Impact of analysis-time tuning on the performance of the DRP-4DVar approach

ADVANCES IN ATMOSPHERIC SCIENCES, VOL. 28, NO. 1, 2011, 207–216

Impact of Analysis-time Tuning on the Performance

of the DRP-4DVar Approach

ZHAO Juan1,2 (� �), WANG Bin∗1 (� �), and LIU Juanjuan1 (��)

1State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics,

Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029

2Graduate University of the Chinese Academy of Sciences, Beijing 100049

(Received 5 November 2009; revised 5 March 2010)

ABSTRACT

In this study we extend the dimension-reduced projection-four dimensional variational data assimilation(DRP-4DVar) approach to allow the analysis time to be tunable, so that the intervals between analysis timeand observation times can be shortened. Due to the limits of the perfect-model assumption and the tangent-linear hypothesis, the analysis-time tuning is expected to have the potential to further improve analyses andforecasts.

Various sensitivity experiments using the Lorenz-96 model are conducted to test the impact of analysis-time tuning on the performance of the new approach under perfect and imperfect model scenarios, respec-tively. Comparing three DRP-4DVar schemes having the analysis time at the start, middle, and end of theassimilation window, respectively, it is found that the scheme with the analysis time in the middle of thewindow outperforms the others, on the whole. Moreover, the advantage of this scheme is more pronouncedwhen a longer assimilation window is adopted or more observations are assimilated.

Key words: DRP-4DVar, analysis-time tuning, perfect-model assumption, tangent-linear hypothesis

Citation: Zhao, J., B. Wang, and J. J. Liu, 2011: Impact of analysis-time tuning on the performance ofthe DRP-4DVar approach. Adv. Atmos. Sci., 28(1), 207–216, doi: 10.1007/s00376-010-9191-3.

1. Introduction

Some hybrid and coupled schemes combining theadvantages of both the four-dimensional variationaldata assimilation (4DVar) and ensemble Kalman filter(Evensen, 1994; Houtekamer et al., 1996; Houtekamerand Mitchell, 1998, 2001, 2005; Snyder and Zhang,2003) approaches have emerged in recent years, suchas the ensemble Kalman smoother (Evensen and VanLeeuwen, 2000), 4DEnKF (Hunt et al., 2004; Fertig etal., 2007), En4D-Var (Liu et al., 2008), and E4DVar(Zhang et al., 2009), among others.

Wang et al. (2010) proposed a new approach knownas DRP-4DVar to implement 4DVar economically,using the technique of dimension-reduced projection(DRP). On one hand, as a retrospective assimila-tion algorithm DRP-4DVar maintains the advantagesof standard 4DVar, by not only best fitting the ob-servations but by also having good consistency with

model dynamics and physics. On the other hand,DRP-4DVar is informed by the ensemble Kalman fil-ter (Fisher, 1999) approach in that the backgrounderror covariance as estimated from an ensemble ofshort-term forecasts is always flow-dependent. In addi-tion, DRP-4DVar abandons the tangent-linear modeland the adjoint model that are necessary for standard4DVar (Lewis and Derber, 1985; Le Dimet and Ta-lagrand, 1986; Courtier and Talagrand, 1987, 1990;Mahfouf and Rabier, 2000; Rabier et al., 2000), soDRP-4DVar is easier to implement and is less com-putationally intensive than standard 4DVar. The per-formance of DRP-4DVar has been tested by conduct-ing two observation system simulation experiments us-ing the Penn State/NCAR fifth generation mesoscalemodel (MM5) (Wang et al., 2010).

