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A revised scheme to compute horizontal covariances in an oceanographic 3D-VAR assimilation system

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a r X i v : 1 4 0 4 . 5 7 5 6 v 1 [ c s . N A ] 2 3 A p r 2 0 1 4
A Revised Scheme to Compute HorizontalCovariances in an Oceanographic 3D-VARAssimilation System
R. Farina
a
, S. Dobricic
a
, A. Storto
a
, S. Masina
a
, and S. Cuomo
b
a
Centro Euro-Mediterraneo sui Cambiamenti Climatici,Bologna, Viale Aldo Moro 44, Italy.Email: raﬀaele.farina@cmcc.it - Telephone: +39 051 378267
b
Department of Mathematics and Applications, University of Naples ”Federico II”.Via Cinthia, 80126, Napoli, Italy.Email: salvatore.cuomo@unina.it - Telephone: +39 081675624
Abstract
We propose an improvement of an oceanographic three dimensional varia-tional assimilation scheme (3D-VAR), named OceanVar, by introducing arecursive ﬁlter (RF) with the third order of accuracy (3rd-RF), instead of aRF with ﬁrst order of accuracy (1st-RF), to approximate horizontal Gaussiancovariances. An advantage of the proposed scheme is that the CPU’s timecan be substantially reduced with beneﬁts on the large scale applications.Experiments estimating the impact of 3rd-RF are performed by assimilatingoceanographic data in two realistic oceanographic applications. The resultsevince beneﬁts in terms of assimilation process computational time, accuracyof the Gaussian correlation modeling, and show that the 3rd-RF is a suitabletool for operational data assimilation.
Keywords:
Data assimilation, Recursive Gaussian Filter and NumericalOptimization.
1. Introduction
Ocean data assimilation is a crucial task in operational oceanography, re-sponsible for optimally combining observational measurements and a priorknowledge of the state of the ocean in order to provide initial conditionsfor the forecast model. How the informative content of the observations is
Preprint submitted to Ocean Modelling April 24, 2014
spread horizontally in space depends on the operator used to model horizon-tal covariances. The three-dimensional variational data assimilation schemecalled OceanVar (Dobricic and Pinardi, 2008) represents horizontal covari-
ances of background errors in temperature and salinity by approximate Gaus-sian functions that depend only on the horizontal distance between the twomodel points. In the framework of the Optimal Interpolation, where theanalysis is found by using only the nearest observations, the calculation of the Gaussian function can be made directly from the distances between themodel point and the typically small number of nearby observations. Thiskind of solution may be impractical in the variational framework where it isnecessary to calculate the covariances between each pair of model points inthe horizontal. Instead, variational schemes often use linear operators thatapproximate the Gaussian function (e.g., Weaver et al., 2003).
In meteorology, Lorenc (1992) approximated the Gaussian function by apply-
ing one-dimensional recursive ﬁlters (RF) with the ﬁrst-order accuracy suc-cessively in the two perpendicular directions. In oceanography, Weaver et al.(2003) used the explicit integration of the two-dimensional diﬀusion equation.Purser et al. (2003) developed higher order recursive ﬁlters for use in atmo-
spheric models. The OceanVar scheme described in Dobricic and Pinardi(2008) used the RF of the ﬁrst order (1st-RF) with imaginary sea points forthe processing on the coast. Mirouze and Weaver (2010) implemented the
implicit integration of one-dimensional diﬀusion equations.Successive applications of one-dimensional recursive ﬁlters or implicit inte-grations of the diﬀusion equation in the two perpendicular directions aremuch more computationally eﬃcient than the explicit integration of the dif-fusion equation (Mirouze and Weaver, 2010). However, the ﬁrst order accu-
rate operators used in most of these schemes still require several iterations toapproximate the Gaussian function. For example, the 1st-RF in OceanVargenerally applies 5 iterations and is the computationally most demandingpart.Recursive ﬁlters with higher order accuracy require more operations for eachiteration, but only one iteration might be enough to accurately approximatethe Gaussian function. Generally high order recursive ﬁlters can be obtain bymeans of diﬀerent strategies e.g. in Purser et al. (2003), Young and Van Vliet
(1995) and Deriche (1987). The main diﬀerence among them is the math-
ematical methodology used to obtain the ﬁlter coeﬃcients. In meteorology,Purser et al. (2003) resolve an inverse problem with exponential matrix of
ﬁnite diﬀerences operator approximating the second derivative
d
2
/dx
2
on a2
line grid of uniform spacing
δx
. They use truncated Taylor expansion to ap-proximate the exponential matrix and obtain the ﬁlter coeﬃcients throughthe
LL
T
factorization of the result approximation. The degree of the Taylorpolynomial is the order of RF obtained.In this study we develop a RF of the third order accuracy (3rd-RF),stillwith the use of the imaginary points for treatment on the coast, that needsonly one iteration to approximate the Gaussian function and allows diﬀerentlength scales. Our approach, based on Young and Van Vliet (1995), deter-
