slant2range

Convert slant range to propagated range

Since R2022b

Syntax

r = slant2range(sr,anht,tgtht)

r = slant2range(sr,anht,tgtht,Name=Value)

[r,t_el,k] = slant2range(___)

Description

r = slant2range(sr,anht,tgtht) returns the propagated range, r, between a target and sensor as a function of the target slant range sr, antenna height anht, and target height tgtht, assuming a Curved Earth Model with an effective radius factor of 4/3. sr is the straight-line geometric distance between the target and the sensor, whereas r is the distance along the slightly curved propagated path that results from atmospheric refraction.

r = slant2range(sr,anht,tgtht,Name=Value) specifies additional inputs using name-value pair arguments. For example, you can specify a curved Earth model with a given radius or a CRPL Exponential Reference Atmosphere Model with custom values.

[r,t_el,k] = slant2range(___) also returns the elevation angle between the target and the propagated path r, t_el, and the effective earth radius factor, k.

If the outputs are returned as r = NaN, t_el = NaN, and k = 1, the propagation path does not exist or cannot be computed for the geometry specified by sr, anht, and tgtht.

Examples

Calculate Propagated Range using Default Parameters

Calculate the propagated range from a slant range of 100 km, antenna height of 1 km, and target height of 2 km. Use default parameter values.

R = slant2range(100000,1000,2000)

R = 
1.0001e+05

Calculate Propagated Range with Effective Earth Radius

Compute the range between an antenna and a target. Start with a slant range of 250 km, an antenna height of 1 km, and target height of 5 km. Using the default 'Curved' Method, set an effective earth radius factor of 1.2.

Re = physconst('EarthRadius');
effactor = 1.2;
R = slant2range(250000,1000,5000,'EffectiveEarthRadius',effactor*Re)

R = 
2.5002e+05

Verify Target Height Calculation

Calculate the propagated range and elevation angle of a target at a slant range of 300 km, an antenna height of 100 m, and target height of 5 km. Use the CRPL method and assume the surface refractivity is equal to 400 N-units. From the calculated propagated range and elevation angle, convert back to target height. Verify that the estimated target height from the range2height function matches the expected 5 km.

sr = 300000;
ha = 100;
ht = 5000;
Ns = 400;
rexp = refractionexp(Ns);

Calculate propagated range and elevation angle.

[R,el] = slant2range(sr,ha,ht,'Method','CRPL', ...
        'SurfaceRefractivity',Ns,'RefractionExponent',rexp)

R = 
3.0009e+05

el = 
0.1286

Convert propagated range and elevation angle back to target height. Verify that calculated value matches 5000 meters.

htest = range2height(R,ha,el,'Method','CRPL', ...,
    'SurfaceRefractivity',Ns,'RefractionExponent',rexp)

htest = 
5.0000e+03

Compare Effective Earth Radius Factors

Compare the effective Earth radius factors calculated from the CRPL, the average radius of curvature, and 4/3 Earth models. Assume the slant range is 100000 m, the antenna heights range from 1 to 10 km, and the target is on the surface at zero altitude.

sr = 100000;
ha = linspace(1,10,50).*1000;
ht = 200;
[~,kAvgCurv] = effearthradius(sr,ha,ht);
[~,~,kCRPL] = slant2range(sr,ha,ht,'Method','CRPL');

Plot the effective earth radius factor as a function of antenna height.

plot(ha*1e-3,kCRPL)
hold on
plot(ha*1e-3,kAvgCurv)
yline(4/3,'--k')
grid on
legend('CRPL','Average Curvature','4/3 Earth')
ylabel('Effective Earth radius factor k')
xlabel('Antenna height (km)')

Input Arguments

sr — Slant range (m)

nonnegative scalar | nonnegative length-M row vector

Slant range, or straight-line geometric distance between sensor and target, specified as a nonnegative scalar or length-M row vector. If sr is a vector, it must have the same size as the other vector input arguments, anht and tgtht. Units are in meters.

Example: 5000.0

Data Types: double

anht — Sensor height (m)

nonnegative scalar | nonnegative length-M row vector

Sensor height specified as a nonnegative scalar or length-M row vector. If anht is a vector, it must have the same size as the other vector input arguments of slant2range. Heights are referenced to the ground. Units are in meters.

Data Types: double

tgtht — Target height (m)

nonnegative scalar | nonnegative length-M row vector

Target height specified as a nonnegative scalar or length-M row vector. If tgtht is a vector, it must have the same size as the other vector input arguments of slant2range. Heights are referenced to the ground. Units are in meters.

