Horizon visibility range
The line observed in the sea, along which the sea seems to connect with the sky, is called the visible horizon of the observer.
If the observer's eye is at a height e M above sea level (i.e. A rice. 2.13), then the line of sight running tangentially to the earth’s surface defines a small circle on the earth’s surface ahh, radius D.
Rice. 2.13. Horizon visibility range
This would be true if the Earth were not surrounded by an atmosphere.
If we take the Earth as a sphere and exclude the influence of the atmosphere, then from a right triangle OAa follows: OA=R+e
Since the value is extremely small ( For e = 50m at R = 6371km – 0,000004 ), then we finally have:
Under the influence of earthly refraction, as a result of the refraction of the visual ray in the atmosphere, the observer sees the horizon further (in a circle bb).
(2.7)
Where X– coefficient of terrestrial refraction (» 0.16).
If we take the range of the visible horizon D e in miles, and the height of the observer's eye above sea level ( e M) in meters and substitute the value of the Earth's radius ( R=3437,7 miles = 6371 km), then we finally obtain the formula for calculating the range of the visible horizon
(2.8)
For example:1) e = 4 m D e = 4,16 miles; 2) e = 9 m D e = 6,24 miles;
3) e = 16 m D e = 8,32 miles; 4) e = 25 m D e = 10,4 miles.
Using formula (2.8), table No. 22 “MT-75” (p. 248) and table No. 2.1 “MT-2000” (p. 255) were compiled according to ( e M) from 0.25 m¸ 5100 m. (see table 2.2)
Visibility range of landmarks at sea
If an observer whose eye height is at the height e M above sea level (i.e. A rice. 2.14), observes the horizon line (i.e. IN) at a distance D e(miles), then, by analogy, and from a reference point (i.e. B), whose height above sea level h M, visible horizon (i.e. IN) observed at a distance D h(miles).
Rice. 2.14. Visibility range of landmarks at sea
From Fig. 2.14 it is obvious that the visibility range of an object (landmark) having a height above sea level h M, from the height of the observer's eye above sea level e M will be expressed by the formula:
Formula (2.9) is solved using table 22 “MT-75” p. 248 or table 2.3 “MT-2000” (p. 256).
For example: e= 4 m, h= 30 m, D P = ?
Solution: For e= 4 m ® D e= 4.2 miles;
For h= 30 m® D h= 11.4 miles.
D P= D e + D h= 4,2 + 11,4 = 15.6 miles.
Rice. 2.15. Nomogram 2.4. "MT-2000"
Formula (2.9) can also be solved using Applications 6 to "MT-75" or nomogram 2.4 “MT-2000” (p. 257) ® fig. 2.15.
For example: e= 8 m, h= 30 m, D P = ?
Solution: Values e= 8 m (right scale) and h= 30 m (left scale) connect with a straight line. The point of intersection of this line with the average scale ( D P) and will give us the desired value 17.3 miles. ( see table 2.3 ).
Geographic visibility range of objects (from Table 2.3. “MT-2000”)
Note:
The height of the navigational landmark above sea level is selected from the navigational guide for navigation "Lights and Signs" ("Lights").
2.6.3. Visibility range of the landmark light shown on the map (Fig. 2.16)
Rice. 2.16. Lighthouse light visibility ranges shown
On navigation nautical charts and in navigation manuals, the visibility range of the landmark light is given for the height of the observer’s eye above sea level e= 5 m, i.e.:
If the actual height of the observer’s eye above sea level differs from 5 m, then to determine the visibility range of the landmark light it is necessary to add to the range shown on the map (in the manual) (if e> 5 m), or subtract (if e < 5 м) поправку к дальности видимости огня ориентира (DD K), shown on the map for the height of the eye.
(2.11)
(2.12)
For example: D K= 20 miles, e= 9 m.
D ABOUT = 20,0+1,54=21,54miles
Then: DABOUT = D K + ∆ D TO = 20.0+1.54 =21.54 miles
Answer: D O= 21.54 miles.
Problems for calculating visibility ranges
A) Visible horizon ( D e) and landmark ( D P)
B) Opening of the lighthouse fire
Conclusions
1. The main ones for the observer are:
A) plane:
Plane of the observer's true horizon (PLI);
Plane of the true meridian of the observer (PL).
The plane of the first vertical of the observer;
b) lines:
The plumb line (normal) of the observer,
Observer true meridian line ® noon line N-S;
Line E-W.
2. Direction counting systems are:
Circular (0°¸360°);
Semicircular (0°¸180°);
Quarter note (0°¸90°).
