1. Introduction
Tropical cyclones (TCs) and their associated wind and water hazards result in the largest number of fatalities, and over 50% of the total estimated economic loss per year, of all U.S. weather and climate disasters since 1980 (Smith 2020), making their accurate prediction of clear importance. To the credit of significant research efforts, TC track forecasts have improved significantly over the past several decades (Cangialosi 2019). The prediction of TC track has improved largely due to improved analysis and prediction of the large-scale atmospheric conditions, resulting from the combined advancements in numerical weather prediction (NWP) models, increased observing capabilities, and advancements in data assimilation (DA) systems (e.g., Heming et al. 2019). In contrast, TC intensity forecasts have improved minimally over the past several decades (Cangialosi 2019). The prediction of TC intensity is inherently more challenging than TC track, because of rapidly growing errors in initial conditions within the TC inner-core region (e.g., Van Sang et al. 2008; Zhang and Sippel 2009; Hakim 2013; Brown and Hakim 2013; Emanuel and Zhang 2017), the strong influence of large-scale atmospheric and oceanic conditions (e.g., Emanuel et al. 2004; DeMaria et al. 2005; Zhang and Tao 2013; Tao and Zhang 2015; Emanuel and Zhang 2016; Nystrom et al. 2018), and errors in model physics (e.g., Bao et al. 2002; Davis et al. 2008; Judt et al. 2015; Zhang et al. 2018; Nystrom and Zhang 2019; Chen et al. 2019).
Previous efforts to improve NWP model physics and initial conditions (ICs) have often been done separately, by either attempting to 1) improve the model physics through comparisons of retrospective model simulations with observations or 2) improve the model ICs through advancements in DA methodologies and/or assimilation of new observations. Conducting these separate efforts aimed at improving model physics or model ICs in isolation has likely limited the effectiveness of both efforts toward improving TC prediction. Efforts to tune NWP model parameters in order to minimize forecast errors, via systematic retrospective model simulations, do not account for the potential errors in ICs and are also extremely time consuming given the large number of model parameters and the regular updates to NWP systems. Additionally, such model tuning efforts run the risk of overfitting model parameters and/or incorrectly accounting for other sources of model error. On the other side, systematic errors in model physics, if untreated, bias the first guess, or prior estimate, used during DA and often degrade the resulting analysis (e.g., Romine et al. 2013, 2014). The errors in model physics, if untreated, not only degrade the DA update, but can drag the forecast away from the true state of the system, and can result in an ensemble forecast being underdispersive, or overconfident. Methods routinely used to treat the ensemble underdispersion resulting from model error include stochastically perturbing the model physics tendencies (e.g., Shutts 2005; Judt 2014; Judt et al. 2015) or utilizing a multiphysics ensemble (e.g., Kieu et al. 2014; Melhauser et al. 2017). However, while such methods have helped to improve the accuracy and reliability of ensemble forecasts, they do not help to improve the underlying model physics directly.
One set of physical processes known to fundamentally influence TC intensity and structure includes the surface fluxes of enthalpy and momentum, which are partially governed by the surface-exchange coefficients of enthalpy (Ck) and momentum (Cd), respectively. The critical role of air–sea heat transfer in supplying energy to TCs has been known for at least 70 years (Riehl 1950; Malkus and Riehl 1960). Furthermore, in one of the earliest numerical simulations of a TC, Ooyama (1969) demonstrated that the maximum wind speed of a simulated TC increases with increasing Ck and decreasing Cd. Since those early studies, numerous studies have demonstrated that the intensity in numerical simulations of idealized and real TCs is sensitive to the model representation of the surface-exchange coefficient (e.g., Rosenthal 1971; Emanuel 1986; Rotunno and Emanuel 1987; Bryan and Rotunno 2009a,b; Montgomery et al. 2010; Emanuel and Rotunno 2011; Emanuel 2012; Bryan 2012; Smith et al. 2014; Green and Zhang 2014; Torn 2016; Peng et al. 2018; Nystrom and Zhang 2019; Nystrom et al. 2020a,b).
While the basic influences of Ck and Cd on TC intensity and structure are well known, the values of Ck and Cd remain highly uncertain, especially at high wind speeds (e.g., Donelan et al. 2004; Bell et al. 2012; Takagaki et al. 2016; Hsu et al. 2017; Donelan 2018; Komori et al. 2018). At wind speeds less than ~30 m s–1, observations from various field campaigns have demonstrated with good confidence that Cd increases approximately linearly with wind speed (e.g., Large and Pond 1981; Black et al. 2007; Bell et al. 2012). However, estimates of Ck and Cd for wind speeds greater than ~30 m s–1 are few and highly uncertain. Accurate estimates of Ck and Cd at high wind speeds have been a longstanding challenge due to the dangers and difficulties in obtaining observations at the air–sea interface under high-wind-speed conditions in nature and in the laboratory with spatially limited tanks. Additionally, under high-wind-speed conditions, defining the air–sea interface is not trivial, as breaking waves and sea-spray complicate the problem. With these challenges in mind, the current leading hypothesis is that Cd saturates for wind speeds greater than ~33 m s–1 and is either constant with wind speed thereafter or decreases with wind speed as the boundary layer flow becomes decoupled from the ocean waves (e.g., Donelan et al. 2004; Troitskaya et al. 2012; Chen and Yu 2016; Donelan 2018). Overall, it is estimated that the current uncertainties in Ck and Cd for wind speeds greater than 33 m s–1 may be as large as 40% (Bell et al. 2012).
