## 1. Introduction

Data assimilation is now a mature and important area of research and application, with the advent of variational and ensemble Kalman filter techniques in the past few decades. Three-dimensional variational data assimilation (3DVar) and four-dimensional variational data assimilation (4DVar) have been greatly advanced since their introductions into simple and complex numerical weather prediction (NWP) models in the late 1980s and early 1990s, respectively (Lewis and Derber 1985; Le Dimet and Talagrand 1986; Parrish and Derber 1992; Rabier et al. 1993; Zupanski 1993, 1997; Courtier et al. 1998).

Variational data assimilation techniques have been used operationally at many forecast centers due to their substantial benefits for better forecast ever since a new set of background errors was introduced in 1997 (Derber and Bouttier 1999). The first successful implementation of variational data assimilation was at the European Centre for Medium-Range Weather Forecasts (ECMWF) in 1997 (Rabier et al. 2000). With the development of variational assimilation, the increase in computing power and the development of methods to reduce computational cost, variational data assimilation techniques have been implemented in data assimilation systems for various NWP models at many operational centers throughout the world (Lorenc et al. 2000; Rabier 2005; Rawlins et al. 2007; Huang et al. 2009; Gustafsson et al. 2012; Sun and Wang 2013; Panteleev et al. 2015).

In recent years, hybrid data assimilation has been proposed, which is based on variational data assimilation and ensemble Kalman filter data assimilation. Hybrid covariance has been introduced to 3DVar (Hamill and Snyder 2000; Wang et al. 2008a,b; Bishop et al. 2011; Kleist and Ide 2015) and to 4DVar (Zhang and Zhang 2012; Clayton et al. 2013). Moreover, 3DVar or 4DVar ensemble–variational data assimilation (3DEnVar or 4DEnVar, respectively) with localization has been introduced (Liu et al. 2008) and tested using NWP models (Buehner et al. 2010a,b); it was also compared with other hybrid methods (Schwartz et al. 2013; Pan et al. 2014; Buehner et al. 2015; Lorenc et al. 2015; Auligné et al. 2016).

Generally, analysis results of data assimilation are not well balanced dynamically due to observation deficiency, model errors, background errors, and simplified background error covariance. Initialization methods that remove spurious gravity waves from the analysis field have been introduced (Daley 1991). Although 4DVar seeks the model trajectory that best fits the observation, it is inevitable that all the errors mentioned above will be present in the analysis field. As a result, spurious high-frequency noise also exists (Courtier and Talagrand 1990). Thus, initialization is necessary even for a 4DVar system (Polavarapu et al. 2000). Some researchers have focused on using weak constraints to reduce the dynamic imbalance between model variables based on the idea that unbalanced initial conditions often generate high-frequency noise with amplitude larger than that observed in nature.

One of the initialization methods used in variational data assimilation is digital filter initialization (DFI), which was proposed by Lynch and Huang (1992) and was used in 4DVar systems by Polavarapu et al. (2000) and by Wee and Kuo (2004). DFI is realized by a digital filter during model integrations (both forward and backward) to remove high-frequency noise (Huang and Lynch 1993), and it improves short-term precipitation forecast (Chen and Huang 2006). With the development of variational data assimilation, the DFI method was incorporated into the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) (Chen and Huang 2006), the WRF Data Assimilation System (WRFDA) (Huang et al. 2007), the ensemble Kalman filter scheme (Ancell 2012), the Rapid Refresh (RAP) scheme (Peckham et al. 2008, 2016), and even in hybrid ensemble–variational data assimilation (Lorenc et al. 2015). Considering the physical mechanisms of DFI are not sufficiently clear, another approach has been taken that involves adding physical constraints to the cost function. The proposed physical constraints include the steady advective–diffusive equation in McIntosh and Veronis (1993); the mass continuity and smoothness in Gao et al. (1999); the temporal and spatial smoothness penalty functions in Sun and Crook (2001); the column model physics in Hou and Zhang (2007); and in particular the dynamic constraints in Brasseur (1991), Ishikawa et al. (2001), Kleist et al. (2009), Ngodock and Carrier (2014), Li et al. (2015), and Vendrasco et al. (2016). In these approaches, physical constraints act as filters to remove high-frequency noise as DFI does. In this study, these methods are referred to as physical filter initialization (PFI).

