Abstract A prestack time migration is presented that is simple, efficient, and provides detailed velocity information. It is based on Kirchhoff prestack time migration and can be applied to both 2-D and 3-D data. The method is divided into two steps: the first is a gathering process that forms common scatterpoint (CSP) gathers; the second is a focusing process that applies a simplified Kirchhoff migration on the CSP gathers, and consists of scaling filtering, normal moveout (NMO) correction, and stacking. A key concept of the method is a reformulation of the double square-root equation (of source-scatterpoint-receiver traveltimes) into a single square root. The single square root uses an equivalent offset that is the surface distance from the scatterpoint to a educated source and receiver. Input samples are mapped into offset bins of a CSP gather, without time shifting, to an offset defined by the equivalent offset. The single square-root reformulation gathers scattered energy to hyperbolic paths on the appropriate CSP gathers.

Abstract Objectives: Recognize how data acquisition produces a distorted view of the subsurface Realize that the distortion is removed by the migration process Become aware of time or depth migration Know that scatterpoints are useful in defining the acquisition response Know that post-stack migration deals with semicircles Know that prestack migration deals with ellipses Identify the three main migration methods Kirchhoff F-K Downward continuation

Abstract Objectives Construct a post stack or zero-offset section using a compass Know that seismic“dips”are defined by a, and geological dips by β Understand modelling by summing diffractions, F-K methods, ray-tracing methods, and the exploding-reflector model Use the“ exploding reflector ”model of 2-D data in (x, z) with the added dimension of time to define the 3-D volume with dimensions (x, z, t)

Abstract Objectives Define and review the geometric shapes of the semicircle, ellipse, and hyperbola from the perspective of modelling and migration. Define and evaluate instantaneous, interval, average, stacking, RMS, DMO, and migration velocities. Gain insight to the Fourier transform and aliasing. Become acquainted with the wave equation and be exposed to some methods of applying it to seismic data. Identify the differences between time and depth migrations.

Abstract Objectives Identify the three main methods of migration: Kirchhoff, FK, and downward continuation. Estimate migration results with a compass . Understand the principles behind Kirchhoff migration, and recognize the range of algorithms from very simple to complex. Identify the features of FK migration and know how and why the parameters are defined. Recognize the principles of downward continuation migration, distinguish between a time and depth migration, and identify different algorithms. Know the difference between one pass and two-pass 3-D migration algorithms. Recognize that all migration algorithms should be tested .

Abstract Objectives Compare and evaluate various migration results. A picture is worth a thousand words. This chapter contains images relating to migration in both time, depth, and transformed space. It is intended that the figure titles describe the content of this chapter however a brief summary text is included at the end of the chapter. All seismic sections are in the space-time domain unless otherwise indicated.

Abstract Objective Identify problems with side swipe and oblique reflectors. Understand Fresnel zones and how they relate to migration. Know that migration lowers the frequency of dipping events. Define recording times and aperture length for structured sections. Gain insight into techniques that deal with large elevation changes that require some form of wave equation datuming. Know that migration should be computed as close to the surface as possible.

Abstract There are a number of ways to derive the wave equation, and many forms in which it is expressed. The approach taken in these notes will be based on the works by Claerbout [23], [294], Yilmaz [83], Stolt [21], and Brysk [100]. The main objective is to define the paths to the solutions used in seismic processing for modelling and migration. The main emphasis will be on the downward continuation method of seismic migration. Since this is a collection from the above authors, the notations used will reflect that used by the specific author to enable comparison to with their results. Consequently some parameters may change, specifically when moving from the solutions of the wave equations to the finite difference solutions. It is assumed that the density is constant, and that the velocity varies in a vertical (z) and horizontal (x) manner. The 2-D seismic model is assumed to be 3-D with axis x, z, and time (t), with the pressure amplitude defined at any point in the volume as P(x, z, t). The zero offset or stacked section is defined by the surface P(x, z=0, t), and the geological cross-section or desired depth migration as P(x, z, t=0). The interval velocity v(x, z) is assumed to be isotropic (independent of direction). Note that some solutions use modified forms of the velocities such as RMS velocities (i.e. Kirchhoff time) and other use velocities defined in time as v(x, t).

