1. Introduction

The upward radiance, just beneath the water surface, due to backscattered solar radiation in natural waters, $L_u^{-}\left({\theta }_w,{\varphi }_w\right)$ propagating in a direction specified by the angles (θ_w, ϕ_w), contains information regarding the constituents of the water [1]. As such it is of considerable interest to the remote sensing community. This subsurface radiance leads to an above-surface radiance exiting the water, the water-leaving radiance $L_u^{+}\left({\theta }_a,{\varphi }_a\right)$, propagating in the direction (θ_a, ϕ_w). It is well established in the literature [2–4] that these above and below upwelling radiances are related by

In a recent publication in this journal Dev and Shanmugan [5] called Eq. (1) into doubt and made the claim that the transmittance factor (referred to there as τ_w,a) should be replaced by a factor that depended on the optical properties the water, rather than on just n_w and (θ_w, ϕ_w). In support of that claim they examined a hypothetical situation in which radiance is incident on a water body having no absorption, so that all photons entering the water body eventually escape. Their conclusion was, that because of multiple surface reflections of photons within the water, in this particular case the transmittance factor must be unity rather than given by Eq. (1). We challenge this conclusion.

To demonstrate that Eq. (1) is correct, we will apply it to a situation in which radiative transfer in a water body is modeled in a simple manner. We assume a totally transparent atmosphere. The water body is modeled as a thin, totally transparent (non-absorbing and non-scattering) layer of water with refractive index n_w bounded below by a lambertian-reflecting surface of albedo (irradiance reflectivity) R. The lambertian-reflecting surface is employed to simulate the bulk of the water body, which reflects downwelling radiance and produces upwelling radiance. The purpose of the thin transparent layer is to provide the refractive index discontinuity between the water and the air and to allocate a region in which multiple reflections between the bulk of the water body and the surface can be enumerated. Multiple reflections play a central role in our arguments and in those of [5].

In natural waters, the irradiance ratio $R=E_u^{-}/E_d^{-}$ can be expressed in terms of the inherent optical properties of the medium according to R ≈ fb_b/a, where a is the absorption coefficient and b_b the backscattering coefficient of the water plus constituents [1]. The quantity f depends somewhat on the scattering phase function of the particles and on the solar zenith angle. For small solar zenith angles and most oceanic waters, f ≈ 0.33 [1]. The assumption of the lambertian-nature of the hypothetical surface means that the radiance distribution reflected from the bulk of the water body is uniform. Generally it is not, but this assumption is made to simplify the computations here.

Now, consider illuminating the water from above by irradiance $E_d^{+}$ incident on the surface. For R = 1, the situation is identical to that considered in [5], Section 2.1.1, i.e., all photons will eventually exit the water. We will compute the radiance exiting the surface by explicitly keeping track of the contribution of multiple reflections from the interface to $L_u^{-}\left({\theta }_w,{\varphi }_w\right)$ and then use Eq. (1) to propagate the resulting radiance through the air-water interface. From $L_u^{+}\left({\theta }_a,{\varphi }_a\right)$ we then determine the associated upwelling irradiance $E_u^{+}$, and show that all photons do in fact exit the water when R = 1, i.e., energy is conserved if Eq. (1) is employed as written.

2. Radiative transfer computation

First, the incident irradiance $E_d^{+}$ is transmitted through the interface via the Fresnel transmittance $t_f^{+}$ evaluated at normal incidence. This transmitted irradiance then reflects from the Lambertian surface to yield an upward radiance $L_{u0}^{-}=E_d^{+}{t}_f^{+}R/\pi$, where the factor π results from the fact that the upward irradiance is totally diffuse. This is the component of $L_u^{-}\left({\theta }_w,{\varphi }_w\right)$ that has not yet been internally reflected from the interface (indicated by the subscript 0). It is then reflected downward leading to a downward radiance $L_{u0}^{-}{r}_f^{-}\left({\theta }_d\right)$, where (θ_d, ϕ_d) is the downward direction of the reflected radiance incident from the direction (θ_w, ϕ_w) and $r_f^{-}$ is the Fresnel reflectance. This leads to a downward irradiance $E_{d1}^{-}={\displaystyle {\int }_{{\Omega }_d}{L}_{u0}^{-}{r}_f^{-}\left({\theta }_d\right)\cos {\theta }_{d}d{\Omega }_d=\pi \overline{r}{L}_{u0}^{-}}$, where $\overline{r}\equiv 2{\displaystyle {\int }_0^{\pi /2}{r}_f^{-}\left({\theta }_d\right)\sin {\theta }_d\cos {\theta }_{d}d{\theta }_d}$, is the internal reflectance for diffuse irradiance and has a value of approximately 0.475 for water. This downward irradiance is then reflected upward leading to a second upwelling radiance $L_{u1}^{-}=\left(\overline{r}R\right)L_{u0}^{-}$: the upward radiance due to photons that have been reflected once from the interface. Continuing this process, $L_{u2}^{-}=\left(\overline{r}R\right)L_{u1}^{-}={\left(\overline{r}R\right)}^2{L}_{u0}^{-}$, etc. The total radiance incident on the underside of the surface is then

3. Discussion

After accounting for all orders of multiple reflections of upwelling radiance from the water side of the air-water interface in this simple example, we have shown that application of Eq. (1) yields an above-water radiance, that when integrated, provides the correct result for the reflected irradiance — specular and diffuse — in the two limiting cases R = 0 and R = 1.

Where was the error that the authors of [5] made in concluding that Eq. (1) was inadequate? We believe they incorrectly interpreted the radiance on the right-hand-side of Eq. (1) as being the radiance before multiple reflections, i.e., $L_{u0}^{-}$ of Eq. (2). We note that $L_u^{-}$ is the actual radiance that would be measured by a radiometer placed just beneath the water surface. The quantity $L_{u0}^{-}$ cannot be directly measured, it must be calculated from Eq. (2).

4. Conclusion

Our conclusion is that Eq. (1) is correct, and there is no relation between the radiance transmittance of the interface, i.e., Eq. (1), and the optical properties of the water, other than n_w. If an investigator measures $L_u^{-}\left({\theta }_w,{\varphi }_w\right)$, then he/she can be confident that $L_u^{+}\left({\theta }_a,{\varphi }_a\right)$ is determined by Eq. (1).