Liang et al. (2007) proposed a model-constrained 3DVar (MC-3DVar) technique to apply the full physics and dynamics of a numerical model as constraints to 3DVar. MC-3DVar can dramatically reduce high-frequency noise in the analysis fields and has yielded satisfactory results. However, 3DVar cannot assimilate observations at different times synchronously. Despite its high computational cost, the 4DVar system has three key advantages over the 3DVar system: the capability to use observations within the assimilation window, the flow-dependent forecast error covariances, and the use of a forecast model as a constraint to enhance the dynamic balance of the analysis field (Huang et al. 2009).

In this study, the MC-3DVar technique is extended to 4DVar, and the new technique is implemented in the 4DVar system of the Weather Research and Forecasting (WRF) Model (WRFDA-4DVar) (Huang et al. 2009), by adding a penalty term to the cost function as a weak constraint. This approach is referred to as the 4DVar with the PFI scheme (PFI-4DVar). The impact of the penalty term is tested by using two single-observation experiments. Then, the capability of PFI-4DVar in precipitation prediction is verified using real-data assimilation experiments.

This paper has the following structure. An overview of WRFDA-4DVar and the introduction of PFI-4DVar are included in section 2. In section 3, the analysis increments of two single-observation experiments are compared between 4DVar and PFI-4DVar. The forecasting capability of 4DVar with and without the PFI scheme in terms of precipitation is verified by 10 precipitation cases in section 4. Finally, a summary and conclusions are given in section 5.

## 2. PFI-4DVar technique

Equation (2.9) also shows that the high-frequency noise in PFI-4DVar is eliminated by adjusting the balance between variables. This differs from the time-filtering scheme of the model or the DFI scheme: The function of the time-filtering scheme is to dissipate the energy of high-frequency waves directly, while the DFI scheme removes high-frequency waves directly and indiscriminately.

## 3. Single-observation experiments

4DVar is used to assimilate the observations that are distributed in an assimilation window. Generally, the background error covariance and the model are not perfect, and observations also have errors, which will induce high-frequency noise in the analysis field. Observations also include several types of frequencies. As a result, the analysis field of 4DVar contains both low- and high-frequency noise. In a traditional 4DVar, the physics and dynamics of the model have strong constraints. The physics and dynamics of the model can contain both low and high frequencies, so we cannot filter out only the high-frequency noise in the form of a strong constraint. Instead, the model constraints can be applied in 4DVar as a penalty term to filter out high-frequency noise, as shown in (2.2). For the expression of the penalty term shown in (2.3),

Two experiments were carried out to demonstrate the filtering capability of the PFI scheme in 4DVar: a traditional 4DVar experiment (Exp CTL) and a 4DVar experiment using the PFI scheme (Exp PFI). The background field and boundary conditions were from the Final (FNL) Operational Global Analysis data of the National Centers for Environmental Prediction (NCEP) at 0000 UTC 11 July 2015. Here, we defined an almost southerly wind at 0100 UTC, with a *υ* component of 4.86 m s^{−1}, as the observation at one point (39.73°N, 117.28°E; elevation of 5285.1 m), namely, a single observation at one model grid point. The single observation was assimilated into both experiments. The difference between the observation and background fields is primarily in the zonal direction with a slight northerly increment, with values of the *u* and *υ* components approximately −8.80 and 0.59 m s^{−1}, respectively. The assimilation time window of the experiments was 0000–0200 UTC, namely, two time slots.

In Exp CTL, the cost function dropped quickly, as shown in Fig. 1. The distance between the analysis and background fields was close to zero (not shown). In the cost function, the fit to the observation played the leading role. After the single observation was assimilated into the model, the wind and height increment fields became unbalanced, because of the imperfect background error covariance and gravity waves resolved by the model. The height field changed abruptly in response to the wind observation. In this process, high-frequency noise was excited in the analysis field. However, the cost function in Exp PFI dropped more slowly, due to the weak constraint of the penalty term. The cost of the penalty term increased rapidly during initial iterations and then slowly declined with time until convergence was reached. The increase in

The analysis increments of the two experiments at the initial time and the model integration results during the first 6 h are shown in Figs. 2 and 3, respectively. In Exp CTL, the increment of the height field showed a large variation with time, especially at the beginning of the model integration. This was due to the fact that high-frequency noise was excited in the analysis field and propagated outward with time. The spatial distributions of the increments of the height and wind fields in Exp CTL were unchanged after just a few hours. Moreover, the increment of the horizontal wind field showed a pair of symmetric vortexes. In contrast, the increment of the height field was relatively small and changed little with time in Exp PFI. The vortexes in the increment wind field were asymmetric with respect to the height field. In other words, the increments in Exp PFI were balanced and showed a flow-dependent pattern due to the penalty term filtering out the high-frequency noise excited by the assimilated observation at the initial time.

