## 1. Introduction

[2] Variations in the depth of penetration of solar radiation play an important role in the physics and biology of coastal water columns. Examples include the solar heating of surface layers [*Zaneveld et al*., 1981; *Kirk*, 1988; *Rochford et al*., 2001], control of the depth to which net phytoplankton production can be sustained [*Behrenfeld and Falkowski*, 1997], the effectiveness of visual prey detection [*Aksnes and Giske*, 1993], and the degree of illumination of benthic communities [*Gattuso et al*., 2006]. These phenomena have traditionally been studied in isolation, but recent work indicates that there are strong interactions between changes in water transparency, thermal effects, biological production, and biogeochemical cycling [*Cahill et al*., 2008; *Zhai et al*., 2011]. Solar illumination at the sea surface is conventionally measured as spectrally resolved downward planar irradiance, *E _{d}* (

*λ*), and the attenuation of this quantity with depth (

*z*) can be described [

*Mobley*, 1994] by the diffuse attenuation coefficient

*K*(

_{d}*z,λ*), defined as

[3] Consequently, in order to determine values of *K _{d}* (

*z,λ*) in situ,

*E*(

_{d}*z*,

*λ*) should be measured at depths which are sufficiently closely spaced that a further reduction in Δ

*z*would have no significant effect on the calculated

*K*(

_{d}*z,λ*) value. However practical limitations, including the finite data acquisition rate of profiling radiometers and the relatively wide spacing of instruments on moored arrays, mean that

*E*(

_{d}*z,λ*) is often measured over intervals that are wider than those required to satisfy equation (1). This makes it necessary to introduce a slightly different quantity, , which describes the attenuation of downward planar irradiance over a finite interval

[4] *Lee et al*. [2005a] introduced the notation for the case where is calculated from immediately beneath the surface to the depth where *E _{d}* (

*λ*) falls to 10% of its surface value. Since

*K*(

_{d}*z,λ*) is an apparent optical property (

*AOP*) which depends not only on the inherent optical properties (

*IOPs*) of the water column, but also on the angular structure of the light field which changes with depth, is not necessarily equal to the average value of over the same depth interval [

*Gordon*, 1989]. Early attempts to derive approximate relationships between and

*IOPs*employed expressions involving the coefficients of absorption,

*a*(

*λ*), and scattering,

*b*(

*λ*) [

*Gordon et al*., 1975;

*Kirk*, 1984], but the use of the backscattering coefficient (

*b*) rather than the total scattering coefficient (

_{b}*b*) would permit a more direct link to remote sensing observations. Two slightly different approaches to this problem have been published by

*Gordon*[1989] and

*Lee et al*. [2005a]. Both are based on the statistical fitting of empirical expressions to extensive radiative transfer calculations, and their validity depends on the extent to which the assumptions incorporated in these calculations are representative of real water columns.

[5] The *Gordon* [1989] model can be written as

where *D _{0}* is the distribution function for downward irradiance just below the sea surface.

*D*varies from 1.02 to 1.35 depending on wavelength, solar angle and cloud cover, with solar angle being the most important determining factor. The constants

_{0}*k*

_{1},

*k*

_{2}, and

*k*

_{3}in equation (3) have values of 1.3197, −0.7559, and 0.4655 respectively. The model proposed by

*Lee et al*. [2005a] employs a two-flow formulation of the radiative transfer problem to partially separate the contributions of

*a*(

*λ*) and

*b*(

_{b}*λ*), resulting in the expression

where *θ _{a}* is the solar zenith angle. In a subsequent paper,

*Lee et al*. [2005b] derived from remote sensing reflectance (

*R*) in two stages by (i) estimating the coefficients

_{rs}*a*and

*b*from

_{b}*R*and (ii) using these

_{rs}*a*and

*b*values to estimate . This two-stage procedure performed well in Case 2 waters, providing indirect support for the model proposed by

_{b}*Lee et al*. [2005a] relating to

*a*and

*b*. To date, however, no independent comparisons of equations (3) and (4) with in situ measurements in optically complex waters have been published. Such comparisons are necessary for validation purposes because radiative transfer calculations inevitably make simplifying assumptions about poorly measured variables such as volume scattering functions, the specific optical properties of individual optically significant constituents, and the distribution of direct and indirect solar irradiances. In this paper, therefore, we compare predictions of derived from equations (3) and (4) with measurements made in situ in a range of water types located off the west coast of the United Kingdom. The data set assembled for this purpose also made it possible to investigate the relationship between measured values of and the depth of penetration of spectrally integrated photosynthetically available radiation (

_{b}*PAR*) in these waters.