impact of parameter estimation on the performance of the

JULY 1999 1497Z H U A N D N A V O N

q 1999 American Meteorological Society

Impact of Parameter Estimation on the Performance of the FSU Global Spectral ModelUsing Its Full-Physics Adjoint

YANQIU ZHU*

Geophysical Fluid Dynamics Institute, The Florida State University, Tallahassee, Florida

I. M. NAVON

Department of Mathematics and Supercomputer Computations Research Institute, The Florida State University,Tallahassee, Florida

(Manuscript received 1 December 1997, in final form 9 July 1998)

ABSTRACT

The full-physics adjoint of the Florida State University Global Spectral Model at resolution T42L12 is appliedto carry out parameter estimation using an initialized analysis dataset. The three parameters, that is, the biharmonichorizontal diffusion coefficient, the ratio of the transfer coefficient of moisture to the transfer coefficient ofsensible heat, and the Asselin filter coefficient, as well as the initial condition, are optimally recovered from thedataset using adjoint parameter estimation.

The fields at the end of the assimilation window starting from the retrieved optimal initial conditions and theoptimally identified parameter values successfully capture the main features of the analysis fields. A number ofexperiments are conducted to assess the effect of carrying out 4D Var assimilation on both the initial conditionsand parameters, versus the effect of optimally estimating only the parameters. A positive impact on the ensuingforecasts due to each optimally identified parameter value is observed, while the maximum benefit is obtainedfrom the combined effect of both parameter estimation and initial condition optimization. The results also showthat during the ensuing forecasts, the model tends to ‘‘lose’’ the impact of the optimal initial condition first,while the positive impact of the optimally identified parameter values persists beyond 72 h. Moreover, the authorsnotice that their regional impacts are quite different.

1. Introduction

A numerical weather forecast model involves a num-ber of parameters that are determined empirically. Someof these parameters, which are very common in physicalparameterization schemes, contain information aboutthe flow’s properties and characteristics or originatefrom the simplification of the physical parameteriza-tions. Other parameters are introduced due to numericalstability considerations. The values of the parametersdirectly or indirectly impact upon the performance ofthe model. Generally, the values of some parameters aredetermined by trial and error; that is, there is no ob-jective criterion to choose ‘‘optimal’’ values of the pa-rameters. In this study, we will focus on the optimal

* Current affiliation: Data Assimilation Office, NASA/GSFC, Sea-brook, Maryland.

Corresponding author address: Dr. I. Michael Navon, Dept. ofMathematics and SCRI, The Florida State University, Tallahassee,FL 32306-4052.E-mail: [email protected]

estimation of several parameters that are known to im-pact the performance of a numerical weather prediction(NWP) model. Namely, we wish to identify the optimalbiharmonic horizontal diffusion coefficient k, the ratiog of the transfer coefficient of moisture to the transfercoefficient of sensible heat, and the Asselin filter co-efficient e using adjoint parameter estimation and studyhow such optimized parameters impact on the ensuingmodel forecasts.

The parameter estimation procedure is aimed atchoosing an optimal parameter in an admissible param-eter set, so that the model solutions corresponding tothis parameter fit the observations as a minimizationproblem of an output-error criterion (Chavent 1974).Research on adjoint parameter estimation has been car-ried out using relatively simple models in the last 20years, first on topics such as aquifer behavior undertransient and steady-state conditions in the field of hy-drology (Carrera and Neumann 1986a–c), bottom dragcoefficient identification in a tidal channel (Panchangand O’Brien 1990), wind stress coefficient estimationalong with the estimation of the oceanic eddy viscosityprofile (Yu and O’Brien 1991), and nudging coefficientestimation in the National Meteorological Center [now

1498 VOLUME 127M O N T H L Y W E A T H E R R E V I E W

known as the National Centers for Environmental Pre-diction (NCEP)] adiabatic version of the spectral model(Zou et al. 1992a), and others. The issue of the adjointparameter estimation was also addressed by LeDimetand Navon (1988). Wergen (1992) recovered both theinitial conditions and a set of parameters from obser-vations using a 1D shallow-water equation model. Hisresults showed that even with noisy observations, theparameters were recovered to an acceptable degree ofaccuracy. For a detailed survey of the state of the artof parameter estimation in meteorology and oceanog-raphy, see Navon (1998). However, very few numericalexperiments have been conducted using a sophisticatedfull-physics model, and very little attention has beenpaid to the impact of the parameter estimation on thethe model forecast although the performances of thephysical parameterization and/or numerical schemes arealso a main factor in determining the model forecastskill. Recently, more and more effort has been focusedon improving the initial condition (Thepaut and Courtier1991; Navon et al. 1992), and its important impact onreducing the forecast error has been demonstrated. Whatis the effect of the parameter estimation on the modelcompared to that of the initial condition? In this study,we intend to explore this question and expect that theperformances of the physical parameterization and/ornumerical schemes can be improved by tuning the pa-rameters involved.

This paper is organized as follows: the model usedand the characteristics of the parameters to be optimallyidentified are described in section 2. The methodologyof adjoint parameter estimation is presented in section3. The detailed parameter estimation procedure, alongwith the minimization algorithm and the optimal pa-rameter values, are provided in section 4. Section 5provides the description of the forecast experiments andresults using both the optimal initial condition and theoptimally identified parameter values. Section 6 de-scribes the impact of the optimal parameters alone onensuing 24-h forecasts. The model’s ‘‘memory’’ (interms of ensuing forecast period results) of the impactof either the optimal initial condition or the optimalparameter values is discussed in section 7. Finally, sum-mary and future research are presented in section 8.

2. The model and characteristics of theparameters

The full-physics Florida State University GlobalSpectral Model (FSU GSM) (Krishnamurti and Dignon1988) and its adjoint model are employed to recoverboth the optimal initial condition and optimal parametervalues from the observations. The FSU GSM has beensuccessfully applied for numerical weather forecasts,especially for the Tropics. The model is a multilevel (12vertical levels) primitive equation model with s coor-dinate. All variables are expanded horizontally in a trun-cated series of spherical harmonic functions (at reso-

lution T42) and the transform technique is applied,which makes it possible and efficient to perform thehorizontal operators and to calculate the physical pro-cesses in real space. The finite difference schemes inthe vertical and semi-implicit scheme for time integra-tion are employed. The full physical packages includeorography, planetary boundary layer, dry adjustment,large-scale precipitation, moist-convection, horizontaldiffusion, and radiation processes.

