Research Article  Open Access
Xiaoyu Gao, Shanhong Gao, "Impact of Multivariate Background Error Covariance on the WRF3DVAR Assimilation for the Yellow Sea Fog Modeling", Advances in Meteorology, vol. 2020, Article ID 8816185, 19 pages, 2020. https://doi.org/10.1155/2020/8816185
Impact of Multivariate Background Error Covariance on the WRF3DVAR Assimilation for the Yellow Sea Fog Modeling
Abstract
Numerical modeling of sea fog is highly sensitive to initial conditions, especially to moisture in the marine atmospheric boundary layer (MABL). Data assimilation plays a vital role in the improvement of initial MABL moisture for sea fog modeling over the Yellow Sea. In this study, the weather research and forecasting (WRF) model and its threedimensional variational (3DVAR) data assimilation module are employed for sea fog simulations. Two kinds of background error (BE) covariances with different control variables (CV) used in WRF3DVAR, that is, CV5 and multivariate BE (CV6), are compared in detail by explorative case studies and a series of application experiments. Statistical verification metrics including probability of detection (POD) and equitable threat scores (ETS) of forecasted sea fog area are computed and compared for simulations with the implementations of CV5 and CV6 in the WRF3DVAR system. The following is found: (1) there exists a dominant negative correlation between temperature and moisture in CV6 near the sea surface, which makes it possible to improve the initial moisture condition in the MABL by assimilation of observed temperature; (2) in general, the performance of the WRF3DVAR assimilation with CV6 is distinctly better, and the results of 10 additional sea fog cases clearly suggest that CV6 is more suitable than CV5 for sea fog modeling. Compared to those with CV5, the average POD and ETS of forecasted sea fog area using 3DVAR with CV6 can be improved by 27.6% and 21.0%, respectively.
1. Introduction
Sea fog occurs within the marine atmospheric boundary layer (MABL). It is a type of fog that forms over an ocean surface and spreads over coastal areas. It can seriously affect many maritime activities (e.g., transport, navigation, fishery, and rescue) due to its low horizontal visibility [1, 2]. The Yellow Sea has the largest number of foggy days among the four seas of China [3–5], with a long fog season starting from March and ending in September. The Yellow Sea is located to the north of the Kuroshio Current and under influences of warmmoist air mass that is transported northward as the East Asia summer monsoon gradually intensifies. Due to the cold sea surface temperature (SST), an atmospheric inversion layer can easily develop in the MABL over the Yellow Sea, leading to the formation of advection sea fog in the inversion layer [3]. Therefore, the sensitivity of sea fog modeling to the initial condition is actually largely related to temperature and moisture in the MABL [6–12].
After the textbook Sea Fog published in 1985 by Wang [4], Koračin and Dorman [2] published a book in 2017, in which they summarize the challenges and advancements in observations, modeling, and forecasting of marine fog over the world. Modeling and forecasting have become an important research content of sea fog. In order to mitigate socioeconomic impacts of sea fog over the Yellow Sea and meet increasing practical demands, it is urgent to improve the forecast skill of sea fog. Undoubtedly, a highquality initial condition is very beneficial to the numerical forecasting for sea fog. A vital way to improve the initial condition in numerical modeling is data assimilation [13–19]. For the assimilation of given observations, different data assimilation methods yield different assimilation effects. In general, better effect can be obtained when using more advanced data assimilation method. For instance, in their modeling study of sea fog using the weather research and forecasting (WRF) model, Gao et al. [20] demonstrated that ensemble Kalman filter (EnKF) performs better than threedimensional variational (3DVAR) assimilation, which is attributed to the flowdependent background error (BE) covariance used in the EnKF. However, the EnKF requires much more computing resources compared to the 3DVAR, which is an issue that has to be considered for realtime forecast of sea fog. The WRF3DVAR [18] has been widely used for sea fog modeling over the Yellow Sea due to its low computational cost [21–24].
3DVAR minimizes the analysis error variance by minimizing the cost function, which is defined by the weighted sum of analysis increment and observation innovation [25], respectively, with BE and observational error (OE) covariance. A better estimate of BE can definitely lead to a better assimilation analysis. Thus, how to conduct reasonable statistics of BE is an important issue, which has spawned a lot of related research works [26–31]. Currently, the National Meteorology Center (NMC; now called the National Centers for Environmental Prediction, NCEP) method [32] is the most commonly used to generate BE for the 3DVAR. It first conducts transformation of model variables to a set of control variables and then estimates the relationships between different variables by linear regression. The NMC method is implemented in the WRF3DVAR [18]. There are two types of BE with different control variables (CV): one defines the control variables in the physical space (CV3), and the other defines the control variables in eigenvector space (CV5, CV6, and CV7). CV3 uses a vertical recursive filter to model the vertical covariance, while the others use an empirical orthogonal function (EOF) to represent the vertical covariance. CV3 is a global BE and can be used for any regional domain, while CV5, CV6, and CV7 are domaindependent and therefore must be generated based on forecast or ensemble data over the same domain. Due to the poor performance of CV3, it is not suitable for sea fog modeling. In CV5, the moisture control variable is not related to any other variables. In CV6, the moisture control variable is the unbalanced portion of the pseudorelative humidity, and six additional correlation coefficients in the definition of the balanced part of analysis control variables are introduced. Namely, CV6 is a multivariate background error covariance, especially for the moisture. CV7 is similar to CV5, but it uses a different set of control variables and there exist no correlations between the moisture and other control variables.