The height field of the background field (contour, gpm), analysis increments of the height field (shading, gpm), and the horizontal wind field (vector, m s^{−1}) at 500 hPa in Exp CTL, integrated from 0000 to 0600 UTC. The time for each snapshot is shown in the upper-left corner of each panel, and the time interval between two neighboring panels varies.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

The height field of the background field (contour, gpm), analysis increments of the height field (shading, gpm), and the horizontal wind field (vector, m s^{−1}) at 500 hPa in Exp CTL, integrated from 0000 to 0600 UTC. The time for each snapshot is shown in the upper-left corner of each panel, and the time interval between two neighboring panels varies.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

The height field of the background field (contour, gpm), analysis increments of the height field (shading, gpm), and the horizontal wind field (vector, m s^{−1}) at 500 hPa in Exp CTL, integrated from 0000 to 0600 UTC. The time for each snapshot is shown in the upper-left corner of each panel, and the time interval between two neighboring panels varies.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

As in Fig. 2, but for Exp PFI.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

As in Fig. 2, but for Exp PFI.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

As in Fig. 2, but for Exp PFI.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

After integrating for several hours, the increments of the height and wind fields in Exp CTL gradually became consistent with those in Exp PFI. An unbalanced analysis field of Exp CTL, as an initial condition, would certainly excite high-frequency noise at the initial time and could be dissipated during the process of integration over several hours. Therefore, the forecast of the first several hours generally contains high-frequency noise introduced by the data assimilation process without any initialization scheme. In comparison, the PFI scheme can achieve a well-balanced analysis field and requires less time to dissipate the noise; that is, it shortens the spinup time.

The (a) RMS of the SLP analysis increment (hPa) and (b) its temporal variation (hPa) between the analysis field and the background field in Exp CTL (solid line) and Exp PFI (dashed line) as a function of integration time (min).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

The (a) RMS of the SLP analysis increment (hPa) and (b) its temporal variation (hPa) between the analysis field and the background field in Exp CTL (solid line) and Exp PFI (dashed line) as a function of integration time (min).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

The (a) RMS of the SLP analysis increment (hPa) and (b) its temporal variation (hPa) between the analysis field and the background field in Exp CTL (solid line) and Exp PFI (dashed line) as a function of integration time (min).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

## 4. Real-data experiments

### a. Case selection and model configuration

In this study, 10 precipitation cases during August 2015 over the region of north China were selected to assess the capability of the PFI scheme in precipitation prediction. The dates of the cases are 1, 3, 4, 5, 7, 18, 19, 22, 23, and 30 August. Two sets of experiments using the WRFDA-4DVar system with and without the PFI scheme were performed for the 10 cases.

For each of these 10 cases, three experiments were run: the simulation test without data assimilation (Exp noDA), WRFDA-4DVar with real data but without the PFI scheme (Exp CTL-RD), and WRFDA-4DVar with real data and with the PFI scheme (Exp PFI-RD). The time window of assimilation was set to 3 h (because the interval of surface synoptic observation is 3 h generally) each over seven slots, and 24-h forecasts were thus produced. The penalty term was calculated at every time step (every 50 s) of the model integration within the assimilation window. The background field and boundary conditions were interpolated from the NCEP Global Forecast System (GFS) 6-h forecast. The model domain has 100 × 80 grid points horizontally and 50 levels vertically. The horizontal resolution is 9 km. The model top is at 50 hPa, and the domain center is at 40°N, 114°E. The following physics and parameterization schemes were used in all simulations: the Thompson microphysics scheme (Thompson et al. 2008), the RRTM longwave radiation scheme (Mlawer et al. 1997), the Dudhia shortwave radiation scheme (Dudhia 1989), the revised MM5 Monin–Obukhov surface layer scheme (Jiménez et al. 2012), a thermal diffusion land surface model, and the Yonsei University planetary boundary layer scheme (Hong et al. 2006).

