Linearized Physics for Data Assimilation at ECMWF - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 253

If M is the model describing the time evolution of the model state x as:

x .t i C 1 / D MΠx .t i / (11.4)

then the time evolution of a small perturbation x can be estimated to the first order

approximating by the tangent linear model M (derived from the NL model

ı

M ):

ı x .t i C 1 / D M Œ x .t i /ı x .t i /

.t i C 1 / D @MŒ

x .t i /

ı x

x ı

x .t i / (11.5)

@

The verification of the correctness of the TL model is first performed through the
classical Taylor formula:

M. x C ı x / M. x /

lim
! 0

D 1 (11.6)

M .ı x /

This examination of asymptotic behaviour, using perturbations the size of which
becomes infinitesimally small, is performed to check the numerical correctness of
the TL code.
For practical applications, it is also important to investigate the accuracy of
TL models for finite-amplitude perturbations (typically perturbations of the size of
analysis increments). The results from applications of tangent-linear and adjoint
models are only useful when the linearized approximation is valid for such
perturbations. Therefore, for the validation of the tangent-linear approximation, the
accuracy of the linearization of a parametrization scheme is studied with respect to
pairs of non-linear results. The difference between two non-linear integrations (one
starting from a background field, x b , and the other from an analysis, x a ) run with
the full NL model,

, is compared to time evolution of the analysis increments
( x a x b ) obtained by integrating the TL model, M , with the trajectory taken from
the background field.
For a quantitative evaluation of the impact of linearized schemes, their relative
importance is evaluated using mean absolute errors between tangent-linear and non-
linear perturbations as:

M

ˇ ˇ ˇ

x b / ˇ ˇ ˇ

x a x b / M.

" D M . x a / M. (11.7)

As a reference for the comparisons, an absolute mean error for the TL model without
physics, " exp is defined as the absolute mean error of the TL model

with the different physical schemes included, then an improvement coming from the
inclusion of more physics in the TL model is expressed as

" ref , is taken. If

" exp <" ref . The relative

errors, r er , and relative improvements, , are also computed as: