PROBLEMS OF GEOCOSMOS

Proceedings of the 7th International Conference "Problems of Geocosmos" (St. Petersburg, Russia, 26-30 May 2008)
where k and n are defined according to (1); r = min( n, k)
; Λ is the diagonal matrix with elements λ i ; u i
and v i are columns of orthogonal matrixes U and V .
The pseudo inverse matrix H and the pseudo-solution of the system (4) are:
-1
T
H=VΛ U ,
m= Hd.
For a regularization solution of an inverse problem (4) let us introduce a parametric functional
(Tikhonov and Arsenin, 1977),
( , ) ( ) ( )
T T
P α
m d = Am−d Am− d + α m m.
The minimization of the functional P ( , )
α m d gives the solution (Yanovskaya and Porokhova, 2004)
m= H d, (7)
where λi are elements of diagonal matrix A.
reg
⎛ λ ⎞
i T
Hreg = Vdiag⎜ 2 ⎟U
,
λi+ α
⎝ ⎠
3. Algorithms of one- and two-dimensional inversion
In the Fig. 1 it is shown the unified scheme of 1D and 2D inversion algorithms which distinguish
only by the components.
data
d
rms
m (0)
Φ(m (0) )
Fig.1. The scheme of 1D and 2D inversion algorithms.
In case of 1D inversion as initial data there are taken either apparent resistivities ρ a , or impedance
phases arg Z with one or several values of apparent resistivities. In the second case if all values of ρ a and
arg Z are used we obtain the joint interpretation:
1-D Algorithm
( T ) it = nt
Z ( T ) ρ ( T )
d : 1) ρ , 1,2, ...,
a it
In case of 2D inversion as initial data there are taken
2) arg and one or more values .
it a it
E
ρ a , or
2-D Algorithm
( T ) it = nt
( ) =
( ) ( )
E
d : 1) ρ , 1,2, ...,
a it
A AΔm=Δd m (s) =m (s-1) +Δm (s)
Φ(m (s) rms>δ
)
H
2) ρ T , it 1,2, ..., nt
a it
H
ρ a , or
E
ρ a and
E H
3) ρa Tit and ρa
Tit , it = 1,2, ..., nt.
The building of the working net for the solution of the forward problem is the same as in the work
(Vagin, 2004).
3. Testing of algorithms
326
rms
model
H
ρ a :
(6)