Abstract

The problem of optical ray propagation in a nonuniform random half-plane lattice is considered. An external source radiates a planar monochromatic wave impinging at an angle θ on a half-plane random grid where each cell can be independently occupied with probability qj=1pj,j being the row index. The wave undergoes specular reflections on the occupied cells, and the probability of penetrating up to level k inside the lattice is analytically estimated. Numerical experiments validate the proposed approach and show improvement upon previous results that appeared in the literature. Applications are in the field of remote sensing and communications, where estimation of the penetration of electromagnetic waves in disordered media is of interest.

© 2006 Optical Society of America

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  1. G. Franceschetti, S. Marano, and F. Palmieri, "Propagation without wave equation toward an urban area model," IEEE Trans. Antennas Propag. 47, 1393-1404 (1999).
    [CrossRef]
  2. G. Grimmett, Percolation (Springer-Verlag, 1989).
  3. D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, 1985).
    [CrossRef]
  4. R. M. Ross, Stochastic Processes (Wiley, 1983).
  5. S. Marano, F. Palmieri, and G. Franceschetti, "Statistical characterization of ray propagation in a random lattice," J. Opt. Soc. Am. A 16, 2459-2464 (1999).
    [CrossRef]
  6. S. Marano and M. Franceschetti, "Ray propagation in a random lattice: a maximum entropy, anomalous diffusion process," IEEE Trans. Antennas Propag. 53, 1888-1896 (2005).
    [CrossRef]
  7. M. Franceschetti, J. Bruck, and L. J. Schulman, "A random walk model of wave propagation," IEEE Trans. Antennas Propag. 52, 1304-1317 (2004).
    [CrossRef]
  8. M. Franceschetti, "Stochastic rays pulse propagation," IEEE Trans. Antennas Propag. 52, 2742-2752 (2004).
    [CrossRef]
  9. T. K. Sarkar, J. Zhong, K. Kyungjung, A. Medouri, and M. Salazar-Palma, "A survey of various propagation models for mobile communication," IEEE Antennas Propag. Mag. 45, 51-82 (2003).
    [CrossRef]
  10. H. L. Bertoni, W. Honcharenko, L. Rocha Maciel, and H. H. Xia, "UHF propagation prediction for wireless personal communications," Proc. IEEE 82, 1333-1359 (1994).
    [CrossRef]
  11. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).
  12. A. Ishimaru, "Wave propagation and scattering in random media and rough surfaces," Proc. IEEE 79, 1359-1366 (1991).
    [CrossRef]
  13. Special issue on "Wave propagation and scattering in random media," J. Opt. Soc. Am. A 2, 2062-2404 (1985).
  14. J. R. Norris, Markov Chains (Cambridge U. Press, 1998).
  15. A. Martini, M. Franceschetti, and A. Massa are preparing a paper to be called "'Electromagnetic wave propagation in nonuniform percolation lattices--theory and experiments."

2005 (1)

S. Marano and M. Franceschetti, "Ray propagation in a random lattice: a maximum entropy, anomalous diffusion process," IEEE Trans. Antennas Propag. 53, 1888-1896 (2005).
[CrossRef]

2004 (2)

M. Franceschetti, J. Bruck, and L. J. Schulman, "A random walk model of wave propagation," IEEE Trans. Antennas Propag. 52, 1304-1317 (2004).
[CrossRef]

M. Franceschetti, "Stochastic rays pulse propagation," IEEE Trans. Antennas Propag. 52, 2742-2752 (2004).
[CrossRef]

2003 (1)

T. K. Sarkar, J. Zhong, K. Kyungjung, A. Medouri, and M. Salazar-Palma, "A survey of various propagation models for mobile communication," IEEE Antennas Propag. Mag. 45, 51-82 (2003).
[CrossRef]

1999 (2)

G. Franceschetti, S. Marano, and F. Palmieri, "Propagation without wave equation toward an urban area model," IEEE Trans. Antennas Propag. 47, 1393-1404 (1999).
[CrossRef]

S. Marano, F. Palmieri, and G. Franceschetti, "Statistical characterization of ray propagation in a random lattice," J. Opt. Soc. Am. A 16, 2459-2464 (1999).
[CrossRef]

1998 (1)

J. R. Norris, Markov Chains (Cambridge U. Press, 1998).

1997 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

1994 (1)

H. L. Bertoni, W. Honcharenko, L. Rocha Maciel, and H. H. Xia, "UHF propagation prediction for wireless personal communications," Proc. IEEE 82, 1333-1359 (1994).
[CrossRef]

1991 (1)

A. Ishimaru, "Wave propagation and scattering in random media and rough surfaces," Proc. IEEE 79, 1359-1366 (1991).
[CrossRef]

1989 (1)

G. Grimmett, Percolation (Springer-Verlag, 1989).

1985 (2)

D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, 1985).
[CrossRef]

Special issue on "Wave propagation and scattering in random media," J. Opt. Soc. Am. A 2, 2062-2404 (1985).

