74 Andrew C. Lorenc
more information in the model state, from previous observations, than there is in a
new batch at a single synoptic time. Thus it is important to preserve this in the
assimilation process; it is not just a question of titting the new data. Since alI information has to be represented within the model, it is important that the model should
be of sufficiently high resolution, with physicalIy realistic detail, to represent the
information observed.
Some research is investigating non-sequential data assimilation methods, especialIy four-dimensional variational assimilation. This paper is designed to provide
the groundwork for this, without going into detail.
5.3 Useful prior knowledge about the atmosphere
We need tirst to decide what aspects of the atmosphere we are modelling. Certain scales, including of course alI those not resolved by the model, and types of
motion, e.g. sound waves, are not represented. We detine our "true" state, the target of an ideal assimilation, as the atmospheric state with these filtered out. They
then contribute to the error ofrepresentativeness ofthe model, but not to the analysis error (i.e. if the analysis fits our "true" state, its analysis error is zero, even if it
does not represent alI the detail ofthe real atmosphere).
For the scales considered in NWP, the atmosphere is usually smooth, slowly
varying, and close to horizontal non-divergence. We can observe these properties
and quantify them. The smoothness can be described by a correlation function, or
power spectrum. The slowly varying property leads to useful balance relationships.
For instance by assuming that the rate of change ofwind, and other residual tenns,
are negligible compared to the pressure gradient and Coriolis forces, we get the
geostrophic relationship. Unfortunately none of these simple relationships is exact,
and imposing them would probably damage our prior estimate of the atmospheric
state, :x? To avoid this, we apply the same balance arguments to:x?, and subtract.
The residual terms which lead to inaccuracy in the simple relationships then
approximately cancel. It is thus appropriate to apply the smoothness, geostrophic
balance, and non-divergence relationships to the residuals, x-:x? They are used in
modelling the structure ofthe background covariance B.
The observation that the atmosphere (for those aspects we are trying to represent)
is slowly varying, can be used to derive balance relationships involving diabatic
processes, such as latent heating, and the circulations of the associated weather systems. However these effects are not easily described mathematicalIy. An alternative approach is to use directly the "slowly varying" property by initialising a
numeric al model so that its initial evolution is slowly varying.
By far the most useful and accurate prior knowledge that we have, is ofthe equations governing the atmosphere's evolution in time, enabling us to build NWP
models. The simplest use of models is to carry information forward in time from a
past analysis, to provide the background for a new analysis. This process is the core
of alI operational NWP systems; without it analyses and forecasts would be sub-
more information in the model state, from previous observations, than there is in a
new batch at a single synoptic time. Thus it is important to preserve this in the
assimilation process; it is not just a question of titting the new data. Since alI information has to be represented within the model, it is important that the model should
be of sufficiently high resolution, with physicalIy realistic detail, to represent the
information observed.
Some research is investigating non-sequential data assimilation methods, especialIy four-dimensional variational assimilation. This paper is designed to provide
the groundwork for this, without going into detail.
5.3 Useful prior knowledge about the atmosphere
We need tirst to decide what aspects of the atmosphere we are modelling. Certain scales, including of course alI those not resolved by the model, and types of
motion, e.g. sound waves, are not represented. We detine our "true" state, the target of an ideal assimilation, as the atmospheric state with these filtered out. They
then contribute to the error ofrepresentativeness ofthe model, but not to the analysis error (i.e. if the analysis fits our "true" state, its analysis error is zero, even if it
does not represent alI the detail ofthe real atmosphere).
For the scales considered in NWP, the atmosphere is usually smooth, slowly
varying, and close to horizontal non-divergence. We can observe these properties
and quantify them. The smoothness can be described by a correlation function, or
power spectrum. The slowly varying property leads to useful balance relationships.
For instance by assuming that the rate of change ofwind, and other residual tenns,
are negligible compared to the pressure gradient and Coriolis forces, we get the
geostrophic relationship. Unfortunately none of these simple relationships is exact,
and imposing them would probably damage our prior estimate of the atmospheric
state, :x? To avoid this, we apply the same balance arguments to:x?, and subtract.
The residual terms which lead to inaccuracy in the simple relationships then
approximately cancel. It is thus appropriate to apply the smoothness, geostrophic
balance, and non-divergence relationships to the residuals, x-:x? They are used in
modelling the structure ofthe background covariance B.
The observation that the atmosphere (for those aspects we are trying to represent)
is slowly varying, can be used to derive balance relationships involving diabatic
processes, such as latent heating, and the circulations of the associated weather systems. However these effects are not easily described mathematicalIy. An alternative approach is to use directly the "slowly varying" property by initialising a
numeric al model so that its initial evolution is slowly varying.
By far the most useful and accurate prior knowledge that we have, is ofthe equations governing the atmosphere's evolution in time, enabling us to build NWP
models. The simplest use of models is to carry information forward in time from a
past analysis, to provide the background for a new analysis. This process is the core
of alI operational NWP systems; without it analyses and forecasts would be sub-