a. GSI hybrid 3D-EnVar analysis

Although the hybrid EnVar shows better performance over 3D-Var through better representation of the cross-variable covariances and spatial correlation, the lack of dynamical constraints in its framework still brings in some important imbalances to the final analysis. In this study, we focus on mass and moisture conservations. Mass residual is defined as the sum of change of dry air mass and column-integrated mass flux divergence; the moisture residual is the sum of column-integrated moisture tendency, moisture flux divergence, and precipitation, minus surface evaporation. As an example, for the case study to be discussed later, in the background (BK), the average of the absolute values of mass residual is around 8 hPa h⁻¹, while in the original hybrid 3D-EnVar analysis, the average increases to 15 hPa h⁻¹ if only conventional observations are assimilated, and jumps to 35 hPa h⁻¹ if radar radial winds are also assimilated. This sudden increase of mass flux divergence can introduce high-frequency oscillation to the surface pressure in the first few hours of the subsequent forecast that is initialized using the analysis. For the moisture residual, the average of its absolute values goes from 2.7 mm h⁻¹ in the background to 2.9 and 3.5 mm h⁻¹ when conventional and radial winds are successively included.

b. Dynamical constraints: Mass and moisture

c. Implementation of the constrained algorithm

Similar to the background error covariance $\mathsf{B}$ in GSI that is not explicitly calculated but constructed by several components, matrix $\mathsf{K}$ in Eqs. (11) and (12) is also not explicitly computed during the minimization. It is derived through 5 successive steps as follows:

1. Matrix $\mathsf{K}$₁ is used to convert the control variables x′ (streamfunction, velocity potential, relative humidity, surface pressure, and virtual temperature) to the state variables (u/υ winds, specific humidity, surface pressure, and temperature) as increments.

2. Matrix $\mathsf{K}$₂ is used to convert the specific humidity to mixing ratio. Mixing ratio w holds the following relationship with specific humidity q: $w^g+w=\left(q^g+q\right)/\left[1-\left(q^g+q\right)\right]$, where the variables with and without the superscripts g represent the guess field and the analysis increments. Following the Taylor expansion, the mixing ratio increment is approximated through $w\approx q/{\left(1-q^g\right)}^2$, with q^g updated by the outer loops.

3. Matrix $\mathsf{K}$₃ is used to convert the surface pressure to the dry air mass in the column. In GSI, the surface pressure ($p_s^a=p_s^g+p_s$, and the superscript a represents analysis) consists of the hydrostatic pressure of the dry atmosphere at the surface (${\mu }_d^a+p_T$, and p_T is the pressure at the model top) and column-integrated water vapor pressure (${\mu }_d^a{\displaystyle {\int }_0^1{w}^a\hspace{0.17em}d\eta }$). We therefore have the dry air mass within the column as ${\mu }_d^a=\left(p_s^a-p_T\right)/\left(1+{{\displaystyle \int }}_0^1{w}^a\hspace{0.17em}d\eta \right)$. Appling the Taylor expansion, the increment for the dry air mass is approximated as ${\mu }_d\approx \left[p_s/\left(1+{{\displaystyle \int }}_0^1{w}^{g}d\eta \right)\right]-\left[\left(p_s^g-p_T\right)/{\left(1+{{\displaystyle \int }}_0^1{w}^{g}d\eta \right)}^2\right]{\displaystyle {\int }_0^{1}wd\eta }$.

