PSI - Issue 44

Elisa Bassoli et al. / Procedia Structural Integrity 44 (2023) 1554–1561 E. Bassoli et al./ Structural Integrity Procedia 00 (2022) 000 – 000

1558

5

n

m

n

m

n

m

; D D       = = = = = = = = = = = = 0; 0; 0; 0; , z j D mD , z i D nD , y j , y i , x j , x i z D D

;

(11)

z

1

1

1

1

1

1

i

j

i

j

i

j

n

n

m

m

; 2 ,     = = = = =  =  =  y x j x y i y x i D m D n D n ; ; 2 2 , 2 2 , 2

2

2 x

;

(12)

, y j D m

= 

1

1

1

1

i

i

j

j

where  x and  y are the radius of gyration of the building plan. These assumptions result in the following variances:

2 H

H 2





( )   = 2 M x

( )   = 2 M z

) ;

(13)

(

(

)

2 x

2

2

x 2

2

2

cos

cos

sin

sin

n

m

n

m



 +





 +



a

d

a

d

2

( )   = 2 M y

 +  

) ( y H 2

(14)

(

)

D n 2

2

2

2

2

sin

sin

cos

cos

m

n

m

 +

 +



z

a

d

a

d

As regards the rotations around x and z axes, variances of DInSAR results are inversely related to the building extension in north-south direction. Note that a more extended plan dimension is also associated with a greater number of PSs, which produces an even more limited variance 2 M  . On the other hand, the variance of the rotation around y is inversely proportional to 2 z D . The dependence from the latter means that, as expected, an increased building height entails a more accurate estimation of y rotations. 3.2. Persistent scatterer positioning errors The propagation error theory applied to Eq. (4) allows to calculate the effect of PS positioning uncertainties ( ) D Σ on the rigid motion uncertainty ( ) Σ θ P : ( ) ( ) T D T P J J D J Σ J Σ θ   =  =  2 (15) where J is the 5-by-2( n+m ) Jacobian matrix that collects the derivatives of the motion vector θ with respect to the distances D x,i , D y,i , D x,j and D y,j . In this paper the building height is assumed to be known, thus the Jacobian matrix does not contain derivatives with respect to D z,i and D z,j . Moreover, ( ) D Σ is assumed to be a diagonal matrix with non-zero elements pairs to 2 Dx  and 2 Dy  . The latter are here supposed to be equal (i.e. 2 2 2 Dy D Dx  =  =  ), as in the case of the square grid resolution that characterizes COSMO-SkyMed data. However, the procedure might be further generalized to account for a non-square grid resolution. In line with the intentions of the paper, each rigid motion component is individually treated. To provide a priori estimations of the output uncertainties due to errors in PS positioning, conditions (i) and (ii) are applied (for their definitions see Section 3.1). The variances of rigid motion parameters caused by errors in the PS positioning result: ( ) ( ) 0; 0; ,G ,G = =   P z P x v v (16)

(

)

(

)

4

4

4

4

cos

cos

sin

sin

n

m

n

m

 +



 +



( )   =  2 P x

( )   =  2 P z

2

2

2 D

2 D

a

d

a

d

(17)

;

;





(

)

(

)

x

z

2

2

2 y

2

2

2 y

2

2

cos

cos

sin

sin

n

m

n

m



 +





 +



a

d

a

d

(

) ( 2 z

)

2 x

4

4

2

2

2

2

cos

cos

sin

cos   +

sin

cos  

( )   =  2 P y

n

m

D n

m



 +

 +

2

2

a

d

a

a

d

d

(18)







(

) ( 2 z

)

D

y

2

2 x

2

2

2

2

cos

cos

sin

sin

n

m

D n

m



 +

 +

 +



a

d

a

d

Note that, as expected, displacements turn out to be singularly not affected by errors in the PS positioning. Indeed,