Spectral dependence of the seawater-air radiance transmission coefficient

Abstract

The transmission coefficient, T_L, commonly used to propagate the upwelling nadir radiance, just below the ocean surface, to above the surface has been assumed to be a constant value of 0.543 in seawater. Because the index of refraction of seawater varies with wavelength, salinity, and temperature, the variation of T_L with these parameters should be taken into account, especially if low uncertainty is required for the quantities derived using T_L. In particular the wavelength dependence of this factor is important. For example at a salinity of 35 g/kg and a temperature of 26° C, T_L will be 1.3% lower at 380 nm and 1.1 % higher at 700 nm than the constant value (0.543) and should be taken into account when calculating the water leaving radiance and normalized water leaving radiance from in-water measurements.

1. Introduction

In ocean color satellite measurements, system vicarious calibration (SVC) is used to improve the total system performance and retrieval of the satellite derived water leaving radiance, L_w(λ) and the normalized water leaving radiance, L_wn(λ). The surface ground truth values for L_w(λ) and L_wn(λ) used in the SVC process must necessarily be of the highest possible quality, and the uncertainties involved in producing these quantities must be understood. For in-water systems, which provide an estimate of the upwelling nadir water leaving radiance just below the surface, L_u(0^2212;, λ), a factor must be introduced to propagate L_u(0⁻, λ) through the surface to form L_w(λ) and L_wn(λ). The equation relating L_u(0⁻, λ) to L_w(λ) is usually written as (Austin 1974, Mueller 2003a):

$$L_w\left(\lambda \right)=\frac{1-\rho }{n^2}{L}_u(0^{-},\lambda )=T_L{L}_u(0^{-},\lambda )$$ (1)

where n is the index of refraction of water, ρ is the Fresnel reflectance at the air-ocean surface, and T_L combines these factors for simplification. The index of refraction of air is taken to be 1. T_L is described in the current literature (Austin 1974, Mueller 2003a) as a constant value of 0.543.

The goal of SVC data is to have the total uncertainty in L_w(λ), or L_wn(λ), be less than 5% (Zibordi and Voss 2014). The Marine Optical Buoy (MOBY, Clark et al. 1997) data set has been used for vicarious calibration of many of the international ocean color satellites (Franz et al. 2007, Wang et al. 2016, Melin 2011). At this time the MOBY data set is following the current protocol (Mueller 2003a) and has used a constant value for T_L (0.543). In developing an uncertainty estimate for the MOBY data set, we have been investigating the uncertainties in each factor that goes into deriving L_w(λ) and L_wn(λ). An example of these sources of uncertainties include calibration errors (Brown et al. 2007), instrument self-shadowing errors (Mueller 2003b), errors in the calculation of the upwelling radiance attenuation coefficient (Voss 2017), and errors in the cosine response of the downwelling surface irradiance collector (Zibordi and Bulgarelli 2007). The value of T_L=0.543 has been experimentally verified (Wei et al. 2015), but with an uncertainty of 10%, which is much larger than we can use in the MOBY uncertainty budget. To reduce uncertainty in T_L, we need to look closely at the factors that go into this parameter.

2. Discussion

Morel et al. (2002) defined R, a term that combines all of the effects due to reflection and refraction at a wind-disturbed, wavy air-sea interface. This term includes the effects on downwelling irradiance, when propagating from air into the water, and the effects on the upwelling radiance as it propagates upwards through the interface. While investigating R, Gordon (2005) stated that the transmittance of upwelling radiance through the air sea interface was within 1% of the flat surface Fresnel transmittance, for solar zenith angles less than 60 degrees and for wind speeds less than 16 m/s. But this 1% constraint is still larger than it needs to be because Gordon (2005) showed that because of reciprocity, several constraints could be made on the relationship between reflectance and transmittance of the air-water interface. In particular, the sum of the reflectance in a given direction, ξ̂, of uniform radiance incident from above, defined as r₊(−ξ̂), and the transmittance of uniform radiance, in the same direction, incident from below, defined as t₋(−ξ̂), is equal to unity.

