The Adjoint Sensitivity Guidance to Diagnosis and Tuning of Error Covariance Parameters - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 213

d D @e
@

(9.57)

u

identifies the direction of small parameter variations u , from the current DAS

configuration, that will be of largest forecast benefit and provides a first-order
optimality diagnostic. For applications to parameter tuning an additional search
must be performed to determine an optimal step length along the descent direction.
Daescu and Todling ( 2010 ) provided an illustration of iterative gradient-based
tuning of observation error variances. Whereas the optimal parameter values may
not be inferred from the derivative information alone, valuable insight may be
gained by monitoring the forecast error sensitivity to the specification of the error
covariance parameters.
A proof-of-concept is given with the Lorenz 40-variable model ( Lorenz and
Emanuel 1998 )

ı

d x j

d

D .x j C 1 x j 2 /x j 1 x j C F; j D 1 W n (9.58)

t

where .

The system ( 9.58 ) is integrated with the standard fourth-order explicit Runge-Kutta
method and a constant time step

n D 40;x n C j D x j , and the forcing constant is specified as F D 8

that is identified with a 6-h time period
to produce a reference trajectory (“the truth”) x t .
The adjoint-DAS sensitivity guidance to diagnosis and tuning of error covariance
parameters is illustrated using an idealized data assimilation system (DAS-I) and
a suboptimal data assimilation system (DAS-II). Observations are assumed to be
available at each grid point, with unbiased and uncorrelated observation errors taken
from a normal distribution with the standard deviation of

t D 0:05

o;t D 0:5 at locations

1-20 and of at locations 21-40. In both DAS-I and DAS-II the assigned

o values are consistent with the true observation error statistics,

o;t D 1

o D o;t ,and
distinction between the experiments is made through the B -matrix specification.
DAS-I provides an optimal analysis by implementing a full Extended Kalman Filter
(EKF) to update in time the background error covariance. Figure 9.3 ashowsthe
average over

N D 7;200 analysis cycles ( a 5-year period) of the background

error covariance in DAS-I,

N X

B D 1
N

B .t i / (9.59)

i D 1

Deficiencies are introduced in DAS-II by ignoring the background error correlations
and the flow dependence of the B matrix. In DAS-II, the background error
covariance is specified as a diagonal matrix ( C b D I ), frozen in time, with the

b diagonal entries taken from ( 9.59 ),asshowninFig. 9.3 b.

The B -sensitivity associated to the Euclidean norm of the 24-h forecast errors is
monitored in each of DAS-I and DAS-II and time averaged results are displayed
in Fig. 9.3 c and in Fig. 9.3 d, respectively. In the optimal system DAS-I, the B -
sensitivity matrix displays no particular structure and this is an indication that
no systematic deficiency in the B matrix specification has been identified. In the