In a recent study examining the practical predictability of Hurricane Patricia (2015), a very intense TC in the eastern Pacific, Nystrom and Zhang (2019) suggested that the intensity predictability was limited by current uncertainties in model representation of the surface exchange coefficients, highlighting a need to reduce their current uncertainties in order to improve the prediction of intense TCs. However, reducing current uncertainties in Ck and Cd will require either development of new observation platforms capable of observing the air–sea interface at high wind speeds or the development of new techniques that can make use of existing observations that may be directly or indirectly related to processes at the air–sea interface, the latter of which will be explored in this study.
One method developed and demonstrated to successfully reduce uncertainty in model parameters is parameter estimation (e.g., Navon 1997; Ruiz et al. 2013a). In parameter estimation with a DA system, optimal model parameters are estimated, analogous to the atmospheric state, using available observations. Effective parameter estimation methodologies have been previously developed for both variational (e.g., Navon 1997; Gong et al. 1998; Zhu and Navon 1999; Bocquet 2012) and ensemble-based (e.g., Anderson 2001; Hacker and Snyder 2005; Aksoy et al. 2006a,b; Tong and Xue 2008; Hu et al. 2010) DA systems. One challenge in parameter estimation using a variational DA system is that the required adjoint model must include the sensitivity to the model parameters to be estimated. In contrast, parameter estimation with an ensemble-based DA system relies solely on the ensemble-estimated covariance between model parameters and state variables—which is obtained from a short-term ensemble forecast—to update and reduce the prior uncertainty estimate of the unknown parameters. Parameter estimation using an ensemble-based DA system has been demonstrated in previous studies to be a promising technique capable of accurately constraining model parameters related to the boundary layer (Aksoy et al. 2006a,b; Hu et al. 2010), microphysics (Tong and Xue 2008; Jung et al. 2010), and cumulus parameterization schemes (Schirber et al. 2013; Kotsuki et al. 2020). Successful parameter estimation has also been shown to improve the subsequent state estimate and improve the accuracy of forecasts.
In order for parameter estimation to be effective, the parameters to be estimated must be identifiable (Navon 1997), meaning they have a detectible influence on the model state. Parameters are deemed to be identifiable and suitable for parameter estimation in the context of an ensemble-based DA system if there are strong correlations between the parameters and observable state variables that are detectible in the presence of uncertainty in other parameter values and in observations (Nielsen-Gammon et al. 2010). The potential for simultaneous state and parameter estimation (SSPE) related to Ck and Cd has been previously suggested by Green and Zhang (2014, hereafter GZ14), in which parameters controlling the behavior of Ck and Cd were found to be generally suitable for parameter estimation. Using a multiparameter ensemble forecast of hurricane Katrina (2005), GZ14 demonstrated that model parameters controlling the representation of Ck and Cd have a significant, and identifiable, influence on the TC intensity point metrics (e.g., maximum 10-m wind speed and minimum central pressure) and the storm structure.
The primary objective of the current study is to examine the sensitivity of simulated TC intensity and structure to Ck and Cd, and demonstrate the potential to use an ensemble-based DA system to constrain parameters controlling the wind speed-dependent behavior of Ck and Cd. The potential for parameter estimation related to Ck and Cd will be investigated in this study using a series of observing system simulation experiments (OSSEs). Using a series of OSSEs, as opposed to real data, allows us to determine if the parameter estimation is successfully converging toward the true parameter values—something that will not be possible with real data. This study presents the first-known attempt to systematically test SSPE related to Ck and Cd. This study also presents the first-known attempts at using SSPE methodologies to improve TC prediction and the first to explore the potential applications of all-sky satellite DA for the purposes of parameter estimation.
The remainder of this manuscript is organized as follows: section 2 will describe the basic modeling and parameter estimation methodologies used in this study, section 3 will discuss the sensitivity of simulated TC intensity and structure to the parameters to be updated and their suitability for parameter estimation, section 4 will describe the results from the SSPE experiments, and section 5 will provide a brief summary, discussion on potential limitations, and future opportunities.
2. Methodology
a. Overview of parameter estimation and the methodology used in this study
Ensemble-based parameter estimation combines differences between the model state and observations (innovation) and the normalized covariance between the parameters and the model state resulting from the ensemble forecasts (Kalman gain) to update the parameter values. As a result, one consequence is that parameter estimation may account for other error sources in the system through the parameters being updated, which may or may not be desirable. In other words, parameter estimation will update the parameters as long as there is a difference between the model state and observations, and the parameters are correlated with the model, regardless of the reason for the differences between the model state and observations.
The parameters to be updated are appended to the state vector as an initially 2D field and horizontal localization is performed, analogous with the state variables, as in Aksoy et al. (2006a) and Kotsuki et al. (2020). A horizontal radius of influence of 300 km for the synthetic hurricane position and intensity (HPI; Chen and Snyder 2007) observation and 30 and 200 km are used for brightness temperature observations thinned every 12 and 18 km, respectively, as in Zhang et al. (2019). Following the assimilation of all observations and the covariance relaxation, the 2D EnKF analysis field of the parameter values is averaged within 100 km of the TC center position in order to obtain a single parameter value to be used during the forecast step. The primary reason we do not append a single global parameter value to the augmented state vector is that we are focused on estimating the model representation of the surface-exchange coefficients related to TCs and we do not wish for observations far away from the TC inner-core region to potentially erroneously update the parameter values.
