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An application of a data assimilation method based on the diffusion stochastic process theory using altimetry data in Atlantic

  • Konstantin P. Belyaev , Andrey A. Kuleshov EMAIL logo and Clemente A. S. Tanajura


A data assimilation (DA) method based on the application of the diffusion stochastic process theory, particularly, of the Fokker-Planck equation, is considered. The method was introduced in the previous works; however, it is substantially modified and extended to the multivariate case in the current study. For the first time, the method is here applied to the assimilation of sea surface height anomalies (SSHA) into the Hybrid Coordinate Ocean Model (HYCOM) over the Atlantic Ocean. The impact of assimilation of SSHA is investigated and compared with the assimilation by an Ensemble Optimal Interpolation method (EnOI). The time series of the analyses produced by both assimilation methods are evaluated against the results from a free model run without assimilation. This study shows that the proposed assimilation technique has some advantages in comparison with EnOI analysis. Particularly, it is shown that it provides slightly smaller error and is computationally efficient. The method may be applied to assimilate other data such as observed sea surface temperature and vertical profiles of temperature and salinity.

MSC 2010: 65C20


Authors would like to thank Prof. Jiang Zhu of the Institute for Atmospheric Physics, Chinese Academy of Sciences for his invitation to visit Beijing and for many useful advices concerning the content of the present work. The third author would like to acknowledge PETROBRAS and the Brazilian oil regulatory agency ANP (Agencia Nacional de Petroleo, Gas Naturale Biocombustiveis), within the special participation research project Oceanographic Modeling and Observation Network (REMO), and Ministry of Education of Brazil.


This research was supported by the Russian Science Foundation, grant 14-11-00434.


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Appendix A Numerical solution of the 2D Fokker-Planck (the second Kolmogorov) equation (1.7)

In standard notations, where x = u, y = v, equation (1.7) can be rewritten as


Let (A1) be defined in the domain Ω= {0 ≼ xl1, 0 ≼ yl2} on the horizontal plane with the boundary conditions p|Γ = 0, b11|Γ = bjj|Γ = 0. Let also the initial condition be p(0, x, y) =p°(x, y). Let the time-grid be introduced with time-step τ as tn+i = tn+ τ, n = 0, 1, 2,..., and on the domain Ω the grid ωh be introduced with the steps h1, h2 in x and y directions, respectively. Equation (A1) is solved with respect to a finite-difference splitting scheme which consists of three stages in each of time-intervals [tn, tn+1]


In (A2)-(A4) all coefficients are known at the time-moment tn. The standard notations of the difference approximations of the derivatives [23] are used as

            ψx°,ij=ψi+1,jψi1,j2h1,ψy°,ij=ψi,j+1ψi,j12h2ψx¯x,ij=ψi+1,j2ψij+ψi1,jh12,ψy¯y,ij=ψi,j+12ψij+ψi,j1h22              ψx°y°,ij=ψi+1,j+1ψi1,j+1ψi+1,j1+ψi1,j14h1h2.

In canonic form the system of equations (A2) with fixed index can be written as


This system has a tridiagonal matrix and is solved with the tridiagonal matrix algorithm (TDMA-method) [23]. The condition of stability of the TDMA-method |Ci| | Ai | +|Bi| holds if


This inequality yields


Let the coefficients of equation (1.7)) satisfy the conditions


Then the similar conditions hold for approximations on each time-interval


Therefore the stability condition (A5) has the following form


The system of linear equations (A3) is solved similarly with the same condition of stability. The system (A4) is solved by the simple iteration method which can be effectively parallelized. As a consequence one may state that the proposed numerical scheme for equation (1.7) has an order of approximation O(τ+h12+h22) and it is stable when conditions (A6)-(A8) hold.

Also, if initially Ωp0dx  dy=1 then from (1.7) and boundary conditions one may have

tΩp(t,x,y)dx  dy=Ωptdx  dy=Ω{(a1p)x(a2p)y+12[2(b112p)x2+22(b0p)xy+2(b222p)y2]}dx  dy=0.

Then Ωpdx  dy=1 for any t > 0, i.e., to the conservative of the initial integral. In conjunction with the stability and approximation of the finite-difference scheme this leads to the conservation of the corresponding calculated integral sums, which is needed to the correct calculation of covariance followed from the solution of the Fokker-Plank (or Kolmogorov) equation.

Received: 2015-9-28
Accepted: 2016-3-10
Published Online: 2016-5-28
Published in Print: 2016-6-1

© 2016 Walter de Gruyter GmbH, Berlin/Boston

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