The computation of the Sommerfeld Integral tails by using the Generalized Pencil of Functions method is here presented. Suppose we want to solve the Sommerfeld-type integral:

(1)

143 APPENDIX B

The computation of the Sommerfeld Integral tails by using the Generalized Pencil of Functions method is here presented. Suppose we want to solve the Sommerfeld-type integral:

( ) ( )

0 , | ' _n ,

SI = ^∞ ∫ ψ k _ρ z z J k _ρ ρ k dk _ρ _ρ ^(B.1) where ψ(k ρ ,z|z’) is the generic spectral domain Dyadic Green’s Functions, J n is the first kind Bessel function of order n, ρ is the radial distance between source and observation point in a cylindrical reference system, k ρ

is the wavenumber in the ρ-direction while z and z’ is the vertical coordinates of the field and source point respectively.

Suppose we have M samples of the spectral domain DGF’s ψ k , through the GPOF method we are able to express each sample as:

1 0,1, , ,

i

M k

k i i

i s t i

b k N

e ^δ

ψ γ

γ

=

= =

=

∑ _ _(B.2)

where b i , s i and δt are the complex residues, the complex poles and the sampling rate respectively.

In order to evaluate the complex residues and poles, we define the following set of matrices:

0 1 1 1 2

3 2 2 1

1 2 1

1 1

.

L L

N L N N L N

N L N L

ψ ψ ψ ψ ψ ψ

ψ

ψ ψ ψ ψ

ψ

ψ ψ ψ ψ

ψ ψ

−

+

− − − −

− − − +

   

   

Λ = Γ =

   

   

 

   

 

( B.3)

Looking into the underling structure of the matrices Λ and ^Γ , it follows that:

, Λ = Ζ⋅Β⋅Π

Γ = Ζ⋅Β⋅Ω⋅Π (B.4)

(2)

GENERALIZED PENCIL OF FUNCTIONS

144 where:

1 1 1

1 2

1 1 1

1 2

1 1

2 2

1 1 1

1

1 0 0 0 0 .

0 0 0 0

L

M

L

N L N L N L

M

M M

b b

b

γ γ

γ γ γ

γ γ

γ γ γ

γ γ

γ

−

− − − − − −

   

   

 

Ζ =       Π =      

   

   

Ω = Β =

   

   

 

    

   

 

       

 

(B.5)

To illustrate the use of the GPOF algorithm we can write:

+ + ,

Λ ⋅Γ = Π ⋅Ω⋅Π (B.6)

where the superscript + denotes the matrix pseudo-inverse. It can be shown from (B.6) that exists a vector p i with i=1,2,…,M such that:

i i i . p γ p

Λ ⋅Γ ⋅ = + (B.7)

To compute the pseudo-inverse, we can use the Singular Value Decomposition:

1 ,

H

U D V

V D U

+ −

Λ = ⋅ ⋅

Λ = ⋅ ⋅ (B.8)

where D is the singular value diagonal matrix, while U and V are left and right singular matrices respectively.

It is important to underline that for noise data, we can choose the M largest singular values of D and the resulting matrix Λ is called truncated ⁺ pseudo inverse.

Since:

H ,

V V I

Λ ⋅ Λ = ⋅ + = (B.9)

(3)

GPOF METHOD

145 where I is the identity matrix, substituting the (B.8) into the (B.9) and left multiplying by V ^H , we obtain:

The computation of the Sommerfeld Integral tails by using the Generalized Pencil of Functions method is here presented. Suppose we want to solve the Sommerfeld-type integral:

143

APPENDIX B