Table 3. Four-step procedure for initial values determination of irradiance reflectance spectra in
deep water.
Step
Determine Algorithm Description
1
CL, CS
analytical
Estimate CL and CS from an analytic equation at a
wavelength in the near infrared.
2
Y, C0
analytical
Estimate Y and C 0 from analytic equations at two
wavelengths; for CL and CS the values from step 1 are
taken.
3
C L , C S , Y
Determine initial values of C L , C s and Y by fit. C 0 is kept
constant at the value from step 2; C L , C S and Y are
initialized using the values from steps 1 and 2.
4
C 0 , Y, S
f it
Determine initial values of C 0 , Y and S by fit. C L is kept
constant at the value from step 3, Y is initialized using the
value from step 3, S is initialized by the user-setting.
Step 1. Suspended matter backscattering, B 0 = b b – b b,W , can be calculated analytically
from R at any wavelength Ȝ IR for which absorption a(Ȝ IR ) is known. For Ȝ IR a
wavelength in the near infrared is chosen since absorption of water constituents is
generally very low compared to absorption of pure water at Ȝ > 700 nm (Babin and
Stramski 2002). Ideally, Ȝ IR > 750 nm should be used, since phytoplankton absorption
a 0 *(Ȝ) is zero above 750 nm. The equation of determination is obtained from eqs. (8)
and (14):
).
(
b
)
(
R
f
)
(
R
)
(
a
B
IR
W
,
b
IR
IR
IR
0
λ
−
λ
−
λ
⋅
λ
=
(29)
f is calculated using the selected f model, e.g. eq. (15), with the user-defined initial
values as parameter values. If a B 0 -dependent f model is selected, B 0 is calculated in
two iterations, i.e. the B 0 value from the first iteration is taken to calculate f again, and
using this f B 0 is calculated a final time.
Conversion from optical units B 0 to gravimetric concentrations C L , C S is based
on eq. (4) assuming b L (Ȝ) = 1. Accordingly it is B 0 = C L · b b,L * + C S · b b,S * · (Ȝ/Ȝ S җ )
n
.
If C S = 0, it is C L = B 0 / b b,L *, otherwise the user-defined ratio r SL = C S /C L is
retained, and C L and C S are calculated as follows:
.
C
r
C
;
b
r
b
B
C
L
SL
S
n
S
IR
*
S
,
b
SL
*
L
,
b
0
L
⋅
=
¸ ¸
¹
·
¨ ¨
©
§
λ
λ
⋅
⋅
+
=
(30)
If Ȝ IR > 750 nm, the accuracy of the analytically estimated parameters C L and C S
depends only on Ȝ IR , Gelbstoff absorption at Ȝ IR , and on C L and C S themselves. The
dependence of the relative C L error on Y and C L is shown in Fig. 5 for two values of
Ȝ IR , 750 and 870 nm. It was calculated as 100 · (C L /C L
fwd – 1), where C L
fwd is the value
of forward calculation and C L the value obtained from eq. (30). Values of S = 0.014 nm
-
1 and C S = 0 were used. As Figure 5 shows, C L can be determined using eqs. (29) and
(30) with an accuracy of about 1 %.
f it
99
Inverse Modeling of Spectral Measurements
deep water.
Step
Determine Algorithm Description
1
CL, CS
analytical
Estimate CL and CS from an analytic equation at a
wavelength in the near infrared.
2
Y, C0
analytical
Estimate Y and C 0 from analytic equations at two
wavelengths; for CL and CS the values from step 1 are
taken.
3
C L , C S , Y
Determine initial values of C L , C s and Y by fit. C 0 is kept
constant at the value from step 2; C L , C S and Y are
initialized using the values from steps 1 and 2.
4
C 0 , Y, S
f it
Determine initial values of C 0 , Y and S by fit. C L is kept
constant at the value from step 3, Y is initialized using the
value from step 3, S is initialized by the user-setting.
Step 1. Suspended matter backscattering, B 0 = b b – b b,W , can be calculated analytically
from R at any wavelength Ȝ IR for which absorption a(Ȝ IR ) is known. For Ȝ IR a
wavelength in the near infrared is chosen since absorption of water constituents is
generally very low compared to absorption of pure water at Ȝ > 700 nm (Babin and
Stramski 2002). Ideally, Ȝ IR > 750 nm should be used, since phytoplankton absorption
a 0 *(Ȝ) is zero above 750 nm. The equation of determination is obtained from eqs. (8)
and (14):
).
(
b
)
(
R
f
)
(
R
)
(
a
B
IR
W
,
b
IR
IR
IR
0
λ
−
λ
−
λ
⋅
λ
=
(29)
f is calculated using the selected f model, e.g. eq. (15), with the user-defined initial
values as parameter values. If a B 0 -dependent f model is selected, B 0 is calculated in
two iterations, i.e. the B 0 value from the first iteration is taken to calculate f again, and
using this f B 0 is calculated a final time.
Conversion from optical units B 0 to gravimetric concentrations C L , C S is based
on eq. (4) assuming b L (Ȝ) = 1. Accordingly it is B 0 = C L · b b,L * + C S · b b,S * · (Ȝ/Ȝ S җ )
n
.
If C S = 0, it is C L = B 0 / b b,L *, otherwise the user-defined ratio r SL = C S /C L is
retained, and C L and C S are calculated as follows:
.
C
r
C
;
b
r
b
B
C
L
SL
S
n
S
IR
*
S
,
b
SL
*
L
,
b
0
L
⋅
=
¸ ¸
¹
·
¨ ¨
©
§
λ
λ
⋅
⋅
+
=
(30)
If Ȝ IR > 750 nm, the accuracy of the analytically estimated parameters C L and C S
depends only on Ȝ IR , Gelbstoff absorption at Ȝ IR , and on C L and C S themselves. The
dependence of the relative C L error on Y and C L is shown in Fig. 5 for two values of
Ȝ IR , 750 and 870 nm. It was calculated as 100 · (C L /C L
fwd – 1), where C L
fwd is the value
of forward calculation and C L the value obtained from eq. (30). Values of S = 0.014 nm
-
1 and C S = 0 were used. As Figure 5 shows, C L can be determined using eqs. (29) and
(30) with an accuracy of about 1 %.
f it
99
Inverse Modeling of Spectral Measurements