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

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: - Telephone: +39 051 378267  b Department of Mathematics and Applications, University of Naples ”Federico II”.Via Cinthia, 80126, Napoli, Italy.Email: - 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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