Fog Visibility Distance Calculator

Understanding Fog and Horizontal Visibility

Fog forms when countless tiny water droplets are suspended near the ground, turning the air into a kind of light-scattering screen. As light from distant objects passes through this droplet-filled layer, it is scattered and partially absorbed, so the contrast between the object and its background fades. Once the contrast drops below what the human eye can detect, the object effectively disappears from view, even if it is still physically present.

This loss of contrast defines meteorological visibility: the greatest distance at which a large, dark object can just be distinguished against the horizon sky. In everyday terms, it answers the question, “How far can I realistically see in these foggy conditions?”

Drivers, pilots, mariners, and outdoor planners all rely on visibility estimates to decide whether conditions are safe enough to operate or travel. A common operational threshold for fog is visibility below about 1 km, but dense ground fog can reduce visibility to tens of metres or less.

Koschmieder’s Law and the Visibility Formula

To link what we see to measurable properties of the atmosphere, meteorologists often use Koschmieder’s law. It relates visibility to the extinction coefficient, a parameter that describes how strongly the atmosphere attenuates (weakens) light per unit distance through scattering and absorption.

The basic form of Koschmieder’s law for visibility V is:

V = 3.912 / β

where:

• V is the meteorological visibility (in kilometres), and
• β (beta) is the extinction coefficient (in 1/km).

The constant 3.912 comes from assuming that the minimum contrast the average human eye can detect is about 2%. In mathematical terms, it is derived from the natural logarithm of 1/0.02. A more formal expression of the relationship between contrast and distance uses an exponential decay model:

Here C is the contrast after travelling distance x through the atmosphere, C₀ is the initial contrast at the source, β is the extinction coefficient, and e is the base of natural logarithms. Setting C equal to the threshold contrast of the human eye and solving for x yields the familiar visibility formula.

Typical values of β vary widely:

• Clean, dry air: β ≈ 0.02–0.05 1/km (very long visibility, tens of kilometres or more)
• Haze or light pollution: β ≈ 0.1–0.3 1/km
• Dense fog or heavy pollution: β > 0.3 1/km (visibility can drop below 1 km, sometimes to tens of metres)

Relative humidity is closely linked to fog formation because moist air is more likely to reach saturation and condense into droplets. However, humidity alone does not fully determine β; droplet number, size, and additional aerosols (such as smoke or dust) all matter as well.

How This Fog Visibility Calculator Works

This calculator uses a simplified application of Koschmieder’s law to estimate how far you can see in foggy or hazy conditions. You provide:

• Relative humidity (%): a general indicator of how close the air is to saturation. Higher values (often > 90%) commonly accompany fog or mist.
• Extinction coefficient (1/km): a measure of how strongly the atmosphere reduces light per kilometre. This is the primary input used in the visibility formula.

The calculator then uses the relationship V = 3.912 / β to estimate visibility in kilometres, optionally presenting the result in metres as well for very low-visibility situations.

Finding or estimating β:

• If you have access to instrument data (e.g., from a visibility sensor, transmissometer, or a detailed weather report), you may be given β directly or a related parameter that can be converted.
• If you only know an approximate visibility from a nearby weather station, you can invert Koschmieder’s law: β ≈ 3.912 / V.
• When no measurements are available, you can use typical ranges such as those in the comparison table below as rough guidance.

Worked Example

Suppose you are planning an early-morning drive and conditions are very moist with a reported relative humidity of 95%. Local weather information indicates that the atmosphere is quite hazy, and you estimate an extinction coefficient of β = 0.3 1/km.

Using Koschmieder’s law:

V = 3.912 / 0.3 ≈ 13.0 km

This means that, under the simplified assumptions of a horizontally uniform atmosphere and a 2% contrast threshold, the maximum distance at which a large, dark object can just be distinguished from the horizon sky is around 13 km. Near the ground, however, actual operational visibility (for example along a roadway) may be lower because fog can be thicker in low-lying areas, over water, or within local pockets.

If you instead estimated a much denser fog with β = 1.0 1/km, then:

V = 3.912 / 1.0 ≈ 3.9 km

And for extreme cases such as heavy ground fog, β can exceed 2 1/km, reducing theoretical visibility to under 2 km, with practical driving visibility sometimes well under 500 m.

Reference Table: Extinction, Visibility, and Conditions

The table below gives approximate relationships between β, the resulting visibility from Koschmieder’s law, and qualitative descriptions of the conditions. These are illustrative, not strict thresholds.

Assumptions and Limitations

This calculator is designed for educational and planning purposes, not for safety-critical decision-making. It relies on a simplified representation of a very complex atmosphere, so it is important to understand the underlying assumptions.

• Single extinction coefficient: The model assumes a single, constant β value along the entire line of sight. Real fog layers often vary with height, terrain, and distance.
• Horizontally uniform conditions: Koschmieder’s law assumes that the atmosphere has uniform properties. In practice, fog can form patches, banks, and layers, leading to sharp visibility changes over short distances.
• Fixed contrast threshold: The 3.912 constant is based on an average human contrast threshold of 2%. Individual eyesight, lighting conditions, and object properties can significantly change the actual detection distance.
• Idealised geometry: The formula is derived for a large, dark object against the horizon sky. Real-world objects may be smaller, lighter, moving, or illuminated in ways that deviate from this ideal.
• Humidity as a proxy: Relative humidity is included as a helpful context variable, but it is not a complete predictor of β. Two situations with the same humidity can have very different visibilities depending on aerosols, droplet sizes, and temperature structure.
• Rapidly changing conditions: Fog and low cloud can form, deepen, or dissipate quickly due to wind, solar heating, or changes in air mass. Any estimate is only valid for the conditions at the time of input.

Practical Use and Safety Disclaimer

Use the output of this calculator as a rough guide to understand how extinction and humidity relate to visibility, to compare scenarios, or to support educational discussions about meteorology and optics.

Do not use the results as your only source of information for operating vehicles, aircraft, or vessels. Visibility estimates from this tool are for informational and educational purposes only and are not a substitute for:

• official weather forecasts and visibility reports,
• aviation, marine, or road authority guidance, and
• local regulations and safety procedures.

Always follow official guidance and err on the side of caution when fog or poor visibility is present.