## Abstract

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

## 1. Introduction

The backscattered solar radiance exiting natural waters contains information regarding the water’s constituents such as phytoplankton, other suspended material, dissolved organic material, etc [1]. 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 [14]. 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 [18]. The simple replacement of the beam attenuation coefficient by the sum of the absorption and backscattering coefficients – *quasi*-single scattering [19] – 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 [20]. 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* = cos*θ*, and *θ* measured from the + *z* axis. The downward (*E _{d}*), upward (

*E*), and scalar (

_{u}*E*

_{0}) irradiances are defined according to [15]

*x*=

*d*,

*u*, or 0.

The radiance is governed by the radiative transfer equation [15]

*c*(

*z*,

*λ*) is the beam attenuation coefficient profile in the water body, $\beta (z,u\text{'}\to u,\varphi \text{'}\to \varphi \text{'},\lambda )$ is the volume scattering function for scattering from the primed direction to the unprimed direction, ${\beta}_{I}(z,u\text{'}\to u,\varphi \text{'}\to \varphi ,{\lambda}_{E}\to \lambda )$ is the inelastic (e.g., Raman) volume scattering function for scattering from the primed directions at

*λ*to the unprimed direction at

_{Ε}*λ*, and

*L*(

*z*,

*u*′,

*ϕ*′,

*λ*) is the radiance in the primed direction at

_{E}*λ*. We write the last term as

_{Ε}*dτ*=

*c*(

*z*,

*λ*)

*dz*,

*P =*4π

*β*/

*b*and

*ω*

_{0}(

*z*,

*λ*) =

*b*(

*z*,

*λ*)/

*c*(

*z*,

*λ*), where

*b*is the scattering coefficient at

*λ*. Then, the RTE (Eq. (2)) can be written

*λ*is known, so $Q=Q(z,u,\varphi ,\lambda )$ is a known function of its arguments. This equation must be solved subject to the boundary condition of incident radiance at

_{Ε}*z*= 0 of

*L*(0,

_{Inc}*u*,

*ϕ*,

*λ*) for

*u*positive. Actually, since we are interested only in the inelastic contribution to the radiance at

*λ*, we will take the surface boundary condition to be

*L*(0,

_{Inc}*u*,

*ϕ*,

*λ*) = 0 for

*u*positive. Were we interested in the elastic component as well, the surface boundary condition would be the radiance incident at

*λ*from the sun and sky.

## 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;

*z*= 0 for

*u*positive, and forming the sum $L={L}^{(0)}+{L}^{(1)}+{L}^{(2)}\text{\hspace{0.17em}}\text{etc}\text{.}$ Each of these equations is of the formwhere

*f*

^{(}

^{n}^{)}(

*τ*) is a known function of

*τ,*derived either from

*Q*or

*L*

^{(}

^{n}^{-1)}(

*τ*). The solution to each differential equation in the set Eq. (6) can be developed by introducing an integrating factor (exp[

*τ*/

*u*]) and performing integration from

*τ*to

_{a}*τ*:

_{b}*τ*(

*u*> 0) we take

*τ*= 0 and

_{a}*τ*=

_{b}*τ*, while for upward radiance, we take

*τ*= infinity and

_{a}*τ*=

_{b}*τ*, with ${L}^{(n)}(\infty )=0.$ The first of these equations (

*n*= 0) yields

*λ*. Use of this lowest order solution represents the single scattering approximation to the inelastically-generated radiance.

For Raman scattering, *β _{I}* =

*β*, the dependence of the scattering on direction is similar to that for Rayleigh scattering but with a different depolarization factor [4]:

_{R}*β*over all solid angles is

_{R}*b*, the Raman scattering coefficient. Raman scattering at

_{R}*λ*is excited by a narrow band of wavelengths (Δ

*λ*) near

_{E}*λ*, where ${\lambda}_{E}=\lambda /(1+3.357\times {10}^{-4}\lambda )$ with the wavelengths given in nm. The Raman scattering coefficient for the entire Raman band is [4,11,12]

_{E}^{−1}.

The quantities related to the light field that are most often measured experimentally are vertical profiles of zenith propagating radiance (*L _{up}*) and upward (

*E*) and downward (

_{u}*E*) propagating irradiance. Let us examine the Raman contribution to

_{d}*L*first. In this case,

_{up}*u*= −1,

*ϕ*is irrelevant, and the expression for

*Q*is easily found by replacing the integral over

*λ*by simple multiplication by Δ

_{E}*λ*:

_{E}*E*

_{0}(

*z, λ*) at the excitation wavelength (the last entry in Eq. (1)). Then

_{E}*μ*, the downward irradiance

_{d}*E*(

_{d}*z, λ*), divided by

_{E}*E*

_{0}

*(*

_{d}*z, λ*), the downward scalar irradiance given by

_{E}*E*

_{0}

*(*

_{u}*z, λ*), is defined through a similar equation but with the integration limits on

_{E}*u*going from −1 to 0 rather than 0 to 1 in Eq. (15).

The lowest order approximation (Eq. (3)) to the desired upward radiance is then

*E*

_{0}(

*z, λ*). These could be available through detailed radiative transfer computations at

_{E}*λ*, experimental measurements at

_{E}*λ*, or some combination of the two.

