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  • View in gallery
    Fig. 1.

    Composite NEXRAD radar reflectivity at (a) 0600 and (b) 1200 UTC 20 May 2011. The red circle is the SGP Central Facility (CF) site.

  • View in gallery
    Fig. 2.

    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.

  • View in gallery
    Fig. 3.

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

  • View in gallery
    Fig. 4.

    (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.

  • View in gallery
    Fig. 5.

    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.

  • View in gallery
    Fig. 6.

    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.

  • View in gallery
    Fig. 7.

    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).

  • View in gallery
    Fig. 8.

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

  • View in gallery
    Fig. 9.

    FSS − FSSuseful 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.

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A Constrained Data Assimilation Algorithm Based on GSI Hybrid 3D-EnVar and Its Application

Jia WangaSchool of Marine and Atmospheric Sciences, Stony Brook University, Stony Brook, New York

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Minghua ZhangaSchool of Marine and Atmospheric Sciences, Stony Brook University, Stony Brook, New York

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Abstract

Data assimilation (DA) at mesoscales is important for severe weather forecasts, yet the techniques of data assimilation at this scale remain a challenge. This study introduces dynamical constraints in the Gridpoint Statistical Interpolation (GSI) three-dimensional ensemble variational (3D-EnVar) data assimilation algorithm to enable the use of high-resolution surface observations of precipitation to improve atmospheric analysis at mesoscales. The constraints use the conservations of mass and moisture. Mass constraint suppresses the unphysical high-frequency oscillation, while moisture conservation constrains the atmospheric states to conform with the observed high-resolution precipitation. We show that the constrained data assimilation (CDA) algorithm significantly reduced the spurious residuals of the mass and moisture budgets compared to the original data assimilation (ODA). A case study is presented for a squall line over the Southern Great Plains on 20 May 2011 during Midlatitude Continental Convective Clouds Experiment (MC3E) of the Atmospheric Radiation Measurement (ARM) program by using ODA or CDA analysis as initial condition of forecasts. The state variables, and the location and intensity of the squall line are better simulated in the CDA experiment. Results show how surface observation of precipitation can be used to improve atmospheric analysis through data assimilation by using the dynamical constraints of mass and moisture conservations.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jia Wang, jia.wang.1@stonybrook.edu

Abstract

Data assimilation (DA) at mesoscales is important for severe weather forecasts, yet the techniques of data assimilation at this scale remain a challenge. This study introduces dynamical constraints in the Gridpoint Statistical Interpolation (GSI) three-dimensional ensemble variational (3D-EnVar) data assimilation algorithm to enable the use of high-resolution surface observations of precipitation to improve atmospheric analysis at mesoscales. The constraints use the conservations of mass and moisture. Mass constraint suppresses the unphysical high-frequency oscillation, while moisture conservation constrains the atmospheric states to conform with the observed high-resolution precipitation. We show that the constrained data assimilation (CDA) algorithm significantly reduced the spurious residuals of the mass and moisture budgets compared to the original data assimilation (ODA). A case study is presented for a squall line over the Southern Great Plains on 20 May 2011 during Midlatitude Continental Convective Clouds Experiment (MC3E) of the Atmospheric Radiation Measurement (ARM) program by using ODA or CDA analysis as initial condition of forecasts. The state variables, and the location and intensity of the squall line are better simulated in the CDA experiment. Results show how surface observation of precipitation can be used to improve atmospheric analysis through data assimilation by using the dynamical constraints of mass and moisture conservations.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jia Wang, jia.wang.1@stonybrook.edu

1. Introduction

The demands for high quality atmospheric analysis at mesoscales have become more prominent in the last decade. Such data are needed as initial conditions to improve the accuracy of short-range forecasts of weather events, especially severe weather events. They are also needed to evaluate and improve the physical parameterizations in weather forecasting and climate models. The quality of the analysis depends on the coverage and precision of the observations, data assimilation techniques, error statistics, and the fidelity of the numerical model.

