Post on 16-Mar-2020
IEEE New Hampshire SectionRadar Systems Course 1Radar Cross Section 1/1/2010 IEEE AES Society
Radar Systems Engineering Lecture 7 –
Part 1
Radar Cross Section
Dr. Robert M. O’DonnellIEEE New Hampshire Section
Guest Lecturer
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Block Diagram of Radar System
Transmitter
WaveformGeneration
PowerAmplifier
T / RSwitch
Antenna
PropagationMedium
TargetRadarCross
Section
Photo ImageCourtesy of US Air ForceUsed with permission.
PulseCompressionReceiver Clutter Rejection
(Doppler Filtering)A / D
Converter
General Purpose Computer
Tracking
DataRecording
ParameterEstimation Detection
Signal Processor Computer
Thresholding
User Displays and Radar Control
This lecture
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Definition -
Radar Cross Section (RCS or σ)
Radar Cross Section
(RCS) is the hypothetical area, that would intercept the incident power at the target, which if scattered isotropically, would produce the same echo power at the radar, as the actual target.
Figure by MIT OCW.
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Factors Determining RCS
Figure by MIT OCW.
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Threat’s View of the Radar Range Equation
Transmitted Pulse
Received Pulse
Target Cross Section σ
Antenna Gain GTransmit Power PT
Figure by MIT OCW.
Distance from Radar to Target
Radar Range Equation
R
SN =
Pt G2 λ2
σ
(4π)3
R4
k TS
Bn
L
Cannot Control
Can ControlCannot Control
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Outline
• Radar cross section (RCS) of typical targets– Variation with frequency, type of target, etc.
• Physical scattering mechanisms and contributors to the RCS of a target
• Prediction of a target’s radar cross section– Measurement– Theoretical Calculation
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Radar Cross Section of Artillery Shell
M107 Shellfor
155mm HowitzerAspect Angle (degrees)
0 20 40 60 80 100 120 140 160
180
Rad
ar C
ross
Sec
tion
(dB
sm)
0
-60
-20
-30
-40
-50
-10
Typical Artillery Shell
RCS vs. Aspect Angle of an Artillery Shell
Courtesy US Marine Corps
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Radar Cross Section of Cessna 150L
S BandV V
Polarization
Measured at RATSCAT (6585th
Test Group) Holloman AFB for FAA
Courtesy of Federal Aviation Administration
Aspect Angle (degrees)0 90 180 270 360
Rad
ar C
ross
Sec
tion
(dB
sm)
0
-20
20
40
Cessna 150L (in takeoff) Cessna 150L (in flight)
Scott Studio Photography with permissionScott Studio Photography with permission
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Aspect Angle Dependence of RCSCone Sphere Re-entry Vehicle (RV) Example
Figure by MIT OCW.
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Examples of Radar Cross Sections Square meters
Conventional winged missile
0.1Small, single engine aircraft, or jet fighter
1Four passenger jet
2Large fighter
6Medium jet airliner
40Jumbo jet
100Helicopter
3
Small open boat
0.02Small pleasure boat (20-30 ft)
2Cabin cruiser (40-50 ft)
10Ship (5,000 tons displacement, L Band)
10,000
Automobile / Small truck
100 -
200Bicycle
2Man
1Birds (large -> medium)
10-2 - 10-3
Insects (locust -> fly)
10-4 - 10-5
Radar Cross Sections of Targets Span at least 50 dBAdapted from Skolnik, Reference 2
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Outline
• Radar cross section (RCS) of typical targets– Variation with frequency, type of target, etc.
• Physical scattering mechanisms and contributors to the RCS of a target
• Prediction of a target’s radar cross section– Measurement– Theoretical Calculation
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RCS Target Contributors
• Types of RCS Contributors– Structural
(Body shape, Control surfaces, etc.)– Avionics
(Altimeter, Seeker, GPS, etc.)– Propulsion (Engine inlets and exhausts, etc.)
