## 1. Introduction

Visibility (

VIS) is an indicator used to distinguish different intensities of fog based on the grade of fog forecast in the World Meteorological Organization (WMO) guide [

1]. The decrease in atmospheric

VIS, associated with the formation and development of fog weather, especially the explosive growth of fog, causes a severe impact on land, marine and, air transportation, and often cause traffic accidents such as car collisions in high-speed vehicles which can endanger people’s lives and property [

2,

3,

4,

5]. Fog is the most common and severe low–visibility weather occurrence, receiving much attention [

6,

7,

8]. The number of articles including the word “fog” in Journals of American Meteorological Society alone was around 4700 until 2007 [

9], with the addition of a further 4268 articles from 2007 to 2021 when searching them in the same way (

https://journals.ametsoc.org, accessed on 1 December 2021), indicating that there is substantial interest in this subject.

Until now, our knowledge on the physics of fog remains limited, including the numerous physical processes influencing fog formation, development, and decay. Although the physical processes of fog, such as droplet microphysics [

10,

11,

12,

13,

14,

15,

16], aerosol physics and chemistry [

17,

18,

19,

20,

21], radiation [

22,

23], turbulence [

24,

25,

26,

27], large/small-scale dynamics [

28,

29,

30,

31], and surface conditions [

32,

33,

34,

35,

36] have been widely investigated, the uncertainty of typical numerical forecast models estimating

VIS is higher than 50% [

37,

38,

39]. As to

VIS estimation methods in meteorological operations, some forecasting methods provide pure mathematical statistical fitting without the explicit consideration of physical processes [

40], such as climatological statistical methods [

41], the rule-based statistical method [

42,

43], numerical model ensemble [

39,

44,

45] and machine learning methods [

43,

46]. However, other methods, based on physical factors [

47], e.g., extinction coefficient, relative humidity (

RH) [

48], liquid/ice water content (

LWC) [

49,

50], droplet number concentration (

N_{d}) and fog droplets size [

4,

14,

51,

52], which can establish a direct relationship to

VIS, are widely adoptable in atmospheric numerical models [

4,

53,

54,

55,

56,

57,

58,

59,

60,

61]. Therefore, research on the relationship between the impact factors and

VIS, and their application effects are summarized in the review.

In terms of the connection between microphysical parameters and

VIS, many studies focused on the influence of the extinction coefficient, fog droplets size,

N_{d}, and

LWC on

VIS. Among them, the effect of the extinction coefficient on visibility provides the basis of others. The Koschmieder’s law, as we know it, laid the foundation for visibility observation [

1,

62]. Other parameters related to the extinction coefficient were later studied, and

LWC. George [

63] pointed out that the fog droplet spectrum and

LWC provide two crucial parameters that characterized the microphysical characteristics of fog and that an excellent inverse relationship exists between

LWC and

VIS. Eldridge [

64] analyzed the influence of droplet growth on the formation and dissipation of fog, and proposed an empirical relationship between

VIS and

LWC. Through a comparison of the observation results with the conclusions of Houghton and Radford, Eldridge [

65] pointed out that it was necessary to consider the effect of

N_{d} on the relationship between

VIS and

LWC. There were inverse correlations between the microphysical parameters and

VIS, and there were also correlations between the parameters themselves. Niu et al. [

14] showed that when nucleation and condensation growth dominated, a pronounced positive correlation between

N_{d} and

LWC existed. When the

D increased, and

N_{d} was small, the

LWC would also be smaller. Many other studies have discussed the relationship between

VIS and the evolution of microphysical parameters [

4,

52,

58]. The following sections will further categorize and explain the corresponding results and their applications in atmospheric numerical models. The characteristics of fog in a polluted environment are fairly remarkable; thus, parameterization schemes of

VIS in fog that contains chemical composition or concentrations of aerosol are beyond the scope of this article.

There are many factors impacting VIS in fog, and only several common physical elements, including RH, extinction coefficient, LWC, N_{d}, and fog droplets size, are introduced in this review due to the length limit. The following sections will summarize corresponding parameterization schemes of VIS in fog and their applications in meteorological operations.