We should note that, as in standard 4DVar, thenew approach (DRP-4DVar) still works under the as-sumption that the model is perfect, which may con-

∗Corresponding author: WANG Bin, [email protected], [email protected]

© China National Committee for International Association of Meteorology and Atmospheric Sciences (IAMAS), Institute of AtmosphericPhysics (IAP) and Science Press and Springer-Verlag Berlin Heidelberg 2010

208 IMPACT OF ANALYSIS-TIME TUNING ON DRP-4DVAR VOL. 28

taminate the analysis field if the model error is signifi-cant. Another point to make is that the tangent-linearhypothesis, whose validity depends not only on themodel but also on the general properties of the assim-ilation system, such as the length of the assimilationwindow and the intervals between the analysis timesand observation times, among others (Warner et al.,1989; Bouttier and Courtier, 1999). Therefore, on onehand we need to make the assimilation window longenough to create well developed dynamic structuresin the analysis field; on the other hand, we have toensure an appropriate length for the assimilation win-dow, during which the assumptions of a perfect modeland the tangent-linear hypothesis are valid.

In recent years, along with rapid progress of remotesensing techniques involving satellite and radar data,plenty of high-resolution asynoptic observational dataare now available for assimilation. Given the irreg-ular temporal distribution of asynoptic observations,the model errors accumulated in the process of mini-mization will be reduced when the analysis time canbe tuned flexibly according to the temporal distribu-tion of observations within the assimilation window.For example, if the analysis time is close to the time ofmost observations, the time intervals between the anal-ysis time and observation times are shortened. As aresult, the model errors accumulated in the forward in-tegration by the tangent-linear model and in the back-ward integration by the adjoint model will be reduced.Moreover, the impact of the tangent-linear hypothesiswill also be alleviated. As a result, analysis-time tun-ing may have the potential to allow for longer assim-ilation windows and to assimilate observational infor-mation more effectively. However, implementation ofanalysis-time tuning in standard 4DVar requires thebackward integration of the tangent-linear model andthe forward integration of the adjoint model, neither ofwhich may exist at all. Thus, it is quite difficult to im-plement a tunable analysis time feature as a part of astandard 4DVar system. Using a new approach, DRP-4DVar, implementation of the tangent-linear modeland the adjoint model is avoided (Wang et al., 2010),potentially providing a good choice for accomplishinganalysis-time tuning.

In this study we have extended DRP-4DVar tomake the analysis time tunable. The Lorenz-96 modelis applied as a pilot of the new assimilation systemwith tunable analysis time. We have designed vari-ous sensitivity experiments to investigate the impactof the analysis-time tuning upon the analysis and fore-casts and to try to find the optimal configuration forthe analysis-time tuning.

The organization of this paper is as follows: thetheoretical background of the extended DRP-4DVar

approach is introduced in section 2. The experimen-tal design is described in section 3.1, and results areshown in section 3.2. Section 4 provides a summaryand discussion.

2. Theoretical background

4DVar produces an optimal analysis of the initialstate by minimizing the cost function⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

J(x(t0))=12

[x(t0)−xg(t0)]T B−1(t0) [x(t0)−xg(t0)]+

12

[y(x(t0))−yobs]T

O−1 [y(x(t0))−yobs] ,

yobs =(y

(1)obs , y

(2)obs , · · · , y

(NUM)obs

)T

,

y(x(t0)) =[y(1)(x(t0)) , y(2)(x(t0)), · · · ,

y(NUM)(x(t0))]T

,

y(i)(x(t0)) = HiMt0→ti [x(t0), τ ] ,

(1)

where the superscript T denotes the transpose, thebold-faced variables represent column vectors or ma-trices, x(t0) is the initial state at the beginning ofthe assimilation window in the Lx-dimensional modelspace, and xg(t0) is the first guess of the initial state.In addition, B is the background error covariance ma-trix (Lx×Lx), yobs is an Ly-dimensional observationvector, NUM is the total number of observation timeswithin the assimilation window [t0, tN ] and N is thenumber of hours within the assimilation window, Hi

and Mt0→ti are the observation operator and the fore-cast model, respectively, and O is the observation errorcovariance matrix (Ly×Ly) of yobs.

