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Smoothing error pitfalls

Smoothing error pitfallsAtmospheric Measurement Techniques, 7, 3023-3034, 2014Author(s): T. von ClarmannThe difference due to the content of a priori information between a
constrained retrieval and the true atmospheric state is usually represented
by a diagnostic quantity called smoothing error. In this paper it is shown
that, regardless of the usefulness of the smoothing error as a diagnostic
tool in its own right, the concept of the smoothing error as a component of
the retrieval error budget is questionable because it is not compliant with
Gaussian error propagation. The reason for this is that the smoothing error
does not represent the expected deviation of the retrieval from the true
state but the expected deviation of the retrieval from the atmospheric state
sampled on an arbitrary grid, which is itself a smoothed representation of
the true state; in other words, to characterize the full loss of information
with respect to the true atmosphere, the effect of the representation of
the atmospheric state on a finite grid also needs to be considered. The idea of a
sufficiently fine sampling of this reference atmospheric state is problematic
because atmospheric variability occurs on all scales, implying that there is
no limit beyond which the sampling is fine enough. Even the idealization of
infinitesimally fine sampling of the reference state does not help, because
the smoothing error is applied to quantities which are only defined in a
statistical sense, which implies that a finite volume of sufficient spatial
extent is needed to meaningfully discuss temperature or concentration.
Smoothing differences, however, which play a role when measurements are
compared, are still a useful quantity if the covariance matrix involved has
been evaluated on the comparison grid rather than resulting from
interpolation and if the averaging kernel matrices have been evaluated on a
grid fine enough to capture all atmospheric variations that the instruments are
sensitive to. This is, under the assumptions stated, because the undefined
component of the smoothing error, which is the effect of smoothing implied by
the finite grid on which the measurements are compared, cancels out when the
difference is calculated. If the effect of a retrieval constraint is to be
diagnosed on a grid finer than the native grid of the retrieval by means of
the smoothing error, the latter must be evaluated directly on the fine grid,
using an ensemble covariance matrix which includes all variability on the
fine grid. Ideally, the averaging kernels needed should be calculated
directly on the finer grid, but if the grid of the original averaging kernels
allows for representation of all the structures the instrument is sensitive to, then their
interpolation can be an adequate approximation.

Atmospheric Measurement Techniques 2014/09/20 - 18:38 Czytaj