4. Matrix $\mathsf{K}$₄ is used to calculate the divergence. The background field used for GSI is on WRF grids, in which the grid sizes are constant in the computational space, but the physical distances between grids vary with the grid positions and depend on the map scale factors, so the divergence calculation takes the map scale factors into consideration:

   ${\nabla }_{\eta }\left({\mu }_d{\mathbf{V}}_{\eta }\right)=m_x{m}_y\left({\displaystyle \frac{\partial \hspace{0.17em}{\displaystyle \frac{u{\mu }_d}{m_y}}}{\partial X}}+{\displaystyle \frac{\partial \hspace{0.17em}{\displaystyle \frac{\upsilon {\mu }_d}{m_x}}}{\partial Y}}\right),$(13)

   in which, m_x and m_y is the map scale factor in x and y direction, respectively, and ∂X and ∂Y is the grid distance in the computational space. To facilitate the construction of $\mathsf{K}$^T, $\mathsf{K}$ needs to be linearized, including the divergence calculation. Taking the mass flux divergence in Eq. (8) as an example, it could be linearized as

   ${\nabla }_{\eta }\left({\mu }_d^a{\mathbf{V}}_{\eta }^a\right)\approx {\nabla }_{\eta }\left({\mu }_d^g{\mathbf{V}}_{\eta }^g\right)+{\nabla }_{\eta }\left({\mu }_d^g{\mathbf{V}}_{\eta }\right)+{\nabla }_{\eta }\left({\mu }_d{\mathbf{V}}_{\eta }^g\right),$(14)

   where ${\mathbf{V}}_{\eta }^g$ and ${\mu }_d^g$ are kept as constants during the inner loops, and updated by the increments at the end of each outer loop, and the calculation ${\nabla }_{\eta }\left({\mu }_d^g{\mathbf{V}}_{\eta }^g\right)$ is included in term b of Eq. (11). The same linearization approximation is applied to the moisture flux divergence.

5. Matrix $\mathsf{K}$₅ is used to integrate the divergence, either mass or moisture flux divergence, along the η-coordinate from top to bottom.

Term b in Eq. (11) has three more components for the moisture constraint: the time tendency of water content, the evaporation and precipitation. In our initial implementation, all of these are provided as input into the constrained data assimilation. Ideally, the column integrated time tendency should be from surface or satellite microwave measurements, but it is currently calculated using the total column water content from the hourly fifth generation of European Centre for Medium-Range Weather Forecasts (ECMWF) atmospheric reanalysis (ERA5) (Hersbach and Dee 2016), while the evaporation is calculated using the accumulated evaporation from ERA5 hourly forecast. Both are gridded at 0.25° horizontally. The precipitation comes from the hourly bias-corrected National Severe Storm Laboratory (NSSL) next-generation National Mosaic and Multisensor Quantitative precipitation estimate product (Q2) (Zhang et al. 2011), with horizontal resolution at 0.01°. These three components are calculated at or extracted from their original grids, and interpolated to the grid of the background field. Since the constraints use hourly averaged data, the column-integrated conservation equations are valid when averaged over an hour.

a. Forecast model

The fully compressible and nonhydrostatic WRF Model version 4.1 is used as the forecast model. A single domain, which is centered around the SGP Central Facility (CF) site in north-central Oklahoma, is adopted. The domain is configured by 227 × 198 grids, with 5-km horizontal grid spacing in the region shown in Fig. 1. The model top is set at 50 hPa, with 51 vertical levels. Unless stated otherwise, the Kain–Fritsch scheme (Kain 2004) is used for cumulus parameterization, while Mellor–Yamada–Janjić scheme (Janjić 2001) is used as planetary boundary layer scheme coupled with Monin–Obukhov surface layer model. Rapid Radiative Transfer Model shortwave and longwave schemes (Iacono et al. 2008) are used. The land surface model is Noah scheme (Chen and Dudhia 2001). The microphysical scheme we use is the WRF single-moment 6-class microphysics scheme (Hong and Lim 2006).