1983 (1)

R. M. Ross, Stochastic Processes (Wiley, 1983).

Bertoni, H. L.

H. L. Bertoni, W. Honcharenko, L. Rocha Maciel, and H. H. Xia, "UHF propagation prediction for wireless personal communications," Proc. IEEE 82, 1333-1359 (1994).
[CrossRef]

Bruck, J.

M. Franceschetti, J. Bruck, and L. J. Schulman, "A random walk model of wave propagation," IEEE Trans. Antennas Propag. 52, 1304-1317 (2004).
[CrossRef]

Franceschetti, G.

G. Franceschetti, S. Marano, and F. Palmieri, "Propagation without wave equation toward an urban area model," IEEE Trans. Antennas Propag. 47, 1393-1404 (1999).
[CrossRef]

S. Marano, F. Palmieri, and G. Franceschetti, "Statistical characterization of ray propagation in a random lattice," J. Opt. Soc. Am. A 16, 2459-2464 (1999).
[CrossRef]

Franceschetti, M.

S. Marano and M. Franceschetti, "Ray propagation in a random lattice: a maximum entropy, anomalous diffusion process," IEEE Trans. Antennas Propag. 53, 1888-1896 (2005).
[CrossRef]

M. Franceschetti, J. Bruck, and L. J. Schulman, "A random walk model of wave propagation," IEEE Trans. Antennas Propag. 52, 1304-1317 (2004).
[CrossRef]

M. Franceschetti, "Stochastic rays pulse propagation," IEEE Trans. Antennas Propag. 52, 2742-2752 (2004).
[CrossRef]

A. Martini, M. Franceschetti, and A. Massa are preparing a paper to be called "'Electromagnetic wave propagation in nonuniform percolation lattices--theory and experiments."

Grimmett, G.

G. Grimmett, Percolation (Springer-Verlag, 1989).

Honcharenko, W.

H. L. Bertoni, W. Honcharenko, L. Rocha Maciel, and H. H. Xia, "UHF propagation prediction for wireless personal communications," Proc. IEEE 82, 1333-1359 (1994).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

A. Ishimaru, "Wave propagation and scattering in random media and rough surfaces," Proc. IEEE 79, 1359-1366 (1991).
[CrossRef]

Kyungjung, K.

T. K. Sarkar, J. Zhong, K. Kyungjung, A. Medouri, and M. Salazar-Palma, "A survey of various propagation models for mobile communication," IEEE Antennas Propag. Mag. 45, 51-82 (2003).
[CrossRef]

Marano, S.

S. Marano and M. Franceschetti, "Ray propagation in a random lattice: a maximum entropy, anomalous diffusion process," IEEE Trans. Antennas Propag. 53, 1888-1896 (2005).
[CrossRef]

S. Marano, F. Palmieri, and G. Franceschetti, "Statistical characterization of ray propagation in a random lattice," J. Opt. Soc. Am. A 16, 2459-2464 (1999).
[CrossRef]

G. Franceschetti, S. Marano, and F. Palmieri, "Propagation without wave equation toward an urban area model," IEEE Trans. Antennas Propag. 47, 1393-1404 (1999).
[CrossRef]

Martini, A.

A. Martini, M. Franceschetti, and A. Massa are preparing a paper to be called "'Electromagnetic wave propagation in nonuniform percolation lattices--theory and experiments."

Massa, A.

A. Martini, M. Franceschetti, and A. Massa are preparing a paper to be called "'Electromagnetic wave propagation in nonuniform percolation lattices--theory and experiments."

Medouri, A.

T. K. Sarkar, J. Zhong, K. Kyungjung, A. Medouri, and M. Salazar-Palma, "A survey of various propagation models for mobile communication," IEEE Antennas Propag. Mag. 45, 51-82 (2003).
[CrossRef]

Norris, J. R.

J. R. Norris, Markov Chains (Cambridge U. Press, 1998).

Palmieri, F.