## 2. Relationships between VIS and Extinction Coefficient

As early as the 1920s, based on the interference effects of fog and haze on the horizontal visual range, the Koschmieder’s law was proposed [

62]. The theory assumes that the atmosphere is uniform, the horizontal extinction coefficient (

β_{ext}) of the atmosphere is constant, and the flat sky is used as the background black body target during the day. Then the brightness contrast threshold (

C) between the target and the background changes with the distance (

VIS), and the relationship is as follows:

which can be transformed as

where

${\beta}_{ext}$ is measured in units of inverse kilometers, the constant

C is a physical quantity related to the human eye, there are two values for the contrast threshold

C, the value recommended by the International Civil Aviation Organization (ICAO) is 0.05, and the value recommended by WMO is 0.02. Therefore, as long as the atmospheric extinction coefficient is obtained, the

VIS value can be obtained. The daytime-target visual range theory proposed by this law has been the basis of manual

VIS observation during daytime for many years. The most significant contribution of the Koschmieder’s law is that it first links

VIS to the atmospheric extinction coefficient, which has become the theoretical basis for studying atmospheric

VIS. To this day, this law is still the basic principle of various optical

VIS measuring instruments. Inverse proportionality between

VIS and

β_{ext} is only applicable under very minimal conditions: the atmosphere must be illuminated homogeneously, the extinction coefficient and the scattering function are not allowed to vary with space, the object should ideally be black and viewed against the horizon, and the eye of the observer must have a constant contrast threshold. Horvath et al. [

66] proposed a general formula, taking the facts above into account. Through the proper selection of the

VIS markers, it is possible to use the Koschmieder’s formula to calculate the extinction coefficient from observed visibilities with an error of less than about 10 percent. Using radiative transfer theory, Lee et al. [

67] point out that the Koschmieder’s model is workable only in situations where a common-sized object can be viewed tens of kilometers away, but not applicable for viewable distances of hundreds of meters when the angular dimension of an object is significantly greater than the eye resolution of the human being. Lee et al.’s research advocates for the measurement and distribution of detectability in bad weather.

The scattering theory of particles, proposed by Mie [

68], is the basis for calculating the extinction coefficient. Since the diameter of the particles is equivalent to the wavelength of the light, the forward-scattered light is stronger than the backward-scattered light, and the scattering intensity is much larger than that of Rayleigh scattering. The

β_{ext} in Mie scattering theory is given as follows:

where

a_{n} and

b_{n} are functions related to the Bessel function and Hankel function, and

x is the radius of the droplet. In 1971, according to Beer’s law, Koening [

69] pointed out that brightness was a function of the microphysical characteristics of the fog, which is due to the dependence of the extinction coefficient on the concentration and radius of the fog droplets. That is,

β_{ext} is related to the

N_{d}, droplet radius, visible light wavelength, etc. Kunkel [

70] pointed out that if the drop-size distribution is known, then

β_{ext} can be readily determined from the following equation (Equation (4))

where

Q_{ext} is the extinction efficiency (normalized extinction cross-section),

n is the

N_{d}, and

r is the droplet radius. Moreover, If the drop-size distribution is unknown, then an empirical formula must be used to relate the

LWC to

β_{ext}, and related content is discussed in detail in the next section.

The total extinction coefficient is a sum of components from clean air, aerosol, cloud, and precipitation. The extinction coefficient for aerosols contains a contribution from different aerosol species, such as sea salt, dust, black carbon, organic matter, sulfates, and so on [

71]. The extinction coefficient of clean air is small and has little practical value, so it is taken to be equivalent to a

VIS of 100 km (10

^{5} m), which defines the maximum

VIS that can be diagnosed [

72], that is

${\beta}_{air}=\left(lne\right)/{10}^{5}$.

The parameterization scheme based on Equation (2) was generally adopted by subsequent numerical research, providing a feasible scheme for the numerical forecast of horizontal

VIS [

73,

74,

75]. This scheme strongly relies on

β_{ext}. Koening’s research [

69] shows that the scheme is determined by multiple factors, which will lead to certain errors in calculations and measurements. For example, Kunkel [

70] compared the extinction coefficient

β_{c}, calculated through the droplet spectral distribution and the actually observed

β_{m}, showing that the calculated extinction coefficient

β_{c} is larger than the observed

β_{m}. The results of Vali et al. [

76] also showed that there was a deviation between the calculated value of the extinction coefficient and the measured value. The correction method proposed by Kunkel [

70] is as follows:

There is significant uncertainty in calculating the extinction coefficient β_{c} through the droplet spectrum distribution. Therefore, if the VIS is calculated by Equation (2), certain errors will inevitably occur.

