Eur. Phys. J. Special Topics 222, 875–883 (2013) © EDP Sciences, Springer-Verlag 2013 DOI: 10.1140/epjst/e2013-01890-3
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
Data assimilation using Ensemble Transform Kalman Filter (ETKF) in ROMS model for Indian Ocean Md. Nurujjaman1,a , A. Apte2 , and P. Vinayachandran3 1 2 3
Department of Physics, National Institute of Sikkim, Ravangla 737139, Sikkim, India Centre for Applicable Mathematics, TIFR, Sharada Nagar, Bangalore 560065, India Centre for Atmospheric and Ocean Sciences, Indian Institute Sciences, Bangalore, India Received 22 March 2013 / Received in final form 3 May 2013 Published online 11 July 2013 Abstract. Study of Oceans dynamics and forecast is crucial as it influences the regional climate and other marine activities. Forecasting oceanographic states like sea surface currents, Sea surface temperature (SST) and mixed layer depth at different time scales is extremely important for these activities. These forecasts are generated by various ocean general circulation models (OGCM). One such model is the Regional Ocean Modelling System (ROMS). Though ROMS can simulate several features of ocean, it cannot reproduce the thermocline of the ocean properly. Solution to this problem is to incorporates data assimilation (DA) in the model. DA system using Ensemble Transform Kalman Filter (ETKF) has been developed for ROMS model to improve the accuracy of the model forecast. To assimilate data temperature and salinity from ARGO data has been used as observation. Assimilated temperature and salinity without localization shows oscillations compared to the model run without assimilation for India Ocean. Same was also found for u and v-velocity fields. With localization we found that the state variables are diverging within the localization scale.
1 Introduction Oceans play a crucial role in regulating the regional climate, and other marine activities range from conventional fishing to high-tech oil and natural gas exploration. Forecasting oceanographic states like sea surface currents, Sea surface temperature (SST) and mixed layer depth at different time scales is extremely important for these activities. These forecasts are generated by various Ocean general circulation models namely Regional Ocean Modelling System (ROMS) [1, 2]. As these models are nonlinear and do not capture all the physics of the Ocean, errors grow to an unacceptable level after some time however accurate the initial state is. Figure 1(a) and (b) shows the salinity and temperature profile from observation (dashed line) and model run (solid line). It shows that temperature and salinity profile from model run can not reproduce the thermocline as well as halocline. It also shows that the model has bias a
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(a)
(b)
−500 −600 34.2 34.4 34.6 34.8 35 35.2 Salinity
Depth (m)
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15 20 25 o Temperature ( C)
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Fig. 1. (a) Salinity profile from ARGO data shown using −− line, (b) Temperature profile from ROMS model run shown using solid line. Both the profile have been shown at (64.7◦ E, 8.5◦ N).
Fig. 2. Comparison of model (first) and WOA05 climatology (second) sea surface temperature (SST) ◦ C during June. Third one represents climatology minus model SST in area of the east Arabian Sea. Isotherms have a spacing of 0.5 ◦ C. Difference plot shows 1.5 ◦ C difference in temperature.
(of 5 ◦ C). Figure 2(a) and (b) and (c) shows the SST at Southeastern Arabian sea from model run, WOA05 climatology and the difference between climatology and model respectively. It shows that there is significant differences between model and climatology. Hence the models need to be corrected and reinitialized time to time with new observations, and the process of reinitialization is called data assimilation (DA). In DA, one uses the model to forecast the current state, using a prior state estimate (which incorporates information from past data) as the initial condition, then uses current observation to correct the prior forecast. We have developed a DA system using Ensemble Transform Kalman Filter (ETKF) [3, 4], which is used in ROMS model to improve the accuracy of the model forecast. For this purpose we have used ARGO
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data [5,6] as observations. In this paper we will present SST with and without assimilation, and the discuss of the implementations of DA using ROMS model in Indian Ocean.