G. Franceschetti, S. Marano, and F. Palmieri, "Propagation without wave equation toward an urban area model," IEEE Trans. Antennas Propag. 47, 1393-1404 (1999).
[CrossRef]

S. Marano, F. Palmieri, and G. Franceschetti, "Statistical characterization of ray propagation in a random lattice," J. Opt. Soc. Am. A 16, 2459-2464 (1999).
[CrossRef]

Rocha Maciel, L.

H. L. Bertoni, W. Honcharenko, L. Rocha Maciel, and H. H. Xia, "UHF propagation prediction for wireless personal communications," Proc. IEEE 82, 1333-1359 (1994).
[CrossRef]

Ross, R. M.

R. M. Ross, Stochastic Processes (Wiley, 1983).

Salazar-Palma, M.

T. K. Sarkar, J. Zhong, K. Kyungjung, A. Medouri, and M. Salazar-Palma, "A survey of various propagation models for mobile communication," IEEE Antennas Propag. Mag. 45, 51-82 (2003).
[CrossRef]

Sarkar, T. K.

T. K. Sarkar, J. Zhong, K. Kyungjung, A. Medouri, and M. Salazar-Palma, "A survey of various propagation models for mobile communication," IEEE Antennas Propag. Mag. 45, 51-82 (2003).
[CrossRef]

Schulman, L. J.

M. Franceschetti, J. Bruck, and L. J. Schulman, "A random walk model of wave propagation," IEEE Trans. Antennas Propag. 52, 1304-1317 (2004).
[CrossRef]

Stauffer, D.

D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, 1985).
[CrossRef]

Xia, H. H.

H. L. Bertoni, W. Honcharenko, L. Rocha Maciel, and H. H. Xia, "UHF propagation prediction for wireless personal communications," Proc. IEEE 82, 1333-1359 (1994).
[CrossRef]

Zhong, J.

T. K. Sarkar, J. Zhong, K. Kyungjung, A. Medouri, and M. Salazar-Palma, "A survey of various propagation models for mobile communication," IEEE Antennas Propag. Mag. 45, 51-82 (2003).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

T. K. Sarkar, J. Zhong, K. Kyungjung, A. Medouri, and M. Salazar-Palma, "A survey of various propagation models for mobile communication," IEEE Antennas Propag. Mag. 45, 51-82 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

S. Marano and M. Franceschetti, "Ray propagation in a random lattice: a maximum entropy, anomalous diffusion process," IEEE Trans. Antennas Propag. 53, 1888-1896 (2005).
[CrossRef]

M. Franceschetti, J. Bruck, and L. J. Schulman, "A random walk model of wave propagation," IEEE Trans. Antennas Propag. 52, 1304-1317 (2004).
[CrossRef]

M. Franceschetti, "Stochastic rays pulse propagation," IEEE Trans. Antennas Propag. 52, 2742-2752 (2004).
[CrossRef]

G. Franceschetti, S. Marano, and F. Palmieri, "Propagation without wave equation toward an urban area model," IEEE Trans. Antennas Propag. 47, 1393-1404 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

Proc. IEEE (2)

A. Ishimaru, "Wave propagation and scattering in random media and rough surfaces," Proc. IEEE 79, 1359-1366 (1991).
[CrossRef]

H. L. Bertoni, W. Honcharenko, L. Rocha Maciel, and H. H. Xia, "UHF propagation prediction for wireless personal communications," Proc. IEEE 82, 1333-1359 (1994).
[CrossRef]

Other (6)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

G. Grimmett, Percolation (Springer-Verlag, 1989).

D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, 1985).
[CrossRef]

R. M. Ross, Stochastic Processes (Wiley, 1983).

J. R. Norris, Markov Chains (Cambridge U. Press, 1998).

A. Martini, M. Franceschetti, and A. Massa are preparing a paper to be called "'Electromagnetic wave propagation in nonuniform percolation lattices--theory and experiments."

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Figures (11)

Fig. 1
Fig. 1

Example of ray propagation in a random lattice. Left-hand side, the ray is reflected back in the above empty half-plane before reaching level k. Right-hand side, the ray goes beyond level k.

Fig. 2
Fig. 2

Markov chain. The ray propagation in the nonuniform random half-plane lattice is modeled as a Markov process.

Fig. 3
Fig. 3

Examples of propagating rays. Left-hand side, θ < 45 ° ; the ray is more likely to travel back through the same cells whenever a reflection occurs. Right-hand side, high density of scatterers; the ray tends to travel over and over on the same sequence of cells.

