Quantifying and reducing the uncertainty of model parameterizations using observations is evaluated for tropical cyclone (TC) intensity prediction. This is accomplished using a nonlinear inverse modeling technique that produces a joint probability density function (PDF) for a set of parameters. The dependence of estimated parameter values and associated uncertainty on two types of observable quantities is analyzed using an axisymmetric hurricane model. When the observation is only the maximum tangential wind speed, the joint PDF of parameter estimates has large variance and is multimodal. When the full kinematic field within the inner core of the TC is used for the observations, however, the joint parameter estimates are well constrained. These results suggest that model parameterizations may not be optimized using the maximum wind speed. Instead, the optimization should be based on observations of the TC structure to improve the intensity forecasts.
It is widely recognized that the skill of tropical cyclone (TC) track forecasts has improved considerably during the past decade, whereas the skill of intensity forecasts has not improved as much (e.g., Rappaport et al. 2009). Although multiple factors have been hypothesized to explain the lack of measurable improvement in TC in- tensity prediction, one of the main challenges is the uncertainty in parameterized representations of physi- cal processes in numerical weather prediction models used for TC intensity (e.g., Rogers et al. 2006).
Recent studies have demonstrated large sensitivity of numerical TC intensity forecasts to the choice of model parameterizations (e.g., Braun and Tao 2000; Zhu and Zhang 2006; Li and Pu 2008; Pattnaik et al. 2011; Green and Zhang 2013). For example, Li and Pu (2008) found significant differences in the intensity and structure forecasts of a rapidly deepening hurricane due to differ- ences in the storm structure resulting from various cloud microphysical and planetary boundary layer parame- terizations. Similarly, Green and Zhang (2013) showed variations in TC intensity forecasts due to the use of different surface flux parameterizations in their numeri- cal simulations. Although these and other studies have demonstrated a sensitivity of TC intensity to the choice of parameterization, it is difficult to ascertain which param- eterizations (or parameter values within them) would be optimal for improving the accuracy of TC intensity forecasts. Among other factors, the difficulty arises from the lack of a formal measure of optimality for repre- senting the uncertainties in the model parameterizations.
In this study, the estimation of parameter uncertainty is investigated using an optimal estimation approach. As an introductory experiment, we employ a simplified two-dimensional model and perform the estimation of parameter uncertainty using a nonlinear inverse model- ing method. This paper proceeds as follows. The optimal estimation approach along with the experimental setup for the introductory study is briefly explained in the next section. Results obtained from using two different set of observable variables are discussed in section 3, followed by the conclusions of this study in section 4.
a. Inverse estimation approach
This study employs the general stochastic inverse problem theory as introduced by Mosegaard and Tarantola (2002). Previous studies (Vukicevic and Posselt 2008; Vukicevic et al. 2010; Coddington et al. 2012) demonstrated the utility of this theoretical formulation in diagnostic analyses of nonlinear estimation problems with atmospheric models containing a small number of control parameters. The formulation allows for explicit computation of a joint posterior probability density function (PDF) of the parameters, given a nonlinear model and observations with their associated stochastic uncertainties. The posterior PDF is computed by con- junction of a numerically determined model-based PDF, in a joint space of parameter and observation values, with an observation-based PDF. The model-based PDF in- cludes explicitly computed transfer functions between parameters and observation quantities, as well as a PDF representation of model solution with uncertainty in the observation space. This uncertainty reflects the presence of modeling errors that are not associated with the con- trol parameters. The transfer function explicitly accounts for the variability due to these parameters. The method is diagnostic because testing the impact of different ob- servable quantities and error characteristics of the model on the posterior PDF estimate of parameter values does not involve model integration other than the computation of the transfer functions. The numerical algorithm for computing the posterior PDF of parameter estimates based on this formulation is described in detail in Vukicevic and Posselt (2008).
In the current study the method is adapted for the estimation of parameter uncertainty using an axisym- metric hurricane model. The estimation is performed for two parameters within the parameterization of un- resolved processes that are known to have significant impact on intensity prediction (described in the next section). Our goal is to evaluate the likelihood of joint parameter values simulating the observable quantities that are relevant to hurricane intensity, given the un- certainty in both the observations and model.
b. Experimental setup
The model employed in this study is the Axisymmetric Simplified Pseudoadiabatic Entropy Conserving Hurri- cane (ASPECH) model (Tang and Emanuel 2012). The variable-resolution grid stretching technique was used, with grid spacings of 4-8 and 0.5-0.8 km, in the radial and vertical directions, respectively, within a domain of 1000km 3 24 km. The model was initialized with an idealized vortex with a maximum surface tangential wind speed of 20 m s21 and an environmental profile with 70% relative humidity and 298C sea surface temperature. To ensure a robust representation of idealized hurricane evolution for the estimation of parameter uncertainty experiments, the model was first spun up for 48 h. The model state at 48 h was then used to initialize the sim- ulations for computing the transfer functions and the reference ''true'' observations.
