1 A Simple Relationship:

A widely published relationship between irradiance reflectance (just beneath the air-water interface) and the inherent optical properties a (absorption coefficient) and bb (backscatter coefficient) is expressed as:

(1)

where rp is a dimensionless parameter whose computed value is compared with the AOMC model results. The relationship in equation (1) has shown to be valid only for optically deep waters whose backscattering coefficient is much less than the absorption coefficient (Morel and Prieur, 1977). Typical values of RP (including those calculated by the AOMC) are shown in the following table:

Author(s)	RP
Gordon et al (1975)	0.3244
Morel and Prieur (1977)	0.33
Kirk (1981)	0.328
Sathyendranath and Platt (1997)	0.343
AOMC	0.329

The AOMC model was run using 8 different scattering coefficient values ranging from 0 to 30. The absorption coefficient was set to a value of 1 giving a range of scattering coefficient values between 0 and 30. The average of Petzold's three San Diego Harbor volume scattering functions was used for these simulations. The averaged VSF results in a Backscatter (B) value of 0.019. The light source is assumed 100% unidirectional (point source) at a sun zenith angle of 0 degrees. The resulting reflectance values are then plotted against the bB/a (bB = bb) ratios and a regression (with intercept 0) is fitted to the data with the slope taken to be the coefficient RP This results in a coefficient of 0.3299 (see aforementioned table).

2 Model Closure

In this problem, a simple homogeneous water body with a solely absorbing ( a = 1 m-1 ) environment is simulated. The light source is a single point in the sky with a zenith angle of 60° resulting in a (below water) refracted angle of 40.5°. The bottom boundary is at a depth of 1 m and has a perfectly specular reflecting bottom with Rb = 1.0. This simulated environment prevents the diffusion (due to scattering) of the ray of light in the aquatic medium. This allows for an analytical solution of the rate of attenuation of the beam of light in the downward and upward directions. In the absence of scattering, the following equation can be used to calculate Ed(z) knowing Ed(z - dz) and the rate of vertical attenuation of Ed:

(2)

Since the model computes Ed at different depths, equation (2) can be solved for k:

(3)

For this simulation, the only attenuating factor is a therefore it is expected that k = a for both the upwelling and downwelling light stream. Results follow:

3 Simulation of a Spectral Pivot Point

Natural waters contain various constituents with differing inherent optical properties (IOP). It is this difference that allows each constituent within a water body to acquire a unique radiometric spectral signature such as surface irradiance reflectance (R). The spectral signature varies as a function of concentration of the constituent. For example, Bukata et al. (1995) show how for a two-component water body (water and chlorophyll), there exists a wavelength at which the bulk subsurface irradiance reflectance is independent of chlorophyll concentration. This feature is called a pivot point. Using an analytical form of the radiative transfer equation, they identify the pivot point feature at around 497 nm. Using the specific absorption and scattering coefficients from Bukata et al. (1995) and the Rayleigh and particle VSF for the water and chlorophyll components respectively, spectral irradiance reflectance signatures were generated for concentrations of chlorophyll of 0.0, 0.5, 1.0, 10.0 and 20.0 mg m-3.
The purpose of this simulation is to evaluate the model's ability to generate realistic spectral signatures. The results of these simulations are shown below . This simulation shows a pivot point at around 495 nm which is in very close agreement to Bukata et al.'s value of 497 nm.

4 Mobley et al.'s Canonical Problems

The next set of simulations uses four of Mobley et al.'s (1993) canonical problems. Mobley et al.'s problems have the following parameters:
· refractive index of water equals 1.340,
· air-water interface is flat,
· water is vertically homogeneous,
· light source is that from a point at a zenith angle of 60 degrees,
· the sun provides a spectral irradiance just above the surface of magnitude 1 W m-2 nm-1 (perpendicular to the sun's rays) on the water surface,
· light source is monochromatic.
Mobley et al. compare the outputs of seven different models (2 analytical and 5 stochastic) for each one of these problems and report the variability between them in terms of a coefficient of variation. The results of the AOMC are compared to the mean values from Mobley et al.'s multiple model runs and presented in terms of a percent relative error in the table at the bottom of this page.

4.1 Rayleigh SPF (Mobley et al.'s problem 1)

Mobley et al.'s first test problem uses a Rayleigh SPF. They present the results for three different apparent optical properties: Ed, Eou, and Lu. These results are listed below. The simulations are run for two different b/a ratios: 0.25 and 0.9. These ratios are translated to absorption coefficients of 1.0 m-1 (for b/a=0.25), 1.0 m-1(for b/a=0.9), and scattering coefficients of 0.25 m-1(for b/a=0.25) and 9.0 m-1 (for b/a=0.9). In all cases, the relative error between the AOMC model and mean values from Mobley et al.'s multi-model simulation fall within the coefficient of variation between Mobley et al.'s seven models.

4.2 Petzold's SPF (Mobley et al.'s problem 2)

Mobley et al.'s second canonical problem replaces the Rayleigh type SPF with that typical of coastal ocean waters which they calculate from three of Petzold's San Diego harbor measurements. The AOMC simulation was performed using the average of these three measured VSF for San Diego Harbor (Petzold, 1972). The table below compares the AOMC results to those from Mobley et al. (1993). Once again, the results from the AOMC simulation fall within the range of the other models in Mobley et al.'s simulations. The radiance values between the AOMC model and Mobley et al.'s are also in good agreement as shown in the following graph (AOMC result is shown in a solid red line, the mean result from Mobley et al.'s models is shown in dashed black lines):

4.3 Stratified water (Mobley et al.'s problem 3)

Mobley et al.'s problem 3 simulates a stratified water body where the concentrations of total particulate matter vary as a function of depth. This problem tests the AOMC model's ability to adjust the photon's pathlength as a function of the concentration of the attenuating medium. Mobley et al. use an analytical equation to describe the vertical profile of the IOPs. For this simulation, tabulated values of a and b are calculated from the analytical equations provided by Mobley et al. at 0.5 meter intervals. The agreement between the AOMC model simulation outputs and the mean values from Mobley et al.'s simulations are reasonably good at the 5 m depth, but diverges at increasing depth where a relative error value for Eou as high as 16.9% is found at a depth of 60 m. This divergence in simulated values is most likely due to the interval resolution used in the vertical profile of the IOPs. AOMC results are shown in the table below. The following graph plots the AOMC result (red solid line) and the results presented in Mobley et al.'s paper (black dashed lines).

4.4 Bottom effect (Mobley et al.'s problem 6)

Mobley et al.'s problem 6 simulates an optically homogeneous water body with a finite depth bottom. The bottom is a Lambertian reflector with an irradiance reflectance of 0.5. This problem tests the AOMC model's bottom boundary interaction. The relative errors for Ed and Eou between the AOMC model and Mobley et al.'s models falls within Mobley et al.'s CV with the exception for the near surface values of Eou and Lu. These differences could be a result of stochastic noise resulting from a low number of upward traveling photons making their way to the surface. Another cause for the error is the possible difference in bin sizes used in logging the angular direction of travel of the photon at each recording depth.