The effort to establish a 4D variational (4D Var) dataassimilation system for the FSU GSM was initiated in1993. The adjoint code of the dry-adiabatic version ofthe FSU GSM was developed by Wang (1993). Later,Tsuyuki (1996) incorporated moisture variables, thesmoothed parameterization of moist processes, horizon-tal diffusion, and a simplified surface friction. We con-tinued their efforts by incorporating radiation and plan-etary boundary layer (including vertical diffusion) pro-cesses into the 4D Var data assimilation system due tothe important roles they play in simulating various large-scale and mesoscale phenomena, especially in tropicalweather systems, thus obtaining the full-physics adjointmodel (Zhu and Navon 1997). This effort renders theadjoint model consistent with the nonlinear forecastmodel. If the two models (the forward model and itsadjoint) are inconsistent, this may imply a negative im-pact on the process of adjoint optimal parameter esti-mation.

The horizontal diffusion term is usually incorporatedin a numerical weather prediction model to parameterizethe effects of motions on the unresolved scales and toinhibit spectral blocking, that is, the growth of smallscales in the dynamic model variables due to the ac-cumulation of energy at high wavenumbers. It is alsoemployed to eliminate the aliasing effect (Phillips 1959;Hamming 1973). The presence of any dissipation, phys-ical or computational, can attenuate the amplitude ofthe short wavelengths very significantly; in this case,the errors introduced by aliasing are minimal. Mac-Vean’s study (1983) showed that, without dissipation,integrations exhibited grossly physically unrealistic fea-tures after about several days even with very high meshresolution, indicating the crucial role played by the dis-sipation in nonlinear baroclinic development. Consid-erable effort has been made by various groups in tuningthe dissipation parameterizations in their general cir-culation or forecast models. For instance, Phillips(1956), Lilly (1965), Leith (1965), and Richard and Ger-man (1965), among others, applied the eddy diffusionterms in their numerical models; Navon (1969) includedthe lateral viscosity in a two-level general circulationmodel and carried out a 62-day integration, comparingits impact to that of using the Matsuno dissipativescheme. Kanamitsu et al. (1983, 1989) and Gordon andStern (1982) employed a biharmonic horizontal diffu-sion in their experiments. Presently, the biharmonic hor-izontal diffusion is used worldwide in NWP models dueto its better scale selectivity. Its limitation is the lack


of a sound physical foundation. Some other horizontaldiffusion schemes are also used, such as the schemedeveloped by Leith (1971) based on turbulence theory,which is now implemented in the NCEP spectral model(Kanamitsu et al. 1991).

The biharmonic horizontal diffusion k¹4 is used forvorticity, divergence, dewpoint depression, and virtualtemperature in the FSU GSM to selectively control small-scale noise without affecting large scales. The e-foldingdiffusive decay time at total wavenumber n is given by

41 at(n) 5 , (1)

2 2k n (n 1 1)

where a is the radius of the earth. The physical signif-icance of the diffusion coefficient k is not directly in-tuitive. Its effect may be understood in terms of thetimescale t at the smallest spatial scale resolved. Themodel dissipation should remove energy from the endof the spectrum at a rate sufficient to prevent a spuriousaccumulation of energy there, while not affecting themedium and large scales. In the Geophysical Fluid Dy-namics Laboratory spectral model, the coefficients ofthe eddy diffusion for ¹4 are determined by trial anderror, using the quality of the medium-range 500-mbgeopotential height forecast as a criterion (Gorden andStern 1982). The value they used is 1 3 1016 m2 s21 or2.5 3 1016 m2 s21 for T30, that is, a value of t of about53 or 21 h [based on Eq. (1)]. The value used in FSUGlobal Spectral Model T42 is 6 3 1015 m2 s21, that is,yielding a value of t corresponding to about 23 h.

In addition to the spatial diffusion, filters are alsocommonly used in numerical models to remove high-frequency noise that cannot be resolved at the givenmodel resolution. In the FSU GSM, a supplementarytime filter of the form

F(t) 5 F(t) 1 e[F(t 2 1) 2 2F(t) 1 F(t 1 1)]

is used. The characteristics of this filter are describedin detail by Asselin (1972). With an adequately chosencoefficient e, this filter damps the spurious computa-tional frequencies and a significant part of the spectrumof the external and internal gravity waves. The Rossbymotions of comparable horizontal dimensions are muchless affected due to their low frequency. This filter withe 5 0.25 removes or filters 2Dt waves and reduces theamplitude of 4Dt waves by one-half, but has little effecton longer-period waves; that is, it acts as a low-passfilter in time. However, despite the advantage of im-mediately suppressing the 2Dt wave, values of e lessthan 0.25 are preferable since the stability conditionrequires a progressively smaller Dt as e increases. More-over, repeated use of even a weak filter eventually damp-ens the lower frequencies (Haltiner and Williams 1980).Skamarock and Klemp (1992) discussed how the As-selin filter affected stability for a nonhydrostatic model.Robert (1966) used this filter in a general circulationprimitive equation spectral model with centered differ-ences to control instability. The values he used are e 5

0.02 and Dt 5 20 min. A number of experiments carriedout by Krishnamurti showed that e 5 0.05 is the bestvalue for the Asselin filter for the model forecast. Wewill carry out an optimal parameter estimation experi-ment to obtain the optimal value of e in this study. Thelower bound of e is set to be zero, while its upper boundis set to be 0.1.