3. Any direction on the Earth's surface can be measured by an angle in the plane of the true horizon, taking the line of the observer's true meridian as the origin.
4. True directions (IR, IP) are determined on the ship relative to the northern part of the observer’s true meridian, and CU (heading angle) - relative to the bow of the longitudinal axis of the ship.
5. Range of the observer's visible horizon ( D e) is calculated using the formula:
.
6. The visibility range of a navigation landmark (in good visibility during the day) is calculated using the formula:
7. Visibility range of the navigation landmark light, according to its range ( D K), shown on the map, is calculated using the formula:
, Where 
.
The geographic range of visibility of objects in the sea D p is determined by the greatest distance at which the observer will see its top above the horizon, i.e. depends only on geometric factors connecting the height of the observer’s eye e and the height of the landmark h at the refractive index c (Fig. 1.42):
where D e and D h are the distances of the visible horizon from the height of the observer’s eye and the height of the object, respectively. That. the visibility range of an object calculated from the height of the observer’s eye and the height of the object is called geographic or geometric visibility range.
Calculation of the geographic range of visibility of an object can be made using the table. 2.3 MT – 2000 according to arguments e and h or according to table. 2.1 MT – 2000 by summing the results obtained by entering the table twice using arguments e and h. You can also obtain Dp using the Struisky nomogram, which is given in MT - 2000 under number 2.4, as well as in each book “Lights” and “Lights and Signs” (Fig. 1.43).
On marine navigation charts and in navigation manuals, the geographic range of visibility of landmarks is given for a constant height of the observer’s eye e = 5 m and is designated as D k - the visibility range indicated on the map.
Substituting the value e = 5 m into formula (1.126), we obtain:
To determine D p it is necessary to introduce a correction D D to D k, the value and sign of which are determined by the formula:
If the actual height of the eye is more than 5 m, then DD has a “+” sign, if less - a “-“ sign. Thus:
. (1.129)
The value of Dp also depends on visual acuity, which is expressed in the angular resolution of the eye, i.e. is also determined by the smallest angle at which the object and the horizon line are distinguished separately (Fig. 1.44).
In accordance with formula (1.126)
But due to the resolution of the eye g, the observer will see an object only when its angular dimensions are not less than g, i.e. when it is visible above the horizon line by at least Dh, which from the elementary DA¢CC¢ at angles C and C¢ close to 90° will be Dh = D p × g¢.
To get D p g in miles with Dh in meters:
where D p g is the geographic range of visibility of an object, taking into account the resolution of the eye.
Practical observations have determined that when the beacon is opened, g = 2¢, and when hidden, g = 1.5¢.
Example. Find the geographic range of visibility of a lighthouse with a height of h = 39 m, if the height of the observer’s eye is e = 9 m, without and taking into account the eye resolution g = 1.5¢.
Influence of hydrometeorological factors on the visibility range of lights
In addition to geometric factors (e and h), the visibility range of landmarks is also influenced by contrast, which allows the landmark to be distinguished from the surrounding background.
The visibility range of landmarks during the day, which also takes into account contrast, is called daytime optical visibility range.
To ensure safe navigation at night, special navigation equipment with light-optical devices is used: beacons, illuminated navigation signs and navigation lights.
Sea lighthouse - This is a special permanent structure with a visibility range of white or colored lights associated with it of at least 10 miles.
Glowing marine navigation sign- a capital structure that has a light-optical apparatus with a visibility range of white or colored lights reduced to it less than 10 miles.
Marine navigation light- a lighting device installed on natural objects or structures of non-special construction. Such aids to navigation often operate automatically.
At night, the visibility range of lighthouse lights and luminous navigation signs depends not only on the height of the observer’s eye and the height of the luminous aid to navigation, but also on the strength of the light source, the color of the fire, the design of the light-optical apparatus, as well as on the transparency of the atmosphere.
The visibility range that takes into account all these factors is called night optical visibility range, those. this is the maximum visibility range of the fire at a given time for a given meteorological visibility range.
Meteorological visibility range depends on the transparency of the atmosphere. Part of the luminous flux of lights of illuminated aids to navigation is absorbed by particles contained in the air, therefore, a weakening of the luminous intensity occurs, characterized by atmospheric transparency coefficient t:
where I 0 is the light intensity of the source; I 1 - luminous intensity at a certain distance from the source, taken as a unit (1 km, 1 mile).
The atmospheric transparency coefficient is always less than unity, so the geographic visibility range is usually greater than the actual one, except in anomalous cases.
