Migration/DMO - Halliburton

239 ProMAX® Reference

Migration/DMO

This chapter outlines the parameter information needed toexecute the processes relating to migration and DMO. Wepreface this section with an overview of the current migrationtechniques employed for seismic imaging and a summary ofthe reasons and results of running DMO.

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In This Chapter

➲ Migration Overview 2268

➲ DMO Overview 2292

➲ 3D Kirchoff Amplitude Weighting

➲ Azimuth Moveout

➲ Common Offset F-K DMO 2304

➲ DMO to Gathers 3D 2316

➲ DMO Stack 3D 2326

➲ Dip Decomposition DMO 2334

➲ Ensemble DMO in the T-X Domain 2338

➲ Explicit FD 3D Depth Migration 2342

➲ Explicit FD 3D Depth Migration (SMP) 2352

➲ Explicit FD 3D Time Migration 2362

➲ Explicit FD 3D Time Migration (SMP) 2372

➲ Explicit FD Depth Migration 2382

➲ Fast Explicit FD Time Migration 2390

➲ Implicit FD Depth Migration 2396

➲ Implicit FD Time Migration 2404

➲ Stolt or Phase Shift 2D Migration 2412

➲ Stolt or Phase Shift 3D Migration 2425

➲ Kirchhoff Depth Migration 2439

➲ Kirchhoff Time Migration 2447

➲ Poststack Kirchoff 2D Time Migration

➲ PostStack Kirchoff 3D Time Migration

➲ PSPC 3D Depth Migration 2451

➲ PSPC 3D Depth Migration (SMP) 2459

➲ PSPC 3D Time Migration

➲ PSPC 3D Time Migration (SMP) 2469

➲ Prestack Curved Ray Kirchoff 2D Time Migration

➲ Prestack Curved Ray Kirchoff 3D Time Migration

➲ Prestack FD Shot Migration 2487

➲ Prestack Kirchoff 2D Time Migration



➲ Prestack Kirchhoff Depth Migration 2501

➲ Prestack Kirchhoff Time Migration 2513

➲ Prestack Kirchhoff 3D Time Mig. 2519

➲ Prestack Plane-wave Depth Migration

➲ Prestack Time Migration 2546

➲ Reverse Time T-K Migration 2552

➲ Steep Dip Explicit FD Time Migration 2558

➲ VSP Kirchhoff Depth Migration 2570

➲ VSP-CDP Transform 258


Migration Overview242 ProMAX® Reference

Migration Overview

This document provides an overview of the migrations in thecategories of:

➲ Poststack 2D

➲ Pseudo Prestack 2D

➲ Prestack 2D

➲ Poststack 3D

The initial discussion is intended for typical users ofmigrations, who may need a good description of generalmigration techniques and to better understand theterminology used in association with the ProMAX®algorithms:

➲ F-K Migration

➲ Phase shift Migration

➲ Finite Difference Migration

➲ Kirchhoff Migration

➲ Reverse-Time Migration

Unofficial timing tests are included to give you a relative feelfor the resource commitment for each migration approach.Graphics are also included to give you a comparison of thedifferent ProMAX® processes.

Discussion

The purpose of this section is to provide an overview of thecurrent migration techniques employed for seismic imaging.A non-mathematical approach is taken when describing themigration techniques. You are encouraged to use the listedreference material at the end of this document formathematical details and expanded descriptions, which arebeyond the scope of our present focus.

What is migration?

According to R.E. Sheriff, “Migration is an inversion operationinvolving rearrangement of seismic information elements so



that reflections and diffractions are plotted at their truelocations.”

Migrations come in a wide variety of categories, and can beapplied to prestack or poststack data. We will start bydiscussing the broad categories of migration, and thendiscuss each category in more detail.

Migration techniques are categorized in a variety of ways,relating to algorithm implementation. An algorithm whichuses RMS velocities analytically and does not account for raybending at interfaces is characterized as a time migration. Onthe other hand, an algorithm which uses interval velocitiesand accounts for ray bending at interfaces is characterized asa depth migration.

The mathematical expression which accounts for ray bendingis known as the thin-lens term, leading to O. Yilmaz’salternate definition of depth migration, “A migration methodthat includes the effects of the thin lens term is called depthmigration...”