1.1 Data assimilation for ocean dynamics DA is a powerful and versatile method for combining partial, noisy observational data of a system with its dynamical model, generally numerically implemented, to generate state estimates of nonlinear, chaotic systems. Assimilation of the observations in the model improves the model state, and provides the most probable state of the system, given the uncertainties in the observations and model forecast. Thus DA is used for state estimation in many practical applications. A variety of data assimilation methods, broadly separated into deterministic or probabilistic methods, have been developed over the past few decades and used mainly in the earth sciences [7–13]. In ocean DA produces a regular, physically consistent, initial conditions for the ocean from a irregular collection of in situ temperature and salinity observations from ARGO floats. The floats are irregularly placed in Indian Ocean and hence sample the data irregularly in space and time. In a forecast model, the collection of numbers needed to represent the atmospheric state of the model is collected as a column matrix called the state vector X f . How the vector components relate to the real state depend on the choice of discretization, which is mathematically equivalent to a choice of basis. The best possible representation of reality as a state vector, which is called the true state (X t ). Data assimilation is a way to keep the model state close to the nature by assimilating observations. Modern methods of data assimilation are a maximum likelihood estimation using all available information, namely, forecasts X f and observations y ◦ . We look for an analysis state X a by a linear weighted mean between X f and y ◦ based on their respective error covariances so that the linear combination of all the available information has the smallest possible total error covariance, and this linear combination can be obtained by Kalman Filter [12]. The equations for Kalman filter are follows: X a = X f + K(y ◦ − HX f )
(1)
P a = (I − KH)P f
(2)
where, X a and X a represent the analysis and forecast state respectively. K = P f H T (HP f H T + R)−1 is the Kalman gain. H is the observation operator which takes the forecast (X f ) to observational space. R is the observational error covariance and is defined as R = [y 0 − y¯0 ][y 0 − y¯0 ]T . Forecast error covariance (P f ) and analysis error covariance (P a ) are defined as P f = [X f − X¯f ][X f − X¯f ]T and P f = [X a − X¯a ][X a − X¯a ]T respectively. Here over-bar (−) represents the mean of the quantity. The above filter diverges for nonlinear system and is not also useful for high dimensional system. In case of Ocean, the state dimension is of the order of ∼ 107 . So to solve above two problems, ensemble formulation, which is basically the modification of above (Eq. (2)), is introduced and the modified filter is termed as ensemble transform Kalman filter (ETKF) [3,4]. Each member of ensemble is basically the out put of one model run and is represented by a column vector of the ensemble matrix. If we subtract the ensemble mean from the ensemble matrix, the resultant matrix is Zef , where e denotes the ensemble. Similarly, Zea represents analysis matrix obtained
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from Zef by some matrix transformation, and the transformation matrix (A) arises from the Eqs. (2) for higher dimensional system. As the forecast (Pef ) and analysis error covariance (Pea ) matrices are symmetric positive-definite matrices, they can be represented as Pef = Zef Zef T and Pea = Zea ZeaT , where, Zef = {δxf1 δxf2 ... δxfN } and Zea = {δxa1 δxa2 ... δxaN }, where, δxfi and δxai s are the mean subtracted ensemble members. Now our main goal is to find out A such that Zea = A.Zef such that Pea = (I − Ke H)Pef , where Kalman gain, Ke = Pef H T (HPef H T + R)−1 . Now Pea = (I − Ke H)Pef
Zea ZeaT = [I − Pef H T (HPef H T + R)−1 H]Pef
= Zef [I − Zef T H T (HZef Zef T H T + R)−1 HZef ]Zef T = Zef (I − V D−1 V T )Zef T
where, V = (HZef )T and D = V T V + R. ˜ f , where, Then the analysis perturbation ensemble is calculated from Z a = AZ A˜ = CΛ1/2 . C and λ can be obtained from the singular value decomposition of (I − V D−1 V T ), i.e., CΛC T = SV D[(I − V D−1 V T )] So forecast is updated as and mean is updated as
˜ ef Zea = AZ
(3)
¯ f + Ke (y − H X ¯ f ). ¯ ea = X X e e
(4)
2 Regional Ocean Modeling system (ROMS) 2.1 Equations of motion The model used for Indian Ocean is the Regional Ocean Modelling System (ROMS) [1,2]. The primitive equations in Cartesian coordinates are given here. The momentum balance in the x- and y-directions are: ∂φ ∂ ∂u ∂u + v · ∇u − f v = − − + Fu + D u uw −ν (1) ∂t ∂x ∂z ∂z ∂φ ∂ ∂v + v · ∇v + f u = − − ∂t ∂y ∂z
v w − ν
∂v ∂z
+ F v + Dv .
(2)
Evolution of a S, T and nutrients (C(x, y, z, t)) is governed by the advective-diffusive equation: ∂ ∂C ∂C + v · ∇C = − + F C + DC . C w − νθ (3) ∂t ∂z ∂z
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Fig. 3. The primitive equations solved in the domain of 30◦ S to 30◦ N and 30◦ E to 120◦ E.
The equation of state is given by: ρ = ρ(T, S, P ).
(4)
Boussinesq approximation: density variations are neglected in the momentum equations except in their contribution to the buoyancy force in the vertical momentum equation. Hydrostatic approximation: vertical pressure gradient balances the buoyancy force: ρg ∂φ =− · ∂z ρo
(5)
The final equation expresses the continuity equation for an incompressible fluid: ∂w ∂u ∂v + + =0 ∂x ∂y ∂z
(6)
where – Du , Dv , DC : diffusive terms; Fu , Fv , FC : forcing terms. – f (x, y): Coriolis parameter; g: acceleration of gravity. – h(x, y): bottom depth; ν, νθ : molecular viscosity and diffusivity; P : total pressure P ≈ −ρo gz. – φ(x, y, z, t): dynamic pressure φ = (P/ρo ). – ρo + ρ(x, y, z, t): total in situ density. – S(x, y, z, t): salinity; t: time. – T (x, yz, t): potential temperature. – u, v, w: the (x, y, z) components of vector velocity v. – x, y: horizontal coordinates; z: vertical coordinate. – ζ(x, y, t): the surface elevation. The model domain for the simulation is 30◦ S to 30◦ N and 30◦ E to 120◦ E of Indian Ocean as shown in Fig. 3.