The transparency of the atmosphere in points is assessed according to the visibility scale of Table 5.20 MT - 2000 depending on the state of the atmosphere: rain, fog, snow, haze, etc.
Since the optical range of lights varies greatly depending on the transparency of the atmosphere, the International Association of Lighthouse Authorities (IALA) has recommended the use of the term “nominal range”.
Nominal fire visibility range is called the optical visibility range at a meteorological visibility range of 10 miles, which corresponds to the atmospheric transparency coefficient t = 0.74. The nominal visibility range is indicated in navigation manuals of many foreign countries. Domestic maps and navigation manuals indicate the standard visibility range (if it is less than the geographic visibility range).
Standard visibility range The fire is called the optical visibility range with a meteorological visibility range of 13.5 miles, which corresponds to the atmospheric transparency coefficient t = 0.8.
In the navigation manuals “Lights”, “Lights and Signs”, in addition to the table of the range of the visible horizon and the nomogram of the range of visibility of objects, there is also a nomogram of the optical range of visibility of lights (Fig. 1.45). The same nomogram is given in MT - 2000 under number 2.5.
The inputs to the nomogram are luminous intensity, or nominal or standard visual range, (obtained from navigation aids), and meteorological visual range, (obtained from the meteorological forecast). Using these arguments, the optical range of visibility is obtained from the nomogram.
When designing beacons and lights, they strive to ensure that the optical visibility range is equal to the geographic visibility range in clear weather. However, for many lights the optical visibility range is less than the geographic range. If these ranges are not equal, then the smaller of them is indicated on charts and in navigation manuals.
For practical calculations of the expected fire visibility range during the day It is necessary to calculate D p using the formula (1.126) based on the heights of the observer’s eye and the landmark. at night: a) if the optical visibility range is greater than the geographic one, it is necessary to take a correction for the height of the observer’s eye and calculate the geographic visibility range using formulas (1.128) and (1.129). Accept the smaller of the optical and geographical calculated using these formulas; b) if the optical visibility range is less than the geographical one, accept the optical range.
If on the map there is a fire or lighthouse D k< 2,1 h + 4,7 , то поправку DД вводить не нужно, т.к. эта дальность видимости оптическая меньшая географической дальности видимости.
Example. The height of the observer's eye is e = 11 m, the visibility range of the fire indicated on the map is D k = 16 miles. The nominal visibility range of the lighthouse from the navigation manual “Lights” is 14 miles. Meteorological visibility range 17 miles. At what distance can we expect the lighthouse to fire?
According to the nomogram Dopt » 19.5 miles.
By e = 11m ® D e = 6.9 miles
D 5 = 4.7 miles
DD =+2.2 miles
D k = 16.0 miles
D n = 18.2 miles
Answer: You can expect to open fire from a distance of 18.2 miles.
Nautical charts. Map projections. Transverse equiangular cylindrical Gaussian projection and its use in navigation. Perspective projections: stereographic, gnomonic.
A map is a reduced distorted image of the spherical surface of the Earth on a plane, provided that the distortions are natural.
A plan is an image of the earth’s surface on a plane, not distorted due to the smallness of the depicted area.
A cartographic grid is a set of lines depicting meridians and parallels on a map.
Map projection is a mathematically based way of depicting meridians and parallels.
A geographic map is a conventional image of the entire earth's surface or part of it constructed in a given projection.
Maps vary in purpose and scale, for example: planispheres - depicting the entire Earth or hemisphere, general or general - depicting individual countries, oceans and seas, private - depicting smaller spaces, topographic - depicting details of the land surface, orographic - relief maps, geological - occurrence of layers, etc.
Nautical charts are special geographic maps designed primarily to support navigation. In the general classification of geographic maps, they are classified as technical. A special place among nautical charts is occupied by MNCs, which are used to plot the course of a ship and determine its place in the sea. A ship's collection may also contain auxiliary and reference charts.
Classification of map projections.
According to the nature of distortions, all cartographic projections are divided into:
- Conformal or conformal - projections in which the figures on the maps are similar to the corresponding figures on the surface of the Earth, but their areas are not proportional. The angles between objects on the ground correspond to those on the map.
- Equal or equivalent - in which the proportionality of the areas of the figures is preserved, but at the same time the angles between the objects are distorted.
- Equidistant - preserving the length along one of the main directions of the ellipse of distortions, i.e., for example, a circle on the ground on a map is depicted as an ellipse in which one of the semi-axes is equal to the radius of such a circle.
- Arbitrary - all others that do not have the above properties, but are subject to other conditions.