Inlet
Control SurfacesAltimeter
SeekerBody Shape
Exhaust
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Single and Multiple Frequency RCS Calculations with the FD-FD Technique
• RCS Calculations for a Single Frequency– Illuminate target with incident sinusoidal wave– Sequentially in time, update the electric and magnetic fields, until
steady state conditions are met– The scattered wave’s amplitude and phase can the be calculated
• RCS Calculations for a Multiple Frequencies– Illuminate target with incident Gaussian pulse– Calculate the transient response– Calculate to Fourier transforms of both:
Incident Gaussian pulse, and Transient response
– RCS at multiple frequencies is calculated from the ratios of these two quantities
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Scattering Mechanisms for an Arbitrary Target
Tip Diffraction atAircraft Nose
Diffraction atCorner
ReturnFrom
Engine Cavity
Edge Diffraction
Tip Diffraction from
Fuel Tank
Specular Surface
ReflectionMultiple
Reflection
Wave Echo from Traveling Wave
Gap, Seam, orDiscontinuity
Echo
Backscatter from
Creeping Wave
CurvatureDiscontinuity
Return
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Measured RCS of C-29 Aircraft Model
Full Scale C-29 BAE Hawker 125-800
Aspect Angle (degrees)
RC
S (d
Bsm
)
0 60 120 180 240 300 360
20
-20
-10
0
10
-30
1/12 ScaleModel
Measurement
X-BandHH PolarizationWaterline Cut
Wing LeadingEdge
FuselageSpecular
Courtesy of Arpingstone
Adapted from Atkins, Reference 5Courtesy of MIT Lincoln Laboratory
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Outline
• Radar cross section (RCS) of typical targets– Variation with frequency, type of target, etc.
• Physical scattering mechanisms and contributors to the RCS of a target
• Prediction of a target’s radar cross section– Measurement– Theoretical Calculation
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Techniques for RCS Analysis
Scaled Model MeasurementsTheoretical Prediction
Full Scale Measurements
Courtesy of MIT Lincoln LaboratoryUsed with Permission
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Full Scale Measurements
Target on Support
• Foam column mounting– Dielectric properties of Styrofoam close to those of free space
• Metal pylon mounting– Metal pylon shaped to reduce radar reflections– Background subtraction can be used
Derived from: http://www.af.mil/shared/media/photodb/photos/050805-F-0000S-003.jpg
Courtesy of MIT Lincoln LaboratoryUsed with Permission
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Full Scale Measurement of Johnson Generic Aircraft Model (JGAM)
RATSCAT Outdoor Measurement Facility at Holloman AFB Courtesy of MIT Lincoln Laboratory
Used with Permission
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Compact Range RCS MeasurementRadar Reflectivity Laboratory (Pt. Mugu) / AFRL Compact Range (WPAFB)
Courtesy of U. S. Navy.
Target
MainReflector
FeedAntenna
PlaneWave
Low RCSPylon
Sub-Reflector
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Scale Model Measurement
SubscaleMeasure at frequency S x F
Scale FactorS
(Reduced Size)
Full ScaleMeasure at frequency f
MQM-107 Drone in 0.29, 0.034, and 0.01 scaled sizes
Courtesy of MIT Lincoln LaboratoryUsed with Permission
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Scaling of RCS of TargetsScale Factor
S
Length
L
L´
= L / S
Wavelength
λ λ´ = λ
/ S
Frequency f
f´
= S f
Time
t
t´
= t / S
Permittivity
ε ε´ = ε
Permeability
μ μ´ = μ
Conductivity
g
g´
= S g
Radar Cross Section
σ σ´ = σ
/ S2
Quantity Full Scale Subscale
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Outline
• Radar cross section (RCS) of typical targets– Variation with frequency, type of target, etc.