b. Assimilation configurations

The background error covariance $\mathsf{B}$ determines how the observation information is spread across the domain and different control variables. In the static error covariance of the hybrid method, the horizontal and vertical length scales are derived using the National Meteorological Center (NMC) method and tunable through the scale factors, and they vary with the latitudes, heights, and variables. The horizontal length scales are applied through a combination of three Gaussian distributions in the recursive filters to achieve the “fat tailed” feature in the covariance (Kleist et al. 2009). Due to the “fat tailed” feature, with the default length scales in GSI, the impact radius of a single observation is beyond 500 km, which almost covers the entire domain of this study. With the focus of this paper on mesoscale systems and based upon the sensitivity experiments, the horizontal length scales used in this paper are reduced by a factor of 0.25. The vertical length scales are those from the NMC method. In the flow-dependent ensemble error covariance, localization is used to determine the distance beyond which the observation has little impact and to reduce spurious correlation at large distance due to insufficient ensemble size. Various horizontal localization radii (4–20 km) have been used in past studies with assimilation of radar data (Johnson et al. 2015; Kong et al. 2018; Sobash and Stensrud 2013; Wang and Wang 2017). Generally, the choice of radius depends on the ensemble size, the grid spacing and the event to be studied. Given the findings that the optimal localization radius increases as the ensemble size increases (Houtekamer and Mitchell 1998) and the slightly larger grid spacing (5 km) compared to those used in past studies (1–4 km), the horizontal localization radius is set to be 20 km. For the vertical localization, three scale heights (natural log of the pressure) are tried: 0.3, 0.5 and 1.0. Sensitivity studies reveal that (not shown), for CDA, the forecast deteriorates as the scale height increases; while for ODA, the experiment with 0.5 scale height outperforms that with 0.3, and the performance of the 1.0 scale height experiment is similar to that with 0.5. To facilitate a fair comparison between ODA and CDA, 0.5 scale height is chosen to be the vertical localization.

The percentage contributions of the static and the ensemble error covariance to the full $\mathsf{B}$ is given by the factor of $1/{\beta }_1$ and $1/{\beta }_2$, which satisfies $\left(1/{\beta }_1\right)+\left(1/{\beta }_2\right)=1$. In the Rapid Refresh (RAP) version 3, $1/{\beta }_1$ and $1/{\beta }_2$ are assigned to be 25% and 75%, respectively (Benjamin et al. 2016). These are the same values used in our study.

The ensemble is formed by a series of short-range WRF forecasts. We used two sets of initial conditions from two operational centers. One set is from the Global Ensemble Forecast System (GEFS), which is made up of 20 ensemble members and one control run (Zhou et al. 2017). The other set is from Ensemble of Data Assimilations (EDA) system in ERA5 data products, which includes 10 ensemble members, plus the ERA5 reanalysis (Hersbach and Dee 2016). To form the ensemble at the analysis time, which is 0600 UTC 20 May 2011, we initialized the WRF at 0600, 1200, 1800, and 2400 UTC 19 May and integrated it for 24, 18, 12, and 6 h, respectively. The forecasts at the analysis time constitute 128 ensemble members, which is comparable to the 80-member ensemble used in RAP version 3 (Benjamin et al. 2016).

Considering the accuracy and spatial resolution of the constraint dataset (precipitation, time tendency and evaporation), we divided the analysis grids into two categories, the precipitation region and little-to-none precipitation region using 0.1 mm h⁻¹ as the threshold. The precipitation is interpolated from Q2 that has 1-km resolution to the analysis grid at 5-km resolution. For precipitation region, considering precipitation is averaged from high resolution to low resolution, the standard deviation could be calculated within the average square, which is about 1.71 mm h⁻¹ and could be considered as the representativeness error. While bearing other errors in mind, such as the measurement error and the approximation error, for the precipitation area, E_Moist is set to be 3 mm h⁻¹ to more realistically represent the error statistics. For the little-to-none precipitation region, based on sensitivity experiments (not shown), E_Moist is set to be 12 mm h⁻¹. In future, more cases will be selected to confirm the optimal choices of these standard deviation values.

