1. Introduction
Lightning is a common discharge phenomenon in convective weather. Electric charges are carried by ions and hydrometeors. It is well established that the non-inductive process is the dominant electrification mechanism in convective clouds [
1,
2]. In the supercooled environment in convective clouds, small ice crystals grow owing to vapor diffusion and accretion of supercooled water droplets. During elastic collisions between more or less rimed particles in the supercooled environment, charges are exchanged between the colliding particles. As the non-inductive process occupies the leading role, charges of opposite polarities separate in rebounding collisions between growing graupel pellets. Charges are then separated at the cloud scale by sedimentation, and vertical motions. The strong relationship among lightning, cloud dynamics and microphysical processes means that lightning can be used to monitor the occurrence and development of mesoscale convective systems [
3,
4,
5,
6]. Under the influence of thermodynamics, dynamics, or terrain, an ascending air mass can form thunderstorm clouds in a convectively unstable environment. Inside the thunderstorm clouds, hydrometeors with different velocities continually collide and separate. Charge transfer results in hydrometeors carrying electric charges of different polarity, and lightning occurs as a result of the accumulation of these electric charges. Therefore, lightning is an outcome of sufficiently severe convection.
Lightning network data are a new type of observation in meteorology and their application and investigation are subjects of current research. Lightning data are relatively unaffected by geographical constraints, and are available at higher temporal and spatial resolution than meteorological radar observations. Lightning data assimilation is a key topic in contemporary research. The difficulty is that lightning flash rate, electric field and charge density are not modeled or prognostic variables in most existing models. Hence, the concept of assimilating lightning data is to find a suitable observation operator that links lightning network data with a model or diagnosed variable, or to obtain a proper lightning data proxy which is physically sound, and then to use different methods to assimilate this variable. In past decades, many researchers have tried to find a reliable relationship between lightning data and other meteorological variables based on the microphysical mechanism in convective clouds, such as convective precipitation rate [
7,
8,
9,
10], convective available potential energy (CAPE) [
11,
12], maximum vertical velocity [
13], proxy radar reflectivity [
14,
15,
16,
17], graupel mass [
18], ice mass flux product [
19,
20], and updraft volume [
21,
22].
Several methods have been used to assimilate lightning data. Alexander et al. [
23] demonstrated the benefit of lightning data assimilation. They combined lightning flash observations together with a classic image processing technique to achieve a continuous time series of rain rate, which is challenging over data-sparse regions. Their results show that the lightning data have a greater positive impact on forecasts than that from passive microwave sensors and infrared sensors. Chang et al. [
24] also confirmed that using lightning as a continuous proxy can improve weather forecasts. A different approach used by some researchers is to transform lightning data into convective precipitation rate and then adjust the latent heat profile by nudging. Papadopoulos et al. [
25] used cloud-to-ground lightning data to nudge model-generated humidity profiles towards empirical profiles. Their results show that assimilation of lightning data can significantly improve the prediction of convective precipitation up to 12 h ahead. Stefanescu et al. [
11] treated convective available potential energy as a proxy for lightning data. They selected 1D +
nD-VAR (one dimensional +
n dimensional variational) (
n = 3, 4) method to assimilate ENTLN (the Earth Networks Total Lightning Network) lightning data and evaluated its performance in two severe storm cases. 1D-VAR technique was firstly utilized to obtain temperature increments, which were then added into the background to achieve column temperature retrievals. The column temperature retrievals were then associated with observed lightning data and assimilated into the WRFDA (the Weather Research and Forecasting model Data Assimilation system)
nD-VAR (
n = 3, 4) framework as conventional data. The results showed that the best improvement was with the 1D + 4D-VAR technique, which decreased the precipitation root mean square errors in both case studies. Qie et al. [
26] utilized an empirical formula from Fierro et al. [
27] and substituted the water vapor mixing ratio with the ice-phase mixing ratio to connect total lightning flash rate with ice-phase particles. They then added the nudging function into the WSM6 (the WRF Single-Moment 6-Class) microphysical scheme of the WRF (the Weather Research and Forecasting model)to adjust the mixing ratio of ice-phase particles between the 0 °C and −20 °C isotherms. They found that this method could be used for improving the short-term precipitation forecasting of Mesoscale Convective Systems (MCSs) with high or moderate lightning flash rates. However, adding ice-phase particles in severe convective regions may augment the vertical downdraft that would be an obstacle for the lateral development of convective systems. Thus, their scheme may perform poorly when applied to a wide range of thunderstorm cases. Some researchers have tried treating lightning as an index to control parameterization schemes. Gallus and Segal [
28] used the National Centers for Environmental Prediction (NCEP) Eta Model with a 10 km horizontal grid spacing to simulate 20 MCS cases. They found that adding humidity information in regions where the radar echo is high with low humidity can improve forecasting skill, and that the Kain–Fritsch (KF) parameterization scheme performs better than the Betts–Miller–Janjic (BMJ) parameterization in a cold pool. Mansell et al. [
29] tested the effect of lightning data assimilation on KF cumulus parameterization in a Coupled Ocean–Atmosphere Mesoscale Prediction System. They utilized lightning as an index to control the KF parameterization scheme for the presence or absence of deep convection. With this method, the lightning data assimilation was successful in generating cold pools that were present in surface observations at forecast initialization, and the improvement was obvious in the first few hours. Lightning data were also utilized as a proxy to control the activation of the convective parameterization scheme in the MM5 (Mesoscale Model 5) non-hydrostatic model by Lagouvardos et al. [
30]. The assimilation of lightning was shown to have a positive impact on the representation of the precipitation field, and provided more realistic positioning of precipitation maxima, mainly during the second day of the event. Mansell [
31] designed a set of observing system simulation experiments to demonstrate the potential benefit of assimilating total lightning flash mapping data using the ensemble Kalman filter (EnKF) method. In his assimilation algorithm, a linear relationship between flash rate and graupel echo volume was utilized as the observation operator. In Rapid Refresh, which is the continental-scale National Oceanic and Atmospheric Administration (NOAA) hourly-updated assimilation/modeling system operational at NCEP, lightning data are considered a good supplement to radar data to improve convective coverage off the coast. In this system, lightning data are transformed into proxy radar reflectivity, which is then assimilated into the numerical weather prediction (NWP) model [
14,
15,
16,
17].
