1. Introduction

A numerical weather forecast model involves a number of parameters that are determined empirically. Some of these parameters, which are very common in physical parameterization schemes, contain information about the flow’s properties and characteristics or originate from the simplification of the physical parameterizations. Other parameters are introduced due to numerical stability considerations. The values of the parameters directly or indirectly impact upon the performance of the model. Generally, the values of some parameters are determined by trial and error; that is, there is no objective criterion to choose “optimal” values of the parameters. In this study, we will focus on the optimal estimation of several parameters that are known to impact the performance of a numerical weather prediction (NWP) model. Namely, we wish to identify the optimal biharmonic horizontal diffusion coefficient κ, the ratio γ of the transfer coefficient of moisture to the transfer coefficient of sensible heat, and the Asselin filter coefficient ϵ using adjoint parameter estimation and study how such optimized parameters impact on the ensuing model forecasts.

The parameter estimation procedure is aimed at choosing an optimal parameter in an admissible parameter set, so that the model solutions corresponding to this parameter fit the observations as a minimization problem of an output-error criterion (Chavent 1974). Research on adjoint parameter estimation has been carried out using relatively simple models in the last 20 years, first on topics such as aquifer behavior under transient and steady-state conditions in the field of hydrology (Carrera and Neumann 1986a–c), bottom drag coefficient identification in a tidal channel (Panchang and O’Brien 1990), wind stress coefficient estimation along with the estimation of the oceanic eddy viscosity profile (Yu and O’Brien 1991), and nudging coefficient estimation in the National Meteorological Center [now known as the National Centers for Environmental Prediction (NCEP)] adiabatic version of the spectral model (Zou et al. 1992a), and others. The issue of the adjoint parameter estimation was also addressed by LeDimet and Navon (1988). Wergen (1992) recovered both the initial conditions and a set of parameters from observations using a 1D shallow-water equation model. His results showed that even with noisy observations, the parameters were recovered to an acceptable degree of accuracy. For a detailed survey of the state of the art of parameter estimation in meteorology and oceanography, see Navon (1998). However, very few numerical experiments have been conducted using a sophisticated full-physics model, and very little attention has been paid to the impact of the parameter estimation on the the model forecast although the performances of the physical parameterization and/or numerical schemes are also a main factor in determining the model forecast skill. Recently, more and more effort has been focused on improving the initial condition (Thépaut and Courtier 1991; Navon et al. 1992), and its important impact on reducing the forecast error has been demonstrated. What is the effect of the parameter estimation on the model compared to that of the initial condition? In this study, we intend to explore this question and expect that the performances of the physical parameterization and/or numerical schemes can be improved by tuning the parameters involved.

This paper is organized as follows: the model used and the characteristics of the parameters to be optimally identified are described in section 2. The methodology of adjoint parameter estimation is presented in section 3. The detailed parameter estimation procedure, along with the minimization algorithm and the optimal parameter values, are provided in section 4. Section 5 provides the description of the forecast experiments and results using both the optimal initial condition and the optimally identified parameter values. Section 6 describes the impact of the optimal parameters alone on ensuing 24-h forecasts. The model’s “memory” (in terms of ensuing forecast period results) of the impact of either the optimal initial condition or the optimal parameter values is discussed in section 7. Finally, summary and future research are presented in section 8.

2. The model and characteristics of the parameters

The full-physics Florida State University Global Spectral Model (FSU GSM) (Krishnamurti and Dignon 1988) and its adjoint model are employed to recover both the optimal initial condition and optimal parameter values from the observations. The FSU GSM has been successfully applied for numerical weather forecasts, especially for the Tropics. The model is a multilevel (12 vertical levels) primitive equation model with σ coordinate. All variables are expanded horizontally in a truncated series of spherical harmonic functions (at resolution T42) and the transform technique is applied, which makes it possible and efficient to perform the horizontal operators and to calculate the physical processes in real space. The finite difference schemes in the vertical and semi-implicit scheme for time integration are employed. The full physical packages include orography, planetary boundary layer, dry adjustment, large-scale precipitation, moist-convection, horizontal diffusion, and radiation processes.

