rta_sa: Radiative transfer semi-analytical approximations
In AlexCast/rho: R Hydrology Optics

Description Usage Arguments Details Value References See Also Examples

This function calculates the subsurface bi-hemispherical and hemispherical-directional water-leaving reflectance over optically deep or optically shallow waters.

1	rta_sa(a, bb, ...)

`a`	Total absorption coefficient (1/m).
`bb`	Total back-scattering coefficient (1/m).
`rho_b`	Bottom bi-hemispherical reflectance (unitless).
`depth`	Bottom depth, positive downwards (m).
`theta_s`	Sun zenith angle refracted underwater (radians).
`theta_v`	Observation nadir angle refracted underwater (radians).
`wsp`	Wind speed (m/s). Only for model "Albert-Mobley03".
`bbp`	Particle back-scattering (1/m). See details.
`aop`	Desired reflectance. One of: 'rho' or 'rrs'. See details.
`model`	Semi-analytical parametrization to use. Possible values are: "Albert-Mobley03" and "Lee98". See details.

In Hydrology Optics, the bi-hemispherical and hemispherical- -directional reflectances are commonly named irradiance refelctance ("R" or "Rho") and remote sensing reflectance ("Rrs"), respectively. Those are two apparent optical properties of the medium. This function uses semi-analytical parametrizations to calculate their subsurface values, i.e., at depth 0 just below the water surface. The lower case names are used to indicate the subsurface quantities. Those quantities can be propagated into air, just above the water surface, with the function propagate_r.

An optically deep medium is equivalent of a semi-infinite medium, i.e., the bottom depth is infinite and has no contribution to the water-leaving reflectance. Optically shallow water are defined as the complement, when there is "measurable" influence of bottom albedo (bi-hemispherical reflectance).

Model "Albert-Mobley03" is used as default. It implements the equations and coefficients provided in Albert & Mobley (2003). The model includes an extended range of water medium properties, the effect of wind speed and include specific coefficients for "rho" and "rrs".

Model "Lee98" is also available. It implements the equations and coefficients provided in Lee et al. (1998), with the corrections of Lee et al. (1999). The model is specifically parametrized for "rrs", and "rho" is calculated (with a warning) by assuming that the diffuse subsurface hemispherical-directional water-leaving reflectance is Lambertian.

Both models can be used to calculate 'rrs' at off-nadir observation. The model "Lee98" requires that the backscattering of particles (bbp) be provided separately in case of off-nadir observation. The "Albert-Mobley03" also allows to calculate off-zenith illumination, while the model of "Lee98" is an average fit to three Sun zenith angles (0, 30 and 60 degrees, in air).

If the arguments passed to the function are outside the range used for parametrization of the specific model, values will be return with a warning of extrapolation beyond model domain.

Note that although the base data for the parametrization of both models includes the inelastic scattering (vibrational Raman from water and fluorescence from pigments and dissolved organic carbon), the semi-analytical approximations do not model those processes. Simulations therefore should present higher errors when those processes are important contributors, particularly pigment fluorescence will produce a spectrally localized error.

A numeric vector with the subsurface bi-hemispherical ('r', unitless) or hemispherical-directional ('rrs', 1/sr) water-leaving reflectance.

Albert, A.; Mobley, C. D. 2003. An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters. Optics Express 11, 22, 2873-2890. DOI: 10.1364/oe.11.002873

Lee, Z.-P.; Carder, K. L.; Mobley, C. D.; Steward, R. G.; Patch, J. S. 1998. Hyperspectral remote sensing for shallow waters. I. A semianalytical model. Applied Optics 37, 27, 6329-6338. DOI: 10.1364/AO.37.006329

Lee, Z.-P.; Carder, K. L.; Mobley, C. D.; Steward, R. G.; Patch, J. S. 1999. Hyperspectral remote sensing for shallow waters: 2. Deriving bottom depths and water properties by optimization. Applied Optics 38, 18, 3831-3843. DOI: 10.1364/AO.38.003831

propagate_r

# Calculate the subsurface hemispherical-directional water-leaving 
# reflectance of average pure seawater in the visible range for optically 
# deep condition:
a  <- a_water(lambda = 400:700, S = 34.72, Tc = 3.5)
b  <- b_water(lambda = 400:700, S = 34.72, Tc = 3.5)
na <- n_air(lambda = 400:700)
nw <- n_water(lambda = 400:700, S = 34.72, Tc = 3.5, type = "complex")
theta_sw <- snell_decomp(snell(rad(30), ni = na, nr = nw), nr = nw)[, 1]
r_am     <- rta_sa(a = a, bb = b / 2, theta_s = theta_sw, 
                   model = "Albert-Mobley03", aop = "rrs")
r_lee    <- rta_sa(a = a, bb = b / 2, theta_s = theta_sw, model = "Lee98", 
                   aop = "rrs")
plot(400:700, r_am, type = 'l')
lines(400:700, r_lee, lty = 2)