In aviation applications, not only horizontal but also vertical

VIS has a significant impact on aircraft take-off and landing. Stoelinga and Warner [

77] believed that the maximum horizontal

VIS for aviation applications was 10 km, which was smaller than the horizontal grid spacing set in the NWP (such as 36 km and 12 km). Therefore, it could be considered that the extinction coefficient has a fixed value, and Equation (2) could be used to calculate the horizontal

VIS. However, this assumption was no longer valid when calculating the vertical

VIS, because the vertical maximum height (2500 m) that aviation considered was significantly greater than the vertical grid spacing (50–500 m) in the model. The extinction coefficient, which was closely related to the atmospheric environment, varied greatly in different model levels. Therefore, the extinction coefficient should be a function of height

z, and the expression must be integrated layer by layer in an upward direction (replacing

x with z) to determine the ceiling

z_{clg} [

77]:

## 3. Relationships between VIS and RH

Fog is a weather phenomenon with a horizontal

VIS of less than 1 km due to water vapor near the ground condensing into tiny water droplets or ice crystals and becoming suspended in the air [

1]. Air saturation is the prerequisite for the appearance of fog weather in a clean atmosphere. Since human activities produce many aerosols which absorb moisture and also contribute to poor

VIS, pollution fog affected by industrialization often occurs when the atmosphere does not reach saturation conditions [

78,

79].

VIS decaying in the unsaturated atmosphere is closely related to increasing

RH. Therefore, summarizing the empirical relationship between

VIS and

RH, which can be used in the atmospheric numerical models to output the final

VIS value, has certain practicability. In 1976, Hanel [

80] proposed an empirical formula between

VIS and

RH, and the empirical equation is as follows:

Equation (7) is valid under the condition of 58% <

RH < 97%. Due to only a single factor being involved in the empirical formula, the simple, clear formula can reflect the changing trend of unsaturated atmospheric

VIS well and can guide for forecasting. Based on Hanel’s work, Smirnova et al. [

81] further improved the

VIS-RH empirical formula, following Equation (8),

the condition of Equation (8) is 30% <

RH ≤ 100%. The parameterization scheme is applied using the Rapid Update Cycle (

RUC) model of the National Environmental Forecast Center of the United States. Based on the Fog Remote Sensing And Modeling (

FRAM) at Pearson Airport, and the Alliance Icing Research Study (AIRS 2) at Mirabell Airport in Canada, Gultepe et al. [

82] pointed out that Smirnova’s scheme used in the RUC is not applicable in Canada, and further identified a significant issue in that when the

RH was close to 100%, the calculated

VIS using the Smirnova’s scheme was approximately twice of the observed value. Based on the ground observation data of two airports, the more suitable

VIS parameterization schemes for forecasting local

VIS are provided in Equation (9),

and Equation (10)

The newly proposed scheme, applied in the numerical model, showed better application performance and the proved to be more suitable for local

VIS forecasting. However, the

VIS-

RH schemes proposed by Gultepe [

82] are not applicable in other regions. Cao et al. [

83] studied the

VIS parameterization scheme in the fog model of Dalian, China, and also found that under high humidity conditions, the calculation value of Smirnova’s scheme was significantly greater than the actual observation value. For example, when

RH = 100%,

VIS_{RUC} = 4.2 km; moreover, approximate 95.3% of the measured

VIS ≤ 1 km. If this scheme is used to calculate visibility, when RH is high, the calculated result will be significantly larger than the observed value. A more suitable

VIS-

RH relationship was proposed based on the Dalian ground observation data. The newly built scheme [

83] had greatly enhanced the local low

VIS forecasting ability and is represented in Equation (11).

It can be seen from Equation (11) that when

RH = 100%,

VIS = 0.63 km, when

RH = 95%,

VIS = 3.56 km, and when

RH < 80%,

VIS > 10 km. The revised

VIS-RH parameterization scheme has greatly improved the ability to predict local low visibility compared to RH-VIS formula developed by Smirnova [

81] and Gultepe [

82].

In 2009, Gultepe et al. [

84] further analyzed the relationship between

VIS and

RH, and pointed out that the

VIS-

RH relationship, based only on observational data fittings cannot be well used for

VIS calculations, because for the same

RH, the corresponding

VIS value varied greatly. Therefore, Gultepe et al. [

84] proposed a probability method, that is, a method in which all

VIS values for the same

RH are sorted by value, and the top 5%, 50%, and 95% of the data sets are used to fit, so as to meet different needs, respectively. For example, due to the concern of extremely poor

VIS at an airport, it is more meaningful to obtain a possible minimum

VIS value than to know the most likely

VIS value. Therefore, the forecast scheme with 5% of the data is more practical, meaning that 95% of the data points have a higher

VIS value. Gultepe et al. suggested the replacement of the deterministic forecast with the probability method, and the established parameterization scheme was found to be more suitable for the actual local meteorology operation. Lin et al. [

85] performed the local application of the above probability method in Sichuan Province. The prediction effect of the mesoscale Weather Research and Forecasting Model (WRF) on the

RH was evaluated, and the

VIS-

RH parameterization scheme by the measured

VIS and

RH data from Chengdu Shuangliu airport was obtained. The test results showed that the

VIS values of the dense fog calculated by the

VIS-

RH parameterization scheme, which accounted for 5% of the data, were the most accurate, that is, the fitting curve with a probability of 5% was the closest to the low

VIS both in trend and magnitude.

