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# effearthradius

Effective earth radius

Since R2021a

## Syntax

``Re = effearthradius``
``Re = effearthradius(refgrad)``
``Re = effearthradius(R,ha,ht)``
``Re = effearthradius(R,ha,ht,'SurfaceRefractivity',ns)``
``Re = effearthradius(R,ha,ht,___,'BreakPointAltitude',altbp)``
``Re = effearthradius(R,ha,ht,___,'BreakPointRefractivity',npb)``
``[Re,k] = effearthradius(___)``

## Description

example

````Re = effearthradius` returns the effective radius `Re` of a spherical earth computed from the gradient of the index of refraction of the atmosphere. The radius is in meters. This syntax uses the default value of `-39e-9` for the gradient, making the effective radius approximately 4/3 of the actual earth radius. For more information about the computation, see Effective Earth Radius from Refractivity Gradient.```

example

````Re = effearthradius(refgrad)` computes the effective radius from the specified gradient of the refractivity, `refgrad`, of the atmosphere.```

example

````Re = effearthradius(R,ha,ht)` returns the effective Earth radius, `Re`, using the average radius of curvature method (see[1]). `R` is the line-of-sight range to the target. `ha` is the radar altitude above mean sea level (MSL). `ht` is the target altitude above MSL. See Effective Earth Radius from Average Radius of Curvature.```

example

````Re = effearthradius(R,ha,ht,'SurfaceRefractivity',ns)` also specifies the scalar surface refractivity, `ns` for the average radius of curvature method. See Effective Earth Radius from Average Radius of Curvature.```

example

````Re = effearthradius(R,ha,ht,___,'BreakPointAltitude',altbp)` also specifies the altitude of the convergence point, `altbp`, for the average radius of curvature method.```

example

````Re = effearthradius(R,ha,ht,___,'BreakPointRefractivity',npb)` also specifies the refractivity at the convergence point, `npb`, for the average radius of curvature method.```

example

````[Re,k] = effearthradius(___)` also outputs the effective radius factor, `k`. Use this option with any of the syntaxes described above. See Effective Earth Radius.```

## Examples

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Return the default effective earth radius due to atmospheric refraction.

`re = effearthradius`
```re = 8.4774e+06 ```

Compute the ratio of the effective earth radius to the actual earth radius.

```r = physconst('EarthRadius'); disp(re/r)```
``` 1.3306 ```

Compute the effective earth radius from a specified refractivity gradient, `-40e-9`.

```rgrad = -40e-9; re = effearthradius(rgrad)```
```re = 8.5498e+06 ```

Calculate the effective Earth radii for a radar positioned at sea level aimed at two targets. The first target is at 8000 meters above sea level at a range of 100 km. The second target is at 9000 meters altitude at a range of 200 km.

```rng = [100e3,200e3]; ha = [0]; ht = [8.0e3, 9.0e3]; re = effearthradius(rng,ha,ht)```
```re = 1×2 106 × 7.4342 7.3525 ```

Calculate the effective Earth radii for a radar positioned at sea level and aimed at two targets. The first target is at 8000 meters above sea level at a range of 100 km. The second target is at 9000 meters altitude at a range of 200 km. Specify the surface refractivity as 100.0 N-units.

```rng = [100e3,200e3]; ha = [0,0]; ht = [8.0e3,9.0e3]; re = effearthradius(rng,ha,ht,'SurfaceRefractivity',100)```
```re = 1×2 106 × 6.3582 6.3582 ```

Calculate the effective Earth radii for a radar positioned at sea level aimed at two targets. The first target is at 8000 meters above sea level at a range of 100 km. The second target is at 9000 meters altitude at a range of 200 km. The breakpoint altitude is 10000.0 meters and the surface refractivity is 350 N-units.

```rng = [100e3,200e3]; ha = [0,0]; ht = [8.0e3,9.0e3]; re = effearthradius(rng,ha,ht,'SurfaceRefractivity',350.0, ... 'BreakPointAltitude',10000.0)```
```re = 1×2 106 × 7.5877 7.4917 ```