The issue on the strengths and shortcomings of CV5 and CV6 has been discussed. Based on the work to use multiple linear regressions to estimate multivariate analysis of humidity in a limitedarea model, Berre [33] pointed that the relationships between forecast errors of humidity and those of mass and wind fields cannot be negligible. Dhanya and Chandrasekar [34, 35] found that the forecasts of rainfall caused by monsoon depressions and tropical cyclones can be improved by applying data assimilation with CV6, which is better than CV5. However, among existing research work on sea fog over the Yellow Sea involving numerical modeling [24, 36–40], quite a few studies, especially those works with focus on the mechanism of sea fog evolution, have little or no mention of data assimilation. Even if data assimilation is employed in the numerical simulations for sea fog, CV5 is generally used [22, 23, 41–43]. The reason might be that CV5 is the default option except CV3 in the WRF3DVAR, which results in the fact that little attention has been paid to CV6 in the application of data assimilation for sea fog modeling.
As stated previously, a successful sea fog modeling definitely requires the assimilation of observations over the sea [22]. Unfortunately, in spite of a large amount of satelliteretrieved data available over the sea, the accuracy of the retrieved water vapor profiles is unsatisfactory. Divakarla et al. [44] performed an evaluation of satelliteretrieved data using global radiosonde observations and found that the root mean square difference is larger than 15% for relative humidity at 1000 hPa. Such big observational errors lead to a malfunction of the retrieved humidity in data assimilation. The approach to correct errors in the observed humidity field and make it more realistic by increments of other control variables is probably an efficient way to modify the MBL moisture structure. Hence, the purpose of the present study is to investigate the impact of multivariate background error covariance on the WRF3DVAR assimilation for fog modeling over the Yellow Sea. The assimilation effects with CV5 and CV6 are examined.
The remaining content is arranged as follows. Section 2 briefly describes the multivariate BE of the WRF3DVAR and the data assimilation scheme, and Section 3 outlines numerical experiments, including data, sea fog cases, model configurations, and numerical experiments. Results, analysis, and evaluation are presented in detail in Section 4. Finally, summary and conclusions are provided in Section 5.
2. Methodology
2.1. Multivariate BE
The cost function for 3DVAR is defined bywhere x and are, respectively, the model state and the background field, y is the observation, and H is the observation operator matrix that transforms data from model space to observation space. In equation (1), analysis increment and observation innovation are, respectively, weighed by BE (B) and OE (R). OE is determined by instrument errors and representation errors caused by the transformation of data from model space to observation space; BE is determined by model forecast errors. The analysis represents a posterior maximum likelihood estimate of the true state of the atmosphere, and it can be obtained if the cost function J(x) is minimized. The WRF3DVAR system uses QuasiNewton method [45] to realize the minimization process.
Assuming that the model has m degrees of freedom and the assimilated observation number is p, x and are mdimensional vectors, y is an pdimensional vector, H is a p × mdimensional matrix, B is a m × mdimensional matrix, and R is a p × pdimensional matrix that is diagonal because observational errors are usually supposed to be uncorrelated. In practice, the amount of calculation of is much larger than that of , since R is diagonal and p is far less than m.
Usually, a numerical model has m degrees of freedom with a typical value of ; thus it is almost impossible to explicitly solve , which requires calculations. However, in the model space, the correlations between different variables or between the same variables at different grid points are available only within a limited range, and thus BE is a sparse matrix. As a result, by decomposing BE, a lot of computational cost can be greatly reduced. This process is known as control variable transform; that is, the model variables x are transformed to control variables , and can then be calculated by v instead of by x. Let , so . Note that U is the transform matrix defined in a way that satisfies . Using the increment formulation [46], equation (1) can be rewritten as
The WRF3DVAR system takes three steps to complete the control variable transform: the horizontal transform , the vertical transform , and the physical transform . Therefore, . uses recursive filter [47, 48] to calculate the correlations between data on different grid points at the same level, implements empirical orthogonal function (EOF) decomposition to obtain the correlations between data at different model levels, and does linear regression to get the correlations between different variables.
In the physical transform , the increments of model variables, that is, zonal wind u, meridional wind , temperature T, pressure P, and mixing ratio Q, are first converted to a new set of variables, that is, stream function , velocity potential , temperature T, surface pressure , and pseudorelative humidity . These new variables are then divided into balanced and unbalanced components: , where can be any variable. The balanced part is related to other control variables by linear regression ; that is, . The unbalanced part is independent and serves as input to the vertical transform. By the above procedures, the correlated model variables are transformed into uncorrelated variables.
In the WRF3DVAR, the correlations are available only within a subset of the variables. CV5 has three basic coefficients: , , and , and these coefficients link , T, and to :where i and j denote the horizontal dimension index, k and l indicate the vertical levels, n is the total number of vertical levels, and the regression coefficients are usually supposed to be horizontally homogeneous.