The enthalpy exchange coefficient Ck and inflated latent heat of vaporization Lyo parameters were chosen1 for the estimation problem. The first parameter is known to influence TC intensity (e.g., Emanuel 1995), yet its reference value in the ASPECH model reflects a consen- sus of previous studies that have estimated this quantity (Tang and Emanuel 2012, and references therein). The second parameter was chosen because the model uses an inflated version of the latent heat of vaporization to compensate for the neglect of liquid-water entropy, as suggested by Bryan (2008). To compute the transfer functions, 30 discrete values for each parameter within the prescribed ranges were used (Table 1). Consequently, a total of 302 5 900 simulations were produced, one for each possible combination of Ck and Lyo. By doing this, we have assumed that all other parameters in the model are perfect, except for Ck and Lyo. The reference true observations for the estimation were derived from the simulation with the parameter-value pair, as in the stan- dard model configuration (see Table 1). It is important to note that the reference true observations are not real observations, but rather the model solution using the default values of Ck and Lyo. The default pair of values was not used for the transfer-function ensemble.
The standard sensitivity result (i.e., time series of an intensity metric, which in this case is the maximum tan- gential wind speed) exhibits large sensitivity to varying values of Ck and Lyo (Fig. 1). Even though all cases start with the same tropical storm intensity, some cases strengthen substantially and others weaken throughout the forecast period. The variance of the ensemble, as well as the difference between the ensemble mean and the reference, increases with time. The deviation of the ensemble mean from the reference is large, suggesting that the response to parameter perturbations is nonlinear. In addition, many different combinations of parameters produce similar intensity. These properties imply that identifying a single optimal pair of parameters using the sensitivity results is unfeasible. Instead, an optimal subset of values should be determined. This is readily achievable by application of the optimal estimation method.
3. Optimal estimation results
The optimal estimates of joined values of Ck and Lyo are analyzed for two types of observations of the axi- symmetric hurricane winds: the maximum tangential wind speed (ymax) and the total wind field (i.e., tangen- tial, radial, and vertical wind) within the inner core of the simulated TC.
a. Maximum wind speed observation
We define the observation as y max at a certain time. The transfer function for such observations is ymax as a function of paired parameter values for each selected time. The transfer function for 24- and 48-h observation times is displayed in Fig. 2. The values shown are the same as in Fig. 1 for the corresponding times, but now shown in the parameter space. Several properties of interest are evident: 1) small values of Lyo correspond to low intensity irrespective of Ck; 2) for moderate to high values of Lyo (above 2.75 3 106 Jkg21) the intensity tends to increase with increasing Lyo, but at a variable rate depending on the value of Ck; and 3) the change of intensity is not linear with respect to either Lyo or Ck. Overall, consistent with the sensitivity result in Fig. 1, the transfer functions indicate that the impact of the param- eters is mutually dependent and nonlinear. These prop- erties were evident during other forecast times as well.
Using the transfer functions and estimates of obser- vation and model errors, as outlined in the previous section, the joint posterior PDF of parameters is com- puted for each observation time. The errors associated with the observation of ymax are assumed Gaussian with a standard deviation of s o 5 5ms21. This value repre- sents the expected uncertainty associated with the Na- tional Hurricane Center's operational estimates of TC intensity (
The joint posterior PDFs were computed for every 6-hourly forecast, but only the PDFs corresponding to the 24- and 48-h observation times will be discussed here as other times showed similar characteristics. As ex- pected, Fig. 3 shows that these PDFs are similar in shape to the transfer functions within the range of parameter values that is determined by convolution of the obser- vation and model PDFs in the observation space. This is consistent with the findings of Vukicevic and Posselt (2008). The posterior PDFs for both observation times exhibit multiple maxima and large variance for both parameters. The absolute maximum of each PDF is in the neighborhood of the reference true solution; how- ever, because of the large variance and multimodality of the PDF its likelihood is small. The results suggest that the optimal values of the parameters cannot be uniquely estimated when using the values of ymax as the observable quantity. Consistent with the sensitivity studies, they point to the need to use an ensemble of model parame- terizations for TC intensity prediction. These results also indicate that the ensemble should be based on the opti- mal estimation in order to include realistic ranges and mutually dependent parameter perturbations between different processes.