Calculate the effective Earth radii for a radar positioned at sea level and aimed at two targets. The first target is at 8000 meters above sea level at a range of 100 km. The second target is at 9000 meters altitude at a range of 200 km. The breakpoint altitude is 10000.0 meters, the breakpoint refractivity is 300 N-units, and the surface refractivity is 375 N-units.

```rng = [100e3,200e3]; ha = 0; ht = [8.0e3, 9.0e3]; re = effearthradius(rng,ha,ht,'SurfaceRefractivity',375, ... 'BreakPointAltitude',10e3,'BreakPointRefractivity',300)```
```re = 1×2 106 × 6.6962 6.6930 ```

Calculate the effective Earth radius factors for a radar positioned at sea level aimed at two targets. The first target is at 8000 meters above sea level at a range of 100 km. The second target is at 9000 meters altitude at a range of 200 km. The break point altitude is one kilometer, the breakpoint refractivity is 300.0 N-units, and the surface refractivity is 350.0 N-units.

```rng = [100e3,200e3]; ha = [0,0]; ht = [8.0e3,9.0e3]; [re,k] = effearthradius(rng,ha,ht,'SurfaceRefractivity',350.0, ... 'BreakPointAltitude',1000.0,'BreakPointRefractivity',300.0)```
```re = 1×2 106 × 7.7113 7.5724 ```
```k = 1×2 1.2104 1.1886 ```

## Input Arguments

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Refractivity gradient, specified as a scalar. Units are in N-units/meter.

Data Types: `double`

Line-of-sight range to the target from the radar, specified as a positive scalar or a 1-by-M vector of positive values. M must be the same for `R`, `ha`, and `ht`. However, if one of `R`, `ha`, and `ht` is a scalar and another is a 1-by-M vector, the scalar is expanded into a 1-by-M vector. Units are in meters.

Data Types: `double`

Radar altitude above mean sea level, specified as a scalar or a 1-by-M vector. M must be the same for `R`, `ha`, and `ht`. However, if one of `R`, `ha`, and `ht` is a scalar and another is a 1-by-M vector, the scalar is expanded into a 1-by-M vector. Units are in meters.

Data Types: `double`

Target altitude above mean sea level, specified as a scalar or an M-length vector. M must be the same `R`, `ha`, and `ht`. However, if one of `R`, `ha`, and `ht` is a scalar and another is a 1-by-M vector, the scalar is expanded into a 1-by-M vector. Units are in meters.

Data Types: `double`

Scalar surface refractivity, specified as a positive scalar. Units are N-units.

#### Dependencies

To enable this argument, use the syntax specifying `'SurfaceRefractivity'`.

Data Types: `double`

Convergence point altitude, specified as a scalar. The convergence point altitude defaults to 12192 meters when any of the input altitudes specified in `ha` or `ht` are greater than 9144 meters. Otherwise, it defaults to 9144 meters. Setting the `'BreakPointAltitude'` and `'BreakPointRefractivity'` values can be used to tune the output to measured refraction values. For more information, see Effective Earth Radius from Average Radius of Curvature. Units are in meters.

#### Dependencies

To enable this argument, use the syntax specifying `'BreakPointAltitude'`.

Data Types: `double`

Convergence point refractivity, specified as a scalar. The refractivity defaults to 66.65 N-units when any of the input altitudes specified in `ha` or `ht` are greater than 9144 meters. Otherwise, the default is 102.9. Setting the `'BreakPointAltitude'` and `'BreakPointRefractivity'` values can be used to tune the output to measured refraction values. For more information, see Effective Earth Radius from Average Radius of Curvature. Units are N-units.

#### Dependencies

To enable this argument, use the syntax specifying `'BreakPointRefractivity'`.

Data Types: `double`

## Output Arguments

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Effective earth radius, returned as a positive scalar. Units are in meters.

Effective earth radius factor, returned as a positive scalar. The effective earth radius factor is the ratio of the effective earth radius to the physical earth radius. Units are dimensionless.