Fig. 4
Fig. 4

Martingale approach. The propagation process is modeled as the sum of many vectorial variables. The nth element of the stochastic process { r n , n 0 } is the vertical component of the vector r ¯ n . Under some assumptions, the process m = 1 n x m behaves as a martingale with respect to the sequence { x m } (Ref. [1]).

Fig. 5
Fig. 5

Uniform random lattice with q = 0.05 . We plot Pr { 0 k } versus k for different values of θ. Crosses denote reference data; solid and dashed curves represent reconstructions obtained by the MKV approach and the MTG approach, respectively.

Fig. 6
Fig. 6

Uniform random lattice with q = 0.15 We plot the prediction error δ k versus k for different incidence angles θ. Left-hand side, MKV approach; right-hand side, MTG approach.

Fig. 7
Fig. 7

Uniform random lattice with q = 0.25 and q = 0.35 . We plot the prediction error δ k versus k for different incidence angles θ. Left-hand side, MKV approach; right-hand side, MTG approach.

Fig. 8
Fig. 8

Density profiles q ( x ) versus the lattice depth x. Left-hand side, linear profiles; right-hand side, double-exponential profiles.

Fig. 9
Fig. 9

Linear density profiles. Estimated values of Pr { 0 k } versus k. Crosses denote reference data; solid and dashed line curves represent predictions obtained by the MKV approach and the MTG approach, respectively.

Fig. 10
Fig. 10

Double-exponential density profiles. Estimated values of Pr { 0 k } versus k. Crosses denote reference data; solid and dashed curves represent reconstructions obtained by the MKV approach and the MTG approach, respectively.

Fig. 11
Fig. 11

Linear and double-exponential profiles worst cases. We consider the two density profiles with the worst prediction error. We plot the mean error δ and the maximum error δ max versus θ for the MKV approach and the MTG approach.

Tables (1)

Tables Icon

Table 1 Parameters of the Density Profiles

Equations (38)

