1 Introduction

Visibility is the maximum distance between motorists and an object that can be recognized. This distance may change due to the light scattering and absorption of particulates like snow particles in the atmosphere. In cold snowy regions, heavy snowfall and blowing snow due to strong wind blind a driver to directional cues, and sometimes such weather conditions cause traffic accidents including chain-reaction collisions.

In the United States, for instance, Federal Highway Administration published road weather management strategies for various weather threats including snow which focused on visibilities [1]. Harada et al. [2] recorded that road closure times in the specific area of north-eastern Hokkaido, Japan, corresponded well with the snow transport rate, i.e., as a function of visibility in snow [3]. Harada et al. [2] proposed five categories of warning levels regarding visibility, referring to the literature [4], with emphasis on contributing to road management during heavy snowstorm-related disasters. The visibility suggested warning levels in heavy snowstorm-related disasters should become an important index for road weather management in snowy regions in Japan. Therefore, it is necessary to understand the visibility obstruction during snowy weather conditions. Takeuchi [5] attempted to precisely express visibility in snow and proposed that afterimages of suspended snow particles affect the visibility of the human eye to determine a driver’s visual range during blowing snow. However, to our knowledge, there is not a clear understanding of the accurate visibility information during blowing snow due to the difficulty of handling the phenomenon of afterimages of particles and a lack of quantitative studies. This study aims to clarify the effect of afterimages of suspended snow particles during blowing snow on the visibility of the human eye, not only to understand visibility in snow quantitatively but also to provide accurate visual information as a contribution to road management and motorists. In this paper, we first discuss previous studies of visibility in snow and the duration of images by the human eye, and introduce the following two experiments; 1) Examination of the visual range during blowing snow by varying the exposure time of digital video cameras lined up on the windshield of weather observation vehicles, 2) Construction of a numerical model taking the visual duration of afterimages of snow particles into consideration to reproduce the results of the visibility of the human eye as previously observed in blowing snow [6].

2 Visibility in Snow

2.1 Fundamentals of Visibility

According to the International Meteorological Organization (WMO), visibility essentially refers to the degree of atmospheric turbidity near the earth's surface in the daytime and is the maximum distance at which the human eye can distinguish between black objects against a sky background [7]. In general, we usually use the definition of visibility for fog and aerosols (solid and liquid particles floating in the atmosphere), and visibility is expressed as the critical value of contrast (ε) at which the human eye can distinguish between the object and the background, and the distance at which light penetrates the atmosphere and decays to the observer's discrimination limit [7]. The light penetration distance (y) can be expressed as the visibility using ε and the extinction coefficient (λ).

$$y=\frac{1}{\lambda }\mathrm{ln}\frac{1}{\varepsilon }$$
(1)

Here, the visibility generally expressed by Koschmieder’s Eq [5, 8] is valid when the atmosphere is uniformly distributed due to the nature of Eq. 1 and represents the penetration of light due to the scattering of fog and aerosol particles. Note, fog and aerosols are usually treated as not visually recognizable particles but snow particles can be visible. The size of the suspended snow, snowfall, and blowing snow, is usually 50 to 500 μm [9, 10].

The WMO has established the use of meteorological optical range (MOR) for instrumental range measurements. This MOR is the distance at which the parallel beam of an incandescent lamp with a color temperature of 2,700 K is scattered and absorbed by the atmosphere, reducing the intensity of the light to 5% of its original intensity. This value of 5% is consistent with the visual distance. However, MOR is a concept that is valid under conditions of uniform atmospheric turbidity. Since suspended snow particles have a large diameter and can be visually recognized by observers, it is unclear at this point whether it is appropriate to use MOR, which deals with the intensity of light scattering and absorption, to describe the appearance of blowing snow and snowfall.

2.2 Previous Studies of Visibility in Snow

In an early study (Mellor, 1966) [11], the visibility concept was applied to snow measured only spatiotemporally as uniform snowfall, i.e., steady snowfall without any wind, and good results with hardly any dispersion were obtained. In blowing snow, on the other hand, the amount of suspended snow particles crossing the field of view varies from time to time and is perceived by observers to have the coarse appearance of suspended snow particles. The cause of traffic accidents in blowing snow depends on the ability to identify objects that are tens or even several meters away. It is therefore important to focus on the behavior of suspended snow particles when discussing visibility during snowfall and blowing snow.

Comparing the visibility under windless conditions [11] with that under blowing snow conditions, it is suggested that at the same concentration of blowing snow, the visibility under blowing snow conditions is 10 times lower than that under windless conditions.