The observation dataset was from the operational regional numerical prediction system the Beijing–rapid update cycle forecasting system (BJ-RUC) (Chen et al. 2009), which is based on WRF. The conventional observations consisted of surface synoptic observations (SYNOP; every 3 h), automatic weather station (AWS) observations in Beijing, China (every hour), and aircraft reports (AIREP; every 0.5 h). In addition to the radiosondes (RTEMP at 0000 UTC, RTEMP is the keyword for searching radiosondes dataset), the global positioning system precipitable water (GPSPW, every hour) of Beijing, Tianjin, Hebei province, and Shanxi province (all in China) was assimilated. The initial time of each case was 0000 UTC, as mentioned above; the time window of assimilation was 0000–0300 UTC, and the subwindow length was 30 min. In summary, various types of observations were assimilated into the model every 30 min from 0000 to 0300 UTC.

### b. Selection of weight coefficient

Following Liang et al. (2007), in the cost function of the PFI scheme, a weight coefficient was chosen to balance the two parts of the cost function: one is the distance from a model variable to the observation field (*A* − *O*) or to the background field, and the other is the penalty term. If the weight coefficient is set to zero, then the PFI-4DVar is equivalent to the traditional 4DVar. In Fig. 5, the three panels all pertain to the case of 4 September 2015 but with different weight coefficients. When a small weight was given to the penalty term (i.e., *A* − *O*) part of the cost function and the smaller the high-frequency noise in the analysis field. However, a large weight (i.e.,

Changes in the cost function with different weights (shown in the upper-left corner) for both Exp CTL and Exp PFI. Note the iteration time (*x* axis) varies in each panel.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Changes in the cost function with different weights (shown in the upper-left corner) for both Exp CTL and Exp PFI. Note the iteration time (*x* axis) varies in each panel.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Changes in the cost function with different weights (shown in the upper-left corner) for both Exp CTL and Exp PFI. Note the iteration time (*x* axis) varies in each panel.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

### c. Case study of 18 August 2015 event

To examine the performance of the PFI scheme in precipitation prediction, the case on 18 August 2015 was investigated in detail. This case was chosen because there were a convective system and light rain in most parts of the model domain, which is indicated by the box in Fig. 6. The upper-level trough and the lower-level wind shear were the dominant dynamic systems related to this precipitation process (Fig. 6). The whole layer was moistened at 0000 UTC and the upper layer gradually dried up after 0600 UTC. The upper-layer dry condition and lower-layer wet condition generated unstable energy, which was favorable for precipitation.

The geopotential height at 500 hPa (contour, gpm) and wind field at 850 hPa (vector, m s^{−1}), both at 0600 UTC, and the observed 6-h accumulated precipitation from 0000 to 0600 UTC (gray dots, mm).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

The geopotential height at 500 hPa (contour, gpm) and wind field at 850 hPa (vector, m s^{−1}), both at 0600 UTC, and the observed 6-h accumulated precipitation from 0000 to 0600 UTC (gray dots, mm).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

The geopotential height at 500 hPa (contour, gpm) and wind field at 850 hPa (vector, m s^{−1}), both at 0600 UTC, and the observed 6-h accumulated precipitation from 0000 to 0600 UTC (gray dots, mm).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Figure 7 shows the hourly accumulated precipitation distribution of the observation (first column), Exp noDA (second column), Exp CTL-RD (third column), and Exp PFI-RD (last column), starting from 0000 UTC. It can be seen that the precipitation in the first 6 h in Exp noDA was very little compared to the observation, and the precipitation increased after 0600 UTC. The precipitation forecast in Exp CTL-RD was better than that in Exp noDA, as the rainfall appeared after a 1-h integration in Exp CTL-RD. However, there were big discrepancies in terms of rainfall distribution. In this regard, the PFI scheme improved the precipitation forecast. This scheme predicted the rainfall from the first hour of the simulation and the forecasted distributions were more consistent with the observations. The benefit of shortened spinup time is reflected in the precipitation prediction shown in Fig. 7.

