Method of forming a model representative of the distribution of a physical quantity in an underground zone, free of the effect of correlated noises contained in exploration data

• US 7,092,823 B2
• Filed: 07/31/2003
• Issued: 08/15/2006
• Est. Priority Date: 08/05/2002
• Status: Expired due to Fees

• Current Assignee: Institut Francais Du Petrole (IFP Energies Nouvelles)
• Original Assignee: Institut Francais Du Petrole (IFP Energies Nouvelles)
• Inventors: Renard, François, Delprat-Jannaud, Florence, Pelle, Laure, Lailly, Patrick
• Primary Examiner(s): Barlow, John
• Assistant Examiner(s): Taylor, Victor J.
• Application Number:
  US10/630,690
• Publication Number:
  US 20050075791A1
• Time in Patent Office: 1,111 Days
• Field of Search: 702/14, 702/16, 702/6, 250/455.11, 703/5, 703/10, 367/57, 367/73
• US Class Current: 702/14
• CPC Class Codes: G01V 1/34 Displaying seismic recordings or visualisation of seismic data or attributes
• CPC Class Codes: G01V 1/36 Effecting static or dynamic corrections on records, e.g. correcting spread; Correlating seismic signals; Eliminating effects of unwanted energy
• CPC Class Codes: G01V 2210/32 Noise reduction

Abstract

The invention is a method of estimating, from data obtained by exploration of a zone of a heterogeneous medium, a model representative of a distribution, in the zone, of at least one physical quantity, the model being free of a presence of correlated noises that may be contained in the data which has application to determining the distribution in an underground zone of acoustic impedance, propagation velocities and permeabilities, etc. The method includes acquiring measurements giving information about physical characteristics of the zone; specifying a noise modelling operator which associates, with a model of each physical quantity, synthetic data that constitute a response of the model; selecting a noise modelling operator which associates a correlated noise with a noise-generating function belonging to a predetermined space of the noise-generating functions; specifying a norm or of a semi-norm in the data space; specifying a semi-norm in the space of the noise-generating functions; defining a cost function; and adjusting the model and of the noise-generating functions.

Claims

1. A method of estimating, from data obtained by exploration of a zone of a heterogeneous medium, a model representative of a distribution, in the zone, of at least one physical quantity, the model being free of a presence of correlated noises that may be contained in the data, comprising:

a) acquiring measurements giving information about physical characteristics of the zone by following a predetermined experimental protocol;

b) specifying a noise modelling operator which associates, with a model of each physical quantity, synthetic data that constitute a response of the model, the measurements and the synthetic data belonging to a data space;

c) selecting, for each correlated noise referenced by a subscript j ranging from 1 to J, a noise modelling operator which associates a correlated noise with a noise-generating function belonging to a predetermined space of the noise-generating functions (B_j);

d) specifying a norm or of a semi-norm in the data space;

e) specifying a semi-norm in the space of the noise-generating functions for which each noise modelling operator establishes substantially an isometric relation between the space of noise-generating functions and the data space;

f) defining a cost function quantifying a difference between the measurements on one hand and a superposition of a model response and of the correlated noise associated with the noise-generating function on the other hand; and

g) adjusting the model and of the noise-generating functions by minimizing the cost function, by means of an algorithmic method taking advantage of isometry of properties of the noise modelling operators.

2. A method as claimed in claim 1, wherein:

a distribution as a function of depth of an acoustic impedance in the medium is sought, the correlated noises affecting the data are tube waves each identified by parameters characterizing their propagation, the measured data are VSP data obtained by means of pickups which detect displacement of particles in the medium in response to a localized seismic excitation, the location of the pickups and a recording time and time sampling points being defined, and the selected noise modelling operator associates the synthetic data with an acoustic impedance distribution as a function of an evaluated depth in travel time and with vertical stress measured as a function of time at a depth of the first pickup.

3. A method as claimed in claim 2, wherein:

the cost function quantifying the difference is a square of the semi-norm of the difference in the data space.

4. A method as claimed in claim 2, wherein:

adjustment of the model and of the noise-generating functions is obtained by means of a block relaxation method for eliminating unknowns corresponding to each correlated noise generating function, the block relaxation method being implemented within iterations of a quasi-Newtonian algorithm for calculation of the model.

5. A method as claimed in claim 3, wherein:

adjustment of the model and of the noise-generating functions is obtained by means of a block relaxation method for eliminating the unknowns corresponding to each correlated noise generating function, this the block relaxation method being implemented within the iterations of a quasi-Newtonian algorithm for calculation of the model.

6. A method as claimed in claim 2, wherein:

numerical calculation of the image of a model by the modelling operator is carried out by a numerical solution of a 1D wave equation for the model, by selecting values taken by displacement of the particles at locations of pickups and at previously defined time sampling points, and by applying an operator likely to compensate for spherical divergence and attenuation effects.

7. A method as claimed in claim 3, wherein:

numerical calculation of the image of a model by the modelling operator is carried out by a numerical solution of the a 1D waves equation for the model considered, by selecting values taken by the displacement of the particles at the locations of pickups and at the previously defined time sampling points, and by applying an operator likely to compensate for the spherical divergence and attenuation effects.

8. A method as claimed in claim 4, wherein:

numerical calculation of the image of a model by the modelling operator is carried out by a numerical solution of the a 1D waves equation for the model considered, by selecting values taken by the displacement of the particles at the locations of pickups and at the previously defined time sampling points, and by applying an operator likely to compensate for the spherical divergence and attenuation effects.