Substantial progress has been made in the last two decades to enhance the coverage of atmospheric measurements, in particular those from radars, satellites, and surface Mesonet stations. The increasing availability of rainfall observations from the dense network of surface measurements has created opportunities to improve the data assimilation product at mesoscales. However, assimilating precipitation into atmospheric models remains a challenge because of its asynoptic nature. Previous studies to assimilate rainfall can be grouped into three categories. The first is to assimilate hydrometeor-affected parameters, such as microwave radiances by using a scattering radiative transfer model (Bauer et al. 2010, 2006a,b; Geer et al. 2018), or radar reflectivity by augmenting the control variables to include precipitating hydrometeor mixing ratios and number concentrations (Carley 2012; Wang et al. 2013; Wang and Liu 2019; Wheatley et al. 2015) or reflectivity itself (Duda et al. 2019; Wang and Wang 2017). The second is to use observed precipitation to constrain the total latent heating in the assimilation system (Benjamin et al. 2016; Lin et al. 2005; Weygandt and Benjamin 2007). The third is to directly assimilate instantaneous rain rate or accumulated rainfall by using various types of precipitation observation operators, including those used in the four-dimensional variational (4D-Var) scheme (Ban et al. 2017; Hou et al. 2004; Lopez 2011; Marécal and Mahfouf 2006; Sun et al. 2020).

Each of these approaches has its advantages and shortcomings. Assimilation of microwave radiances or radar reflectivity depends heavily on the accuracy of the parameterization of cloud microphysics and precipitation. Using precipitation to constrain latent heating relies on the vertical heating profile which depends on the convection and stratiform condensation schemes. Direct assimilation of precipitation using an observational operator rests on the accuracy of the operator which also depends on the model physical parameterization schemes.

In this study, we describe an algorithm to assimilate precipitation by adding dynamical constraints into the Gridpoint Statistical Interpolation (GSI) hybrid three-dimensional ensemble variational (3D-EnVar) system. We use precipitation to enforce column-integrated conservation of mass and total water. The importance of these conservations has been demonstrated in the observation objective analysis (Zhang and Lin 1997; Zhang et al. 2001) and simple model experiments (Janjić et al. 2014; Ruckstuhl and Janjić 2018). The constraints can produce a similar effect as the observation operator. Different from the standard 4D-Var algorithm, this operator does not depend on the physical parameterization of the assimilation system. Relative to 4D-Var, using the two sets of dynamical constraints instead of the whole model dynamics and physics packages simplifies the construction and reduces the computational cost. This enables the algorithm to be more easily used beyond research communities. The constrained data assimilation (CDA) utilizes the hybrid framework to benefit from the flow-dependent ensemble error covariance.

The outline of this paper is as follows: section 2 describes the constrained algorithm and its implementation. Section 3 presents the experimental design. Section 4 shows a case study of a squall line on 20 May 2011 over the Southern Great Plains (SGP) during the Atmospheric Radiation Measurement (ARM) Midlatitude Continental Convective Clouds Experiment (MC3E) field campaign, including evaluation of the analysis and the forecast results. The last section contains a summary and discussion.

2. Constrained hybrid 3D-EnVar based on GSI

a. GSI hybrid 3D-EnVar analysis

Different from the three-dimensional variational (3D-Var) algorithm that uses a static background error covariance, the hybrid EnVar incorporates flow-dependent ensemble error covariances (Hamill and Snyder 2000; Wang 2010). The inclusion of the flow-dependent ensemble covariance on top of the static one has demonstrated its advantages in several studies and has been implemented in the operational centers (Benjamin et al. 2016). Similar to 3D-Var, the hybrid EnVar finds the optimal analysis through minimizing the cost function, which is formulated as
J(x1,a)=Jb+Jo=β112x1TBS1x1+β212aTA1a+12(Hxd)TR1(Hxd),
where the analysis increment x′ is a sum of the static x1 and ensemble component, superscript T stands for a transpose, BS represents the static background error covariance, vectors a are the extended control variables, A is a block-diagonal matrix that modulates the ensemble covariance, d is the observation innovation, H is the linearized observation operator, and R denotes the observation error covariance. The full background error covariance could be considered as a combination of the static and the ensemble covariance with the inverse of β1 and β2 representing the weight assigned to each of them.