mines the ﬁlter coeﬃcients of a 3rd-RF by the matrix-vector multiplicationof gaussian operator for a input ﬁeld, using a known rational approximationof the gaussian function. Note that this strategy has been so far exploitedonly in signal processing, and represents a completely novel methodology ingeophysical data assimilation. Furthermore, we compare the 3rd-RF perfor-mance with those of the existing 1st-RF on two diﬀerent conﬁgurations of OceanVar: Mediterranean Sea and Global Ocean. The new ﬁlter should beat least as accurate as the existing one and it should execute more rapidly onmassively parallel computers. Tests on parallel computer architectures areespecially important because higher order accurate ﬁlters compute the solu-tion from several nearby points, and as a consequence, transfer more dataamong processors eventually becoming less eﬃcient.Section 2 gives a general description of the existing OceanVar data assimila-tion scheme. Section 3 demonstrates in detail the development of the 3rd-RF.It also provides all numerical values and the method to calculate the coeﬃ-cients of the ﬁlter with diﬀerent length scales. Moreover we give an estimateof approximation error between the result of a RF and the real Gaussian con-volution. In Sections 4 and 5 the ﬁlter is applied in the operational version of OceanVar used respectively in the Mediterranean Sea (Pinardi et al., 2010)
and Global Ocean (Storto et al., 2011). Its performance is compared to theperformance of 1st-RF. In Section 5 we present the conclusions and indicatethe future directions of the development of the operator for the horizontalcovariances.3
2. General Description
2.1. The OceanVar Computational Kernel
The computational kernel of the OceanVar data assimilation scheme is basedon the following regularized constrained least square problem:min
D
{
J
(
x
) = 12
x
−
x
b
2
B
−
1
+ 12
y
−
H
(
x
)
2
R
−
1
/x
∈
D
}
(1)where
D
is a grid domain in
R
3
. In equation (1) the vector
x
=
T,S,η,u,v
⊤
is an ocean state vector composed by the temperature
T
, the salinity
S
, sealevel
η
and horizontal velocity ﬁeld (
u,v
). The vector
x
b
is the backgroundstate vector, achieved by numerical solution of an ocean forecasting modeland is an approximation of the ”true” state vector
x
t
. The diﬀerence betweenbackground
x
b
and any state vector
x
is denoted by
δx
:
x
b
=
x
+
δx
(2)The vectors
x
and
x
b
are deﬁned on the same space called
physical space
. Thevector
y
in (1) is the observational vector deﬁned on a diﬀerent space called
observational space
and the function
H
is a non linear operator that convertsvalues deﬁned in the physical space to values deﬁned in the observationalspace. An ocean state vector
x
is related to observations
y
by means thefollowing relation:
y
=
H
(
x
) +
δy
(3)where
δy
is an eﬀective measurement error. In (1) the matrix
R
=
δy
t
δy
t
⊤
,with
δy
t
=
y
t
−
H
(
x
t
), is the observational error matrix covariance and itis assumed generally to be diagonal, i.e. observational errors are seen asstatistically independent. The
B
=
δx
t
δx
t
⊤
, with
δx
t
=
x
b
−
x
t
, is thebackground error matrix covariances and is never assumed to be diagonal inits representation.Problem (1) is solved by minimizing the following explicit form of cost func-tion
J
(
x
):
J
(
x
) = 12(
x
−
x
b
)
⊤
B
−
1
(
x
−
x
b
) + 12(
y
−
H
(
x
))
⊤
R
−
1
(
y
−
H
(
x
))
.
(4)It is often numerically convenient to exploit the weak non linearity of
H
byapproximating
H
(
x
), for small increments
δx
, with a linear approximationaround the background vector
x
b
:4
H
(
x
)
≈
H
(
x
b
) +
H
δx.
(5)where the linear operator
H
is the
H
’s Jacobian evaluated at
x
=
x
b
. The costfunction
J
, using (5), is approximated by the following quadratic function:
J
(
δx
) = 12
δx
⊤
B
−
1
δx
+ 12(
d
−
H
δx
)
⊤
R
−
1
(
d
−
H
δx
) (6)deﬁned on increment space. In (6) the vector
d
=
y
−
H
(
x
b
) is the misﬁt.The minimum of the cost function
J
(
δx
) on the increment space may be jus-tiﬁed by posing
∇
J
(
δx
) = 0. Then we obtain, as also shown in Haben et al.(2011), the following preconditioned system:
I
+
BH
T
R
−
1
H
δx
=
BH
T
R
−
1
d
(7)To solve the linear equation system (7) iterative methods able to convergetoward a practical solution are needed. Generally, the OceanVar model usesthe Conjugate Gradient Method (Byrd et al., 1995).The iterative minimizer schema is based essentially on matrix-vector opera-tion of some vector
v
=
H
T
R
−
1
H
δx
with the covariance matrix
B
. Thiscomputational kernel is required at each iteration and its huge computationalcomplexity is a bottleneck in practical data assimilation. This problem canbe overcome by decomposing the covariance matrix
B
in the following form:
B
=
VV
T
(8)However due to its still large size, the matrix
V
is split at each minimiza-tion iteration as a sequence of linear operators (Weaver et al., 2003). Moreprecisely, in OceanVar the matrix
V
is decomposed as:
V
=
V
D
V
uv
V
η
V
H
V
V
(9)where the linear operator
V
V
transforms coeﬃcients which multiply verti-cal EOFs into vertical proﬁles of temperature and salinity deﬁned at themodel vertical levels,
V
H
and
V
η
apply respectively the gaussian ﬁlteringto the ﬁelds of temperature and salinity, and sea surface.
V
uv
calculatesvelocity from sea surface height, temperature and salinity, and
V
D
applies adivergence damping ﬁlter on the velocity ﬁeld. A more detailed formulationof each linear operator is described in Dobricic and Pinardi (2008). In thispaper we focus on operator
V
H
.5

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