# US 6,295,505 B1 · Method of filter generation for seismic migration using Remez algorithm

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- Number: US 6,295,505 B1
- Document ID: US6295505B1
- Status: Granted
- Kind: B1
- Application number: 08/999,900
- Filed: 1997-12-17
- Granted: 2001-09-25
- Earliest claimed: 1995-01-10
- Assignees: Schlumberger Technology Corporation
- Inventors: Steven Brent Assa; Jeffrey Mark Rutledge
- Web page: https://patentgrep.com/patent/US6295505B1

## CPC

- G01V 1/364: {Seismic filtering (G01V1/37 takes precedence)}
- G01V 2210/27: Other pre-filtering
- G01V 2210/32: Noise reduction
- G01V 2210/57: Trace interpolation or extrapolation, e.g. for virtual receiver; Anti-aliasing for missing receivers
- G01V: GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS (means for indicating the location of accidentally buried, e.g. snow-buried, persons A63B29/02)

## Abstract

A method for efficiently and accurately producing filter coefficients for wavefield extrapolation which is particularly useful for migrating seismic data using a multiprocessor parallel computer as provided. In 3D seismic migration, the techniques have been developed for 3D processing using 1D extrapolation filters. The present invention uses a Remez-type approximation for developing such extrapolation filters which increases image quality or reduces processing time, or both. Additionally, the method provides an automated method for producing such extrapolation filters which is particularly useful in parallel seismic migration because of the large number of extrapolation filters which must be produced. The automated method of generating filters sets a frequency transition band where the amplitude is specified to be monotonically decreasing which insures stable filter design.

## Claims

### Claim 1

A method of migrating seismic data using a parallel computer having a number of processing components each with local memory, where the seismic data is downwardly continued in the frequency domain for all depths of interest, the method comprising the steps of:
i) assigning processing components to respective subgrids in the x,y depth plane;
ii) determining a table of extrapolation filters which varies according to lateral (x,y) velocity variations by approximating the extrapolation filters using a Remez-type approximation with an error less than a certain amount;
iii) loading said table into local memory of each processing component;
iv) operating said processing components independently to extrapolate the seismic data for an x,y depth plane, each processing component accessing its table in local memory according to the velocity in its assigned subgrid.

### Claim 2 (depends on 1)

The method of claim 1, said determining step ii) including the substeps of setting a desired maximum dip angle and selecting a wavenumber frequency transition zone.

### Claim 3 (depends on 1)

The method of claim 1, said determining step ii) including selecting a frequency transition zone in which the amplitude is monotonically decreasing.

### Claim 4 (depends on 1)

The method of claim 1, said table comprising a one dimensional array indexed by normalized frequency ω ⁢ ⁢ Δ ⁢ ⁢ x v .

### Claim 5 (depends on 1)

The method of claim 1, said extrapolation filters each comprising N filter coefficients hn.

### Claim 6 (depends on 1)

The method of claim 1, including repeating the step of operating said processing components independently to extrapolate successive (x,y) depth planes.

### Claim 7 (depends on 1)

The method of claim 1, wherein said processing components are assigned to the same spatial subgrid in each (x,y) depth plane.

### Claim 8 (depends on 1)

The method of claim 1, said operating step iv) comprising convolving the McClellan transform of the filter for its assigned subgrid with the seismic data.

### Claim 9 (depends on 6)

The method of claim 6, including performing steps i-iv for a single frequency plane and repeating for successive frequency planes.

### Claim 10

A method of seismic wavefield depth extrapolation comprising the steps of:
i) approximating the complex 1D depth extrapolation operator D(k) with an ordinary Fourier cosine series having N coefficients hn;
ii) setting a desired maximum migration dip angle θ and desired N;
iii) computing a table of extrapolation filters hn for a range of normalized frequencies ω ⁢ ⁢ Δ ⁢ ⁢ x v ,
where
for a given normalized frequency ω ⁢ ⁢ Δ ⁢ ⁢ x v ,
define a frequency transition band width and
determine the N coefficients hn for the given normalized frequency ω ⁢ ⁢ Δ ⁢ ⁢ x v
using a Remez-type approximation where the amplitude is monotonically decreasing in the transition band;
iv) extrapolating a seismic wavefield for a lateral region at a given depth using the N filter coefficients hn from said table corresponding to the normalized frequency for the lateral region.

### Claim 11 (depends on 10)

The method of claim 10, where said extrapolation step iv) determines the N filter coefficients hn by linear interpolation between the nearest normalized frequency values in the table.

### Claim 12 (depends on 10)

The method of claim 10, where the maximum amplitude is less than a certain amount.

### Claim 13 (depends on 10)

The method of claim 10, where the maximum amplitude in the transition band and pass band is less than 1.001.

### Claim 14 (depends on 10)

The method of claim 10, including the step of increasing the width of the transition band provided the amplitude in the pass and transition bands does not exceed a certain amount.

### Claim 15 (depends on 10)

The method of claim 10, the setting a desired N substep including the substep of decreasing the number N provided the amplitude in the transition band decreases monotonically.

### Claim 16 (depends on 10)

The method of claim 10, the setting a desired dip angle θ step ii) including the substep of decreasing the maximum dip angle θ provided the amplitude in the pass and transition bands do not exceed a specified amount.

### Claim 17

A method of migrating seismic data, for converting stacked seismic data into a geologic image, using a parallel computer having a number of processing components, each with local memory, where the seismic data is downwardly continued in the frequency domain for all depth planes of interest, including the steps of:
i) assigning processing components to respective lateral subgrids of seismic data in x,y depth plane;
ii) determining a table of extrapolation filters which varies according to lateral (x,y) subgrids of seismic data by approximatingthe extrapolation filters using a Remez-type approximation with an error less than a certain amount;
iii) loading said table into local memory of each processing component; and
iv) operating said processing components independently to extrapolate the seismic data for the x,y depth plane, each processing component accessing its table in local memory according to the subsurface acoustic velocity in its assigned subgrid.