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Pr { 0 + k + 1 } = p 1 p 2 1 + p 1 p 2 i = 0 k 3 q k i p k i p k i 1 , k 1 .
Pr { 0 + k + 1 } = p 2 ( k 2 ) q + 1 , k 1 ,
lim q 0 Pr { 0 + k + 1 } = 1 ,
lim q 1 Pr { 0 + k + 1 } = 0 .
Pr { ( j 1 ) + j + 1 } = p j p j + q j Pr { ( j 1 ) 1 ( j 1 ) + } , j 2 .
Pr { ( j 1 ) + j + 1 } = p j + q j Pr { ( j 1 ) ( j 1 ) + 1 } × Pr { ( j 1 ) + j + 1 } , j 2 ,
Pr { ( j 1 ) + j + 1 } = p j 1 q j Pr { ( j 1 ) ( j 1 ) + 1 } , j 2 .
Pr { ( j 1 ) + j + 1 } = p j p j + q j Pr { ( j 1 ) 1 ( j 1 ) + } , j 2 .
Pr { j 1 j + } = p j 1 Pr { ( j 1 ) 1 ( j 1 ) + } p j + q j Pr { ( j 1 ) 1 ( j 1 ) + } , j 2 .
Pr { j 1 j + } = Pr { A } + Pr { B } , j 2 .
Pr { A } = p j 1 Pr { ( j 1 ) 1 ( j 1 ) + } ,
Pr { B } = p j 1 Pr { ( j 1 ) ( j 1 ) + 1 } × Pr { ( j 1 ) + 1 j + } .
Pr { B } = p j 1 [ ( 1 Pr { ( j 1 ) 1 ( j 1 ) + } ) × ( 1 Pr { ( j 1 ) + j + 1 } ) ] = p j 1 [ q j Pr { ( j 1 ) 1 ( j 1 ) + } ( 1 Pr { ( j 1 ) 1 ( j 1 ) + } ) p j + q j Pr { ( j 1 ) 1 ( j 1 ) + } ] ,
Pr { j 1 j + } = p j 1 Pr { ( j 1 ) 1 ( j 1 ) + } p j + q j Pr { ( j 1 ) 1 ( j 1 ) + } j 2 .
Pr { 0 + k + 1 } = Pr { 0 + ( k 1 ) + 1 } × Pr { ( k 1 ) + k + 1 } , k 2 ,
Pr { ( k 1 ) + k + 1 } = p k 1 p k p k 1 p k + q k Pr { 0 + ( k 1 ) + 1 } , k 2 .
Pr { ( k 1 ) + k + 1 } = p k p k + q k Pr { ( k 1 ) 1 ( k 1 ) + } = p k 1 p k p k 1 p k + p k 1 q k Pr { ( k 1 ) 1 ( k 1 ) + } , k 2 .
Pr { ( k 1 ) 1 ( k 1 ) + } = Pr { 0 + ( k 1 ) + 1 } p k 1 , k 2 .
Pr { k 1 k + } = p k 1 Pr { ( k 1 ) 1 ( k 1 ) + } p k + q k Pr { ( k 1 ) 1 ( k 1 ) + } , k 2 .
Pr { k 1 k + } = Pr { 0 + ( k 1 ) + 1 } p k + q k Pr { ( k 1 ) 1 ( k 1 ) + } , k 2 .
Pr { 0 + k + 1 } = p k Pr { 0 + ( k 1 ) + 1 } p k + q k Pr { ( k 1 ) 1 ( k 1 ) + } , k 2 ,
Pr { k 1 k } = Pr { 0 + k + 1 } p k , k 2 ,
r n = r 0 + m = 1 n x m , n 0 ,
x m = r m r m 1 , m 1 .
Pr { 0 k } = i Pr { 0 k r 0 = i } Pr { r 0 = i } ,
Pr { r 0 = i } = { q 1 , i = 0 , q e 1 + ( j = 1 i 1 p e j + ) , i 0 , )
Pr { 0 k r 0 = i } { 0 , i = 0 , i k , 0 < i < k , 1 , i k .
Pr { 0 k } = i = 1 k 1 i k p 1 q e i + j = 1 i 1 p e j + + p 1 j = 1 k 1 p e j + ,
δ k Pr R { 0 k } Pr P { 0 k } max k [ Pr R { 0 k } ] × 100 , ( Prediction Error ) ,
δ 1 K max k = 1 K max δ k , ( Mean Error ) ,
δ max = max k { δ k } , ( Maximum Error ) ,
q ( x ) = q + α ( x 1 ) ,
q ( x ) = { α exp [ ( x L ) τ ] , x L , α exp [ ( L x ) τ ] , x > L ,
Pr { 0 + k + 1 } = Pr { 0 + ( k 1 ) + 1 } × p k 1 p k p k 1 p k + q k Pr { 0 + ( k 1 ) + 1 } , k 2 .
Pr { 0 + ( k 1 ) + 1 } = p 1 p 2 1 + p 1 p 2 i = 0 k 4 q ( k 1 ) i p ( k 1 ) i p ( k 1 ) i 1 , k 2 ,
Pr { 0 + k + 1 } = p 1 p 2 1 + p 1 p 2 i = 0 k 4 q k 1 i p k 1 i p k 2 i p k 1 p k p k 1 p k + q k [ ( p 1 p 2 ) ( 1 + p 1 p 2 i = 0 k 4 q k 1 i p k 1 i p k 2 i ) ] = p 1 p 2 1 + p 1 p 2 i = 0 k 4 q k 1 i p k 1 i p k 2 i p k 1 p k ( 1 + p 1 p 2 i = 0 k 4 q k 1 i p k 1 i p k 2 i ) p k 1 p k + p k 1 p k i = 0 k 4 q k 1 i p k 1 i p k 2 i + q k p 1 p 2 = p 1 p 2 1 + p 1 p 2 i = 0 k 4 q k 1 i p k 1 i p k 2 i + q k p 1 p 2 p k 1 p k = p 1 p 2 1 + p 1 p 2 i = 1 k 4 q k 1 i p k 1 i p k 2 i = p 1 p 2 1 + p 1 p 2 i = 0 k 3 q k i p k i p k i 1 , k 2 .
Pr { 0 k } = i = 1 k 1 i k p 1 q e i + ( j = 1 i 1 p e j + ) + i = k p 1 q e i + ( j = 1 i 1 p e j + ) .
i = k p 1 q e i + j = 1 i 1 p e j + = i = k p 1 j = 1 i 1 p e j + i = k p 1 p e i + j = 1 i 1 p e j + = p 1 j = 1 k 1 p e j + + p 1 i = k + 1 j = 1 i 1 p e j + p 1 i = k j = 1 i p e j + = p 1 j = 1 k 1 p e j + + p 1 i = k + 1 j = 1 i 1 p e j + p 1 i = k + 1 j = 1 i 1 p e j + = p 1 j = 1 k 1 p e j .

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