effearthradius

Effective earth radius

Since R2022b

Syntax

Re = effearthradius(atmos)

Re = effearthradius(atmos,slr,ha,ht)

[Re,k] = effearthradius(___)

Description

Re = effearthradius(atmos) returns the effective radius Re of a spherical earth where atmos is an atmosphere of a radarScenario object. The effective radius is computed using the atmosphere and its associated properties. This syntax generally applies for altitudes less than 2 km. For more information about the computation, see Effective Earth Radius from Refractivity Gradient.

Re = effearthradius(atmos,slr,ha,ht) returns the effective Earth radius, Re, using the average radius of curvature method (see [1]). slr 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.

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

Examples

Compute Effective Earth Radius in Radar Scenario

Define an atmosphere in a radar scenario using an effective Earth radius model with a default 4/3 Earth radius.

scenario = radarScenario('IsEarthCentered',true);
atmos = atmosphere(scenario,'EffectiveEarth');
Re = effearthradius(atmos)

Re = 
8.4774e+06

Compare Effective Earth Factors in Radar Scenario

Compare the effective Earth factors calculated from the CRPL and 4/3 Earth models. Assume the slant range is 100 km, the antenna heights range from 1 to 10 km, and the target altitude is at the surface.

 SR = 100e3;
 ha = linspace(1,10,50).*1e3;
 ht = 0;

Create a radar scenario and a CRPL atmosphere.

scenario = radarScenario('IsEarthCentered',true);
atmos = atmosphere(scenario,'CRPL');
[~,kCRPL] = effearthradius(atmos,SR,ha,ht);

Plot the computed k-factor and a vertical 4/3 line.

semilogy(kCRPL,ha*1e-3)
hold on
xline(4/3,'-.r')
xline(1,'--k')
xlim([0.99 1.37])
grid on
legend('CRPL','4/3 Earth','True Earth')
xlabel('Effective Earth Radius Factor k')
ylabel('Altitude (km)')
hold off

Input Arguments

atmos — Atmosphere

-39e-9 (default) | scalar

Atmosphere belonging to a radarScenario object.

Data Types: double

slr — Line-of-sight range to target

positive scalar | 1-by-M vector of positive values

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 slr, ha, and ht. However, if one of slr, 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

ha — Radar altitude above mean sea level

scalar | 1-by-M vector

Radar altitude above mean sea level, specified as a scalar or a 1-by-M vector. M must be the same for slr, ha, and ht. However, if one of slr, 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

ht — Target altitude above mean sea level

scalar | M-length vector

Target altitude above mean sea level, specified as a scalar or an M-length vector. M must be the same slr, ha, and ht. However, if one of slr, 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

Output Arguments

Re — Effective earth radius

4/3 actual earth radius (default) | positive scalar

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

k — Effective earth radius factor

4/3 (default) | positive scalar

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

Effective Earth Radius

The effective earth radius method is an approximation used for modeling 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 R_e 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 effective Earth radius is based on the radarScenario atmosphere. All atmosphere types output an effective Earth radius. There are four ways to specify it. It can be free space, effective Earth radius (k or effective radius), refractivity gradient, or the CRPL atmosphere.

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 $$-39\times {10}^{-9}{\text{ m}}^{-1}$$.

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 R2022b