Abstract

In this paper and its companion [J. Opt. Soc. Am. A. 23, 2251 (2006) ], the problem of ray propagation in nonuniform random half-plane lattices is considered. Cells can be independently occupied according to a density profile that depends on the lattice depth. An electromagnetic source external to the lattice radiates a monochromatic plane wave that undergoes specular reflections on the occupied sites. The probability of penetrating up to level k inside the lattice is analytically evaluated using two different approaches, the former applying the theory of Markov chains (Markov approach) and the latter using the theory of Martingale random processes (Martingale approach). The full theory concerned with the Martingale approach is presented here, along with an innovative modification that leads to some improved results. Numerical validation shows that it outperforms the Markov approach when dealing with ray propagation in dense lattices described by a slowly varying density profile.

© 2007 Optical Society of America

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References

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  1. G. Grimmett, Percolation (Springer-Verlag, 1989).
  2. D. Stauffer, Introduction to Percolation Theory (Taylor and Francis, 1985).
    [CrossRef]
  3. A. Martini, M. Franceschetti, and A. Massa, "Ray propagation in nonuniform random lattices," J. Opt. Soc. Am. A 23, 2251-2261, 2006.
    [CrossRef]
  4. J. R. Norris, Markov Chains (Cambridge U. Press, 1998).
  5. 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]
  6. R. M. Ross, Stochastic Processes (Wiley, 1983).

2006 (1)

1999 (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]

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]

Franceschetti, M.

Grimmett, G.

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

Marano, S.

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.

Massa, A.

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]

Ross, R. M.

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

Stauffer, D.

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

IEEE Trans. Antennas Propag. (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]

J. Opt. Soc. Am. A (1)

Other (4)

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

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Vectorial representation of the stochastic process modeling ray propagation inside nonuniform random lattices. The n th element of the stochastic process r n is the vertical component of the vector r ¯ n .

Fig. 2
Fig. 2

The two mutually exclusive situations at reflection n = N , (a), (b) N being min {n: r n k or r n 0 } and (c), (d) N being min {n: r n k or r n 1 }.

Fig. 3
Fig. 3

Uniform random lattices. Mean error δ for different q values when θ = 45 o .

Fig. 4
Fig. 4

Linear density profiles, q j = q α ( j 1 ) , q = 0.35 ; α values specified in Table 1. Prediction error δ k versus k when θ = 45 o . Left, MTG approach; right, MKV approach.

Fig. 5
Fig. 5

Uniform random lattices with q = { 0.3 , 0.35 , 0.4 } . Global mean error δ for different incidence angles θ.

Fig. 6
Fig. 6

Uniform random lattices with q = { 0.3 , 0.35 , 0.4 } . Plots of the global prediction error δ k versus k for different incidence angles θ. Left, MTG approach; right, MKV approach.

Tables (3)

Tables Icon

Table 1 Linear Density Profiles a

Tables Icon

Table 2 Double-Exponential (DE) and Pseudo-Gaussian (PG) Density Profiles a

Tables Icon

Table 3 Nonuniform Density Profile L1, DE, and PG a

Equations (48)

