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Data assimilation concepts
and methods
March 1999
By F. Bouttier and P.
Courtier
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6 . Models of error covariances
A correct specification of observation and background error covariances is crucial to the quality of the analysis, because they determine to what extent the background fields will be corrected to match the observations. The essential parameters are the variances, but the correlations are also very important because they specify how the observed information will be smoothed in model space if there is a mismatch between the resolution of the model and the density of the observations. In the framework of Kalman filtering and 4D assimilation with model as a weak constraint, a third kind of covariances to specify is
, the model error covariances (see the relevant section below).6.1 Observation error variances
They are mainly specified according to the knowledge of instrumental characteristics, which can be estimated using collocated observations, for instance. As explained before, they should also include the variance of representativeness errors which is not negligible when analysing phenomena which cannot be well represented in model space. It is wrong to leave observation biases as a contribution to the observation error variances because it will produce biases in the analysis increments; whenever observation biases can be identified, they should be removed from the observed value or from the background fields, depending on whether one thinks they are caused by problems in the model or in the observation procedure (unfortunately we do not always know what to decide).
6.2 Observation error correlations
They are often assumed to be zero, i.e. one believes that distinct measurements are affected by physically independent errors. This sounds reasonable for pairs of observations carried out by distinct instruments. This may not be true for sets of observations performed by the same platform, like radiosonde, aircraft or satellite measurements, or when several successive reports from the same station are used in 4D-Var. Intuitively there will be a significant observation error correlation for reports close to one another. If there is a bias it will show up as a permanent observation error correlation. The observation preprocessing can generate artificial correlations between the transformed observations e.g. when temperature profiles are converted to geopotential, or when there is a conversion between relative and specific humidity (correlation with temperature), or when a retrieval procedure is applied to satellite data. If the background is used in the observation preprocessing, this will introduce artificial correlations between observations and background errors which are difficult to account for: moving the observation closer to the background may make the observation and background errors look smaller, but it will unrealistically reduce the weight of the originally observed information. Finally, representativeness errors are correlated by nature: interpolation errors are correlated whenever observations are dense compared to the resolution of the model. Errors in the design of the observation operator, like forecast model errors in 4D-Var, are correlated on the same scales as the modelling problems.
The presence of (positive) observation error correlations can be shown to reduce the weight given to the average of the observations, and thus give more relative importance to differences between observed values, like gradients or tendencies. Unfortunately observation error correlations are difficult to estimate and can create problems in the numerics of the analysis and quality control algorithms. In practice it often makes sense to try to minimize them by working on a bias correction scheme, by avoiding unnecessary observation preprocessing, by thinning dense data and by improving the design of the model and observation operators. Most models of
covariances used in practice are diagonal or
almost.6.3 Background error variances
They are usually estimates of the error variances in the forecast used to produce
. In the
Kalman filter they are estimated automatically using the tangent-linear
model, so they do not need to be specified (although this means that the
problem is moved to the specification of the model error 
and the tuning of approximated algorithms that are
less costly than the complete Kalman filter). A crude estimate can be obtained
by taking an arbitrary fraction of climatological variance of the fields
themselves. If the analysis is of good quality (i.e. if there are a lot
of observations) a better average estimate is provided by the variance of
the differences between the forecast and a verifying analysis. If the observations
can be assumed to be uncorrelated, much better averaged background error
variances can be obtained by using the observational method explained
below. However, in a system like the atmosphere the actual background errors
are expected to depend a lot on the weather situation, and ideally the background
errors should be flow-dependent. This can be achieved by the Kalman filter,
by 4D-Var to some extent, or by some empirical laws of error growth based
on physical grounds. If background error variances are badly specified,
it will lead to too large or too small analysis increments. In least-squares
analysis algorithms, only the relative magnitude of the background and observation
error variances is important. However, the absolute values may be important
if they are used to make quality-control decisions on the observations (it
is usually desirable to accept more easily the observations with a large
background departure if the background error is likely to be large).6.4 Background error correlations
They are essential for several reasons:
Information spreading.
In data-sparse areas, the shape of the analysis increment is completely
determined by the covariance structures (for a single observation
it is given by  ). Hence the
correlations in  will perform the spatial spreading of
information from the observation points (real observations are usually
local) to a finite domain surrounding it. |
Information smoothing.