Fog is sensitive to meteorological factors. Even under the same weather condition, fog formation is still a probability event. The method proposed by Gultepe et al. [

84]. can provide the probability of low

VIS under the same

RH condition, but the results are limited by the sampling. By the LEPS (Local Ensemble Prediction System), based on ensemble forecast, which can directly output the probability of LVP (Low

VIS Procedure) events, Roquelaure et al. [

86] carried out low

VIS prediction assessment of Paris Charles de Gaulle Airport. The category of LVP can be obtained according to the established comparison table of event probability and categories. The results show that the system can reduce false alarms by 50–60%.

Results from some studies investigating the

VIS-

RH relationship are listed in

Table 1. Using

VIS and

RH observation data from the automatic weather station(AWS) from 2016 to 2017 in Tianjin urban meteorological observation (Tianjin), a localized

VIS-RH scheme based on the probability method of 5%, proposed by Gultepe et al. [

84], was made. The results show that T5, T10, T9, and the localized fitting curve can represent low

VIS under the condition of high

RH, but T5 overestimates

VIS for all

RH, and T10 also overestimates

VIS under the condition of

RH > 95% and

RH < 40%. Comparatively speaking, the localized fitting curve can denote a better

VIS-RH relationship (

Figure 1).

Even though the

VIS-

RH schemes vary in different regions and have obvious regional characteristics, the

VIS-

RH relationship has been widely used in various models, including Numerical Weather Prediction (NWP) and fog models [

87,

88,

89]. The

RH value is easily obtained through the atmospheric numerical model, and it is also a direct observational element, which is convenient for

VIS-RH verification. For the same

RH value, the calculated

VIS values vary in a wide range. Moreover, there are significant differences between

VIS parameterization schemes, and the calculation accuracy cannot meet the needs of refined forecasting services. At present, most NWP and fog models no longer use

VIS-RH schemes separately.

## 4. Relationships between VIS and LWC

The

VIS-

LWC scheme, based on the Koschmieder’s law [

62], is relatively common in

VIS parameterization schemes. The scheme calculates the

VIS using

LWC, and Equation (17) is as follows:

where

LWC is provided in g/m

^{3}. A large number of studies have shown that the relationship between

β_{ext} and

LWC satisfies the power–function relationship mentioned above. The values of empirical coefficients

a and

b from various regions vary greatly. The size distributions of droplets are affected by many factors such as the observation range of droplet size, experiment design, air particulates, and fog types. For example, in 1966 and 1971, Eldridge [

65,

90] conducted a comparative analysis of different droplet size ranges through experiments, and the obtained empirical values of

a and

b varied across a broad range. When the droplet size is between 0.6 and 16 μm,

$\beta =163LW{C}^{0.65}$, and when the upper limit of the droplet size range increases,

$\beta =91LW{C}^{0.65}$. In 1976, Tomasi and Tampieri [

91] obtained empirical values of a and b for different types of fog. Under the warm and humid fog conditions

$\beta =65LW{C}^{2/3}$ was obtained, while

$\beta =115LW{C}^{2/3}$ was obtained under cold fog conditions. In the existing research, the empirical coefficient ranges from 65 to 178, and b ranges from 0.63 to 0.96 [

58]. It can be seen that the performance of the

VIS-

LWC parameter scheme is analogous to that of the

VIS-

RH scheme, which is also affected by many other factors and has a strong regional character.

At present, the most commonly used

VIS-

LWC program is the K84 program. In 1984, Kunkel [

70] found that the correlation coefficient between the extinction coefficient and

LWC reached 95% in the observational study of advection fog. Compared with the research results of other studies [

90,

91,

92], there is a higher correlation between the two parameters. Kunkel [

70] proposed the formula for calculating the extinction coefficient with the

LWC in the fog, which is given as follows

Substituting this formula into the Koschmieder’s law (Equation (2)), the K84 scheme is obtained as follows:

Some models work with

LWC, and the K84 scheme provides a convenient solution for relating

LWC to

VIS. So, the K84 scheme is widely used in numerical models to calculate the values of

VIS [

54,

73,

74,

75,

89,

93,

94]. However, the K84 scheme was still improved as following Equation (20) by Gultepe [

58] using Radiation and Aerosol Cloud Experiment (RACE) observation in 1995 in the eastern Canada area.

