Figure 8A compares the HYDROLIGHT and WASI spectra at 2 m and 10 m bottom
depth for C L = 2 mg/l. The differences can be attributed to Gelbstoff absorption and
chlorophyll a fluorescence, which is accounted for in HYDROLIGHT but not in WASI.
Both sets of spectra were inverted using WASI (eq. 19). During inversion only the
bottom depth z B was a fitted parameter; and all other parameters were fixed at their
correct values. Figure 8B shows the relative errors 100 · (z B
fit / z B –1) as a function of z B
for C L = 1, 2, 3, 4, 5 mg/l, where z B
fit are the results from inverting the HYDROLIGHT
spectra. The influence of the bottom albedo on remote sensing reflectance decreases
with increasing depth and increasing concentration of suspended matter. Thus, the
accuracy of z B determination decreases accordingly. The corresponding errors from
inverting the WASI spectra were close to zero and are not shown. Since only z B was
unknown, the errors in Figure 8B demonstrate the lower limit for model errors. More
realistic error estimates are obtained by fitting additional variables such as C 0 , Y and C L
along with z B. However, the errors obtained are a mixture of errors from the model and
error propagation (see 4.2.4). These two effects can be separated only by performing
more detailed studies.
4.2.3 Errors from input data
Each spectrum type has a specific set of input data. Since all input data have
uncertainties, it is useful to study the influence of their anticipated errors on the
retrieved parameters. Three types of input data can be distinguished: (1) input spectra
from a data base, (2) input spectra from actual field measurements, and (3) input
parameters. Examples of input data with potentially significant errors are: a i *(Ȝ), a W (Ȝ),
b L (Ȝ), a n (Ȝ), t A (Ȝ), and t C (Ȝ) for type (1); L s (Ȝ), E d (Ȝ), and R(Ȝ) for type (2); and S, b b,L *,
b b,S *, ı, ı L , f, Q, B n , and v for type (3).
The implications of input data errors on the accuracy of data analysis are studied
by simulating measurements using a certain set of input data, and then inverting these
simulations with altered input data. The method of altering input data depends on its
type: for type (1) other files must be used, for type (2) the input measurements can be
simulated, and for type (3) individual parameters have to be changed.
An example is given in Figure 9. Irradiance reflectance was calculated for shallow
water using eq. (16). The following water constituent values were used: C 0 = 2 µg/l, C L
= 2 mg/l, Y = 0.3 m
–1
, and S = 0.014 nm
–1
. For Figure 9A the wavelength-independent
spectrum of bottom albedo, a 0 (Ȝ) = 0.1, was chosen. Its absolute value was changed
from 0.1 to 0.3 during forward calculation by iterating the parameter f 0 of eq. (22) from
1 to 3, but kept constant at 0.2 during inversion. In this way relative errors of the
bottom albedo from -50 % to 100 % were simulated. During inversion only the bottom
depth z B was a fit parameter. Its relative error is shown in Figure 9A as a function of z B
and of the relative albedo error. For Figure 9B, the silt spectrum provided with WASI
was used as bottom albedo for forward calculation, and the five other bottom types
were used during inversion. No fit of the incorrect parameters (areal fractions of the
bottom types) was allowed. As expected, the errors depend on the bottom type and
decrease with z B .
105
Inverse Modeling of Spectral Measurements
depth for C L = 2 mg/l. The differences can be attributed to Gelbstoff absorption and
chlorophyll a fluorescence, which is accounted for in HYDROLIGHT but not in WASI.
Both sets of spectra were inverted using WASI (eq. 19). During inversion only the
bottom depth z B was a fitted parameter; and all other parameters were fixed at their
correct values. Figure 8B shows the relative errors 100 · (z B
fit / z B –1) as a function of z B
for C L = 1, 2, 3, 4, 5 mg/l, where z B
fit are the results from inverting the HYDROLIGHT
spectra. The influence of the bottom albedo on remote sensing reflectance decreases
with increasing depth and increasing concentration of suspended matter. Thus, the
accuracy of z B determination decreases accordingly. The corresponding errors from
inverting the WASI spectra were close to zero and are not shown. Since only z B was
unknown, the errors in Figure 8B demonstrate the lower limit for model errors. More
realistic error estimates are obtained by fitting additional variables such as C 0 , Y and C L
along with z B. However, the errors obtained are a mixture of errors from the model and
error propagation (see 4.2.4). These two effects can be separated only by performing
more detailed studies.
4.2.3 Errors from input data
Each spectrum type has a specific set of input data. Since all input data have
uncertainties, it is useful to study the influence of their anticipated errors on the
retrieved parameters. Three types of input data can be distinguished: (1) input spectra
from a data base, (2) input spectra from actual field measurements, and (3) input
parameters. Examples of input data with potentially significant errors are: a i *(Ȝ), a W (Ȝ),
b L (Ȝ), a n (Ȝ), t A (Ȝ), and t C (Ȝ) for type (1); L s (Ȝ), E d (Ȝ), and R(Ȝ) for type (2); and S, b b,L *,
b b,S *, ı, ı L , f, Q, B n , and v for type (3).
The implications of input data errors on the accuracy of data analysis are studied
by simulating measurements using a certain set of input data, and then inverting these
simulations with altered input data. The method of altering input data depends on its
type: for type (1) other files must be used, for type (2) the input measurements can be
simulated, and for type (3) individual parameters have to be changed.
An example is given in Figure 9. Irradiance reflectance was calculated for shallow
water using eq. (16). The following water constituent values were used: C 0 = 2 µg/l, C L
= 2 mg/l, Y = 0.3 m
–1
, and S = 0.014 nm
–1
. For Figure 9A the wavelength-independent
spectrum of bottom albedo, a 0 (Ȝ) = 0.1, was chosen. Its absolute value was changed
from 0.1 to 0.3 during forward calculation by iterating the parameter f 0 of eq. (22) from
1 to 3, but kept constant at 0.2 during inversion. In this way relative errors of the
bottom albedo from -50 % to 100 % were simulated. During inversion only the bottom
depth z B was a fit parameter. Its relative error is shown in Figure 9A as a function of z B
and of the relative albedo error. For Figure 9B, the silt spectrum provided with WASI
was used as bottom albedo for forward calculation, and the five other bottom types
were used during inversion. No fit of the incorrect parameters (areal fractions of the
bottom types) was allowed. As expected, the errors depend on the bottom type and
decrease with z B .
105
Inverse Modeling of Spectral Measurements