Compared to CV5, the multivariate BE (CV6) contains six extra coefficients, that is, . also consists of balanced and unbalanced components. The control variable transform can be expressed by
Formula (4d) indicates that an increment of four other variables can lead to an increment of in CV6. As mentioned in Introduction, moisture information in the initial condition is vital for sea fog modeling. Note that the analysis of moisture is retrieved from the analysis of , background temperature, and pressure together. The moisture increment is directly proportional to increment because background data are recognized as constant.
2.2. Data Assimilation Scheme
A cycling3DVAR scheme [21] is employed to do data assimilation (DA) in this study. The scheme is shown in Figure 1. CV5/6 presents CV5 or CV6, and “obs” stands for observations. Individual assimilation cycles are connected by the WRF integration (i.e., wrf.exe), and the final analysis is taken as the initial condition for the WRF simulation. In this study, all WRF simulations are designed with two domains in twoway nesting.
As shown in Figure 1, the cycling3DVAR scheme includes three 3DVAR updates. Each update is only performed on the outer WRF model domain with its BE, and initial condition of the inner domain is obtained by interpolation of analysis field for the outer domain after its 3rd update. In the case study, the data assimilation period is 6 hours with a DA interval of 3 hours.
The NMC method [32] is used to generate CV5 and CV6, assuming that the BEs are well approximated by the averaged differences between 12hour forecasts and 24hour forecasts, which are expressed bywhere and are 12hour and 24hour forecasts, respectively. These forecasts can be obtained from a period of time (e.g., 15−30 days) or can be got from an ensemble prediction system. The former way is adopted here, and thus equation (5) is rewritten aswhere T_{n} is the time period for statistics. Here, the forecasts are collected within 15 days centered at the initial day of the case simulation.
3. Numerical Experiments
3.1. Data
Initial and lateral boundary conditions for numerical modeling were derived from the NCEP Final Analysis (FNL) Data (1° × 1°; 6 hourly). Bottom boundary conditions were extracted from the daily SST dataset (0.25° × 0.25°) of the NorthEast Asian Regional Global Ocean Observing System (NEARGOOS). Observational data for the 3DVAR data assimilation include conventional surface and upper air measurements from the Global Telecommunication System (GTS), buoy and island measurements, and AIRS (Atmospheric Infrared Sounder) retrieved temperature and humidity profiles.
Surface synoptic maps were downloaded from the Korea Meteorological Administration (KMA) to show weather situations of the sea fog cases in the present study. The observed fog areas were empirically retrieved following the approach of Wang et al. [22], which uses the multichannel data of Japanese Multifunctional Transport Satellite (MTSAT), including albedo, infrared, and visible cloud imageries. The NCEP ship observations in PrepBUFR format, which can be downloaded from the NCAR Computational and Information Systems Laboratory (CISL) Research Data Archive, were used to evaluate model results.
3.2. Sea Fog Cases
Two typical sea fog cases over the Yellow Sea, one spreading narrowly over the eastern Yellow Sea along the Korean Peninsula coast (Figure 2(b)) and the other occupying a broad area over the southern Yellow Sea (Figure 3(b)), were selected for case studies. Additionally, extra 10 sea fog cases were chosen for quantitative evaluation on sea fog area forecast with the application of multivariate background error covariance.
The two typical sea fog cases occurred on 31 March 2011 and 10 April 2009 (hereafter referred to as CaseA and CaseB), respectively. Looking at the correspondence between spatial distributions of weather system and fog patches shown in Figures 2 and 3 (cf. Figure 2(a) with Figure 2(b); cf. Figure 3(a) with Figure 3(a)), both cases formed over the areas controlled by a highpressure system. However, the SST conditions are different for the two cases. CaseA occurred along the west coast of the Korean Peninsula, where the SST was about 5°C (Figure 2(c)). The fog area was under the control of a weak highpressure system with the center located at the Korean Peninsula (Figure 2(a)). Warmmoist air mass was transported from the central Yellow Sea to the cold SST area near the coast by the southwesterly wind (Figure 2(c)). CaseB occurred over an ocean area, where a warm SST tongue and a cold SST tongue (Figure 3(c)) coexisted with large SST gradients. Such an SST distribution is usually favorable for the formation of sea fog [49]. In addition, easterly winds developed over the southern Yellow Sea (Figure 3(c)) due to the influence of the high pressure (Figure 3(a)), transporting huge amounts of warmmoist air mass to the fog area.
3.3. Model and Its Configuration
The Advanced Research core of the WRF (ARW, version 3.8.1) and its corresponding WRF3DVAR were employed in the present study for numerical simulations and data assimilations. Based on the previous work related to sea fog simulation over the Yellow Sea fog [22–24, 42], the Yonsei University (YSU) planetary boundary layer (PBL) scheme and the Lin microphysics scheme (LIN) were selected for the present study. Appropriate, vertical levels were specified as well. Detailed model configurations are listed in Table 1. Due to the distinctly different occurrence locations and areas between the two sea fog cases, the model domains and resolutions were separately designed for the two cases (Figure 4). Twoway nesting was implemented in the simulations.