The effort to establish a 4D variational (4D Var) data assimilation system for the FSU GSM was initiated in 1993. The adjoint code of the dry-adiabatic version of the FSU GSM was developed by Wang (1993). Later, Tsuyuki (1996) incorporated moisture variables, the smoothed parameterization of moist processes, horizontal diffusion, and a simplified surface friction. We continued their efforts by incorporating radiation and planetary boundary layer (including vertical diffusion) processes into the 4D Var data assimilation system due to the important roles they play in simulating various large-scale and mesoscale phenomena, especially in tropical weather systems, thus obtaining the full-physics adjoint model (Zhu and Navon 1997). This effort renders the adjoint model consistent with the nonlinear forecast model. If the two models (the forward model and its adjoint) are inconsistent, this may imply a negative impact on the process of adjoint optimal parameter estimation.

The horizontal diffusion term is usually incorporated in a numerical weather prediction model to parameterize the effects of motions on the unresolved scales and to inhibit spectral blocking, that is, the growth of small scales in the dynamic model variables due to the accumulation of energy at high wavenumbers. It is also employed to eliminate the aliasing effect (Phillips 1959;Hamming 1973). The presence of any dissipation, physical or computational, can attenuate the amplitude of the short wavelengths very significantly; in this case, the errors introduced by aliasing are minimal. MacVean’s study (1983) showed that, without dissipation, integrations exhibited grossly physically unrealistic features after about several days even with very high mesh resolution, indicating the crucial role played by the dissipation in nonlinear baroclinic development. Considerable effort has been made by various groups in tuning the dissipation parameterizations in their general circulation or forecast models. For instance, Phillips (1956), Lilly (1965), Leith (1965), and Richard and German (1965), among others, applied the eddy diffusion terms in their numerical models; Navon (1969) included the lateral viscosity in a two-level general circulation model and carried out a 62-day integration, comparing its impact to that of using the Matsuno dissipative scheme. Kanamitsu et al. (1983, 1989) and Gordon and Stern (1982) employed a biharmonic horizontal diffusion in their experiments. Presently, the biharmonic horizontal diffusion is used worldwide in NWP models due to its better scale selectivity. Its limitation is the lack of a sound physical foundation. Some other horizontal diffusion schemes are also used, such as the scheme developed by Leith (1971) based on turbulence theory, which is now implemented in the NCEP spectral model (Kanamitsu et al. 1991).

The biharmonic horizontal diffusion κ∇⁴ is used for vorticity, divergence, dewpoint depression, and virtual temperature in the FSU GSM to selectively control small-scale noise without affecting large scales. The e-folding diffusive decay time at total wavenumber n is given by

View Expanded

where a is the radius of the earth. The physical significance of the diffusion coefficient κ is not directly intuitive. Its effect may be understood in terms of the timescale τ at the smallest spatial scale resolved. The model dissipation should remove energy from the end of the spectrum at a rate sufficient to prevent a spurious accumulation of energy there, while not affecting the medium and large scales. In the Geophysical Fluid Dynamics Laboratory spectral model, the coefficients of the eddy diffusion for ∇⁴ are determined by trial and error, using the quality of the medium-range 500-mb geopotential height forecast as a criterion (Gorden and Stern 1982). The value they used is 1 × 10¹⁶ m² s⁻¹ or 2.5 × 10¹⁶ m² s⁻¹ for T30, that is, a value of τ of about 53 or 21 h [based on Eq. (1)]. The value used in FSU Global Spectral Model T42 is 6 × 10¹⁵ m² s⁻¹, that is, yielding a value of τ corresponding to about 23 h.

The third parameter we consider is the ratio γ of the transfer coefficient of moisture to the transfer coefficient of sensible heat, which arises from the parameterization of the surface fluxes in the boundary layer. Accurate heat and moisture flux parameterizations are very important since the surface heat and moisture fluxes exert a strong influence on the surface energy budget and precipitation rate. Similarity theory has successfully provided a framework for the description of the atmosphere surface layer, the lowest 50 m or so of the boundary layer where the Coriolis force can be ignored and the fluxes can be assumed to be constant with height. The flux profiles and other properties of the flow are reasonably parameterized via this theory. However, certain empirical parameters or constants evolving from the theory need to be determined by experiments, for instance, the von Kármán constant and constants associated with the stability dependence of the flux-profile formulations. In this study, however, our goal is not to determine the parameter values from physical soundness. We will instead aim to obtain an optimal value of γ for this particular model via a variational parameter identification approach in order to improve the model forecast skill. Dyer (1967) found that, over the range of the ratio of height z to the Monin–Obukhov length L (0.02 < |z/L| < 0.6) both the transfer coefficient of sensible heat ϕ_H and the transfer coefficient of moisture ϕ_Q varied approximately as |z/L|^−1/3; for |z/L| > 0.2, ϕ_H is found to vary as |z/L|^−1/2, but insufficient data limited the value of the corresponding analysis for ϕ_Q. Since reasonable agreement is found between the ϕ_H and ϕ_Q data, many numerical weather prediction models adopt a simple relationship of the type ϕ_Q = ϕ_H; that is, γ = 1.0. Recently, Smith et al. (1996) reviewed the 25 years of progress on air–sea fluxes. Theory and observations indicate that the ratio should depend on the surface characteristics. However, since the ratio is considered as a constant in the original FSU GSM and also numerical experience with the model shows that the model tends to produce larger precipitation, we will still take the parameter as a constant and set its upper and lower bounds to be 1.8 and 0.3, respectively, in this study. Its optimal value will be estimated via numerical fitting of the model to the observations. The specification of the γ value directly impacts the moisture flux. We expect to improve the performance of its corresponding physical process by tuning this parameter.