In order to make the analysis time tunable, we in-troduce two additional notations, x̂(t0) and x̂g(t0),which respectively represent the initial state and itsfirst guess at time t0. Then, the cost function in Eq.(1) can be reformulated as⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

J̃(x(ta))=12[x(ta)−xg(ta)]TB−1(ta)[x(ta)−xg(ta)]+

12[y(x(ta)) − yobs]TO−1[y(x(ta)) − yobs] ,

x(ta) = Mt0→ta [x̂(t0)] , xg(ta) = Mt0→ta [x̂g(t0)] ,

y(i)(x(ta)) = HiMta→ti [x(ta), τ ] .(2)

Note that the control variable x(ta), first guess xg(ta),and the background error covariance matrix B(ta)in Eq. (2) are all specified for the analysis timeta (t0�ta�tN ), which can be any time within the

NO. 1 ZHAO ET AL. 209

assimilation window. The new control variablex(ta) = Mt0→ta [x̂(t0)] is derived from the initial statex̂(t0) by the nonlinear forecast model, and the newfirst guess xg(ta) = Mt0→ta(x̂g(t0)) is the state at theanalysis time ta as predicted by the nonlinear modelinitialized with x̂g(t0) at the beginning of the assimi-lation window.

Using the incremental approach of 4DVar (Courtieret al., 1994), the cost function in Eq. (2) can be rewrit-ten as

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

J̃(x′(ta)) =12[x′(ta)]TB−1(ta)[x′(ta)]+

12[y′(x′(ta)) − y′

obs]TO−1×

[y′(x′(ta)) − y′obs] ,

x′(ta) = x(ta) − xg(ta)

= Mt0→ta [x̂g(t0) + x̂′(t0)]−Mt0→ta [x̂g(t0)] ≈ M ′

t0→ta x̂′(t0) ,

x̂′(t0) = x̂(t0) − x̂g(t0) ,

y′(i)(x′(ta)) = y(i)(x(ta)) − y(i)g (xg(ta)) ,

y′(i)obs = y

(i)obs − y

(i)g (xg(ta)) ,

(3)

where x′(ta) is the analysis increment at time ta. Thisincrement is supposed to be approximately linearlycorrelated with x̂′(t0), which is the difference betweenthe two parameters x̂(t0) and x̂g(t0). The vectory′(i)obs = y

(i)obs − y

(i)g (xg(ta)) is the innovation vector

at observation time ti. In deducing the incrementalformula in Eq. (3), a linear approximation has beendefined as

y′(i)(x′(ta)) = HiMta→ti [x(ta), τ ]

−HiMta→ti [xg(ta) , τ ]

≈ H ′iM

′ta→ti

x′(ta) ,

(4)

where H ′i is the tangent-linear operator of Hi, and

M ′ta→ti

is the tangent-linear model of Mta→ti .To obtain the optimum solution to the cost func-

tion described in Eq. (3), we extend the DRP-4DVarapproach to make the analysis time tunable and ex-plore the impact of analysis-time tuning.

First, a group of initial perturbation samplesx̂′

1(t0), x̂′2(t0), · · · , x̂′

m(t0) are prepared, which stemfrom a series of short-term model forecasts or a groupof Monte Carlo random perturbations. m is the totalnumber of the initial perturbation samples. Then, weobtain the corresponding perturbation samples at theanalysis time ta by using the nonlinear forecast model.

These are defined as

x′j(ta) = Mt0→ta [x̂g(t0) + x̂′

j(t0)]−Mt0→ta [x̂g(t0)]

≈ M ′t0→ta x̂

′j(t0) .

(5)

Meanwhile, the initial perturbation samples x̂′1(t0), x̂′

2

(t0) , · · · , x̂′m(t0) are also employed to generate NUM

groups of samples y′1(ti), y′

2(ti) , · · · , y′m(ti) , where

y′j(ti) = HiMt0→ti [x̂g(t0) + x̂′

j(t0)]

−HiMt0→ti [x̂g(t0)]

≈ H ′iM

′t0→ti

x̂′j(t0) .