Abstract We have defined the Fourier transform pairs on an infinite time and frequency scale. In the practical world of computers, the time and frequency domain data are assumed to be at discrete sample locations, and with a finite number of samples. In this realm, the Fourier transform becomes the discrete Fourier series where the input and transformed functions must be considered to be periodic, and attention given to wrap-around and aliasing. Applying the derivative in the frequency domain becomes a very simple procedure. When the data has been Fourier transformed, the derivative is found by applying a 90 degree phase shift (the “j” part), and multiplying the amplitudes of the frequency values by the ramp type function ω or 2πf. The inverse Fourier transform then gives the exact derivative. Note that derivatives found by subtracting two samples is only an approximation. Higher order (n) derivatives are found in a similar manner by applying n90 degrees, and multiplying the amplitudes by ω n . Integral are found very simply by dividing the spectrum by jω. The exponential of a complex value may be expressed as e Jθ = cos(θ) + j sin(θ) It has a magnitude of one, i.e. |e jθ | = 1 The phase = φ may be found from The above time shift t 0 is a linear phase shift in the frequency domain θ= ω t 0 . Non-integer static shifts (say t 0 ) can be made accurately in the frequency domain with a simple linear phase shift.

Abstract Repeated acquisition experiments when the elevation of the surface is lowered to one scatterpoint Diffraction estimation at various depths using surface measurements. Assume the source and receiver arrays are lowered into the earth at the levels indicated on the left side of the geological model. Sketch the seismic image on the following time sections for the appropriate depths. Copy the zero-time energy to the corresponding depth section below. Compare the migrated depth section with the original geology. Note that: the data always remains as a time section only the top edge of the time section is copied to the migrated section this top edge is referred to as the imaging condition . Estimating the downward layer diffraction from the surface diffraction using vertical elevation corrections. Generalized diffraction estimation using all possible ray path correction.

Abstract This volume, SEG Course Notes Series No. 13, is designed to give the practicing geophysicist an understanding of the principles of poststack migration, presented with intuitive reasoning rather than laborious math. Modeling is introduced as a natural process that starts with a geologic model and then builds seismic data. Migration is then described as the reverse process that uses seismic data to find the geologic model. Many other topics are covered relating to the quality of the migrated section, such as aliasing, rugged topography, or use of the correct velocity. Significant new material has been added in this revised edition of the original 1997 book, especially algorithms based on the phase-shift method, such as PSPI and the omegaX method.

Abstract Know that the incident and reflected rays follow different paths, but maintain equal angles of incidence and reflection. Define and become familiar with the various ways of displaying prestack data such as source gathers, constant offset sections, or CMP gathers in the prestack volume ( x, h, t ). Construct a source gather and a constant offset section. Know that conventional stacking is only valid for horizontal reflectors. Become aware that care must be taken when forming limited-offset stacks from a range of offsets to create a constant-offset section. Know that diffractions don’t stack. A scatterpoint forms a surface in the prestack volume ( x, h, t ) which is referred to as Cheops pyramid.

Abstract Know that prestack migration is equivalent to DMO and poststack migration for constant velocities. Know that DMO (constant velocity) enables all dips to be MO’d with the same (RMS) velocity i.e. NMO. Know that DMO (constant velocity) eliminates dip smear. Know that constant velocity assumption may be extended to areas with smoothly varying velocities. Understand the basic DMO algorithms. Become aware that some DMO algorithms are available for variable velocities. Be able to identify the different types of processes referred to as DMO.

Abstract Know that prestack migration can eliminate errors of conflicting dips. Know that NMO, DMO, and poststack migration is considered by some to be prestack migration. Identify the main methods of prestack migration. Understand the iterative nature of prestack migrations that are based on CMP concepts. Understand the process of GDMO-PSI. Introduction to EOM (covered in detail in Chapter 11 ). Know that constant velocity 3-D prestack migration may be accomplished with 2-D DMO and 3-D poststack migration.

Abstract Recognize properties of the DMO operator. Know how to test a DMO operator. Identify areas where DMO benefits data. Identify areas where DMO may harm data. Know that test results only evaluate the implementation of an algorithm, not the algorithm. Identify the differences of time and depth migrations when they are applied to structured data.