Data Types: double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: r = slant2range(sr,anht,tgtht,Method="CRPL",SurfaceRefractivity=300,RefractionExponent=0.15)

Method — Earth model

"Curved" (default) | "CRPL"

Earth model used for computation, specified as "Curved" or "CRPL".

• "Curved" — Assumes a Curved Earth Model with an effective radius of 4/3 times the actual Earth radius, which is a commonly used approximation for modeling refraction effects in the troposphere. To specify another value for the effective Earth radius, use the EffectiveEarthRadius name-value pair argument.

• "CRPL" — Assumes a curved Earth model with the atmosphere defined by the CRPL Exponential Reference Atmosphere Model with a refractivity of 313 N-units and a refraction exponent of 0.143859 km^–1. To specify other values for the refractivity and the refraction exponent, use the SurfaceRefractivity and RefractionExponent name-value arguments. This CRPL Exponential model accounts for refraction at elevation angles greater than approximately 10 millirad (about 0.573 degrees) and heights above approximately 1 km. For more information, see CRPL Model Geometry.

Data Types: char | string

EffectiveEarthRadius — Effective Earth radius (m)

4/3 × Earth radius (default) | positive scalar

Effective Earth radius, specified as a positive scalar. By default, the effective Earth radius is calculated using a refractivity gradient of –39 × 10^–9 N-units/meter, which results in approximately 4/3 times the actual Earth radius. See effearthradius for more information. Units are in meters.

Dependencies

To enable this argument, set the Method name-value pair argument to "Curved".

Data Types: double

SurfaceRefractivity — Surface refractivity

313 (default) | nonnegative scalar

Surface refractivity in N-units, specified as a nonnegative scalar. The surface refractivity is a parameter of the CRPL Exponential Reference Atmosphere Model. This quantity is dimensionless.

Dependencies

To enable this argument, set the Method name-value pair argument to "CRPL".

Data Types: double

RefractionExponent — Refraction exponent

0.143859 (default) | nonnegative scalar

Refraction exponent, specified as a nonnegative scalar. The refraction exponent is a parameter of the CRPL Exponential Reference Atmosphere Model. This quantity is dimensionless.

Dependencies

To enable this argument, set the Method name-value pair argument to "CRPL".

Data Types: double

MaxNumIterations — Maximum number of iterations for the CRPL method

10 (default) | nonnegative integer

Maximum number of iterations for the CRPL method, specified as a nonnegative integer. This input acts as a safeguard to preempt long iterative calculations.

If MaxNumIterations is set to 0, a faster but less accurate non-iterative CRPL calculation is performed. The non-iterative calculation has a maximum height error of 0.056388 m (0.185 ft) at a target height of 30,480 m (100,000 ft) and an elevation angle of 0 degrees. The height error for the non-iterative method decreases with decreasing target height and increasing elevation angle. This quantity is dimensionless.

Dependencies

To enable this argument, set the Method name-value pair argument to "CRPL".

Data Types: double

Tolerance — Numerical tolerance for the CRPL method

1e-6 (default) | positive scalar

Numerical tolerance for the CRPL method, specified as a positive scalar. The iterative process terminates when the numerical tolerance is achieved.

Dependencies

To enable this argument, set the Method name-value pair argument to "CRPL" and set the MaxNumIterations name-value pair argument to be greater than 0. This quantity is dimensionless.

Data Types: double

Output Arguments

r — Propagated range (m)

scalar | length-M row vector

Propagated range between the target and sensor, returned as a scalar or length-M row vector. If r is a vector, it has the same size as the vector input arguments of slant2range. The propagated range is the distance along the slightly curved propagated path that results from atmospheric refraction over long distances and is the range that would be retrieved from measurements acquired by a sensor with pointing geometry defined by sr, anht, and tgtht.Units are in meters.

Data Types: double

t_el — Elevation angle (deg)

scalar | length-M row vector

Elevation angle between the target and propagated path r, returned as a scalar or length-M row vector. At long slant ranges, t_el differs from the elevation angle of the sensor due to atmosphere refraction. Units are in degrees.

Data Types: double

k — Effective earth radius factor

scalar | length-M row vector

Effective earth radius factor, returned as a scalar or length-M row vector. This quantity is dimensionless.

Data Types: double

More About

Curved Earth Model

Atmospheric refraction bends radar signals, which causes a slightly curved propagation path that is longer than the straight-line slant path to the target. In other words, over long distances, the propagated range is greater than the slant range due to refraction. The curved Earth model leverages the fact that the index of refraction of air depends on height and uses an effective Earth radius that is larger than the actual value to approximate the propagated path as a straight line. Therefore, in this model, the propagated range is the length of the slant range that is determined using a larger effective Earth radius.