(6)

Note that y′j(ti) is supposed to be linearly correlated

with x̂′j(t0). According to the approximately linear re-

lationship between x′j(ta) and x̂′

j(t0) shown in Eq. (5),it is easy to prove that y′

j(ti) is approximately linearlyrelated to x′

j(ta):

y′j(ti) ≈ H ′

iM′ta→tix

′j(ta) . (7)

Similarly to Wang et al. (2010), x′(ta) and y′ can beprojected to reduced ensemble spaces, and are ex-tended by x′

1(ta), x′2(ta), · · · , x′

m(ta) and y′1, y′

2,· · · , y′

m, respectively:⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

x′(ta) = α1x′1(ta) + α2x

′2(ta) + · · · + αmx′

m(ta)

= Pxα ,

Px = [x′1(ta), x′

2(ta), · · · , x′m(ta)] ,

α = (α1, α2, · · · , αm)T ,

(8)and

{y′ = α1y

′1 + α2y

′2 + · · · + αmy′

m = Pyα ,

Py = (y′1, y

′2, · · · , y′

m) ,(9)

where Px, Py are the projection matrices, α is theprojection parameter from the model space to an en-semble space, and y′

j is defined as:

y′j = (y′

j(t1), y′j(t2), · · · , y′

j(tNUM))T . (10)

In this way, we can obtain the optimum solution tothe cost function by using the DRP-4DVar approachdirectly. This means that DRP-4DVar can be ex-tended to use the tunable analysis time.

3. Experiments with the Lorenz-96 model

3.1 Experimental design

As an initial attempt to test and understand theimpact of analysis-time tuning, we apply the extended


DRP-4DVar to the relatively simple and manageablemodel of (Lorenz, 1996):

dxj

dt= −xi−2xi−1+xi−1xi+1−xi+F , i = 1, I , (11)

with periodic boundary conditions. Similar to the con-figuration for the Lorenz-96 model used in other stud-ies (Hunt et al., 2004; Fertig et al., 2007), we choosethe external forcing to be F=8.0 and the number ofspatial elements to be I=40. A fourth-order Runge-Kutta scheme is adopted for temporal integration witha time step of 0.05 units, equivalent to 6h in the at-mosphere (Lorenz, 1996).

A long integration of the model from an arbitraryinitial condition is considered as the true trajectory.The “true” simulation is used to generate simulatedobservations and is also used as the reference to evalu-ate the performance of the assimilation system. Noisyobservations are produced by adding uncorrelated ran-dom noise with a standard Gaussian distribution (zeromean and variance of 1) to the true state at each modelgrid point. In this pilot study, observations are uni-formly temporally distributed. For example, the ob-servations may be made available at each time step(once per 6 h) or every four steps (once per 24 h)within a 48-h assimilation window.

The initial ensemble is obtained by adding Gaus-sian random perturbations to the true state. One ofthe ensemble members is chosen as the first guess forthe initial condition, while the others are employed togenerate the initial perturbation samples. As a con-ceptual study, we employ a relatively large number ofensemble members (500–1500) to reduce the samplingerror in the absence of any localization or inflationtechnique.

The experimental design in this study closely fol-lows that of Zhang et al. (2009). We conduct vari-ous sensitivity experiments in two kinds of scenariosby assuming either a perfect or imperfect model, re-spectively. As discussed in section 1, theoretically theassumption of a perfect model and the tangent-linearhypothesis would affect the quality of the optimum so-lution of the cost function in Eq. (3). So, we attemptto separately examine the influences of these two fac-tors in the two scenarios. In addition, we conduct 300assimilation cycles to obtain steady and reliable re-sults. To avoid reusing the observations, we use 6hintervals (one time step) between two adjacent assim-ilation windows.

To test the impact of analysis-time tuning onthe performance of DRP-4DVar, we choose the start,middle, and end of the assimilation window as theanalysis time, respectively (denoted as ASS−start,ASS−middle, and ASS−end experiments). Then, the

performances of the three schemes in the cycling ex-periments are compared and analyzed.