The

LWC, N

_{d} from the fog droplet spectral observation and

VIS from AWS during 2016–2017 in TianJin [

16] were used to validate

VIS parameter formulas from literatures and to fit the local formula. The observation

VIS and

LWC ranged from 0 to 8.2 km and 0 to 0.25 g/m

^{3}, respectively. The VIS_K84 [

70] and VIS_Gultepe schemes [

58] were verified in

Figure 2 with a logarithmic plot. It should be pointed out that only data for

VIS that were less than 1 km were adopted in

Figure 2a, while the full range of observation data was adopted in

Figure 2b.

The VIS_K84 and VIS_Gultepe schemes simulation show a sharply decreasing VIS trend with LWC increasing. However, both of them are obviously larger than observation VIS especially for VIS less than 1 km in fog events. Deviations between simulated and observed VIS shows that something need to do to improve current VIS-LWC relationships. The local relationship of VIS and LWC was fitted as$VIS=0.0618LW{C}^{-0.126}\mathrm{and}\text{}VIS=0.0813LW{C}^{-0.126}$when using the VIS data less than 1 km and the full range of data, respectively. The local VIS-LWC shows a flatter decline than the VIS_K84 and VIS_Gultepe schemes, which are denoted by the black dashed line. It is due to a larger number of VIS observations being located at numerical intervals of less than 1 km. Evidently, the local fitting formula does not have the ability to express a VIS of larger than 1 km when LWC exists. So, no satisfactory corresponding relationship has been found between VIS and LWC as of yet.

Most of the visibility values simulated by the visibility parameterization schemes are greater than the observed values, as

Figure 2 shows. Some widely used

VIS parameterization schemes are conducted where air pollution is commonly not serious. When applied to fog forecast/simulation in the polluted environments, these schemes may overestimate

VIS and underestimate fog intensity due to the absence of aerosol extinction [

95,

96].

So far, the three parameterization schemes of

VIS-

RH,

VIS-

β, and

VIS-

LWC, which correspond to the three key elements of

RH,

β and

LWC respectively, have been discussed in this study. There are certain correlations that exist between the three elements, and research on the relationship between

β and

LWC has been introduced in this section. Some studies demonstrate a negative correlation between

RH and

LWC. For example, Gonser et al. [

97] first revealed the inverse relationship between

RH and

LWC in topographic fog through mountain cloud and fog observation experiments in the Chilan Mountains of Taiwan, and pointed out that, in principle, this situation can be explained by the cohesion growth theory of droplets containing soluble or insoluble substances, but the reasons for this need to be further studied. In addition, compared with the smaller droplets, larger diameters droplets can exist in a lower

RH environment, but whether it can be used to explain the significant changes in

RH and

LWC is still unclear. The local imbalance between the droplet and the air mass during the turbulent transport process may also be a potential cause of the inverse relationship. Therefore, more research that includes the chemical properties of droplets, and microphysical modeling, are required to further explain the correlation between

RH and

LWC.

## 5. Relationships between VIS, N_{d}, and Fog Droplets Size

In 1980, Meyer et al. [

98] proposed that

VIS was negatively related to the fog-droplets number concentration and the square of the diameter, and that it varied with the concentration of fog. Therefore, the parameterization schemes of

VIS under the thick fog (

VIS_{MH} ≤ 1 km) and the thin mist (

VIS_{ML} > 1–2 km), are obtained, respectively, as follows

and

where the

N_{d}, which is given in PCS m

^{−3}, both equations can be applied to the droplet with a diameter of larger than 0.5 μm. Meyer’s observational experiments also showed that the average droplet size essentially remains constant in thin fog, while

VIS decreased with increasing droplet size in thick fog, and the formula under the condition of visibility in 1–2 km was

$VIS=1.46\times {10}^{-4}{\left({D}_{e}^{2}\right)}^{-0.49}$, where

D_{e} was the effective diameter of the droplet. Assuming that the scattering coefficient is a constant in the spectrum, the

N_{d}·

D_{e} is proportional to the extinction coefficient. Combined with the relationship between

VIS and extinction coefficient,

was obtained, with the slope parameter approaching −1.0. Furthermore, a tiny variation in index results in significant changes of

VIS, which might be due to the assumption that the scattering coefficient is a constant and the assumption is valid only for sufficiently large droplets.