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

To reduce the mass and moisture residuals in the final analysis and potentially improve the subsequent forecasts, dynamical constraints are added during the data assimilation. The constraints are implemented on the native grids used by the background field. The background field is from the Weather Research and Forecasting (WRF) Model. It uses the terrain-following hydrostatic-pressure vertical coordinate η defined (Skamarock et al. 2008) as follows:
η=pdhpdhtμd=pdhpdhtpdhspdht,
where μd represents the mass of the dry air per unit area within the column, and pdh is the hydrostatic pressure of the dry atmosphere, while pdhs and pdht is the hydrostatic pressure at the surface and at the top of dry atmosphere, respectively.
The mass and moisture equations in η coordinate can be written as
μdt+3(μdV)=0,
μdqmt+3(μdVqm)=Fqm,
with kinematic boundary conditions (Laprise 1992) as follows:
η˙|η=1=η˙|η=0=0,whereη˙=dηdt.
Here, V is the velocity vector denoted as (u,υ,η˙), qm are the mixing ratios for water vapor, cloud, rain, ice, snow, graupel, and Fqm represents source and sink terms arising from model physics. The formula of the flux divergence is 3(Va)=(ua/x)+(υa/y)+(η˙a/η).
Vertical integration of Eq. (3) together with the boundary conditions Eq. (5) yields the column-integrated mass conservation equation:
μdt+01η(μdVη)dη=0,
where Vn is the horizontal wind component, and ∇η represents horizontal divergence.
Combining the moisture equations Eq. (4) for water vapor and all hydrometeors leads to the prognostic equation for the total water content. After vertical integration and applying the boundary condition Eq. (5), we can write the column-integrated total moisture equation as follows:
01μdqtdηt+01η(μdVηqt)dη=EP,
where qt is the mixing ratio for water vapor and all hydrometeors, E is the surface evaporation rate, and P is the precipitation rate.
In our initial implementation, we approximated the above constraints by the following:
01η(μdVη)dη0,
01μdqtdηt+01η(μdVηqυ)dηEP.
The zero divergence assumption in Eq. (8) is based on the fact that the spurious mass residual in original analysis is much larger than the observed change of surface pressure. Because hydrometeor mixing ratios are not used as control variables in the current system, the water vapor mixing ratio qυ is used as a substitute for the mixing ratio of total water content qt in the divergence term in Eq. (9). The impact of this substitution will be studied further in future.

c. Implementation of the constrained algorithm

The dynamical constraints are introduced as the penalty terms into the cost function:
J=Jb+Jo+JMass+JMoist,
where the third and fourth term JMass and JMoist corresponds to the mass and moisture constraint, respectively. They are applied as weak constraints, so the penalty terms are written in the quadratic form:
JM=12(Kxb)TΛ(Kxb),
where Λ denotes a diagonal matrix with each element acting as the weighting factor for the constraint at each column, and its values could be uniform for the whole domain or varying according to the importance and the uncertainties of the constraints. The expression (Kx′ − b) is the dynamical constraint, and will be explained in detail later in this section.
The constrained algorithm still follows the preconditioned conjugate gradient framework in GSI, but with modifications in the gradient calculation accordingly. Based on the modified cost function, the gradients are changed into
xJ=(x1JaJ)=B1x+CTHTR1(HCxd)+CTKTΛ(KCxb),
where C acts as a conversion matrix between the control variables and the analysis increments, x=C(x1a)=Cx. The formulas for the searching directions and step size remain the same, with the gradients substituted using counterparts from Eq. (12).

Similar to the background error covariance B in GSI that is not explicitly calculated but constructed by several components, matrix 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 K1 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 K2 is used to convert the specific humidity to mixing ratio. Mixing ratio w holds the following relationship with specific humidity q: wg+w=(qg+q)/[1(qg+q)], 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 wq/(1qg)2, with qg updated by the outer loops.