The third parameter we consider is the ratio g of thetransfer coefficient of moisture to the transfer coefficientof sensible heat, which arises from the parameterizationof the surface fluxes in the boundary layer. Accurateheat and moisture flux parameterizations are very im-portant since the surface heat and moisture fluxes exerta strong influence on the surface energy budget andprecipitation rate. Similarity theory has successfullyprovided a framework for the description of the atmo-sphere surface layer, the lowest 50 m or so of the bound-ary layer where the Coriolis force can be ignored andthe fluxes can be assumed to be constant with height.The flux profiles and other properties of the flow arereasonably parameterized via this theory. However, cer-tain empirical parameters or constants evolving fromthe theory need to be determined by experiments, forinstance, the von Karman constant and constants as-sociated with the stability dependence of the flux-profileformulations. In this study, however, our goal is not todetermine the parameter values from physical sound-ness. We will instead aim to obtain an optimal value ofg for this particular model via a variational parameteridentification approach in order to improve the modelforecast skill. Dyer (1967) found that, over the rangeof the ratio of height z to the Monin–Obukhov lengthL (0.02 , |z/L| , 0.6) both the transfer coefficient ofsensible heat f H and the transfer coefficient of moisturef Q varied approximately as |z/L|21/3; for |z/L| . 0.2, f H

is found to vary as |z/L|21/2, but insufficient data limitedthe value of the corresponding analysis for f Q. Sincereasonable agreement is found between the f H and f Q

data, many numerical weather prediction models adopta simple relationship of the type f Q 5 f H; that is, g5 1.0. Recently, Smith et al. (1996) reviewed the 25years of progress on air–sea fluxes. Theory and obser-vations indicate that the ratio should depend on the sur-face characteristics. However, since the ratio is consid-ered as a constant in the original FSU GSM and alsonumerical experience with the model shows that themodel tends to produce larger precipitation, we will stilltake the parameter as a constant and set its upper andlower bounds to be 1.8 and 0.3, respectively, in thisstudy. Its optimal value will be estimated via numericalfitting of the model to the observations. The specifi-cation of the g value directly impacts the moisture flux.We expect to improve the performance of its corre-sponding physical process by tuning this parameter.

3. The methodology of parameter estimationIn this study, we aim to perform optimal parameter

estimation via a variational approach, that is, to obtainan optimal value a8 of the parameter vector a such that


J(a8) , J(a) ∀a,

where J is a cost function that measures the discrepancybetween the observations and the corresponding modelforecast variables. Hence, the optimal parameter can beretrieved by fitting the model forecast fields to the ob-servations.

Given constrained parameters whose values vary be-tween certain bounds, for instance, when the parameterai satisfies ai ∈ [ai, bi], where a and b denote the lowerand upper bounds, respectively, the cost function forparameter estimation may assume the following form:

tR1obs obsJ(X, a) 5 ^W(X 2 X ), (X 2 X )& dtE2 t0

T1 l g(a), (2)

where l is the penalty coefficient vector, W is the di-agonal weighting matrix defined as in Navon et al.(1992), ^ & denotes spatial integration, X represents thestate variable vector, Xobs the observation vector, and t0

and tR denote the assimilation window. The second termconsists of a penalty function. The value of l is deter-mined such that the penalty term is of the same orderof magnitude as the other terms in the cost function,and g(a) is defined as

12(a 2 b ) if a $ bi i i i2

g (a) 5 0 if a , a , b (3)i i i i

12 (a 2 a ) if a # a ,i i i i2

where gi(ai) is a function only of the violated con-straints. The first derivative of this function is

a 2 b if a $ bi i i i]gi 5 0 if a , a , b (4)i i i]a i a 2 a if a # a . i i i i

Another type of penalty effective in transforming aconstrained optimization problem into an unconstrainedone is the barrier method, which imposes a penalty forreaching the bound of an inequality constraint. Typi-cally, a logarithmic barrier function is of the form

tR1obs obsJ(X, a) 5 ^W(X 2 X ), (X 2 X )& dtE2 t0

T2 m log(h(a)), (5)

where m is the barrier coefficient and h is the constraintfunction. The barrier methods are strictly feasible meth-ods; that is, the iterates lie in the interior of the feasibleregion and create a ‘‘barrier’’ to keep iterates away fromboundaries of the feasible region (Nash and Sofer 1996).

There are two different purposes for the inclusion ofthe second term in Eqs. (2) or (5). One is to ensure thatthe retrieved parameter lies within the prescribed

bounds. A penalty term of the form in Eq. (2) or alogarithm of the box constraints as in Eq. (5) is efficientto achieve this. The other purpose is to increase theconvexity of the cost function by adding a positive valuel to the Hessian matrix, thereby increasing its positivedefiniteness.

Suppose that the forward model is perfect and is givenin the form

]X5 F(X, a, t). (6)

]t

Its corresponding tangent linear model is defined as

]dX ]F(X, a, t) ]F(X, a, t)5 dX 1 da. (7)1 2 1 2]t ]X ]a

Then the adjoint model can be expressed in the form

T]P ]F(X, a, t)

obs2 2 P 5 W(X 2 X ), (8)1 2]t ]X

where P represents the adjoint variables. The gradientsof the cost function with respect to the initial conditionand the parameter a are, respectively,

= J 5 P(0) (9)X0

TtR ]F ]gT= J 5 P dt 1 l . (10)a E 1 2[ ]]a ]at0

The adjoint model is of the same form as that whereonly the initial condition is considered as the controlvariable. Hence, the problem of parameter estimationvia the adjoint method does not result in an additionalcomputational effort when the number of parameters tobe estimated is small. We may expect that the parameterestimation process will provide us with both optimallydetermined parameters and initial condition simulta-neously. The gradient of the cost function with respectto both the initial condition and parameters is written as

=J 5 ( =aJ)T.= J,X0(11)

4. Parameter estimation procedure and results

In this study, the FSU Global Spectral Model and itsfull-physics adjoint model are employed to optimallyidentify the aforementioned three parameters separatelyalong with initial condition retrieval. An arbitrary ini-tialized European Centre for Medium-Range WeatherForecasts (ECMWF) analysis dataset is used in ourstudy. A 6-h assimilation window is used, that is, from0000 to 0600 UTC 3 September 1996. The initializedECMWF analysis data at the beginning and end of theassimilation window are taken as the observations dur-ing the 6-h assimilation window period.

The parameter estimation procedure is carried out asfollows.

1) Take the 6-h forecast starting from the initialized


FIG. 1. The divergence fields at 200 mb at 0000 UTC 3 Sep 1996: (a) the observation, (b) theinitial guess. The contour interval is 5.0 3 1026 s21. Dashed values are negative.

analysis at 1800 UTC 2 September 1996 as the initialguess of the initial condition. Given a positive def-inite initial approximation to the inverse Hessian ma-trix H0 (generally taken as the identity matrix I), weintegrate the full-physics FSU GSM 6 h and thencalculate the cost function using Eq. (2). The initialguess of a, the penalty coefficient value l, and theupper and lower bounds where the parameters mayvary are specified here.

