurements in clear waters off Hawaii, we computed the Raman contribution to the radiance for a phytoplankton pigment concentration of 0.3 mg/m3 and a solar zenith angle of 60 deg. Then using Eq. (16), along with the required quantities at the excitation wavelength, we computed L(0) just beneaththe surface (z = 0). Table 2 provides the results for 450 and 550 nm (excitation at 391 and 464 nm, respectively) computed using Eq. (16) directly (the single scattering approximation) and by replacing c by a + bb in Eq. (16) (the quasi-single scattering approximation [19,20]), which approximately accounts for the multiple scattering of Raman-generated radiance. Note that, in the quasi-single scattering approximation, , which would be available from , i.e., measurements of the downwelling irradiance at the emission wavelength (near the surface, it is only weakly influenced by Raman scattering). The fraction of the Raman contribution to the total radiance at the surface was 7.7% and 14.5%, respectively, at 450 and 550 nm (note, however, that the elastic contribution is strongly dependent on the bio-optical model). The quantity Lup(0)(Approx.) refers to the same computation when E0 is replaced by its estimate from Ed and Lup, and the average squared cosine is replaced by (u0w)2. Such replacements would be required in a typical experimental scenario. The error using quasi-single scattering is reduced by a large factor compared to single scattering, with an error of 9% at 450 nm and 1.6% at 550 nm. The larger error at 450 nm is to be expected because the single scattering albedo is significantly larger there (0.71) than at 550 nm (0.44), so the quasi-single scattering approximation is less effective. The error in Lup(0)(Approx.) is larger than it is when the correct excitation quantities are used, but still not excessive (13.4% at 450 nm and 5.2% at 550 nm). The relative error (using Eq. (16) with the correct E0 and μd) increases as a function depth: reaching 16% at 20 m and 30% at 100 m for 450 nm; and 3.2% at 20 m and <5% at 100 m for 550 nm. This increase is mainly due to the ineffectiveness of the QSSA as the depth increases.
4. Concluding remarks
We have provided a simple, surprisingly accurate, estimate of the contribution of Raman scattering of water molecules to several commonly measured properties of the in-water light field. It can be used both to compute the Raman component based on simulations of the light field at the excitation wavelength and to estimate the Raman contribution to field measurements of light field properties given similar measurements at the excitation wavelength. In particular, if Ed(z, λ) and Eu(z, λ) or Ed(z, λ) and Lup are measured throughout the spectrum, then the Raman contribution can be estimated in a manner similar to Lup(0)(Approx.) and Eu(0)(Approx.) in Table 1 and Lup(0)(Approx.) in Table 2. Of course, the attenuation coefficient c(z, λ) must be known or estimated at the wavelength of interest. The solution provided is equivalent to single scattering with the appropriate source function. Should more accuracy be needed, L(0)(z,u,ϕ,λ) can be inserted into Eq. (6) and L(1)(z,u,ϕ,λ) computed. This is an arduous task; however, as in Table 2, one can employ the quasi-single scattering approximation for this purpose which involves the simple replacement of c(z, λ) in Eqs. (15)–(21) by a(z, λ) + bb(z, λ) or u0wKd(z,λ).
Although, in the cases provided here, the solutions are in error by ~1-10%, the accuracy is still sufficient to assess the contribution of Raman scattering in most situations. Full radiative transfer simulations  using a bio-optical model relating the phytoplankton pigment concentration to the inherent optical properties suggest, that for pigment concentrations varying from 0 to 0.3 mg/m3, the Raman contribution to Lup varies from ~10 to 12% at 440 nm and 25 to 11% at 550 nm. Thus, even with an error of 10% in Eq. (16), the error in retrieval of the elastic component of Lup from the total would not exceed 2%, over and above any error in the measurement of Lup.
To understand the impact of Raman scattering effects to ocean color remote sensing, one must consider the actual application. The traditional empirical algorithms relating water-leaving radiance to pigment concentrations were all based on measurements that included Raman effects [1,16,17] (even though it was not recognized at the time). Thus, to a certain extent the Raman correction is already “built in” to such algorithms and their more modern counterparts. However, recent algorithms, particularly those that relate the water’s backscattering coefficient to phytoplankton biomass , rely on being able to estimate the backscattering coefficient from remote measurements. If the Raman augmentation of the radiance is not removed, then the retrieved backscattering coefficient will be too large roughly by the ratio of the Raman water-leaving radiance to the total water-leaving radiance which in can be as high as 25% at wavelengths > 500 nm in low-chlorophyll waters. The biomass will then be overestimated roughly by the same ratio, i.e., there would be a positive bias in biomass estimates, which would vary with the pigment concentration .
A second inelastic process of interest in oceanic optics is fluorescence of dissolved organic material or of chlorophyll a contained in phytoplankton. For fluorescence, the emission is isotropic, i.e., βI = βF is independent of the direction of the incident and scattered photons, so The equations developed here can be adapted to handle fluorescence as well, by replacing 0.55 by 0 and 1.1833 by 1 in Eq. (4) and all subsequent equations. When these changes are made the resulting equations agree with those developed earlier  for the chlorophyll a fluorescence near 683 nm. Raman scattering by water is typically a more important inelastic process than fluorescence, as it is almost always much larger than fluorescence except in specific regions of the spectrum (e.g., near 683 nm at high chlorophyll concentrations).
It is understood that the Raman contribution can be accurately assessed using available radiative transfer codes and the vertical profiles of the absorption coefficients, the scattering coefficients, and the particulate scattering phase function at both the excitation and emission wavelength [14,15,27]. The principal values of the simple expressions developed here is in allowing rapid estimates of the Raman contribution in experimental measurements which allows separation of the elastic and inelastic components of the light field. Pedagogically the derived expressions also isolate the important parameters of the problem and provide a simple way to explore the influence of various optical properties of the medium on the Raman contribution.
References and links
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