 
= 1.0000, 0.9975, 0.9925, 0.9850, 0.9775, 0.9700, 0.9540, 0.9340, 0.9090, 0.8800, 0.8506, 0.8212, 0.7918, 0.7625, 0.7084, 0.6573, 0.6090, 0.5634, 0.5204, 0.4798, 0.4415, 0.4055, 0.3716, 0.3397, 0.3097, 0.2815, 0.2551, 0.2303, 0.2071, 0.1854, 0.1651, 0.1461, 0.1284, 0.1118, 0.0965, 0.0822, 0.0689, 0.0566, 0.0452, 0.0346, 0.0249, 0.0159, 0.0076, and 0.0000. 
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3.4. Experimental Design
The experiments in this study were divided into 3 categories: singleobservation experiments, case study experiments, and evaluation experiments. The singleobservation experiments were conducted based on CaseA, with a focus on RHs in CV6. The case study experiments include six experiments, which were conducted to explore the impacts of CV6 on the 3DVAR assimilation. The evaluation experiments were conducted for the purpose of evaluating the effect of CV6 application.
Table 2 lists the case study experiments for CaseA and CaseB. The initial times of these simulations are presented in Table 1, and all the simulations ran out to 24 hours. The assimilation of observations collected at different platforms may have complicated superposition effects, which is not favorable for clear analysis on the assimilation impact. Therefore, only the GTS routine observations were used for assimilation. For the coastal sea fog case, only observations over the southern Korean Peninsula (Figure 4(a)) were assimilated and the data collected at other far away observational sites were excluded. For the widespread sea fog case, all the GTS observations in D1 were assimilated (Figure 4(b)).

A suite of experiments were conducted for 10 additional sea fog cases that occurred over the Yellow Sea during 2007–2014. In order to produce better initial conditions for the WRF model, the assimilation window was extended from 6 hours for the case study to 12 hours for these cases, and the observations assimilated include the GTS routine observations, buoy and ship observations and island measurements, and AIRS retrieved data. As with the BEs of those experiments listed in Table 2, CV5 and CV6 were generated in advance separately for each case using the method described in Section 2.2. The model configurations are the same as those used for the widespread sea fog case (see the description for CaseB in Table 1 and the domains in (Figure 4(b)). The experiments consist of two groups: GroupCV5 running with CV5 and GroupCV6 running with CV6, respectively.
4. Results
4.1. SingleObservation Experiments
4.1.1. Relationship between and
As mentioned in Introduction, there is no relationship between moisture control variable and any other variable in CV5. However, such relationships exist in CV6. Equation (4a) shows that, in CV6, is related to four other variables, and the relationships between them are decided by the regression coefficients. In order to investigate how close these variables are related, contribution rates of other control variables to were calculated from CV6 generated in advance via the NMC method for CaseA. The results are shown in Figure 5(a). At low levels of the model, the balanced component accounts for 30–40% of the total , among which the term contributes up to 20%–30% of the total. Other individual terms can only make a contribution of less than 5%. As a result, the term accounts for more than ∼70% of the balanced component of , indicating that the temperature increment is much more important for the humidity increment than the wind and pressure increments. Similar results are also found for CaseB (figure not shown).
Figure 5(b) presents the distribution of regression coefficient of at level k2 to calculate at level k1 (see equation (4a)). The horizontal/vertical axis stands for k2/k1. The coefficients along the diagonal of Figure 5(b) are all negative, and their absolute values are much larger than other data on the same row. As a result, the humidity increment at any specific point is basically decided by temperature increment at the same point, and a negative increment of always causes a positive increment of .
4.1.2. Contribution Rate of to
A series of singleobservation experiments were designed based on CaseA with CV5/CV6, and the single observation is marked by the red dot shown in Figure 4(a). Each experiment was conducted at different vertical model level with a temperature observation of 1 K lower than the background. These experiments are denoted by ExpSk, where k (from 1 to 20) stands for the vertical level number.
Since is negative, decrease of temperature with a maximum value of about −0.3 K (Figure 6(a)) produces an increase of mixing ratio with a maximum value of about 0.06 g/kg (Figure 6(c)). The maximum values of temperature and mixing ratio increments at levels 1–20 from ExpS1–20 are illustrated in Figures 6(b) and 6(d). Through vertical transform, the negative temperature increment spreads to several vertical levels and causes increases of mixing ratio there. Figure 6(d) has a very similar pattern to that of Figure 6(b), which again shows that the humidity change mainly results from the change of temperature at the same level, though there exist some slight influences from neighboring levels. Compared with CV6, although similar patterns and a little bit weaker intensities of temperature assimilation result are obtained for CV5 (cf. Figure 6(e) with Figure 6(f) and Figure 6(a) with Figure 6(b), respectively), there is not obviously any gain for mixing ratio (Figures 6(g) and 6(h)) due to the fact that there is no relationship between moisture control variable and any other variable.