$$r_{+}(-\hat{\xi })+t_{-}(-\hat{\xi })=1$$ (2)

Although the actual upwelling radiance distribution in the water is not uniform, the departure from uniformity is not large enough to seriously affect the use of this observation. In addition, r₊(−ξ̂) is equal to the irradiance reflectance (spectral albedo) of the surface from a parallel beam incident from the −ξ̂ direction. Figure 18 in Preisendorfer and Mobley (1986) shows the result of a calculation of the spectral albedo and provides a more stringent constraint. This result shows that for incident angles less than 10 degrees, and wind speeds up to 20 m/s (neglecting whitecaps and breaking waves), there is no difference in r₊(−ξ̂) between a wind roughened surface and a flat surface. Therefore t₋(−ξ̂) is also the same for a wind roughened surface and a flat surface, through the reciprocity condition of Gordon (2005), and we can calculate this transmittance exactly using the Fresnel Equation for transmittance at normal incidence through an air-sea interface.

The full parameter including the index of refraction of air, T_L’, is given by the Fresnel transmittance (the first part of the term on the right side of Eq. 3, below) and invariance of L/n² (the second part of the term on the right side of Eq. 3, below):

$$T_{\mathrm{L}}\text{'}=\frac{4n_a{n}_w}{{(n_a+n_w)}^2}{\left(\frac{n_a}{n_w}\right)}^2$$ (3)

Where n_a is the index of refraction of air and n_w is the index of refraction of water.

When one calculates T_L’, using a nominal value of the index of refraction of seawater at 500 nm, 35 g/kg salinity and 25° C (Austin and Halikas 1976), this gives the nominal value of T_L’=0.543. However, if one uses the wavelength dependence of the index of refraction of seawater, as parameterized by Quan and Fry (1995), and shown to be valid over the range 300 nm – 800 nm by Huibers (1997), it can be seen that for MOBY data (380 nm – 700 nm) this factor can vary from 0.536 – 0.549 for a salinity of 35 g/kg and a temperature of 25° C. Thus a constant value of 0.543 is biased high by 1.3% at 380 nm, and low by 1.1% at 700 nm. Figure 1 shows T_L’ and the error resulting from using the constant value over the 300 nm to 800 nm spectral range. To be used in SVC, the uncertainty goal for L_w(λ) and L_wn(λ) is 5%. There are many factors which go into this total uncertainty including radiometric response uncertainty (2–4%), instrument self-shadowing (1–12%) and others (Brown et al. 2007). In addition, a bias such as this is not reduced through averaging multiple data sets, as opposed to other factors which randomly vary. Thus if the goal is data suitable for SVC, over a large wavelength range, it is important that a spectrally varying factor is used for T_L’. If Eq 3 is used to derive T_L’, using the true values of the index of refraction of seawater for each wavelength, the uncertainty in T_L’ is reduced to the uncertainty in the knowledge of the salinity and temperature of the specific measurement of L_w(λ). In the case of the MOBY site, off of the island of Lanai in Hawaii, our salinity record indicates a range of salinities from 33 g/kg to 36 g/kg (mean salinity is 34.85 g/kg ±0.18 g/kg), and surface water temperatures of 23° C to 30° C (mean temperature is 25.9° C ± 1.0° C). Using a 7 year record (2001–2009) of salinities and temperatures at the MOBY site we calculated T_L’ for each individual data point. For this 7 year period, the uncertainty in using the spectrally varying average value for T_L’ is reduced to 0.1%.

Figure 1.

Graph showing the spectral dependence of the transmission factor, T_L’, and the error in L_w(λ) caused by assuming T_L’ is a constant value of 0.543. This calculation was done for a salinity of 35 g/kg and a temperature of 25° C.

The final conclusion is that ignoring the wavelength dependence of T_L’ introduces an unnecessary spectrally varying bias in the calculation of L_w(λ) and L_wn(λ). For sites with more salinity and temperature variations, or for measurements in many locations, it would be best to have contemporaneous salinity and temperature values with which to calculate a specific and spectrally varying T_L’ for that data set.