How Do You Calculate Refractivity?

Refractivity, often expressed through the refractive index (n), can be calculated for atmospheric applications using formulas that account for changes with altitude. Based on the provided reference, one method uses an exponential model relating the refractive index to height.

## Understanding Atmospheric Refractivity

Atmospheric refractivity describes how the refractive index of air deviates from 1. This property is crucial for understanding how electromagnetic waves, like radio waves, propagate through the atmosphere, affecting communications, radar, and navigation systems. The refractive index varies primarily with pressure, temperature, and humidity, which change significantly with height.

## The Calculation Formula

According to the provided reference, the refractive index **n(h)** at a specific height **h** can be calculated using the following formula:

**n(h) = 1 + N0 × 10–6 × exp (–h/h0)**

This formula models the refractive index's dependency on altitude using an exponential decay relative to a sea-level value.

## Key Variables Explained

The formula utilizes several key variables to determine the refractive index at a given height:

| Variable | Description                                                          | Unit(s) |
| :------- | :------------------------------------------------------------------- | :------ |
| **n(h)** | The refractive index of the atmosphere at height **h**.              | Unitless |
| **h**    | The height above sea level for which the refractivity is calculated. | km      |
| **N0**   | The average value of atmospheric refractivity extrapolated to sea level. | Unitless |
| **h0**   | The scale height, representing the rate at which refractivity decreases with height. | km      |

**N0** and **h0** are not universal constants. As noted in the reference, they are typically **determined statistically for different climates and locations**. This means their values will vary depending on the typical atmospheric conditions (temperature, pressure, humidity profiles) of a specific region.

## Applying the Formula

To calculate the refractive index n(h) at a specific height h using this model:

1.  Determine the appropriate values for **N0** and **h0** based on the climate or location of interest. These values are usually derived from historical atmospheric data for that region.
2.  Identify the height **h** (in kilometers) at which you want to calculate the refractive index.
3.  Plug the values of **N0**, **h0**, and **h** into the formula:
    `n(h) = 1 + N0 × 10–6 × exp (–h/h0)`
4.  Calculate the result to obtain the refractive index **n(h)**.

### Example

Suppose for a specific climate, statistical data provides:
*   **N0** = 310
*   **h0** = 7.5 km

To calculate the refractive index at a height of **h** = 3 km:

n(3) = 1 + 310 × 10–6 × exp (–3 / 7.5)
n(3) = 1 + 310 × 10–6 × exp (–0.4)
n(3) ≈ 1 + 310 × 10–6 × 0.6703
n(3) ≈ 1 + 207.793 × 10–6
n(3) ≈ 1.000207793

So, the refractive index at 3 km height in this climate would be approximately 1.000208.

This formula provides a simplified but often effective way to model the vertical profile of the atmospheric refractive index for various applications.