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Pr { 0 k } = p 1 p 2 1 + p 1 p 2 i = 0 k 3 q k i p k i p k i 1 .
Pr { 0 k } = p 2 ( k 2 ) q + 1
r n = r 0 + m = 1 n x m , n 0 ,
x n = r n r n 1 , n 1
N = min { n : r n k or r n 0 } .
Pr { r N k } = i = 0 Pr { r N k r 0 = i } Pr { r 0 = i } ,
Pr { r 0 = 0 } = q 1 ,
Pr { b } = j = 1 i 1 p e j + ,
Pr { c } = q i s = 0 tan θ 1 p i s + q i + 1 p i tan θ = 1 p e i + = q e i + .
Pr { r 0 = i } = { q 1 , i = 0 p 1 q e i + j = 1 i 1 p e j + , i 1 } .
r n = r n r 0 = m = 1 n x m , n 1 .
Pr { r N k r 0 = i } = r N r N r 0 r N r N k r 0 r N r N r 0 i k ,
Pr { r N k r 0 = i } { 0 , i = 0 i k , 0 < i < k 1 , i k } .
Pr { r N k } = p 1 [ i = 1 k 1 i k q e i + j = 1 i 1 p e j + + j = 1 k 1 p e j + ] .
Pr { r N k r 0 = i } Δ A 1 Δ A 1 + Δ A 2 = i k ,
Pr { r N k r 0 = i } Δ A 1 Δ A 1 + Δ A 2 = i 1 k 1 .
Pr { 0 k } = Pr { 0 1 0 } Pr { 1 k 1 } ,
Pr { 0 k 0 } = { p 1 , k = 1 p 1 p 2 [ i = 2 k 1 i 1 k 1 q e i + j = 2 i 1 p e j + + j = 2 k 1 p e j + ] , k > 1 } ,
Pr { 0 k 0 } = { p , k = 1 p 2 [ 1 p e ( k 1 ) ] q e ( k 1 ) , k > 1 }
Pr { x n = i } = { Pr { x n = 0 x n = x n + } Pr { x n = x n + } + Pr { x n = 0 x n = x n } Pr { x n = x n } , i = 0 Pr { x n = i x n = x n + } Pr { x n = x n + } , i > 0 Pr { x n = i x n = x n } Pr { x n = x n } , i < 0 } .
Pr { x n = 0 x n = x n + } = q e r ( n 1 ) + .
Pr { x n = 0 x n = x n } = q r ( n 1 ) s = 0 tan θ 1 p r ( n 1 ) s + p r ( n 1 ) tan θ q r ( n 1 ) 1 = 1 p r ( n 1 ) tan θ p r ( n 1 ) 1 = 1 p e r ( n 1 ) = q e r ( n 1 ) ,
Pr { x n = i , i > 0 x n = x n + } = q e r ( n 1 ) + i + s = r ( n 1 ) r ( n 1 ) + i 1 p e s + ,
Pr { x n = i , i < 0 x n = x n } = q e r ( n 1 ) + i s = r ( n 1 ) + i + 1 r ( n 1 ) p e s ,
Pr { x n = x n + } = i = 0 , even n a 1 = 1 n i + 1 a 2 = a 1 + 1 n i + 2 a i = a i 1 + 1 n c n Pr { c n } s = 1 i ξ h [ a s , r ( a s 1 ) ] b = 1 , b a 1 , . . , a i n ξ v [ b , r ( b 1 ) ] .
ξ h [ j , r ( j 1 ) ] = { q r ( j 1 ) + 1 tan θ q r ( j 1 ) + q r ( j 1 ) + 1 , x ( j 1 ) = x ( j 1 ) + q r ( j 1 ) 1 tan θ q r ( j 1 ) + q r ( j 1 ) 1 , x ( j 1 ) = x ( j 1 ) } .
ξ v [ j , r ( j 1 ) ] = { tan θ q r ( j 1 ) tan θ q r ( j 1 ) + q r ( j 1 ) + 1 , x ( j 1 ) = x ( j 1 ) + tan θ q r ( j 1 ) tan θ q r ( j 1 ) + q r ( j 1 ) 1 , x ( j 1 ) = x ( j 1 ) } .
ξ h [ j , r ( j 1 ) ] 1 tan θ + 1 = ξ h ,
ξ v [ j , r ( j 1 ) ] tan θ tan θ + 1 = ξ v .
Pr { x n = x n + } i = 0 , even n ξ h i ξ v n i a 1 = 1 n i + 1 a 2 = a 1 + 1 n i + 2 a i = a i 1 + 1 n c n Pr { c n } ,
Pr { x n = x n + } i = 0 , even n ( n i ) ξ h i ξ v n i = 1 2 [ 1 + ( ξ v ξ h ) n ] = ̂ α n ,
Pr { x n = x n } 1 α n .
Pr { x n = i } { α n q e r ( n 1 ) + + ( 1 α n ) q e r ( n 1 ) , i = 0 α n q e r ( n 1 ) + i + s = r ( n 1 ) r ( n 1 ) + i 1 p e s + , i > 0 ( 1 α n ) q e r ( n 1 ) + i s = r ( n 1 ) + i + 1 r ( n 1 ) p e s , i < 0 } .
δ k 1 I i = 1 I ( δ k ) i , ( Global Prediction Error ) ,
δ 1 K max k = 1 K max δ k , ( Global Mean Error ) ,
S = q MAX q MIN Δ j ,
q j = q α ( j 1 ) ,
q j = { α exp [ ( j L ) ] , j L α exp [ ( L j ) τ ] , j > L } ,
q j = α exp { ( j L ) 2 σ 2 } ,
{ r n < r n + 1 x n , x n 1 , , x 1 = r n } . ( A 1 ) ( A 2 )
r n i = 1 n x i ,
x i x i MAX ,
x i MAX = p e MAX q e MAX .
r n i = 1 n x i i = 1 n x i MAX = n p e MAX q e MAX < .
i = 1 n x i < i = 1 n x i MAX ,
r n < .
r n + 1 x n , x n 1 , , x 1 = r n + x n + 1 x n , x n 1 , , x 1 = r n x n , x n 1 , , x 1 + x n + 1 x n , x n 1 , , x 1 = r n + x n + 1 = r n .
Pr { r 0 = i } = { q 1 , i = 0 p 1 q e i + j = 1 i 1 p e j + , i 1 . }

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