INTERPOLATION OF PERIODIC DATA

• US 20110191032A1
• Filed: 01/29/2010
• Published: 08/04/2011
• Est. Priority Date: 01/29/2010
• Status: Active Grant

• Current Assignee: WesternGeco LLC (Schlumberger NV)
• Original Assignee: WesternGeco LLC (Schlumberger NV)
• Inventors: Moore, Ian
• Granted Patent: US 8,265,875 B2
• Time in Patent Office: Days
• US Class Current: 702/14
• CPC Class Codes: G01V 1/28 Processing seismic data, e.g. for interpretation or for event detection G01V1/48 takes precedence
• CPC Class Codes: G01V 2210/57 Trace interpolation or extrapolation, e.g. for virtual receiver; Anti-aliasing for missing receivers
• CPC Class Codes: G06F 17/17 Function evaluation by approximation methods, e.g. inter- or extrapolation, smoothing, least mean square method G06F17/18 takes precedence ; interpolation for numerical control G05B19/18

Abstract

A method for interpolating data. The method includes receiving data acquired at one or more locations where the data represents subterranean formations in the earth. The method also includes selecting one or more of the locations such that the selected locations are within a vicinity of an interpolation location. Next, the method includes forming a matrix of interpolation coefficients based on a first function having one or more interpolation variables, a bandwidth for an interpolation operator and one or more differences between the selected locations, wherein at least one of the interpolation variables is periodic. The method then includes forming a differences vector based on a second function having the interpolation variables, the bandwidth and one or more differences between the selected locations and the interpolation location. Using the matrix of interpolation coefficients and the differences vector, the method then forms an interpolation operator vector by applying an inverse of the matrix of interpolation coefficients to the differences vector. The method then estimates a data value for the interpolation location using the interpolation operator vector.

Claims

1. A method for interpolating seismic data, comprising:

receiving data acquired at one or more locations, wherein the data represents subterranean formations in the earth;

selecting one or more of the locations, wherein the selected locations are within a vicinity of an interpolation location;

forming a matrix of interpolation coefficients based on a first function having one or more interpolation variables, a bandwidth for an interpolation operator and one or more differences between the selected locations, wherein at least one of the interpolation variables is periodic;

forming a differences vector based on a second function having the interpolation variables, the bandwidth and one or more differences between the selected locations and the interpolation location;

forming an interpolation operator vector by applying an inverse of the matrix of interpolation coefficients to the differences vector; and

estimating a data value for the interpolation location using the interpolation operator vector.

2. The method of claim 1, further comprising receiving the bandwidth for the interpolation operator before forming the matrix of interpolation coefficients.

3. The method of claim 1, further comprising:

computing a mean square error for the interpolation operator vector;

comparing the mean square error for the interpolation operator vector with a predetermined acceptable mean square error; and

if the mean square error for the interpolation operator vector is less than or equal to the predetermined acceptable mean square error, estimating the data value for the interpolation location using the interpolation operator vector.

4. The method of claim 3, further comprising:

if the mean square error for the interpolation operator vector is greater than the predetermined acceptable mean square error, reducing the bandwidth for the interpolation operator;

forming a second matrix of interpolation coefficients based on a third function having the interpolation variables, the reduced bandwidth and the differences between the selected locations;

forming a second differences vector based on a fourth function having the interpolation variables, the reduced bandwidth and the differences between the selected locations and the interpolation location;

forming a second interpolation operator vector by applying an inverse of the second matrix of interpolation coefficients to the second differences vector; and

estimating the data value for the interpolation location using the second interpolation operator vector.

5. The method of claim 3, further comprising saving the mean square error for the interpolation operator vector as a quality control measure.

6. The method of claim 1, wherein selecting the locations comprises:

computing a difference between each of the locations and the interpolation location;

identifying one or more differences lower than a predetermined value; and

selecting a portion of the locations based on the identified differences.

7. The method of claim 1, wherein selecting the locations comprises:

subtracting a multiple of a period from each of the locations to create one or more modified locations;

computing a difference between each of the modified locations and the interpolation location;

identifying one or more differences lower than a predetermined value; and

selecting a portion of the locations based on the identified differences.

8. The method of claim 7, wherein the period is 180 degrees or 360 degrees.

9. The method of claim 7, wherein the multiple is an integer.

10. The method of claim 1, wherein the vicinity comprises an area within a predetermined distance from the interpolation location.

11. The method of claim 1, wherein the interpolation variables represent time, azimuth or temperature.

12. The method of claim 1, wherein the first function is defined by:

13. The method of claim 1, wherein the second function is defined by:

14. The method of claim 4, wherein the third function is defined by:

15. The method of claim 4, wherein the fourth function is defined by:

16. The method of claim 1, further comprising defining the first function as:

17. A method for interpolating seismic data, comprising:

receiving seismic data acquired at one or more locations, wherein the seismic data represents subterranean formations in the earth;

selecting one or more of the locations, wherein the selected locations are within a vicinity of an interpolation location;

selecting a bandwidth for an interpolation operator;

forming a matrix of interpolation coefficients based on a first function having one or more interpolation variables, the bandwidth and one or more differences between the selected locations, wherein at least one of the interpolation variables is periodic;

forming a differences vector based on a second function having the interpolation variables, the bandwidth and one or more differences between the selected locations and the interpolation location;

forming an interpolation operator vector by application of the inverse of the matrix of interpolation coefficients to the differences vector;

computing a mean square error for the interpolation operator vector;

comparing the mean square error for the interpolation operator vector with a predetermined acceptable mean square error; and

if the mean square error for the interpolation operator vector is less than or equal to the predetermined acceptable mean square error, estimating a seismic data value for the interpolation location using the interpolation operator vector.

18. The method of claim 17, further comprising:

if the mean square error for the interpolation operator vector is greater than the predetermined acceptable mean square error, reducing the bandwidth for the interpolation operator;

forming a second matrix of interpolation coefficients based on a third function having the interpolation variables, the reduced bandwidth and the differences between the selected locations;

forming a second differences vector based on a fourth function having the interpolation variables, the reduced bandwidth and the differences between the selected locations and the interpolation location;

forming a second interpolation operator vector by applying an inverse of the second matrix of interpolation coefficients to the second differences vector; and

estimating the seismic data value for the interpolation location using the second interpolation operator vector.

19. A computer system, comprising:

a processor; and

a memory comprising program instructions executable by the processor to;

receive data acquired at one or more locations, wherein the data represents subterranean formations in the earth;

select one or more of the locations, wherein the selected locations are within a vicinity of an interpolation location;

select a bandwidth for an interpolation operator;

form a matrix of interpolation coefficients based on a first function having on one or more interpolation variables, the bandwidth and one or more differences between the selected locations, wherein at least one of the interpolation variables is periodic;

form a differences vector based on a second function having the interpolation variables, the bandwidth and one or more differences between the selected locations and the interpolation location;

form an interpolation operator vector by applying an inverse of the matrix of interpolation coefficients to the differences vector; and

estimate a data value for the interpolation location using the interpolation operator vector.

20. The computer system of claim 19, wherein the program instructions executable by the processor to select the locations comprises program instructions executable by the processor to:

subtract a multiple of a period from each of the locations to create one or more modified locations;

compute a difference between each of the modified locations and the interpolation location;

identify one or more differences lower than a predetermined value; and

select a portion of the locations based on the identified differences.