A Direct Spectral Domain Method for Near-Ground Microwave Radiation by a Vertical Dipole Above Earth in the Presence of Atmospheric Refractivity

8/13/2019 A Direct Spectral Domain Method for Near-Ground Microwave Radiation by a Vertical Dipole Above Earth in the Presence of Atmospheric Refractivity

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A Direct Spectral Domain Method for

Near-Ground Microwave Radiation by a Vertical

Dipole Above Earth in the Presence of

Atmospheric Refractivity

Raj Bhattacharjea

Department of Electrical and Computer EngineeringGeorgia Institute of Technology

Atlanta, GA 30332, USA

August 12, 2013


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Outline

Background and Motivation

Environmental and Mathematical Models

The Spectral Domain Greens FunctionsSommerfeld Integrals

Pole ExtractionHybrid Asymptotic / Numerical Evaluation

Numerical Experiments

Future Research

R. Bhattacharjea, GTA Direct Spectral Domain Method for Near-Ground Microwave Radiation by a Vertical Dipole Above Earth in the Presence of Atmospheric Refractivity 2/19


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Motivation: Real World Applications

Mesh topology low power networks

Wireless sensor networks

Unattended ground sensor

networksApplications

Intrusion detection

Localization

TomographyAgricultural Sensing

Cooperative Cognitive Radio

R. Bhattacharjea, GTDirect Spectral Domain Method for Near-Ground Microwave Radiation by a Vertical Dipole Above Earth in the Presence of Atmospheric Refractivity 3/19


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Motivation: Real World Applications

Mesh topology low power networks

Wireless sensor networks

Unattended ground sensor

networksApplications

Intrusion detection

Localization

TomographyAgricultural Sensing

Cooperative Cognitive Radio



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Refractivity



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Refractivity

In non-magnetic media, n=r

In air, n is a weak function of pressure, temperature, and air composition(H2O is dominant term, followed by CO2) as

N= (n 1) 106 =K1 PT

+K2e

T +K3

e

T2



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Refractivity

In non-magnetic media, n=r

In air, n is a weak function of pressure, temperature, and air composition(H2O is dominant term, followed by CO2) as

N= (n 1) 106 =K1 PT

+K2e

T +K3

e

T2

IfP,T, eare functions of height zabove the earth, then so is N.

Refractive effects are known to be significant (dominant propagationmechanism in some cases) in classical radar scenarios, i.e. tens ofkilometers of range and kilometers of height over the ocean.



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Near-Ground Temperature and Humidity

Dynamics: boundary-layer fluid mechanics + solar forcing.



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Near-Ground Temperature and Humidity

Dynamics: boundary-layer fluid mechanics + solar forcing.

We use some measurements

(P(z),T(z), e(z)) N=K1 PT

+K2e

T +K3

e

T2



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Maxwells Equations in Plane-Stratified Media



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Time-harmonic Maxwells equations(exp(jt)), Lorenz gaugepotentials, in multilayered media:



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Time-harmonic Maxwells equations(exp(jt)), Lorenz gaugepotentials, in multilayered media:

2 +k2 A= 0JTangential E,Hcontinuous at zSommerfeld radiation condition

= {1, 2, . . . L+ 1}



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The Spectral Domain Greens Function

2 +k2 Az= 0 Iz0(z z)Tangential E,H continuous at zSommerfeld radiation condition

= {1, 2, . . . L+ 1}



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= {1, 2, . . . L+ 1}

AzFx,y

Az(x, y)

Fx,y (kx, ky)(2 +k2 )

Fx,y 2z2

+k2zk2z=k

2 k2 , k2 =k2x+k2y



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= {1, 2, . . . L+ 1}

AzFx,y

Az(x, y)

Fx,y (kx, ky)(2 +k2 )

Fx,y 2z2

+k2zk2z=k

2 k2 , k2 =k2x+k2y

Reduces the PDE to an ODE

2Azz2

+k2zAz=

0 Iz0 (zz

)42 , source layer

0, other layers



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= {1, 2, . . . L+ 1}

AzFx,y

Az(x, y)

Fx,y (kx, ky)(2 +k2 )

Fx,y 2z2

+k2zk2z=k

2 k2 , k2 =k2x+k2y

Reduces the PDE to an ODE

2Azz2

+k2zAz=

0 Iz0 (zz

)42 , source layer

0, other layers

With solution

Az=j0 Iz0

82

ejkz|zz|

kz+R+ e

jkz(zz1) +R ejkz(zz), source layer

R+ ejkz(zz1) +R e

jkz(zz), other layers



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Spectral Domain Boundary Conditions