3. The methodology of parameter estimation

Given constrained parameters whose values vary between certain bounds, for instance, when the parameter α_i satisfies α_i ∈ [a_i, b_i], where a and b denote the lower and upper bounds, respectively, the cost function for parameter estimation may assume the following form:

View Expanded

where λ is the penalty coefficient vector, W is the diagonal weighting matrix defined as in Navon et al. (1992), 〈 〉 denotes spatial integration, X represents the state variable vector, X^obs the observation vector, and t₀ and t_R denote the assimilation window. The second term consists of a penalty function. The value of λ is determined such that the penalty term is of the same order of magnitude as the other terms in the cost function, and g(α) is defined as

View Expanded

where g_i(α_i) is a function only of the violated constraints. The first derivative of this function is

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Another type of penalty effective in transforming a constrained optimization problem into an unconstrained one is the barrier method, which imposes a penalty for reaching the bound of an inequality constraint. Typically, a logarithmic barrier function is of the form

View Expanded

where μ is the barrier coefficient and h is the constraint function. The barrier methods are strictly feasible methods; that is, the iterates lie in the interior of the feasible region and create a “barrier” to keep iterates away from boundaries of the feasible region (Nash and Sofer 1996).

There are two different purposes for the inclusion of the second term in Eqs. (2) or (5). One is to ensure that the retrieved parameter lies within the prescribed bounds. A penalty term of the form in Eq. (2) or a logarithm of the box constraints as in Eq. (5) is efficient to achieve this. The other purpose is to increase the convexity of the cost function by adding a positive value λ to the Hessian matrix, thereby increasing its positive definiteness.

Suppose that the forward model is perfect and is given in the form

View Expanded

Its corresponding tangent linear model is defined as

View Expanded

Then the adjoint model can be expressed in the form

View Expanded

where P represents the adjoint variables. The gradients of the cost function with respect to the initial condition and the parameter α are, respectively,

View Expanded

The adjoint model is of the same form as that where only the initial condition is considered as the control variable. Hence, the problem of parameter estimation via the adjoint method does not result in an additional computational effort when the number of parameters to be estimated is small. We may expect that the parameter estimation process will provide us with both optimally determined parameters and initial condition simultaneously. The gradient of the cost function with respect to both the initial condition and parameters is written as

∇J∇_X0J,∇_αJ^T(11)

4. Parameter estimation procedure and results

In this study, the FSU Global Spectral Model and its full-physics adjoint model are employed to optimally identify the aforementioned three parameters separately along with initial condition retrieval. An arbitrary initialized European Centre for Medium-Range Weather Forecasts (ECMWF) analysis dataset is used in our study. A 6-h assimilation window is used, that is, from 0000 to 0600 UTC 3 September 1996. The initialized ECMWF analysis data at the beginning and end of the assimilation window are taken as the observations during the 6-h assimilation window period.

The parameter estimation procedure is carried out as follows.

1. Take the 6-h forecast starting from the initialized analysis at 1800 UTC 2 September 1996 as the initial guess of the initial condition. Given a positive definite initial approximation to the inverse Hessian matrix H₀ (generally taken as the identity matrix I), we integrate the full-physics FSU GSM 6 h and then calculate the cost function using Eq. (2). The initial guess of α, the penalty coefficient value λ, and the upper and lower bounds where the parameters may vary are specified here.