Given the effective Earth radius R₀, the antenna height h_a, and the initial elevation angle θ₀, the model relates the target height h_T and the slant range R_T by

$${\left(R_0+h_T\right)}^2={\left(R_0+h_a\right)}^2+R_T^2+2R_T\left(R_0+h_a\right)\sin {\theta }_0.$$

In this equation, R_T approximates the propagated range because R₀ is larger than the actual Earth radius. Similarly, the initial elevation angle θ₀ is considered to be the elevation angle between the target and the propagated path.

The equation can be rearrange to solve for the target height,

$$h_T=\sqrt{{\left(R_0+h_a\right)}^2+R_T^2+2R_T\left(R_0+h_a\right)\sin {\theta }_0}-R_0.$$

To compute the ground range G, use

$$G=R_0\varphi =R_0\arcsin \frac{R_T\cos {\theta }_0}{R_0+h_T}.$$

A standard propagation model uses an effective Earth's radius that is 4/3 times the actual value. This model has two major limitations:

1. The model implies a value for the index of refraction near the Earth's surface that is valid only for certain areas and at certain times of the year. To mitigate this limitation, use an effective Earth's radius based on the near-surface refractivity value.

2. The model implies a value for the gradient of the index of refraction that is unrealistically low at heights of around 8 km. To partially mitigate this limitation, use an effective Earth's radius based on the platform altitudes.

For more information, see effearthradius.

CRPL Exponential Reference Atmosphere Model

Atmospheric refraction evidences itself as a deviation in an electromagnetic ray from a straight line due to variation in air density as a function of height. The Central Radio Propagation Laboratory (CRPL) exponential reference atmosphere model treats refraction effects by assuming that the index of refraction n(h) and the refractivity N decay exponentially with height. The model defines

$$N=\left(n\left(h\right)-1\right)\times {10}^6=N_{\text{s}}{e}^{-R_{\text{exp}}h},$$

where N_s is the atmospheric refractivity value (in units of 10^–6) at the surface of the earth, R_exp is the decay constant, and h is the height above the surface in kilometers. Thus

$$n(h)=1+\left(N_{\text{s}}\times {10}^{-6}\right)e^{-R_{\text{exp}}h}.$$

The default value of N_s is 313 N-units and can be modified using the SurfaceRefractivity name-value argument in functions that accept it. The default value of R_exp is 0.143859 km^–1 and can be modified using the RefractionExponent name-value argument in functions that accept it. This CRPL Exponential model accounts for refraction at elevation angles greater than approximately 10 millirad (about 0.573 degrees) and heights above approximately 1 km.

CRPL Model Geometry

When the refractivity of air is incorporated into the curved Earth model, the ray paths do not follow a straight line but curve downward. (This statement assumes standard atmospheric propagation and nonnegative elevation angles.) The true elevation angle is different from the initial . The actual range , which is the distance along the curved path , is different from the slant range .

Given the Earth's radius , the antenna height , the initial elevation angle , and the height-dependent index of refraction with value at , the modified model relates the target height and the actual range by

When Method is specified as "CRPL", the integral is solved using from CRPL Exponential Reference Atmosphere Model.

To compute the ground range , use

N-units

N-units are a convenient way to express the index of refraction. Because the index of refraction is very close to unity, N-units express just the deviation from unity. The refractivity N in N-units is related to the index of refraction n by

$$N=(n-1)\times {10}^6\text{.}$$

For example, an index of refraction of 1.000313 becomes 313 in N-units. N-units are dimensionless.

References

[1] Barton, David K. Radar Equations for Modern Radar. Norwood, MA: Artech House, 2013.

[2] Bean, B.R., and G.D. Thayer. "Central Radio Propagation Laboratory Exponential Reference Atmosphere." Journal of Research of the National Bureau of Standards, Section D: Radio Propagation 63D, no. 3 (November 1959): 315. https://doi.org/10.6028/jres.063D.031.

[3] Blake, Lamont V. "Ray Height Computation for a Continuous Nonlinear Atmospheric Refractive-Index Profile." Radio Science 3, no. 1 (January 1968): 85–92. https://doi.org/10.1002/rds19683185.

[4] Doerry, A. W. "Earth Curvature and Atmospheric Refraction Effects on Radar Signal Propagation." Sandia National Laboratories, SAND2012-10690 (Jan. 2013).

Extended Capabilities

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Version History

Introduced in R2022b