b. Model formulation of the surface-exchange coefficients and the model parameters
Model representations of (a)–(c) Cd, (d)–(f) Ck, and (g)–(i) Ck/Cd as a function of 10 m wind speed and (a),(d),(g) α; (b),(e),(h) Vc; and (c),(f),(i) β. The CNTL Cd and Ck representation is shown in black in all panels. Blue and red curves denote α or β values less than and greater than CNTL, respectively. Increasing Vc values, up until CNTL, are depicted with increasingly dark red curves.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
c. Numerical model and OSSE configurations
All model simulations in this study use WRF V3.9.1 (Skamarock et al. 2008) and three two-way nested domains with horizontal grid spacing of 27 (378 × 243), 9 (297 × 297), and 3 km (297 × 297), respectively, and 42 vertical levels. With the exception of the surface exchange coefficients, all physics options are identical to the PSU WRF EnKF real-time hurricane analysis and prediction system (Weng and Zhang 2016). Namely, all simulations use the WSM6 microphysics (Hong and Lim 2006), YSU planetary boundary layer scheme (Hong et al. 2006), RRTM longwave radiation (Mlawer et al. 1997), and Dudhia shortwave radiation (Dudhia 1989). The ICs for the single and multiparameter ensemble forecasts discussed in section 3 and the “Truth” runs for the parameter estimation experiments in section 4 are from the 1800 UTC 21 October EnKF analysis mean of Nystrom and Zhang (2019) which assimilated all available conventional and airborne observations at an hourly frequency.
Synthetic HPI and GOES-13 channel-3 (6.5 μm water vapor channel) radiance observations are generated from the “Truth” simulations for the OSSEs, as in Zhang et al. (2016), by using the Community Radiative Transfer Model (CRTM) for the generation of the brightness temperature observations. Observation errors are assumed to be 3 hPa for the HPI observations and 3 K for the simulated GOES-13 radiances, both as in Zhang et al. (2016). The simulated all-sky GOES-13 observations are assimilated by invoking the adaptive observation error inflation (AOEI) methodology developed in Minamide and Zhang (2017) and the adaptive background error inflation (ABEI) method developed in Minamide and Zhang (2018).
OSSEs are conducted both with and without IC uncertainties in section 4. In the OSSEs without IC uncertainty, the ICs are identical to the “Truth” simulation at the beginning of the spinup period and vary subsequently as determined by the unknown parameters. The primary purpose of these experiments is to demonstrate that the OSSEs are capable of improving the estimates of the unknown parameters under the simplest scenario when the ensemble variability is driven entirely by the parameter uncertainty. This is an important first step as assimilation of the primary observation source in this study, the GOES-13 radiances, involves a nonlinear forward operator, the CRTM, and therefore may limit the effectiveness of parameter estimation. These experiments also provide an upper bound on the potential for parameter estimation related to Cd and Ck for this case. In more realistic OSSEs with IC uncertainty, the ensemble is initialized from the aforementioned EnKF analysis mean at 1800 UTC 21 October, and random ensemble perturbations are generated through sampling of the climatological background covariance matrix, which is calculated using the National Meteorological Center (NMC) method and 24- and 48-h GFS forecast differences (WRFDA CV option 3; Barker et al. 2004). The ensemble parameter values for the parameters to be estimated are initially generated by randomly sampling within the same range as the multiparameter experiment above, and other parameter/s are set to their “True” value. In all OSSEs a 6-h spinup is conducted prior to beginning SSPE at 0000 UTC 22 October. The cycling EnKF is continued until 1800 UTC 23 October, after which the simulated storm begins to rapidly weaken. Multiple “True” values for each parameter are tested in order to investigate the ability to converge toward the correct parameter value, regardless of where it falls in the parameter space.
3. Sensitivity of tropical cyclone intensity and structure to surface-exchange coefficient parameters
a. Single parameter sensitivity
To better understand the model sensitivity to the chosen parameters (α, Vc, β), a set of single-parameter ensemble forecasts is first conducted in which each parameter is varied independently. The single-parameter ensembles are conducted using the following values: α = 0.5, 0.75, 1.0, 1.5, 2.0; Vc = 30, 41, 52, 63, 74 m s−1; and β = 0.5, 0.75, 1.0, 1.5, 2.0. The values of the parameters not varied are set to those of the CNTL Cd and Ck representations. The single parameter forecasts are conducted with no IC uncertainty and all forecasts are run until 1200 UTC 24 October, which is after the simulated landfall time. The primary purpose of these single parameter ensemble forecasts is to understand how each of the parameters independently influence the intensity and structure of the simulated TC; section 3b will address the multiparameter sensitivity to these parameters.