Overall, for CV6, regression coefficients of the same level are all negative and their absolute values are usually much larger than those of neighboring levels, leading to an increase of mixing ratio when a lower temperature observation is assimilated. For 3DVAR with CV5, assimilation of temperature observations cannot change humidity analysis, since is an independent variable in CV5.
4.2. Case Study 1: The Coastal Sea Fog
4.2.1. Verification
Simulated sea fog is defined to those areas where the following criteria are met [22]: cloud liquid water (CLW) at the model’s lowest level is 0.016 g/kg or cloud top is 400 m. CLW 016 g/kg is equivalent to a visibility 1 km. The cloud top is calculated based on the threshold of CLW 0.016 g/kg; and observations indicate that advection sea fogs are deeper than other types of fog but rarely exceed 400 m [56].
The fog occurred over a narrow ocean area along the western coast of the Korean Peninsula (Figures 7(a)–7(e)). As mentioned in Section 3.4, only observations over the southern Korean Peninsula Figure 4(a) were assimilated, because we want to explore the impact of assimilating observations that are close to the fog area. The fog areas forecasted by ExpA_CV5 and ExpA_CV6 are shown in Figure 7. It is apparent that ExpA_CV6 performs better than ExpA_CV5, because the former can well capture the sea fog evolution, while the latter completely fails to reproduce the sea fog (cf. Figures 7(f)–7(j) and 7(k)–7(o) with 7(a)–7(e), respectively). Note that the difference of 925 hPa wind speeds between these two experiments is less than 1 m/s (figure not shown), which indicates that CV6 makes a trivial change for wind field.
The agreement between the initial conditions and the radiosonde observations (see the big dots in Figure 4(a) for their locations) over the Korean Peninsula was examined. Figure 8 shows the average vertical profiles of bias and root mean square error (RMSE) of temperature and mixing ratio for ExpA_CV5 and ExpA_CV6. Below the height of 600 m, the initial temperatures of both experiments are higher than observations (Figure 8(a)), while most of the initial mixing ratios are smaller than observations (Figure 8(b)). Nevertheless, the mixing ratio bias of ExpA_CV6 is much smaller than that of ExpA_CV5. The same is true for the RMSE. It means that the initial moisture condition of ExpA_CV6 is much wetter than that of ExpA_CV5, which results in the better forecasted sea fog area by ExpA_CV6.
In addition, ship measurements (not assimilated; see their sites in Figure 9(a) were used to verify the moisture condition near the sea surface. Biases between model output and the measurements were calculated and results are shown in Figure 9(b). It is obvious that the initial moisture conditions of both ExpA_CV5 and ExpA_CV6 are drier than the observed ones. However, the dry condition is significantly improved in ExpA_CV6 compared to that in ExpA_CV5. Note that the sites S6, S7, and S8 are close to the Korean Peninsula. Compared to ExpA_CV5, the improvements at these three sites in ExpA_CV6 are, respectively, 48.7%, 76.7%, and 100%, and the corresponding biases are reduced by 0.39, 0.23, and 0.47 g/kg, respectively.
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4.2.2. Analysis Increments
ExpA_CV5 performs poorly. As shown in Figures 7(f)–7(j), it produces almost no sea fog during the whole forecast period. In contrast, ExpA_CV6 successfully simulates the sea fog event (Figures 7(k)–7(o)). From Figure 9, it is found that the initial condition of ExpA_CV5 is quite dry. For example, the moisture bias at the site of S6 is about −0.8 g/kg. Note that this dry bias is corrected in ExpA_CV6 and the bias is reduced to 0.4 g/kg. The reason why ExpA_CV6 can gain such a significant moisture improvement is discussed below.
During the analysis update of the WRF3DVAR, the minimization of the cost function (equation (1)) produces the analysis field . is determined by the balance of analysis increment and observation innovation under the joint constraint of OE and BE. Because OE is usually prescribed regardless of a specific sea fog case, BE plays the key role. There are three updates for in the designed data assimilation scheme (Figure 1), and the 3rd update yields the initial condition for the experiments. Figure 10 demonstrates analysis fields of temperature, wind, and mixing ratio at the lowest model level for three DA updates in ExpA_CV5 (upper row) and ExpA_CV6 (lower row). It is clearly seen that the analysis fields of both temperature and wind are almost the same between ExpA_CV5 and ExpA_CV6. The analysis fields of mixing ratio are obviously different, especially at the 3rd update (cf. Figure 10(c) with Figure 10(f); see bold lines with the value of 5.5 g/kg). Water vapor is much less near surface over sea area west of Korean Peninsula in ExpA_CV5 than that in ExpA_CV6, which leads to its failure. Analysis fields at upper levels are also checked and significant difference of mixing ratio between ExpA_CV5 and ExpA_CV6 exists below ∼600 m (not shown). Next, we try to investigate in detail the difference between the observation innovations of ExpA_CV5 and ExpA_CV6 at the 3rd update, as well as the consequence of the difference on analysis increment.
