1. Introduction
Due to their impact on severe weather, mesoscale convective systems (MCSs) and their forecasts have long been the focus of the operational centers and research communities (Peters and Schumacher 2014; Schumacher and Johnson 2005). Apart from model dynamics and physics (Feng et al. 2018; Lin et al. 2019; Lu and Wang 2020), data assimilation (DA) also plays a major role in the improvement of MCS forecasts. Data assimilation combines information from the observations and the background field to provide optimal estimates of the initial conditions for these short-range forecasts, which is mainly an initial-value problem (Bauer et al. 2015).
Conventional observations, such as those from surface stations and rawinsondes have been extensively used in the data assimilation (Benjamin et al. 2010). But because of the short lifetime span and fast evolving characteristics of the convections, the conventional observing networks are often too coarse in time and space to sample the structures of the convective systems. Remote sensing networks implemented in recent decades can fill the spatial and temporal gaps if the high-resolution data can be properly assimilated (Heiss et al. 1990; Jean-Noël 2003).
Among the remote sensing observations, precipitation from radars (Ban et al. 2017; Lopez 2011) and satellites (Hou et al. 2004) are particularly valuable for convective scale forecasts. The extensive coverage of precipitation provides the potential to constrain the high-resolution atmospheric analysis. Many previous research efforts have studied the impact of precipitation assimilation on weather forecasts. Multiple approaches have been developed to: Assimilate hydrometeor-affected fields like radiances (Bauer et al. 2006a,b, 2010; Geer et al. 2018) and reflectivity (Chen et al. 2021; Duda et al. 2019), assimilate rainfall information through the latent heating nudging (Benjamin et al. 2016; Jones and Macpherson 1997; Macpherson 2001; Mesinger et al. 2006; Weygandt and Benjamin 2007), or assimilate precipitation directly (Ban et al. 2017; Sun et al. 2020). Various schemes have been implemented to facilitate the assimilation, including variational (Lopez 2011; Sun et al. 2020) and ensemble methods (Lien et al. 2016). Variational methods are convenient to implement constraints, while the flow-dependent error covariances in the ensemble methods are more suitable to represent high variability related to the convective systems.
Wang and Zhang (2021) described a constrained data assimilation (CDA) algorithm based on Gridpoint Statistical Interpolation (GSI) three-dimensional ensemble variational (EnVar) method, which not only restores the balance in the analysis by enforcing mass and moisture conservations, but also assimilates precipitation through the moisture constraint. Regarding the precipitation assimilation, zero-rain issues have been reported when four-dimensional variational (4D-Var) method is used (Lopez 2011): when observation shows precipitation, but the background has no precipitation, there is no sensitivity of the method with respect to the observation, and no increment of any state variables corresponding to the observed precipitation (Bannister et al. 2020; Vobig et al. 2021). Since the CDA does not depend on the physical parameterizations, it circumvents the zero-rain issues. This paper is a follow-on of Wang and Zhang (2021). In this paper, the CDA performance is evaluated using more cases in a larger domain with a larger set of observations. The effects of the physical constraints are not only assessed using short-range forecasts when all available observations are assimilated, but also examined through the forecast skill loss in radial-wind denial experiments. In addition, we extend the Wang and Zhang (2021) algorithm by calculating the time tendencies in the constraints based on the analysis ensembles.
The paper is organized as follows. Section 2 gives descriptions of the selected cases, the constrained data assimilation algorithm, observations, and the forecast model. Evaluation of the analyses from the algorithm based on short-range forecasts is given in section 3. Section 4 assesses the impact of the physical constraints of the algorithm on the mitigation of forecast skill losses when radar winds are absent. Section 5 describes the impact of alternate calculations of the time tendencies in the constraints on the analysis. A summary and discussion are given in the last section.
2. Cases and methodology
a. Case descriptions
Three convective events during the Midlatitude Continental Convective Clouds Experiment (MC3E) field campaign of the Atmospheric Radiation Measurement (ARM) program, occurring on 20 May, 11 May, and 25 April 2011, are selected to assess the assimilation system (Fig. 1), which are referred to as case0520, case0511, and case0425, respectively. The analysis times for case0520, case0511, and case0425 are 0600 UTC 20 May, 1800 UTC 11 May, and 1800 UTC 25 April 2011, respectively. These cases have been described in Jensen et al. (2016). The case0520 and case0511 were squall lines that had developed in the warm zone ahead of cold fronts, while case0425 were mesoscale convective systems developed in the warm sector of a stationary front. For all the three cases, the southerly or southeasterly winds blowing from the Gulf of Mexico brought abundant moisture into the precipitating area. They are selected based on the heavy precipitation criteria to make the effect of the moisture constraint clearer.