3.2 Results

3.2.1 Perfect-model experimentsThe forecast model is assumed to be perfect, and

the same forcing coefficient (F=8.0) is adopted for themodel producing both the forecast and the “true” sim-ulation from which observations are taken. In thisideal scenario, the aforementioned perfect-model as-sumption is applicable. In order to examine the im-pact of the tangent-linear approximation, we calculatethe function

φ(α) =Hk {Mt0→tk

[xg(t0) + αδxa(t0)]}α

−

Hk {Mt0→tk[xg(t0)]}

α,

(12)

where α is a tunable coefficient for perturbationδxa(t0) and tk is the prediction time by using the fore-cast model M .

According to Eq. (7), this function is related tothe tangent-linear model as

φ(α) =Hk {Mt0→tk

[xg(t0) + αδxa(t0)]}α

−

Hk {Mt0→tk[xg(t0)]}

α

≈ H ′kM ′

t0→tkδxa(t0) .

(13)

Therefore, the tangent-linear approximation will beacceptable when the function φ(α) is invariant withrespect to the coefficient α. We calculate the valuesof φ(α) when α equals 1.0, 0.1, 0.01 and 0.001, withintegration time (tk − t0) of 12 h, 24 h, 36 h, and 48 h,

Fig. 1. Values of φ(α) with respect to different valuesof the coefficient α for 12 h (dots), 24 h (crosses), 36 h(circles), and 48 h (squares) assimilation windows.


Fig. 2. Time series (300 assimilation cycles) of domain-averaged root mean square errors (RMSEs) for CTRL(black), ASS−start (green), ASS−middle (red), andASS−end (blue) experiments at the end of the assimi-lation window.

respectively (Fig. 1). Figure 1 illustrates that thevalues of φ(α) are almost constant with integrationtime of 12 h or 24 h, indicating that the tangent-linearapproximation is feasible for a 24-h-long assimilationwindow at the utmost. However, when the integrationtime length reaches 36 h or 48 h, the function showssome nonlinearity. In fact, the longer the integrationtime becomes, the stronger nonlinearity φ(α) shows(not shown). So, the length of the assimilation windowis limited by the validity of the tangent-linear hypoth-esis. From this point of view, analysis-time tuning hasthe potential to improve the analysis and forecast byshortening the intervals between the analysis time andthe observation times.

Figure 2 compares the performances of the con-trol simulation (denoted by CTRL) without data as-similation and three assimilation runs (ASS−start,ASS−middle, and ASS−end) with an assimilation win-dow length of 48 h (8 time steps) and ensemble sizeof 1200. Note that the ASS−start and ASS−middleanalyses are integrated to the end of the assimilationwindow and then are verified against the true simula-tion. It is clear from Fig. 2 that without model errorand given a large number of ensemble members, all thethree assimilation schemes show satisfactory perfor-mance in terms of the long-term evolution of domain-averaged root mean square errors (RMSEs). In orderto visualize the differences between the three schemes,we perform subtraction of the RMSEs at all verifica-tion times (Figs. 3a and 3b). From the differences be-tween the RMSEs of ASS−start and ASS−middle (Fig.

3a), we find that at the end of the assimilation win-dow, ASS−middel performs significantly better thanASS−start. Moreover, the ASS−middle forecasts atthe end of the assimilation window are comparable tothe ASS−end analyses and even show a little improve-ment over the ASS−end analyses (Fig. 3b). A recentstudy by Liu et al. (2009) reaches a similar conclusionusing the WRF En4D-Var system when the majorityof the observations are close to the middle of the as-similation window.

