Numerical modelling

The presence of a deployment structure at a measurement site is source of three- dimensional (3D) inhomogeneities and introduces abrupt medium changes within a predominantly plane-parallel system. Consequently, the simulation of superstructure shading effects on in–water optical radiometric measurements requires the 3D radia- tive transfer modelling of the ocean–atmosphere system. This is achieved by simu- lating radiative transfer processes using Monte Carlo methods to produce a solution of the generalized radiative transfer equation.

a. Principles for MC simulations

The Photon Transport (PHOTRAN) MC code, developed for the ocean–atmosphere system (Bulgarelli and Doyle, 2004; Doyle and Rief, 1998) was used to quantify the tower shading perturbations on in–water radiometric measurements. Radiances and irradiances at a specific point in the modeled system were computed using an imple- mentation (Gordon, 1985) of Case’s reciprocity relationship (Case, 1957).

Within PHOTRAN the ocean–atmosphere system is modeled on a 3D grid which delimits the macroscopic volumes (cells) each containing a medium of uniform optical properties. In each cell of the grid, the optically active components (air or water molecules, aerosols, hydrosols, etc.) are specified, and their IOPs are assigned (e.g.,

c, ω0, and ˜β). Cell boundaries are spectrally characterized by transmittance and reflectance, and by the associated transmission and reflection angular distribution functions (ADFs). Specific ADFs are defined both for the source and for the detector. Photons detected by a radiometer are a fraction of those emitted by the sun and reach the sensor after absorption, scattering, reflection, and refraction processes in the

Chapter 7 Measurement Perturbations

ocean–atmosphere system. Accounting for the invariance of time-reversal processes characterizing the propagation of photons, backward MC methods are efficiently ap- plied in the specific problem. Virtual photons initially having unitary statistical weight, are released from the detector within its field-of-view according to the prede- fined ADF. A free-flight optical distance to the next collision point is sampled (Lux and Koblinger, 1991), possible cell-boundary crossing processes are considered to de- fine flight direction modifications, and finally, photon trajectory is computed taking into account possible changes in IOPs along the trajectory. At the collision point (defined as the point where the sampled optical distance is exhausted): (i) a scat- terer is sampled and the virtual photon is re-weighted using ω0; and (ii) the flight direction of the re-weighted virtual photon is determined by retrieving the scattering angle from a random sampling of ˜β. The latter is adequately modeled into an equal probability interval table (Lux and Koblinger, 1991). If virtual photons encounter a purely absorbing medium (i.e., the tower structure), their weight is zeroed. The single process is stopped when photons exit from the atmosphere. The weight of those virtual photons travelling in the direction of sun is then accounted for to quantify simulated measurements.

By tracking so-called twin virtual photons, one interacting with the tower and the other not, a correlated sampling scheme (Spanier and Gelbard, 1969) is produced which minimizes the score variance of differences between tower-perturbed and un- perturbed signals. The number of initiated twin virtual photons defines the estimated statistical relative error on simulated data.

Chapter 7 Measurement Perturbations

b. Simulations for the AAOT

The simulation frame consists of one large 3D box that encloses the grid defining the geometrical features of the system. The atmosphere, the ocean, and the bound- aries (top of the atmosphere, sea surface, and sea floor) are modelled as horizontally plane-parallel. This plane-parallel symmetry is broken by introducing, at a specific location within the reference frame, a geometrical object schematically representing the AAOT with completely absorbing surfaces. The reference system for the simula- tion frame uses Cartesian orthonormal coordinates (x, y, z), centered at a tower leg. The schematic of the tower structure and the relevant 3D features introduced in the PHOTRAN code are shown in Fig. 7.7.

The radiometer, located at a specific point in the reference frame, is described by its field-of-view and the associated ADF. The ADF for the direct source, as seen in a forward MC perspective, is formulated by a Dirac δ centered on the sun zenith,

θ0, and sun azimuth, φ0. By modelling the atmosphere, sea surface, water column and sea floor following Zibordi et al. (1999), radiance is simulated assuming an in– water 20 degrees full-angle field-of-view with a unitary collection ADF. Irradiance is simulated assuming a 2π sr field-of-view with a cosine collection ADF.