The NEXRAD composite reflectivity at the analysis time (a) 0600 UTC 20 May, (b) 1800 UTC 11 May, and (c) 1800 UTC 25 Apr 2011. The red circle in (a)–(c) symbolizes the Southern Great Plains (SGP) Central Facility (CF) cite. The radar acronyms in (b) represent locations of the assimilated radar sites. The red triangles in (c) indicate the sounding arrays deployed during the Midlatitude Continental Convective Clouds Experiment (MC3E) campaign.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
b. Constrained data assimilation algorithm
Due to the inevitable uncertainties in the observation measurements, without proper adjustments, the observations do not satisfy the column-integrated conservation of mass, moisture, heat, momentum. The magnitude of spurious residuals could be comparable with other leading components (Zhang and Lin 1997). Wang and Zhang (2021) showed that the successive assimilation of conventional and radar radial wind observations increases the mass and moisture residuals in the analysis. These imbalances, either inherited from the observation or brought in during the assimilation procedure, motivated us to incorporate physical constraints in the data assimilation system by developing the CDA algorithm. The importance of physical conservation in the data assimilation has been demonstrated in some idealized case studies (Janjić et al. 2014; Ruckstuhl and Janjić 2018) and also in our previous work (Wang and Zhang 2021).
The CDA is built upon the GSI EnVar framework, which incorporates the ensemble error statistics through extended control variables (Wang 2010). The value of the flow-dependent error covariance has been shown in multiple studies, through comparing the performances of the EnKF/EnVar method with the three-dimensional variational (3D-Var) approach (Carley 2012; Gao et al. 2019; Kong et al. 2021; Tong et al. 2020; M. Zhang et al. 2011). Its effectiveness is even more prominent when radar data are assimilated, since the advantage from radar data assimilation persists longer when the ensemble instead of the static error covariance is used (Johnson et al. 2015).
For the calculation of the ensemble background error covariance, a series of short-range WRF forecasts have been conducted to form the ensemble. To consider uncertainties in observations and model physics, we use initial conditions from two operational centers, with 21 members from the Global Ensemble Forecast System (GEFS) (Zhou et al. 2017) of National Centers for Environmental Prediction (NCEP), and 10 members from the Ensemble of Data Assimilations (EDA) system (Hersbach and Dee 2016) and 1 member from the ERA5 reanalysis (Hersbach et al. 2020) of European Centre for Medium-Range Weather Forecasts (ECMWF). To increase the effective ensemble size, we used the time-lagged methodology (Hoffman and Kalnay 1983), which has been shown to positively impact the analysis in previous studies (Caron et al. 2019; Lorenc 2017; Wang et al. 2017). Forecasts with different lead times but valid at the same analysis time form the ensemble in this study. Four batches of ensemble forecasts, with 24-, 18-, 12-, and 6-h lead time, provide a total of 128 ensemble members.
The algorithm described in Wang and Zhang (2021) assumes the time tendency of dry air mass to be zero in the mass constraint and uses ERA5 reanalysis to calculate the time tendency of the total water content in the moisture constraint. In this study, we performed tests to make the tendency calculations as part of the analysis, in which the tendencies in both the mass and moisture constraint will be computed using the ensemble which has been used to generate the ensemble covariance. Results from these two types of implementations will be compared in section 5.
c. Observations
For the three cases studied in the paper, we have assimilated observations from the surface Mesonet stations, wind profilers, Global Positioning Satellite (GPS) integrated precipitable water (IPW), and aircraft data that are typically used in the DA experiments. These data are accessible from https://rda.ucar.edu/datasets/ds337.0/. Since the analysis time in our study is either at 0600 or 1800 UTC, which is not the regular launching hour for radiosondes, upper air sounding data are assimilated from only the six-site radiosonde network (red triangles in Fig. 1c) deployed during the MC3E campaign (Jensen et al. 2015). The MC3E sounding data are available from the ARM data discovery (https://adc.arm.gov/discovery/). The radial velocity datasets from the operational Weather Surveillance Radar-1988 Doppler (WSR-88D) (Fig. 1b), available in the National Climate Data Center (NCDC, https://www.ncdc.noaa.gov/nexradinv/), are also assimilated after velocity dealiasing by using the region-based algorithm in the Python ARM Radar Toolkit (Py-ART) (Helmus and Collis 2016) and visual inspection. Following Aksoy et al. (2009) and Duda et al. (2019), only the radial winds with reflectivity above 10 dBZ are assimilated to eliminate possible contamination from nonprecipitation sources. The radar-derived and gauge-corrected Q2 precipitation dataset is assimilated via the moisture constraint in the CDA.