The advantage of ASS−middle is also observedfrom the sum of variances over 300 cycles at the endof the assimilation window (Table 1). Various ex-periments are conducted with different configurations(varying the length of the assimilation window, the ob-servation frequency, and the ensemble size) to explorethe sensitivity of DRP-4DVar to analysis-time tuning.The relative error reductions (RER) are defined as

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

RER(middle|start)=E(middle)−E(start)

E(start)×100% ,

RER(end|start) =E(end) − E(start)

E(start)× 100% ,

RER(middle|end) =E(middle) − E(end)

E(end)× 100%,

(14)

where E represents the sum of variances over 300 cy-cles at the verification time. RER(middle|start) isthe relative error reduction of ASS−middle with re-spect to ASS−start, RER(end|start) is the relative er-ror reduction of ASS−end with respect to ASS−start,and RER(middle|end) is the relative error reductionof ASS−middle with respect to ASS−end.

Compared to ASS−start and ASS−end, the posi-tive effect of ASS−middle can be seen in all the sensi-tivity experiments (Table 1). Moreover, ASS−middleshows larger RER values with a longer assimilationwindow, which is reasonable according to the afore-mentioned limitations caused by the tangent-linear hy-pothesis. For example, in the 36 h-window experi-ment, the longest time interval between analysis andobservations for ASS−start and ASS−end reaches 36h, for which the validity of the tangent-linear approx-imation is degraded. However, for ASS−middle thelongest interval is just 18 h, more adequately satis-fying the tangent-linear hypothesis. Stronger nonlin-earity would result in more significant degradation forASS−start and ASS−end as the assimilation windowis lengthened.

Given a certain length of assimilation window andcomparing the results between different observationfrequencies, we see larger forecast and analysis errors


Table 1. The total sum of variances in 300 cycles for the three schemes (ASS−start, ASS−middle, and ASS−end)with different experimental configurations (different lengths of assimilation windows, observation frequencies, and en-semble sizes). Numbers in parentheses indicate the relative error reduction of ASS−middle with respect to ASS−start[RER(middle|start)] and ASS−end [RER(middle|end)], respectively. NUM is the total number of observations and m isthe number of ensemble members.

24 h-window 36 h-window 48 h-window

NUM=120 NUM=200 NUM=120 NUM=280 NUM=120 NUM=360m=500 m=800 m=500 m=1000 m=500 m=1200

ASS−start 15.097 16.875 12.872 16.730 12.351 18.252(−0.55%) (−0.59%) (−3.49%) (−4.25%) (−10.4%) (−13.79%)

ASS−middle 15.014 16.774 12.422 16.019 11.066 15.735ASS−end 15.176 16.978 12.868 16.467 11.445 16.029

(−1.07%) (−1.20%) (−3.46%) (−2.72%) (−3.3%) (−1.83%)

at the end of the assimilation window when assimi-lating more frequent observations (available at eachtime step). This indicates that the as similation sys-tem does not always perform more effectively whenthere are more frequent observations available withinthe assimilation window. Nevertheless, the advan-tage of ASS−middle in comparison to ASS−start ismore pronounced when more observations are as-similated. More specifically, ASS−middle has anRER(middle|start) value as low as −13.79% for the48h-window with NUM = 360 observations as com-pared to ASS−start.

Furthermore, the performance of long-term fore-casts (144 hours after the end of the assimilation win-dow) initialized with the ASS−start, ASS−middle, andASS−end analyses are compared and analyzed. Fromthe forecast RMSEs of the three assimilation runs,the relative error reduction of ASS−middle with re-spect to ASS−start [RER(middle|start)] and the rel-

ative error reduction of ASS−end with respect toASS−start [RER(end|start)] are calculated at the ver-ification times, respectively. Figures 4a and 4b de-scribe the temporal evolution of RER(middle|start)and RER(end|start) in the case of NUM = 120 andan ensemble size of 500. The curves (Figs. 4a, b)show that ASS−middle and ASS−end perform con-sistently better than ASS−start throughout the 144hours (6 days). Comparing the relative error reduc-tion values for 24 h-, 36 h-, and 48 h-length assimila-tion windows, the results indicate that the superiorityof ASS−middle and ASS−end is more significant whena longer assimilation window is used. This is especiallytrue for the 48h-window experiment, with several rel-ative error reduction values smaller than −10.0%. Atthe forecast time of 144 hours, ASS−middle still hasan RER(middle|start) value of as small as −9.10% forthis configuration. Thus, we conduct some longer-termforecasts (20 days) from the ASS−start, ASS−middle,

Fig. 3. Difference between RMSEs of the ASS−middle forecast and (a) RMSEs of the ASS−startforecast; (b) RMSEs of the ASS−end analyses at the end of assimilation window during 300 assim-ilation cycles. The black (grey) lines represent differences that are positive (negative).


