• Acta Optica Sinica
  • Vol. 42, Issue 12, 1200004 (2022)
Zhongping Lee*
Author Affiliations
  • State Key Lab of Marine Environmental Science, College of Ocean and Earth Sciences, Xiamen University, Xiamen 361102, Fujian, China
  • show less
    DOI: 10.3788/AOS202242.1200004 Cite this Article Set citation alerts
    Zhongping Lee. Beauty of Math in Ocean Optics: Two-Stream Equations of Åas[J]. Acta Optica Sinica, 2022, 42(12): 1200004 Copy Citation Text show less

    Abstract

    Solar radiation in the visible domain can penetrate aquatic environment, which drives photon-related processes including phytoplankton photosynthesis and heating of the upper water column. In addition, the scattered light in the water column can emerge (escape) from water, which forms the bases to sense properties in aquatic environments using sensors onboard satellites. Thus, an understanding of the processes and properties related to the propagation of solar radiation in-and-out of water is a basic requirement in ocean optics and ocean color remote sensing. The spatial (and spectral for inelastic scattering) variation of radiance is governed by the radiative transfer equation, which is not directly applicable to infer in-water optical properties or to describe the relationships between the optical properties measured in the field and inherent optical properties related to environmental properties. Through simple mathematical derivations, or manipulations, of the radiative transfer equation (RTE), ?as transferred the RTE into a set of two equations describing the change of upwelling and downwelling irradiance with depth, and further obtained concise analytical relationships between the apparent and inherent optical properties. These equations not only form the basic theoretical relationships in ocean optics, but also lay the foundation of semi-analytical algorithms in ocean color remote sensing.

    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)[1], and for radiance in the aquatic environment, it can be written as:

    dL(z,θ,φ)dzcosθ=-c(z)L(z,θ,φ)+β(z,θ',φ'θ,φ)L(z,θ',φ')dΩ',(1)

    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)[2]. Note that in this equation, it is assumed that there are no internal sources such as fluorescence and Raman scattering, otherwise a third term should be included for radiance from such processes. Also, the variable wavelength (λ) is omitted for brevity.

    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[2] and IOPs, Åas[3] worked on Eq. (1) by integrating both sides of Eq. (1) over the 4π solid angle. The left side becomes:

    ddzΩ=04πL(z,θ,φ)cosθdΩ=ddzΩ=02πdL(z,θ,φ)cosθdΩ-Ω=02πuL(z,θ,φ)cosθdΩ=ddz[Ed(z)-Eu(z)],(2)

    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:

    -cΩ=04πL(z,θ,φ)dΩ+Ω=04πβ(z,θ',φ'θ,φ)L(z,θ',φ')dΩ'dΩ=-cEo(z)+Ω'=04πL(z,θ',φ')β(θ',φ'θ,φ)dΩdΩ'=-cEo(z)+bEo(z),(3)

    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.,

    c=a+b,(4)

    Eq. (2) and Eq. (3) suggest that:

    ddz[Ed(z)-Eu(z)]=-aEo(z).(5)

    This is the Gershun equation, which is significantly simpler than Eq. (1), but obtained completely differently many decades earlier[4]. This equation suggests that, if the three quantities (Eo, Ed, and Eu) can be accurately measured in the field, a profile of the absorption coefficient can then be calculated.

    In a similar manner, but not integrating over the 4π solid angle, rather the upper and lower hemispheres separately, Åas[3] obtained a set of equations after introducing shape factors rd and ru:

    dEd(z)dz=-aμd(z)Ed(z)-rd(z)bbμd(z)Ed(z)+ru(z)bbμu(z)Eu(z),(6)dEu(z)dz=-aμu(z)Eu(z)-ru(z)bbμu(z)Eu(z)+rd(z)bbμd(z)Ed(z),(7)

    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:

    rd(z)=1bbEod(z)×Ω'=02πdΩ=02πuβ(θ',φ',θ,φ)dΩL(z,θ',φ')dΩ',(8)ru(z)=1bbEou(z)×Ω'=02πuΩ=02πdβ(θ',φ',θ,φ)dΩL(z,θ',φ')dΩ',(9)

    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[5].

    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[5-6], which leave the change of Ed and Eu mainly governed by a and bb.

    3 Applications of two-stream equations

    Define the diffuse attenuation coefficient of downwelling irradiance (Kd) and irradiance reflectance (R), respectively, as:

    Kd=-1EddEddz,(10)R=EuEd,(11)

    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:

    Kd(z)=aμd(z)+rd(z)μd(z)-ru(z)R(z)μu(z)bb,(12)

    which, for simplicity, maybe written as:

    Kd(z)=m(z)a+ν(z)bb.(13)

    This shows that, conceptually, since the four distribution parameters vary in a narrow range[5-6], the variation of Kd is mainly driven by a and bb. Further, in principle, since the two scaling parameters (m, ν) of a and bb to Kd do not equal, Eq. (12) and Eq. (13) indicate that the weightings of a and bb to Kd are not the same, contrary to commonly adopted approximations.