2) Integrate the full-physics adjoint model of the FSUGSM backward in time with null initial condition toobtain the gradient of J with respect to the controlvariable Y 5 (X0, a)T,

g0 5 g(Y) 5 ( =aJ)T,= J,X0

and the search direction

d0 5 2H0g0.

Generally, physical processes involve many on–offswitches that may introduce discontinuities in theparameterization schemes. In the full-physics adjointmodel of the FSU GSM, since we hope to keep theoriginal parameterization characteristics unchangedas much as possible, only the discontinuities thatmost impact the tangent linear approximation andthe convergence of the minimization algorithm areconsidered for removal by using smooth function andcubic spline interpolation (Zhu and Navon 1997).

3) For k 5 0, 1, 2, . . . , minimize J(Yk 1 bkdk) withrespect to b $ 0 to obtain Yk11 as

Yk11 5 Yk 1 bkdk,

where bk is a positive scalar, the step size beingobtained by a line search so as to satisfy a sufficientrate of decrease (see Gill et al. 1981).

4) Compute the new gradient and search direction, re-spectively,

gk11 5 =J(Yk11) dk11 5 2Hk11gk11.

5) Check whether the solution converges. If the con-vergence criterion

\gk11\ # e9Max(1, \ Yk11 \)

is satisfied, where e9 is a user-supplied small number,then the algorithm terminates with Yk11 as the op-timal solution; otherwise go back to step 3.

The 6-h forecast from the initialized analysis at 1800UTC 2 September 1996, which serves as the initial guessof the initial condition, shows important underestimatesof the tropical divergence field, particularly in the Pa-cific (Fig. 1).

The minimization procedure is terminated after 60iterations. Although the convergence rate is expected tobe much slower than that of the adiabatic version, dueto the high nonlinearity of the model including the fullphysical processes, a sufficient decrease in the cost func-tion and its gradient for each experiment is achievedafter 60 iterations. At the end of the minimization pro-cedure, both the optimal initial condition and optimalvalue of the parameter are recovered.

The initial guess (estimated value) of the horizontaldiffusion coefficient k is taken to be 6 3 1015 as usedroutinely in the FSU GSM T42L12. The penalty param-


FIG. 2. The variation of the cost function and the gradient normwith respect to the iteration number when both the optimal initialcondition and the biharmonic horizontal diffusion coefficient k areretrieved.

FIG. 3. The evolution of the horizontal diffusion coefficient.

eter lk for this parameter is taken to be 1.0 3 10212.The upper and lower bounds where k varies are takento be 3.0 3 1016, that is, five times the value used inthe original forecast model, and 2.0 3 1015, respectively.The variations of the cost function and the gradient normwith respect to the number of iterations are presentedin Fig. 2. We see that the cost function decreases froman initial value of 3643 to 575.3, that is, it decreases toabout 15.8% of its original value, while the norm of thegradient value decreases to 23.6% of its original value.Figure 3 displays the evolution of the horizontal dif-fusion coefficient during the minimization procedure.This coefficient experiences a rapid increase until the35th iteration, reaching its peak at the 44th iteration,then experiences a slight decrease. The optimal valueobtained for the horizontal diffusion coefficient is1.1124 3 1016. This value is almost twice the valueused in the original forecast model. We also notice thatthe optimal parameter value is not very sensitive to thevalue of lk as long as the penalty term is of the sameorder of magnitude as the other terms in the cost func-tion.

The issue of the sensitivity of the optimal parametersvalues to the values of the penalty coefficients deservessome attention. Research work by Craven and Wahba(1979) and by Lewis and Grayson (1972) has shownstrong dependence of the analysis results on the spec-ification of the penalty coefficients.

In our case the optimal parameter values are not foundto be very sensitive to the value of the penalty coeffi-cient as long as the penalty parameter attains a certainthreshold value.

To understand why this happens, one should considerthe basics of the penalty technique as a method to solvea sequence of unconstrained optimization problemswhose solution is usually infeasible to the original con-strained problem. A penalty for violation of the con-straints is incurred and as this penalty is increased, theiterates are forced toward the feasible region.

The experience of Zou et al. (1992b, 1993) in apply-ing a penalty method for controlling gravity oscillationsin variational data assimilation shows that the effect ofthe penalty parameter is ineffective until it attains athreshold value. Thereafter it is effective in constrainingthe minimizers toward the feasible region and filteringthe gravity waves. Increasing the penalty parameters byan order of magnitude or more over the threshold valuedoes not change this effect, but slows down the mini-mization due to the ill-conditioning of the related Hes-sian of the cost functional (see Nash and Sofer 1996;Gill et al. 1981; Fletcher 1987).

A higher sensitivity of minimization results to thevalue of the penalty parameter is typical for cases when,from the start, the iterates are rather far from the feasibleregion and reaching the effective threshold value re-quires several iterations.

In the present case we are from the starting point verynear to the feasible region, hence the perceived lack ofsensitivity of the optimal parameter values to the valuesof the penalty coefficients when these attain their thresh-old value. When we increase the penalty parametersfurther (i.e., by an order of magnitude or more), thereis no perceived change except for a serious slowdown


TABLE 1. Experiments designed to assess the impact of optimal parameter estimation.

Expt Initial condition Parameter value Integration (h)

C1C2O1O2O3O4

Initial guessInitial guessOptimal initial conditionOptimal initial conditionOptimal initial conditionOptimal initial condition

Estimated valueOptimal valueEstimated valueOptimal valueEstimated valueOptimal value

6666

2424

in the rate of convergence of the minimization due tothe aforementioned ill-conditioning.

Similar variations of the cost function and the gra-dient norm with respect to the iteration number are ob-tained when the ratio g and the Asselin filter coefficiente are retrieved from the observations, respectively. Theinitial guess (estimated value) of the Asselin filter co-efficient e is set to be 0.05, which is the value used inthe original forecast model. The upper and lower boundsfor the variation of this parameter are taken as 0.1 and0.0, respectively, and the penalty parameter le for thisparameter is set to be 1.0 3 108. The optimal e valueobtained is 0.0487, which is very close to the initialguess, which means that either the initial guess is a fairlygood guess or the model is not very sensitive to theparameter. Further study with the initial guess awayfrom 0.05 still yields the same result, which indicatesthat the initial guess is fairly good. The initial guess(estimated value) of the ratio g of the transfer coefficientof moisture to the transfer coefficient of sensible heatis initially set to be 1.0, which is the value used in theoriginal FSU GSM T42L12. The penalty parameter lg

for g is set to be 2.0 3 105, while the upper and lowerbounds where g may vary are taken to be 1.8 and 0.3,respectively. The optimal parameter estimation for gyields a value of 0.4974, which is only half of the initialguess value, due to numerical fitting of the model tothe observations. We should be aware that the optimalvalue retrieved here includes the possibility of the com-pensating for errors in the physical parameterizationschemes of the forecast model. It does not represent thetrue physical value; it is only the optimal value for thisparticular model for improving the model forecast.