c. Observations

Both the ODA and CDA assimilate the same set of the conventional and radar radial winds observation. The conventional observations include the surface stations, wind profilers, aircraft reports, radar derived VAD winds, satellite wind data, et cetera, and could be downloaded from https://rda.ucar.edu/datasets/ds337.0/. Since the analysis time is 0600 UTC, no radiosonde observation within the domain is available at this time. The radial wind datasets are publicly accessible from the National Climatic Data Center (NCDC, https://www.ncdc.noaa.gov/nexradinv/), and they are stored as raw data in which velocity aliasing exists. Although the large volume of radial wind observations is beneficial to the data assimilation, the erroneous wind observation caused by aliasing could have undesirable effects. Therefore, before data assimilation, the radial velocity is de-aliased using a region-based algorithm in the Python ARM Radar Toolkit (Py-ART) (Helmus and Collis 2016). After that, winds are further visually inspected and suspicious winds are manually removed.

a. Analyses

We first show the spatial distribution of the mass and moisture residuals in the background field (Figs. 2a,b), ODA (Figs. 2c,d), and the CDA (Figs. 2e,f) at 0600 UTC. Different from ODA, which increases the average of the absolute values of mass residual from 8 to 35 hPa h⁻¹ and moisture residual from 2.7 to 3.5 mm h⁻¹, CDA largely reduces both residuals, with mass and moisture residual decreased to 3 hPa h⁻¹ and 0.6 mm h⁻¹, respectively.

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Mass and moisture residual for the (a),(b) background; (c),(d) ODA analysis; and (e),(f) CDA analysis at 0600 UTC 20 May 2011, respectively. The hatched area is the area where precipitation is larger than 1 mm h⁻¹.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0052.1

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We next show the moisture flux divergence in the background, ODA and CDA analyses (Fig. 3). The area with precipitation (observed from 0500 to 0600 UTC) exceeding 1 mm h⁻¹ is hatched. Three rainfall regions are shown in the observation: a northern rainband expanding from Nebraska into northern Kansas, some convective cells in Oklahoma, and a southern rainband stretching from southwest Oklahoma to Texas. In the background (Fig. 3a), the moisture flux convergence is positioned ahead of the two observed rainbands, and the convective cell in central Oklahoma is collocated with divergence, both of which are not physically reasonable, since precipitation should collocate with the column-integrated moisture flux convergence. This is a reflection of the inability of the forecasting model to capture the observed precipitation.

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Vertically integrated moisture flux divergence for the (a) background, (b) ODA analysis, and (c) CDA analysis at 0600 UTC 20 May 2011, respectively.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0052.1

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In the ODA analysis (Fig. 3b), the magnitude of moisture flux divergence and convergence have both been increased, especially the convergence with its extreme going from 16.5 to 59.3 mm h⁻¹. The increase could be partially attributed to the boost in the mass flux divergence/convergence due to data assimilation. For some parts of the observed rainfall, convergence has been pushed toward the observed rainfall locations, but overall it still does not align well with the precipitation, which results in the increase of moisture residuals. This moisture imbalance, without proper treatments such as nonprecipitation echo assimilation (Gao et al. 2021, 2018), will impact the performance of the subsequent forecast, causing spatially misplaced simulated rainfall, which will be discussed in section 4. In the CDA analysis (Fig. 3c), the moisture flux convergence is collocated with the observed precipitation in all three precipitation centers, so the moisture residual is largely reduced compared to that of the analysis from ODA.

To show the impact of the constraints on the analysis, we compare the increments of potential temperature, water vapor mixing ratio, and wind and moisture flux divergence at 825 hPa from ODA and CDA in Fig. 4. In the background, a warm and wet tongue of air intrudes northeast from Texas to Oklahoma, within which the southern rainband is located. The rainband crosses the region where the moisture gradient is large. Meanwhile, the moisture flux convergence is slightly ahead of the observed.