    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:

    R=μuμdKdμd-aa+μuKd.(14)

    Further, since Eq. (12) indicates Kdμd-a is proportional to bb, the above expression indicates that there is:

    R=μuμd1-μdμururdRrdbba+μuKd.(15)

    Considering the ratio ru/rd is in a narrow range[5-6] and the value of R is small (generally less than a few percent for oceanic waters), the above equation approximates:

    Rμuμdrdbba+μuKd,(16)

    or,

    Rbbxa+ybb.(17)

    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[7] through a quasi-single-scattering approximation. This relationship shows that to the first order, irradiance reflectance is proportional to the ratio of bb/(a+bb) (after model parameters x and y are approximated as equal), with bb appearing in both nominator and denominator to reflect its positive and negative effects in the propagation of the two streams of irradiance. One important implication of bb in both nominator and denominator is that when bb is significantly greater than a (for instance, at some wavelengths for waters with extremely high load of suspended sediments), R will approach an asymptotic value, instead of increasing proportionally (linear or nonlinear) with the increase of concentrations of suspended sediments.

    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[8] separated the integration term in the right side of Eq. (1) into the upper and lower hemispheres, which becomes:

    Ω'=04πβ(θ',φ'0,0)L(z,θ',φ')dΩ'=φ'=02πθ'=0π/2β(θ',φ'0,0)L(z,θ',φ')sinθ'dθ'dφ'+φ'=02πθ'=π/2πβ(θ',φ'0,0)L(z,θ',φ')sinθ'dθ'dφ'.(18)

    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[8] utilized the observations that the volume scattering function in the backscattering domain and the upwelling radiance do not vary greatly for different angles, and then wrote Eq. (18) as:

    Ω'=04πβ(θ',φ'0,0)L(z,θ',φ')dΩ'=fb(z,0,0)bbφ'=02πθ'=0π/2L(z,θ',φ')sinθ'dθ'dφ'+fL(z,0,0)Lu(z)φ'=02πθ'=0π/2β(θ',φ'0,0)sinθ'dθ'dφ',(19)

    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):

    fb(z,0,0)=1bbEod(z)φ'=02πθ'=0π/2β(θ',φ'0,0)L(z,θ',φ')sinθ'dθ'dφ',(20)fL(z,0,0)=1bfLu(z,0,0)φ'=02πθ'=π/2πβ(θ',φ'0,0)L(z,θ',φ')sinθ'dθ'dφ'.(21)

    Based on the definition of Eod and forward scattering coefficient (bf), Eq. (19) leads to:

    Ω'=04πβ(θ',φ'0,0)L(z,θ',φ')dΩ'=fb(z,0,0)bbEod(z)+fL(z,0,0)Lu(z,0,0)bf.(22)

    Thus, for upwelling radiance going to zenith [Lu(z,0,0)], applying both Eq. (1) and Eq. (22), there is:

    -dLu(z,0,0)dz=-cLu(z,0,0)+fb(z,0,0)bbEod(z)+fL(z,0,0)Lu(z,0,0)bf.(23)

    Define ratio (rs, in sr-1) of upwelling radiance to downwelling scalar irradiance and the diffuse attenuation coefficient of upwelling radiance (KLu), respectively, as:

    rs(z,0,0)=Lu(z,0,0)Eod(z),(24)KLu(z,0,0)=-dLu(z,0,0)Lu(z,0,0)dz,(25)

    and rs can then be written as

    rs(z,0,0)=fb(z,0,0)bbKLu(z,0,0)+c-fL(z,0,0)bf.(26)

    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:

    rs(z,0,0)=fb(z,0,0)bb(1+m')a+(1+v')bb+[1-fL(z,0,0)]bf.(27)

    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:

    rrs(z)=1μd(z)fb(z)bb(1+m')a+(1+v')bb+[1-fL(z)]bf.(28)

    Again, this is simply a mathematical re-write of the RTE, as Zaneveld[8] pointed out, “it is an exact solution”. What remain unknown are the values of the modeling parameters (fb, fL, m', v'). Further, these parameters vary in a narrow range, such as fL being in a range of 1.0--1.1[9], and thus the physics meaning of this formulation is very clear: remote sensing reflectance is mainly driven by the absorption and backscattering coefficients.