Fig. 4. The temporal evolution of relative error reduction of (a) ASS−middle with re-spect to ASS−start, (b) ASS−end with respect to ASS−start in the perfect model sce-nario with the assimilation window lengths of 24 h (cross-line), 36 h (circle-line), and 48h (square-line), respectively.

Fig. 5. The temporal evolution of the relative error re-duction of ASS−middle with respect to ASS−start (solidline) and the relative error reduction of ASS−end withrespect to ASS−start (dotted line) in the perfect modelscenario with an assimilation window length of 48 hours.

and ASS−end analyses with the 48 h-length assimi-lation window. From the RER(middle|start) andRER(end|start) values shown in Fig. 5, we find thatASS−middle outperforms ASS−start almost consis-tently throughout the 480 hours (20 days), althoughbasically the RER(middle|start) value grows as the in-tegration time increases. Comparing the two curves ofRER(middle|start) and RER(end|start), it can be seenthat ASS−end even performs better than ASS−middlefrom 24 h to 120 h. After that period, ASS−middleshows smaller RER values than ASS−end and main-

tains an advantage over ASS−start until the 20th day.

3.2.2 Experiments with model errorIn these experiments, the forecast model for assim-

ilation runs uses a forcing coefficient (F=9.0) differentfrom that used in the true simulation (F=8.0). Inorder to illustrate the impact of model error, we com-pare two 72 h (12 time steps) forecasts initialized withidentical initial condition by using the perfect model(F=8.0) and imperfect model (F=9.0), respectively.Figure 6 shows the domain-averaged RMSEs of theimperfect model forecast at each time step, verifiedagainst the assumed perfect simulation. It is clear thatthe forecast error caused by model error increases with

Fig. 6. Evolution of forecast errors (RMSEs) caused bythe imperfect forecast model.


Table 2. Same as in Table 1, except for the imperfect model (F=9.0).

24 h-window 36 h-window 48 h-windowNUM=120 NUM=200 NUM=120 NUM=280 NUM=120 NUM=360

m=500 m=800 m=500 m=1000 m=500 m=1200

ASS−start 15.414 17.149 16.085 18.884 23.535 27.029(1.16%) (0.66%) (−3.26%) (−4.94%) (−16.2%) (−22.86%)

ASS−middle 15.592 17.262 15.561 17.950 19.715 20.850ASS−end 16.450 18.024 17.270 19.481 19.913 21.073

(−5.21%) (−4.23%) (−9.89%) (−7.86%) (−1.0%) (−1.06%)

integration time in a nonlinear fashion. Furthermore,the rate of error increase also grows with time. Re-markably, the forecast errors increase threefold from24 h to 48 h. Thus, the model error would have moresignificant impact on the analyses and forecasts whena longer assimilation window is adopted. On the otherhand, long-term model integration is of benefit to de-veloping dynamic structures in the analysis field.

In this subsection, various possible scenarios arealso tested under the assumption of an imperfectmodel (Table 2). The experiments still perform well,albeit with significant model error (not shown); eachresult has a larger total sum of variance than thecorresponding perfect-model experiment (Table 1 vs.Table 2). When the length of the assimilation win-dow is longer, the degradation is more pronounced,

Fig. 7. Same as in Fig. 3, except for the imperfect model.