• Physical scattering mechanisms and contributors to the RCS of a target
• Prediction of a target’s radar cross section– Measurement– Theoretical Calculation
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Radar Cross Section Calculation Methods
• Introduction– A look at the few simple problems
• RCS prediction– Exact Techniques
Finite Difference-
Time Domain Technique (FD-TD) Method of Moments (MOM)
– Approximate Techniques Geometrical Optics (GO) Physical Optics (PO) Geometrical Theory of Diffraction (GTD) Physical Theory of Diffraction (PTD)
• Comparison of different methodologies
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Radar Cross Section of Sphere
Rayleigh Regionλ >> aσ = k / λ4
Mie or ResonanceRegion
OscillationsBackscattered wave interferes with creeping wave
Optical Regionλ
<< aσ = π a2
Surface and edge scattering occur
Circumference/ wavelength = 2πa / λ
Rad
ar C
ross
Sec
tion
/ πa2
10
1
10-3
10-2
10-1
0.1 0.2 0.4 0.7 1 2 4 7 10 20
RayleighRegion
Resonance or MieRegion
OpticalRegion
a
Higher Wavelengths Lower Wavelengths
Figure by MIT OCW.
λ
>> a
λ << a
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Radar Cross Section Calculation Issues
• Three regions of wavelength Rayleigh (λ
>> a) Mie / Resonance (λ
~ a) Optical
(λ
<< a)
• Other simple shapes– Examples: Cylinders, Flat Plates, Rods, Cones, Ogives– Some
amenable to relatively straightforward solutions in some
wavelength regions
• Complex targets:– Examples: Aircraft, Missiles, Ships)– RCS changes significantly with very small changes in frequency
and / or viewing angle See Ref. 6 (Levanon), problem 2-1 or Ref. 2 (Skolnik) page 57
• We will spend the rest of the lecture studying the different basic methods of calculating radar cross sections
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High Frequency RCS Approximations (Simple Scattering Features)
Scattering Feature
Orientation
Approximate RCS
Corner Reflector
Axis of symmetry along LOS
Flat Plate
Surface perpendicular to LOS
Singly Curved Surface
Surface perpendicular to LOS
Doubly Curved Surface
Surface perpendicular to LOS
Straight Edge
Edge perpendicular to LOS
Curved Edge
Edge element perpendicular to LOS
Cone Tip
Axial incidence
Where:
22eff /A4 λπ
22 /A4 λπ
22 /A4 λπ
21 aaπ
πλ /2
2/a λ
)2/(sin42 αλ
Adapted from Knott is Skolnik Reference 3
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Radar Cross Section Calculation Issues
• Three regions of wavelength Rayleigh (λ
>> a) Mie / Resonance (λ
~ a) Optical
(λ
<< a)
• Other simple shapes– Examples: Cylinders, Flat Plates, Rods, Cones, Ogives– Some
amenable to relatively straightforward solutions in some
wavelength regions
• Complex targets:– Examples: Aircraft, Missiles, Ships)– RCS changes significantly with very small changes in frequency
and / or viewing angle See Ref. 6 (Levanon), problem 2-1 or Ref. 2 (Skolnik) page 57
• We will spend the rest of the lecture studying the different basic methods of calculating radar cross sections
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RCS Calculation -
Overview
• Electromagnetism Problem– A plane wave with electric field, , impinges on the target of
interest and some of the energy scatters back to the radar antenna
– Since, the radar cross section is given by:
– All we need to do is use Maxwell’s Equations to calculate the scattered electric field
– That’s easier said that done
– Before we examine in detail these different techniques, let’s review briefly the necessary electromagnetism concepts and formulae, in the next few viewgraphs
IEr
2
I
2
S2
r E
Er4lim r
r
π=σ∞→
SEr
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Maxwell’s Equations
• Source free region of space:
• Free space constitutive relations:
0)t,r(B0)t,r(D
t)t,r(D)t,r(H
t)t,r(B)t,r(E
=⋅∇
=⋅∇
∂∂
=×∇
∂∂
−=×∇
→
→
rr