d. Forecast model
All simulations are conducted using the fully compressible and nonhydrostatic Advanced Research Weather Research and Forecasting (WRF) Model version 4.1. A single domain is adopted and covers most of the Southern Great Plains (SGP). The number of horizontal grid points is 357 × 328, with 5-km grid spacing. In total, 51 vertical levels are used with the model top at 50 hPa. Unless otherwise specified, the following physics parameterizations are chosen: the WRF single-moment 6-class microphysics scheme (Hong and Lim 2006), the Kain–Fritsch cumulus parameterization (Kain 2004), the Mellor–Yamada–Janjić planetary boundary layer scheme (Janjić 2001) coupled with the Monin–Obukhov surface layer model, the Rapid Radiative Transfer Model shortwave and longwave schemes (Iacono et al. 2008), and the Noah land surface model (Chen and Dudhia 2001).
3. Performance evaluation
In this section, we compare the performance between the original and constrained data assimilation system, which are referred to as the ODA and CDA experiments, respectively, for each case. Both experiments assimilate all available conventional and radar radial wind observations. Except the constraints used in the CDA experiments, all other assimilation configurations are the same for these two experiments to facilitate a fair comparison. For the ensemble covariance, the horizontal and vertical localization radius are 20 km and 0.5 scale height (natural log of the pressure), respectively. The error covariances in these experiments are entirely from the ensemble covariance.
The evaluation is based on the influence of the analyses on the performances of the ensuing forecasts. The analyses from ODA and CDA are used to initialize the 6-h WRF forecasts. Due to the relative coarse resolution of the analysis grid (5 km), the one-way nesting-down method from the 5- to 1.666-km grids is employed, in which the initial and lateral boundary conditions of the 1.666-km runs are obtained from the 5-km runs every 1 h, and no cumulus parameterization is used in the 1.666-km runs. All evaluations are conducted using the 1.666-km simulations. The forecasted state variables are checked against the radiosonde observations including the sounding array from the MC3E campaign labeled in Fig. 1c, while the simulated precipitation is verified against the Q2 product using the Model Evaluation Tool (MET).
a. Forecasted state variables
Although the analysis time for the three cases is not at the routine launching hours (0000 and 1200 UTC) of radiosondes, their 6-h forecasts happen to be valid at these regular hours. Together with the sounding array from the MC3E campaign, there are up to 24 radiosondes within the nested domain to examine the relative large-scale features across the vertical levels. The root-mean-square errors (RMSEs) of the WRF forecasts at 6-h lead time against the radiosondes are calculated for each case and then averaged, and the results are plotted in Figs. 2a–d. For the u/υ wind (Figs. 2a,b), the CDA experiment shows improvements over the ODA for most levels, and the large error reductions are located around low- to midlevels. For the temperature (Fig. 2c), the CDA slightly outperforms the ODA for most levels. For the specific humidity (Fig. 2d), below 750 hPa, the CDA performs better than the ODA, while between 750 and 550 hPa, the CDA shows slightly larger RMSEs. The relatively better depiction of the large-scale structure in the CDA could be a result of better-simulated mesoscale convections.
Vertical profiles of the root-mean-square errors (RMSEs) of WRF forecasts at 6-h lead time against the radiosonde measurements further averaged over all three cases for the ODA and CDA experiments for (a) the u wind component, (b) the υ wind component, (c) the temperature, and (d) the specific humidity. (e) The vertical profile of the number of available radiosondes.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
b. Simulated precipitation
Both subjective and objective assessments against the Q2 dataset are applied to the simulated precipitation. For the objective evaluation, the fractions skill score (FSS) based on the neighborhood method is utilized to examine the hourly precipitation for the thresholds of 0.1, 0.5, 1.0, 2.5, 5.0, 7.5, 10.0, and 20.0 mm h−1. Unlike the traditional metrics, FSS allows forecasts within a certain neighborhood of the observation to be deemed skillful. Similar to Roberts and Lean (2008) and Stratman et al. (2013), forecasts with skills larger than a target value, FSS > FSSuseful, are considered to be useful. The target skill is defined as FSSuseful = 0.5 + BASER/2, and BASER symbolizes the base rate, which is the ratio of the area where observed precipitation exceeds the threshold to the domain.