Based on the method of constructing projections, they are divided into:
|
, orthographic - when the point of view is removed to infinity, external - the point of view is at a finite distance further than the opposite pole of the sphere, central or gnomonic - when the point of view is in the center of the sphere. Perspective projections are neither conformal nor equivalent. Measuring distances on maps constructed in such projections is difficult, but the arc of a great circle is depicted as a straight line, which is convenient when plotting radio bearings, as well as courses when sailing along the DBC. Examples. Maps of the circumpolar regions can also be constructed in this projection. Depending on the point of contact of the picture plane, gnomonic projections are divided into: normal or polar - touching at one of the poles transverse or equatorial - touching at the equator 
horizontal or oblique - touching at any point between the pole and the equator (meridians on the map in such a projection are rays diverging from the pole, and parallels are ellipses, hyperbolas or parabolas. 
The spherical radius ВС of a small circle with с¹с²с³ is called the theoretical range of the visible horizon.
The value of the spherical radius depends on the height of the observer's eye above sea level.
So, if the observer's eye is at point A1 at a height BA¹ = e¹ above sea level, then the spherical radius Bc" will be greater than the spherical radius Bc.
To determine the relationship between the height of the observer’s eye and the theoretical range of his visible horizon, consider the right triangle AOC:
Ac² = AO² - Os²; AO = OB + e; OB = R,
Then AO = R + e; Os = R.
Due to the insignificance of the height of the observer's eye above sea level compared to the size of the Earth's radius, the length of the tangent Ac can be taken equal to the value of the spherical radius Bc and, denoting the theoretical range of the visible horizon through D T, we obtain
D 2T = (R + e)² - R² = R² + 2Re + e² - R² = 2Re + e²,
Rice. 8
Considering that the height of the observer's eye e on ships does not exceed 25 m, and 2R = 12,742,220 m, the ratio e/2R is so small that it can be neglected without compromising accuracy. Hence,
since e and R are expressed in meters, then Dt will also be in meters. However, the actual range of the visible horizon is always greater than the theoretical one, since the ray coming from the observer’s eye to a point on the earth’s surface is refracted due to the unequal density of the atmospheric layers in height.
In this case, the ray from point A to c does not go along the straight line Ac, but along the curve ASm" (see Fig. 8). Therefore, to the observer, point c appears visible in the direction of the tangent AT, i.e., raised by an angle r = L TAc , called the angle of terrestrial refraction. The angle d = L HAT is called the inclination of the visible horizon. And in fact, the visible horizon will be a small circle m", m" 2, tz", with a slightly larger spherical radius (Bm" > Вс).
The magnitude of the angle of terrestrial refraction is not constant and depends on the refractive properties of the atmosphere, which vary with temperature and humidity, and the amount of suspended particles in the air. Depending on the time of year and the date of the day, it also changes, so the actual range of the visible horizon compared to the theoretical one can increase up to 15%.
In navigation, the increase in the actual range of the visible horizon compared to the theoretical one is assumed to be 8%.
Therefore, denoting the actual, or, as it is also called, geographical, range of the visible horizon through D e, we obtain:
To obtain De in nautical miles (taking R and e in meters), the radius of the earth R, as well as the height of the eye e, is divided by 1852 (1 nautical mile is equal to 1852 m). Then
To get the result in kilometers, enter the multiplier 1.852. Then
to facilitate calculations for determining the range of the visible horizon in table. 22-a (MT-63) shows the range of the visible horizon depending on e, ranging from 0.25 to 5100 m, calculated using formula (4a).
If the actual height of the eye does not coincide with the numerical values indicated in the table, then the range of the visible horizon can be determined by linear interpolation between two values close to the actual height of the eye.
Visibility range of objects and lights
The visibility range of an object Dn (Fig. 9) will be the sum of two ranges of the visible horizon, depending on the height of the observer’s eye (D e) and the height of the object (D h), i.e.It can be determined by the formula
where h is the height of the landmark above the water level, m.
To make it easier to determine the visibility range of objects, use the table. 22-v (MT-63), calculated according to formula (5a): To determine from this table at what distance an object will open, you need to know the height of the observer’s eye above the water level and the height of the object in meters.
The visibility range of an object can also be determined using a special nomogram (Fig. 10). For example, the height of the eye above the water level is 5.5 m, and the height h of the setting sign is 6.5 m. To determine D n, a ruler is applied to the nomogram so that it connects the points corresponding to h and e on the extreme scales. The point of intersection of the ruler with the middle scale of the nomogram will show the desired visibility range of the object D n (in Fig. 10 D n = 10.2 miles).