Distributions of (first column) observed precipitation (mm), and predicted precipitation (mm) from (second column) Exp noDA, (third column) Exp CTL-RD, and (fourth column) Exp PFI-RD during (a) the first 6 h and (b) the second 6 h.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Distributions of (first column) observed precipitation (mm), and predicted precipitation (mm) from (second column) Exp noDA, (third column) Exp CTL-RD, and (fourth column) Exp PFI-RD during (a) the first 6 h and (b) the second 6 h.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Distributions of (first column) observed precipitation (mm), and predicted precipitation (mm) from (second column) Exp noDA, (third column) Exp CTL-RD, and (fourth column) Exp PFI-RD during (a) the first 6 h and (b) the second 6 h.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

To identify the reasons for the precipitation prediction improvement, the analysis increment of the wind field, the divergence field, and the water vapor mixing ratio in the three tests are compared (Fig. 8). The first column (Figs. 8a and 8d) is the analysis increment of the horizontal wind field (vector) in Exp CTL-RD. We can see that an anticyclonic circulation was superimposed onto the background wind field in Exp CTL-RD. And the lower-layer wind shear was enhanced after assimilation. The second column (Figs. 8b and 8e) is the difference in the horizontal wind field (vector) and divergence field (shading) between Exp PFI-RD and Exp CTL-RD (Exp PFI-RD minus Exp CTL-RD). In the third column (Figs. 8c and 8f), the contours are the differences of the water vapor mixing ratio between the two real-data tests (Exp PFI-RD minus Exp CTL-RD). Corresponding to the precipitation amount and distribution in Fig. 7, there were obviously enhanced convergence at 850 hPa and enhanced divergence at 500 hPa over the precipitation area in Exp PFI-RD. Moreover, the water vapor mixing ratio of Exp PFI-RD increased at 850 hPa in the region of convergence and decreased at 500 hPa in the region of divergence, compared to Exp CTL-RD. In other words, the upper-layer divergence and lower-layer convergence of the analysis field obtained from using the PFI scheme provided dynamic conditions favorable for initiating precipitation. Lower-level moisture was greater and converged over the precipitation region, providing a favorable water vapor condition for precipitation. Adding less water vapor in the upper layer produced an unstable stratification for precipitation. As a result, the forecast of precipitation in Exp PFI-RD achieved the best result (Fig. 7), especially during the first 6 h.

(a),(d) The analysis increment of horizontal wind (arrow, m s^{−1}) of Exp CTL-RD. (b),(e) The difference in horizontal wind (arrow, m s^{−1}) and the divergence field (shading, ×10^{−5} s^{−1}) between Exp PFI-RD and Exp CTL-RD (Exp PFI-RD minus Exp CTL-RD). (c),(f) The difference of water vapor mixing ratio (contour, g kg^{−1}) between Exp PFI-RD and Exp CTL-RD (Exp PFI-RD minus Exp CTL-RD). (a)–(c) 500 and (d)–(f) 850 hPa.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

(a),(d) The analysis increment of horizontal wind (arrow, m s^{−1}) of Exp CTL-RD. (b),(e) The difference in horizontal wind (arrow, m s^{−1}) and the divergence field (shading, ×10^{−5} s^{−1}) between Exp PFI-RD and Exp CTL-RD (Exp PFI-RD minus Exp CTL-RD). (c),(f) The difference of water vapor mixing ratio (contour, g kg^{−1}) between Exp PFI-RD and Exp CTL-RD (Exp PFI-RD minus Exp CTL-RD). (a)–(c) 500 and (d)–(f) 850 hPa.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

(a),(d) The analysis increment of horizontal wind (arrow, m s^{−1}) of Exp CTL-RD. (b),(e) The difference in horizontal wind (arrow, m s^{−1}) and the divergence field (shading, ×10^{−5} s^{−1}) between Exp PFI-RD and Exp CTL-RD (Exp PFI-RD minus Exp CTL-RD). (c),(f) The difference of water vapor mixing ratio (contour, g kg^{−1}) between Exp PFI-RD and Exp CTL-RD (Exp PFI-RD minus Exp CTL-RD). (a)–(c) 500 and (d)–(f) 850 hPa.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Next, the RMS of the SLP analysis increment and its temporal change were compared between the two assimilation experiments (not shown). During the first several hours of integration, the RMS in Exp PFI-RD was relatively smaller than that in Exp CTL-RD. As mentioned above, the reasons were that the initial condition (the analysis field of Exp PFI-RD) was well balanced as a result of the filtering effect of the penalty term and that the analysis increment was smaller in Exp PFI-RD. The temporal change of RMS further explains that the amplitude of the RMS of the SLP analysis increment in Exp CTL-RD was larger, especially during the first several hours of integration, primarily due to the high-frequency noise after the observation data were assimilated. The high-frequency noise dissipated gradually during the model integration. As a result, the RMS of the SLP analysis increment changed sharply during the first few hours and became close to the results of Exp PFI-RD later. The high-frequency noise was suppressed effectively in Exp PFI-RD. The wind and pressure fields were balanced dynamically in the model. These results further demonstrate that the PFI-4DVar can shorten the model’s spinup time and improve precipitation prediction effectively.