Fig. 8. Same as in Fig. 4, but for the imperfect model scenario.


due to nonlinear growth of model error. However, asthe length of the assimilation window increases, theASS−middle experiment outperforms ASS−start moreremarkably, achieving a value for RER(middle|start)as low as 22.86% in the case of a 48 h-window withNUM=360 observations. We also observe a signifi-cant positive effect of ASS−middle from the differencein RMSEs between the ASS−start and ASS−middleforecasts at the end of the assimilation window (Fig.7a).

From the RER(middle|start) and RER(end|start)values obtained with respect to time (Figs. 8a, b) wecan see that, although the relative error reduction val-ues increase along with integration time and approachzero at the 144th hour, ASS−middle still outperformsASS−start throughout the 144 hours for the 48-h-window experiment. It is appropriate to note that atthe end of the assimilation window (0000h in Figs.8a, b) the RER(middle|start) and RER(end|start)values are positive for the 24-h-window experiment.The results that at the end of assimilation window,ASS−start has a little smaller sum of variance thanASS−end and ASS−middle (Table 2) also indicate thatASS−start performs the best at the end of the assim-ilation window for the 24 h-window experiment. Thismay be related to a model spin-up problem. As men-tioned earlier, the assimilation window needs to belong enough to ensure a fully developed dynamic struc-ture in the analysis field. On one hand, the largestmodel error accumulated throughout a 24 h-length as-similation window just reaches a value of just 0.1236(Fig. 6), which is almost acceptable compared to theerror of the initial condition. On the other hand, at theend of the assimilation window, the forecasts from theASS−start analyses are more compatible with modeldynamics and physics than the ASS−middle forecasts,not to mention the ASS−end analyses.

4. Summary and discussion

Through various sensitivity experiments, thisstudy exploits the potential of tuning analysis timeto further improve the performance of DRP-4DVar.The extended DRP-4DVar approach, which allows fora tunable analysis time, is implemented in the Lorenz-96 model, which has strong nonlinear characteristics,to assimilate the “observations” derived from a truesimulation.

First of all, it is found that whatever experimentaldesign is chosen, the extended DRP-4DVar shows sat-isfactory performance when verified against the truesimulation. The results also show that the analysesand forecasts are sensitive to the configuration used

for the analysis time, when the start, middle, or endof the assimilation window is taken as the analysis time(ASS−start, ASS−middle, ASS−end).

Under the perfect model scenario, the forecastsfrom the analyses at the middle and end of the assim-ilation window are superior to those produced fromanalysis at the start of the assimilation window. Onthe whole, the experiment with the analysis time at themiddle of the window (ASS−middle) has the best per-formance, and this configuration mainly benefits fromthe shorter intervals between the analysis time andthe observation times, ameliorating the degradationcaused by the limitations of the tangent-linear hypoth-esis. In addition, the advantages of the ASS−middleanalysis and forecasts are more pronounced when alonger assimilation window is adopted or more obser-vations are assimilated.

In addition, the ASS−middle and ASS−end ex-periments still show more improvement than theASS−start experiment when the forecast model hassignificant error. Combined with the characteristics ofmodel error growth, we can easily understand how theASS−middle outperforms ASS−start more remarkablyas the length of the assimilation window increases.

This study shows preliminary results concerningthe effect of analysis-time tuning when assimilatingsimulated observations that are uniformly temporallydistributed, using a simplified model. In future study,we will transplant the extended DRP-4DVar to theWeather Research and Forecasting (WRF) model. Weshould not expect issues elucidated in the context ofthe simplified model to simply recur for more complexmodels or real observation networks. But the extendedDRP-4DVar would still have the potential to improvethe analysis and forecasts by tuning the analysis timeflexibly according to the temporal and spatial distri-butions of observations. Moreover, it may help to im-plement a longer assimilation window.

Acknowledgements. This research was supported

by the Special Project of the Meteorological Sector pro-

gram [Grant No. GYHY(QX) 200906011] and the 973

project (Grant No. 2004CB418304).

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Impact of analysis-time tuning on the performance of the DRP-4DVar approach

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