spectral resolutions of spectral channels; radiometric resolution; accuracy of radio
metric and spectral calibration; noise; and drift.
For estimating the limits of accuracy caused by the sensor, spectra of the given
sensor are simulated for different combinations of model parameters, and spectra of a
“perfect” sensor are calculated for the same combinations. Both sets of spectra are
inverted, and the accuracies of the fit parameters are compared.
Sensor characteristics which are readily defined and changed easily using WASI
are number and center wavelengths of the channels, radiometric resolution, and
statistical noise. For studying the influence of spectral resolution and calibration
accuracy, WASI can be used to calculate spectra which are sensor-adjusted with respect
to the number and center wavelengths of the channels, radiometric resolution, and
statistical noise. These spectra are then modified according to the instrument
characteristics: the spectra are smoothed if the instrument has a lower spectral
resolution than the input data, error spectra are added and/or multiplied to the spectra if
radiometric calibration errors are investigated, and the wavelength values are changed if
spectral calibration errors are analyzed. This is done using separate software, such as a
spread sheet program. Finally, the sensor-adjusted spectra are inverted using WASI.
An example is given in Figure 7. Absorption spectra of water constituents were
simulated for 4 sensors which differ in the number of spectral channels and in the noise
level: spectra a WC (Ȝ) were calculated using eq. (1) from 400 to 800 nm for wavelength
intervals of 2 and 20 nm and statistical noise of 0.002 and 0.02 m
–1 standard deviation.
The Gelbstoff parameters chosen were Y = 0.2 m
–1 and S = 0.014 nm
–1
. Concentrations
C 1 , ..., C 5 and X were set to zero. Phytoplankton chlorophyll concentration C 0 was
changed from 0.1 to 100 µg/l in 51 steps. 20 spectra a WC (Ȝ) were calculated for each C 0
value by applying a simple trick: a parameter not used in the actual model was iterated
during forward calculation. Each of the 51 · 20 = 1020 spectra for each sensor was
inverted with C 0 and Y as fit parameters. Figure 7 compares the C 0 -dependency of the
relative C 0 errors for the 4 sensor specifications. C 0 errors are more sensitive to noise
than to the number of channels in this example.
4.2.2 Errors from the model
The radiative transfer equation for an absorbing and scattering medium like water
cannot be solved analytically, hence observations which depend on the radiation field
(AOPs) can only be approximated. WASI uses analytic approximations based on
parameters which can be measured with relative ease. Advantages are that inversion is
relatively simple, altered input data sets are included quickly, and computing is fast.
From a numerical point of view, any desired accuracy can be achieved by using
converging methods such as Monte Carlo, invariant imbedding, matrix operator,
successive order of scattering, finite elements, etc. However, high accuracy is at the
expense of computing time, which, for these methods, is by far too long for inverting a
set of spectra. In order to estimate errors introduced by the approximations of WASI, a
set of spectra must be calculated using a numerically exact program, and for the same
conditions a second set of spectra is calculated using WASI. Both sets of spectra are
inverted, and the differences of the fit parameters reflect the errors introduced by the
simplified model.
An example is given in Figure 8. Remote sensing reflectance spectra of shallow
water were calculated using both HYDROLIGHT (Mobley et al., 1993, Mobley, 1994)
and WASI (eq. 19). All input data and parameters for the two models were identical.
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Inverse Modeling of Spectral Measurements
metric and spectral calibration; noise; and drift.
For estimating the limits of accuracy caused by the sensor, spectra of the given
sensor are simulated for different combinations of model parameters, and spectra of a
“perfect” sensor are calculated for the same combinations. Both sets of spectra are
inverted, and the accuracies of the fit parameters are compared.
Sensor characteristics which are readily defined and changed easily using WASI
are number and center wavelengths of the channels, radiometric resolution, and
statistical noise. For studying the influence of spectral resolution and calibration
accuracy, WASI can be used to calculate spectra which are sensor-adjusted with respect
to the number and center wavelengths of the channels, radiometric resolution, and
statistical noise. These spectra are then modified according to the instrument
characteristics: the spectra are smoothed if the instrument has a lower spectral
resolution than the input data, error spectra are added and/or multiplied to the spectra if
radiometric calibration errors are investigated, and the wavelength values are changed if
spectral calibration errors are analyzed. This is done using separate software, such as a
spread sheet program. Finally, the sensor-adjusted spectra are inverted using WASI.
An example is given in Figure 7. Absorption spectra of water constituents were
simulated for 4 sensors which differ in the number of spectral channels and in the noise
level: spectra a WC (Ȝ) were calculated using eq. (1) from 400 to 800 nm for wavelength
intervals of 2 and 20 nm and statistical noise of 0.002 and 0.02 m
–1 standard deviation.
The Gelbstoff parameters chosen were Y = 0.2 m
–1 and S = 0.014 nm
–1
. Concentrations
C 1 , ..., C 5 and X were set to zero. Phytoplankton chlorophyll concentration C 0 was
changed from 0.1 to 100 µg/l in 51 steps. 20 spectra a WC (Ȝ) were calculated for each C 0
value by applying a simple trick: a parameter not used in the actual model was iterated
during forward calculation. Each of the 51 · 20 = 1020 spectra for each sensor was
inverted with C 0 and Y as fit parameters. Figure 7 compares the C 0 -dependency of the
relative C 0 errors for the 4 sensor specifications. C 0 errors are more sensitive to noise
than to the number of channels in this example.
4.2.2 Errors from the model
The radiative transfer equation for an absorbing and scattering medium like water
cannot be solved analytically, hence observations which depend on the radiation field
(AOPs) can only be approximated. WASI uses analytic approximations based on
parameters which can be measured with relative ease. Advantages are that inversion is
relatively simple, altered input data sets are included quickly, and computing is fast.
From a numerical point of view, any desired accuracy can be achieved by using
converging methods such as Monte Carlo, invariant imbedding, matrix operator,
successive order of scattering, finite elements, etc. However, high accuracy is at the
expense of computing time, which, for these methods, is by far too long for inverting a
set of spectra. In order to estimate errors introduced by the approximations of WASI, a
set of spectra must be calculated using a numerically exact program, and for the same
conditions a second set of spectra is calculated using WASI. Both sets of spectra are
inverted, and the differences of the fit parameters reflect the errors introduced by the
simplified model.
An example is given in Figure 8. Remote sensing reflectance spectra of shallow
water were calculated using both HYDROLIGHT (Mobley et al., 1993, Mobley, 1994)
and WASI (eq. 19). All input data and parameters for the two models were identical.
-
103
Inverse Modeling of Spectral Measurements