Compressive sensing

• US 9,632,193 B2
• Filed: 10/31/2014
• Issued: 04/25/2017
• Est. Priority Date: 11/01/2013
• Status: Active Grant

• Current Assignee: Shearwater Geoservices Software Incorporated
• Original Assignee: Conocophillips Company (ConocoPhillips)
• Inventors: Kaplan, Sam T., Li, Chengbo, Mosher, Charles C., Brewer, Joel D., Keys, Robert G.
• Primary Examiner(s): Murphy, Daniel L
• Application Number:
  US14/529,690
• Publication Number:
  US 20150124560A1
• Time in Patent Office: 907 Days
• Field of Search: 367 14
• CPC Class Codes: G01V 1/003 Seismic data acquisition in general, e.g. survey design G01V1/3808, G01V1/42 take precedence
• CPC Class Codes: G01V 1/30 Analysis G01V1/50 takes precedence
• 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 2210/169 Sparse arrays
• CPC Class Codes: G01V 2210/57 Trace interpolation or extrapolation, e.g. for virtual receiver; Anti-aliasing for missing receivers
• CPC Class Codes: G01V 2210/60 Analysis
• CPC Class Codes: G01V 2210/614 Synthetically generated data

Abstract

Computer-implemented method for determining optimal sampling grid during seismic data reconstruction includes: a) constructing an optimization model, via a computing processor, given by min_u∥Su∥₁s.t. ∥Ru−b∥₂≦σ wherein S is a discrete transform matrix, b is seismic data on an observed grid, u is seismic data on a reconstruction grid, and matrix R is a sampling operator; b) defining mutual coherence as

$\mu \le \sqrt{\frac{C}{S}\frac{m}{{\left(\log n\right)}^6}},$
wherein C is a constant, S is a cardinality of Su, m is proportional to number of seismic traces on the observed grid, and n is proportional to number of seismic traces on the reconstruction grid; c) deriving a mutual coherence proxy, wherein the mutual coherence proxy is a proxy for mutual coherence when S is over-complete and wherein the mutual coherence proxy is exactly the mutual coherence when S is a Fourier transform; and d) determining a sample grid r_*=arg min_r μ(r).

Claims

1. A computer-implemented method for determining optimal sampling grid during seismic data reconstruction, the method comprising:

a) constructing an optimization model, via a computing processor, given by min_u∥

Su∥

₁ s.t. ∥

Ru−

b∥

₂≦

σ

wherein S is a discrete transform matrix, b is seismic data on an observed grid, u is seismic data on a reconstruction grid, σ

represents noise level in observed data, and matrix R is a sampling operator;

b) defining mutual coherence as

2. The method of claim 1, wherein the sample grid is determined via randomized greedy algorithm method.

3. The method of claim 2, wherein the randomized greedy algorithm method finds local minimum.

4. The method of claim 1, wherein the sample grid is determined via stochastic global optimization method.

5. The method of claim 1, wherein r_*=arg min_r μ

(r) is non-convex.

6. The method of claim 1, wherein the mutual coherence proxy is derived using fast Fourier transform.