1) Qualitative evaluation
For case0520, there are three convective systems involved in the simulation period from 0600 to 1200 UTC (Fig. 3a): the southern rainband stretching from Texas into Oklahoma, the northern rainband striding across the boundary between the Kansas and Nebraska, and convection cells within Oklahoma. They are labeled as “A,” “B,” and “C” at 0700 UTC in the observation. The northern rainband slowly moved toward the northeast direction. The southern rainband interacted and merged with convections in Oklahoma between 0700 and 0800 UTC, forming a single squall line after 0900 UTC.
Hourly precipitation (mm) for (a)–(c) case0520 from 0700 to 1200 UTC 20 May, (d)–(f) case0511 from 1900 UTC 11 May to 0000 UTC 12 May, and (g)–(i) case0425 from 1900 UTC 25 Apr to 0000 UTC 26 Apr 2011. The Q2 observations are shown in (a), (d), and (g); the ODA simulations are shown in (b), (e), and (h); and the CDA simulations are shown in (c), (f), and (i). In each panel, the four-digit number at the top-left corner is the UTC-referenced forecasting time. Markers “A,” “B,” and “C” represent convection areas of interest.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
Comparing the forecasts from the ODA and CDA experiments with the observation, both simulations underestimate the precipitation at 0700 UTC (Figs. 3b,c). But relative to the ODA forecast, the forecast from the CDA experiment has a better organized northern rainband; it also has a better coverage, magnitude, especially the linear structure of the southern rainband, and slightly suppresses the spurious rainfall in central Texas compared to the ODA experiment at 0700 UTC. In the CDA (Fig. 3c), similar to the observation, the southern band interacted and merged with convections in Oklahoma, even though this occurs two hours later (0900–1000 UTC) and the simulated southern rainband does not extend long and propagate fast enough. In the ODA (Fig. 3b), convections developed close to the southern boundary of Oklahoma (0800 UTC) and then accompanied by the originally north–south-oriented rainband developed along the west boundary of Oklahoma (0900 UTC), with the two catching up around 1200 UTC. This development process is different from what had been observed or simulated in the CDA experiment. There is an area of spurious precipitation in western Arkansas from 1000 to 1200 UTC in the ODA experiment. This spurious precipitation is largely suppressed in the forecast using the CDA.
For case0511, there were two convective features in interest at 1900 UTC (Fig. 3d): the two lines of convection in the central Oklahoma and Texas, which are labeled as “A” and “B” at 1900 UTC, respectively. The convective line in Oklahoma propagated in the northeast direction, and by 2300 UTC, it merged with the southern linear convection and formed a single squall line stretching from Oklahoma to Texas.
Both the ODA and CDA simulation captures the convection line in Texas (B) at 1900 UTC, but with magnitudes smaller compared to the observation (Figs. 3e,f). In the following hours, both the ODA and CDA simulate the convection B at roughly similar locations as in the observation. While both experiments overestimate it around 2200–2300 UTC, the overestimation in the CDA is a little higher. Only the CDA experiment reproduces the linear precipitation pattern in central Oklahoma (A) at 1900 UTC, although still underestimates its strength (Fig. 3f). In the ODA experiment (Fig. 3e), the observed convection line (A) is not presented in the simulation until 2200–2300 UTC, and it has a more north–south orientation rather than the northeast–southwest orientation in the observation. In the CDA experiment (Fig. 3f), the simulated convective line (A) persists through the forecast, and catches up with the southern rainband (B) around 0000 UTC, which bears more resemblance to the observation.
For case0425, the dominant feature was the development of two northeast–southwest-oriented rainbands, labeled as A and B in the observation at 1900 UTC in Fig. 3g. These two lines eventually merged around 0000 UTC, forming a long rainband extending from southern Illinois to northeast Texas.