2. Integrate the full-physics adjoint model of the FSU GSM backward in time with null initial condition to obtain the gradient of J with respect to the control variable Y = (X₀, α)^T,

   g₀gY∇_X0J,∇_αJ^T

   and the search direction

   d₀H₀g₀

   Generally, physical processes involve many on–off switches that may introduce discontinuities in the parameterization schemes. In the full-physics adjoint model of the FSU GSM, since we hope to keep the original parameterization characteristics unchanged as much as possible, only the discontinuities that most impact the tangent linear approximation and the convergence of the minimization algorithm are considered for removal by using smooth function and cubic spline interpolation (Zhu and Navon 1997).
3. For k = 0, 1, 2, . . . , minimize J(Y_k + β_kd_k) with respect to β ≥ 0 to obtain Y_k+1 as

   Y_k+1Y_kβ_kd_k

   where β_k is a positive scalar, the step size being obtained by a line search so as to satisfy a sufficient rate of decrease (see Gill et al. 1981).
4. Compute the new gradient and search direction, respectively,

   g_k+1∇JY_k+1d_k+1H_k+1g_k+1
5. Check whether the solution converges. If the convergence criterion

   g_k+1ϵY_k+1

   is satisfied, where ϵ′ is a user-supplied small number, then the algorithm terminates with Y_k+1 as the optimal solution; otherwise go back to step 3.

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

The minimization procedure is terminated after 60 iterations. Although the convergence rate is expected to be much slower than that of the adiabatic version, due to the high nonlinearity of the model including the full physical processes, a sufficient decrease in the cost function and its gradient for each experiment is achieved after 60 iterations. At the end of the minimization procedure, both the optimal initial condition and optimal value of the parameter are recovered.

The initial guess (estimated value) of the horizontal diffusion coefficient κ is taken to be 6 × 10¹⁵ as used routinely in the FSU GSM T42L12. The penalty parameter λ_κ for this parameter is taken to be 1.0 × 10⁻¹². The upper and lower bounds where κ varies are taken to be 3.0 × 10¹⁶, that is, five times the value used in the original forecast model, and 2.0 × 10¹⁵, respectively. The variations of the cost function and the gradient norm with respect to the number of iterations are presented in Fig. 2. We see that the cost function decreases from an initial value of 3643 to 575.3, that is, it decreases to about 15.8% of its original value, while the norm of the gradient value decreases to 23.6% of its original value. Figure 3 displays the evolution of the horizontal diffusion coefficient during the minimization procedure. This coefficient experiences a rapid increase until the 35th iteration, reaching its peak at the 44th iteration, then experiences a slight decrease. The optimal value obtained for the horizontal diffusion coefficient is 1.1124 × 10¹⁶. This value is almost twice the value used in the original forecast model. We also notice that the optimal parameter value is not very sensitive to the value of λ_κ as long as the penalty term is of the same order of magnitude as the other terms in the cost function.

The issue of the sensitivity of the optimal parameters values to the values of the penalty coefficients deserves some attention. Research work by Craven and Wahba (1979) and by Lewis and Grayson (1972) has shown strong dependence of the analysis results on the specification of the penalty coefficients.

In our case the optimal parameter values are not found to be very sensitive to the value of the penalty coefficient as long as the penalty parameter attains a certain threshold value.

To understand why this happens, one should consider the basics of the penalty technique as a method to solve a sequence of unconstrained optimization problems whose solution is usually infeasible to the original constrained problem. A penalty for violation of the constraints is incurred and as this penalty is increased, the iterates are forced toward the feasible region.

The experience of Zou et al. (1992b, 1993) in applying a penalty method for controlling gravity oscillations in variational data assimilation shows that the effect of the penalty parameter is ineffective until it attains a threshold value. Thereafter it is effective in constraining the minimizers toward the feasible region and filtering the gravity waves. Increasing the penalty parameters by an order of magnitude or more over the threshold value does not change this effect, but slows down the minimization due to the ill-conditioning of the related Hessian of the cost functional (see Nash and Sofer 1996; Gill et al. 1981; Fletcher 1987).

A higher sensitivity of minimization results to the value of the penalty parameter is typical for cases when, from the start, the iterates are rather far from the feasible region and reaching the effective threshold value requires several iterations.

