range2height
Description
returns the target height tgtht
= range2height(r
,anht
,el
)tgtht
as a function of the propagated range
r
, the sensor height anht
, and the local
elevation angle el
assuming a Curved Earth Model with a 4/3 effective Earth
radius.
specifies additional inputs using name-value arguments. For example, you can specify a flat
Earth model, a curved Earth model with a given radius, or a CRPL Exponential Reference Atmosphere Model with custom
values.tgtht
= range2height(r
,anht
,el
,Name=Value
)
Examples
Target Height from Propagated Range
Determine the target height in meters given a range of 300 km, a sensor height of 10 meters, and an elevation angle of 0.5 degrees. Assume a curved Earth with an effective radius equal to 4/3 times the Earth's actual radius.
R = 300e3; anht = 10; el = 0.5; range2height(R,anht,el)
ans = 7.9325e+03
Target Height Using Different Earth Models
Compute target heights in meters using different Earth models and compare the values you obtain. Assume a range of 200 km and an antenna height of 100 meters. Use a range of elevation angles from 0 to 5 degrees.
R = 200e3; anht = 100; el = (0:0.1:5)';
Compute the target height for the given parameters assuming a flat Earth.
tgthtFlat = range2height(R,anht,el,Method="Flat");
Compute the target height for the given parameters assuming free-space propagation with a curved Earth.
r0 = physconst("EarthRadius"); tgthtFS = range2height(R,anht,el,Method="Curved", ... EffectiveEarthRadius=r0);
Compute the target height for the given parameters assuming a 4/3 effective Earth radius.
tgthtEffRad = range2height(R,anht,el);
Compute the target height for the given parameters assuming the CRPL atmospheric model.
tgthtCRPL = range2height(R,anht,el,Method="CRPL");
Plot the results.
plot(el,[tgthtFlat(:) tgthtFS(:) tgthtEffRad(:)], ... el,tgthtCRPL,'--',LineWidth=1.5) grid on xlabel("Elevation Angle (degrees)") ylabel("Target Height (m)") legend(["Flat" "Free Space" "4/3 Earth" "CRPL"],Location="best") title("Target Height Estimation")
Input Arguments
r
— Propagated range
real-valued scalar | real-valued vector
Propagated range between the target and the sensor in meters, specified as a
real-valued scalar or vector. If r
is a vector, it must have the
same size as the other vector input arguments of
range2height
.
Data Types: double
anht
— Sensor height
nonnegative real-valued scalar | nonnegative real-valued vector
Sensor height in meters, specified as a nonnegative real-valued scalar or vector. If
anht
is a vector, it must have the same size as the other
vector input arguments of range2height
. Heights are referenced to
the ground.
Data Types: double
el
— Local elevation angle
real-valued scalar | real-valued vector
Local elevation angle in degrees, specified as a real-valued scalar or vector. The local elevation angle is the initial elevation angle of the ray leaving the sensor. If el
is a vector, it must have the same size as the other vector input arguments of range2height
.
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.
Example: Method="CRPL",SurfaceRefractivity=300,RefractionExponent=0.15,MaxNumIterations=8,Tolerance=1e-7
Method
— Earth model
"Curved"
(default) | "Flat"
| "CRPL"
Earth model used for the computation, specified as "Curved"
,
"Flat"
, or "CPRL"
.
"Curved"
— Assumes a Curved Earth Model with a 4/3 effective Earth radius, which is an approximation used for modeling refraction effects in the troposphere. To specify another value for the effective Earth radius, use theEffectiveEarthRadius
name-value argument."Flat"
— Assumes a Flat Earth Model. In this case, the effective Earth radius is infinite."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 theSurfaceRefractivity
andRefractionExponent
name value arguments. This method requiresel
to be positive. For more information, see CRPL Model Geometry.
Data Types: char
| string
EffectiveEarthRadius
— Effective Earth radius
4/3 of Earth's radius (default) | positive scalar
Effective Earth radius in meters, specified as a positive scalar. If this argument is
not specified, range2height
calculates the effective Earth radius
using a refractivity gradient of –39 × 10–9 N-units/meter,
which results in approximately 4/3 of the real Earth radius. This argument applies only
if Method
is specified as "Curved"
.
Data Types: double
SurfaceRefractivity
— Surface refractivity
313
(default) | real-valued scalar
Surface refractivity in N-units, specified as a nonnegative real-valued scalar. The surface
refractivity is a parameter of the CRPL Exponential Reference Atmosphere Model used by
range2height
. This argument applies only if
Method
is specified as "CRPL"
.
Data Types: double
RefractionExponent
— Refraction exponent
0.143859
(default) | real-valued scalar
Refraction exponent, specified as a nonnegative real-valued scalar. The refraction exponent is
a parameter of the CRPL Exponential Reference Atmosphere Model used by
range2height
. This argument applies only if
Method
is specified as "CRPL"
.
Data Types: double
MaxNumIterations
— Maximum number of iterations for the CRPL method
10
(default) | nonnegative scalar integer
Maximum number of iterations for the CRPL method, specified as a nonnegative
scalar integer. This input acts as a safeguard to preempt long iterative calculations.
This argument applies only if Method
is specified as
"CRPL"
.
If MaxNumIterations
is set to 0
,
range2height
performs a faster but less accurate noniterative
CRPL calculation. The noniterative 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.
The height error for the noniterative method decreases with decreasing target height
and increasing elevation angle.
Data Types: double
Tolerance
— Numerical tolerance for the CRPL method
1e-6
(default) | positive real scalar
Numerical tolerance for the CRPL method, specified as a positive real scalar. The
iterative process terminates when the numerical tolerance is achieved. This argument
applies only if Method
is specified as "CRPL"
and MaxNumIterations
is greater than 0
.
Data Types: double
Output Arguments
tgtht
— Target height
nonnegative real-valued scalar | nonnegative real-valued row vector
Target height in meters, returned as a nonnegative real-valued scalar or row vector.
If tgtht
is a vector, it has the same size as the vector input
arguments of range2height
. The height is referenced to the
ground.
More About
Flat Earth Model
The flat Earth model assumes that the Earth has infinite radius and that the index of refraction of air is uniform throughout the atmosphere. The flat Earth model is applicable over short distances and is used in applications like communications, automotive radar, and synthetic aperture radar (SAR).
Given the antenna height ha and the initial elevation angle θ0, the model relates the target height hT and the slant range RT by
so knowing one of those magnitudes enables you to compute the other. The actual range R is equal to the slant range. The true elevation angle θT is equal to the initial elevation angle.
To compute the ground range G, use
Curved Earth Model
The fact that the index of refraction of air depends on height can be treated approximately by using an effective Earth's radius larger than the actual value.
Given the effective Earth's radius R0, the antenna height ha, and the initial elevation angle θ0, the model relates the target height hT and the slant range RT by
so knowing one of those magnitudes enables you to compute the other. In particular,
The actual range R is equal to the slant range. The true elevation angle θT is equal to the initial elevation angle.
To compute the ground range G, use
A standard propagation model uses an effective Earth's radius that is 4/3 times the actual value. This model has two major limitations:
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.
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
where Ns is the atmospheric refractivity value (in units of 10–6) at the surface of the earth, Rexp is the decay constant, and h is the height above the surface in kilometers. Thus
The default value of Ns is 313
N-units and can be modified using the SurfaceRefractivity
name-value
argument in functions that accept it. The default value of
Rexp is 0.143859 km–1 and can be modified using the RefractionExponent
name-value argument in functions that accept it.
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
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.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Version History
Introduced in R2021b
See Also
Apps
Functions
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