In the forecasts at 1900 UTC, while both the ODA (Fig. 3h) and CDA (Fig. 3i) underestimates precipitation, they all capture the northeast–southwest-oriented band (“A”) around the southeast corner of Oklahoma. Compared with the ODA experiment, the magnitude and the structure of this A band is better simulated in the CDA experiment at 1900 UTC (Fig. 3i). In the following hours, in the CDA experiment (Fig. 3i), although the A band is overdeveloped around 2200 to 2300 UTC, it persists through the forecast with comparable magnitudes and similar orientation as the observation, which shows better performance than its counterpart in the ODA experiment. For the B rainband, neither of the experiments fully simulated it at 1900 UTC, but the CDA forecast has more precipitation extending from the northeast to central Oklahoma than the ODA one at the supposed location (Fig. 3i). In the CDA experiment, precipitation at the northeast corner of Oklahoma at 1900 UTC develops to the B rainband at 2000 UTC, while in the ODA forecast, precipitation is not organized at the corresponding location until 2200 UTC. The structure of the merged band of precipitation at the end of the forecast period is also better reproduced in the CDA forecast than in the ODA forecast.
2) Quantitative evaluation
We next use the FSS to quantitatively evaluate the impact of constraints on the performance of precipitation forecasts. The differences in FSSs between the ODA and CDA forecasts are plotted in Fig. 4, in which positive values symbolize improvements of the CDA over the ODA, and the red and blue lines represent the neighborhood radii at which the ODA and CDA reaches the target skills, respectively.
The differences in the aggregated FSSs between the ODA and CDA experiments, FSSCDA − FSSODA, for (top) FH 1–3 and (bottom) FH 4–6 for (a),(b) case0520; (c),(d) case0511; and (e),(f) case0425. The thick red and blue line represents the neighborhood radii at which the ODA and CDA experiment obtains the target skills, respectively.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
Among all three cases, case0520 shows the largest improvement at forecast hour (FH) 1–3, with a more than 0.2 increase in FSSs, and for this case, the CDA experiment outperforms the ODA for all neighborhood size and precipitation threshold combinations (Fig. 4a). The CDA experiment reaches the target skills at smaller neighborhood sizes when compared with the ODA experiment. At FH 4–6, although the neighborhood sizes at which the ODA experiment has the useful skills have been brought down compared to those at FH 1–3, the benefits in the precipitation FSSs gained from the constraints still exist for the region bounded by the target skills, in which, CDA’s FSSs remains above those from the ODA (Fig. 4b). The ODA experiment scores better than the CDA for thresholds larger than 5 mm h−1 and radii less than 80 km. This is because the simulated ODA precipitation south of the SGP site overlapped with the observed precipitation around 1000 UTC, despite the development was different.
For case0511, at FH 1–3, the CDA experiment scores better than the ODA one (Fig. 4c). For example, for the 2.5 mm h−1 threshold, the ODA experiment never gains useful skills even at the largest neighborhood size used in the paper, while the CDA experiment scores the target skill around 60 km. At FH 4–6 (Fig. 4d), the CDA experiment shows improvements over the ODA one for most of the neighborhood sizes for precipitation under 10 mm h−1, but a slight degradation for the 20 mm h−1 precipitation threshold and neighborhood radii larger than 60 km, which may be due to the overestimation in the CDA simulation.
For case0425, the CDA experiment performs better than the ODA one at both FH 1–3 and 4–6 (Figs. 4e,f), although the degree of improvement has been reduced from FH 1–3 to FH 4–6, which is a common feature also presenting in the other two cases.
The above results illustrate the positive impact of the dynamical constraints [Eqs. (2)–(3)] on the forecasts of precipitation in all three cases when the constrained analyses are used to initialize the forecasts. This is consistent with Wang and Zhang (2021), in which only one forecasting case was studied on a domain smaller than that used in this paper, and the error covariance was configured differently from this paper with 25% and 75% from the static and ensemble error covariance, respectively, following Benjamin et al. (2016).
4. Assessment of the impact of constraints on the effect of radar radial winds
Due to its high resolution and almost seamless coverage over the continental United States (CONUS), radar data have been proved to be beneficial for models to simulate the location and intensity of the MCSs when they can be assimilated properly (Bachmann et al. 2020; Mazzarella et al. 2020). In the CDA system, the radar estimated precipitation (Q2) is assimilated though the moisture constraint that drives the adjustment of both the moisture and winds in the atmospheric columns, and the radar radial wind is assimilated to directly adjust the three-dimensional wind field. These two datasets are complementary to each other in improving the analysis from the data assimilation, since the precipitation improves both the thermodynamical and dynamical component, while the radial wind improves the dynamical part directly. Unlike in the United States, however, radar coverage in many parts of the world is sparse or lacking. For these regions, precipitation can be obtained from satellite measurements, but radar radial winds are missing. In this section, we demonstrate how the dynamical constraints [Eqs. (2)–(3)] can partially remedy the missing radar radial winds. This is shown by withholding the radar wind data in the ODA and CDA and examining the impact on the forecast performance when the analyses are used as initial conditions.