The impact of including multiple observation times in the posterior estimate is also evaluated. This impact is assessed in two ways: 1) by convolving the posterior PDFs for the individual observation times and 2) by computing the average of these PDFs. The convolution method is equivalent to performing the sequential esti- mation with cycling in time, whereas the average is equivalent to compositing the estimates that correspond to different periods of the TC vortex evolution. For both methods we used an observation frequency of 6 h, be- ginning with the 6-h forecast and ending with the 48-h forecast, inclusive. The convolution resulted in mutually independent estimates for the two parameters (Fig. 3c). The PDF corresponding to Lyo is primarily bimodal, as in the individual observation times, but it exhibits less variance. The Ck PDF, however, is uniformly distrib- uted, thus showing the high uncertainty associated with this parameter. On the other hand, the composite of the PDFs resulted in a weakly correlated joint parameter estimate with slightly better constrained maximum likelihood than for the individual observation times (Fig. 3d). With respect to the optimization of the ensem- ble of parameterizations, these results indicate that the ensemble estimates would be sensitive to the approach for combining information from different observation periods and cases.
b. Kinematic field observations
The possibility of estimating the parameters with re- spect to the observations of TC vortex wind field instead of just the maximum wind speed is explored next. The observations are defined as the radial u,tangentialy,and vertical w wind within the inner 150-km radius, extending from the surface up to an 18-km height. Similar to the experiments for ymax, the error variance for these obser- vations is prescribed based on expected errors in practice (e.g., airborne measurements) and the variances for the model error in the equivalent fields were computed using ASPECH simulations with perturbed initial conditions.
Unlike for the maximum intensity experiment, the posterior solution shows perfect constraint with the ki- nematic field observations (i.e., the posterior PDF consists of a two-dimensional delta function for all observation times; Figs. 4a,b). The singular value estimates are close to, but not exactly equal to, the reference true values. This could be attributed to the low resolution of the para- meter bins that was used to compute the transfer func- tions (i.e., only 30 bins were used for each parameter). Regardless of this limitation, the result suggests that pa- rameterizations could be effectively optimized using the observations of kinematic structure of the TC vortex, which would, in turn, improve the simulations with re- spect to the maximum tangential wind speed.
Similar to the analysis for y max, the cumulative impact of observations from different times was evaluated for the kinematic field observations. To account for the lack of resolution of the parameter bins, the posterior delta-function PDFs were first smoothed by adding a two-dimensional uncorrelated Gaussian error to the posterior parameter estimate at each time indepen- dently. The convolution and averaging were then ap- plied as in the previous section to compute the cumulative posterior PDFs. As in the y max experiment, the convo- lution resulted in uncorrelated estimates for the two parameters but with a well-constrained maximum likelihood that is slightly biased relative to the reference true value (Fig. 4c). The compositing of the posterior PDFs produced correlated joint parameter estimates with a well-defined and more accurate maximum likelihood solution (Fig. 4d). It is worth noticing that the biases (the deviation from the reference solution) for the ymax and kinematic field experiments are very similar for the pa- rameter Lyo, and that for both experiments the biases are reduced when using the composited PDFs.
Additional experiments were carried out using only vertical profiles of u, y , and w at a specific distance from the storm center [e.g., at the radius of maximum wind (RMW), at 2 RMW, etc.] as observations. The PDFs from those experiments are surprisingly similar to those shown in Fig. 4, and also show a perfect constraint of the parameter values at all observation times (not shown).
The potential for quantifying and reducing uncertainty in parameterizations using optimal estimation with observations is evaluated for an idealized case of tropical cyclone intensity prediction. Using the nonlinear inverse estimation method with the Axisymmetric Simplified Pseudoadiabatic Entropy Conserving Hurricane model, it is shown that two parameters affecting the intensity forecast could not be effectively optimized using only the maximum tangential wind speed observations. In con- trast, the joint parameter estimates are well constrained when the observations of the inner vortex core circulation are used. The results suggest that full kinematic field observations, such as Doppler winds measurements, are beneficial for optimizing the parameterizations with respect to the intensity prediction problem. It is also demonstrated that optimal estimation with observations would lead to mutually dependent estimates of the parameters. Such estimates would benefit the design of optimal parameter-based ensemble forecast perturba- tions. Although it would be difficult to compute a non- linear inverse solution for full-physics, three-dimensional models because of the large dimension, more practical methods such as the ensemble
Acknowledgments. This research was conducted un- der the auspices of the Significant Opportunities in At- mospheric Research and Science (SOARS) Program, which is managed by the
1 Sensitivity tests done prior to employing the methodology showed a sufficient sensitivity of TC intensity to these two pa- rameters, although the choice is somewhat arbitrary.
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