Overall, Figs. 2 and 3 demonstrate that α, Vc, and β each have differing time-varying influences on the maximum V10 and Pmin as well as the radial TC structure. First, as α is increased, relative to CNTL, the maximum V10 is decreased at all times (Fig. 2a), as expected when Ck/Cd is decreased [Fig. 1 and Eq. (1)]. However, Pmin is generally similar to, or slightly less than, CNTL when α is increased up until ~48 h, after which the CNTL Pmin is less than when α is increased (Fig. 2d). The similar, or slightly lower, Pmin, relative to CNTL, over the first 48 h when α is increased is likely due to the greater boundary layer inflow (Vr), which helps to spin up the TC secondary circulation more quickly, initially at the expense of the primary circulation, and decrease the radius of maximum wind (RMW; dashed line in Figs. 3a,d). In contrast, when α is decreased, relative to CNTL, Pmin is greater than CNTL at all times (Fig. 2d) and the maximum V10 is increased for the first ~24 h, but is less than CNTL thereafter (Fig. 2a). This time-varying sensitivity of V10, relative to CNTL, is the result of reduced Cd, and therefore surface friction, which appears to allow the pointwise domain maximum in V10 to remain initially greater (Fig. 2a). Furthermore, as α is decreased, relative to CNTL, the magnitude of Vr within the boundary layer is initially reduced and the RMW increased (Figs. 3a,d). The sensitivity of the TC radial structure to α appears to modify the pressure–wind relationship, consistent with GZ14. Within the α ensemble, the peak intensity (greatest V10 and lowest Pmin) is found in the CNTL simulation, not with the largest Ck/Cd. We speculate this is because the intensification rate increases with α (Cd), consistent with Peng et al. (2018), and the simulations with smallest α values do not have enough time to reach their maximum intensity before the simulated TC begins to weaken as a result of land interactions and increased vertical wind shear. The environmental vertical wind shear increases at ~30 h from near 0 m s–1 to almost 10 m s–1 by 48 h (not shown). In addition, while the vertical wind shear increases similarly for all ensemble members, the influences of the vertical wind shear on the simulated TCs may vary because of the different simulated TC structures.
Simulated (a)–(c) maximum 10 m wind speed and (d)–(f) minimum central pressure for single-parameter (a),(d) α; (b),(e) Vc; and (c),(f) β ensembles. The CNTL simulation is shown in black in all panels and warmer or cooler colors depict parameter values greater than or less than the CNTL parameter values, respectively.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
Azimuthally averaged radial profiles of (a)–(c) Vt, (d)–(f) Vr, and (g)–(i) surface latent heat flux (LH) for single-parameter (a),(d),(g) α; (b),(e),(h) Vc; and (c),(f),(i) β ensembles. The CNTL simulation is shown in black in all panels. The dashed and solid curves correspond to 12 and 54 h, respectively.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
While α is found to quickly influence the intensity and structure, sensitivity to Vc is limited until ~24 h (Figs. 2b,e and 3b,e,h), consistent with when V10 begins to exceed 40 m s–1 and Cd and Ck differences begin (Fig. 1). Additionally, and in agreement with the α ensemble, the intensification rate beyond 24 h increases with decreasing Vc (increasing Cd), for at least a brief period, and the peak V10 is found to increase with Vc, in agreement with Eq. (1). The Pmin is found to be fairly insensitive to Vc at all times, suggesting that the pressure–wind relationship is also sensitive to Vc. The sensitivity of the radial structure of the near surface winds and the surface latent heat flux to Vc is found to be generally consistent with the α ensemble. More specifically, the maximum Vt is found with the largest Vc value (Fig. 3b), consistent with the smallest Cd and greatest Ck/Cd, and the greatest Vr is found with the smallest Vc value (Fig. 3e), consistent with the largest Cd.
The final parameter, β, has the largest influence among the three parameters on V10 and Pmin, which are found to increase and decrease with β, respectively, at all times (Figs. 2c,f). Furthermore, Vt is found to steadily increase with β, consistent with the increase in Ck/Cd, and the RMW appears rather insensitive to β (Fig. 3c). Recall that β acts to modify Ck only, as opposed to α and Vc which primarily modify Cd but also cause smaller changes in Ck which partially offset (Fig. 1). Therefore, the greater sensitivity to β, in comparison with α and Vc, is consistent with changes in β resulting in the largest change in Ck/Cd. In addition to the intensity increasing with β, the intensification rate also increases with β, in agreement with that predicted by the analytical model proposed by Emanuel (2012), and is likely related to the greater surface latent heat flux with increasing β (Fig. 3i). Last, while the pressure–wind relationship was found to be sensitive to α and Vc, it is found here to be insensitive to β (not shown), suggesting that the pressure–wind relationship is more sensitive to Cd than Ck, in agreement with GZ14.
In summary, all three parameters are found to influence the simulated TC maximum intensity, intensification rate, and TC structure. However, the time-varying sensitivity of α and Vc may make their identifiability more challenging when all three parameters are varied, as will be examined next. In addition, the sensitivity to Vc may be particularly challenging to detect in this specific case as the sensitivity is limited to a short period of time when V10 is greater than ~40 m s–1.
b. Multiple parameter sensitivity of simulated TC intensity
A multiparameter ensemble forecast is also examined in order to further determine the parameters which are most identifiable and suitable for parameter estimation. The multiparameter ensemble parameter values are generated by randomly sampling the following range for each parameter: α = 0.5–2.0; Vc = 30.0–74.0 m s−1; β = 0.5–2.0.
A 60-member ensemble forecast with random values of α, Vc and β results in large uncertainty in simulated V10 and Pmin, with the maximum V10 ranging from 50 to 110 m s–1 and Pmin from 960 to 840 hPa (Fig. 4), suggesting strong sensitivity to the model representation of Cd and Ck. Over the first ~18 h, moderate negative correlations (greater than 0.5) are found between the maximum simulated V10 and α, suggesting increased V10 with smaller α (Figs. 4a and 5a). Almost immediately, and through the remainder of the forecast, strong negative correlations (less than −0.8) between β and Pmin are found, suggesting that Pmin is strongly influenced by β (Figs. 4f and 5a). Additionally, beyond ~18 h strong positive correlations between β and V10 are also found (Figs. 4c and 5a). The temporal delay between the near-immediate strong correlation between Pmin and β, and the delayed correlation between V10 and β, suggest that β may first act to modify Pmin, with the surface wind field then adjusting to the pressure field, as will be further discussed. Last, moderate positive correlations (near 0.5) between V10 and Vc are only found for a limited time period near peak intensity (Figs. 4b and 5a), suggesting the time period when Vc is identifiable and is thereby suitable for parameter estimation may be temporally limited in this case. During this same time period Vc is also negatively correlated with Pmin (Figs. 4e and 5a), but is less correlated than with V10.
