Applications of the Inverse 3D-Var Method to Data

Applications of the Inverse 3D-Var Method to Data Assimilation Seon Ki Park,1  Eugenia Kalnay,1 2 3 and Zhao-Xia Pu4 ;

; ;

Cooperative Institute for Mesoscale Meteorological Studies and 2 School of Meteorology University of Oklahoma, Norman, OK 73019, U.S.A. 3 National Center for Environmental Prediction, NOAA, Washington, D.C., 20233, U.S.A. 4 Goddard Space Flight Center, NASA, Code 912, Greenbelt, MD 20771, U.S.A. 1

1. Introduction The four-dimensional variational data assimilation (4D-Var) using the adjoint model, in which a cost function (weighted squared distance between model solutions and observations) is minimized via iterative processes, is considered as the most promising tool for tting model solutions to observations and has received considerable attention in recent years (Derber 1989; Courtier et al. 1994; Zupanski 1993; and many other papers). However, a major problems with the adjoint 4D-Var is that it takes a large computational cost to obtain an optimal initial condition because a few hundreds iterations are often required for the minimizations process to converge. Thus implementing the complete adjoint 4D-Var scheme into the operational system is practically infeasible. Recently Wang et al. (1997) suggested the use of inverse model integrations in order to accelerate the convergence of 4D-Var. They developed a new optimization method called adjoint Newton algorithm (ANA) and demonstrated the eciency of the ANA when applied to a simpli ed 4D-Var problem. In the ANA, the Newton descent direction is obtained by a backward integration of the tangent linear model (TLM), in which the sign of timestep is changed (i.e., inverse). Here, the sign of dissipative terms is also changed in order to avoid computational instability (i.e., quasi-inverse). By adopting this \quasi-inverse" approach and assuming the availability of a complete set of observations at nal time, Wang et al. (1997) showed that the ANA converged in an order of magnitude fewer iterations, and to an error level an order of magnitude smaller than the conventional adjoint approach to solve the same (simpli ed) 4D-Var problem. However, their application of the quasi-inverse method was limited to nding the Newton descent direction only, and a complete data set at the end of the assimilation period. The quasi-inverse technique can be applied to either the tangent linear or the full nonlinear model, each of which has advantages for dierent applications. Pu et al. (1997) showed that for the problem of forecast sensitivity, closely related to 4D-Var, a backward integration with the quasi-inverse TLM (Q-ILM) gave results far superior to those obtained using the adjoint model. The Q-ILM has been tested successfully at NCEP in several dierent applications, e.g., forecast error sensitivity analysis and data assimilation (Pu et al. 1997), and adaptive observations (Pu et al. 1998). Kalnay et al. (1999) demonstrated, using the Q-ILMs of simple models, a much faster performance of the quasi-inverse method in data assimilation than the conventional adjoint 4D-Var method. The former is dubbed \Inverse 3D-Var" (I3D-Var) in the sense that the observation is used only once at the end of the assimilation interval. The adjoint 4D-Var requires to run both the adjoint model, to provide the gradient information to the minimization algorithm, as well as the minimization process. In contrast, the I3D-Var does not require to run either the adjoint model or the minimization process. In the following, we provide a brief overview on the theoretical background of the I3D-Var. For a more detailed discussion see Kalnay et al. (1999).

2. The I3D-Var Method Corresponding Author: Dr. Seon Ki Park, Department of Meteorology, University of Maryland, CSS 3433, College Park, MD 20742, U.S.A.; E-mail: [email protected] 

The major goal of I3D-Var is to nd an increment in initial conditions x that \optimally" corrects a perceived forecast error at the nal time t. Here, the cost function includes both data and background distances. In order to maintain the ability to solve the minimization eciently, however, the background term is estimated at the end of the interval, rather than at the beginning as in Lorenc (1986). Assume (for the moment) that data y is available at the end of the assimilation interval t, with x=x ? x , y =y -H(x ). Here x and x are the analysis and rst guess, respectively, and H is the \forward observation operator" which converts the model rst guess into rst guess observations (see Ide et al. 1997 for notation). The cost function that we minimize is the 3D-Var cost function at the end of the interval. It is given by the distance to the background of the forecast at the end of the time interval (weighted by the inverse of the forecast error covariance B), plus the distance to the observations (weighted by the inverse of the observational error covariance R), also at the end of the interval: 1 1 J = (Lx) B ?1 (Lx) + [HLx ? y ] R?1 [HLx ? y ]: (1) 2 2 Here x (the control variable) is the dierence between the analysis and the background (at the present iteration) at the beginning of the assimilation window, L and L are the TLM and its adjoint, respectively, and H is the tangent linear version of the forward observation operator H. If we take the gradient of J with respect to the initial change x, we obtain o

a

b

o

b

a

b

T

T

T

rJ = L fB ?1Lx + H R?1[HLx ? y]g: T

(2)

T

From this equation we see that the gradient of the cost function is given by the backward adjoint integration of the rhs terms in (2). In the adjoint 4D-Var, an iterative algorithm (such as quasi-Newton, conjugate gradient, or steepest descent) is used to estimate an optimum perturbation: x =  rJ ?1 i

i

(3)

i

and the procedure is repeated until after many iterations rJ becomes very small, and the minimum of J is reached. It should be noted that in order to determine an optimal value for the step size  , the i

minimization algorithms such as conjugate gradient or quasi-Newton require additional computations of the gradient rJ ?1 , so that the number of direct and adjoint integrations required by the 4D-Var can be signi cantly larger than the number of iterations. In the I3D-Var, however, we seek to obtain directly the \perfect solution" i.e., the x that makes rJ =0 for small x. From (2) we can eliminate the adjoint operator, and obtain the \perfect" solution given by Lx = (B ?1 + H R?1 H )?1 H R?1 y: (4) Since we have a good approximation of L?1 at hand (the quasi-inverse model obtained by integrating the tangent linear model backward, but changing the sign of frictional terms), we can apply it and obtain i