5. Forecast experiments using both the retrievedinitial condition and parameters

a. Assessing the combined impact of the optimalinitial condition and parameter estimation

Three experiments are carried out to integrate themodel for 6 h in order to obtain the analysis fields atthe end of the assimilation window for the biharmonichorizontal diffusion coefficient k, namely, 1) a controlexperiment (i.e., experiment C1 in Table 1), which isintegrated from the initial guess fields, that is, 6-h fore-cast from the 1800 UTC 2 September 1996 analysis,and uses the estimated parameter k 5 6 3 1015; 2) theoptimal experiment (i.e., experiment O2 in Table 1), so

called since we are using both the optimal initial con-dition and the optimal parameter value k; and 3) a sim-ulation experiment using the estimated parameter k 56 3 1015, where the model is integrated from the ini-tialized 0000 UTC 3 September 1996 analysis to serveas the best simulation without using variational dataassimilation. The differences between the control ex-periment and the optimal experiment reflect the com-bined impact of the optimal initial condition and theoptimal biharmonic horizontal diffusion coefficient pa-rameter value.

The rms errors of the fields at the end of the assim-ilation window are calculated for the aforementionedexperiments and are provided in Fig. 4. The results ob-tained show that the optimal experiment yields the bestresults throughout all of the vertical levels, especiallyfor the divergence field. The rms errors of the logarithmof the surface pressure are 0.140E22, 0.939E23, and0.517E23 for the control experiment, the simulationexperiment, and the optimal experiment, respectively.

Figure 5 presents the divergence analysis fields at 200mb for the aforementioned experiments and the obser-vation at the end of the assimilation, that is, 0600 UTC3 September 1996. Only the optimal experiment suc-cessfully provides a high quality analysis, capturing themain features of the divergence field; both the controlexperiment and the simulation experiment fail to sim-ulate the strong divergence field corresponding to aheavy precipitation event over Indonesia and overesti-mate the divergence field to the west of Africa. Thecontrol experiment also overestimates the divergencefield over northern Africa and underestimates the di-vergence field around 358S, 208W, while the simulationexperiment overestimates the divergence field to the eastof Australia.

The difference fields of the specific humidity at 500mb and of the temperature at 850 mb between the resultsof the aforementioned experiments and the observationfield at the end of the assimilation are displayed in Figs.6 and 7. The improvement obtained from the optimalexperiment over the results obtained from the other ex-periments is obvious, particularly over areas with largeerrors.

Figure 8 displays the rms errors of the meteorologicalvariables for the 24-h forecast control experiment andoptimal experiment (i.e., experiment O4 in Table 1) forthe biharmonic horizontal diffusion coefficient k. Theuse of both the optimal initial condition and the param-


FIG. 4. The rms errors of the vorticity, divergence, virtual temperature, and dewpoint depressionat the end of the assimilation window: solid line, the control experiment; dashed line, the simulationexperiment; dotted line, the optimal experiment.

eter value improves the 24-h forecast fields; however,compared with Fig. 4, the differences between the con-trol experiment and the optimal experiment decrease asthe forecast period increases.

Similar results are also obtained for the other twoparameters.

b. Assessing the impact of optimal parameterestimation

In the previous sections, we recovered three pairs ofthe optimal initial conditions and optimal values of sev-

eral model parameters, and discussed their combinedimpact on the model forecast. An important questionremains to be clarified: how much of the improvementobtained is directly attributable to the optimal initialcondition, and how much of it originates from the op-timal values of the identified parameters. In order toprovide a closer look at this issue, three additional ex-periments are performed to compare with the above-mentioned 6-h forecast control experiment (referred toas C1) and the optimal experiment (referred to as O2)as well as with the 24-h forecast optimal experiment(referred to as O4) for k. The additional experiments


FIG. 5. The divergence fields at 200 mb at the end of the assimilation window: (a) the observation,(b) the control experiment, (c) the simulation experiment, and (d) the optimal experiment, re-spectively. The contour interval is 8.0 3 1026 s21.

are experiment C2, in which the initial guess of theinitial condition and the optimally identified parametervalue are used, experiment O1, in which the optimalinitial condition and estimated parameter value are used,and experiment O3, in which the optimal initial con-dition and estimated parameter value are used (Table 1).

To assess the impact of the optimal parameter esti-

mation, we use the differences between the rms errorsat all vertical levels for vorticity, divergence, virtualtemperature, and dewpoint depression fields. The dif-ference between experiments C1 and C2 reflects theimpact of the optimal parameter values on the forecastwhen the optimal initial condition is not used, while thedifferences between experiments O1 and O2, and be-


FIG. 6. The difference fields of specific humidity at 500 mb between the results of the followingthree experiments and the observation at the end of the assimilation window: (a) the controlexperiment, (b) the simulation experiment, and (c) the optimal experiment. The contour intervalis 0.5 3 1023 g/g.

tween experiments O3 and O4, reflect the impacts ofthe optimal parameter value on the forecast when theoptimal initial condition is applied for the 6-h and 24-hforecasts, respectively. If the difference is negative, itmeans that the rms error of the experiment, in whichthe optimally identified parameter value is used, issmaller than the rms error of the experiment in whichthe estimated parameter value is used. The differencesof the rms errors of the vorticity, divergence, virtualtemperature, and the dewpoint depression at the end ofthe forecast between the experiment pairs C2 and C1,O2 and O1, and O4 and O3 are shown in Fig. 9. Forthe pair C2 and C1, the rms differences are negativeexcept for the dewpoint depression at the lowest twolevels. This indicates a relatively small (compared tothe differences of the rms errors between C1 and O2)but positive impact of the optimal parameter value onthe model simulation. Experiment O1 is comparable to

O2 in terms of rms error, with the impact of the optimalinitial condition dominating that of the optimal param-eter value at the end of the assimilation window. Thisis very reasonable, since in the first few hours of theforecast, the optimal initial condition may reconstructthe dynamic structure while the optimal biharmonic hor-izontal diffusion coefficient value can only adjust it.However, the lower-level divergence fields are largelyimproved in all of the pairs of experiments when theoptimal parameter value k is used. As the length of theforecast period increases, the impact of the optimal pa-rameter value on the model forecast becomes more pro-nounced for the pair O3 and O4. Compared with Fig.8, this result implies the known fact that the effect ofthe optimal initial condition decays as the forecast pe-riod becomes longer.