Simulated (a)–(c) maximum 10 m wind speed and (d)–(f) minimum central pressure for multiple-parameter ensemble forecast colored by (a),(d) α; (b),(e) Vc; and (c),(f) β.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
Ensemble correlations between (blue) α, (green) Vc, or (red) β and (a) maximum 10 m wind speed (solid) or minimum central pressure (dashed) and (b) the average brightness temperature within 100 km (solid) or 50 km (dashed) from the TC center.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
c. Correlations between surface-exchange parameters and potential observations
To examine how identifiable the three parameters are in the context of existing TC observations, azimuthal mean cross sections of the TC wind field and total hydrometeor mixing ratio are examined from the multiparameter ensemble simulations at 12 and 48 h. Observations of the TC wind field can be obtained by dropsonde and airborne radar observations (e.g., Gamache et al. 1995; Weng and Zhang 2016) and the total hydrometeor mixing ratio (qtot) serves as a proxy for the location of clouds, which can be regularly observed via geostationary and polar orbiting satellites (e.g., Zhang et al. 2016; Schmit et al. 2017; Zhang et al. 2019; Kim et al. 2020). The correlation of the three parameters with simulated GOES-13 channel-3 observations is also shown and related back to the model state variables. We note that, at the current time, airborne observations of the TC wind field may be too temporally limited for parameter estimation. However, we still examine the correlations of the parameters with the TC wind field in this study in order to highlight their influences on the TC structure and in hopes of more frequent TC wind field observations in the future.
At 12 h, Vt and Vr are strongly correlated with α near the eyewall and within the boundary layer (Figs. 6a,d). More specifically, the strong negative correlations between Vr and α near the surface suggest that increasing α acts to increase the magnitude of the boundary layer inflow (Fig. 6d). In addition, the dipole pattern across the RMW observed in the correlations between Vt or qtot and α suggests that increasing α acts to decrease the RMW (Figs. 6a,g), consistent with the single parameter ensemble (Fig. 3a). At 12 h, β is strongly correlated with Vt throughout the troposphere and with Vr within the outflow region (Figs. 6c,f). We speculate that the stronger correlations between Vr and β found in the outflow region, in comparison with those within the boundary layer, may be caused by the increased qtot (Figs. 6i) and the increased diabatic heating within the eyewall with increased β (not shown). Additionally, the strong positive correlations between Vr and β above the level of the maximum Vr in the ensemble mean (~15 km) suggest that the outflow, and the associated cloud tops, may be shifted up in the vertical (Figs. 6f). At this time Vc is not well correlated with Vt, Vr, or qtot (Figs. 6b,e,h).
Ensemble correlations between (a),(d),(g) α; (b),(e),(h) Vc; or (c),(f),(i) β and azimuthally averaged (a)–(c) Vt, (d)–(f) Vr, and (g)–(i) qtot at 12 h. The ensemble-mean Vt, Vr, and qtot are contoured in black every 10 m s−1, 2 m s−1, and 0.002 kg kg−1, respectively.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
At 48 h, α and β are increasingly well correlated with the model state and wide spread weaker, but physically consistent, correlations between Vc and the model state are now visible (Fig. 7). Overall, the correlations between α or β and the model state are stronger but generally consistent with those at 12 h. As at 12 h, a dipole pattern in the correlation between α and Vt is found across the RMW at 48 h, suggesting strong sensitivity of the RMW to α (Fig. 7a), and the boundary layer inflow is found to increase with α (Fig. 7d). Additionally, by 48 h, the overall strength of the primary and secondary circulations are both found to increase with increasing β (Fig. 7c,f). As the simulated TCs have now reached the high wind speed regime (V10 > 40 m s−1), the TC structure is also now weakly correlated with Vc by 48 h (average correlation within the eyewall is ~0.25; Figs. 7b,e,h). More specifically, positive correlations within the eyewall between Vt and Vc suggest that the intensity of the storm increases with Vc (Figs. 7b), consistent with the increase in Ck/Cd with increased Vc (Fig. 1). Furthermore, weak positive correlations within the TC outflow region suggest that Vr increases with Vc, and the outflow appears to be shifted up in the vertical as well (average correlation within the outflow is ~0.23; Figs. 7e,h). The dynamically consistent correlations between α, Vc, or β and the TC azimuthal mean wind structure suggest the potential to use airborne wind observations to constrain the aforementioned parameters, although the substantially weaker correlations with Vc makes this parameter less suitable than α and β for this case. However, one additional challenge in using airborne observations for parameter estimation is that they are currently temporally limited, as they are only available when an appropriately equipped aircraft is within the TC.