The area-mean values of hourly accumulated precipitation in both assimilation experiments and observations are compared in Fig. 9. The area-mean value of the model forecast (observation) is the sum of hourly accumulated precipitation at all grid points (stations) in the model domain divided by the number of grids (stations) that received precipitation. To compare these values easily, normalization was first performed by using the maximum of the sequence. We can see that the hourly accumulated precipitation in Exp PFI-RD was obviously higher than that in Exp CTL-RD and was closer to the observation during the first several hours. The precipitation in Exp CTL-RD in the first hour was almost zero. The temporal change pattern in Exp PFI-RD was more similar to the pattern of observations than the pattern in Exp CTL-RD. These results illustrate that the stimulated precipitation with the PFI scheme fit the observation better from the very beginning of the integration, while the spinup problem was obvious in all 4DVar.

Normalized area-mean values of hourly accumulated precipitation (mm) in Exp CTL-RD (solid line), Exp PFI-RD (dashed line), and observation (dotted line) as a function of time (h).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Normalized area-mean values of hourly accumulated precipitation (mm) in Exp CTL-RD (solid line), Exp PFI-RD (dashed line), and observation (dotted line) as a function of time (h).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Normalized area-mean values of hourly accumulated precipitation (mm) in Exp CTL-RD (solid line), Exp PFI-RD (dashed line), and observation (dotted line) as a function of time (h).

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

The TSs and biases for the precipitation thresholds of 0.1, 5.0, 10.0, 25.0, and 50.0 mm are compared in Fig. 10. For the forecast from 0000 to 0600 UTC, the TSs of the PFI-4DVar were higher than those of the traditional 4DVar for all thresholds, while the biases showed higher false alarm rates in both Exp PFI-RD and Exp CTL-RD. For the forecast from 0600 to 1200 UTC, both the 4Dvar and PFI-4DVar improved the TSs but both had larger biases. Figure 7 also shows that the most significant differences between Exp PFI-RD and Exp CTL-RD were in the first few hours (the first 6 h).

TSs and biases of 6-h-accumulated precipitation in the case of 18 Aug 2015: (a) TSs of 0000–0600 UTC, (b) TSs of 0600–1200 UTC, (c) biases of 0000–0600 UTC, and (d) biases of 0600–1200 UTC.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

TSs and biases of 6-h-accumulated precipitation in the case of 18 Aug 2015: (a) TSs of 0000–0600 UTC, (b) TSs of 0600–1200 UTC, (c) biases of 0000–0600 UTC, and (d) biases of 0600–1200 UTC.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

TSs and biases of 6-h-accumulated precipitation in the case of 18 Aug 2015: (a) TSs of 0000–0600 UTC, (b) TSs of 0600–1200 UTC, (c) biases of 0000–0600 UTC, and (d) biases of 0600–1200 UTC.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

The biases in Exp CTL-RD and Exp PFI-RD were much larger than those in Exp noDA. To find out the cause, the number of stations with different precipitation forecast outcomes (hit, false, or miss) in the three tests were examined (Fig. 11). During 0600–1200 UTC, for instance, there were 39 stations with hit prediction, 384 stations with false prediction, and 180 stations with miss prediction for the threshold of 5.0 mm in Exp noDA. However, the number of hit, false, and miss stations were 143, 704, and 76, respectively, in Exp CTL-RD. Clearly, the number of false prediction stations increased in Exp CTL-RD. On the other hand, the cause of larger bias was mainly due to more stations that were miss in Exp noDA became hit in Exp CTL-RD. In (4.2), the number of hit plus miss is always equal to the number of stations with rainfall, but changing from miss to hit and the increased number of false will make the numerator bigger (hit plus false), and lead to a larger bias. It can also be seen from Fig. 7 that Exp CTL-RD had more hit stations but did not decrease the number of false stations compared with Exp noDA. The PFI-4DVar was based on the traditional 4DVar, so it is reasonable that the results in Exp PFI-RD tended to be similar to those in Exp CTL-RD after having integrated for several hours. All in all, the results of this case (and averaged performance of all 10 cases, which will be shown next) indicate that the traditional 4DVar (or the observation data) used in this study can increase the correct precipitation forecast stations but cannot eliminate false forecast stations. Since the focus of this paper is to compare traditional 4DVar with and without the PFI scheme, we will examine this problem in another study.

