Seismic Data Processing Lecture 15

Kirchhoff Migration

Prepared by Dr. Amin Khalil

School of Physics, USM2014

What is Kirchhoff migration?

Kirchhoff-type migration methods are widely used in the current oil and gas industry for both depth imaging and iterative velocity analysis. For migrating a single event on a single trace, a full-aperture KM smears the event energy to all possible subsurface points in the model space. After smearing all samples on all traces, a KM image is obtained by stacking all individual contributions. Both the obliquity factor and the geometric spreading factor are compensated in the KM algorithm.

Kirchhoff Migration of a Single Trace

Given a source and a geophone on the free surface, and a single dipping reflector in a homogeneous acoustic medium, there will be only one primary reflection recorded in the seismic trace (see Figure 2.1). For convenience, multiples and direct waves will be ignored. The arrival time of this event is equal to the traveltime for energy to propagate from the source to the reflection point P and from P to the geophone. The dashed line in Figure 2.1 depicts the associated specular ray. In forward modeling, the reflectivity at point P is convolved with the source wavelet, leading to a waveform other than a unit pulse being observed. Mathematically, modeling is described as d=Lm , where d is the forward modeled seismic data, L is a linear forward modeling operator, and m is the reflectivity model.

The reverse process of seismic forward modeling is seismic migration which back-projects the observed energy to its subsurface reflector. Mathematically, the migrated image is given by m = LT m . To implement migration, we need to know the medium velocity. By applying velocity analysis, we can obtain a reasonable estimate of the velocity distribution. Migration methods can not be performed without the basic knowledge of the medium's velocity distribution.

Think how seismic velocity analysis is important to seismic data processing!!!!

Description

KM steps:

1- Apply ray tracing or solve the eikonal equation to build travel time field for both source and geophone. The travel time obtained is coarse and can be fine tunned via interpolation.

2- The next step in KM is to smear the observed energy to its primary reflection point. This is, however, a blind operation because we know nothing about where the true reflection point is. As a result, we have to migrate the event to all possible reflection points. Such image points are those whose reflection traveltimes are equal to the observed traveltime of the event.

What is ray tracing?

In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces, complicating analysis. Ray tracing solves the problem by repeatedly advancing idealized narrow beams called rays through the medium by discrete amounts. Simple problems can be analyzed by propagating a few rays using simple mathematics. More detailed analyses can be performed by using a computer to propagate many rays.WIKIPEDIA

Realistic Example

In ray tracing each ray is identified by unique parameter called ray parameter. This parameter is taken from snell’ law and equal to:

ovp )sin(

Where is the incident angle of ray, and vo is the starting velocity

Illustration of step 2

Some possible points

Problems With KM

Previous figure depicts a full-aperture KM, where the observed event is migrated to an entire ellipsoid. Image noise is generated because the energy is migrated far away from its specular reflection point, leading to strong far-field migration artifacts. These artifacts can be effectively suppressed by migrating many traces and stacking their individual migrated images.

Figure 2.3 shows that the dipping reflector in Figure 2.1 can be clearly resolved after many traces have been migrated.

The problem is solved?!!!Answer: To some extent, since some misleading information may results in far-field artifacts!!!

This means that at far-field of the KM-ellipsoid we may obtain false reflector, due to violation of the antiphase criteria at certain range of the aperture.

meaning

SolutionPossible solution is to apply anti-alias filter.

More Problems

Generally, KM method is computationally expensive, meaning it requires large time of computation at powerful computers called supercomputers or PowerStation. For post-stack applications the method may take days in computation. For prestack we may need months.

3D Vs 2D

So far we were dealing with 2D seismic profile. Again this is an approximation to the medium, in which we assume that the physical properties change only with spatial coordinates; namely; the horizontal X direction and vertical Z direction. The y direction neglected or in other words the physical parameters is assumed constant in that direction. Is this assumption valid for all cases?

NO !!!

Why 3D?!!!

Because:1- The earth change its properties in three

dimensions.

2- In 2D we may get reflections and diffraction from points outside the plane. Thus all our migration will fail or become misleading.

Comparison

http://www.searchanddiscovery.com/documents/2003/brewer03/images/04.htmCopyright © 2014 Datapages, Inc. All rights reserved

3D migration

Downward Continuation

Artifact

Velocity is needed for applying 3D migration. In the present case we need to know the velocity change in 3D meaning with x, y and z direction which is not feasible. In addition using velocity which change spatially will also complicate the problem.

Thank you