9. A method as claimed in claim 5, wherein:

numerical calculation of the image of a model by the modelling operator is carried out by a numerical solution of the a 1D waves equation for the model considered, by selecting values taken by the displacement of the particles at the locations of pickups and at the previously defined time sampling points, and by applying an operator likely to compensate for the spherical divergence and attenuation effects.

10. A method as claimed in claim 2, wherein:

the noise modelling operator is a finite-difference centered numerical scheme for discretizing a noise transport equation, and a utilized noise generating function has an initial condition along an edge of an observation zone belonging to the space (B_j);

of support time functions in a give time interval.

11. A method as claimed in claim 3, wherein:

the noise modelling operator is a finite-difference centered numerical scheme for discretizing the noise transport equation, and the utilized noise-generating function involved as the function has an initial condition along the an edge of the an observation zone belongs belonging to a the space (B_j) consisting of the support time functions in a give time interval.

12. A method as claimed in claim 4, wherein:

the noise modelling operator is a finite-difference centered numerical scheme for discretizing the noise transport equation, and the utilized noise-generating function involved as the function has an initial condition along the an edge of the an observation zone belongs belonging to a the space (B_j) consisting of the support time functions in a give time interval.

13. A method as claimed in claim 5, wherein:

the noise modelling operator is a finite-difference centered numerical scheme for discretizing the noise transport equation, and the utilized noise-generating function involved as the function has an initial condition along the an edge of the an observation zone belongs belonging to a the space (B_j) consisting of the support time functions in a give time interval.

14. A method as claimed in claim 6, wherein:

the noise modelling operator is a finite-difference centered numerical scheme for discretizing the noise transport equation, and the utilized noise-generating function involved as the function has an initial condition along the an edge of the an observation zone belongs belonging to a the space (B_j) consisting of the support time functions in a give time interval.

15. A method as claimed in claim 7, wherein:

the noise modelling operator is a finite-difference centered numerical scheme for discretizing the noise transport equation, and the utilized noise-generating function involved as the function has an initial condition along the an edge of the an observation zone belongs belonging to a the space (B_j) consisting of the support time functions in a give time interval.

16. A method as claimed in claim 8, wherein:

the noise modelling operator is a finite-difference centered numerical scheme for discretizing the noise transport equation, and the utilized noise-generating function involved as the function has an initial condition along the an edge of the an observation zone belongs belonging to a the space (B_j) consisting of the support time functions in a give time interval.

17. A method as claimed in claim 9, wherein:

the noise modelling operator is a finite-difference centered numerical scheme for discretizing the noise transport equation, and the utilized noise-generating function involved as the function has an initial condition along the an edge of the an observation zone belongs belonging to a the space (B_j) consisting of the support time functions in a give time interval.

18. A method as claimed in claim 2, wherein:

a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and a semi-norm selected for the space of the noise-generating function is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

19. A method as claimed in claim 3, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

20. A method as claimed in claim 4, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

21. A method as claimed in claim 5, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

22. A method as claimed in claim 6, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

23. A method as claimed in claim 7, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

24. A method as claimed in claim 8, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

25. A method as claimed in claim 9, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

26. A method as claimed in claim 10, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

27. A method as claimed in claim 11, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

28. A method as claimed in claim 12, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

29. A method as claimed in claim 13, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating functions space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

30. A method as claimed in claim 14, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

31. A method as claimed in claim 15, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

32. A method as claimed in claim 16, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br><br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br><br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br><br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

33. A method as claimed in claim 17, wherein:

the a semi-norm selected for the data space is;

${<br><br>u<br><br>}_D={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>\underset{i=0}{\overset{I-1}{\sum <br><br>}}<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br><br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left(u_i^{n+1}+u_i^n\right)}^2\right)}^{\frac{1}{2}}$

and the a semi-norm selected for the space of the noise-generating function space is;

${<br><br>\mathrm{\beta <br><br>}<br><br>}_B={\left(\mathrm{\Delta <br><br>}<br><br><br><br>x<br><br><br><br>\mathrm{\Delta <br><br>}<br><br><br><br>t<br><br><br><br>I<br><br>\underset{n=0}{\overset{N-1}{\sum <br><br>}}<br><br><br><br>\frac{1}{{\mathrm{\tau <br><br>}}^{n+\frac{1}{2}}}<br><br>{\left({\mathrm{\beta <br><br>}}^{n+1}+{\mathrm{\beta <br><br>}}^n\right)}^2\right)}^{\frac{1}{2}}.$

34. A method as claimed in claim 1, wherein:

a distribution of disturbances, in relation to a previously selected reference model, of an impedance and of a velocity in the zone of the medium is sought, the correlated noises affecting the data are due to multiple reflections whose kinematics and amplitude variations with an offset have been previously estimated, the measured data are picked up by seismic surface pickups, the location of the pickups, a seismic excitation mode, a recording time and time sampling points being defined, and the modelling operator being defined by linearization of the waves equation around the model representative of the distribution.

35. A method as claimed in claim 34, wherein:

the cost function quantifying the difference is the square of the semi-norm of the difference in the data space.

36. A method as claimed in claim 34, wherein:

adjustment of the model and of the noise-generating functions is obtained by a block relaxation method for eliminating unknowns corresponding to each correlated noise generating function, the relaxation method being implemented within iterations of a conjugate gradient algorithm for calculation of the model.

37. A method as claimed in claim 35, wherein:

adjustment of the model and of the noise-generating functions is obtained by means of a block relaxation method for eliminating the unknowns corresponding to each correlated noise generating function, this the relaxation method being implemented within the iterations of a conjugate gradient algorithm for calculation of the model.