In the present case we are from the starting point very near to the feasible region, hence the perceived lack of sensitivity of the optimal parameter values to the values of the penalty coefficients when these attain their threshold value. When we increase the penalty parameters further (i.e., by an order of magnitude or more), there is no perceived change except for a serious slowdown in the rate of convergence of the minimization due to the aforementioned ill-conditioning.

Similar variations of the cost function and the gradient norm with respect to the iteration number are obtained when the ratio γ and the Asselin filter coefficient ϵ are retrieved from the observations, respectively. The initial guess (estimated value) of the Asselin filter coefficient ϵ is set to be 0.05, which is the value used in the original forecast model. The upper and lower bounds for the variation of this parameter are taken as 0.1 and 0.0, respectively, and the penalty parameter λ_ϵ for this parameter is set to be 1.0 × 10⁸. The optimal ϵ value obtained is 0.0487, which is very close to the initial guess, which means that either the initial guess is a fairly good guess or the model is not very sensitive to the parameter. Further study with the initial guess away from 0.05 still yields the same result, which indicates that the initial guess is fairly good. The initial guess (estimated value) of the ratio γ of the transfer coefficient of moisture to the transfer coefficient of sensible heat is initially set to be 1.0, which is the value used in the original FSU GSM T42L12. The penalty parameter λ_γ for γ is set to be 2.0 × 10⁵, while the upper and lower bounds where γ may vary are taken to be 1.8 and 0.3, respectively. The optimal parameter estimation for γ yields a value of 0.4974, which is only half of the initial guess value, due to numerical fitting of the model to the observations. We should be aware that the optimal value retrieved here includes the possibility of the compensating for errors in the physical parameterization schemes of the forecast model. It does not represent the true physical value; it is only the optimal value for this particular model for improving the model forecast.

5. Forecast experiments using both the retrieved initial condition and parameters

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

A number of 24-h forecast experiments beginning from the 0000 UTC 3 September 1996 analysis are performed in order to examine the impact of each parameter separately as well as the combined impact of all three parameters on the ensuing forecast fields. The experiments are (a) experiment with the estimated parameters, that is, κ = 6.0 × 10¹⁵, ϵ = 0.05, and γ = 1.0; (b) experiment with the estimated ϵ and γ but the optimally retrieved κ value; (c) experiment with the estimated ϵ and κ but the optimally retrieved γ; and (d) experiment using optimally retrieved κ, ϵ, and γ simultaneously. Since the optimal value of ϵ is very close to its estimated value, the impact of this parameter is rather marginal.

The rms errors of the vorticity, divergence, dewpoint depression, and the virtual temperature fields at all vertical levels are calculated for the above experiments with respect to the analysis at 0000 UTC 4 September 1996. Figure 10 displays the differences of the rms errors between experiments b and a, c and a, d and a, which indicate the impacts of optimal values of κ, γ, and their combined impact, respectively. Negative values of the rms differences indicate that the rms errors are less than those of experiment a with the estimated parameter values; that is, the optimal parameter value has a positive impact on the forecast.

The results show that the optimal horizontal diffusion coefficient κ impacts mainly the vorticity and divergence fields, and also has a positive impact on the virtual temperature and dewpoint depression at upper and middle levels but a negative impact at lower levels. On the contrary, in experiment c when only the optimal value of γ is employed, the vorticity and divergence fields are only slightly improved, but the middle and lower levels of the virtual temperature and the lower levels of the dewpoint depression experience a large improvement. The effect of γ is mainly confined to the lower levels of the model. The best forecasts are obtained by experiment d, in which all the optimal parameters values of κ and ϵ as well as γ are used simultaneously. In this sense, experiment d combines all of the advantages of experiments b and c.

7. A study of the model’s “memory” of impact of optimal initial condition and identified parameter values

So far, we have studied either the combined impact of both the optimal initial conditions and the optimal parameter values or the impact of only the optimally identified parameter values on the ensuing forecast. In this section, we discuss the persistence (memory) of the combined impact of the above-mentioned three optimally identified parameter values as well as the optimal initial condition obtained by variational data assimilation on the ensuing forecast.

Three sets of experiments are carried out by integrating the model for 24, 48, and 72 h, respectively, from 0000 UTC 3 September 1996. The first set of experiments (C1, C2, and C3) are control experiments that are integrated from the initial guess of the initial condition (i.e., the 6-h forecast from 1800 UTC 2 September 1996) using the estimated parameter values. The second set of optimal parameter experiments (P1, P2, and P3) start from the initial guess of the initial condition using the optimally identified values of the above-mentioned three parameters. The third set of optimal initial condition experiments (I1, I2, and I3) are integrated from the variationally derived optimal initial condition using estimated parameter values (Table 2).

