Reminiscences of MODE
23
In another experiment starting from a randomly chosen initial eddy field, the
net horizontal pressure force instantaneously exerted on the bottom was calculated in
the model, and a barotropic mean velocity was allowed to develop freely in response,
preserving the total energy of the eddies and mean flow as well as the mean density
gradients. The eddy motion was chaotic, though because of bottom friction and lateral
viscosity the overall intensity decayed slowly. Characteristic statistical properties of
the eddies were monitored during this decay, to be compared in due course with the
results of the MODE field program. They showed larger space and time scales above
the thermocline, a minimum energy in middepths, and some intensification of the
smallest scales near the bottom.
Another aspect of the Baltimore project was to prepare for introducing observational data into the model, either as initial conditions or on an ongoing basis. Mindful
of that visit to the Royal Meteorological Office in Bracknell, it was anticipated that
this might not be straightforward. In particular a complete initial state for the model
required fields of dynamic heights at all grid points at all levels. Given this input data,
the geostrophic velocities and potential vorticity necessary to proceed one time step
could be derived in the model by finite differences, and the dynamic heights at the
next time by solving a form of Poisson’s equation. However, all the observational
data would come from instruments that were at discrete, irregularly spaced locations
rather than model grid points, and the measured variables would be either density or
velocity, which convert into vertical or horizontal gradients of dynamic height, rather
than of dynamic height itself. Thus, a consistent, automated procedure was needed to
interpolate or extrapolate onto the grid.
The method chosen assumes that the underlying physical variables are governed
by a joint normal multivariate probability distribution, with means and covariances
that are known a priori. Then, if a number of observational data are provided containing independent, linearly additive errors that are also normally distributed with known
mean and variance, Bayes theorem permits in principle calculation of the a posteriori
probabilities of both the physical fields and the measurement errors, including the
most probable value for each physical variable at each grid point and the uncertainty
surrounding this value. This most probable value is a maximum likelihood estimate
for the physical variable concerned. The calculation involves solving a potentially
large number of linear simultaneous equations, but appropriate algorithms for doing
so numerically were available. A major benefit of this approach is that it also permits calculation of the mean square error of the estimate, even if no actual data are
available but only the underlying means and covariances. In particular, it provided
an objective, computerized method of mapping in an intellectually consistent manner
a collection of observations of different types onto a regular grid suitable for further
analysis or modeling activities, together with a measure of the probable sampling
uncertainties inherent in this mapping.
When I discussed this concept with Russ Davis, who was also a member of the
Theoretical Panel, he pointed out that, provided attention is restricted to interpolation
23
In another experiment starting from a randomly chosen initial eddy field, the
net horizontal pressure force instantaneously exerted on the bottom was calculated in
the model, and a barotropic mean velocity was allowed to develop freely in response,
preserving the total energy of the eddies and mean flow as well as the mean density
gradients. The eddy motion was chaotic, though because of bottom friction and lateral
viscosity the overall intensity decayed slowly. Characteristic statistical properties of
the eddies were monitored during this decay, to be compared in due course with the
results of the MODE field program. They showed larger space and time scales above
the thermocline, a minimum energy in middepths, and some intensification of the
smallest scales near the bottom.
Another aspect of the Baltimore project was to prepare for introducing observational data into the model, either as initial conditions or on an ongoing basis. Mindful
of that visit to the Royal Meteorological Office in Bracknell, it was anticipated that
this might not be straightforward. In particular a complete initial state for the model
required fields of dynamic heights at all grid points at all levels. Given this input data,
the geostrophic velocities and potential vorticity necessary to proceed one time step
could be derived in the model by finite differences, and the dynamic heights at the
next time by solving a form of Poisson’s equation. However, all the observational
data would come from instruments that were at discrete, irregularly spaced locations
rather than model grid points, and the measured variables would be either density or
velocity, which convert into vertical or horizontal gradients of dynamic height, rather
than of dynamic height itself. Thus, a consistent, automated procedure was needed to
interpolate or extrapolate onto the grid.
The method chosen assumes that the underlying physical variables are governed
by a joint normal multivariate probability distribution, with means and covariances
that are known a priori. Then, if a number of observational data are provided containing independent, linearly additive errors that are also normally distributed with known
mean and variance, Bayes theorem permits in principle calculation of the a posteriori
probabilities of both the physical fields and the measurement errors, including the
most probable value for each physical variable at each grid point and the uncertainty
surrounding this value. This most probable value is a maximum likelihood estimate
for the physical variable concerned. The calculation involves solving a potentially
large number of linear simultaneous equations, but appropriate algorithms for doing
so numerically were available. A major benefit of this approach is that it also permits calculation of the mean square error of the estimate, even if no actual data are
available but only the underlying means and covariances. In particular, it provided
an objective, computerized method of mapping in an intellectually consistent manner
a collection of observations of different types onto a regular grid suitable for further
analysis or modeling activities, together with a measure of the probable sampling
uncertainties inherent in this mapping.
When I discussed this concept with Russ Davis, who was also a member of the
Theoretical Panel, he pointed out that, provided attention is restricted to interpolation