Although, strictly speaking, the thin-lens term only appearsin certain solutions to the scalar wave equation,consideration of the equivalent effects of the thin lens termdoes provide a good discrimination between time and depthmigration.

Migrations are further categorized on the basis of algorithmor technique by which the migration correction is performed.One way to categorize migrations is on the basis of thetransform domain in which the migration is performed, suchas T-X, T-K, F-X, F-K. In this discussion we will use thefollowing classifications: Kirchhoff, F-K, Phase Shift, Finite-Difference, and Reverse Time. The newer F-X migrationtechniques, implemented with spatially-variant convolutionalfilters, are often grouped with Finite-Difference techniques,although mathematical finite differences are not employed.

In this discussion, we will use the following categories in ourdiscussion of migration techniques:

• Migration type, such as Time or Depth.

• Migration implementation, such as Kirchhoff, F-K, PhaseShift, Finite-Difference, or Reverse Time.

• Migration stage, such as Prestack or Poststack.



• Data type, such as 2D or 3D.

The following charts provide an initial introduction to thetypes of ProMAX® migration algorithms, and how they can becategorized.

Migration Name Category Type

Stolt 2D F-K Time

Phase Shift 2D Phase Shift Time

Steep-Dip Explicit FD Time Finite-Difference Time

Fast Explicit FD Time Finite-Difference Time

Explicit FD Depth Finite-Difference Depth

Kirchhoff Depth Kirchhoff Depth

Kirchhoff Time Kirchhoff Time

Reverse-Time T-K Reverse Time Time


Stolt 2D (pseudo) F-K Time

Prestack Time (pseudo) F-K Time

Phase Shift 2D (pseudo) Phase Shift Time

Reverse-Time T-K (pseudo) Reverse Time Time

Prestack FD Shot Finite-Difference Depth

Prestack Kirchhoff Depth Kirchhoff Depth

Prestack Kirchhoff Time Kirchhoff Time


Stolt 3D F-K Time

Phase Shift 3D Phase Shift Time

PSPC 3D Depth Phase Shift Depth

Explicit FD 3D Depth Finite-Difference Depth

Explicit FD 3D Time Finite-Difference Time



In the following discussions we will expand upon the majorcharacteristics of the various migration implementations.

F-K Migration

F-K migration is the fastest implementation of migration andis based upon the approach outlined by Stolt (1978). In itssimplest implementation, F-K migration involves a FourierTransform from the T-X domain to the F-K domain, where amapping operation is performed using a single velocity value.The migrated image is generated by performing an inverseFourier Transform from the F-K domain to the T-X domain.

To accommodate vertical variation in the velocity field (and insome implementations, lateral velocity variation), a StoltStretch is applied to the T-X data prior to migration. Theamount of stretch applied to the data is dependent upon theratio between the local RMS velocity and the lowest RMSvelocity on the data section. This lowest RMS velocity value isthen used to perform the F-K constant-velocity migration.Following the inverse Fourier Transform, an inverse StoltStretch is performed to arrive at the final T-X migratedsection.

F-K migration is very accurate up to 90 degrees of dip, whenthe constant velocity assumption is valid, which is a likelycontradiction. When vertical and/or lateral velocity variationsexist, this method becomes quite inaccurate. While the StoltStretch technique does offer some compensation, thiscorrection is only true at the apexes of diffraction hyperbolae,and not on the flanks.

The mapping operation of the F-K techniques is performedfrom the frequency value (omega) to the wavenumber value indepth (kz), using a constant velocity value. This mappingoperation requires an interpolation, which may require thatthe sampling interval in frequency be very small. A poorlysampled dataset, in frequency, can result in a final migratedproduct where only the flat events will be imaged, while thedipping events are attenuated. Interpolation has beendesigned to provide accurate results and avoid thesesampling issues.



Phase Shift Migration

Phase shift migration is based upon the approach outlinedby Gazdag (1978). Data are migrated using a downwardcontinuation approach, which equates to phase shifts in theF-K domain. Phase shift migration is applied in two steps:wavefield extrapolation and imaging. The wavefieldextrapolation step consists of downward continuing therecorded data into a form corresponding to a repositioning ofthe recording plane, through use of the scalar wave equation.The imaging step consists of outputting a portion of migrateddata, corresponding to the zero traveltime of the repositioneddataset. The data are recursively migrated by using theoutput of one wavefield extrapolation for the next.