T

T

x = L?1 (B ?1 + H R?1 H )?1 H R?1 y: T

T

(5)

This can be interpreted as starting from the 3D-Var analysis increment at the end of the interval and integrating backwards with the TLM or an approximation of it. If we do not include the forecast error covariance term B ?1 , (5) reduces to the ANA algorithm of Wang et al. (1997) except that we do not require line minimization. We have tested the I3D-VAR with the ARPS model and found that for this reason, the I3D-Var is computationally about twice as fast as Wang et al. (1997) ANA scheme. Kalnay et al. (1999) proved that the I3D-Var is equivalent to solving the minimization problem (at each time level) using the full-Newton method. The results of Wang et al. (1997) and Pu et al. (1997) support considerable optimism for this method. For a quadratic function, the Newton algorithm (and the equivalent I3D-Var) converges in a single iteration. Since the cost functions used in the 4D-Var are close to quadratic functions, the I3D-Var can be considered equivalent to perfect preconditioning of the simpli ed 4D-Var problem.

3. Results and discussions Figure 1 shows the performance of the I3D-Var and the 4D-Var (using LBFGS minimization algorithm) for a simple advection-diusion problem (Burgers' equation) including a passive scalar transport, q. Using the I3D-Var in which observations are available only at the end of the interval, the cost function converges to 10?13 of its original value after 3 iterations. However, the 4D-Var with data at the end of the interval requires 44 equivalent model integrations (both forward and adjoint) for the cost function to converge to 10?10 of the original value. If we provide the 4D-Var with observations for every time step, the cost function converges to the same order in 12 time integrations. ADJ (ALL OBS) ADJ (END OBS) INV (END OBS)

1e+00

Normalized Cost Function

1e-02 1e-04 1e-06 1e-08 1e-10 1e-12 0

5

10

15

20

25

30

35

40

45

ICALL

Fig. 1: Convergence rate of normalized cost functions for 4D-Var and I3D-Var methods as a function of ICALL

(number of calls for the nonlinear model and adjoint or inverse model). For the 4D-Var, the case with observations only at the end of the assimilation window is compared with the case with observations at all time steps. The assimilation window length is 81, diusion coecient is 10?3, and the initial error magnitude is 10% for u and 100% for q (from Kalnay et al. 1999).

If there are data at dierent time levels, we can use the Q-ILM to bring the increments to the same initial time where they can be averaged with weights that may depend on the time increment or the type of data. For applications in which \knowing the future" is allowed, such as reanalysis, the observational increments could be brought to the center of an interval, and used for the nal analysis. Kalnay et al. (1999) compared forecasts beyond the assimilation window, started from initial conditions using multiple time levels of noisy observations. For the I3D-Var approach, the noisy observational increments from multiple time levels were brought to the initial time and averaged to yield the initial conditions. The results of the forecasts with one iteration of the I3D-Var were comparable to those of 20 iterations of the 4D-Var, whereas 3 iterations of the I3D-Var resulted in much better forecasts. The I3D-Var also has some potential disadvantages. One of them is the growth of noise that projects on decaying modes during the backward integration (Kalnay et al. 1999). However, those errors decay again during the subsequent forward integration. We also found that the performance of I3D-Var is dependent upon the magnitude of dissipation (Park and Kalnay 1999). We believe that these problems can be overcome with further development and experimentation. At least the I3D-Var might serve as a preconditioner when carrying minimization in the framework of the 4D-Var. To test this idea, a preliminary experiment using the Burgers' equation is performed. In Fig. 2, preconditioning with one iteration of the I3D-Var, the 4D-Var showed much better performance in minimizing the cost function. The result shows a possibility of using the I3D-Var as a preconditioner of the 4D-Var including full physics (see Park and Kalnay 1999). The use of the I3D-Var may deal eciently with the part of the spectrum of the Hessian of the cost function related to the dynamics part of the model. This will allow the minimization to focus only on the nonlinearities associated with the physics and result in a large computational economy (Navon 1999; personal communication).

1E+01

Cost Function (J)

I3D-Var 4D-Var (PRECON 0) 4D-Var (PRECON 1)

1E+00

1E-01

0

5

10

15

20

25

30

ICALL

Fig. 2: Variations in the cost function from the I3D-Var and the 4D-Var starting from \no preconditioning" (PRECON 0) and from preconditioning using one iteration of the I3D-Var (PRECON 1). The assimilation period is 101 and the diusion coecient is 10?3. The initial error magnitude is 40% in u and each observation includes random errors with maximum magnitude of 10%. Observations are provided at 51 and 101 (from Park and Kalnay 1999). Currently the I3D-Var scheme is under development for a storm-scale model called the Advanced Regional Prediction System (ARPS: Xue et al. 1995). One of the major errors in stormscale numerical prediction (e.g., squall lines) is the phase error. With preliminary tests using the adiabatic version of ARPS, the I3D-Var proved to be a useful tool for correcting phase errors. We are extending this idea to a full-physics model by developing simpli ed reversible microphysics.

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