To overcome this potential hurdle, regarding the need for regularly available observations to constrain the parameter values, our attention is shifted toward geostationary infrared satellite observations (e.g., those from GOES-16/-17), which are available continuously at high spatial and temporal resolution (Schmit et al. 2017). By 12 h, moderate negative correlations are found within ~50 km between α and simulated GOES-13 channel-3 brightness temperature (BT) observations, suggesting colder cloud tops with increased α (Fig. 8a). Additionally, β is also negatively correlated with BT over a large region within ~100 km from the TC center, suggesting colder cloud tops with increased β (Fig. 8c), consistent with the outflow being found at higher levels (Fig. 6f). By 48 h, all three parameters are well correlated with BT (Figs. 8e–f). As found at 12 h, α and β are negatively correlated with BT within the inner-core region, again suggesting colder cloud tops with increased α or β. A small region of positive correlations near the TC center between β and BT is associated with the clear sky within the eye, thereby suggesting a stronger warm core with increased β. In addition, by 48 h, negative correlations are also found between Vc and BT (Fig. 8e), consistent with the higher outflow with increased Vc suggested by Fig. 7e.
Ensemble correlations between (a),(d) α; (b),(e) Vc; or (c),(f) β and simulated GOES-13 channel-3 brightness temperature (BT) at (a)–(c) 12 and (d)–(f) 48 h. The ensemble-mean sea level pressure is contoured in black every 10 hPa.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
A time series of the correlation between the average BT within 50 or 100 km of the TC center and each of the parameters suggests that the parameters are each correlated with simulated BT for extended time periods and therefore have the potential to be estimated by assimilating GOES-13 BTs (Fig. 5b). In particular, β is found to quickly develop strong correlations with BT within 100 or 50 km from the TC center, which persist throughout the forecast period. In contrast, α quickly develops persistent negative correlations with BT within 50 km from the TC center, and correlations which weaken with time within 100 km from the TC center. Vc is best negatively correlated with BT later in the forecast period when V10 is greater. However, Vc is still found to have the weakest correlations among the three parameters. Based on the encouraging correlations found between BT and α, Vc, or β, and their regular availability, we next test the ability to constrain the model representation of Cd and Ck using GOES-13 simulated BTs.
4. Simultaneous state and parameter estimation experiments
a. Parameter estimation with a single unknown parameter
To test the potential to correctly constrain the model representations of Cd and Ck through SSPE, single parameter OSSEs are first examined in which simulated GOES-13 BTs and HPI observations are assimilated every 6 h. The focus is placed on estimating α and β, since the greatest sensitivity of the state was found to these parameters within section 3 and the time period for which Vc is most identifiable is limited in this case. A cycling frequency of 6 h is initially chosen to allow adequate time for the ensemble correlations with the parameters to spin up, and a covariance relaxation coefficient of 0.5 was used for the parameter relaxation.
First tested is the ability of SSPE to accurately constrain β, which was found to most strongly influence the TC intensity and should therefore be the easiest to estimate. When β is the only unknown parameter, the other parameters are known perfectly, SSPE quickly corrects the estimate of β to the “True” value, regardless of if the “True” β value is less than or greater than the initial ensemble estimate (Figs. 9a and b). In these experiments, the RMSE of β following the eighth and final DA cycle is ~0.03 for the OSSE with β initially underestimated and ~0.01 for the OSSE with β initially overestimated. Additionally, when estimating β, it does not appear that much, if any, parameter uncertainty remains following the final DA cycle. The ensemble spread of β following the eighth DA cycle is ~0.02 for the OSSE with β initially underestimated and ~0.01 for the OSSE with β initially overestimated. Furthermore, the spread–error relationship is still excellent (ratio of parameter spread to the parameter error is ~1), suggesting that covariance relaxation is sufficient for maintaining adequate parameter spread in this case.
Posterior EnKF updated ensemble estimation of (a) and (b) two cases with different “Truth” values and β as the only unknown parameter, (c) and (d) two cases with different “Truth” values and α as the only unknown parameter, and (e) and (f) a case with α and β as simultaneous unknown parameters. The “Truth” value for each parameter and experiment is shown in the black horizontal line.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
In addition to the promise toward estimating β through SSPE, OSSEs with a single unknown parameter (α) also demonstrate encouraging results for either a “True” α value less than or greater than the initial ensemble estimate (Figs. 9c and d). Similar to the single parameter β OSSEs, SSPE is able to quickly correct the α estimate toward the “True” value and reduce the RMSE of α. The RMSE of α following the eighth DA cycle is ~0.07 for the OSSE with α initially overestimated (Fig. 9c) and ~0.2 for the OSSE with α initially underestimated (Fig. 9d). While the initial α RMSE is reduced in both case—90% when α is initially overestimated and 74% when α is initially underestimated—the time-varying performance of SSPE appears different depending on whether the “True” α is less than or greater than CNTL (α = 1.0).
In the case where the “True” α value is much smaller than CNTL (initially overestimated by the ensemble), α is quickly corrected to the “True” value and following cycles further reduce the ensemble spread, the ensemble spread is ~0.02 following the eighth and final DA cycle (Fig. 9c). In contrast, when α is greater than CNTL (initially underestimated by the ensemble), the SSPE adjusts α more slowly toward the “True” value and does not converge to the precise “True” α value (Fig. 9d). In this case, a bias in the background ensemble at later cycles causes a bias in the estimation of α through the ensemble estimated covariance–the truth in state space lies outside of the background ensemble. This result demonstrates that bias in the state can corrupt the estimation of a parameter. As a result, the ensemble appears to overconstrain α, as the RMSE diverges from the parameter spread following the fourth cycle and the final ensemble spread of α (~0.03) is ~7 times smaller than the RMSE after the final DA cycle (Fig. 9d). In addition, it is possible that the optimal parameter ensemble spread is sensitive to the parameter (e.g., α or β) and, for α, the specific “Truth” value. Nevertheless, the initial estimates of both α and β were improved during the single parameter estimation experiments.