Figure 11 illustrates the observation innovations and analysis increments at the lowest model level for ExpA_CV5 and ExpA_CV6. For the temperature innovation, there is nearly no difference between ExpA_CV5 and ExpA_CV6 (cf. Figure 11(a) with Figure 11(e)). Meanwhile, the difference in moisture innovation is very small, and the moisture innovation in the area denoted by a reddashed frame (Figure 11(f)) is a little bit larger in ExpA_CV6 than in ExpA_CV5 (Figure 11(b)). However, ExpA_CV6 obtains much more moisture increment than ExpA_CV5 (cf. Figure 11(c) with Figure 11(g)). Does this moisture increment still result from temperature innovation through the way revealed in the singleobservation experiments of Section 4.1? To answer this question, two extra experiments were conducted, and the value of (see equations (3a)–(3c) and (4a)–(4d)) is set to 0 in the 3rd cycle of the WRF3DVAR; namely, the temperature information is not used in the assimilation process. Results of the two extra experiments show that the difference in moisture increment between ExpA_CV6 and ExpA_CV5 is very small (cf. Figure 11(d) with Figure 11(h)), which confirms that the larger moisture increment shown in Figure 11(g) is due to the negative temperature innovation shown in Figure 11(e) by the function of CV6 in ExpA_CV6. Note that the area of positive moisture increments in Figure 11(g) only covers the fog area (see Figure 7). Therefore, the success of ExpA_CV6 is undoubtedly the consequence of the moisture improvement in its initial condition, and CV6 is vital to guarantee this improvement.
4.3. Case Study 2: The Widespread Sea Fog
4.3.1. Verification
The results of forecasted sea fog area for ExpB_CV5 and ExpB_CV6 are shown in Figure 12. Similar to the results of CaseA, the sea surface wind fields of the two experiments are almost the same (cf. Figures 12(f)–12(j) with 12(k)–12(o), respectively). Although the two experiments both underestimate the sea fog areas, it is clear that ExpB_CV6 still performs better than ExpB_CV5. Compared with that produced by ExpB_CV5, the fog area produced by ExpB_CV6 is much closer to the observed one (cf. Figures 12(f)–12(j) and 12(k)–12(o) with 12(a)–12(e), respectively). Similar to the coastal sea fog, the difference in winds between the two experiments in the MABL is quite small (figure not shown).
The temperature and moisture structures of the MABL in the initial conditions of ExpB_CV5 and ExpB_CV6 were compared with six radiosonde soundings (red dots in Figure 4(b). Vertical profiles of RMSE and bias of temperature and mixing ratio for ExpA_CV5 and ExpA_CV6 are presented in Figure 13. Similar to Figure 8, the initial temperatures of both experiments are higher than observations (Figure 13(a)). However, the initial mixing ratios are smaller than observations below 300 m see the biases in (Figure 13(b)). Compared to ExpB_CV5, the drier condition is alleviated by 0.2 g/kg in ExpB_CV6, which results in better performance. Ship measurements (not assimilated; see their sites in (Figure 14(a)) were also employed to check the moisture condition near the sea surface. The biases between model outputs and the measurements are displayed in (Figure 14(b)), which shows clearly that ExpB_CV6 outperforms ExpB_CV5 at six out of eight sites and the maximum improvement is up to 36.7%.
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4.3.2. Differences in Initial Condition
Since ExpB_CV5 and ExpB_CV6 share the same model configuration, their different performances are attributed to differences between their initial conditions. Figure 15 shows the differences in initial condition near the surface between the two experiments (ExpB_CV6 minus ExpB_CV5). The differences in pressure and wind are not presented because they are very small. The difference in moisture is relatively larger than that in temperature. The differences in moisture and temperature over the sea fog area (see the sea fog in Figures 12(a), 12(f), and 12(k)) are quite small. For example, the difference in temperature is within the range of about −0.2–+0.2°C, and the differences in mixing ratio and relative humidity are within the ranges of about −0.2–+0.2 g/kg and 0–5%, respectively. Note that there are two areas with large positive and negative differences in mixing ratio, respectively (Figure 15(b)). The area of positive difference is located over the northwestern Yellow Sea (see the red shading around Shandong Peninsula marked in Figure 2(b)), and the area of negative difference is located over the southern Korea Peninsula. The area of positive mixing ratio difference stretches to the southeast, while positive difference in relative humidity occurs over almost the entire Yellow Sea (Figure 15(c)). Different from the positive analysis increment of mixing ratio over the Yellow Sea in ExpB_CV6, there is almost no positive gain of mixing ratio in ExpB_CV5 (cf. Figures 16(a) and 16(b)). The above results demonstrate that the initial condition of ExpB_CV6 is wetter than that of ExpB_CV5 in the MABL, which leads to better performance.