    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[10-11], many approximations or numerical parameterizations have been proposed, which include:

    1) Irradiance reflectance

    Equation (16) or (17) has been commonly simplified to:

    R=fbba+bb,(29)

    with f approximated as 0.33 in Ref. [12], while Morel and Gentili[13] developed a Look-Up-Table (LUT) for “Case-1” waters. On the other hand, from more than 22000 Hydrolight simulations, Albert and Mobley[14] proposed a formulation for R as:

    R=p1(1+p2u+p3u2+p4u3)×1+p51cosθs(1+p6U)u,(30)

    with u for

    u=bba+bb,(31)

    where U represents surface wind speed, and θs represents subsurface solar zenith angle. Values of p1--p6 can be found in Table 3 in Ref. [14].

    Separately, for reflectance at a wavelength where bb is much greater than a (roughly>2, i.e., high scattering, weak absorption condition)[15], it is found that the Kubelka-Munk model is also applicable[16], where R is described as

    R=bb/a1+bb/a+1+2bb/a.(32)

    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[17].

    2) Remote sensing reflectance

    Based on Eq. (28), also for simple parameterization, rrs has been commonly approximated as a function of u:

    rrs=gu,(33)

    with the variation of g further expressed as a function of u by Gordon et al.[18]:

    g=g0+g1u.(34)

    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[18]. In addition to this quadratic formulation for rrs, Albert and Mobley[14] proposed to use 4-th order polynomials:

    rrs=p1(1+p2u+p3u2+p4u3)×1+p51cosθs(1+p6U)1+p71cosθvu,(35)

    where values of p1--p7 can be found in Table 3 of Albert and Mobley[14], with θv for sensor's viewing angle in water.

    To account for the different phase functions of molecular scattering and particle scattering, it was proposed to express rrs using two separate terms,

    rrs(λ,Ω)=gw(Ω)bbw(λ)a(λ)+bb(λ)+gp(λ,Ω)bbp(λ)a(λ)+bb(λ),(36)

    with

    gw=0.113,gp=0.1971-0.636exp-2.552bbpa+bb,(37)

    for nadir-viewing rrs after Hydrolight simulations[19]. Here bbw and bbp are the backscattering coefficients of pure (sea) water and particles, respectively (bb=bbw+bbp), while gw and gp represent different weightings of molecule and particle backscatterings contributing to rrs.

    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.[20] proposed the following approximation:

    K-d=(1+0.005θa)a+4.26(1-0.265ηw)(1-0.52e-10.8a)bb,(38)

    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[14] obtained:

    Kd(0)=1.055a+bbcosθs,(39)

    which is similar to that found by Gordon[21] from Monte Carlo simulations. In this kind of formulations, however, the weightings of a and bb are considered equal, which is not exactly matching that derived from the radiative transfer equation [see Eq. (12)], although a numerical estimation of Kd may not differ much.

    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.[22] obtained a relationship of radiance due to chlorophyll fluorescence (Lf) as:

    Lf(λem)=14πϕQa*(λem)Cf(λem)λ=400700aph(λ)Eo(λ,0)K(λ)+af(λem)dλ,(40)

    where ϕ is the quantum yield of chlorophyll fluorescence, Qa*is the portion of emitted fluorescence not reabsorbed within the cell, Cf is the proportionality factor that converts fluorescence at λem to the whole fluorescence band, aph is the absorption coefficient of phytoplankton, K is the diffuse attenuation coefficient of scalar irradiance, and af is the attenuation coefficient of upwelling fluorescence radiance. Ratio of Lf to Ed(0-) provides the subsurface remote sensing reflectance due to fluorescence radiance.

    Further, for radiance from Raman scattering and applying a single scattering approximation along with an assumption of homogeneous water, Westberry et al.[23] obtained a formulation for nadir-viewing remote sensing reflectance due to Raman scattering as:

    Rrs,Raman(λem)=t2n2βr(θsπ)br(λem)Ed(0+,λex)[Kd(λex)+KL(λem)]Ed(0+,λem)×1+bb(λex)μu[Kd(λex)+κ(λex)]+bb(λem)2μuκ(λem),(41)

    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[24].

    7 Concluding remarks

    Through pure algebraic derivations, Åas[3] and Zaneveld[8] obtained relationships that show the fundamental dependence of AOPs (irradiance reflectance, diffuse attenuation coefficient, and remote sensing reflectance) on IOPs (in particular, absorption and backscattering coefficients). Although the exact values of the introduced variables could not be derived from the radiative transfer equation, these relationships provide a clear physics guide on the most important properties and the way of dependences. While it appears today that more and more practices use data or “big data” to answer science questions, the algebraic manipulations presented by Åas and Zaneveld highlight the power of math and physics in finding the core relationships governing properties in the natural environment.

    References

    [3] Åas E. Two-stream irradiance model for deep waters[J]. Applied Optics, 26, 2095-2101(1987).

    Zhongping Lee. Beauty of Math in Ocean Optics: Two-Stream Equations of Åas[J]. Acta Optica Sinica, 2022, 42(12): 1200004
    Download Citation