_{E}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 *λ _{E}* = 464 nm. Monte Carlo simulations of the light field at 464 nm show that in this case ${\u3008{u}^{2}\u3009}_{E}$ is essentially independent of depth and that

*E*

_{0}(

*z, λ*) decays exponentially with depth: ${E}_{0}(z,{\lambda}_{E})={E}_{0}(0,{\lambda}_{E})\mathrm{exp}[-{K}_{0}({\lambda}_{E})z].$ When this is the case (and

_{E}*c*is constant) the integration can be carried out yielding

*K*

_{0}at the excitation wavelength. Comparison between the predictions of the above relationship (at

*z*= 0) and results of solving the full RTE (Eq. (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

*L*

_{up}^{(0)}(0,

*λ*) using Eq. (16) or (17) when the light field at the excitation wavelength is known. If we want the estimate

*L*

_{up}^{(0)}(0,

*λ*) in an experimental situation, usually the only properties of the excitation light field that would be measured are

*E*(

_{d}*z, λ*) and

_{E}*L*

_{u}^{(0)}(0,

*λ*) or

_{E}*E*(

_{u}*z, λ*). Thus, we need to estimate

_{E}*K*

_{0}(

*z, λ*),

_{E}*E*

_{0}(0,

*λ*) and ${\u3008{u}^{2}\u3009}_{E}$. (Note however, that instrumentation capable of measuringthe entire radiance distribution exists [21,22], and has recently been miniaturized [23], so increasingly in the future these needed quantities at the excitation wavelength will be directly available.) The estimate of

_{E}*K*

_{0}(

*z, λ*) can be effected by approximating it with

_{E}*K*(

_{d}*z, λ*). The scalar irradiance is given bywhere

_{E}*μ*is the average cosine of the upwelling light field, i.e,

_{u}*E*(

_{u}*z, λ*) /

_{E}*E*

_{0}

*(*

_{u}*z, λ*). Given that the radiance at

_{E}*λ*is strongly peaked in the direction of the refracted incident solar beam, we expect${\mu}_{d}\approx {u}_{0w},$ where

_{E}*u*

_{0}

*= cos(*

_{w}*θ*

_{0}

*) with*

_{w}*θ*

_{0}

*the polar angle of the refracted solar beam. Further, since*

_{w}*E*(0) >>

_{d}*E*(0), a precise approximation to

_{u}*μ*is not particularly important, and an asymptotic value of 0.4 is used [15]. If

_{u}*L*(0,

_{up}*λ*) is measured rather than

_{E}*E*(0,

_{u}*λ*), Eq. (18) with these approximations to

_{E}*μ*and

_{d}*μ*, can still be used with

_{u}*E*(0,

_{u}*λ*) replaced by π

_{E}*L*(0,

_{up}*λ*). This assumes that

_{E}*L*is independent of

_{u}*u*. The final quantity required can be approximated through ${\u3008{u}^{2}\u3009}_{E}\approx {u}_{0w}^{2},$ which is similar to the approximation ${\mu}_{d}\approx {u}_{0w}.$ The 4th column in Table 1 provides

*L*

_{up}^{(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 $\mu =-u$ the upward radiance in the direction (*μ,ϕ*) is

^{2}α yields terms constant in $\varphi $, proportional to $\mathrm{cos}(\varphi \text{'}-\varphi )$ and to $\mathrm{cos}2(\varphi \text{'}-\varphi )$ allowing the $\varphi $and the integrations to be carried out easily. The result is

*z*, and ${E}_{0}(z,{\lambda}_{E})={E}_{0}(0,{\lambda}_{E})\mathrm{exp}[-{K}_{0}({\lambda}_{E})z]$, the

*z*integral in Eq. (7) can be carried out analytically yielding

*K*

_{0}at the excitation wavelength serving as the decay coefficient. Table 1 compares the value of ${E}_{u}^{(0)}(0,\lambda )$ 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

*λ*, to which must be added the upwelling Raman irradiance at

_{E}*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

*E*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

_{d}*E*.

_{d}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 *K*_{0} depends strongly on depth, especially when the solar zenith angle is large. Using a bio-optical model similar to that in Ref [14], 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/m^{3} 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* + *b _{b}* 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, $a(\lambda )+{b}_{b}(\lambda )={u}_{0w}{K}_{d}(\lambda )$, which would be available from ${E}_{d}(z,\lambda )$, 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

*L*

_{up}^{(0)}(Approx.) refers to the same computation when

*E*

_{0}is replaced by its estimate from

*E*and

_{d}*L*, and the average squared cosine is replaced by (

_{up}*u*

_{0}

*)*

_{w}^{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

*L*

_{up}^{(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

*E*

_{0}and

*μ*) 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.

_{d}## 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 *E _{d}*(

*z*,

*λ*) and

*E*(

_{u}*z*,

*λ*) or

*E*(

_{d}*z*,

*λ*) and

*L*are measured throughout the spectrum, then the Raman contribution can be estimated in a manner similar to

_{up}*L*

_{up}^{(0)}(Approx.) and

*E*

_{u}^{(0)}(Approx.) in Table 1 and

*L*

_{up}^{(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*) +

*b*(

_{b}*z*,

*λ*) or

*u*

_{0}

*(*

_{w}K_{d}*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 [14] 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/m^{3}, the Raman contribution to *L _{up}* 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

*L*from the total would not exceed 2%, over and above any error in the measurement of

_{up}*L*.

_{up}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 [24], 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 [25].

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., *β _{I}* =

*β*is independent of the direction of the incident and scattered photons, so ${\beta}_{F}={b}_{F}(z,{\lambda}_{E}\to \lambda )/4\pi .$ 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 [26] for the chlorophyll

_{F}*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.

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