Applying the concept of visibility to blowing snow, Takeuchi and Fukuzawa (1975) [12] showed that Koschmieder's equation can be extended to blowing snow if the visibility is 100 m or more, but Koschmieder's equation is not valid for low visibility ranges, such as 5 to 100 m. They suggested that this was due to the afterimage effect since suspended snow particles passing close to the human eye are identified and perceived as lines or tails. Takeuchi and Fukuzawa (1976) [13] derived a better relationship between the mass flux of snow and visibility than the spatial density of snow. It has also been noted that the effect of afterimages appears even during windless snowfall [14]. However, there are only a few studies that have examined the effect of afterimages on visibility during blowing snow.

Blowing snow is a phenomenon in which suspended snow particles are blown through the air by the wind, and the number of suspended snow particles moving at a given height can be expressed by the mass flux of the blowing snow q (z) (g m−2 s−1),

$$q\left(z\right)=N\left(z\right)u\left(z\right)$$
(2)

where N(z) (g m−3) and u(z) (m s−1) are the spatial density of suspended snow and the windspeed at the given height z(m), respectively. As mentioned above, the visibility Vis(z)(m) of blowing snow can be expressed as a function of q(z) e.g., [3, 13, 15, 16]. In our previous study, [6] we compared the results of visibility measurements between measuring instruments and the human eye. Regarding the relationship between Vis(z) and q(z), the following equation was derived,

$$\mathrm{log}\left(Vis(z)\right)=-0.0886\mathrm{log}\left(q\left(z\right)\right)+2.648$$
(3)

where z is usually set at the height of passenger vehicles, for instance, a height of 1.5 m. They found that at distances of less than 100 m, the visibility of the human eye is lower than the value in Eq. 3 (see Fig. 9). This is presumably because the higher q is, the more pronounced the afterimage effect of suspended snow particles becomes. However, the effect of the afterimage on the visibility of the human eye is still not clear.

2.3 Afterimages and the Aim of this Study

It is known that a psychological afterimage occurs several milliseconds after the light source is offset relative to the visual stimulus. The temporal and spatial characteristics of the afterimage with fixed eyesight have been extensively studied e.g., [17], and an image of the same size as the distance traveled by the light spot is perceived for a visual duration ranging from approximately several tens to 200 ms. The visual duration of the saccade-induced afterimages is approximately the same as the fixed eyesight of 120 ms [18]. Note, that the saccade is a quick, simultaneous movement of both eyes between two or more phases of fixation in the same direction.

In the meantime, the texture and appearance of images captured by a digital video camera depend on the frame rate (fps: frames per second) and exposure time τ (s). The fps is the number of frames per unit of time (usually seconds) and τ is the visual duration of one frame. We assume that the exposure time and the visual duration of the afterimage can be treated equally, and believe it is possible to reproduce the human eye.

In this study, we investigated the visual range with a digital video camera considering its exposure time, and a visibility model of the human eye with afterimages of suspended blowing snow particles was established to compare it with the visibility of the human eye measured during our previous observations.

In recent years, automated driving technology has made remarkable progress, and cameras and 3D laser range sensors are widely used to control vehicles [19]. Clarifying the visibility of the human eye in such weather conditions will be useful for future automated driving technology and intelligent transportation systems.

Please note, in this study, the maximum distance that can be seen by the digital video camera is defined as the visual range, which is different from the definition of visibility of the human eye.

3 Measurement of the Visual Range with Exposure

We conducted two observations in a field study. The first observation period was from 17:00 to 23:00 on January 29, 2021, and the location was national route R238 in northern Hokkaido. The second observation period was from 10:00 to 15:00 on February 23, 2021, and the location was national route R337 near Sapporo, Hokkaido.

With generally used video cameras, exposure time cannot be set manually, while it is possible to set the exposure time arbitrarily with the industrial camera we used in this study (WAT-2400S, WATEC Co. Ltd.). The frame rate of the industrial camera is fixed at 30 fps, and the exposure time can be changed from the usual 0.04 (1/25) second to 3.9 × 10–5 (1/25600) second with the electrical frequency of 50 Hz (northern Japan). It is possible to change the exposure time to longer than 1/25 s but we used 1/25 s as the longest in this study. Note that white balance and backlight compensation, etc., are set automatically or we use them as the default. Two industrial cameras were lined up on the windshield of the weather observation vehicles (Fig. 1). These two industrial cameras compare the difference in how a visual distance changes and how we see suspended blowing snow particles when we use different exposure times with each camera.