By defining the tower shading error ²T

<(λ) as the percent difference between the true and shading contaminated values for the specific radiometric quantity <(λ), an extensive theoretical sensitivity analysis was carried out. PHOTRAN computations were performed assuming typical values of sea-water IOPs at the AAOT (Zibordi et al., 1999), and two extreme illumination conditions representing overcast and ideal clear sky (i.e., τa = 0), with varying θ0 and φ0 = 180 degrees, and sensor distance from the superstructureχ = 7.5 m. The value forχwas chosen to ensure exploration

Chapter 7 Measurement Perturbations

Sun

x

y

z

Underwater

Radiometer

Tower

Sea Surface

Sea Floor

q

f

Figure 7.7: Schematic of the AAOT as defined in the PHOTRAN code (after Zibordi et al. (1999)).

of the AAOT shading effects at the location of the WiSPER radiometers; the value of φ0 was chosen to simulate the typical measurement conditions in which WiSPER is deployed from the sunny side of the tower.

Simulated downward irradiance ²T

Ed(λ) and upwelling radiance ²TLu(λ) errors, are

summarized in Tab. 7.2 for overcast sky and in Fig. 7.8 for clear sky conditions at

Chapter 7 Measurement Perturbations

Table 7.2: Computed tower shading errors for downwelling irradiance Ed and up-

welling radiance Lu at 0− depth and 7.5 m distance from the AAOT for a diffuse

light source assuming typical values of inherent optical properties, at different wave- lengths (in nm). Confidence intervals are given in parentheses (after Zibordi et al. (1999)). Parameter Unit 443 555 665 ²T Ed % 19.8 (±0.2) 19.9 (±0.2) 20.1 (±0.2) ²T Lu % 19.5 (±0.1) 18.5 (±0.1) 19.8 (±0.1)

The data in Tab. 7.2 show values almost independent of wavelength for both

²T

Ed(λ) and ²TLu(λ). In fact, by assuming that for an overcast sky most of the pertur-

bations are induced by interaction of the diffuse irradiance field with the superstruc- ture of the tower, the shading error becomes closely proportional to the perturbed above–water irradiance. Consequently, ²T

Ed(λ) — and analogously ²TLu(λ) values —

are very close and do not exhibit any significant dependence on λ. Values of ²T

Ed(λ) and ²TLu(λ) in Fig. 7.8, show a strong dependence on θ0 and

λ. The ²T

Ed(λ) data show values increasing with θ0, while ²TLu(λ) data show a more

complex dependence on θ0 exhibiting minima close to 30 and 40 degrees, at 443 and 555 nm, respectively. The dependence on θ0 can be explained by the different perturbations induced by the AAOT on the nearby light field. During overcast sky conditions, the diffuse sky irradiance,Ei(λ), is solely responsible for the in-water light

field and the tower perturbation is uniformly cast in all directions. During clear sky conditions, when both the Ei(λ) and the direct sun irradiance, Es(λ), contribute to

the in-water light field, the perturbation associated withEi(λ) adds to that associated

with Es(λ). The latter perturbation results in a pronounced tower shadow projected

Chapter 7 Measurement Perturbations

T T

Figure 7.8: Simulated tower-shading errors²T

Edand²TLuat 0−depth and 7.5 m distance

from the tower as a function of θ0 assuming φ = 180 degrees and ideal clear-sky conditions. The vertical bars show the confidence limits for the simulation (after Zibordi et al. (1999)).

the tower. Then, during clear sky conditions, excluding cases characterized by very low values of θ0,Es(λ) creates perturbations that are confined at some distance from

the measurement point. These perturbations produce shading errors which decrease with increasing θ0 as shown in Fig. 7.9 (the ²TEd(λ) and ²TLu(λ) values shown in

Fig. 7.9 were computed analogously to those proposed in Fig. 7.8, but assuming no atmosphere). These errors add to those produced byEi(λ), which increase withθ0(as a result of an increase ofEi(λ) with θ0). Consequently, these errors vary significantly both withθ0 and with the ratio Ir(λ), exhibiting the trends displayed in Fig. 7.8 for

²T

Ed(λ) and ²TLu. The ²EdT (λ) values shown in Fig. 7.9, contrary to the ²TLu(λ) values,

Chapter 7 Measurement Perturbations

T T

Figure 7.9: Simulated tower-shading errors²T

Edand²TLuat 0−depth and 7.5 m distance

from the tower as a function ofθ0 assumingφ = 180 degrees and no atmosphere. The vertical bars show the confidence limits for the simulation (after Zibordi et al. (1999)). is assigned to the Ed sensor. This depth induces the observed nonzero ²TEd at 0− m,

which in any case has little weight on the ²T

Ed(λ) values displayed in Fig. 7.8.