An additional experiment is conducted with the fixedinitial condition 0000 UTC 3 September 1996 and where


FIG. 7. The difference fields of temperature at 850 mb between the results of the following threeexperiments and the observation at the end of the assimilation window: (a) the control experiment,(b) the simulation experiment, and (c) the optimal experiment. The contour interval is 1 K.

the parameter k is considered as the sole control vari-able. An identical cost function and initial parametervalue to those used before are employed. The cost func-tion reaches a constant value while the norm of gradientdecreases by three orders of magnitude in five iterations.The optimal biharmonic horizontal diffusion coefficientvalue obtained is 1.1946 3 1016, which is very closeto that obtained when we recovered both the optimalinitial condition and the optimal parameter value. Asimilar positive impact is observed when comparing theforecasts using the estimated parameter value and theoptimal parameter value, respectively.

Similar experiments (related to the experiments de-tailed in Table 1) are also conducted separately for theparameters e and g. Since the optimal value of Asselinfilter coefficient e is very close to its estimated value,the impact of this parameter on the ensuing forecasts ismarginal. A positive impact of the optimal ratio g onthe forecasts is also observed while the impact of the

optimal initial condition is found to be dominant duringthe first several hours of the forecast.

6. Impact of the optimal parameters alone onensuing 24-h forecasts

A number of 24-h forecast experiments beginningfrom the 0000 UTC 3 September 1996 analysis are per-formed in order to examine the impact of each parameterseparately as well as the combined impact of all threeparameters on the ensuing forecast fields. The experi-ments are (a) experiment with the estimated parameters,that is, k 5 6.0 3 1015, e 5 0.05, and g 5 1.0; (b)experiment with the estimated e and g but the optimallyretrieved k value; (c) experiment with the estimated eand k but the optimally retrieved g; and (d) experimentusing optimally retrieved k, e, and g simultaneously.Since the optimal value of e is very close to its estimatedvalue, the impact of this parameter is rather marginal.


FIG. 8. The rms errors of the vorticity, divergence, virtual temperature, and dewpoint depressionat the end of the 24-h forecast: solid line, the control experiment; dotted line, the optimal experiment.

The rms errors of the vorticity, divergence, dewpointdepression, and the virtual temperature fields at all ver-tical levels are calculated for the above experiments withrespect to the analysis at 0000 UTC 4 September 1996.Figure 10 displays the differences of the rms errors be-tween experiments b and a, c and a, d and a, whichindicate the impacts of optimal values of k, g, and theircombined impact, respectively. Negative values of therms differences indicate that the rms errors are less thanthose of experiment a with the estimated parameter val-ues; that is, the optimal parameter value has a positiveimpact on the forecast.

The results show that the optimal horizontal diffusioncoefficient k impacts mainly the vorticity and diver-

gence fields, and also has a positive impact on the virtualtemperature and dewpoint depression at upper and mid-dle levels but a negative impact at lower levels. On thecontrary, in experiment c when only the optimal valueof g is employed, the vorticity and divergence fields areonly slightly improved, but the middle and lower levelsof the virtual temperature and the lower levels of thedewpoint depression experience a large improvement.The effect of g is mainly confined to the lower levelsof the model. The best forecasts are obtained by ex-periment d, in which all the optimal parameters valuesof k and e as well as g are used simultaneously. In thissense, experiment d combines all of the advantages ofexperiments b and c.


FIG. 9. The differences of the rms errors of the vorticity, divergence, virtual temperature, and thedewpoint depression between the three pairs at the end of the forecast: solid line, C2 and C1; dashedline, O2 and O1; dotted–dashed line, O4 and O3.

7. A study of the model’s ‘‘memory’’ of impact ofoptimal initial condition and identifiedparameter values

So far, we have studied either the combined impactof both the optimal initial conditions and the optimalparameter values or the impact of only the optimallyidentified parameter values on the ensuing forecast. Inthis section, we discuss the persistence (memory) of thecombined impact of the above-mentioned three opti-mally identified parameter values as well as the optimalinitial condition obtained by variational data assimila-tion on the ensuing forecast.

Three sets of experiments are carried out by inte-

grating the model for 24, 48, and 72 h, respectively,from 0000 UTC 3 September 1996. The first set ofexperiments (C1, C2, and C3) are control experimentsthat are integrated from the initial guess of the initialcondition (i.e., the 6-h forecast from 1800 UTC 2 Sep-tember 1996) using the estimated parameter values. Thesecond set of optimal parameter experiments (P1, P2,and P3) start from the initial guess of the initial con-dition using the optimally identified values of the above-mentioned three parameters. The third set of optimalinitial condition experiments (I1, I2, and I3) are inte-grated from the variationally derived optimal initial con-dition using estimated parameter values (Table 2).


FIG. 10. The differences of the rms errors of the 24-h forecast vorticity, divergence, virtual tem-perature, and the dewpoint depression between the following pairs: dashed line, expts b and a; dot–dot–dashed line, expts c and a; solid line, expts d and a.

TABLE 2. Experiments carried out by integrating the model for 24, 48, and 72 h from 0000 UTC 3 Sep 1996, respectively.

Expt Initial condition Parameter values Length of forecast (h)

C1C2C3P1P2P3I1I2I3

Initial guessInitial guessInitial guessInitial guessInitial guessInitial guessOptimal initial conditionOptimal initial conditionOptimal initial condition

Estimated valuesEstimated valuesEstimated valuesOptimal valuesOptimal valuesOptimal valuesEstimated valuesEstimated valuesEstimated values

244872244872244872


FIG. 11. The percent differences of the rms errors of the ensuing forecast fields vorticity, divergence,virtual temperature, and the dewpoint depression between when the optimally identified parametervalues are used and when the estimated parameter values are used: solid line, 24-h forecast; long,short dashed line, 48-h forecast; dot–dot–dashed line, 72-h forecast.