b. Parameter estimation with multiple unknown parameters
As a next step, the methodologies are extended to examine the potential for simultaneous estimation of α and β, first with all ensemble uncertainty driven only by the uncertainty in α and β (i.e., no IC uncertainty, as in section 4a). When uncertainty to α and β both exist simultaneously, SSPE still quickly adjusts both α and β toward their “True” values within a few cycles and thereafter reduces the ensemble spread while maintaining a nearly constant parameter value (Figs. 9e and f). More quantitatively, the initial RMSE of α and β are reduced by 84% and 95%, respectively. Similarly, the initial ensemble spread of α and β are reduced by 92% and 94%, respectively. The results of this multiparameter SSPE OSSE suggest that if IC uncertainty can be constrained enough, such that 1) α and β are identifiable (i.e., the sensitivities to α and β are detectable within the ensemble estimated covariance) and 2) the ensemble state is not biased relative to observations because of other model errors or state errors, then accurate estimates of α and β can be obtained through assimilation of HPI and all-sky IR observations.
A final set of OSSEs are conducted that includes climatological IC uncertainty and uncertainty in α and β. In addition, we explore the sensitivity of the parameter estimation to the initial values of α and β by conducting five experiments that differ only by the initial parameter values. The initial mean α and β values were randomly drawn and vary from 0.85 to 1.36 and from 0.94 to 1.45, respectively. The initial standard deviation in α and β vary from 0.34 to 0.50 and from 0.38 to 0.45, respectively. These OSSEs use a covariance relaxation coefficient of 0.75 for the parameters and 0.5 for the model state. Additional SSPE experiments with a larger covariance relaxation for the model state variables reduce the state uncertainty less quickly and likewise adjust the unknown parameters more slowly (not shown). We suspect this is because α and β are less identifiable if too much uncertainty exists in the model state variables. In other words, it is likely that it takes a longer time frame for the state to build sensitivity to the parameters when significant initial state uncertainty exists. In these experiments we update the model state at an hourly frequency, using a 1-h ensemble forecast from the previous analysis, and the parameters at a 6-hourly frequency, using a 6-h ensemble forecast from the previous parameter analysis. The state is updated hourly in hopes of constraining the rapidly growing state errors associated with a rapidly intensifying TC, while the parameters are updated every 6 h to avoid overfitting the parameters and also to allow adequate time for the state to build sensitivity to the parameters. In addition, we also relocate the TC location in each ensemble member only when performing the parameter update in order to remove the influence of position spread which acts to substantially weaken the ensemble estimated correlations with the parameters.
When more realistic IC uncertainty is considered, the parameter estimation reduced the average initial ensemble RMSE of α and β by 44% and 63%, respectively (Fig. 10). However, only β is consistently adjusted (increased) across each experiment. In other words, it appears that the estimation of β is more successful for this case and this set of observations. One key difference, in comparison with the OSSE without IC uncertainty (Figs. 9e and f), is that the RMSE and ensemble spread of α and β are reduced more slowly when more realistic IC is considered (Fig. 10). The much smaller parameter updates found in the first few cycles suggest that the IC uncertainty may need to first be reduced during the first few EnKF updates before the parameters become identifiable. Although the ensemble-mean estimates of α and β do move closer to their “True” values in all cases, the resulting average parameter spread of α and β is ~5 times larger than in the experiment without IC uncertainty (Figs. 9e,f and Fig. 10). In addition, there is clear sensitivity of the final estimated parameter values to their initial estimates, in agreement with Aksoy et al. (2006a). Across our experiments, the reduction of RMSE in the estimates of α and β varies from 29% to 72% and from 44% to 75%, respectively. It is possible that additional EnKF update cycles may allow further reduction in the parameter spread and also reduce the differences between the experiments with different initial parameter values, recall that the current TC begins to weaken rapidly beyond the eighth cycle and therefore limits extending the parameter estimation.
Summary of five experiments for simultaneous estimation of (a) α and (b) β with initial-condition uncertainty and with five different initial parameter values. The mean parameter values for the EnKF analysis mean for each experiment are denoted by the thin gray lines and black numerals. The black horizontal lines denote the “Truth” value for α and β for all experiments. The red line denotes the mean parameter value for all experiments and the blue-shaded region depicts the standard deviation for each parameter averaged across all experiments.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
Another challenge is that the differences in the final parameter estimates are related to errors in the model state, which depend on the nonlinear interactions between the initial parameter errors and the initial state errors. As a result of the fast-growing errors during the rapid intensification of TCs, as is the case for the TC in this study, it is likely that errors in the model state will be caused by both the initial state uncertainty and the parameters. This error in the forecasted model state has the potential to bias the resulting parameter estimates. We leave a more detailed examination of the influence of the initial parameter error and uncertainty on the final parameter estimates to future work and note that regardless of the aforementioned challenges, the average initial (NoPE) model representations of Cd, Ck, and Ck/Cd are adjusted closer to the “True” representation by the final parameter updates (PE) in the OSSEs in which α and β are both unknown (Fig. 11).
(top) Cd, (middle) Ck, and (bottom) Ck/Cd representation colored by DA cycle number averaged across the five multiple unknown parameter estimation experiments shown in Fig. 10. The color-shaded region corresponding to each curve depicts the average ensemble spread (±1σ) of Cd, Ck, and Ck/Cd. The “Truth” representation of Cd, Ck, and Ck/Cd is shown in black.