The experiments with and without radial wind assimilation are referred to as the control and data denial experiments, respectively. As in section 3, short-range WRF forecasts are conducted after the data assimilation. Whether radial winds are assimilated or not, it directly affects the horizontal wind analysis through the observation operator, and indirectly impacts other variables’ analysis through the background error statistics, and in turn influences prognostic dynamical and thermodynamical variables in the forecast. Since precipitation could be considered as one reflection of all these influences, performances are assessed using the FSSs of hourly precipitation. Figures 5 and 6 shows the differences of FSSs in the first three forecasting hours between the experiments with and without radial wind assimilation under the ODA and CDA configurations, respectively, with each column for each case.
The differences in the FSSs between the control and radial wind denial experiment under the ODA configuration, FSSdenial − FSScontrol, for (top) FH 1; (middle) FH 2; and (bottom) FH 3 for (a)–(c) case0520, (d)–(f) case0511, and (g)–(i) case0425. The thick red and blue line represents the neighborhood radii at which the control and denial experiment obtains the target skills, respectively.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
As in Fig. 5, but under the CDA configuration.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
Under the ODA configuration, it is seen that the denial of radial wind assimilation worsens the precipitation forecast considerably (Fig. 5). The FSSs decrease when radial winds are not assimilated, and this deterioration is especially clear for case0520 (Figs. 5a–c) and case0511 (Figs. 5d–f).
In the CDA configuration, the impacts of the missing radial wind are much smaller than those in the ODA for all three cases (Fig. 6). Taking case0520 for example, the removal of radial wind assimilation brings down the FSSs by a maximum of 0.3 through FH 1–3 under the ODA configuration (Figs. 5a–c), while in the CDA experiment, the loss of radial winds only causes a maximum decrease of 0.1 in FSSs (Figs. 6a–c). Figure 6 also shows that in all three cases, FSSs even increase for the precipitation thresholds larger than 2.5 mm h−1 in FH 1 when the radial winds are not assimilated. This is likely because the simultaneous assimilation of the radial wind on top of rainfall data reduces the large moisture increments through the background error covariance and lowers the forecast skill of large precipitation, as has been also reported in Sun et al. (2020) in the 4D-Var scheme.
The impact of the missing radar wind described above can be seen more clearly in Fig. 7 which uses the forecasted precipitation in the first hour (0700 UTC) for case0520 as an example. Figures 7a and 7b are the simulated precipitation from the ODA and CDA experiments without assimilating the radar winds, respectively. These precipitation distributions can be compared with the observed precipitation and the simulation ones from the corresponding experiments discussed earlier with the radial wind assimilation (Figs. 3a–c).
The first hour precipitation forecast (0600–0700 UTC) from the (a) ODA and (b) CDA radar data denial experiment, respectively, for case0520. The red ellipse indicates the location of the southern rainband in the observation at 0700 UTC 20 May 2011.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
For the ODA experiments, when the radial wind assimilation is deprived, there is no forecast precipitation at all in the circled area which roughly corresponds to the observed southern rainband (Fig. 7a). And the northern rainband is also too light and not well defined in the simulation (Fig. 7a). When the radial winds are assimilated (Fig. 3b), the magnitude of the northern rainband is increased, although the linear structure is still not clear, but still better than the denial experiment. And in the circled region, precipitation starts to appear and bears some similarity to the observation, although the magnitude and areal coverage is much smaller (Fig. 3b).
For the CDA experiment, even without the radial wind assimilation, it simulates both the northern and southern rainband at the correct locations, with lower magnitude and less coverage comparing to the observation (Fig. 7b). When radial winds are assimilated (Fig. 3c), it reduces the simulated large precipitation a little bit in both rainbands, which explains the increases in FSSs in the denial experiment for higher precipitation thresholds at FH 1 (Fig. 6a). But still, the precipitation patterns between the experiments with and without the radial wind assimilation appear similarly, which is a manifest that the constraints in the CDA system could alleviate the negative impacts resulting from the loss of radial wind observations.