Tangential continuity ofE,H+ Sommerfeld radiation condition:Az= Az(+1)

z=z

R+1 = 0, R

L+1 = 0

1

n2

Azz

= 1

n2+1

Az(+1)z

z=z



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z=z

R+1 = 0, R

L+1 = 0

1

n2

Azz

= 1

n2+1

Az(+1)z

z=z

0 1 2 3 4 5 6 7 8 9

0

1

2

3

4

5

6

7

8

9

atr x o umn

MatrixRow



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z=z

R+1 = 0, R

L+1 = 0

1

n2

Azz

= 1

n2+1

Az(+1)z

z=z

0 1 2 3 4 5 6 7 8 9

0

1

2

3

4

5

6

7

8

9

atr x o umn

MatrixRow



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Sommerfeld Integral and Quadrature

Fourier integral simplifies to Sommerfeld integral: (kx,ky)

Az(kx, ky)exp(j(kxx+kyy)) dkxdky

0

Az(k)J0(k)kdk



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0

Az(k)J0(k)kdk

Two issues to handle for numerical integration



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0

Az(k)J0(k)kdk


Singularities in complex-k plane



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0

Az(k)J0(k)kdk


Singularities in complex-k plane Oscillatory nature ofJ0(k)



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Singularity Extraction and Modal Solutions



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log(Az(k)), z=

5, z = 3.1123



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S S


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0

f(z)

k2

k2pJ0(k) k dk =

j

2 f(z)H

(1)0 (kp)


H dli O ill I l


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Handling Oscillatory Integrals



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H dli O ill t I t l


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Usual ApproachesContour deformation to less oscillatory paths

Curve fit Azwith functions that have closed form Sommerfeld integrals.


H dli O ill t I t l


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Usual ApproachesContour deformation to less oscillatory paths

Curve fit Azwith functions that have closed form Sommerfeld integrals.

New approach: account for the oscillation in the quadrature rule weights.Takes numerical samples and returns asymptotic () results:Filon-Clenshaw-Curtis Quadrature.


Filon Clenshaw Curtis Quadrature


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Filon-Clenshaw-Curtis Quadrature

Az(k) M

m=0

amTm(k)




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Az(k) M

m=0

amTm(k)

Az(k)J0(k)kdkM

m=0

am Tm(k)J0(k)kdk



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Az(k) M

m=0

amTm(k)

Az(k)J0(k)kdkM

m=0

am Tm(k)J0(k)kdkwm() =

Tm(k)J0(k)kdk

There is a recurrence relationship among Tm polynomials (and thus

among the weights). Solving the recurrence numerically for any M and is its own problem, solved by V. Domnguez in the FCC package.




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Az(k) M

m=0

amTm(k)

Az(k)J0(k)kdkM

m=0

am Tm(k)J0(k)kdkwm() =

Tm(k)J0(k)kdk

There is a recurrence relationship among Tm polynomials (and thus

among the weights). Solving the recurrence numerically for any M and is its own problem, solved by V. Domnguez in the FCC package.

Numerically stable implementation for

Az(k) exp(jk)dk, but

J0(k)

12k

ejkj

4 +ejk+j

4

for large


Putting it All Together: Numerical Results 1


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Range [0]

Height[0

]

Dipole radiation in an exponential gradient above lossy dielectric material

0 10 20 30 40 50 60 70 80 90 100

5

0

5

10

15

60

40

20

0

Range [0]

He

ight[0

]

Dipole radiation over PEC with no gradient

0 10 20 30 40 50 60 70 80 90 100

5

0

5

10

15

60

40

20

0




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Range [0]

Height[

0]

Dipole guided modes at 5.8 GHz in a linear gradient above copper

50 100 150 200 250 300 350 400 450 500

0

10

20

dB

80

60

40

20

0

Range [0]

Height[

0]

Raytracing of guided modes in a linear gradient above copper

50 100 150 200 250 300 350 400 450 500

0

10

20

IndexofRefraction

1

1.02

1.04

1.06

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Future Research


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Closed form expressions for the quadrature weights

Horizontal Dipoles and Magnetic DualsScattering formulation based on surface integral equations for terrain

Validation against measurements


Acknowledgements


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g

Dr. V. Domnguez of Universidad Publica de Navarra

This work is sponsored by the US Office of Naval Research


Extra Material


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Environmental Model: Plane-Stratified Lossy Dielectric


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Documents

A Direct Spectral Domain Method for Near-Ground Microwave Radiation by a Vertical Dipole Above Earth in the Presence of Atmospheric Refractivity