Abstract
Keywords
1 Background
Ocean optics belongs to the category of environmental optics. Compared to classical optics or laser optics, where the subject under study is an individual photon or a light beam, ocean optics studies light (or radiance) in a three-dimensional (3D) space, or the diffuse light. For a light beam, the size of a measurement sensor is significantly greater than the width of this light beam, but in ocean optics, it is completely the opposite where sensor's size is incomparable to the radiance environment; consequently, a completely different set of “laws” or relationships must be developed in order to adequately describe, and understand, the variation of radiance in this 3D space.
The quantities that can be adequately measured with a spectroradiometer in this 3D environment are radiance or irradiance, where the latter is an integration of radiance over a pre-defined angular range, which can also be viewed as a “broad-angle radiance.” The propagation of radiance (L) is governed by the radiative transfer equation (RTE)[
where z (in m) is depth from surface and positive, θ is the zenith angle, φ is azimuth angle, and dΩ' is an infinitesimal solid angle around angle (θ',φ'). c (in m-1) is the beam attenuation coefficient, with β (in m-1·sr-1) for water's volume scattering function, and both c and β are inherent optical properties (IOPs)[
Equation (1) provides a fundamental law regarding the loss and gain for radiance in a direction after an infinitesimal distance; but, due to its complexity, this equation is not directly applicable for the understanding of irradiance reflectance in water, nor for the inversion of water's inherent optical properties from a reflectance spectrum. A simplified relationship, but with a root in the radiative transfer equation, is required.
2 Simplification with clever algebraic derivations
To establish applicable relationships between apparent optical properties[
with Ed for downwelling irradiance, Eu for upwelling irradiance, and 2πd and 2πu for the solid angles in the downward and upward hemispheres, respectively. The vertical profiles of both Ed and Eu can be adequately measured with a planar irradiance spectroradiometer in the field.
The integration of the right side of Eq. (1) becomes:
where Eo is the scalar irradiance, and b (in m-1) is the scattering coefficient. Note that starting from here, the variation of IOPs with z is omitted for simplicity.
Since the beam attenuation coefficient is a sum of the absorption coefficient (a) and b, i.e.,
Eq. (2) and Eq. (3) suggest that:
This is the Gershun equation, which is significantly simpler than Eq. (1), but obtained completely differently many decades earlier[
In a similar manner, but not integrating over the 4π solid angle, rather the upper and lower hemispheres separately, Åas[
where bb (in m-1) is the backscattering coefficient, and μd and μu are the average cosine of downwelling and upwelling irradiance, respectively. Shape parameters rd and ru are defined as:
with Eod and Eou for the downwelling and upwelling scalar irradiance, respectively. Basically rd and ru reflects normalized reflectance coefficients of downward and upward scalar irradiance, respectively. For chlorophyll concentration in a range of 0.01--10.0 mg/m3 and assume the optical properties of other constituents co-vary with that of chlorophyll, the values of rd and ru were found in a range of ~1.2--20, with ru/rd ratio of roughly 1.4--2.2[
Equation (6) and equation (7) are the famous two-stream equations that describe the vertical variations of Ed and Eu, which show that the consequence of absorption is always a loss for both Ed and Eu, but the backscattering affects both positively (gain) and negatively (loss) for the propagation of these irradiances. Considering that the solar radiation comes from above the sea surface, Eu would be 0 (or waters will be black) if there is no backscattering. More importantly, the variations of the four radiance-distribution-related parameters (μd, μu, rd, and ru) vary in a much narrow range in natural aquatic environments[
3 Applications of two-stream equations
Define the diffuse attenuation coefficient of downwelling irradiance (Kd) and irradiance reflectance (R), respectively, as:
where Kd and R (as well as remote sensing reflectance described below) are the most important AOPs in ocean optics. Divide both side of Eq. (6) by Ed, we can get:
which, for simplicity, maybe written as:
This shows that, conceptually, since the four distribution parameters vary in a narrow range[
Separately, based on Eq. (5), after expanding Eo to Eod and Eou, and omitting the vertical variation of R with depth (for homogeneous waters), there is:
Further, since Eq. (12) indicates Kdμd-a is proportional to bb, the above expression indicates that there is:
Considering the ratio ru/rd is in a narrow range[
or,
This is an important and basic relationship in ocean optics that is achieved completely from the radiative transfer equation, which is echoed by Sathyendranath and Platt[
4 Extension to remote sensing reflectance
In a similar fashion and focusing on the upwelling radiance pointing to zenith [Lu(z,0,0)], a quantity can be measured by a remote sensor, and Zaneveld[
