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An inversion model applied to DC soundings interpretation

C . DEL GIUDICE *

Received on September 10, 1982

ABSTRACT

The inversion technique is used in DC soundings intepretation to determine the

thicknesses and the true resistivities of the layers, starting from the field apparent

resistivities.

The relationship between the predicted apparent resistivities and the earth

parameters is not linear. Starting from this relationship a methodology is described

to obtain a wellposed system of M linear equations. This system permits to

calculate, by means of an iterative procedure, the earth parameters that minimize

the differencies (error) between the field and the predicted apparent resistivities.

Three different iterative procedures are described. Practical examples have

shown that all the iterative procedures are reliable and give comparable results in

terms of minimum error reached and CPU time.

RIASSUNTO

La tecnica dell'inversione è usata nella interpretazione di sondaggi elettrici

verticali per determinare gli spessori e le resistività vere degli elettrostrati

(parametri del terreno) partendo dalle resistività apparenti. La relazione tra le

resistività apparenti ed i parametri del terreno non è lineare.

* Presently employed by GEOMATH - Via Cavour, 43 - PISA (Italy).

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C. DEL GIUDICE

Partendo da questa relazione viene descritta una metodologia già nota per

ottenere un sistema ben posto di M equazioni lineari.

Questo sistema permette di calcolare, per mezzo di una procedura iterativa, i

parametri del terreno che minimizzano la differenza tra le resistività apparenti di

campo e quelle calcolate. Vengono inoltre descritte tre diverse procedure iterative.

Gli esempi pratici hanno mostrato che tutte le procedure iterative utilizzate sono

attendibili e paragonabili tra loro per quanto riguarda i risultati del procedimento

di minimizzazione e i tempi di calcolo utilizzati.

1. STATMENT OF PROBLEM

The apparent resistivities in ohm- m versus the AB/2 distances

in M represent the field data. In terms of the voltage V,, currents I,

and geometric factor K, (i = 1, N) where N is the number of

samples, the apparent resistivity is:

K,

AV,

I,

We want to find an earth model, consisting of a distribution of

true resistivities and thicknesses that minimize the error between

the field apparent resistivities and the predicted ones.

The observed apparent resistivities are, in vectorial form:

Pa,

P a2

If we call P' the resistivities predicted by a forward earth model P

consisting of the true resistivities and the thicknesses of the ^ ^ ^

layers

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AN INVERSION MODEL APPLIED, ECC.

167

The above values are functions of the particular earth model P,

(/ = 1, M) where generally M < N.

Pi"

P2

Then

P' = f(P)

[1]

The relationship [1] between P and P' is not linear. By expanding

the equation [1] in a Taylor series and by keeping only the linear

terms, we obtain:

- - - - 8«P) _

Pi = f(P + A P) = f (P) + - y ^ - AP

where AP is the model improvement.

We need the P' to fit the observed data:

P., = P'

Or:

öf(P) _

P o = f(P) + " T T " AP

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C. DEL GIUDICE

Calling d = pu — f (P), this is:

a f (P) -

d = AP

dP

Or:

d = [A) AP

[2]

where [A] is the sensitivity matrix and

ter correction (Jupp, Vozof 1975, Lanczos 1961, Marquardt 1963).

Writing out the matrices of equation [2] we have:

AP is the vector parame-

d,

8 f, (P)

¿ P ,

d2 •

•

•

•

•

•

•

•

• •

d f.v (P)

0P,

8f, (P)

d P „

3 t\, (P)

3PM

A P ,

A P2

A P.

[3]

This is a system of N equations in M unknowns with N > M. The

above system of equations is then over-determined and generally

ill-posed in the sense that small changes in the data lead to large

changes in the solutions. To solve this redundant system of

equations, we can apply the method of least squares.

Calling e = d —[A] AP we have:

N •

S

; - i

«

e t e

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AN INVERSION MODEL APPLIED, ECC.

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where eT is the transpose of e or:

e2 =[ d —[A] AP]T • [d — [A] AP]

For e2 to be a minimum; its derivative with respect to AP must

be zero.

Then differentiating the above equation we have:

[A]T [A] AP = [A]t d [4]

We have transformed the old system [2] of N equations in M

unknowns into an M x M system [4], From the above expression

[4] we have:'

AP = [ AT A ] "1 [ A ]T d [5]

2 . ITERATIVE PROCEDURES

In the iterative procedure we do not use the algorithm [5] but

its modification made by Marquardt (Marquardt 1963):

AP = [AT A -I- k21]"1 [A]T d [6]

where k2 is called «Marquardt parameter» and I is the unit

matrix.

The system of equations [6] is well-posed.

The algorithm [6] has the advantage over [5] that the region of

convergence is greater and the amplitude of the parameter correc-

tion APis smaller.

The general expression for the iterative procedure is:

pimi _ p (m-1) ^ p (m)

[7]