In data-dense areas, one can show that in the presence of discrete
observations (which is the usual case) the amount of smoothing1
of the observed information is governed by the correlations in  , which can
be understood by remarking that the leftmost term in  is  . The smoothing of the increments is important in ensuring
that the analysis contains scales which are statistically compatible
with the smoothness properties of the physical fields. For instance,
when analysing stratospheric or anticyclonic air masses, it is desirable
to smooth the increments a lot in the horizontal in order to average
and spread efficiently the measurements. When doing a low-level analysis
in frontal, coastal or mountainous areas, or near temperature inversions,
it is desirable on the contrary to limit the extent of the increments
so as not to produce an unphysically smooth analysis. This has to
be reflected in the specification of background error correlations. |
| Balance properties. There are often more degrees of freedom in a model than in reality. For instance, the large-scale atmosphere is usually hydrostatic. It is almost geostrophic, at least there is always a large amount of geostrophy in the extratropics. These balance properties could be regarded as annoying constraints on the analysis problem, and enforced brutally e.g. using an a posteriori normal-mode initialization. On the other hand, they are statistical properties that link the different model variables. In other words, they show up as correlations in the background errors because the existence of a balance in the reality and in the model state will imply that there is a (linearized) version of the balance that exists in the background error covariances, too. This is interesting for the use of observed information: observing one model variable yields information about all variables that are balanced with it, e.g. a low-level wind observation allows one to correct the surface pressure field by assuming some amount of geostrophy. When combined with the spatial smoothing of increments this can lead to a considerable impact on the quality of the analysis, e.g. a temperature observation at one point can be smoothed to produce a correction to geopotential height around it, and then produce a complete three-dimensional correction of the geostrophic wind field (Fig. 7 ). The relative amplitude of the increments in terms of the various model fields will depend directly on the specified amount of correlation as well as on the assumed error variance in all the concerned parameters. |
Figure 7 . Example of horizontal structure functions commonly used in meteorology: the horizontal autocorrelation of height (or pressure) has an isotropic, gaussian-like shape as a function of distance (right panel). In turn, geostrophy implies that wind will be cross-correlated with height at distances where the gradient of height correlation is maximum. Hence, an isolated height observation will generate an isotropic height "bump" with a rotating wind increment in the shape of a ring.
Ill-conditioning of the
assimilation. It is possible to include into the control
variables some additional parameters which are not directly observed,
like model tuning parameters or bias estimates. This can be an efficient
indirect parameter estimation technique if there is a realistic coupling
with the observed data, usually through the design of the observation
operator or of the model (in a 4-D assimilation). It may not be possible
or sensible to specify explicit correlations with the rest of the
model state in  . However, one
must be careful to specify a sensible background error for all parameters
of the control variable, unless it is certain that the problem is
over- determined by the observations. A too small error variance will
obviously prevent any correction to the additional parameters. A too
large variance may on the other hand make the additional parameters
act like a sink of noise, exhibiting variations whenever it improves
the fit of the analysis to observations, even if no such correction
of the additional parameters is physically justified. This can create
genuine problems because some implicit analysis coupling is often
created by variable dependencies in the observation operators or in
the model (in 4D-Var). Then, the specification of background errors
for additional parameters will have an impact on the analysis of the
main model state. They should reflect the acceptable amplitude of
the analysis corrections. |
Flow-dependent structure
functions. If enough is known about the dynamics of the problem,
one can make  depend on the uncertainty of the previous
analysis and forecast, not only in terms of background error variances,
but also in the correlations. In geophysical fluids there is not just
a loss of predictability during the forecast, there are waves that
follow specific patterns, and these patterns are expected to be found
in the background errors. For instance, in an area prone to cyclogenesis,
one expects the most likely background errors to have the shape (or
structure function) of the most unstable structures, perhaps
with a baroclinic wave tilt, and anticorrelations between the errors
in the warm and in the cold air masses. This is equivalent to a balance
property, and again if the relevant information can be embedded into
the correlations of  , then the observed information can
be more accurately spread spatially and distributed among all model
parameters involved. Such information can be provided in the framework
of a Kalman filter or 4D-Var. |
| ref: Courtier et al. 1998 |
6.5 Estimation of error covariances
It is a difficult problem, because they are never observed directly, they can only be estimated in a statistical sense, so that one is forced to make some assumptions of homogeneity. The best source of information about the errors in an assimilation system is the study of the background departures (
) and they can be used in a variety of ways. Other indications can
be obtained from the analysis departures, or from the values of the cost
functions in 3D/4D-Var. There are some more empirical methods based on the
study of forecasts started from the analyses, like the NMC method or the
adjoint sensitivity studies, but their theoretical foundation is rather
unclear for the time being. A comprehensive and rigorous methodology is
being developed under the framework of adaptive filtering which
is too complex to explain in this volume. Probably the most simple yet reliable
estimation method is the observational method explained below.