The size spectrum of each hydrometeor category is often described by a three-parameter gamma distribution function,

${N}_{d}={N}_{0}{D}^{\alpha}{e}^{-\lambda D}$. Two-moment schemes generally treat

N_{0} and λ as prognostic parameters while maintaining the shape parameter

α constant. Milbrandt et al. [

99] analyzed the influence of shape parameter

α on sedimentation and microphysical growth rate using different schemes. The results show that

α plays an important role in determining the rate of size sorting. Kunkel [

70] analyzed more than 1400 droplet size samples in 1983, finding a good correlation between the droplet terminal velocity and c (

LWC^{2}/N

_{d})

^{d} (parameter c and d are both fitting coefficients). Under the condition of a fixed

LWC value, air pollutants interact with water vapor to form a mass of liquid drops, which increase the

N_{d}. At the same time, a smaller droplet radius decreases the droplet terminal velocity, which results in the deposition rate of liquid water being reduced.

The extinction coefficient increases, and the

VIS decreases, due to the increasing average number concentration of droplets associated with the above two physical processes and other chemical processes. Kunkel [

70] also indicated that the effect of pollutant concentration in fog should be taken into account. Therefore, the appropriate formulas and droplet terminal velocity should be selected for different polluted conditions. The

VIS in fog is affected by the extinction of fog droplets [

69,

70,

76,

84], and based on Mie scattering theory, the extinction coefficient is closely related to the Nd. Therefore, the

N_{d} is considered as one of the impact factors of

VIS.

In 2006, Gultepe [

82] also pointed out that the

VIS in fog was not only related to

LWC, but also relied on Nd. Result showed that there were differences in the

VIS-

N_{d} relationship of the ice fog and liquid fog. The fog was classified into either ice fog (

T < −1 °C) or liquid fog (

T ≥ −1 °C) on the basis of temperature threshold. Approximate formulas between the

VIS, ice fog number concentration (

N_{i}) and liquid fog number concentration (

N_{d}) were obtained through observational analysis and the relationships are as follows:

and

where the unit of

N_{i} is PCS L

^{−1}, and the unit of

N_{d} is PCS cm

^{−3}, and PCS is the abbreviation of pieces. Gultepe [

82] also noted that the results for

VIS > 50 km were invalid due to uncertainties in the observation of small droplet by existing instruments. Moreover, on account of the properties of logarithmic relationships, the

VIS_{Ni}, which should be treated cautiously, varied greatly for a given

N_{i}. Based on the observational data from the forward-Scattering Spectrometer in the same year, a new relationship between

VIS and

N_{d} was developed by Gultepe [

58] as follows:

Compared with that calculated using Meyers’s expression, the VIS calculated using the new relationship reduced at a faster rate than with the increase of number concentration. Gultepe suggested that the discrepancy may be due to the uncertainty of the number concentration observations in earlier studies or impact factors such as conducting observations in low clouds. The N_{d} should be treated as an independent variable in parameterization schemes of VIS. Furthermore, in order to establish a more rational parameterization scheme, the accurate monitoring of the number concentration is required.

Based on the observational

Nd and VIS_obs data, which were obtained using droplet spectrometer (DMT, FM-120) and

VIS meter (Vaisala, PWD 10) in Tianjin [

16] from 2016 to 2017, the relationship between

VIS and

N_{d} in Tianjin was obtained as the fitting formula:

$VIS=0.2522{N}_{d}^{-0.121}$ (

Figure 3). The constants and exponents parameter of local formula for

VIS-N_{d} relationship is largely different to others. Although a generally decreasing power relationship exists between

VIS and

N_{d}, there is still large uncertainty in various regions.

Both the droplet size and the number concentration were not used as a direct output for many previous numerical models. For the droplet size, introducing droplet types and calculating droplet spectra will increase the complexity significantly, while using empirical expressions can simplify the model. For the number concentration, there are no microphysical schemes for the near-surface fog. Even though some numerical models contain microphysical schemes that can directly predict the number concentration, the output of the cloud droplet concentration cannot be treated as the fog-droplets number concentration under the condition of high clouds. The distribution of cloud droplet spectrum is different from that of fog-droplet spectrum; therefore, it is necessary to use the empirical statistical method to estimate the droplets number concentration. For example,

N_{d} is usually given a constant value. Fu G. [

6] discussed the performance of the parameterization scheme when

N_{d} = 300 PCS cm

^{−3}, and showed that the obtained results are significantly smaller than those that do not considering the number concentration.