In our previous work, we attempted to assimilate lightning data utilizing radar proxy reflectivity transformed in Gridpoint Statistical Interpolation (GSI) code using diverse methods such as physical initialization and cloud analysis [
32,
33,
34]. However, each method had shortcomings; for example, cloud analysis provided stronger precipitation in short-term forecasts after assimilation than was observed [
33]. The physical initialization method provided a better precipitation forecast, but the improvement was short-lived [
32]. These limitations provide motivation for an improved scheme for linking lightning data with other variables. Fierro et al. [
27] suggested a simple relationship between flash rate and the water vapor mixing ratio at 9 km resolution that appears useful. In their case study, the nudging method was used primarily for analysis rather than forecast. This simple and computationally inexpensive assimilation technique can be easily implemented into the WRF model and showed promising improvements in the representation of the convection and cold pools at the analysis time to provide more accurate analysis and forecasts of storm structure and evolution. Marchand and Fuelberg [
35] put forward a method that warmed the most unstable low levels of the atmosphere at locations where lightning was observed but deep convection was not simulated due to the absence of graupel. They compared their simulation results with the Fierro et al. [
27] nudging method. The new method performed better at simulating isolated thunderstorms and other weakly forced deep convection, while the Fierro et al. [
27] nudging method performed better for cases with strong synoptic forcing. In Fierro et al. [
36], the smooth nudging method was also compared with a three-dimensional variational (3DVAR) technique which assimilated radar reflectivity and radial velocity data. A suite of sensitivity experiments revealed that lightning assimilation was better able to capture the position and intensity of reflectivity up to 6 h into the forecast. The computationally inexpensive lightning data nudging method was evaluated with more case studies in Fierro et al. [
37]. The performance of the accumulated precipitation forecast for 67 days in the 2013 USA warm season was evaluated. The results showed that the nudging method had considerable promise for routinely improving short-term (≤6 h) forecasts of high-impact weather with convection-allowing forecast models. The Fierro et al. [
27] nudging method is efficient for initializing convection where lightning is detected, but has limited ability to suppress spurious convection or to modulate convection.
Nudging is a type of economical four-dimensional data assimilation method. The advantage of the nudging method is that it can incorporate assimilated data at proper time even at several integration steps. It relaxes the model state to the observations during the assimilation period by adding a non-physical diffusive-type term to the model equations [
38]. However, the approach is only a fitting procedure between observed data and model variables and lacks a solid theoretical foundation; it does not consider physical process and physical balances between analyzed variables. In variational methods, 3DVAR and 4DVAR are widely used in many operational platforms. The 3DVAR method is popular as it has a high computational efficiency and can directly assimilate various observations (e.g., conventional data, radar data, bogus data, etc.). The basic concepts of 3DVAR and 4DVAR methods are the same except that the 4DVAR method employs an additional set of prognostic equations as a strong constraint [
39,
40]. The 4DVAR method utilizes an optimal control approach based on the adjoint model integration process to obtain the gradient of the cost function, with respect to the control variables for the minimization procedure [
38]. One difference between 3DVAR and 4DVAR is the background error covariance. The 3DVAR framework utilizes a static background error covariance, while in the 4DVAR framework, the background error covariance varies with flow patterns, which may achieve a more detailed analyzed field. Nevertheless, calculating adjoint with high resolution in a 4DVAR framework needs great computational power, which is not available in some real-time operational platforms. As lightning data are a good supplement to radar data, using the 3DVAR method to assimilate lightning data can easily be implemented in many existing operational model platforms, which utilize the same method to assimilate radar data and conventional data for daily real time forecasts. Thus, in this study, a 3DVAR method that can adjust variables within an influence radius is used in the assimilation of total lightning data, and is tested based on the relationship between total flash rate and water vapor.
Based on a large number of case studies tested in Fierro et al. [
37], the lightning data assimilation algorithm in this paper converts lightning data to water vapor mixing ratio via the simple smooth continuous function given in Fierro et al. [
27]. In their study, a9 km gridded resolution total flash rate and simulated graupel mixing ratio were used as input variables. The water vapor mixing ratio is then transformed into relative humidity in the form of sounding data. Finally, the proxy relative humidity is assimilated into the background field utilizing the 3DVAR method in WRFDA. The benefits of assimilating lightning data are demonstrated in a series of experiments using data from a strong convection event that affected Beijing, Tianjin, Hebei and Shandong Province, on 31 July 2007. The remainder of this paper is organized as follows.
Section 2 describes the methodology,
Section 3 presents results and discussion, and
Section 4 summarizes the findings.