used. Thus, the search for the minimum can be optimized by adjusting the construction
law to the inversion problem.
Algorithms. The user has the choice between 6 algorithms to calculate ǻ:
2
N
1
i
i
i
i
f
m
g
N
1 ¦
=
−
⋅
=
∆
(28a)
¦
=
−
⋅
=
∆
N
1
i
i
i
i
f
m
g
N
1
(28b)
2
N
1
i
i
i
i
m
f
1
g
N
1 ¦
=
−
⋅
=
∆
(28c)
2
N
1
i
i
i
i
)
f
ln(
)
m
ln(
g
N
1 ¦
=
−
⋅
=
∆
(28d)
¦
=
−
⋅
=
∆
N
1
i
i
i
i
)
f
ln(
)
m
ln(
g
N
1
(28e)
2
N
1
i
i
i
i
)
m
ln(
)
f
ln(
1
g
N
1 ¦
=
−
⋅
=
∆
(28f)
The residuum is a weighted sum over N spectral channels. The subscript i indicates the
channel number, m i denotes the measured value, f i the fit value, and g i the weight. Eq.
(28a) in combination with g i = 1 is the classical least-squares fit. During inversion the f i
values are changed, but not the m i and g i values.
The impact of residuum algorithm selection on the shape of the surface in the
parameter space is illustrated in the example of Figure 4. Two contour plots of the
residuum are shown for an inversion of absorption spectra, which were calculated using
eq. (1) by summing the absorptions of phytoplankton and Gelbstoff. In this example the
concentrations C 1 ,...,C 5 and X were set to zero. Phytoplankton chlorophyll
concentration C 0 was set to 2 µg/l during forward and inverse calculation. The
Gelbstoff parameters were set to S = 0.014 nm
–1 and Y = 0.3 m
–1 during forward
calculation and then were iterated from 0.01 nm
–1 S 0.02 nm
–1 and 0 Y 0.6 m
–1
during inversion. No fit was performed during inversion; only the residuum was
calculated for each parameter combination with equal weights g i = 1.
The major difference between the two plots of Fig. 4 is the orientation of the valley
which forms the minimum: the valley is almost parallel to the S axis for the classical
least-squares fit (eq. 28a), i.e. the inversion cannot determine S reliably. When the
logarithms of the m i and f i are taken (eq. 28d), the valley is oriented diagonally in the SY-plane, and thus fitting of both S and Y is feasible. For the concentrations chosen, eq.
28d is more appropriate because the absorption spectrum is dominated by the
exponential function of Gelbstoff.
97
Inverse Modeling of Spectral Measurements
law to the inversion problem.
Algorithms. The user has the choice between 6 algorithms to calculate ǻ:
2
N
1
i
i
i
i
f
m
g
N
1 ¦
=
−
⋅
=
∆
(28a)
¦
=
−
⋅
=
∆
N
1
i
i
i
i
f
m
g
N
1
(28b)
2
N
1
i
i
i
i
m
f
1
g
N
1 ¦
=
−
⋅
=
∆
(28c)
2
N
1
i
i
i
i
)
f
ln(
)
m
ln(
g
N
1 ¦
=
−
⋅
=
∆
(28d)
¦
=
−
⋅
=
∆
N
1
i
i
i
i
)
f
ln(
)
m
ln(
g
N
1
(28e)
2
N
1
i
i
i
i
)
m
ln(
)
f
ln(
1
g
N
1 ¦
=
−
⋅
=
∆
(28f)
The residuum is a weighted sum over N spectral channels. The subscript i indicates the
channel number, m i denotes the measured value, f i the fit value, and g i the weight. Eq.
(28a) in combination with g i = 1 is the classical least-squares fit. During inversion the f i
values are changed, but not the m i and g i values.
The impact of residuum algorithm selection on the shape of the surface in the
parameter space is illustrated in the example of Figure 4. Two contour plots of the
residuum are shown for an inversion of absorption spectra, which were calculated using
eq. (1) by summing the absorptions of phytoplankton and Gelbstoff. In this example the
concentrations C 1 ,...,C 5 and X were set to zero. Phytoplankton chlorophyll
concentration C 0 was set to 2 µg/l during forward and inverse calculation. The
Gelbstoff parameters were set to S = 0.014 nm
–1 and Y = 0.3 m
–1 during forward
calculation and then were iterated from 0.01 nm
–1 S 0.02 nm
–1 and 0 Y 0.6 m
–1
during inversion. No fit was performed during inversion; only the residuum was
calculated for each parameter combination with equal weights g i = 1.
The major difference between the two plots of Fig. 4 is the orientation of the valley
which forms the minimum: the valley is almost parallel to the S axis for the classical
least-squares fit (eq. 28a), i.e. the inversion cannot determine S reliably. When the
logarithms of the m i and f i are taken (eq. 28d), the valley is oriented diagonally in the SY-plane, and thus fitting of both S and Y is feasible. For the concentrations chosen, eq.
28d is more appropriate because the absorption spectrum is dominated by the
exponential function of Gelbstoff.
97
Inverse Modeling of Spectral Measurements