Figure 8 . Schematic representation the observational method. The (observation - background) covariance statistics for a given assimilation system are stratified against distance, and the intercept at the origin of the histogram provides an estimate of the average background and observation error variances for these particular assimilation and observation systems.
The observational (or Hollingworth-Lonnberg) method. This method2 relies on the use of background departures in an observing network that is dense and large enough to provide information on many scales, and that can be assumed to consist of uncorrelated and discrete observations. The principle (illustrated in Fig. 8 ) is to calculate an histogram (or variogram) of background departure covariances, stratified against separation (for instance). At zero separation the histogram provides averaged information about the background and observation errors, at non-zero separation it gives the averaged background error correlation: if
and 
are two
observation points, the background departure covariance 
can be calculated empirically and it is equal to
|
If one assumes that there is no correlation between observation and background errors, the last two terms on the second line vanish. The first term is the observation error covariance between
and 
, the second term is the background error
covariance interpolated at these points, assuming both are homogeneous over
the dataset used. In summary,
|
• if  ,  , the sum of the observation and the background error variances, |
• if  and the
observation errors are assumed to be uncorrelated,  , the background error covariance between  and  . (If there are observation error correlations,
it is impossible to disentangle the information about  and  without additional data) |
• Under the same assumption,
if  and  are very
close to each other without being equal, then  , so that by determining the intercept for zero separation of  , one can determine  . |
• Then, one gets  and the background error correlations are given by  (we have assumed that the background error variances are homogeneous
over the considered dataset). |
In most systems the background error covariances should go to zero for very large separations. If this is not the case, it is usually the sign of biases in the background and/or in the observations and the method may not work correctly (Hollingsworth and Lonnberg 1986.).
6.6 Modelling of background correlations
As explained above the full
matrix is usually
too big to be specified explicitly. The variances are just the n
diagonal terms of 
, which are
usually specified completely. The off-diagonal terms are more difficult
to specify. They must generate a symmetric positive definite matrix, so
one must be careful about the assumptions made to specify them. Additionally
is often required to have some physical
properties which are required to be reflected in the analysis:| • the correlations must be smooth in physical space, on sensible scales, |
| • the correlations should go to zero for very large separations if it is believed that observations should only have a local effect on the increments, |
| • the correlations should not exhibit physically unjustifiable variations according to direction or location, |
| • the most fundamental balance properties, like geostrophy, must be reasonably well enforced. |
| • the correlations should not lead to unreasonable effective background error variances for any parameter that is observed, used in the subsequent model forecast, or output to the users as an analysis product. |
| • Correlation models can be specified independently from variance fields, under the condition that the scales of variation of the variances are much larger than the correlation scales, otherwise the shape of the covariances would differ a lot from the correlations, with unpredictable consequences on the balance properties. |
| • Vertical autocorrelation matrices for each parameter are usually small enough to be specified explicitly. |
| • Horizontal autocorrelations cannot be specified explicitly, but they can be reduced to sparse matrices by assuming that they are homogeneous and isotropic to some extent. It implies that they are diagonal in spectral space3. In grid-point space some low-pass digital filters can be applied to achieve a similar result. |
| • Three-dimensional multivariate correlation models can be built by carefully combining separability, homogeneity and independency hypotheses like: zero correlations in the vertical for distinct spectral wavenumbers, homogeneity of the vertical correlations in the horizontal and/or horizontal correlations in the vertical, property of the correlations being products of horizontal and vertical correlations. Numerically they imply that the correlation matrix is sparse because it is made of block matrices which are themselves block-diagonal4 |
| • Balance constraints can be enforced by transforming the model variables into suitably defined complementary spaces of balanced and unbalanced variables. The latter are supposed to have smaller background error variances than the former, meaning that they will contribute less to the increment structures. |
• The geostrophic balance
constraint can be enforced using the classical  -plane or  -plane balance equations, or projections onto
subspaces spanned by so-called Rossby and Gravity normal modes. |
| • More general kinds of balance properties can be expressed using linear regression operators calibrated on actual background error fields, if no analytical formulation is available. |
Two last requirements which can be important for the numerical implementation of the analysis algorithm are the availability of the symmetric square root of
(a matrix 
such that 
) and of its
inverse. They can constrain notably the design of 
.ref: Courtier et al. 1998
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1 There is an equivalence between statistical analysis and the theory of interpolation by splines.
2 named after the authors that popularized it in meteorology, although it was known and used before in geophysics. The widespread kriging method is closely related.
3 This is the Khinchine-Bochner theorem. The spectral coefficients are proportional to the spectral variance of the correlations for each total wavenumber. This is detailed on the sphere in Courtier et al. (1996).
4 It corresponds to the mathematical concept of tensor product.
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03.12.2001 |
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