### Claim 18 (depends on 17)

The method of claim 17, said determining step ii) including the substeps of setting a desired maximum dip angle and selecting a wavenumber frequency transition zone.

### Claim 19 (depends on 17)

The method of claim 17, said determining step ii) including selecting frequency transition zone in which the amplitude is monotonically decreasing.

### Claim 20 (depends on 17)

The method of claim 17, said table comprising a one dimensional array indexed by normalized frequency ω ⁢ ⁢ Δ ⁢ ⁢ x v .

### Claim 21 (depends on 17)

The method of claim 17, said extrapolation filters each comprising N filter coefficients hn.

### Claim 22 (depends on 17)

The method of claim 17, including repeating the step of operating said processing components independently to extrapolate successive (x,y) depth planes.

### Claim 23 (depends on 17)

The method of claim 17, wherein said processing components are assigned to the same spatial subgrid in each (x,y) depth plane.

### Claim 24 (depends on 17)

The method of claim 17, said operating iv) comprising convolving the McClellan transform of the filter for its assigned subgrid with the seismic data.

### Claim 25 (depends on 22)

The method of claim 22, including performing steps i-iv) for a single frequency plane and repeating for successive frequency planes.

### Claim 26

A parallel computer-implemented method of seismic wavefield depth extrapolation, in furtherance of transforming stacked seismic data into a geologic image, comprising the steps of:
i) approximating the complex ID depth extrapolation operator D(k) with an ordinary Fourier cosine having N coefficients hn;
ii) setting a desired maximum migration dip angle Ø and desired N;
iii) computing a table of extrapolation filters hn for a range of normalized frequencies ω ⁢ ⁢ Δ ⁢ ⁢ x v ,
where
for a given normalized frequency ω ⁢ ⁢ Δ ⁢ ⁢ x v ,
define a frequency transition bandwidth and
determine the N coefficients hn for the given normalized frequency ω ⁢ ⁢ Δ ⁢ ⁢ x v
using a Remez-type approximation where the amplitude is monotonically decreasing in the transition band;
iv) extrapolating a seismic wavefield for a lateral region at a given depth using the N filter coefficients hn from said table corresponding to the normalized frequency for the lateral region.

### Claim 27 (depends on 26)

The method of claim 26, where said extrapolation step iv) determines the N filter coefficients hn by linear interpolation between the nearest normalized frequency values in the table.

### Claim 28 (depends on 26)

The method of claim 26, where the maximum amplitude is less than a certain amount.

### Claim 29 (depends on 26)

The method of claim 26, wherein the maximum amplitude in the transition band and pass band is less than 1.001.

### Claim 30 (depends on 26)

The method of claim 26, including the step of increasing the width of the transition band provided the amplitude in the pass and transition bands does not exceed a certain amount.

### Claim 31 (depends on 26)

The method of claim 26, the setting a desired B substep including the substep of decreasing the number N provided the amplitude in the transition band decreases monotonically.

### Claim 32 (depends on 26)

The method of claim 26, the setting a desired dip angle θ step ii) including the substep of decreasing the maximum dip angle θ provided the amplitude in the pass and transition bands do not exceed a specified amount.

### Claim 33

A parallel computer programmed for seismic migration to transform stacked seismic data into a geologic image where the seismic data is downwardly continued in the frequency domain for a number x,y planes of seismic data in an x,y depth plane;
memory associated with each processing component having loaded therein a table of extrapolation filters which vary according to lateral (x,y) subsurface acoustic velocity variations in a depth plane by approximating the extrapolation filters using a Remez-type approximation with an error less than a certain amount;
means for operating said processing components independently to extrapolate the seismic data for the x,y depth plane, each processing component accessing its table in memory according to the velocity in its assigned subgrid.

### Claim 34 (depends on 33)

The computer of claim 33, where the memory comprises a local memory for each processing component with each local memory loaded with said table.

### Claim 35 (depends on 33)

The computer of claim 33, where said operating emans functions to operate said processing components independently to extrapolate other x,y depth planes.

### Claim 36 (depends on 33)

The computer of claim 33, said table of extrapolation filters comprising N filter coefficients hn for a range of normalized frequencies ω ⁢ ⁢ Δ ⁢ ⁢ x v .

### Claim 37

A method of acquiring and processing seismic data comprising the steps of:
i) acquiring seismic data;
ii) applying quality control, filtering and deconvulation on a trace basis to the seismic data;
iii) estimating the velocity of the subsurface formations;
iv) performing D dip moveout correction and stacking to the seismic data; and
v) migrating the seismic data, using a parallel computer having a number of processing components, each with local memory, where the seismic data is downwardly continued in the frequency domain for all depth planes of interest, such that a map of formation locations is created, including the steps of:
a) assigning processing components to respective lateral subgrids of seismic data in an x,y depth plane;
b) determining a table of extrapolation filters which varies according to lateral (x,y) subgrids of seismic data by approximating the extrapolation filters using a Remez-type approximation with an error less than a certain amount;
c) loading the table into local memory of each processing component; and
d) operating the processing components independently to extrapolate the seismic data for the x,y depth plane, each processing component accessing its table in local memory according to the subsurface acoustic velocity in its assigned subgrid.

### Claim 38 (depends on 37)

The method of claim 37 wherein:
the determining step (v)(b) includes the substeps of setting a desired maximum dip angle and selecting a wavenumber frequency transition zone.