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(left) Potential temperature; (center) water vapor mixing ratio; and (right) wind and moisture flux divergence for (a)–(c) the background, and their corresponding increments for (d)–(f) ODA and (g)–(i) CDA analysis at 825 hPa. The contour interval for potential temperature increment in (d) and (g) and water vapor increments in (e) and (h) is 0.4 K and 1.0 g kg⁻¹, respectively.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0052.1

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For the ODA analysis, both the temperature and moisture increments are mainly around the observed precipitation (Figs. 4d,e). The adjustments of temperature are small through the column, which is partially due to lack of upper-air observation at 0600 UTC. Most changes of water vapor are limited below 700 hPa (not shown). The horizontal scales of its adjustments are small (Fig. 4e), which is related to the small length scales chosen for both the ensemble and the static error covariance. The maximum positive increment of water vapor appears at the southern end of the precipitation band with a value of 8.6 g kg⁻¹ at 825 hPa, which is collocated with a strong southerly wind adjustment. Along the linear rainband, moisture flux convergence is generated, but the signals are rather unstructured (Fig. 4f).

For the CDA analysis, except for the southern rainband, the temperature increment is also small (Fig. 4g). Most of the water vapor adjustment appears along and ahead of the precipitation (Fig. 4h). Water vapor is increased around three precipitation regions, with the maximum moisture increment along the linear rainband. This is indication of the importance of the precise location of the moisture front to the water budget. Ahead of the precipitation, water vapor is decreased, with relatively smaller magnitude. The changes are limited to levels below 600 hPa (not shown). Although there are no upper-level humidity observations at 0600 UTC, the moisture adjustments in CDA have a broader spatial extent and structures of the precipitation cells, due to the inclusion of the observed precipitation through the moisture constraint. In contrast, although with the same flow dependent error covariance, the increments in the ODA analysis tend to be more isolated and comparatively spherical (Fig. 4e). Similar to the moisture adjustments, adjustment in moisture flux convergence is generated around the three precipitation centers (Fig. 4i).

b. Forecasts

The ODA and CDA analyses are next used as initial conditions to perform 12-h forecasts. The grid spacing for data assimilation is 5 km, which is a compromise between the domain size and the computational cost. In the forecast, to resolve the small scales of the convective systems, the one-way nesting-down approach is adopted, with the grid spacing for the nested domain 1/3 of the analysis domain. All results showed below are from 1.666-km runs, which use the initial and hourly boundary conditions from the 5-km runs. The forecasted state variables at 1200 UTC are verified against the radiosonde observation. And the simulated precipitations are evaluated using both the traditional and neighborhood metrics. Both metrics are computed using Version 8.1.1 of the Model Evaluation Tool (MET) developed by the National Center for Atmospheric Research (NCAR) Developmental Testbed Center (DTC).

1) State variables

Figure 5 shows the root-mean-square errors (RMSEs) of horizontal wind components, temperature, and specific humidity forecast at 1200 UTC against radiosonde measurements. The number of radiosondes within the domain is 10 at most (Fig. 5e). The RMSEs are calculated by averaging the model forecast over the 3 × 3 grids surrounding the radiosondes at each level. For the horizontal winds (Figs. 5a,b), above 875 hPa, the CDA experiment performs much better than the ODA, with the largest error reduction around 4.8 m s⁻¹, whereas below 875 hPa, it has larger RMSEs than the ODA. For the temperature (Fig. 5c), above 800 hPa, the CDA is superior to the ODA for the most part, and below it, the CDA has larger RMSEs. For the water vapor (Fig. 5d), the CDA also outperforms the ODA above 875 hPa, but underperforms below it. By design of the cost function, the adjustments in the CDA are made to primarily improve the spatial gradients, but there are also clear improvements in the state variables forecasts above 875 hPa in the CDA over ODA.

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Vertical profiles of root-mean-square errors (RMSEs) of the 6-h WRF forecasts valid at 1200 UTC 20 May 2011 against the radiosonde measurements for the ODA (red) and CDA (blue) experiments for (a) the u wind component, (b) the υ wind component, (c) the temperature, and (d) specific humidity. (e) The profile of the number of radiosondes for RMSE calculation.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0052.1

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Figure 6 is the time evolution of the surface pressure at a typical location (34.19°N, 97.09°W) in the ODA and CDA. In the ODA experiment, due to the large mass residual in its analysis, there are high-frequency oscillations close to the beginning of the forecast. The surface pressure jumps a few hectopascals just within a few minutes. Through the inclusion of the mass constraint, these high-frequency oscillations have disappeared in the CDA experiment.