Phase shift migration is accurate to 90 degrees, with theassumption that velocity does not change laterally. In itsbasic implementation, the phase shift technique is used fortime migrations, and accepts a single RMS or interval velocityfunction, that is VRMS(t) or VINT(t). Modifications to the phaseshift technique allow imaging beyond 90 degrees, throughmigration of turning-ray energy. In the case of anaccelerating velocity field in time, energy which approachesthe evanescent region of the F-K domain, can be saved andsubsequently imaged to the T-X domain.

Three general approaches can be used to modify the phaseshift migration method to accommodate lateral velocityvariations. The simplest approach is to apply a Stolt Stretch,as in F-K migration. Another straightforward approachconsists of performing multiple phase shift migrations, withlaterally-constant velocity functions, and interpolating acomposite dataset from these results, which corresponds tothe laterally-varying velocity function.

An alternative method is to apply a traditional phase shiftcorrection with a laterally-constant velocity function, followedby an additional correction to accommodate lateral velocityvariations. The simplest correction, termed the first-orderphase correction (PSPC1), is a first-order correction whichconsists only of a time shift. The PSPC1 correction, whichcorresponds to the thin-lens term commonly described infinite-difference migration literature, is generally asatisfactory correction. The second-order phase correction(PSPC2) incorporates both the PSPC1 correction and asecond term, which depends upon spatial location and dip.The actual implementation of the PSPC2 correction requires



significantly more processing time than the PSPC1correction.

Finite-Difference Migration

Finite-difference migration traditionally encompassedthose techniques which employed finite difference methods tosolve the appropriate wave equation. Recently, the family ofF-X migrations, performed through spatially-variantconvolutional filters, have been often been classified as finite-difference migrations, for simplicity reasons. While we willmaintain this classification generalization, the followingdiscussion will address the differences between traditionalfinite difference and F-X migrations.

The original finite-difference migrations were based upon theapproach outlined by Claerbout (1972). These algorithms canbe characterized as time migrations which use spatially andtime-variant RMS velocity fields, that is V RMS(x,t). Thecorrection for lateral velocity variation is an approximation,as these are time migrations, and does not equate to the thin-lens correction.

Finite-difference migration is applied in two steps: wavefieldextrapolation and imaging. The wavefield extrapolation stepconsists of downward continuing the recorded data, usingthe scalar wave equation, into a form corresponding to arepositioning of the recorded plane. The imaging stepconsists of outputting a portion of migrated datacorresponding to the zero traveltime of the repositioneddataset. The data are recursively migrated by using theoutput of one wavefield extrapolation as input for the next.

Various levels of assumptions can be used to simplify thescalar wave equation, resulting in implementations withvarious levels of dip accuracy, often classified as 15 degrees,45 degree, and 65 degree finite-difference migrations. Thistype of finite-difference migration involves an implicitsolution to the scalar wave equation, in other words, anindirect solution requiring an additional matrix inversion.Implicit finite-difference migrations have the advantage ofstability, but the disadvantages of poor accuracy andsignificant frequency-dispersion effects. Note: Implicitalgorithms are no longer supported within the ProMAX®processing system.



The F-X migrations, in contrast, provide explicit solutions tothe scalar wave equation through application of spatially-variant convolution filters applied in the F-X or F-X-Ydomain. A previously created table of migration convolutionaloperators, for every ratio of omega/VINT, are generallyemployed to reduce run time and ensure stability withrespect to evanescent, that is, non-propagating energy. TheF-X migration approach can be used to implement either timeor depth migrations, which are more accurate and suffermuch less from dispersions effects, compared with traditionalimplicit finite-difference techniques.

Kirchhoff Migration

Kirchhoff Migration is commonly implemented following thetechnique outlined by Schneider (1978) for the integralsolution of the scalar wave equation. Kirchhoff migration iscommonly implemented as a time or depth migration.Kirchhoff time migration calculates traveltimes analyticallyfrom laterally and temporally varying RMS velocities,although the accommodation of lateral velocity variation isapproximate, as for any time migration. In a depthimplementation, Kirchhoff migration estimates traveltimesfrom a velocity model, using ray-tracing or solving anexpression such as the Eikonal equation.