The rms errors of vorticity, divergence, virtual temperature, and dewpoint depression fields are calculated for each experiment, then the percentages of the differences of the rms errors between C1 and P1, C2 and P2, C3 and P3, C1 and I1, C2 and I2, and C3 and I3 are computed. The percentages of the decrease of rms errors due to the use of the three optimally identified parameter values are displayed in Fig. 11. The results show that all of the experiments using optimally identified parameter values exhibit smaller rms errors than those using estimated parameter values through all of the vertical levels except for the virtual temperature at the top vertical level of the model; that is, the combined impact of the three identified parameter values still persists for the 72-h forecast, and probably even further beyond. The largest improvement in the 24-h forecast occurs at the low levels of the dewpoint depression field where the rms error decreases by up to 17%. The overall forecast rms errors further decrease for the 48-h forecast. The impacts of the optimally identified parameter values on the vorticity field and on the middle levels of the divergence, virtual temperature fields as well as the lower levels of the dewpoint depression field decay for the 72-h forecast compared to the results obtained with optimal parameters for the corresponding 48-h forecast; however, the rms errors still decrease by up to 10% compared to the experiment where estimated parameters are used for the same 72-h forecast.

The percentages of the decrease in rms errors for vorticity, divergence, virtual temperature, and dewpoint depression fields by using optimal initial condition are presented in Fig. 12. A very clear trend is observed, namely, the impact of the optimal initial condition decays as the forecast time increases, especially the impact on the divergence field decays very rapidly. This is in agreement 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 identified parameter values exceeds that obtained due to the impact of the optimal initial condition in the ensuing 72-h forecast. Even for the 24-h forecast, the impact of the three optimal parameter values on the lower level of the dewpoint depression is larger than that of the optimal initial condition. The model tends to first “lose” the impact of the optimal initial condition in the ensuing forecast, while the impact of using optimal parameters on the above-mentioned forecast fields lingers even after 72 h.

Figures 13 and 14 show how the rms errors of vorticity and dewpoint depression evolve with forecast time, respectively, in which black bars are for the optimal parameter experiments (P1, P2, and P3) and gray bars are for the optimal initial condition experiments (I1, I2, and I3). The results indicate that the errors of the optimal initial condition experiment become larger than those of the optimal parameter experiment at 72-h forecast.

We would also like to point out that the mechanisms of the impacts resulting from optimal initial condition and from the optimal parameter values are quite different. The impact of the optimally identified parameter values is effective throughout the entire integration period via the corresponding physical parameterization or numerical schemes, while the impact of the optimal initial condition obtained via variational data assimilation is to enhance the model forecast skill by reducing the errors in the initial condition, which might lead to a poor forecast. Therefore, their impacts are quite different. Figure 15 displays the difference fields between experiments C1, P1, and the analysis for specific humidity at 850 mb, respectively, while the difference field of specific humidity at 850 mb between experiments C1 and P1 is displayed in Fig. 16. Comparing Figs. 15a and 15b, we notice that they have almost the same pattern, except that the positive differences between the forecast and the analysis become smaller while the negative differences between the forecast and the analysis become larger over most of the forecast area in experiment P1 when the optimally identified parameter values are used. That is to say, the use of the optimally identified parameter values tends to produce a smaller specific humidity forecast over most of the global domain, since the optimal ratio of the transfer coefficient of the moisture to the transfer coefficient of the sensible heat is only about half of its estimated value. Larger specific humidity forecasts in experiment P1 than those in experiment C1 are observed in only a few regions, such as central South America, which might have been caused by the impact of the optimal biharmonic horizontal diffusion coefficient or by interactions between the impacts of the three optimal parameter values. In this case, the improvement due to the use of the optimally identified parameter values is observed mainly over the overestimated areas with respect to the analysis (in experiment C1), which spread over the land. Figure 17 displays the difference fields of the specific humidity at 500 mb between experiments C1, I1, and the analysis, respectively. Experiment I1 in which the optimal initial condition is used, however, does not exhibit such features as mentioned above, rather the improvement is obtained both in underestimated and overestimated areas compared with the result of experiment C1, especially over areas of large errors.