Fig. 1
figure 1

Weather observation vehicles (lower images) and industrial cameras lined up on the windshields (upper images). We used the left vehicle for the 1st observation and the right one for the 2nd observation

Full size image

Figure 2 shows images captured by the industrial cameras while driving the vehicle during blowing snow at the first observation. The image when the exposure time of “auto” shows blowing snow near the surface of the road (Fig. 2a). In the meantime, the surface of the road can clearly be seen with an exposure time of 1/1000 s (Fig. 2b). Although we are not sure of the exposure time set as “auto,” the difference in images should relate to the exposure time of the industrial camera.

Fig. 2
figure 2

Captured images are taken by the industrial camera during the 1st observation. The exposure time is a) “auto,” b) 1/1000 s

Full size image

Figure 3 shows images captured by the industrial cameras while driving the vehicle during blowing snow in the second observation. The fixed-post delineators (hereinafter, delineator) are lined up at the lane edge. The delineators are usually 50 m from each other. The 1/6400 s image (Fig. 3a) shows two delineators in front of the image at the left side of the lane edge, while the 1/25 s image (Fig. 3b) shows the closer delineator but the distant delineator is unclear. The vehicle in the passing lane is also clear in the 1/6400 s image (Fig. 3a) but is blurred in the 1/25 s image (Fig. 3b). The exact visual distance from the vehicle is unknown, but the delineator is unclear from at least approximately 50 m away in the 1/25 image and approximately 100 m away in the 1/6400 s image.

Fig. 3
figure 3

Captured images are taken by the industrial camera during the 2nd observation. The exposure time is a) 1/6400 s and b) 1/25 s, respectively. a') and b') are enlarged images of the white frames in a) and b), respectively. The arrows in the figures indicate a fixed-post delineator with an arrow-shaped pointer (so-called yabane in Japanese) located at intervals of approximately 50 m between each other

Full size image

These differences in how the images show the distance and the clearness of the view may be explained by the exposure time of the industrial camera. As mentioned above (“Afterimages and the aim of this study” section), the exposure time and the visual duration of afterimages of suspended snow particles by the human eye can be treated equally, and it is possible to reproduce the human eye by varying the visual duration.

4 A model of the Visibility with Afterimages of Snow Particles

WE constructed a model of visibility to clarify the effect of the afterimages of snow particles, considering the difference in visibility depending on their visual duration by the human eye.

Road images captured by the camera, for instance, are centered in the distance with no obstructions, making the center of the image appear farther away and the periphery appear closer.

Figure 4 shows an example of such an image that was taken at our observation field in Ishikari, Hokkaido. The distance z (m) from the camera position to the object (road lights or snow poles) and the road width (3.5 m) is a known distance. In image d (arbitrary unit) is the length of the road width at the same distance as where the object is located. Note, d may change the unit as pixels but here in this study, we use an arbitrary unit for convenience. Figure 4b shows the relationship between z and d based on the regression line in Fig. 4b, namely,

$$d={C}_{1}{e}^{-\alpha z}$$
(4)

where α and \({C}_{1}\) are constants that depend on the angle of view. From Fig. 2b, the constant α is -0.01266, and the value of α should apply only to this study. Since we set d as an arbitrary unit, the constant \({C}_{1}\) can be arbitrary, and therefore we assume here that \({C}_{1}=1\). We extend from a line (1 dimension) to an area (2 dimensions), and the unit area A can be expressed as follows.

Fig. 4
figure 4

Width d (arb. unit) on an image of the road width (3.5 m) and the distance from the camera. a) Image used for calculation, b) relationship between the distance z from the camera and the size d on the image (regression equation: \(d=6.01{e}^{-0.01266z}\), correlation coefficient 0.92)

Full size image
$${A}_{z}={d}^{2}={e}^{-2\alpha z}$$
(5)

The unit area can be portrayed as shown in Fig. 5a. A unit area Az is smaller when the distance z is further from the image, and at several hundred meters, it becomes almost a dot on the image.

Fig. 5
figure 5

Conceptual diagram of the projected area fraction and the distance of the suspended snow particles. a) Relationship between the unit area Az and the distance z. b) Conceptual diagram of the projected area fraction of suspended snow particles (gray area), spatial density (box volume), and mass flux (shaded area)

Full size image

Figure 5b is a conceptual diagram showing suspended snow particles passing through the observer's field of view. We defined the projected area fraction for snow particles R, and the projected area fraction for the afterimage of snow particles R0 as,

$$R=\pi {r}^{2}{N}_{t}$$
(6)
$${R}_{0}=2rx{N}_{t}$$
(7)

where r (m), x (m), and N (m−1) are the snow particle radius, the tail length by afterimage, and the snow particle density of the unit per depth (to be discussed later), respectively. Note, we assume that the radius of the suspended snow particles is 0.04 mm (4 × 10–6 m) [9] in this study. As shown above, the tail distance can be expressed as,

$$x=\tau u$$
(8)

where u (m s−1) is the speed of suspended snow particles, and τ is the visual duration. While it is known that the speed of snow particles differs from the windspeed during blowing snow [10], we hereby assume that the speed of suspended snow particles is the same speed as the wind for the sake of convenience.