Data Types: `double`

## More About

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### Effective Earth Radius

The effective earth radius method is an approximation used for modelling refraction effects in the troposphere. Changing the radius of the earth can account for refraction effects. The effective radius method ignores other types of propagation phenomena such as ducting. A related quantity, the effective earth radius factor, is the ratio of the effective earth radius to the actual earth radius.

`$k=\frac{{R}_{e}}{r}$`

where r is the actual earth radius and Re is the effective earth radius. Commonly, the effective earth radius factor, k, is chosen as 4/3. However, at long ranges and with shallow angles, k can deviate greatly from the 4/3. (With no atmospheric refraction, k = 1. An infinite value for k represents a flat Earth). The `effearthradius` function provides two methods for calculating the effective earth radius: the refractivity gradient method and the average radius of curvature method.

### Effective Earth Radius from Refractivity Gradient

An estimate of the effective earth radius factor, k, can be derived from the refractivity gradient using

`$k=\frac{1}{1+r\cdot refgrad}$`

where r is the actual earth radius in meters. refgrad is the gradient of the index of refraction specified by the `refgrad` argument. The index of refraction for a given altitude is the ratio of the free-space propagation speed of electromagnetic waves to the propagation speed in air at that altitude. The gradient is the rate of change of the index of refraction with altitude. The value of 4/3 corresponds to an index of refraction gradient of .

### Effective Earth Radius from Average Radius of Curvature

Another way of estimating the effective earth radius factor is by using the average radius of curvature method described in [1]. The first step in the method is to compute the average radius of curvature over the signal propagation path

`${\rho }_{avg}=\frac{1}{{h}_{a}-{h}_{t}}{\int }_{{h}_{t}}^{{h}_{a}}\rho dh=\frac{{H}_{b}}{{10}^{-6}{N}_{s}\mathrm{cos}{\psi }_{g}}\frac{{e}^{\left(\frac{{h}_{a}-{h}_{t}}{{H}_{b}}\right)}-1}{\frac{{h}_{a}-{h}_{t}}{{H}_{b}}}$`

where the integral spans the range from the radar altitude (ha) to the target altitude (ht).

The constants in the equation where

• ht is the altitude of the target, specified by the `ht` argument.

• ha is the altitude of the radar, specified by the `ha` argument.

• hb is the altitude of the convergence point or breakpoint, specified by the `altbp` argument.

• Nb is the refractivity measure (in N-units) at the convergence point or breakpoint specified by the `npb` argument.

• Ns is the refractivity measure (in N-units) at the surface.

Altitudes are with respect to mean sea level. The constant Hb is computed from

`${H}_{b}=\frac{{h}_{b}-{h}_{t}}{\mathrm{ln}\frac{{N}_{t}}{{N}_{b}}}$`

Then, the effective earth radius factor is computed from the average radius of curvature using

`$k=\frac{1}{1-\frac{{R}_{e}}{{\rho }_{avg}}}$`

### Refractivity Measure and N-Units

The refractivity measure, N, is related to the index of refraction, n by:

`$n=1+{10}^{-6}N$`

10-6N represents the deviation of the index of refraction from the index of refraction of free space. N is expressed in N-units.

## References

[1] Doerry, Armin. W. "Earth Curvature and Atmospheric Refraction Effects on Radar Signal Propagation", Sandia National Laboratories, SAND2012-10690, January 2013.

[2] Long, Maurice W. Radar Reflectivity of Land and Sea, 2nd Ed. Artech House, 2001.

[3] Mahafza, Bassem R. Radar Signal Analysis and Processing Using MATLAB, CRC Press, 2009.

[4] Skolnik, Merrill I. Introduction to Radar Systems, Third edition, McGraw-Hill, 2001.

[5] Ward, James. "Space-Time Adaptive Processing for Airborne Radar", Lincoln Lab Technical Report, 1994.

## Version History

Introduced in R2021a