A special implementation of Kirchhoff prestack timemigration is Prestack Imaging (PSI), outlined by Gardner et al(1986). The PSI process consists of first performing DMO,then summing constant time slices of the DMO-correctedCMP gathers along circles in the CMP-half-offset domain. Theresulting gathers, which have offset equal to the diameters ofthe circles, are used to perform a velocity analysis. Theprestack migration output is generated by applying NMO tothe PSI gathers and stacking. The strength of PSI is thattraditional NMO velocity analysis is performed aftermigration.

Reverse-Time Migration

Reverse-time migrations use the two-way acoustic waveequation to essentially inverse model the seismic data. Thewavefield is extrapolated backwards in time, using anexploding reflector model, to derive the desired source, that is



reflector distribution in depth. The reverse-time migrationapproach can be used for either time or depth migration.

A full T-X implementation of reverse-time migration willaccurately handle vertical and lateral velocity variations, anddips up to and beyond 90 degrees. However, such analgorithm will be extremely compute-intensive, runningperhaps 20 times longer than an Explicit F-D migration. Anefficient reverse-time migration can be implemented in the T-K domain, resulting in an accurate time migration, forvertical velocity variations.

Poststack 2D Migrations

The following chart provides a listing of Poststack 2DMigrations and their key characteristics:

These tests are not controlled benchmarks, as processing occurred acrossnetworks during peak usage times

Unofficial Timing Tests

Poststack 2D Migration Algorithms

Timing tests were performed with the following dataset:

Migration Name Category Type Velocity V(x) V(t/z) SteepDip

RunTime

Stolt 2D F-K Time VRMS(x,t) Poor Poor Fair 0.2

Phase Shift 2D Phase Shift Time VINT(x,t) None Good Good 1.0

Steep-Dip ExplicitFD Time

F-D (70 deg)(50 deg)

TimeTime

VINT(x,t)VINT(x,t)

FairFair

GoodGood

GoodFair

21.010.0

Fast Explicit FD Time F-D Time VINT(x,t) Fair Good Fair 9.6

Explicit FD Depth F-D Depth VINT(x,z) Good Good Good 21.7

Kirchhoff Depth Kirchhoff/ImExplicitMult. Arr.

DepthDepthDepth

VINT(x,z)VINT(x,z)VINT(x,z)

FairGoodExcel.

GoodGoodExcel.

GoodGoodExcel.

7.312.064.0

Kirchhoff Time Kirchhoff Time VRMS(x,t) Fair Good Good 14.6

Reverse-Time T-K ReverseTime

Time VINT(t) None Good Good 2.5



• Synthetic land dataset, with 200 m of topographychange

• 421 traces

• 1100 samples/trace

• 4 ms sample rate

• Shot spacing - 100 m

• CDP spacing - 50 m

Tests were run on the following equipment:

IBM 370

• 64 Meg memory

• Remote Disk

Sun Sparc 10

• 128 Meg memory

• Remote Disk

Sample Migrations

The following graphics give a visual comparison of Poststack2D Migrations using the stack dataset, described above:



Stolt 2D Migration

Comment: Fastest Poststack 2D Migration.

Phase Shift 2D Migration

Comment: Results similar to Reverse-Time Migration.



Steep-Dip Explicit FD Time Migration

Comment: No padding used. Typically 30% padding is used.

Fast Explicit FD Time Migration

Comment: Maximum dip about 45 degrees.



Explicit FD Depth Migration

Comment: No padding used. Typically 30% padding is used.

Kirchhoff Depth Migration

Comment: Comparable to F-X Depth Migration.



Kirchhoff Time Migration

Comment: New poststack 2D time migration.

Reverse Time T-K Migration

Comment: Results are very similar to Phase Shift Migration.



Pseudo Prestack 2D Migrations

The following chart provides a listing of Pseudo Prestack 2DMigrations, and their key characteristics:

A table of performance run times are provided to give you aqualitative sense of the relative speeds of the variousalgorithms. These tests are not controlled benchmarks, asprocessing occurred across networks during peak usagetimes.