It is also known that forecasts starting from the variationally derived optimal initial conditions are not as good as the forecasts starting from the latest available analysis (Pu et al. 1997). However, we compare the forecasts starting from 0600 UTC 3 September 1996 analysis (the latest available analysis in this study) using both the estimated values of the above-mentioned three parameters and the previously optimally identified parameter values, respectively. Similar improvements as shown in Fig. 11 are observed. The optimally identified values of the biharmonic horizontal diffusion coefficient and the ratio of the transfer coefficient of the moisture to the transfer coefficient of the sensible heat actually improve the performance of the corresponding physical parameterization and/or numerical schemes, so that their impacts on the model forecast persist under similar large-scale atmospheric environment.

Finally, we present the anomaly correlation results for the specific humidity forecasts at 850 mb for the control experiments, optimal parameter experiments, and optimal initial condition experiments as well as the experiments using both optimal initial conditions and optimally identified parameter values. These anomaly correlations are measured both for the tropical belt [defined here to be 40°S–40°N where physical initialization is usually carried out (Krishnamurti 1991)] and for the Northern Hemisphere (Fig. 18). The specific humidity field, which is mainly distributed in the lower troposphere, is very important in the tropical system. Comparing Figs. 18a and 18b, we see that the anomaly correlation for the Northern Hemisphere is consistently higher than that for the tropical belt for the entire 5-day integration period. The results of optimal initial condition experiments appear to be more skillful than the optimal parameter experiments for the first 24-h forecast in the tropical belt and for the first 48-h forecasts in Northern Hemisphere, respectively. However, later in the forecast (i.e., for the period 48–120 h), the optimal parameter experiments yield better skill results than the optimal initial condition experiments both in the tropical belt and Northern Hemisphere. The experiments using the optimal initial condition and optimally identified parameter values simultaneously have the highest anomaly correlations, where for the 120-h forecast the anomaly correlation increases from 65.1% in the control experiment to 70.7% for the tropical belt and from 72.5% in the control experiment to 77.8% for the Northern Hemisphere.

8. Summary and future research

In this study, we recovered both optimal initial conditions and optimal values of the biharmonic horizontal diffusion coefficient, the Asselin filter coefficient, and the ratio of the transfer coefficient of moisture to the transfer coefficient of sensible heat using the full-physics adjoint of the FSU Global Spectral Model. The results obtained are very encouraging. The fields at the end of the assimilation window starting from the retrieved optimal initial condition and the optimal parameter values successfully capture the main features of the analysis fields. Although the impact of optimal initial conditions dominates that of the optimal parameter values at early stages of the forecast, a positive impact due to each optimally estimated parameter value is observed. The ensuing 24-h forecast experiments in which only the optimal parameter values are used further indicate the positive impact of using optimal parameter values. The biharmonic horizontal diffusion coefficient improves vorticity and divergence fields as well as the upper levels of virtual temperature and of dewpoint depression fields, while the ratio of the transfer coefficient of moisture to the transfer coefficient of sensible heat has a large positive impact on the lower levels of the model, especially those of the virtual temperature and the dewpoint depression fields. By combining the three optimal parameter values, we obtain the best forecast results. Further studies of ensuing forecasts using the optimally identified parameter values and the optimal initial conditions, respectively, show that the model tends to first lose the impact of the optimal initial condition while the impact of optimally identified parameter values persists beyond 72 h. The optimally identified values of the biharmonic horizontal diffusion coefficient and the ratio of the transfer coefficient of the moisture to the transfer coefficient of the sensible heat improve the performance of the corresponding physical parameterization schemes. The experiments using the optimal initial condition and optimally identified parameter values simultaneously yield the best performance. However, we should be aware that these results are obtained for a single case study. Further studies should be conducted for different initial conditions in order to draw a 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 account the model forecast error; that is, the forecast model is regarded as a weak constraint, and its effect on the retrieved initial conditions and optimal parameter values should be examined.

The parameters studied in this paper are assumed to be constant in both time and space. In future research, we should address the issue of retrieving values of κ and γ that vary both in space and time. In particular, we would like to focus on the values of γ for different stages of a tropical system and for different regions, and study their impact on the forecasts, especially on the precipitation field. Moreover, for a parameter estimation to be properly specified, the parameter’s seasonal variation should also be taken into account. The issue of the period of validity of optimally estimated parameters and the frequency with which they should be refreshed in an operational model would necessitate further studies. Another question that deserves further study is the issue of the feedbacks between the effects of various optimally identified parameters.