The Hydrolight-based synthetic dataset described in Section 3.2.2 was used to investigate the relationship between above-water Rrsand a+bbb
b, which is of primary
importance to physics-based Rrs inversions. For brevity, the lower-case letter u
is often substituted for the term bb
a+bb following Eq. 3.40. u is used in Figures
3.13a-c. u = bb a + bb . (3.40)
Figure 3.13a shows the Hydrolight-computed relationship between above-water Rrs and u for 400 nm, 550 nm and 700 nm, for the entire range of concentrations
shown in Table 3.3.
There is a slightly non-linear trend apparent in the data. It is also interesting to note that the wavelength dependence of the relationships between Rrs and u
appears to be subtle.
For comparison, Fig. 3.13b shows the subsurface rrsof the Hydrolight simulations
performed at solar zenith angle θs = 0 and sensor nadir view angle θv = 0 at
550 nm, alongside the model fits of Lee et al. (1999) and Gordon et al. (1988). These models both exhibit the functional form shown in Eq. 3.41,
rrs = g0u + g1u2, (3.41)
where lowercase rrs denotes sub-surface remote sensing reflectance, and g0 and g1
are model coefficients (examples of which are shown in Table 3.4).
ID Author g0 g1
G88 Gordon et al. (1988) 0.0949 0.0794 L99 Lee et al. (1999) 0.0840 0.170 L02 Lee et al. (2002) 0.0895 0.1247
*** This study 0.0849 0.1211
Table 3.4: Published polynomial coefficients relating rrs to u. L02 as published
in Lee et al. (2002) was determined from the average of the G88 and L99 sets. All sets overplotted in Fig. 3.13b, labelled according to the ID column.
whereas the Lee et al. (1999) coefficients were optimized for coastal conditions. As a result, Lee et al. (2002) used the average of these two sets of values in order to equally represent coastal and oceanic waters. The bio-optical model Lee et al. (1999) used to generate their ‘coastal’ coefficients relied on chlorophyll-a based particle scattering, and the ‘average particle’ phase function described in Mobley (1994). The results of the present study are derived from concentration specific, but slightly randomized blends of molecular, Mie mono-dispersion (small phyto- plankton), Mie polydispersion (large phytoplankton) and Fornier-Fourand based (mineral and detrital) phase functions in an attempt to better match the particle assemblages found in the GBR. Despite the differences in bio-optical modelling approaches, the coefficients calculated in this study closely match (5.2% for g0
and 2.9% for g1) the averaged coefficients of Lee et al. (2002) (Set L02 in Table
3.4).
The use of the simplistic 2nd order polynomial relationship of Eq. 3.41 may be a consequence of the previously mentioned studies using a blend of only two phase functions (i.e. molecular and either ‘average Petzold’ (Mobley 1994, Petzold 1977) or Kullenberg (1974) for particles). By examining Fig. 3.13c, one can see that, at low values of u, a simple second order polynomial treatment, as in Eq. 3.41, cannot attempt to model the results of the observed synthetic Hydrolight runs, particularly at low u values.
The shortfall of Eq. 3.41 was recognised by Lee et al. (2004), who attributed the spread of data to the relative abundance of molecular scattering as opposed to particulate scattering (using Petzold’s ‘average particle’ phase function). By par- titioning the molecular and particulate scattering components, Lee et al. (2004) was able to determine more accurate parameters which related rrs (subsurface)
to u.
A similar partitioning approach was used in analysing this synthetic data set. However, a polynomial functional form (Eq 3.42) was used for the particulate
a)
b)
c)
Figure 3.13: a) Above water nadir-viewed Rrs versus u for θs = 0 (400 nm
in diamonds, 550 nm in triangles and 700 nm in filled circles). b) Modelled subsurface nadir-viewed rrsversus u(550) for θs = 0. The relationships of Gordon
et al. (1988) (G88), Lee et al. (1999) (L99), Lee et al. (2002) L(02) and the parabolic model fit of this study (***) are shown. c) u(550)-normalised rrs(550)
versus u(550) similar to the plot in Lee et al. (2004), highlighting the deviations from the simplistic 2nd order polynomial approach. The relationships of Gordon et al. (1988) (G88), Lee et al. (1999) (L99), Lee et al. (2002) L(02) and the
contribution and coefficients were determined from above-water Rrs, Rrs= Gw bbw a + bb + G0 bbp a + bb + G1 bbp a + bb 2 + G2 bbp a + bb 3 , (3.42)
where Gw, G0, G1 and G2 are the polynomial model parameters, bbw is the
backscattering coefficient of water molecules, and bbp is the backscattering coef-
ficient of all particles (phytoplankton, detritus and minerals). For brevity, wave- length, geometry and sea state are omitted in the equation, but are described below.
A specific set of Gi (i denotes coefficient subscripts w, 0, 1 and 2) polynomial co-
efficients were determined using Levenberg-Markvardt optimisation (Markwardt 2009) for the modelled Rrs values determined for each wind speed (ws), solar
zenith angle (θs), sensor view nadir angle (θv), sun-relative sensor view azimuth
(φaz) and wavelength (λ) combination. The resultant Gi(λ, ws, θs, θv, φaz) values
were stored in a LUT which can be searched to retrieve the appropriate Gi co-
efficients for a measured Rrs(λ, w, θs, θv, φaz). Example Gi coefficients are shown
in Fig. 3.14a.
For some analytical inversion approaches (i.e. Lee et al. (2002)), it is desirable to have knowledge of
bb
a+bb
based on Rrs alone. In this case, the ratio between bbw
and bbpis unknown, so Eq. 3.42 cannot be used directly. The Hydrolight synthetic
dataset was used to determine an empirical polynomial relationship that could be applied to Rrs directly,
bb
a + bb
= U0+ U1Rrs+ U2R2rs+ U3R3rs+ U4R4rs, (3.43)
where U0, U1, U2, U3 and U4 are the polynomial model parameters, and their
spectral dependence is omitted for brevity.
Example spectral Ui coefficients are shown in Fig. 3.14c. As for the G model
coefficients, the U model coefficients vary with wind speed, solar zenith angle, sensor view nadir angle, sun-relative sensor view azimuth and wavelength. The
a) b)
c) d)
Figure 3.14: a) Spectral Gi model parameters for Eq. 3.42, for θs = 0, φaz = 135,
θv = 40 and ws = 1 ms−1. b) RMS error when determining Rrs with Eq. 3.42
with the aforementioned model parameters and geometry from a). c) Spectral Ui
model parameters for Eq. 3.43, for θs = 0, φaz = 135, θv = 40 and ws = 1ms−1.
d) RMS error when determining bb
a+bb
U model values are also stored in a Look-Up-Table (LUT).
In comparing the RMS errors shown in Fig. 3.14b and d, one may be misled into concluding that the automatic conversion of Rrs into u yields a lower overall
error, however it is important to realize that Rrs is between approximately 5%
to 7% of u in this dataset (see Fig. 3.13a), and as such, will score higher RMS errors, especially in the red wavelengths due to there generally being very low Rrs in these regions.