Regarding the projected area fraction shown in Fig. 5b, the value of the visibility in this model Zvis (m) should be a function of the addition of the projected area fraction to the maximum distance C2 that is visible, and we show it here as the following equation.

$${\sum }_{z=1}^{{Z}_{vis}}R{A}_{z}={C}_{2}$$
(9)

While the human eye has the ability to focus on an object, it becomes blurry if you are not focusing on it [20]. In other words, the human eye can’t focus on all objects in front of a person when looking at a further or near-field object. Therefore, we assume that afterimages are only formed by an object that you focused on. We added the term of a projected area fraction for the afterimage of snow particles to Eq. 9 as follows,

$${R}_{0}{A}_{0}+{\sum }_{z=1}^{{Z}_{vis}}R{A}_{z}={C}_{3}$$
(10)

where A0 is the area on which you focused.

Regarding the snow particle density of unit per depth N, overlapped particles in the volume should increase if the number of suspended particles increases. Here we define the several numbers of particle N* (m−3) existing in the unit per volume to which the parameter is related as suspended snow particles N (g m−3) in Eq. 2, namely,

$${N}^{*}=\frac{N}{\rho \sigma }$$
(11)

where ρ (m3) and σ (g m−3) are the volumes of the snow particle with a radius r, and the density of the snow particle, respectively. Let’s assume the density of the snow particle is 0.9 (g m−3) for the sake of convenience. When we imagine several particles np in a minute volume, the probability of its existence P(np) is considered to follow a Poisson distribution and the equation is as follows,

$$P\left(n_p\right)=e^{\;-\overline n\;\frac{\overline n^{n_p}}{np!}}$$
(12)
$$\overline{n }={N}^{*}v$$
(13)

where v and \(\overline{n }\) are the minute volume and the expected value of the average number of particles in v, respectively. When the suspended snow particles in V are projected onto the area A, the suspended snow particles roughly overlap if there are many particles existing in the minute-volume v. The relation between the projected number of particles Nt and the volume V is,

$${N}_{t}=\frac{{N}^{*}}{{n}_{p}}$$
(14)

and N* in the equation expressed with N from Eq. 11,

$${N}_{t}=\frac{N}{\rho \sigma }\frac{1}{{n}_{p}}.$$
(15)

Figure 6 shows the Poisson distribution (Eq. 12) with varied N from 0.1 to 3.0 g m−3. The distribution is unique, but we used the local maximal value of each N. for the overlapped particle number. Figure 7 shows the relation between N and Nt (Eq. 15). The figures suggest that the overlapped number of particles is constant even if the N increased from 0.3 to 3.0 g m−3, and the number of particles projected onto area At is stably constant when the N value is more than 0.3 g m−3. As a result, we assigned N as a constant value of 1 g m−3 for convenience, because the value of Nt is constant.

Fig. 6
figure 6

Probability of existence P(np) of the number particles np contained in the minute-volume v of N (0.1–3.0 g m−3)

Full size image
Fig. 7
figure 7

Relationship between N and Nt

Full size image

Based on the above considerations, R of Eq. 6 is constant, and R0 of Eq. 7 is a function of the visual duration τ so that the value of τ, which we chose, is 0.12 s and 0.04 s based on the vision studies e.g., [17, 18].

5 Visibility with Afterimage Effect

Regarding C2 and C3, these values can be written by introducing parameters due to actual blowing snow phenomena. The critical wind speed of blowing snow, including layers of saltation and suspension, was previously investigated [21, 22] and the speed was approximately 5 m s−1, for instance, when the ambient temperature was lower than -5 °C. Takechi et al. (2009) [6] suggested that the visibility in the snow is approximately 200 m when the windspeed of 5 m s−1. We assigned the value of visibility in the snow (200 m) to Eq. 9, and found that C2 is equal to 0.11 approximately as a constant from Fig. 8. To express the effect of the afterimage, we assume here that the constant C2 is equal to C3, and Eq. 10 can be replaced as follows,

$$2r\tau uN+\pi r^2N\cdot\sum\nolimits_{z=1}^{Z_{vis}}e^{-2\alpha z}=0.11$$
(16)

where we would like to know the value of Zvis when the result of Eq. 16 shows 0.11 and the variable parameters are τ, u, and z. To make the concept more accessible, we tested the value of C3 in Fig. 8 when τ is 0.04 and 0.12 s, respectively. We defined that C3 is equal to C2, which are constants of 0.11 and we see that the values of Zvis are a shortened distance than with Eq. 9. Note, that other constant values are listed in Table 1.

