It is necessary to consider two strata for a coupled atmosphere-ocean system; one for the atmosphere with real part of the refractive index matm,r and another for the ocean with real part of the refractive index mocn,r . The basic radiance, defined as I / m2, where mr is the real part of the refractive index, is a conserved quantity. Thus, if we neglect reflection losses, the basic radiance will be constant across the interface between the two strata. Assuming for simplicity that the interface is flat and smooth (a calm ocean), then the radiance must satisfy Snell's law and Fresnel's equations. As illustrated in Fig. 9.3, the downward radiation distributed over 2 n sr in the atmosphere will be restricted to a cone less than 2n sr (referred to as region II in Fig. 9.3) after being refracted across the interface into the ocean. Outside the refractive region in the ocean, i.e., in the total reflection region (referred to as region I in Fig. 9.3), the radiation is due to in-water multiple scattering. The demarcation between the refractive region and the total reflection region in the ocean is given by the critical angle dc given by

/uc = cos 6>c = sj 1 -1/ m^ , where mrel = mocn r / matm r is the real part of the refractive index in the ocean relative to that in the atmosphere.

Because the radiation field in the ocean is driven by solar radiation passing through the atmosphere-water interface, Eq. (9.19) applies also in the ocean, but the solar pseudo-sources are different in the two media. In the atmosphere we have:

SL (?,U) = P(J, ~Mo,U) + Ps ("M); mrel) P (J,M^,U)

4n 4n

Here the first term is the usual solar beam pseudo-source, and the second term is due to the reflection occurring at the interface, which is proportional to ps mrel), the specular reflectance. The pseudo-source in the ocean is just the attenuated solar beam refracted through the interface:

s;cn(r, u) = "p-Tb (-^ mrel) p(*, ~Mom,u)e"' / "e" ^ m

Here Tb (-^„; mrel) is the beam transmittance through the interface, and ju0m is the cosine of the solar zenith angle in the ocean, which is related to ju0 by Snell's law /u0m = M0mC«0,mrel) = yj 1 - (1 -^02)/m2el . With the use of the appropriate pseudo-sources for the atmosphere [ S^(r, u)] and ocean [ S*cn(r, u)], Eq. (9.19) can now be solved subject to boundary conditions at the top of the atmosphere and the bottom of the ocean. However, we must also properly account for the reflection from and transmission through the atmosphere-ocean interface by requiring that Fresnel's equations are satisfied.

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