  3. Matrix K3 is used to convert the surface pressure to the dry air mass in the column. In GSI, the surface pressure (psa=psg+ps, and the superscript a represents analysis) consists of the hydrostatic pressure of the dry atmosphere at the surface (μda+pT, and pT is the pressure at the model top) and column-integrated water vapor pressure (μda01wadη). We therefore have the dry air mass within the column as μda=(psapT)/(1+01wadη). Appling the Taylor expansion, the increment for the dry air mass is approximated as μd[ps/(1+01wgdη)][(psgpT)/(1+01wgdη)2]01wdη.

  4. Matrix K4 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:
    η(μdVη)=mxmy(uμdmyX+υμdmxY),
    in which, mx and my 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 KT, 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
    η(μdaVηa)η(μdgVηg)+η(μdgVη)+η(μdVηg),
    where Vηg and μdg are kept as constants during the inner loops, and updated by the increments at the end of each outer loop, and the calculation η(μdgVηg) is included in term b of Eq. (11). The same linearization approximation is applied to the moisture flux divergence.
  5. Matrix K5 is used to integrate the divergence, either mass or moisture flux divergence, along the η-coordinate from top to bottom.

Then matrix K is constructed as K = K5K4K3K2K1. Considering the linearity of these steps, its adjoint KT is a combination of the adjoint of each above step, and written as KT=K1TK2TK3TK4TK5T, and could be verified through the relationship (Kx)T(Kx) = xTKT(Kx).

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.

3. Experimental design

The ARM MC3E field campaign has been described in Jensen et al. (2016) and Xie et al. (2014). The 20 May 2011 squall-line case has been studied in Fan et al. (2017) which showed that the WRF model could not capture the precipitation location in the simulations when it is initialized with the NCEP Final (FNL) Operational Global Analysis data with different microphysics schemes. In this study, two experiments will be compared: one using the original GSI hybrid data assimilation algorithm, referred to as ODA; the other using the constrained algorithm as CDA. The case features a typical squall-line system, with a leading line of heavy convective precipitation and broad trailing stratiform precipitation at its peak at 1200 UTC [Fig. 1, in which, the composite reflectivity is from https://rda.ucar.edu/datasets/ds841.0/ (Bowman and Homeyer 2017)]. The squall line developed in the warm sector ahead of a cold front. The south and southeast wind originating from the Gulf of Mexico brought abundant moisture into Texas and Oklahoma.

Fig. 1.
Fig. 1.

Composite NEXRAD radar reflectivity at (a) 0600 and (b) 1200 UTC 20 May 2011. The red circle is the SGP Central Facility (CF) site.

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

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 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 B is given by the factor of 1/β1 and 1/β2, which satisfies (1/β1)+(1/β2)=1. In the Rapid Refresh (RAP) version 3, 1/β1 and 1/β2 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).

The matrix Λ in Eq. (11) depends on the accuracy of the constraint dataset and the relative weighting of each constraint. The ΛMoist in the moisture constraint is formulated as follows:
ΛMoist=1(EMoist×9.83600)2,
where EMoist is the error standard deviation (mm h−1), the other two parameters are used to convert the unit of EMoist from millimeters per hour (mm h−1) to pascals per second (Pa s−1), and the denominator represents the error variances.

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, EMoist 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), EMoist 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.

Similarly, the weighting factor in the mass constraint is formulated as follows:
ΛMass=1(EMass×1003600)2,
where EMass is the error standard deviation (hPa h−1), and the other two parameters are used to convert the unit of EMass to Pa s−1. In the dynamical constraints, EMass and EMoist determine the relative significance of mass and moisture conservations. To reach a compromise between these two, EMass is set to be 90 hPa h−1.

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.

4. Analyses and forecasts

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.

Fig. 2.
Fig. 2.

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

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.

Fig. 3.
Fig. 3.

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

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.

Fig. 4.
Fig. 4.

(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

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.

Fig. 5.
Fig. 5.

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

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.

Fig. 6.
Fig. 6.

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

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.

Fig. 7.
Fig. 7.

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

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.