The rms errors of vorticity, divergence, virtual tem-perature, and dewpoint depression fields are calculatedfor each experiment, then the percentages of the dif-ferences of the rms errors between C1 and P1, C2 andP2, C3 and P3, C1 and I1, C2 and I2, and C3 and I3are computed. The percentages of the decrease of rmserrors due to the use of the three optimally identifiedparameter values are displayed in Fig. 11. The resultsshow that all of the experiments using optimally iden-tified parameter values exhibit smaller rms errors thanthose using estimated parameter values through all ofthe vertical levels except for the virtual temperature at

the top vertical level of the model; that is, the combinedimpact of the three identified parameter values still per-sists for the 72-h forecast, and probably even furtherbeyond. The largest improvement in the 24-h forecastoccurs at the low levels of the dewpoint depression fieldwhere the rms error decreases by up to 17%. The overallforecast rms errors further decrease for the 48-h forecast.The impacts of the optimally identified parameter valueson the vorticity field and on the middle levels of thedivergence, virtual temperature fields as well as the low-er levels of the dewpoint depression field decay for the72-h forecast compared to the results obtained with op-


FIG. 12. The percent differences of the rms errors of the ensuing forecast fields vorticity, divergence,virtual temperature, and the dewpoint depression between when the optimal initial condition is usedand when the initial guess of initial condition is used: solid line, 24-h forecast; long, short dashedline, 48-h forecast; dot–dot–dashed line, 72-h forecast.

timal parameters for the corresponding 48-h forecast;however, the rms errors still decrease by up to 10%compared to the experiment where estimated parametersare used for the same 72-h forecast.

The percentages of the decrease in rms errors forvorticity, divergence, virtual temperature, and dewpointdepression fields by using optimal initial condition arepresented in Fig. 12. A very clear trend is observed,namely, the impact of the optimal initial condition de-cays as the forecast time increases, especially the impacton the divergence field decays very rapidly. This is inagreement with the results obtained in section 4. The

improvement of the forecast (in terms of rms errors)due to the combined impact of the three optimally iden-tified parameter values exceeds that obtained due to theimpact of the optimal initial condition in the ensuing72-h forecast. Even for the 24-h forecast, the impact ofthe three optimal parameter values on the lower levelof the dewpoint depression is larger than that of theoptimal initial condition. The model tends to first ‘‘lose’’the impact of the optimal initial condition in the ensuingforecast, while the impact of using optimal parameterson the above-mentioned forecast fields lingers even after72 h.


FIG. 13. The histogram of vorticity rms errors for the optimalparameter experiments (black bar) and optimal initial condition ex-periments (gray bar).

FIG. 14. The histogram of dewpoint depression rms errors for theoptimal parameter experiments (black bar) and optimal initial con-dition experiments (gray bar).

Figures 13 and 14 show how the rms errors of vor-ticity and dewpoint depression evolve with forecasttime, respectively, in which black bars are for the op-timal parameter experiments (P1, P2, and P3) and graybars are for the optimal initial condition experiments(I1, I2, and I3). The results indicate that the errors ofthe optimal initial condition experiment become largerthan those of the optimal parameter experiment at 72-hforecast.

We would also like to point out that the mechanismsof the impacts resulting from optimal initial conditionand from the optimal parameter values are quite dif-ferent. The impact of the optimally identified parametervalues is effective throughout the entire integration pe-riod via the corresponding physical parameterization ornumerical schemes, while the impact of the optimal ini-tial condition obtained via variational data assimilationis to enhance the model forecast skill by reducing theerrors in the initial condition, which might lead to apoor forecast. Therefore, their impacts are quite differ-ent. Figure 15 displays the difference fields betweenexperiments C1, P1, and the analysis for specific hu-midity at 850 mb, respectively, while the difference fieldof specific humidity at 850 mb between experiments C1and P1 is displayed in Fig. 16. Comparing Figs. 15a

and 15b, we notice that they have almost the same pat-tern, except that the positive differences between theforecast and the analysis become smaller while the neg-ative differences between the forecast and the analysisbecome larger over most of the forecast area in exper-iment P1 when the optimally identified parameter valuesare used. That is to say, the use of the optimally iden-tified parameter values tends to produce a smaller spe-cific humidity forecast over most of the global domain,since the optimal ratio of the transfer coefficient of themoisture to the transfer coefficient of the sensible heatis only about half of its estimated value. Larger specifichumidity forecasts in experiment P1 than those in ex-periment C1 are observed in only a few regions, suchas central South America, which might have beencaused by the impact of the optimal biharmonic hori-zontal diffusion coefficient or by interactions betweenthe impacts of the three optimal parameter values. Inthis case, the improvement due to the use of the opti-mally identified parameter values is observed mainlyover the overestimated areas with respect to the analysis(in experiment C1), which spread over the land. Figure17 displays the difference fields of the specific humidityat 500 mb between experiments C1, I1, and the analysis,respectively. Experiment I1 in which the optimal initial


FIG. 15. The difference fields between experiments (to be enu-merated below) and the analysis for the specific humidity at 850 mbas follows: (a) the control experiment C1, (b) expt P1. The contourinterval is 0.3 3 1022 g/g.

FIG. 16. The difference field of specific humidity at 850 mb betweenexpt C1 in which the estimated parameter values are used and exptP1 in which the optimally identified parameter values are used. Thecontour interval is 0.8 3 1023 g/g.

FIG. 17. The difference fields between experiments (to be enu-merated below) and the analysis for the specific humidity at 500 mbas follows: (a) the control experiment C1, (b) expt I1. The contourinterval is 0.8 3 1023 g/g.

condition is used, however, does not exhibit such fea-tures as mentioned above, rather the improvement isobtained both in underestimated and overestimated areascompared with the result of experiment C1, especiallyover areas of large errors.