For CaseA described in Section 4.2, the assimilation of the observations over the Korea Peninsula (see their sites in Figure 4(a) can noticeably improve the modeling result of ExpA_CV6, because positive analysis increment of moisture is produced by negative temperature observation innovation (see Figure 11). On the contrary, for CaseB, both ExpB_CV5 and ExpB_CV6 get negative analysis increments of moisture over AreaK (shown in Figure 16), and the increment for ExpB_CV6 is larger than that for ExpB_CV5. This can be explained by the observation innovations of moisture and temperature over AreaK. The moisture innovation is negative and the temperature innovation is positive over AreaK for both ExpB_CV5 and ExpB_CV6 (figures not shown). For ExpB_CV5, the negative increment of moisture is produced because is an independent variable in CV5. For ExpB_CV6, however, since there exist correlations between and other variables, especially temperature (equation (4a)), negative moisture increment and positive temperature increment jointly produce a large negative increment of moisture over AreaK. In contrast, ExpB_CV6 gets positive moisture increment (see the yellow shading in Figure 16(b)) over other areas of the Yellow Sea due to the assimilation of other observations beyond AreaK.
Forced by the sea surface wind field (see the vectors in Figure 12), the southwesterly advection pulls drier air mass from AreaK to the southern Yellow Sea, which is not conducive to the formation and development of sea fog there. Despite this unfavorable situation, ExpB_CV6 still performs better than ExpB_CV5, which is attributed to the fact that its initial condition contains positive moisture increment over the Yellow Sea caused by the correlation between moisture and other control variables in CV6. To further explore the impact of CV6, two sensitivity experiments—ExpB_CV5e and ExpB_CV6e—were, respectively, conducted based on ExpB_CV5 and ExpB_CV6 but without assimilation of the observations over the southern Korea Peninsula (the domain framed by dashed line in Figure 4(b). Figure 17 compares the analysis increments of mixing ratio near the surface between ExpB_CV5 and ExpB_CV5e and initial difference inmixing ratio between them. It can be seen clearly that the assimilation of the observations over the southern Korea Peninsula results in an obvious decrease of mixing ratio in the MABL in the initial condition. Comparative results of the forecasted sea fog area between ExpB_CV5/6 and ExpB_CV5/6e are shown in Figure 18. Compared to ExpB_CV5, the fog area simulated by ExpB_CV5e is significantly improved by 29.0–45.4%. However, the difference between results of ExpB_CV6e and ExpB_CV6 is no more than ∼5%. The above results indicate that the assimilation effect of CV6 is more reliable than CV5.
4.4. Discussion on the Impact of CV6
The results of the above case studies show that CV6 outstrips CV5 in the both cases. It seems that the performance of CV6 is much better in CaseA than in CaseB (see Figures 7 and 12). In CaseA, the sea fog patch is small and very near the western coast of the Korean Peninsula. The assimilation of temperature observations over the Korean Peninsula can affect the area where the sea fog occurred (Figure 11(g)), which leads to relatively great improvement in CaseA. In contrast, the sea fog patch in CaseB is much larger than that in CaseA, and most of the sea fog patch in CaseB is far away from the coast; the assimilation information of observations over land is difficult to spread over the area where sea fog occurred. This results in a relatively poor performance in CaseB.
Due to the negative correlation between temperature and moisture in CV6, assimilation of negative observational innovations of temperature can definitely get positive moisture increments, which leads to improvement of sea fog simulation. However, degree of improvement (i.e., development of sea fog area) is primarily dominated by the localization scales in BE, which limits the spatial spread of observational information. The case studies indicate that the impact of CV6 depends on weather and realtime condition. Actually, this is directly due to the individual localization scales in BEs for CaseA and CaseB.
The localization scales are calculated when BE is generated by the NMC method. As mentioned in Section 2.2, forecast differences during 15 days are used to produce BE in the NMC method. Thus, BE comes from the temporal average of forecast differences, which means that BE does not vary with time and sometimes the localization scales are not suitable for a specific realtime condition, especially that the weather system changes rapidly. If ensemble forecast members are available from an ensemble forecast system, ensemblebased perturbations can be used instead of using forecast differences to generate CV6 in the NMC method. Perhaps this kind of CV6 becomes more effective, because some flowdependent information could be introduced into the BE.
4.5. Quantitative Evaluation
The evaluation method of the simulated sea fog area in case study is a socalled eyeball method, which visually compares forecasted fog and satellite observed fog. This is a qualitative and subjective method, which cannot provide a quantitative statistical result for multiple cases. According to Zhou and Du [56], a more objective and comprehensive method is to remap both the observed and forecasted fog areas onto the same grids, in which pointtopoint comparisons can be conducted. Fog area prediction can be regarded as a binary event (yes or no; 1 or 0). Statistical scores, including the probability of detection (POD), false alarm ratio (FAR), bias, and equitable threat score (ETS), are used for evaluation. They are defined as follows:
Here, H, F, and O refer to the numbers of correctly forecast points (hits), forecast points, and observed points; R = F(O/N) is a random hit penalty, and N is the total number of grid points. The observed fog areas are retrieved from the MTSAT using the method designed by [22], and D2 is taken as the verification domain with a mesh resolution of 0.1°. Grids covered by highaltitude clouds are excluded over both the land and the sea areas. Verification was carried out for all model outputs at 1hour intervals. Quantitative evaluation was conducted using the statistical scores defined above for each case, and the scores were then temporally averaged. The evaluation results are presented in Table 3, in which the cases are arranged chronologically according to initial time.