Two traditional metrics are used: the Gilbert skill score (GSS) and the frequency bias (FBIAS), which are given by
GSS=hitshitsrandomhits+misses+falsealarmshitsrandom,
FBIAS=hits+falsealarmshits+misses,
where hitsrandom represents the number of hits by chance. GSS ranges from −1/3 to 1 with 1 as the perfect score, and FBIAS larger (smaller) than 1 represents events are over (under) forecasted.
The fractions skill score (FSS) from the neighborhood method is computed as follows:
FSS=11NN(PfPo)21N(NPf2+NPo2),
where N is the number of neighborhoods for each neighborhood size, and Pf (Po) is the ratio of the area where the simulated (observed) precipitation exceeds the specified threshold to the neighborhood size. FSS ranges from 0 to 1, with 1 representing a perfect score. Following Roberts and Lean (2008) and Stratman et al. (2013), a target skill which could be considered as the average between a random and a perfect forecast skill is adopted, and calculated by
FSSuseful=0.5+BASER/2,
where BASER represents the base rate, which is the fraction of area at which the field exceeds the threshold over the domain. Forecasts with skills above the target skill (FSS − FSSuseful > 0) are considered to be useful. As indicated by the formulas, GSS assesses the forecast skills grid point by grid point, while FSS allows forecasts within a certain neighborhood of the observation to be considered skillful (Stratman et al. 2013).

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.

Fig. 8.
Fig. 8.

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

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.

Fig. 9.
Fig. 9.

FSS − FSSuseful 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

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.

5. Summary and discussion

We have described a data assimilation algorithm that enforces mass and moisture dynamical constraints in the GSI hybrid EnVar. The algorithm enables high-resolution surface precipitation to be assimilated. Two experiments are conducted as a demonstration of the application. The experiment ODA assimilates conventional and radial wind observations using the original hybrid algorithm, while the experiment CDA assimilates the same set of observations, but using the constrained algorithm with precipitation as input. All other configurations, such as length scales and weights assigned to the static and ensemble covariance, are the same for both experiments. Performance of both assimilation and forecast is investigated using a squall-line case that happened on 20 May 2011 during the MC3E campaign.

We have shown that the mass residual is reduced by an order of magnitude in CDA compared to ODA. The column-integrated moisture flux convergence in the CDA analysis is aligned well with the observed precipitation, which is not the case for ODA analysis. This misalignment in ODA results in larger moisture residuals, and likely causes the misplacement of simulated precipitation in the first few forecast hours. Overall, the increments in temperature and moisture fields are weaker in the ODA analysis than in the CDA analysis. The 6-h forecasts of the state variables valid at 1200 UTC are verified against the radiosonde using RMSEs. The CDA experiment outperforms the ODA above 875 hPa for the most part across variables. The forecast experiments are also evaluated by visual inspections and quantitative metrics of precipitation. Based on the visual comparison, compared to the ODA experiment, the CDA experiment has better convection initiation and more realistic interactions among convective cells through the development. When evaluating the performance using GSS and FSS, CDA outperforms ODA for most of the precipitation thresholds and in the first few forecast hours.

This paper shows only the preliminary results of CDA. Several improvements need to be made in the future. The most obvious is that more case studies are needed to better understand the performance of the algorithm. Second, the ensemble used to provide the ensemble error covariance is formed by a set of short-range forecasts, which makes it straightforward in the construction of the prototype CDA. A more sophisticated ensemble data assimilation and forecast system could be maintained and used to generate ensemble with better error covariance. This ensemble could also be used to produce the time tendency for the moisture constraint at the resolution same as the analysis grid, which could alleviate the problems related to the relatively low-resolution and not real-time ERA5 data. Third, independent data should be used to validate the analysis. Additionally, because hydrometeors are not used as control variables in this GSI variational algorithm, the mixing ratio of water content is approximated by the water vapor mixing ratio. Inclusion of hydrometeors together with observational input of the tendency in the moisture constraint should reduce errors brought in by the approximations in the current implementation of our algorithm. These will be pursued in the future.

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.

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