Citation: Monthly Weather Review 149, 7; 10.1175/MWR-D-20-0259.1
5. Summary and discussion
This study systematically examines the potential to estimate the model representations of Cd and Ck through ensemble data assimilation. To determine the identifiability of the three parameters (α, Vc, and β parameter), and highlight the sensitivity of the simulated TC to each parameter, a series of single parameter ensembles and a multiparameter ensemble are examined. The first parameter, α, primarily acts to modify Cd and strongly influences the overall intensity of the simulated TC—the maximum V10 increases with decreasing α (Cd) and the intensification rate increases with α (Cd)–consistent with Eq. (1) in terms of the maximum V10 and Peng et al. (2018) in terms of the intensification rate. In addition, α is found to strongly influence the size of the TC, specifically the RMW, which generally decreases with increased α, and the IR brightness temperature in cloudy regions, which decreases within the TC inner core with increased α. The second parameter (Vc), which primarily modifies Cd for wind speeds greater than ~40 m s–1, also influences the maximum intensity—the maximum V10 increases with increasing Vc (decreasing Cd). The Vc is also found to be negatively correlated with IR brightness temperature in cloudy regions once the simulated TC intensity exceeds ~40 m s–1. The last parameter, β, which modifies the amplitude of Ck, has the strongest influence on the overall intensity of the simulated TC–the maximum V10 and the intensification rate increase with β (Ck)–consistent with Eq. (1) in terms of the maximum V10 and Emanuel (2012) in terms of the intensification rate. The β is also found to strongly influence the radial TC structure and the IR brightness temperature, which decreases in cloudy regions with increased β. Overall, dynamically consistent correlations between the TC intensity or structure with each of the parameters was found, suggesting that each parameter is identifiable and therefore suitable for parameter estimation. In particular, observations of the TC wind field, such as those available from airborne radar, and all-sky IR brightness temperatures are believed to be observations well suited to successfully estimate the model parameters through parameter estimation.
The potential to successfully constrain the model representation of Cd and Ck using all-sky IR brightness temperature and HPI observations is then explored through a series of OSSEs with one or multiple unknown parameters and with or without IC uncertainty. In all parameter estimation experiments the initial parameter estimates are improved through cycling data assimilation with an EnKF. However, while the results presented here are encouraging, there is clear sensitivity to the initial parameter values and to errors in the model state, especially during the rapid intensification stage. More specifically, the challenges of constraining the fast-growing state errors for a rapidly intensifying TC are a key obstacle that must be overcome in order to avoid state errors from projecting onto the parameter estimates. Overall, these results strongly suggest the need for more frequent and direct observations of the model state and/or more advanced data assimilation methodologies in order to better constrain the model state during rapid intensification. The sensitivity of the parameter estimation to the magnitude of IC uncertainty also highlights the importance of regularly available TC observations, such as all-sky radiances, in order for the chosen parameters to be identifiable and in hopes of limiting any bias to the model state. In addition to the use of all-sky IR brightness temperature observations, the results presented here suggest the potential strong positive impact of airborne observations of the TC wind field on parameter estimation, which have not yet been tested.
Given the sensitivities to the initial parameter values and the state errors, the results presented here suggest it may be necessary to average parameter estimation results over multiple well observed TCs, in order to average out any biases resulting from a single case. In addition, one limitation of the current study is that all OSSEs, with the exception of the parameters to be estimated, use a perfect model. In reality, model errors related to other aspects of model physics, such as the boundary layer parameterization (e.g., Smith et al. 2014; Zhang et al. 2018), will also influence TC prediction and potentially limit the success of parameter estimation. The current OSSEs also all use fixed SST, which may result in errors in the estimation of Cd and Ck for real TCs, as the influences of ocean feedbacks may be accounted for through errors in Cd and Ck. Furthermore, more rigorous treatment of the parameter spread, to determine how much the uncertainty of a given model parameter can be reduced during cycling DA, should be developed, such as that of Ruiz et al. (2013b).
In addition to these remaining challenges, several unexplored opportunities also exist. The strong influences of the surface-exchange coefficients on the simulated ocean–atmosphere interactions (e.g., Nystrom et al. 2020a), in conjunction with the recent advancements in strongly coupled ensemble DA (e.g., Li and Toumi 2018; Chen and Zhang 2019) highlight an additional opportunity to use ocean and atmosphere observations to estimate the model representation of Cd and Ck. Furthermore, recent advancements in unmanned airborne observation platforms (e.g., Cione et al. 2020) present an additional source of high spatial and temporal resolution boundary layer observations that can likely help to constrain the model representation of Cd and Ck. Last, a TC with a longer life cycle and a more prolonged steady-state period would allow for additional opportunities to update the unknown parameters without the additional challenges of a very rapidly evolving state. Nevertheless, the results of this study highlight a potential opportunity to improve TC prediction via systematic state and parameter estimation using an EnKF and existing observations.
Acknowledgments
This work was supported by NASA Grant 17-EARTH17F-184 under the NASA Earth and Space Science Fellowship Program, NOAA Grant NA18NWS4680054, University of Michigan Subcontract 3004628721, and ONR Grant N00014-18-1-2517. Computing was conducted at the Texas Advanced Computing Center (TACC). Due to the large size of the dataset, all data from this study are stored at TACC and can be made freely available upon request. The authors are also grateful for helpful discussions with Yue Ying, Man-Yau Chan, and the entire ADAPT group at Penn State during the course of this research as well as three anonymous reviewers and the editor (Jeffrey Anderson) whose comments were helpful in revising this manuscript.
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