Multisource encoding and decoding using the signal apparition technique

Amundsen, Lasse; Андерссон, Фредрик; Manen, D.J. van; Robertsson, Johan O. A.; Eggenberger, Kurt

doi:10.1190/geo2017-0206.1

Cited by 9 publications

(5 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we generalize the findings of Andersson et al (2017b) to that of M simultaneous sources (Andersson et al 2017a;Amundsen et al 2018). We show that the method of signal apparition results in optimally large regions in the frequency-wavenumber space for exact separation of sources.…”

Section: Introductionsupporting

confidence: 66%

“…Outside the flawless diamonds, the simultaneous source problem is solved using additional constraints to tackle the otherwise ill-posed problem (e.g., [4,6]). In this paper we generalize the findings of [5] to that of M simultaneous sources [2,3]. We show that the method of signal apparition results in optimally large regions in the frequency-wavenumber space for exact separation of sources.…”

Section: Introductionmentioning

confidence: 64%

“…() and Amundsen et al . (). Supposing that the M sources

f_{n}

are sampled in

2 M m

points for some integer m , we define the semi‐discrete Fourier transform of

f_{n}

\begin{matrix} true \\ right {\hat{f}}_{n} (ω, k) & = & left \sum_{j} prefixexp ((), \frac{- 2 π i k j}{2 M m}) \\ left \int_{- \infty}^{\infty} f_{n} (t, {normalΔ}_{x} j) prefixexp (() - 2 π i t ω) d t . \end{matrix}

Using the Poisson summation formula and the condition

supp F (f_{n}) \subset C

, it is straightforward to check that

{\overset{f}{true}}_{n} false(ω, k false) = \frac{1}{Δ_{x}} F false(f_{n} false) (ω, \frac{k}{2 M m {normalΔ}_{x}})

for

(ω, k)

such that

false| ω false| < 2 ω_{0}

and

- M m \leq k \leq M m - 1

.…”

Section: Theorymentioning

confidence: 97%

“…The domains C and D are depicted in Figure 1. We remark that using signal apparition allows for each F(f n ) to be perfectly reconstructed in D, see [2,3]. Supposing that the M sources f n are sampled in 2M m points for some integer m, we define the semi-discrete Fourier transform of f n as…”

Section: Theorymentioning

confidence: 99%

“…; Amundsen et al . ). We show that the method of signal apparition results in optimally large regions in the frequency–wavenumber space for exact separation of sources.…”

Section: Introductionmentioning

confidence: 97%

See 4 more Smart Citations

Perfect partial reconstructions for multiple simultaneous sources

Wittsten¹,

Андерссон

Robertsson

et al. 2019

Geophysical Prospecting

Self Cite

View full text Add to dashboard Cite

A major focus of research in the seismic industry of the past two decades has been the acquisition and subsequent separation of seismic data using multiple sources fired simultaneously. The recently introduced method of signal apparition provides a new take on the problem by replacing the random time‐shifts usually employed to encode the different sources by fully deterministic periodic time‐shifts. In this paper, we give a mathematical proof showing that the signal apparition method results in optimally large regions in the frequency–wavenumber space where exact separation of sources is achieved. These regions are diamond shaped and we prove that using any other method of source encoding results in strictly smaller regions of exact separation. The results are valid for arbitrary number of sources. Numerical examples for different number of sources (three, respectively, four sources) demonstrate the exact recovery of these diamond‐shaped regions. The implementation of the theoretical proofs in the field is illustrated by the results of a conducted field test.

show abstract

Section: Introductionsupporting

confidence: 66%

Section: Introductionmentioning

confidence: 64%

“…() and Amundsen et al . (). Supposing that the M sources

f_{n}

are sampled in

2 M m

points for some integer m , we define the semi‐discrete Fourier transform of

f_{n}

\begin{matrix} true \\ right {\hat{f}}_{n} (ω, k) & = & left \sum_{j} prefixexp ((), \frac{- 2 π i k j}{2 M m}) \\ left \int_{- \infty}^{\infty} f_{n} (t, {normalΔ}_{x} j) prefixexp (() - 2 π i t ω) d t . \end{matrix}

Using the Poisson summation formula and the condition

supp F (f_{n}) \subset C

, it is straightforward to check that

{\overset{f}{true}}_{n} false(ω, k false) = \frac{1}{Δ_{x}} F false(f_{n} false) (ω, \frac{k}{2 M m {normalΔ}_{x}})

for

(ω, k)

such that

false| ω false| < 2 ω_{0}

and

- M m \leq k \leq M m - 1

.…”

Section: Theorymentioning

confidence: 97%

Section: Theorymentioning

confidence: 99%

“…; Amundsen et al . ). We show that the method of signal apparition results in optimally large regions in the frequency–wavenumber space for exact separation of sources.…”

Section: Introductionmentioning

confidence: 97%

See 3 more Smart Citations