3.2.2 Failure to converge
An inversion algorithm doesn’t always find a minimum. Several conditions can
cause such a failure to converge:
• Initial increments are too small. If the initial steps of the search algorithm are
too small, the differences in the residuals are too small to indicate an
improvement for a particular parameter combination compared to others.
Depending on the criterion for termination, this may cause premature
conclusion, or travelling in the wrong direction in the multidimensional
parameter space.
• Acceptable errors are too large. If the criterion for terminating inversion is
chosen as too weak, the inversion algorithm may stop too early, before the
minimum is found.
• Unsuited initial values. If the initial values of the fit parameters are too
different from the correct values, the search for the minimum may start in a
wrong direction. The greater the number of fit parameters, the more difficult it
is to find the correct region in the multidimensional parameter space.
3.3 PROBLEM SOLUTIONS OF WASI
3.3.1 Use of pre-knowledge
In WASI, one can make use of expected parameter values. Typical values of all
parameters and constants are stored in the file WASI.INI, which is read during start up
of the program. The user can change them all. The parameters which are fitted during
inversion can be initialised either with these expected values, or with estimates
calculated by using analytic approximations (see 3.3.3). When a series of spectra is
analyzed, the parameter values may be similar. Thus, the fit results of one measurement
can be taken as start values for the next.
The parameter range can also be modified. The range of possible values is known
in general for each parameter, and within WASI a lower and an upper value is
attributed to each fit parameter. The defaults of these border values are read during
program start from the WASI.INI file, and the user can change them. They are used
whenever the search algorithm attempts to assign an out-of-range value to a parameter.
When a well-known correlation exists between model parameters, it may be useful
to restrict a parameter search to values which depend on the actual values of one
or more other parameters. No general scheme of this method (regularization) is
implemented in WASI. However, some correlations between parameters can be
utilized. For example, suspended matter can be correlated to phytoplankton chlorophyll
by setting C L = C 0 (see 2.2.1); the reflection factor for sky radiance can be related to the
viewing angle using the Fresnel equation (12); and the areal fraction f n of one bottom
type can be related to the fractions of the other types using Ȉ f n = 1 (see 2.7).
3.3.2 Adjust calculation of the residuum
The residuum, ǻ, is the measure of the difference between a measured spectrum
and its fitted curve. The inversion procedure’s task is to find its minimum. The
residuum can be envisaged as a surface in the M+1 dimensional space formed by the M
fit parameters and by ǻ. The shape of that surface depends on which construction law is
96
Gege and Albert
An inversion algorithm doesn’t always find a minimum. Several conditions can
cause such a failure to converge:
• Initial increments are too small. If the initial steps of the search algorithm are
too small, the differences in the residuals are too small to indicate an
improvement for a particular parameter combination compared to others.
Depending on the criterion for termination, this may cause premature
conclusion, or travelling in the wrong direction in the multidimensional
parameter space.
• Acceptable errors are too large. If the criterion for terminating inversion is
chosen as too weak, the inversion algorithm may stop too early, before the
minimum is found.
• Unsuited initial values. If the initial values of the fit parameters are too
different from the correct values, the search for the minimum may start in a
wrong direction. The greater the number of fit parameters, the more difficult it
is to find the correct region in the multidimensional parameter space.
3.3 PROBLEM SOLUTIONS OF WASI
3.3.1 Use of pre-knowledge
In WASI, one can make use of expected parameter values. Typical values of all
parameters and constants are stored in the file WASI.INI, which is read during start up
of the program. The user can change them all. The parameters which are fitted during
inversion can be initialised either with these expected values, or with estimates
calculated by using analytic approximations (see 3.3.3). When a series of spectra is
analyzed, the parameter values may be similar. Thus, the fit results of one measurement
can be taken as start values for the next.
The parameter range can also be modified. The range of possible values is known
in general for each parameter, and within WASI a lower and an upper value is
attributed to each fit parameter. The defaults of these border values are read during
program start from the WASI.INI file, and the user can change them. They are used
whenever the search algorithm attempts to assign an out-of-range value to a parameter.
When a well-known correlation exists between model parameters, it may be useful
to restrict a parameter search to values which depend on the actual values of one
or more other parameters. No general scheme of this method (regularization) is
implemented in WASI. However, some correlations between parameters can be
utilized. For example, suspended matter can be correlated to phytoplankton chlorophyll
by setting C L = C 0 (see 2.2.1); the reflection factor for sky radiance can be related to the
viewing angle using the Fresnel equation (12); and the areal fraction f n of one bottom
type can be related to the fractions of the other types using Ȉ f n = 1 (see 2.7).
3.3.2 Adjust calculation of the residuum
The residuum, ǻ, is the measure of the difference between a measured spectrum
and its fitted curve. The inversion procedure’s task is to find its minimum. The
residuum can be envisaged as a surface in the M+1 dimensional space formed by the M
fit parameters and by ǻ. The shape of that surface depends on which construction law is
96
Gege and Albert