Pseudo Prestack 2D Migration Algorithms

Timing tests were performed with the dataset described below:


• 106 shots

• 421 CDPs




IBM 370

• 64 Meg memory

• Remote Disk

Migration Name Category Type Velocity V(x) V(t/z) Dip RunTime

Stolt 2S F-K Time VRMS(x,t) Poor Poor Fair 0.2

Prestack Time F-K Time VRMS(x,t) Poor Poor Poor 0.4

Phase Shift 2D Phase Shift Time VINT(x,t) None Excel. Good 1.0

Reverse-Time T-K Reverse-Time

Time VINT(t) None Good Good 2.9



Sample Migrations

The following graphics give a visual comparison of PseudoPrestack 2D Migrations using the stack dataset, describedabove:

Stolt 2D Migration

Comment: Fastest pseudo prestack migration option.



Prestack Time Migration

Comment: Macro using Reverse Time Migration.


Comment: Similar results to Reverse Time Migration.



Reverse Time T-K Migration

Comment: Similar results to Phase Shift Migration.

Note: Stolt, Phase Shift, and FK-DMO are the only 2Dmigrations that preserve amplitude with offset.

Prestack 2D Migrations

The following chart provides a listing of Prestack 2DMigrations, and their key characteristics:

A table of performance runtimes are provided to give you aqualitative sense of the relative speeds of the variousalgorithms. These tests are not controlled benchmarks, asprocessing occurred across networks during peak usagetimes.

Migration Name Category Type Velocity V(x) V(t/z) Dip RunTime

Prestack FD Shot F-D Depth VINT(x,z) Good Good Good 44.3

Prestack KirchhoffDepth

Kirchhoff (Imp)Kirchhoff (Mult)

DepthDepth

VINT(x,z)VINT(x,z)

FairExcel

GoodExcel

GoodExcel

0.89.7

Prestack KirchhoffTime

Kirchhoff Time VRMS(x,t) Fair Good Good 3.8




Prestack 2D Migration Algorithms

Poststack 2D timing tests were performed with the dataset described below:


• 106 shots

• 421 CDPs




IBM 370

• 64 Meg memory

• Remote Disk

Sample Migrations

The following graphics give a visual comparison of Prestack2D Migrations using the stack dataset, described above:



Prestack FD Shot Migration

Comment: Every fifth shot record migrated.

Prestack Kirchhoff Depth Migration

Comment: Multiple arrival ray tracing option used.



Prestack Kirchhoff Time Migration

Comment: New prestack time migration.

Poststack 3D Migrations

The following chart provides a listing of Poststack 3DMigrations, and their key characteristics:

A table of performance run times are provided to give you aqualitative sense of the relative speeds of the variousalgorithms. These tests are not controlled benchmarks, asprocessing occurred across networks during peak usagetimes.


Migration Name Category Type Velocity V(x,y) V(t/z) Dip

Stolt 3D F-K Time VRMS(x,y,t) Poor Poor Poor

Phase Shift 3D Phase Shift Time VINT(x,y,t) None Good. Good

PSPC 3D Depth Phase Shift Depth VINT(x,y,z) Fair Good Good

Explicit FD 3D Time F-D Time VINT(x,y,t) Fair Good Good

Explicit FD 3D Depth F-D Depth VINT(x,y,z) Good Good Good



Poststack 3D Migration Algorithms

Poststack 3D timing tests were performed with the dataset described below:

• Synthetic land dataset

• 15,000 traces

• 100 lines by 150 traces




IBM 370

• 64 Meg memory

• Local Disk

Sample Migrations

The following graphics give a visual comparison of Poststack3D Migrations using the stack dataset, described above:

Stolt 3D Migration




PSPC 3D Depth Migration



Explicit FD 3D Depth Migration

Explicit FD 3D Time Migration



References

General Migration

Berkhout, A.J., 1981, Wave Field Extrapolation Techniques in Seismic Migration, aTutorial: Geophysics, 1638-1656 (also in Migration of Seismic Data).

Claerbout, J.F., 1985, Imaging the Earth’s Interior: Blackwell Scientific Publica-tions.

Gardner, H.F. ed., 1985, Migration of Seismic Data: SEG, Tulsa.

Stolt, R.H. and Benson, A.K., 1986, Seismic Migration: Theory and Practice: Geo-physical Press, London.