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3 Implementation of ETKF 3.1 Preprocessing for ROMS model 3.1.1 Formulation of Zef from model output Temperature (T), salinity (S), horizontal velocities (u and v) and dynamics ocean height (ζ) have been taken as prognostic variables, which have been used to form state vector of the DA system. For Indian Ocean these five variables were read from the model output run into the Zef matrix. Each column of the Zef matrix represents different model run at same time using different initial conditions, and land-points are excluded from the matrix. So, ⎛
.. .. . . . ⎜ ⎜ · · · Ti+1,j Ti+1,j+1 ⎜ ⎜ · · · Si+2,j Si+2,j+1 Zef = ⎜ .. .. ⎜ . . ⎜··· ⎜··· ζ ζ i+5,j i+5,j+1 ⎝ .. .. ··· . . ..
⎞ ··· ⎟ ···⎟ ⎟ ···⎟ ⎟· ⎟ ···⎟ ···⎟ ⎠ .. .
3.1.2 Preprocessing of ARGO data Salinity and Temperature from the ARGO data have been used as observations (y ◦ ). Different profiles (p) from different ARGO data has been used as observations at particular time. Hence the observational vector is ⎞ ⎛ .. ⎜ . ⎟ ⎜ Tp+1 ⎟ ⎟ ⎜ ⎟ ⎜ ◦ y = ⎜ ... ⎟ · ⎟ ⎜ ⎜ Sp+1 ⎟ ⎠ ⎝ .. . For the current assimilation, one day data has been used as the data at particular instant. We have used 10% of temperature and salinity as R in the current run. Localization scheme used as shown in Ref. [14, 15]. The model run with data assimilation using ETKF has been shown in the next Section (Sect. 4).
4 Results and discussion Figure 4 shows the comparison the ensemble mean of sixty ensembles of one step assimilation and model run (lower) panel. The figures show in the case of one step update the SST (upper panel) shows oscillations. Here we have not used localization. From these figure it is clear that without localization observations at a particular point disturbs the whole ocean. This is not desirable, so covariance localization has been introduced [14, 15].
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Fig. 4. Lower panel shows the mean of sea surface temperature (SST) of sixty (60) model run, and the upper panel shows the ensemble mean of one step update of sixty ensemble.
Covariance localization is a process, which cut off the long-range correlation in the Pef at a specified distance, and it is performed by applying Schur Product to the forecast error covariance matrix (Pef ) [14, 16]. All the elements of the Pef are multiplied by a correlation function with local support. Detail of the procedure will be found in Ref. [14,16]. Figure 5 shows the SST with localization with scale of 200 Km. Here we have shown the comparison of SST of sixty ensembles assimilation (First column) and forty ensembles assimilation (second column) and their difference in the third column. The white patches in Fig. 5 shows the ARGO float positions in the Indian Ocean. It also shows that the the effect of observations are limited in the radius of 200 km. Inside the influence circle temperature diverges from the normal temperature of the ocean. Almost same kind of behavior was observed in u-velocity. Figure 6 shows that u-velocity also changes considerably around the ARGO float within the radius of influence. First column of Fig. 6 is the sixty ensembles assimilation, second column is the forty ensembles assimilation, and third column difference between first and second column. As without localization assimilation affects whole ocean, we have introduced localization in the assimilation. In both cases we have not yet got ambient results. As the model run also diverged after few assimilation, we had introduced a mean factor ¯a = X ¯ f + Ke (y − H X ¯f) (mf ), which has been introduced in Eq. (4) as follows X e e e ∗ mf .
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Fig. 5. Data assimilation using localization with localization scale 200 km. First column: sea surface temperature (SST) using sixty ensemble; second column: SST using forty ensemble and third column difference between first and second column.
Fig. 6. Data assimilation using localization with localization scale 200 km. First column: u-velocity using sixty ensemble; second column: u-velocity using forty ensemble and third column difference between first and second column.
5 Conclusion In this paper we have presented preliminary results of the implementation of the DA tool ETKF in ROMS model to improve the model forecast. We have modified and implemented the ETKF for large dimensional system. Currently, the model runs for 21 days with the modified the ETKF, and the result of such runs are presented in Sect. 4. In this implementation phase, it is difficult to asses the improvement in the assimilated run, as there are several issues to be fixed to get effective data assimilation system. Few of such problems, which are under study, are (a) the estimation and validation of error covariance from model and observations and time-dependent error covariance estimation from Kalman filter; (b) designing the effective initial perturbation
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fields for ROMS model; and (c) finding the proper covariance localization of DA system for Indian Ocean. Finally, development such DA system for Indian Ocean will improve the forecast of the model, and contribute to Weather/Monsoon/Climate forecast by providing forcing for Atmospheric Models. Authors would like to thank the Indian National Centre for Ocean Information Services for financial support through Project No. INCOIS/93/2007.
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