Distributions of stations with correct (red dots), false alarm (blue dots), or miss (green dots) forecasts in the period from 0600 to 1200 UTC: (a) Exp noDA with the threshold of 0.1 mm, (b) Exp CTL-RD with the threshold of 0.1 mm, (c) Exp noDA with the threshold of 5.0 mm, and (d) Exp CTL-RD with the threshold of 5.0 mm.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Distributions of stations with correct (red dots), false alarm (blue dots), or miss (green dots) forecasts in the period from 0600 to 1200 UTC: (a) Exp noDA with the threshold of 0.1 mm, (b) Exp CTL-RD with the threshold of 0.1 mm, (c) Exp noDA with the threshold of 5.0 mm, and (d) Exp CTL-RD with the threshold of 5.0 mm.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

Distributions of stations with correct (red dots), false alarm (blue dots), or miss (green dots) forecasts in the period from 0600 to 1200 UTC: (a) Exp noDA with the threshold of 0.1 mm, (b) Exp CTL-RD with the threshold of 0.1 mm, (c) Exp noDA with the threshold of 5.0 mm, and (d) Exp CTL-RD with the threshold of 5.0 mm.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

### d. Results of multiple cases

The initial time of the 10 cases was always 0000 UTC, and the integration time was 12 h. The TSs and biases of 6-h-accumulated precipitation from 0000 to 0600 UTC and from 0600 to 1200 UTC of the 10 cases were calculated (Fig. 12). Higher TSs were achieved by Exp PFI-RD. The PFI scheme had positive impacts on the first 6-h-accumulated precipitation prediction for all rainfall thresholds (as mentioned in section 4c) and improved the second 6-h-accumulated rainfall forecast for the on/off threshold (0.1 mm). As mentioned in section 4c, the biases of every 6 h were larger in the assimilation experiments.

As in Fig. 10, but for the average of the 10 cases.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

As in Fig. 10, but for the average of the 10 cases.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

As in Fig. 10, but for the average of the 10 cases.

Citation: Monthly Weather Review 145, 6; 10.1175/MWR-D-16-0274.1

## 5. Conclusions and discussion

In this study, the model-constrained scheme proposed by Liang et al. (2007) was implemented in the WRFDA-4DVar using the PFI scheme (PFI-4DVar). The penalty term in the cost function was defined by the time tendency of the model variable, playing the role of a weak constraint. The tangent linear model and adjoint model were used in the process of calculating the penalty term in the 4DVar system.

Two single-observation assimilation experiments with both traditional 4DVar and PFI-4DVar were carried out to test the filtering scheme. The results showed that a more dynamically balanced analysis field was obtained when using this penalty term. The high-frequency noise in the analysis field and the integration results in Exp PFI were relatively smaller, and the temporal evolution of the analysis increments was stable during the integration, while the noise was obvious in the traditional 4DVar experiment. The high-frequency noise was filtered out effectively by the PFI scheme. In addition, the analysis increments in Exp PFI showed an obvious flow-dependent pattern.

The results from real-data cases indicated that the PFI scheme shortened the spinup time effectively and improved the accuracy of qualitative precipitation prediction. However, both PFI-4DVar and traditional 4DVar experiments had higher false alarm rates. This was because the 4DVar system or observation data used in this study could only increase the number of stations (grid points) with a correct precipitation forecast but could not eliminate a false precipitation forecast at grid points. This is a topic for further research.

The results presented in this paper demonstrated the technique of PFI-4DVar and its impacts on a precipitation forecast, especially on a precipitation nowcast. However, the inclusion of the penalty term increased the computation cost. The emphasis of our further study will be on cutting back on the computation cost of PFI-4DVar and making it practical.

## Acknowledgments

This work is supported by the National Basic Research Program of China (Grant 2013CB430101), the National Key Technology R&D Program of China (Grant 2012BAC22B02), and the National Innovation Project for Meteorological Science and Technology of China: Quality Control, Fusion, and Reanalysis of Meteorological Observations. We thank the anonymous reviewers for their valuable suggestions, which helped to improve our paper, and Dr. Zuojun Yu for editing the paper.

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