Yilmaz, O., 1987, Seismic Data Processing: SEG, Tulsa, 240-353.

F-K Migration

Chun, J.H. and Jacewitz, C.A., 1981, Fundamentals of Frequency Domain Migra-tion: Geophysics, 46, 717-733 (also in Migration of Seismic Data.).

Stolt, R.H., 1978, Migration by Fourier Transform: Geophysics, 43, 23-48 (also inGeophysics, 50, 2219-2244 and Migration of Seismic Data).

Phase Shift Migration

Gazdag, J., 1978, Wave-equation Migration with the Phase Shift Method: Geo-physics, 43, 1342-1351 (also in Migration of Seismic Data.).

Gazdag, J., and Sguazzero, P., 1984, Migration of Seismic Data by Phase ShiftPlus Interpolation: Geophysical Prospecting, 49, 124-131.

Gardner, G.H.F., et al, 1987, Phase Shift-Based Prestack Depth Migration for Lat-erally Varying Velocities, SEG Expanded Abstracts, 737-740.

Hale, D., Hill, N.R., and Stefani, J., 1991, Imaging Salt with Turning SeismicWaves: SEG Annual Meeting, Expanded Abstracts, 1171-1174.

Lhemann, O., 1986, A Superfast 3D Migration with Lateral Variations of Velocitiesfor the Cray XMP: Research Computation Laboratory, Annual Progress Review,2, 1-25.

Finite Difference Migration

Claerbout, J.F. and Doherty, S.M., 1972, Downward Continuation of Moveout-Cor-rected Seismograms: Geophysics, 37, 741-768 (also Geophysics, 50, 2033-2060 and Migration of Seismic Data).

Hale, D., 1991, 3-D Depth Migration via McClellan Transformations: Geophysics,56, 1778-1785.



Hale, D., 1991, Stable Explicit Depth Extrapolation of Seismic Wavefields: Geo-physics, 56, 1770-1777.

Hatton, L., Larner, K.L. and Gibson, B.S., 1980, Migration of Seismic Data FromInhomogenous Media, 46, 751-767 (also in Migration of Seismic Data.).

Holberg, O., 1988, Towards Optimum One-Way Wave Propagation: GeophysicalProspecting, 36, 99-114.

Judson, D.R., et al, 1980, Depth Migration After Stack: Geophysics, 45, 361-375(also in Migration of Seismic Data.).

Kosloff, D.D. and Baysal, E., 1983, Migration with the Full Acoustic Wave Equa-tion: Geophysics, 48, 677-687 (also in Migration of Seismic Data.).

Larner, K.L., et al, 1980, Depth Migration of Imaged Time Sections: Geophysics,46, 734-750 (also in Migration of Seismic Data).

Soubaras, R., 1992, Explicit 3-D Migration Using Equiripple Polynomial Expansionand Laplacian Synthesis: SEG Annual Meeting, New Orleans, ExpandedAbstracts, 905-908.

Kirchhoff Migration

Deregowski, S.M., 1985, Prestack Depth Migration by the 2-D Boundary IntegralMethod: SEG Annual Meeting, Expanded Abstracts, 414-417.

Gardner, G.H.F., et al, Dip Moveout and Prestack Imaging: Offshore TechnologyConference, Houston, 1986, 268-277.

Reshef, M. and Kosloff, D., Migration of Common-shot Gathers: Geophysics, 51,324-331.

Schneider, W.A., 1978, Integral Formulation for Migration in Two and ThreeDimensions: Geophysics, 43, 49-76 (also in Migration of Seismic Data.).

Van Tier, J., and Symes, W.W., 1991, Upwind Finite-difference Calculation of Trav-eltimes: Geophysics, 56, 812-821.

Vidale, J., 1988, Finite-difference Calculation of Travel Times: Bull. Seism. Soc.Am., 78, 2062-2076.

Reverse Time Migration

Baysal, E., Kosloff, D.D. and Sherwood, W.C., Reverse Time Migration: Geophys-ics, 48, 1514-1524 (also in Migration of Seismic Data.).

Hale, D., 1991, Migration in the Time-Wavenumber Domain: CWP Annual Report.

Levin, S.AA., 1984, Principle of Reverse-Time Migration: Geophysics, 49, 581-583.


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Migration/DMO - Halliburton