Compressive sensing
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∥ 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, and matrix R is a sampling operator; b) defining mutual coherence as μ ≤ C S m ( log n ) 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).
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
μ
(
r
)
=
max
l
≠
0
r
^
l
=
max
l
≠
0
∑
k
=
1
n
r
k
ω
kl
wherein r is sampling grid, {circumflex over (r)} 1 are Fourier transform coefficients, ω=exp(−2π√{square root over (−1)}/n), and n is number of elements in r;
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).
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.