DATA ASSIMILATION IN BIOGEOCHEMICAL MODELLING
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Another issue is the functional form of the terms entering the cost
function. The most common approach of using sums of squares in the
cost function gives the same weight to positive and negative misfits, and
minimising least squares corresponds to a maximum-likelihood estimate
only as long as the error distributions are Gaussian. In most cases
this will not be correct. For example, there is strictly zero probability
that nutrient concentrations are negative. This could, for example, be
reflected in the cost functions by terms that go to infinity as simulated
nutrient concentrations approach zero (e.g., by a high negative power of
the nutrient concentration). A further aspect to consider is a possible
correlation of different observations in space and in time. Sometimes it is
attempted to take these into account by weighting different observations
by the number of measurements taken. More frequent observations are
then assumed to be correlated and accordingly downweighted in the cost
function, whereas rare observations get relatively more weight. If prior
information on the parameter values is available, the respective costfunction terms have to be weighted against the model data misfit terms
as well.
When all information is accounted for by appropriately weighted cost
function terms, the resulting cost function defines the metrics that measures the qualitiy of any parameter set. Construction of the cost function
will always involve some subjective elements regarding the functional
forms or weights of the individual terms. This basically reflect that,
as usual in life, different people have different views on what is “best”.
The optimisation results will always depend on the quality of the cost
function which therefore should be crafted as carefully as possible.
5.3
Minimisation methods
In principle, one could just explore the cost function “landscape” in
parameter space by explicitly evaluating J(p) for a large number of different choices of the parameter vector p . In practice, however, this will
usually not be possible. Even for a simple NPZD-type ecosystem model
with, say, 15 parameters, a very coarse sampling of only 10 possible values per parameter would require 10 15 evaluations of the cost function.
As soon as models have more than a handful parameters, more efficient
minimisation methods are needed.
A large variety of such minimisation methods exist, most of which
have been developed outside oceanography. They can be devided into
methods that make use of the cost function’s gradient, i.e., information
about the downhill direction in the cost-function landscape, and into
methods that do not use this information and therefore do not require
537
Another issue is the functional form of the terms entering the cost
function. The most common approach of using sums of squares in the
cost function gives the same weight to positive and negative misfits, and
minimising least squares corresponds to a maximum-likelihood estimate
only as long as the error distributions are Gaussian. In most cases
this will not be correct. For example, there is strictly zero probability
that nutrient concentrations are negative. This could, for example, be
reflected in the cost functions by terms that go to infinity as simulated
nutrient concentrations approach zero (e.g., by a high negative power of
the nutrient concentration). A further aspect to consider is a possible
correlation of different observations in space and in time. Sometimes it is
attempted to take these into account by weighting different observations
by the number of measurements taken. More frequent observations are
then assumed to be correlated and accordingly downweighted in the cost
function, whereas rare observations get relatively more weight. If prior
information on the parameter values is available, the respective costfunction terms have to be weighted against the model data misfit terms
as well.
When all information is accounted for by appropriately weighted cost
function terms, the resulting cost function defines the metrics that measures the qualitiy of any parameter set. Construction of the cost function
will always involve some subjective elements regarding the functional
forms or weights of the individual terms. This basically reflect that,
as usual in life, different people have different views on what is “best”.
The optimisation results will always depend on the quality of the cost
function which therefore should be crafted as carefully as possible.
5.3
Minimisation methods
In principle, one could just explore the cost function “landscape” in
parameter space by explicitly evaluating J(p) for a large number of different choices of the parameter vector p . In practice, however, this will
usually not be possible. Even for a simple NPZD-type ecosystem model
with, say, 15 parameters, a very coarse sampling of only 10 possible values per parameter would require 10 15 evaluations of the cost function.
As soon as models have more than a handful parameters, more efficient
minimisation methods are needed.
A large variety of such minimisation methods exist, most of which
have been developed outside oceanography. They can be devided into
methods that make use of the cost function’s gradient, i.e., information
about the downhill direction in the cost-function landscape, and into
methods that do not use this information and therefore do not require