132 Antonio Navarra
laws. Such relations and laws are usually well accepted and in general there is a
good degree of agreement among scientists on their number and formulation. The
big shock that the original Lorenz (1963) result brought was mainly originated in
the training that scientists have received based on the idea that nature is smooth,
simple and elegant. The sensitivity to initial condition broke with this scenario generating constemation and the impression that the class of equation that include the
atmospheric equations were special and mysterious in some sense, therefore the
name "strange attractors". The result in a nutshell can be seen as the statement that
if one picks arbitrarily a system of ordinary differential equations of more
than 3 variables the probability of choosing a system with a strange attractor is
virtually one. Strange attractors are not strange at all, but they are everywhere, on
the other hand, extremely intricate behavior can be described by powerfullaws and
equations. The sensitivity of the atmospheric motions to initial condition is therefore a very common occurrence typical of systems of several (greater than three)
strongly interacting state variables.
The situation should not be confused with a real stochastic systems, like quantum
mechanical systems, where the stochastic component is intrinsic, i.e. an experiment
with exactly the same initial and boundary condition would end up in different
results. The atmospheric motions are not intrinsically stochastic, because experiment with the same initial condition and boundary condition would result in the
same evolution. However, the atmospheric circulations, and even more so the interlocked atmosphere-ocean-Iand-ecosystem, are largely composed of unobservable
state variables that we will probably never be able to observe or specify in a model.
This level ofuncertainty, coupled to the sensitivity to the initial condition, creates a
situation where is conceptually more convenient and more economic to treat the
system as if it were an intrinsic stochastic system, exploiting the power of statistical and probabilistic concept into its description.
A single simulation or forecast is therefore going to be scarcely representative of
the true behavior of the system, unless in special circumstances. In principle both
initial and boundary conditions are going to be affected by unobserved state-variables, but the sensitivity to initial condition has been greatly investigated and it is
certain1y the most relevant in the classical initial value forecasting problem.
Restricting to the initial condition case, we assume that the initial condition is
really made up of two parts:
(1)
representing the observed part X o and the unobserved part X u that in this formulation contains also the unknown error on the observed variables. The true initial
condition corresponds to a certain unknown value of the unobserved variables.
Since in a prediction experiment we only have available x o ' the evolution of the
forecasts will be dependent on the value of the unobservable variables. A single
forecasts will not have agreat probability ofhitting the right value, to increase our
chances will be necessary to perform multiple forecasts, hopefully filling up a
hyper dimensional sphere that includes the true value.
laws. Such relations and laws are usually well accepted and in general there is a
good degree of agreement among scientists on their number and formulation. The
big shock that the original Lorenz (1963) result brought was mainly originated in
the training that scientists have received based on the idea that nature is smooth,
simple and elegant. The sensitivity to initial condition broke with this scenario generating constemation and the impression that the class of equation that include the
atmospheric equations were special and mysterious in some sense, therefore the
name "strange attractors". The result in a nutshell can be seen as the statement that
if one picks arbitrarily a system of ordinary differential equations of more
than 3 variables the probability of choosing a system with a strange attractor is
virtually one. Strange attractors are not strange at all, but they are everywhere, on
the other hand, extremely intricate behavior can be described by powerfullaws and
equations. The sensitivity of the atmospheric motions to initial condition is therefore a very common occurrence typical of systems of several (greater than three)
strongly interacting state variables.
The situation should not be confused with a real stochastic systems, like quantum
mechanical systems, where the stochastic component is intrinsic, i.e. an experiment
with exactly the same initial and boundary condition would end up in different
results. The atmospheric motions are not intrinsically stochastic, because experiment with the same initial condition and boundary condition would result in the
same evolution. However, the atmospheric circulations, and even more so the interlocked atmosphere-ocean-Iand-ecosystem, are largely composed of unobservable
state variables that we will probably never be able to observe or specify in a model.
This level ofuncertainty, coupled to the sensitivity to the initial condition, creates a
situation where is conceptually more convenient and more economic to treat the
system as if it were an intrinsic stochastic system, exploiting the power of statistical and probabilistic concept into its description.
A single simulation or forecast is therefore going to be scarcely representative of
the true behavior of the system, unless in special circumstances. In principle both
initial and boundary conditions are going to be affected by unobserved state-variables, but the sensitivity to initial condition has been greatly investigated and it is
certain1y the most relevant in the classical initial value forecasting problem.
Restricting to the initial condition case, we assume that the initial condition is
really made up of two parts:
(1)
representing the observed part X o and the unobserved part X u that in this formulation contains also the unknown error on the observed variables. The true initial
condition corresponds to a certain unknown value of the unobserved variables.
Since in a prediction experiment we only have available x o ' the evolution of the
forecasts will be dependent on the value of the unobservable variables. A single
forecasts will not have agreat probability ofhitting the right value, to increase our
chances will be necessary to perform multiple forecasts, hopefully filling up a
hyper dimensional sphere that includes the true value.