To understand the differences in the forecasts, the analysis increments from the experiments without and with radial winds under the ODA and CDA configurations are analyzed. Figure 8 plots the water vapor adjustments at 825 hPa from four experiments. In the ODA denial experiment (Fig. 8a), the humidity increments are sporadic and small-scaled, which is partially due to the small horizontal localization length scale used in the paper. When radial winds are assimilated in the ODA experiment (Fig. 8b), especially along the southern rainband, not only the magnitudes, but also the horizontal extents of water vapor increments are larger compared to those in the denial experiment. Under the CDA configuration, water vapor is increased along the southern rainband even in the experiment without radial winds (Fig. 8c), and there are some similarities in the adjustments between the ODA control experiment and the CDA denial experiment. In the CDA control experiment (Fig. 8d), compared to the CDA denial experiment, it reduces water vapor increments for some parts of the rainbands, which could be the reason of smaller precipitation at 0700 UTC. One thing worth notice is that, in both the ODA and CDA experiment, compared to the denial experiment, the control experiment increases the water vapor in central Texas, which causes spurious precipitation at 0700 UTC in the simulations.
The water vapor mixing ratio increments for (a) the ODA denial, (b) the ODA control, (c) the CDA denial, and (d) the CDA control analysis at 825 hPa for case0520. The contour interval for the increments is 2 g kg−1. The hatched area is where precipitation is larger than 1.0 mm h−1 in the observation.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
5. Alternative calculation of time tendencies in the constraints
For the CDA experiments in sections 3 and 4, the dry air mass time tendency in the mass constraint is approximated as zero, and the time tendency of water content in the moisture constraint is calculated from ERA5 reanalysis with a reduced grid resolution (31 km). The impact of these assumptions is investigated in this section. We use the forecast ensemble, which have been used to generate the ensemble error covariance, to derive the time tendencies in both the mass and moisture constraints. The tendencies are therefore at the same resolution as the analysis grid. One benefit of using this approach is the real-time data assimilation without relying on the ERA5 reanalysis which has a latency of about 5 days.
As mentioned in section 2b, the ensemble is constituted by four batches of WRF forecasts using the time-lagged approach. The time tendencies could be derived from the whole set of 128 members, or just 32 members from a single batch with the shortest forecast length (6 h). Here the 32-member ensemble is used.
As in the previous two sections, short-range WRF forecasts are performed to assess the effect of alterations in time tendencies. The analysis quality is evaluated by using the performance score FSS for the hourly precipitation forecasts. Figure 9 shows the differences of FSSs between the experiments using the ERA5 and ensemble time tendencies for each case. For all three cases, using the ensemble tendencies slightly improve precipitation forecasts for FH 1–3 over the original forecasts.
The differences in the FSSs between the experiments using the ERA5 and ensemble time tendencies in the CDA system, FSSensemble_TEND − FSSERA5_TEND, for (top) FH 1; (middle) FH 2; and (bottom) FH 3 for (a)–(c) case0520, (d)–(f) case0511, and (g)–(i) case0425. The thick red and blue line represents the neighborhood radii at which the experiment with the ERA5 and ensemble tendencies obtains the target skills, respectively.
Citation: Monthly Weather Review 151, 2; 10.1175/MWR-D-22-0144.1
We have conducted additional experiments to use all 128 members as the ensemble to derive the tendencies (not shown). We found that because this larger ensemble has larger spreads, the magnitudes of the time tendencies are weak. As a result, there are small reductions of the FSSs. For example, the domain averaged time tendencies of the column-integrated total water content (absolute values) in ERA5 in the three cases are 0.86, 1.07, and 0.82 mm h−1, respectively. In the 128-member ensemble, these become 0.63, 0.70, and 0.56 mm h−1. This reduction is caused by the large dispersion of the members when the time range of the forecast window is larger. The weak tendencies cause the decrease in FSSs. In the 32-member ensemble from 6-h forecasts, the magnitudes of tendencies are 0.89, 0.96, and 0.74, respectively.
The above discussion shows that it is feasible to use the time tendencies from the ensemble in the Wang and Zhang (2021) data assimilation algorithm to enforce the dynamical constraints. It is possible to conditionally sample the ensemble members based on forecasted precipitation to improve the quality of the tendencies. These are left for future studies.
6. Summary and discussion
The constrained data assimilation system, which incorporates the mass and moisture constraint described in Wang and Zhang (2021), has been assessed for three convective cases during the ARM MC3E field campaign using two methods. One is through the evaluation of short-range forecast performance; the other is through observation denial experiments in which radar radial winds are not assimilated.