## 6. Relationships between VIS, LWC and N_{d}

Based on the descriptions in

Section 3 and

Section 4, both the

LWC and the

N_{d} are considered as the impact factors of

VIS; moreover, there is no simple one-to-one relationship between the two factors. For example, the

N_{d} varies over a wide range for a certain

LWC, resulting in great differences in

VIS. Moreover, the two factors are related to each other. Considering these two main factors at the same time can better reflect the changes of

VIS than considering one of them alone [

70]. In 2006, based on the previous studies, a new parameterization scheme was established by Gultepe [

58], combining the

LWC and

N_{d}, t expressed as Equation (27)

which is suitable for the conditions of 0.005 gm

^{−3} <

LWC< 0.5 gm

^{−3} and 1 cm

^{−3} <

N_{d} < 400 cm

^{−3}. Additionally, a new definition of the fog index (FI) was formulated FI = 1/(

LWC*

N_{d}), Compared with Kunkel’s studies [

70], Gultepe [

58] established a quantitative relational equation, and applied the new scheme to the mesoscale non-hydrostatic model of NOAA. By comparing the results with schemes of K84 and Meyer [

98], the results show Equation (27), in which

LWC and

N_{d} were taken into account, was more accurate in predicting

VIS.

Compared with the K84 scheme, which only considers the

LWC, Equation (27) performed better, with significantly lower uncertainty in

VIS prediction. While Equation (27) is applied to the Tianjin area (

Figure 4), there are still overestimates of

VIS over all

FI ranges, especially an absence of many low

VIS cases in which the

FI is large. According to the form of Equation (27), the localized

VIS-LWC &

N_{d} formula was fitted as

VIS = 0.1418

$F{I}^{0.065}$. The local relationship is far lower than in Equation (27), while it has no ability to express some larger

VIS. That is to say, even though Equation (27) had done well compared to schemes of K84 and Meyer [

98], there is still some work required to promote a parameterized

VIS-LWC &

Nd relationship in the future.

Hu et al. [

100] established a coastal-fog forecast procedure based on the

VIS-

LWC &

N_{d} parameterization scheme proposed by Gultepe et al. [

82]. The

LWC was calculated from the physical quantity

qcloud output by the WRF model, and the

N_{d} was solved using a historical experience statistical method. The

N_{d} was obtained according to the inversion formula of

VIS:

${N}_{d}={\mathrm{e}}^{Tmp}$, where

$Tmp=\frac{1}{0.6437}\mathrm{ln}\left(\frac{1.002}{VI{S}_{obs}}\right)\mathrm{ln}\left(LW{C}_{obs}\right)$, in which

VIS_{obs} and

LWC_{obs} were the

VIS and

LWC of similar cases, respectively. The

VIS forecast value can be obtained by substituting the obtained

LWC and

N_{d} values into the parameterized scheme, which improves the fog forecast accuracy from 61% to 73%, compared with the Stoelinga-Warner scheme [

77]. There is also a correlation between

LWC and

N_{d}. Gultepe et al. [

58] observed that

LWC increased with increasing

N_{d}, while the range of

N_{d} changes was very large for a given

LWC value. Huang et al. [

101] observed and analyzed the microphysical characteristics of sea fog using a droplet spectrometer, finding that an increasing number of droplets with a diameter of more than 10 μm is the main reason for the increase of

LWC, while the increase of

LWC is the main reason for the decrease of atmospheric

VIS under the same

N_{d} interval.

The

VIS-

LWC &

N_{d} parameterization scheme has certain advantages and a rather high accuracy rate, since both the

LWC and

N_{d} are considered, and the microphysical interpretation is relatively more realistic. Similar to the

VIS-

N_{d} scheme, it suffers from the same problem of using empirical statistical methods to estimate the concentration of droplets. In addition, the scheme still has a high degree of uncertainty. The research of Gultepe [

58] showed that the uncertainty of the scheme for the calculation of

VIS in various types of fog still reached 27%. In their study of the characteristics of the fog-droplet spectrum during heavy fog in Tianjin, Liu et al. [

52] found that the effects of

LWC and

N_{d} on

VIS are not the same, and that there was a pronounced negative correlation between

VIS and

N_{d} while not so obvious for

LWC. Therefore, to better apply the

VIS-

LWC &

N_{d} relationship obtained by Gultepe et al. [

58], the scheme also needs to be improved according to the actual situation in various regions. Song [

102] developed a new visibility parameterization by further taking De into the

VIS-

LWC &

N_{d} relationship consideration at a mountain site in Korea, and indicated that this new parameterization showed better performance than the original

VIS-

LWC &

N_{d} relationship obtained by Gultepe et al. [

58] in visibility value prediction when De was larger than 10 μm, while no obvious improvement was observed when De was less than 10 μm.