rr
rrrr
rrrr
)t,r(H)t,r(B
)t,r(E)t,r(D
o
orrrr
rrrr
μ=
ε= = Free space permittivity
= Free space permeability
oε
oμ
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Maxwell’s Equations in Time-Harmonic Form
• Source free region:
• Time dependence
0)r(B
0)r(D
)r(Di)r(H
)r(Bi)r(E
=⋅∇
=⋅∇
ω−=×∇
ω=×∇→
→
rr
rr
rrrr
rrrr
{ }{ }ti
ti
e)r(HRe)t,r(He)r(ERe)t,r(E
ω−
ω−
=
=rrrr
rrrr
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Boundary Conditions
• Tangential components of and are continuous:
• For surfaces that are perfect conductors:
• Radiation condition:
– As
Er
Hr
21
21
Hxn̂Hxn̂
Exn̂Exn̂rr
rr
=
=
0Exn̂ =r
r1)r(Er ∝∞→
rr
2222
1111
HE
HErr
rr
εμ
εμ n̂Surface
BoundaryMedium 1
Medium 2
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Scattering Matrix
• For a linear polarization basis
• The incident field polarization is related to the scattered field polarization by this Scattering Matrix -
S
• For and a reciprocal medium and for monostatic radar cross section:
• For a circular polarization basis
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
HI
VI
HHHV
VHVVikr
HS
VSS E
ESSSS
re
EEE
r
2VHVH
2HHHH
2VVVV
S4
S4
S4
π=σ
π=σ
π=σ
RLLLRR ,, σσσ
HVVH σ=σ
Scattering Matrix -
S
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Radar Cross Section Calculation Methods
• Introduction– A look at the few simple problems
• RCS prediction– Exact Techniques
Finite Difference-
Time Domain Technique (FD-TD) Method of Moments (MOM)
– Approximate Techniques Geometrical Optics (GO) Physical Optics (PO) Geometrical Theory of Diffraction (GTD) Physical Theory of Diffraction (PTD)
• Comparison of different methodologies
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Methods of Radar Cross Section Calculation
RCS Method
Approach to Determine Surface Currents
Finite Difference- Time Domain (FD-TD)
Solve Differential Form of Maxwell’s Equation’s for Exact Fields
Method of Moments (MoM)
Solve Integral Form of Maxwell’s Equation’s for Exact Currents
Geometrical Optics (GO)
Current Contribution Assumed to Vanish Except at Isolated Specular Points
Physical Optics (PO)
Currents Approximated by Tangent Plane Method
Geometrical Theory of Diffraction (GTD)
Geometrical Optics with Added Edge Current Contribution
Physical Theory of Diffraction (PTD)
Physical Optics with Added Edge Current Contribution
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Finite Difference-
Time Domain (FD-TD) Overview
• Exact method for calculation radar cross section
• Solve differential form of Maxwell’s equations– The change in the E field, in time, is dependent on the change in the H
field, across space, and visa versa
• The differential equations are transformed to difference equations– These difference equations are used to sequentially calculate the E
field at one time and the use those E field calculations to calculate H field at an incrementally greater time; etc. etc.
Called “Marching in Time”
• These time stepped E and H field calculations avoid the necessity of solving simultaneous equations
• Good approach for structures with varying electric and magnetic properties and for cavities
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Maxwell’s Equations in Rectangular Coordinates
• Examine 2 D problem –
no dependence:
• Equations decouple into H-field polarization and E-field polarization
ZoXY
YoZX
XoYZ
Et
Hy
Hx
Ht
Ex
Ez
Et
Hz
Hy
∂∂
ε=∂∂
−∂∂
∂∂
μ−=∂∂
−∂∂
∂∂
ε=∂∂
−∂∂
ZoXY
YoZX
XoYZ
Ht
Ey
Ex
Et
Hx
Hz
Ht
Ez
Ey
∂∂
μ−=∂∂
−∂∂
∂∂
ε=∂∂
−∂∂
∂∂
μ−=∂∂
−∂∂
0y
=∂∂
• H-field polarization • E-field polarization
ZXY EEH ZXY HHE
y
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Maxwell’s Equations in Rectangular Coordinates
• Examine 2 D problem –
no dependence:
• Equations decouple into H-field polarization and E-field polarization
0y
=∂∂
• H-field polarizationZXY EEH
