3.1 IMPLEMENTED METHOD
3.1.1 Curve fitting
The fit parameters are determined iteratively using the method of nonlinear curve
fitting. In the first iteration, a model spectrum is calculated using initial values for the
fit parameters. This model spectrum is compared with the measured spectrum by
calculating the residuum as a measure of correspondence. Then, in the further
iterations, the values of the fit parameters are altered, resulting in altered model curves
and altered residuals. The procedure is stopped after the best fit between the calculated
and measured spectrum is found. The best fit corresponds to the minimum residuum,
and these values are the estimates of fit parameters.
3.1.2 Search algorithm
Since an infinite number of possible parameter combinations exists, an effective
algorithm of the iteration process is needed to select a new set of parameter values from
the previous sets. WASI uses the Simplex algorithm (Nelder and Mead, 1965; Caceci
and Cacheris, 1984). It has two advantages compared to other customary algorithms
like Newton-Ralphson and Levenberg-Marquardt: it always converges, and compu
tations are rapid since no matrix operations are required.
In the Simplex algorithm, a virtual space of M+1 dimensions is constructed, where
M dimensions represent the M fit parameters, and one dimension the residuum. Each
model curve corresponds to one point in that space, and the set of all possible model
Table 2. List of WASI parameters.
Symbol
WASI
Units
Description
Ci
C[i]
µg/l
Concentration of phytoplankton class no. i, i = 0..5
CL
C_L
mg/l
Concentration of large suspended particles
CS
C_S
mg/l
Concentration of small suspended particles
X
C_X
m
-1
Concentration of non-chlorophyllous particles
Y
C_Y
m
-1
Concentration of Gelbstoff
S
S
nm
-1
Exponent of Gelbstoff absorption
n
n
-
Exponent of backscattering by small particles
T
T_W
°C
Water temperature
f
f
-
Proportionality factor of reflectance ("f-factor")
Q
Q
sr
Anisotropy factor ("Q-factor")
θ sun
sun
°
Sun zenith angle
θv
view
°
Viewing angle (0 = nadir)
σL
sigma_L
-
Reflection factor of sky radiance
z B
zB
m
Bottom depth
ν
nue
-
Exponent of aerosol scattering
α
alpha
-
Fraction of irradiance due to direct solar radiation
β
beta
-
Fraction of irradiance due to molecule scattering
γ
gamma
-
Fraction of irradiance due to aerosol scattering
δ
delta
-
Fraction of irradiance due to cloud scattering
α*
alpha_s
sr
-1
Fraction of radiance due to direct solar radiation
β*
beta_s
sr
-1
Fraction of radiance due to molecule scattering
γ*
gamma_s
sr
-1
Fraction of radiance due to aerosol scattering
δ*
delta_s
sr
-1
Fraction of radiance due to cloud scattering
fn
fA[n]
-
Areal fraction of bottom surface type no. n, n = 0..5
-
93
Inverse Modeling of Spectral Measurements
3.1.1 Curve fitting
The fit parameters are determined iteratively using the method of nonlinear curve
fitting. In the first iteration, a model spectrum is calculated using initial values for the
fit parameters. This model spectrum is compared with the measured spectrum by
calculating the residuum as a measure of correspondence. Then, in the further
iterations, the values of the fit parameters are altered, resulting in altered model curves
and altered residuals. The procedure is stopped after the best fit between the calculated
and measured spectrum is found. The best fit corresponds to the minimum residuum,
and these values are the estimates of fit parameters.
3.1.2 Search algorithm
Since an infinite number of possible parameter combinations exists, an effective
algorithm of the iteration process is needed to select a new set of parameter values from
the previous sets. WASI uses the Simplex algorithm (Nelder and Mead, 1965; Caceci
and Cacheris, 1984). It has two advantages compared to other customary algorithms
like Newton-Ralphson and Levenberg-Marquardt: it always converges, and compu
tations are rapid since no matrix operations are required.
In the Simplex algorithm, a virtual space of M+1 dimensions is constructed, where
M dimensions represent the M fit parameters, and one dimension the residuum. Each
model curve corresponds to one point in that space, and the set of all possible model
Table 2. List of WASI parameters.
Symbol
WASI
Units
Description
Ci
C[i]
µg/l
Concentration of phytoplankton class no. i, i = 0..5
CL
C_L
mg/l
Concentration of large suspended particles
CS
C_S
mg/l
Concentration of small suspended particles
X
C_X
m
-1
Concentration of non-chlorophyllous particles
Y
C_Y
m
-1
Concentration of Gelbstoff
S
S
nm
-1
Exponent of Gelbstoff absorption
n
n
-
Exponent of backscattering by small particles
T
T_W
°C
Water temperature
f
f
-
Proportionality factor of reflectance ("f-factor")
Q
Q
sr
Anisotropy factor ("Q-factor")
θ sun
sun
°
Sun zenith angle
θv
view
°
Viewing angle (0 = nadir)
σL
sigma_L
-
Reflection factor of sky radiance
z B
zB
m
Bottom depth
ν
nue
-
Exponent of aerosol scattering
α
alpha
-
Fraction of irradiance due to direct solar radiation
β
beta
-
Fraction of irradiance due to molecule scattering
γ
gamma
-
Fraction of irradiance due to aerosol scattering
δ
delta
-
Fraction of irradiance due to cloud scattering
α*
alpha_s
sr
-1
Fraction of radiance due to direct solar radiation
β*
beta_s
sr
-1
Fraction of radiance due to molecule scattering
γ*
gamma_s
sr
-1
Fraction of radiance due to aerosol scattering
δ*
delta_s
sr
-1
Fraction of radiance due to cloud scattering
fn
fA[n]
-
Areal fraction of bottom surface type no. n, n = 0..5
-
93
Inverse Modeling of Spectral Measurements