Basically, in the right side of Eq. (18), the first integration is over radiance going downward, i.e., it is the backscattering of downwelling radiance contributing to Lu(z,0,0). The second integration of the right side of Eq. (18), on the other hand, is over the radiance going upward, i.e., it is the forward scattering of upwelling radiance contributing to Lu(z,0,0). Further, Zaneveld[
with parameters fb and fL defined as (note that here fb has no 2π in the nominator, which then has a unit as sr-1 and is different from the original fb in Zaneveld, but the essence is the same):
Based on the definition of Eod and forward scattering coefficient (bf), Eq. (19) leads to:
Thus, for upwelling radiance going to zenith [Lu(z,0,0)], applying both Eq. (1) and Eq. (22), there is:
Define ratio (rs, in sr-1) of upwelling radiance to downwelling scalar irradiance and the diffuse attenuation coefficient of upwelling radiance (KLu), respectively, as:
and rs can then be written as
Further, as c=a+b=a+bb+bf, and consider the diffuse attenuation coefficient is in general a function of a and bb [see Eq. (13)], we get:
Furthermore, since downwelling scalar irradiance can be converted to downwelling planar irradiance through the average cosine of downwelling irradiance (μd), the in-water remote-sensing reflectance (rrs), defined as the ratio of upwelling radiance to downwelling planar irradiance, is:
Again, this is simply a mathematical re-write of the RTE, as Zaneveld[
5 Numerical parameterizations
The above expressions provide a general guidance between AOPs and IOPs, and it is necessary to parameterize the formulations for the purpose to derive IOPs from AOPs or to estimate AOPs from the measurement of IOPs. Unfortunately, this parameterization could not be derived from the RTE, and must rely on data or numerical simulations. Using data simulated from Monte Carlo or Hydrolight[
1) Irradiance reflectance
Equation (16) or (17) has been commonly simplified to:
with f approximated as 0.33 in Ref. [12], while Morel and Gentili[
with u for
where U represents surface wind speed, and θs represents subsurface solar zenith angle. Values of p1--p6 can be found in
Separately, for reflectance at a wavelength where bb is much greater than a (roughly>2, i.e., high scattering, weak absorption condition)[
However, because the absorption coefficient of aquatic environment is highly spectrally dependent, even for waters with high load of sediments, only some wavelengths meet this bb>2a condition, and thus Eq. (32) may not work well to model an R spectrum from the spectra of bb and a[
2) Remote sensing reflectance
Based on Eq. (28), also for simple parameterization, rrs has been commonly approximated as a function of u:
with the variation of g further expressed as a function of u by Gordon et al.[
For nadir-viewing rrs, values of g0 and g1 are found as 0.0949 sr-1 and 0.0794 sr-1 through Monte Carlo simulations[
where values of p1--p7 can be found in
To account for the different phase functions of molecular scattering and particle scattering, it was proposed to express rrs using two separate terms,
with
for nadir-viewing rrs after Hydrolight simulations[
3) Diffuse attenuation coefficient of downwelling irradiance
The formulation for Kd [Eq. (12) and Eq. (13)] indicates that this property also varies with depth (light field) even for homogeneous waters. For the averaged attenuation between surface and a depth where 10% of surface solar radiation remains, Lee et al.[
with ηw for the ratio of bbw/bb, bbw for the backscattering coefficient of pure (sea)water, and θa for solar zenith angle in air (in degree).
Based on Hydrolight simulations and for the subsurface diffuse attenuation coefficient of downwelling irradiance [Kd(0)], Albert and Mobley[
which is similar to that found by Gordon[
6 Contributions from chlorophyll fluorescence and Raman scattering
The above discussions, including Eq. (1), omitted the contributions from inelastic processes, such as those of chlorophyll fluorescence and Raman scattering. While these two are generally small in the surface layer of the ocean compared to the downwelling irradiance from the Sun and sky, they can be significant for some wavelengths and some waters in the upwelling radiance, and thus can be detected for sensors in remote platforms. For chlorophyll fluorescence induced by solar radiation and considering the water column is homogeneous and focusing on the emission wavelength (λem), after some approximations, Huot et al.[
where ϕ is the quantum yield of chlorophyll fluorescence,
Further, for radiance from Raman scattering and applying a single scattering approximation along with an assumption of homogeneous water, Westberry et al.[
where t is the transmittance across the air-water interface, n is the refractive index of water, βr is the Raman phase function, br is the Raman scattering coefficient, KL is the attenuation coefficient for upwelling radiance at emission wavelength, λex is the excitation wavelength, and κ is the diffuse attenuation coefficient for radiance backscattered at a depth propagating towards the surface[
7 Concluding remarks
Through pure algebraic derivations, Åas[
References
[3] Åas E. Two-stream irradiance model for deep waters[J]. Applied Optics, 26, 2095-2101(1987).
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