• E-field polarizationZXY HHE
ZoXY
YoZX
XoYZ
Et
Hy
Hx
Ht
Ex
Ez
Et
Hz
Hy
∂∂
ε=∂∂
−∂∂
∂∂
μ−=∂∂
−∂∂
∂∂
ε=∂∂
−∂∂
ZoXY
YoZX
XoYZ
Ht
Ey
Ex
Et
Hx
Hz
Ht
Ez
Ey
∂∂
μ−=∂∂
−∂∂
∂∂
ε=∂∂
−∂∂
∂∂
μ−=∂∂
−∂∂
= 0
= 0
= 0
= 0
y
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• H-field polarization:
• Discrete form:
• Electric and magnetic fields are calculated alternately by the marching in time method
Discrete Form of Maxwell’s Equations
( ) ( )
( )t,y,xEx
t,y,xEz
t,y,xHt
Z
XYo
∂∂
−
∂∂
=∂∂
μ−
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ Δ
+−⎟⎠⎞
⎜⎝⎛ Δ
+Δ+Δ
−
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ Δ
+−⎟⎠⎞
⎜⎝⎛ Δ+
Δ+
Δ=
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ Δ
−Δ
+Δ
+−⎟⎠⎞
⎜⎝⎛ Δ
+Δ
+Δ
+Δμ
−
oZ
ooZoZ
oXoZX
ooX
oXoZoX
oXZ
To
Zo
XoY
To
Zo
XoY
T
o
t,2
z,xEt,2
z,xE1
t,z,2
xEt,z,2
xE1
2t,
2z,
2xH
2t,
2z,
2xH
z
xy
Ex
EZ
HY Yee’sLattice(2-D)
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FD-TD Calculations and Absorbing Boundary Conditions (ABC)
• Absorbing Boundary Condition (ABC) Used to Limit Computational Domain– Reflections at exterior boundary are minimized– Traditional ABC’s model field as outgoing wave to estimate field quantities outside
domain– More recent perfectly matched layer (PML) model uses non-physical layer, that
absorbs waves
Ex
EZ
HY
Perfect ConductorLayer Perfectly Matched0H
x21
tc1
txc1
y2
2
2
2
2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
+∂∂
∂2nd
Order ABC
1st
Order ABC 0Htc
1xz2
1y =⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
Total Field
Target
Domain of Computation
Absorbing Boundary
0ETAN =
Scattered Field
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RCS Calculations Using the FD-TD Method
• Single frequency RCS calculations– Excite with sinusoidal incident wave– Run computation until steady state is reached– Calculate amplitude and phase of scattered wave
• Multiple frequency RCS calculations– Excite with Gaussian pulse incident wave– Calculate transient response– Take Fourier transform of incident pulse and transient
response– Calculate ratios of these transforms to obtain RCS at multiple
frequencies
From Atkins, Reference 5Courtesy of MIT Lincoln Laboratory
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Description of Scattering Cases on Video
Finite Difference Time Domain (FDTD) Simulations
Ei Hi
Hi
Ei
15 deg 15 degHi
Ei
Case 1 –
Plate I Case 2 –
Plate II Case 3 –
Plate III
Case 4 –
Cylinder I Case 5 –
Cylinder II Case 6 –
Cavity
Hi
EiHiEiEi Hi
Courtesy of MIT Lincoln Laboratory Used with Permission
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FD-TD Simulation of Scattering by Strip
0.5 m
Ey
4 m
• Gaussian pulse plane wave incidence• E-field polarization (Ey
plotted)• Phenomena: specular reflection
Case 1
Courtesy ofMIT Lincoln LaboratoryUsed with Permission
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Case 1
Courtesy ofMIT Lincoln LaboratoryUsed with Permission
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FD-TD Simulation of Scattering by Strip
Case 1
Courtesy ofMIT Lincoln LaboratoryUsed with Permission
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FD-TD Simulation of Scattering by Cylinder
0.5 mHy
2 m
• Gaussian pulse plane wave incidence• H-field polarization (Hy
plotted)• Phenomena: creeping waveCase 5
Courtesy ofMIT Lincoln LaboratoryUsed with Permission
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Case 5
Courtesy ofMIT Lincoln LaboratoryUsed with Permission
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FD-TD Simulation of Scattering by Cylinder
Case 5
Courtesy ofMIT Lincoln LaboratoryUsed with Permission
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Backscatter of Short Pulse from Sphere
Creeping Wave Return
SpecularReturn
Radius of sphere is equal to the radar wavelength
Figure by MIT OCW.