In navigation manuals - on maps, in directions, in descriptions of lights and signs - the visibility range of objects DK is indicated at an observer's eye height of 5 m (on English charts - 15 feet).
In the case when the actual height of the observer's eye is different, it is necessary to introduce the AD correction (see Fig. 9).
Rice. 9
Example. The visibility range of the object indicated on the map is DK = 20 miles, and the height of the observer’s eye is e = 9 m. Determine the actual visibility range of the object D n using the table. 22-a (MT -63). Solution.
At night, the visibility range of a fire depends not only on its height above the water level, but also on the strength of the light source and on the discharge of the lighting apparatus. Typically, the lighting apparatus and the strength of the light source are calculated in such a way that the visibility range of the fire at night corresponds to the actual visibility range of the horizon from the height of the fire above sea level, but there are exceptions.
Therefore, the lights have their own “optical” visibility range, which can be greater or less than the visibility range of the horizon from the height of the fire.
Navigation manuals indicate the actual (mathematical) visibility range of the lights, but if it is greater than the optical one, then the latter is indicated.
The visibility range of coastal navigation signs depends not only on the state of the atmosphere, but also on many other factors, which include:
A) topographical (determined by the nature of the surrounding area, in particular the predominance of a particular color in the surrounding landscape);
B) photometric (brightness and color of the observed sign and the background on which it is projected);
C) geometric (distance to the sign, its size and shape).
Each object has a certain height H (Fig. 11), therefore the visibility range of the object Dp-MR is composed of the range of the visible horizon of the observer De=Mc and the range of the visible horizon of the object Dn=RC:
Rice. 11.
Using formulas (9) and (10), N. N. Struisky compiled a nomogram (Fig. 12), and in MT-63 the table is given. 22-v “Visibility range of objects”, calculated according to formula (9).
Example 11. Find the visibility range of an object with a height above sea level H = 26.5 m (86 ft) when the height of the observer's eye above sea level is e = 4.5 m (1 5 ft).
Solution.
1. According to the Struisky nomogram (Fig. 12), on the left vertical scale “Height of the observed object” we mark the point corresponding to 26.5 m (86 ft), on the right vertical scale “Height of the observer’s eye” we mark the point corresponding to 4.5 m ( 15 ft); connecting the marked points with a straight line, at the intersection of the latter with the average vertical scale “Visibility range” we get the answer: Dn = 15.1 m.
2. According to MT-63 (Table 22-c). For e = 4.5 m and H = 26.5 m, the value Dn = 15.1 m. The visibility range of the Dk-KR lighthouse lights given in navigation manuals and on sea charts is calculated for the height of the observer’s eye equal to 5 m. If the actual height the observer's eye is not equal to 5 m, then the correction A = MS-KS- = De-D5 must be added to the range Dk given in the manuals. The correction is the difference between the distances of the visible horizon from a height of 5 m and is called the correction for the height of the observer’s eye:
As can be seen from formula (11), the correction for the height of the eye of observer A can be positive (when e> 5 m) or negative (when e
So, the visibility range of the beacon light is determined by the formula
Rice. 12.
Example 12. The visibility range of the lighthouse indicated on the map is Dk = 20.0 miles.
From what distance will an observer see the fire, whose eye is at a height of e = 16 m?
Solution. 1) according to formula (11)
2) according to table. 22-a ME-63 A=De - D5 = 8.3-4.7 = 3.6 miles;
3) according to formula (12) Dp = (20.0+3.6) = 23.6 miles.
Example 13. The visibility range of the lighthouse indicated on the map is Dk = 26 miles.
From what distance will an observer on a boat see the fire (e=2.0 m)
Solution. 1) according to formula (11)
2) according to table. 22-a MT-63 A=D - D = 2.9 - 4.7 = -1.6 miles;
3) according to formula (12) Dp = 26.0-1.6 = 24.4 miles.
The visibility range of an object, calculated using formulas (9) and (10), is called geographical.
Rice. 13.
Visibility range of the beacon light, or optical range visibility depends on the strength of the light source, the beacon system and the color of the fire. In a properly constructed lighthouse, it usually coincides with its geographical range.
In cloudy weather, the actual visibility range may differ significantly from the geographic or optical range.
IN lately Research has established that in daytime sailing conditions, the visibility range of objects is more accurately determined by the following formula:
In Fig. Figure 13 shows a nomogram calculated using formula (13). We will explain the use of the nomogram by solving the problem with the conditions of Example 11.
Example 14. Find the visibility range of an object with a height above sea level H = 26.5 m, with the height of the observer’s eye above sea level e = 4.5 m.
Solution. 1 according to formula (13)