Because ETS is a comprehensive scoring indicator, the result of ETS is analyzed first. It is clearly seen from Table 3 that the average ETS of GroupCV6 is much higher than that of CroupCV5 with an improvement of 21%, and only one case (Case5 in Table 3) gets a worse ETS among the ten cases. It demonstrates that implementing CV6 BE in the WRF3DVAR can obviously improve sea fog modeling. Further casebycase analyses of POD, FAR, bias, and ETS indicate that the improvement of ETS is mainly attributed to the increase of POD. Compared to that for GroupCV5, POD for GroupCV6 improved by 27.6% on average. Meanwhile, the average decreases in FAR and bias were −5.5% and −22.9%, respectively. Similar to the cases studied in the present study, CV6 results in more water vapor in the initial condition compared to CV5 for almost every case here and hence promotes sea fog development and larger fog area (see POD improvements in Table 3). Although the results of FAR and bias show that the sea fog areas are a bit overforecasted in GroupCV6, this is acceptable because the average deterioration of FAR is only 5.5%.
5. Summary
Data assimilation is absolutely necessary for numerical modeling of sea fog, because modeling result is highly sensitive to initial condition, especially to initial moisture status. The WRFARW model and WRF3DVAR module have been widely employed for sea fog modeling, and the WRF3DVAR has already been proven to be able to effectively improve the initial conditions for sea fog simulations over the Yellow Sea. However, the default domaindependent BE of the WRF3DVAR is CV5, which does not consider the relationship between moisture and other control variables. As a result, the improvement of moisture in the initial condition has to be only dependent on the direct assimilation of moisture information. This approach causes some disadvantageous issues; for example, other control variables like temperature can hardly contribute to the improvement of moisture status.
By theoretically comparing CV5 with CV6 (multivariate BE), it is found that CV6 includes the correlation between moisture and temperature, which is absent in CV5. In order to explore the impact of multivariate BE on the 3DVAR assimilation effect over the Yellow Sea fog modeling, two sea fog cases that differ greatly were selected for case study. The advantage of CV6 compared to CV5 has been primarily explained by singleobservation experiments and further revealed in detail based on analysis of the modeling results from the cases study. In addition, the impact of CV6 application in the WRF3DVAR has been evaluated by a series of experiments of extra ten sea fog events over the Yellow Sea. The major conclusions are as follows:(1)The performance of the WRF3DVAR assimilation with CV6 is obviously better than that with CV5. The assimilation with CV6 can significantly improve the forecasted sea fog area, which is clearly demonstrated in the study of the selected cases. Particularly, for the case of spreading narrowly along the coast, the sea fog evolution is successfully reproduced when the assimilation with CV6 is implemented. In contrast, the model completely fails to capture this fog event when using the assimilation with CV5.(2)There exists a dominant negative correlation between temperature and moisture in CV6 near the sea surface, which makes it possible to improve initial moisture condition in the marine atmospheric boundary layer by assimilating temperature observations near the sea surface. However, the primary way to improve moisture in the WRF3DVAR with CV5 is to assimilate moisture observations.(3)Experimental results of the 10 additional sea fog cases clearly indicate that CV6 is more suitable than CV5 in the WRF3DVAR assimilation for sea fog modeling. Compared to that with CV5, the average POD and ETS of forecasted sea fog area using the WRF3DVAR with CV6 can be improved by 27.6% and 21.0%, respectively, and positive impacts are found in 9 out of the 10 cases in the present study.
The present study has demonstrated the importance of improving moisture condition in the initial condition of sea fog modeling and the necessity of using CV6 in the WRF3DVAR to achieve this improvement. This is of reference value for the realtime operational forecast of sea fog when using the WRF3DVAR. Despite encouraging results that have been achieved in this study, further indepth studies are still needed. CV6 is essentially a type of climatological background error covariance, which usually varies on monthly basis and may also have a feature of rapid variation in the lowlevel atmosphere due to transient weather systems. Thus, when ensemble forecast members are available, it is better to use ensemblebased perturbations instead of using forecast differences in the NMC method. This is because some flowdependent information might be introduced into the BE. In addition, the localizations of horizontal and vertical correlation in CV6 represent an issue that needs to be addressed.
Data Availability
FNL data can be downloaded at https://rda.ucar.edu/datasets/ds083.2. NEARGOOS SST data are available at http://ds.data.jma.go.jp/gmd/goos/data. AIRSretrieved temperature and humidity profiles can be accessed at https://rda.ucar.edu/datasets/ds735.0. MTSAT data are available at http://weather.is.kochiu.ac.jp/sat/GAME, including albedo, infrared, and visible cloud imageries. The NCEP ship observations are from https://rda.ucar.edu/datasets/ds337.0. Other data for this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was funded by the National Key Research and Development Program of China (2017YFC1404200 and 2017YFC1404100), the Key Research and Development Program of Shandong Province (2019GSF111066), and the National Natural Science Foundation of China (42075069). The authors acknowledge UCAR/NCAR/CISL/TDD for providing the powerful graphic tool NCL (https://doi.org/10.5065/D6WD3XH5).
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Copyright © 2020 Xiaoyu Gao and Shanhong Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.