Fig. 8
figure 8

Relationship between Zvis and constant C2 and C3 calculated by Eqs. 9 and 10 when u is 5 m s−1 for Eq. 9, and τ are written in the figure

Full size image
Table 1 Parameters that we used in Eq. 16
Full size table

Takeuchi and Fukuzawa (1976) [13] suggested that the mass flux of snow q and visibility Vis show a better relationship than the spatial density of snow N. Therefore, we hereafter use q instead of N from Eq. 2, which represents the function of u. Saito and Shimizu (1971) [14] showed us that the Vis is 200 m if q is 0.2–1.7 g m−2 s−1. We used the minimum of q as 0.2 g m−2 s−1 as an initial value, and we calculated it by varying u from 0.2–30 m s−1 using Eq. 16. Because we set N as 1 g m−3, q varies 0.2 – 30 g m−2 s−1, which is the same value as u, see Eq. 2. The results should be compared with the visibility of the human eye and Koschmieder’s equation (Eq. 1). Takechi et al. (2009) [6] measured visibility in the snow with the human eye and reprinted the results in Fig. 9 with the results of Eq. 16.

Fig. 9
figure 9

Comparison of the visual distance (Eq. 16) with the visual durations of the human eye of 0.04, 0.12, and 0.001 s, incorporating the visibility in snow observed by the human eye (Takechi et al., 2009) [6], and Koschmieder’s equation (Eq. 1)

Full size image

In Fig. 9, the calculated visibility was expressed as a curve along the lower limit of the visibility based on the measurement result by the human eye (measured in 2007) when τ was 0.12 s. The curve along the upper limit of visibility was the measurement result of visibility by the human eye when τ was 0.04 s. The good agreement between visibility in the snow of visualization with the human eye and the calculated visibility results suggests that the effect of the visual duration, i.e., the afterimage of suspended snow particles in a range of at least 0.04 to 0.12 s affects the range of the blowing snow. On the other hand, the 2008 results of the visibility of the human eye could not be reproduced, even when τ was 0.04 and 0.12 s. Takechi et al. (2009) [6] attributed the difference in visibility between 2007 and 2008 to the illuminance at the time of the study (daytime or dusk). The afterimage distance is longer at dusk and shorter during the daytime. Thus, differences by visual duration are indeed obtained with this study, but not to the extent of reproducing the 2008 data.

Blowing snow is not uniformly distributed in space and time, but fluctuates from time to time [5]. It is possible that Spatio-temporal variations in blowing snow could increase the value of visibility because motorists consciously want to see further, or measurement by skilled motorists might provide a shorter afterimage distance. We notice here that the 2008 values could have been reproduced if we had assumed that the afterimage of the visual duration is 0.001 s. Since there is no proof that the afterimage of the visual duration of the human eye is 0.001 s, it is a matter of speculation.

Although N and u were uniquely determined in this numerical experiment, at least our model (Eq. 16), which expressed the visual duration as the afterimage of the visual duration of suspended snow particles, reproduced the visibility in snow observed by the human eye. Note, however, that this result was obtained by varying the windspeed while N remained constant, and could not apply to varying N conditions in this paper. Therefore, we would like to mention that the model is limited to proposing one of the models for visibility in snow during blowing snow that incorporates afterimage effects.

6 Conclusion

The afterimage of the human eye is believed to affect the visibility in snow during blowing snow. In fact, a digital video camera with varying exposure times changes the visual range of the distance and images that change as a function of the exposure time of an industrial camera’s setup. Note, in this study, we assume that the exposure time of the digital video camera and the visual duration of the afterimage is equivalent. Numerical experiments were conducted on visibility in snow during blowing snow considering the visual duration of afterimages to express the human eye during blowing snow. The number of snow particles in a minute volume was assumed to follow a Poisson distribution, and the visibility of blowing snow with afterimage was expressed in terms of the projected area. Considering the overlap of suspended snow particles per minute volume and the visual duration of the human eye as the afterimage, as a result, we believe that this model can express the visibility in snow obtained by the human eye when the visual duration is between 0.04 to 0.12 s. Finally, we believe that the monitoring of blowing snow by using cameras with properly adjusted exposure times will provide accurate visibility information to road administrators and motorists.