The forecasted state variables are evaluated against the radiosonde observations. Compared to the ODA experiment, the CDA experiment exhibits lower RMSEs for u/υ wind and temperature at most vertical levels and for specific humidity under 750 hPa. The convection evolutions are then subjectively evaluated and the simulated precipitation is objectively compared. The CDA outperforms the ODA regarding the precipitation location, intensity and evolution.
The observation denial experiment has shown that the physical constraints in the CDA can greatly mitigate the loss of forecasting skill of convections due to the missing of radar radial winds. This is achieved through the thermodynamical and dynamical adjustments related to the mass and moisture conservation.
We have extended the Wang and Zhang (2021) by alleviating the dependency of the algorithm on the ERA5 reanalysis that has a relatively coarse resolution. The ensemble members, upon which the ensemble covariance is derived, are used to generate the time tendencies in the mass and moisture constraint. We have shown that the ensemble method can give comparable or slightly better forecast skills. In future, for retrospective experiments, higher-resolution analysis, such as those from High-Resolution Rapid Refresh (HRRR, 3 km) (Benjamin et al. 2016), may be used to estimate the time tendencies and evaporation for the moisture constraint, considering that the coarse resolution (31 km) ERA5 reanalysis possibly misses the mesoscale characteristics.
In the current implementation of our algorithm, the analysis domain has been categorized into precipitation and little-to-none precipitation regions, and the weight λM of the moisture constraint is assigned according to this classification. Another possible method to determine the weight has been tried. It uses the standard deviation of the precipitation within the 5 km × 5 km analysis grid as the proxy for the error standard deviation eM, since the precipitation in Eq. (3) is interpolated from the Q2 dataset at 1-km resolution. We have found that the smaller error standard deviation in regions of weaker precipitation makes the gradient of the cost function extremely large, and it significantly slows down the convergence of the minimization of the cost function. In Fig. 6, the FSSs for heavy precipitation are decreased when radial winds are assimilated under the CDA configuration. Although better analysis does not necessarily guarantee the better forecast, it is possible that the weights assigned to the mass and moisture constraint have large influence on the optimality of the system. So, how to best specify the errors in the constraints to improve the optimality of the CDA algorithm and maintain the computational efficiency, remains to be studied.
The quality of the ensemble is of great importance for the EnVar system when a large percentage of the background error covariance is derived from the ensemble. Although the time-lagged method used in this paper to form the ensemble is cost effective and has demonstrated its value in both the ODA and CDA experiments, there is room in the DA system to further improve the quality of the ensemble. For example, to better account for uncertainties in the model physics, perturbations in physical tendencies, which has been adopted in the GEFS (Hou et al. 2006; Zhou et al. 2017) and EDA (Hersbach et al. 2020), could be included in the ensemble. These are left for future research.
The physical constraints used in this paper could also be potentially valuable in the four-dimensional ensemble-variational (4D-EnVar) data assimilation system. Different from the ensemble four-dimensional variational (En4DVar) method, in which the tangent linear (TL) and adjoint (AD) models are required to propagate the innovation, the 4D-EnVar uses ensemble perturbations throughout the assimilation time window to estimate the four-dimensional error covariance and avoids the usage of TL and AD models (Buehner et al. 2010a,b; Liu et al. 2008; Wang and Lei 2014). The 4D-EnVar method has been implemented in several operational centers, such as the Global Data Assimilation System (GDAS) in the NCEP (Wang et al. 2019) and the global and regional DA system in Environment Canada (EC) (Buehner et al. 2015; Caron et al. 2015). Since the 4D-EnVar is a temporal extension of its 3D counterpart, the CDA algorithm could in principle be extended to the 4D-EnVar. This is left for future research.
Acknowledgments.
This study is supported by the Atmospheric Science Research (ASR) Program and the Climate Model Development and Validation (CMDV) Program of the U.S. Department of Energy, Office of Science, Office of Biological and Environmental Research.
Data availability statement.
The observation dataset assimilated in this study is publicly accessible: the conventional observations are from National Center for Atmospheric Research (NCAR) research data archive (https://rda.ucar.edu/datasets/ds337.0/), the radiosonde observations during the MC3E campaign are available from ARM data discovery (https://adc.arm.gov/discovery/), and radial winds from WSR-88D is from NCDC (https://www.ncdc.noaa.gov/nexradinv).
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