In addition, certain correlations still exist among these factors. In order to establish

VIS parameterization schemes that are more applicable for the forecast of local

VIS, some scholars have carried out the “combination schemes”, which are composed of different parameterization schemes based on the observation data. To solve the problem of abrupt changes in

VIS as calculated by

LWC scheme, Cao et al. [

83] used the localized

VIS-

RH scheme value to replace discontinuity points in the

VIS-

LWC scheme. Using 0.05 g kg

^{−1} as the critical lower limit of

LWC, Bao et al. [

103] simulated cluster fog along the Shanghai-Nanjing Expressway, and used Equation (27) to calculate the value of visibility, and when

N_{d} = 0 or

LWC = 0, Equation (27) is found to be no longer applicable, so the

VIS-RH scheme is replaced. The results show that in the early stage of heavy fog, the simulated value is lower than the observed value, but the trend of the simulated value and the observed value is consistent during the maintenance and dissipation of the fog. Long et al. [

104] determined a combination scheme of

VIS-

LWC and

VIS-

RH for the North Coast of Bohai Bay; the results show that if only one scheme is adopted, the simulated

VIS value and the measured

VIS value will deviate significantly when

LWC = 0.03 g/kg. However, the combination scheme behaves well in this condition.

## 7. Conclusions and Discussion

The characteristics and applications of different parameterization schemes, developed based on various impact factors of VIS, such as RH, extinction coefficient, LWC, and N_{d}, have been summarized. It can be seen that the fitted parameters, with precise physical meanings, allow for the calculation of VIS by interfacing with numerical forecast products. Therefore, many models are directly interfaced with the corresponding parameterization schemes while postprocessing VIS.

By reviewing the achievements of the

VIS parameterization schemes, we understand that none of the current formulas, derived from introducing impact factors or environmental factors, can always accurately calculate the

VIS in the fog. It’s important to point out the occurrence and development of fog is the result of multiple processes occurring simultaneously that interact nonlinearly with each other. These interactions likely result in nontrivial sets of key fog parameter values leading to fog formation, while other combinations of values prevent fog formation [

105]. Owing to the specific correlations among these factors, the “combination schemes” have also been adopted by many researchers.

The parameterization scheme, which is based on a statistical analysis, has certain shortcomings due to the incomplete consideration of physical processes, and because the physical factors introduced depend on other factors. In fact, the main impact factors vary in different environments, such as the influences of aerosols, as many widely used

VIS parameterization schemes are conducted in areas where air pollution is commonly not severe. When applied to fog forecast/simulation in polluted areas, these schemes may overestimate

VIS due to the absence of aerosol extinction. For the most commonly used scheme of

VIS-

LWC, there discrepancies exist between the observed

LWC and the model output

LWC. The discrepancies in

LWC lead to deviations in

VIS prediction, and the same problem exists for the observation and data acquisition of

RH,

N_{d}, etc. Gultepe et al. [

8] point out that further modifications in microphysical observations and parametrizations are needed to promote the fog predictability of numerical-weather-prediction models.

Although existing

VIS parameterization schemes have regional limitation, the improvement of the

VIS parameterization scheme in the future still required sustained exploration, and it is hoped that a universal scheme can be applied to the atmospheric forecast operation. In addition, with the continuous innovation of computer technology, further studies on the mechanism of different types of fog [

106,

107], the application of artificial neural network methods [

43,

108,

109,

110], and the rapid advance of machine learning and artificial intelligence technologies [

111,

112], can utilize more tools by which to enhance the accurate numerical calculations of

VIS.

VIS parameterization schemes are often highly dependent on the accuracy of the meteorological elements or trigger conditions provided by microphysical schemes [

96] or numerical models [

95]. When numerical forecast accuracy is not sufficient, even if the VIS parameterization scheme is ideal, the VIS forecast will vary. Some trigger mechanisms or enhancing/limiting processes, such as wind and surrounding buildings, also affect fog prediction accuracy in numerical models [

94,

97]. Furthermore, the vertical resolution setting has certain influence on fog prediction in the numerical model. Fog can be modeled as different types under the condition of various vertical resolutions [

78]. Therefore, it is also important to select the appropriate numerical model for the established parameterization scheme.