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The time evolution of surface pressure in WRF forecasts initialized by the ODA (red) and CDA (blue) analysis at 34.19°N, 97.09°W.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0052.1

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2) Precipitation

Figure 7a displays the Q2 hourly precipitation during this event from 0700 to 1800 UTC. At 0600 UTC (shown earlier in Fig. 1a), there were three mesoscale precipitation systems: two rainbands and some isolated convective cells in central Oklahoma. The southern band merged with some of the convective cells in central Oklahoma from 0600 to 0700 UTC and later fully developed into a mature squall line around 1100–1200 UTC as characterized by a leading edge of heavy convective precipitation and a broad area of trailing stratiform precipitation.

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Hourly precipitation (mm) from 0700 to 1800 UTC from (a) NSSL Q2 observation (the left two columns), (b) ODA simulation (the center two columns), and (c) CDA simulation (the right two columns).

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0052.1

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Figure 7b shows the hourly precipitation forecasts using the ODA analysis as the initial condition. In the first two hours, two precipitation centers developed across the border between Texas and Oklahoma, which was ahead of the observed rainband and with wrong orientations. At 0900–1000 UTC, a thin linear band of precipitation developed close to the western border of Oklahoma, with a largely north–south orientation. Later on, this linear band intensified and caught up with the abovementioned precipitation centers, which had been slowly moving northeastward to central Oklahoma. After the merge (1300–1400 UTC), the band then changed to a southwest–northeast orientation, resembling the observation.

Figure 7c presents the precipitation forecasts using the CDA analysis. In the first hour, the forecast captured convection cells in central Oklahoma and the linear band from southwest Oklahoma to northern Texas. The locations are similar to the observation, but magnitude smaller than the observed. Around 0900 UTC, the southern rainband interacted with the convective cells in central Oklahoma and merged with them later to form the squall line. The precipitation in the northern Kansas was initially weaker than that in observation and dissipated in 0800–0900 UTC, while it lingered in the observation until 1000 UTC. The discrepancy may be because the precipitation is on the domain’s northern boundary, impacted by the lateral boundary conditions.

Based on the visual inspection, CDA outperforms ODA in several aspects: the better convection initiation, the more realistic interactions among cells through the development, and the faster development of the squall line. Some features are not captured well in both forecasts. The broad area of trailing stratiform precipitation at the mature stage was missing, and the propagation in both simulations was slower than the observed. Both of them may be related to the microphysical parameterization and the cold pool simulation as well as biases in the initial conditions (Sobash and Stensrud 2013). The observed rainband spanned deep into Texas, while in both forecasts, the simulated rainband only slightly extended into Texas, which is possibly impacted by inflow lateral conditions from the southern boundary. In the observation, some loosely organized rain cells formed around 1100 UTC in eastern Oklahoma, then strengthened and slowly moved east in the following hours. But in both ODA and CDA experiments, the corresponding cells were misplaced in western Arkansas. Further study is needed to pin down the causes of these errors, whether they are in the initial and boundary conditions, or the model physics.

Quantitative evaluation

To provide objective assessments of the forecast performance, skill scores are computed for the hourly precipitation field using thresholds of 0.1, 0.5, 1.0, 2.5, 5.0, 7.5, 10.0, 20.0, 30.0, 40.0 50.0, and 60.0 mm h⁻¹ for the 6-h forecast length.