It is also known that forecasts starting from the var-iationally derived optimal initial conditions are not asgood as the forecasts starting from the latest availableanalysis (Pu et al. 1997). However, we compare theforecasts starting from 0600 UTC 3 September 1996analysis (the latest available analysis in this study) usingboth the estimated values of the above-mentioned threeparameters and the previously optimally identified pa-rameter values, respectively. Similar improvements asshown in Fig. 11 are observed. The optimally identifiedvalues of the biharmonic horizontal diffusion coefficientand the ratio of the transfer coefficient of the moistureto the transfer coefficient of the sensible heat actuallyimprove the performance of the corresponding physicalparameterization and/or numerical schemes, so that theirimpacts on the model forecast persist under similarlarge-scale atmospheric environment.

Finally, we present the anomaly correlation results

for the specific humidity forecasts at 850 mb for thecontrol experiments, optimal parameter experiments,and optimal initial condition experiments as well as theexperiments using both optimal initial conditions andoptimally identified parameter values. These anomaly


FIG. 18. Anomaly correlation of the specific humidity forecasts at850 mb for the control experiments (solid line), optimal parameterexperiments (dot–dot–dashed line), optimal initial condition experi-ments (dotted line), and the experiments using both the optimal initialcondition and optimally identified parameter values (dashed line):(top) tropical belt, (bottom) Northern Hemisphere.

correlations are measured both for the tropical belt [de-fined here to be 408S–408N where physical initializationis usually carried out (Krishnamurti 1991)] and for theNorthern Hemisphere (Fig. 18). The specific humidityfield, which is mainly distributed in the lower tropo-sphere, is very important in the tropical system. Com-paring Figs. 18a and 18b, we see that the anomaly cor-relation for the Northern Hemisphere is consistentlyhigher than that for the tropical belt for the entire 5-dayintegration period. The results of optimal initial con-dition experiments appear to be more skillful than theoptimal parameter experiments for the first 24-h forecastin the tropical belt and for the first 48-h forecasts inNorthern Hemisphere, respectively. However, later in

the forecast (i.e., for the period 48–120 h), the optimalparameter experiments yield better skill results than theoptimal initial condition experiments both in the tropicalbelt and Northern Hemisphere. The experiments usingthe optimal initial condition and optimally identifiedparameter values simultaneously have the highest anom-aly correlations, where for the 120-h forecast the anom-aly correlation increases from 65.1% in the control ex-periment to 70.7% for the tropical belt and from 72.5%in the control experiment to 77.8% for the NorthernHemisphere.

8. Summary and future research

In this study, we recovered both optimal initial con-ditions and optimal values of the biharmonic horizontaldiffusion coefficient, the Asselin filter coefficient, andthe ratio of the transfer coefficient of moisture to thetransfer coefficient of sensible heat using the full-phys-ics adjoint of the FSU Global Spectral Model. The re-sults obtained are very encouraging. The fields at theend of the assimilation window starting from the re-trieved optimal initial condition and the optimal param-eter values successfully capture the main features of theanalysis fields. Although the impact of optimal initialconditions dominates that of the optimal parameter val-ues at early stages of the forecast, a positive impact dueto each optimally estimated parameter value is observed.The ensuing 24-h forecast experiments in which onlythe optimal parameter values are used further indicatethe positive impact of using optimal parameter values.The biharmonic horizontal diffusion coefficient im-proves vorticity and divergence fields as well as theupper levels of virtual temperature and of dewpoint de-pression fields, while the ratio of the transfer coefficientof moisture to the transfer coefficient of sensible heathas a large positive impact on the lower levels of themodel, especially those of the virtual temperature andthe dewpoint depression fields. By combining the threeoptimal parameter values, we obtain the best forecastresults. Further studies of ensuing forecasts using theoptimally identified parameter values and the optimalinitial conditions, respectively, show that the modeltends to first lose the impact of the optimal initial con-dition while the impact of optimally identified parametervalues persists beyond 72 h. The optimally identifiedvalues of the biharmonic horizontal diffusion coefficientand the ratio of the transfer coefficient of the moistureto the transfer coefficient of the sensible heat improvethe performance of the corresponding physical param-eterization schemes. The experiments using the optimalinitial condition and optimally identified parameter val-ues simultaneously yield the best performance. How-ever, we should be aware that these results are obtainedfor a single case study. Further studies should be con-ducted for different initial conditions in order to drawa general conclusion.

In this study, the nonlinear forecast model is assumed


to be perfect, that is, it is used as a strong constraint.Further studies should be conducted to take into accountthe model forecast error; that is, the forecast model isregarded as a weak constraint, and its effect on the re-trieved initial conditions and optimal parameter valuesshould be examined.

The parameters studied in this paper are assumed tobe constant in both time and space. In future research,we should address the issue of retrieving values of kand g that vary both in space and time. In particular,we would like to focus on the values of g for differentstages of a tropical system and for different regions, andstudy their impact on the forecasts, especially on theprecipitation field. Moreover, for a parameter estimationto be properly specified, the parameter’s seasonal var-iation should also be taken into account. The issue ofthe period of validity of optimally estimated parametersand the frequency with which they should be refreshedin an operational model would necessitate further stud-ies. Another question that deserves further study is theissue of the feedbacks between the effects of variousoptimally identified parameters.

Acknowledgments. Special thanks go to Dr. T. N.Krishnamurti for his encouragement and insightful com-ments throughout this study. The first author also wishesto thank Drs. H. S. Bedi, X.-L. Zou, and W. Han fortheir stimulating discussions. Thanks are also due to Dr.W. Han for providing all the data used in this study andDr. H. S. Bedi for providing the anomaly correlationsubroutine. The second author would like to acknowl-edge personal communication with Dr. J. Sela (NCEP)who suggested using biharmonic horizontal diffusioncoefficients for parameter identification. Stimulatingdiscussions with Dr. Eugenia Kalnay on the topic ofparameter estimation are also acknowledged. The au-thors also wish to thank the two anonymous reviewersfor their useful and stimulating comments and sugges-tions.

The support provided by NSF Grant ATM-9413050managed by Dr. Pamela Stephens is gratefully acknowl-edged. This work is part of satisfying the requirementsof a Ph.D. thesis directed by Prof. I. M. Navon andsupported fully by the above grant. Additional supportis provided by Supercomputer Computations ResearchInstitute at The Florida State University, which is par-tially funded through Contract DE-FC0583ER250000.

The computer facilities are provided by the SpecialSCD Grant 35111089 in NCAR and National ScienceFoundation Grant 35111100. Additional computer sup-port was provided by Supercomputer Computations Re-search Institute at The Florida State University.

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