Method for temporal dispersion correction for seismic simulation, RTM and FWI

Patent metadata

Field	Value
Publication number	US-10185046-B2
Application number	US-201514670085-A
Country	US
Kind code	B2
Filing date	Mar 26, 2015
Priority date	Jun 9, 2014
Publication date	Jan 22, 2019
Grant date	Jan 22, 2019

Abstract

Method for correcting seismic simulations, RTM, and FWI for temporal dispersion due to temporal finite difference methods in which time derivatives are approximated to a specified order of approximation. Computer-simulated seismic data (51) are transformed from time domain to frequency domain (52), and then resampled using a mapping relationship that maps, in the frequency domain, to a frequency at which the time derivative exhibits no temporal dispersion (53), or to a frequency at which the time derivative exhibits a specified different order of temporal dispersion. Alternatively, measured seismic data from a field survey (61) may have temporal dispersion of a given order introduced, by a similar technique, to match the order of approximation used to generate simulated data which are to be compared to the measured data.

First claim

The invention claimed is:

1. A method for prospecting for hydrocarbons, comprising:
obtaining measured seismic data;
generating, with a computer, simulated seismic data using a finite-difference, time-stepping algorithm that approximates a time derivative operator to a selected order of approximation;
performing, with the computer, full waveform inversion or reverse time migration of the measured seismic data with the simulated seismic data, wherein temporal numerical dispersion corresponding to the selected order of approximation is (i) removed from the simulated seismic data or (ii) introduced into the measured seismic data by steps including,
performing, with the computer, a Fourier transform in time on (i) the simulated or (ii) the measured seismic data,
then resampling the transformed seismic data in frequency domain, and then performing an inverse Fourier transform from frequency domain back to time domain,
wherein said resampling utilizes a property of a class of stationary finite-difference operators,
wherein in frequency domain, an aspect of the temporal numerical dispersion is that a desired numerical solution for a given frequency is computed at an incorrect frequency, and said resampling uses a mapping relationship that maps the incorrect frequency to the given frequency; and

prospecting for hydrocarbons with the full-waveform-inverted seismic data or the reverse-time-migrated seismic data.

2. The method of claim 1 , further comprising scaling the resampled frequency-domain seismic data with a frequency-dependent scaling factor before performing the inverse Fourier transform back to time domain.

3. The method of claim 2 , wherein simulation of the seismic data comprises a wave propagation equation with a stationary, finite-difference differential operator and a source term S in frequency domain, and wherein the scaling factor can be expressed as

S
⁡

(

x
,

ω2
⁡

(
ω1
)

)

S
⁡

(

x
,
ω1

)

where ω 1 is the incorrect frequency, ω 2 is the given frequency, and x is spatial location of the source.

4. The method of claim 1 , wherein the time derivative being approximated by the finite-difference algorithm is a centered second derivative, the given frequency is true frequency, and the mapping relationship can be expressed as

ω
approx

⁡

(

n
order

,
ω
,

Δ
⁢

⁢
t

)

=

-

a
0

-

2
⁢

∑

j
=
1

n
order

/
2

⁢

⁢

a
j

⁢

cos
⁡

(

j
⁢

⁢
ωΔ
⁢

⁢
t

)

Δ
⁢

⁢
t

where ω approx is the incorrect frequency, ω is the true frequency, n order is the selected order of approximation, Δt is duration of a time step in the algorithm, a j are coefficients of a symmetric filter used to approximate the second derivative operator, and j is an integer index ranging from 1 to n order /2.

5. The method of claim 1 , wherein said class of stationary finite-difference operators includes any differential operator that operates on a function of spatial position and time to equate to a source term, where the differential operator includes at least one spatial derivative of any order, at least one time derivative of any order, and may vary with position but is constant with time.

6. The method of claim 1 , wherein temporal numerical dispersion is removed from simulated seismic data by using a mapping relationship that maps the incorrect frequency to a true frequency, being a frequency at which the simulation generates a solution for the incorrect frequency with no temporal numerical dispersion.

7. The method of claim 6 , wherein spatial derivatives in the finite difference time stepping algorithm are approximated to order at least 20.

8. The method of claim 1 , wherein temporal numerical dispersion is removed from simulated seismic data by using a mapping relationship that maps the incorrect frequency to a frequency at which the simulation generates a solution having temporal numerical dispersion of a same order of approximation as a spatial derivative approximation in the finite-difference algorithm.

9. The method of claim 1 , wherein temporal numerical dispersion corresponding to the selected order of approximation in the algorithm is introduced into the measured seismic data to

Assignees

• Anderson John E
• Baumstein Anatoly
• Marcinkovich Carey
• Vdovina Tetyana
• Brytik Valeriy
• Exxonmobil Upstream Res Co

Inventors

• Anderson John E
• Baumstein Anatoly
• Marcinkovich Carey
• Vdovina Tetyana
• Brytik Valeriy

Classifications

• G01V2210/44

  F-k domain · CPC title

• G01V1/36 Primary

  Effecting static or dynamic corrections on records, e.g. correcting spread; Correlating seismic signals; Eliminating effects of unwanted energy · CPC title

• G01V2210/51

  Migration · CPC title

• G01V1/28

  Processing seismic data, e.g. for interpretation or for event detection (G01V1/48 takes precedence) · CPC title

• G01V2210/679

  Reverse-time modeling or coalescence modelling, i.e. starting from receivers · CPC title