The GSS skill scores from the two forecasts and their differences are shown in Fig. 8. For the ODA experiment (Fig. 8a), at forecast hour (FH) 1, GSSs at all precipitation thresholds are under 0, representing almost no skills at all. In FH 2–3, the GSS has become slightly positive for thresholds under 2.5 mm h⁻¹, while it remains less or equal to 0 at larger thresholds. After that, GSS increases with time toward FH 6 for thresholds under 5.0 mm h⁻¹.

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GSS for (a) ODA and (b) CDA simulation, and (c) their differences.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0052.1

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For the CDA experiment, GSS is positive starting at the beginning of the forecast (Fig. 8b). The difference of the GSS scores between the two experiments (Fig. 8c) shows systematic higher score in the first 4 h from the CDA experiment. Same as in ODA experiment, GSS at thresholds larger than 10 mm h⁻¹ is also close to 0, which shows that extreme precipitation is still not resolved. The better GSS scores in ODA around FH 4 and 10 mm h⁻¹ threshold is because there is a simulated precipitation center to the south of SGP site that was coincidently collocated with the observed linear rainband, although the initiation progress was different from the observation (0700–1000 UTC in Fig. 7b).

Considering that GSS emphasizes the gridpoint-by-gridpoint match between the observation and the forecast, which is not quite suitable for the mesoscale convection systems given their highly discontinuous nature and the fact that the model physics did not fully capture the evolution of the rainfall propagating eastward, the neighborhood method is also used to better characterize the forecast skills in case of displacement errors in the forecasts.

In FH 1–3, the CDA forecast at 0.5 mm h⁻¹ threshold does not have useful skills until 26-km neighborhood radius, while the forecast at 10 mm h⁻¹ threshold does not have useful skills up to 160-km radius (Fig. 9b). Beyond 10 mm h⁻¹, the forecast does not gain useful skills even at the largest neighborhood size used in this study. At the same period, the ODA experiment shows no useful skills for all the thresholds and neighborhood size combinations (Fig. 9a). For the threshold and neighborhood size combinations encased by the targeted skill, FSSs from CDA are greater than that of ODA by at least 0.25 with maxima larger than 0.5 (Fig. 9c). According to the frequency bias (Fig. 9d), both ODA and CDA experiments underestimate the precipitation for all thresholds, but the underestimation is less severe in the CDA experiment. The frequency bias of the ODA experiment is below 0.4 at 0.1 mm h⁻¹ threshold, and it decreases further as the threshold increases, and remains below 0.2 at thresholds higher than 1.0 mm h⁻¹. For the CDA experiment, at thresholds from 0.1 to 5.0 mm h⁻¹, the frequency bias is around 0.5. The noticeable improvement of CDA over ODA in FSS and FBIAS is likely a result of better initial conditions.

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FSS − FSS_useful of ODA, CDA, and their differences for (a)–(c) FH 1–3, (e)–(g) FH 4–6. The thick red and blue line in (c) and (g) represents the scale at which ODA and CDA reaches the target skill, respectively. FBIAS − 1 of ODA (red bars) and CDA (blue bars) for (d) FH 1–3 and (h) FH 4–6.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-21-0052.1

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In FH 4–6, for the CDA experiment, compared with FH 1–3, the neighborhood size at which the forecast has useful skills has been brought down to around 13 km for 0.5 mm h⁻¹ threshold and 70 km for 10.0 mm h⁻¹ threshold (Fig. 9f). Meanwhile, the ODA experiment starts to show useful skills, and the targeted skill is reached around 30 km for 0.5 mm h⁻¹ threshold and 80 km for 10 mm h⁻¹ thresholds (Fig. 9e). It is clear that the targeted skills for thresholds between 0.1 and 10 mm h⁻¹ are obtained at smaller neighborhood size in the CDA experiment (Fig. 9g). For the area bounded by the target skills, CDA’s FSS values remain above those of ODA for the most part, with maximum differences around 0.09 for neighborhood size less than 90 km. For the frequency bias (Fig. 9h), both of the experiments still underestimate at all thresholds, but in comparison to FH 1–3, not only the underestimation is reduced, but also the differences between these two are decreased.