Environmental and Exploration Geophysics II tom.h.wilson Department of Geology and Geography West Virginia University Morgantown, WV Migration

In today’s lecture we address basic issues associated with the process of migration which attempts to eliminate the geometrical distortions we have become intimately familiar with during the semester. Can we collapse diffractions back to their point of origin? Can we uncross the dipping limbs of synclines, eliminate the reverse branches, move dipping reflectors back up dip … ?

Look over this section for a few minutes - where are the faults?

Briefly look over this section and identify the faults.

Coincident source-receiver view of a horizontal reflector

Two-way travel times are accurately converted to depth when multiplied by V/2. The depth converted surface is referred to as the record surface.

In the above display both the reflector surface and record surface are plotted. The record surface is plotted in a pseudo depths which would be obtained from the travel time transformation V/2.

Note that the actual reflector has dip  whereas the depth converted record surface has dip .

The pseudo depth display provide useful information. The pseudo depths of the ends of the record surface correspond to the path lengths traveled by wavefronts to the the ends of the reflector surface. Note that two triangles can be formed each of which has sides of length SS’ and l 2 - l 1 = . SS’

The ratio of  to SS’ forms the tan  where  is the apparent dip of the record surface and sin  where  is the dip of the reflector surface. Once we know what the actual dip  is we can reconstruct the position of the reflector surface from the distances l 1 and l 2.

From our record section (depth converted time section) we measure , l 1 and l 2.

The reflector surface - whatever is configuration - is a common tangent to the population of wavefronts emitted by the coincident source-receivers along the profile.

The record point appears at a depth l 2 which corresponds to the radius of the wavefront incident on the reflection point B. Derived from a record of arrival time, it plots directly beneath the surface recording point.

The record point from an isolated reflection point could arise anywhere along the wavefront. We are able to determine direction from which the event originated from the dip of the record surface.

From , we compute . l 1 and l 2 are the distances to the actual reflection points. We rotate l 1 and l 2 through the angle  in the updip direction to locate the actual points of reflection B and B’. We now know where the reflector surface is located. B B’

The foregoing approach is referred to as the tan  -sin  method. The relationship between record points, wavefronts and reflection points can also be used to derive the location of the reflector another way.

Each point on the record surface AA’ can be swung out along a circular arc (the wavefront). Find all points tangent to the population of wavefronts and draw a line connecting them. That line is the reflector surface. wavefronts

The last geometrical approach we will consider is referred to as the maximum convexity front approach. Don’t let this fancy name baffle you. This is just what we call a diffraction when it is plotted as a record surface in pseudo depth.

Each point on the record surface lies along on a maximum convexity front whose apex coincides with the actual reflection point. The record surface is the common tangent to the maximum convexity fronts. Maximum convexity front

Time Response

Record section, pseudo depth format - 2/V x V/2 = 1

A A’ The record surface Maximum Convexity Fronts How can we use the maximum convexity front relationship to locate the reflector surface given the record surface?

Points on the record surface that form points of tangency to a maximum convexity front migrate or are relocated to the apex of the maximum convexity front. Thus, in the above, A’ is a point of tangency and B’ - the reflection point - lies at the apex of the maximum convexity front.

Note that the wavefront and the maximum convexity front intersect at two points. One point is located on the reflector surface and the other on the record surface.

Based on the foregoing geometrical argument we can see that reflection events observed in time are simply a superposition of diffractions from points on the reflector surface. In the above simulation, the apex of individual diffractions coincide with the locations of reflection points.

The reflection or record surface represents a zone of constructive interference associated with the superposition of point diffractions arising from individual reflection points. In the limit that the spacing between diffraction points used to represent the reflector surface drops to zero, destructive interference between diffractions eliminates the diffraftion limbs that hang below the record surface.

We have presented three different approaches to migration each of which illustrate basic geometrical interrelationships between the reflector surface and record surface. These methods include I) tan  -sin  migration II) Wavefront migration III) Maximum Convexity Front migration

Class Exercise:You will find the simple section below in your handout. Use the tan  -sin  and wavefront migration methods to determine the location of the reflector surface.

Measure  and for the two record surfaces below and then move the end points for each record-surface to the corresponding end-points on the reflector surface. Use compass provided in class.

Using the compass, trace out wavefronts in the updip direction for each record surface. Use a straight-edge to locate the tangent to the wavefronts. The common tangent corresponds to the reflector surface. Do for both record surfaces and compare the results of the wavefront method to the tan  -sin  method.

The maximum convexity method requires that the interpreter have a set of maximum convexity curves. The maximum convexity curves are easily constructed.

First, measure off distances from surface points to the two diffraction points.

In the record section (and time section) the diffraction arrivals plot vertically beneath the source-receiver locations. The Maximum convexity front or diffraction form hyperbola in the record or time section.

Move the maximum convexity front into points of tangency with individual reflection points. Move the record point to apex of the maximum convexity front.

Look over this section for a few minutes - where are the faults?

After migration fault locations are often easier to identify.

Briefly look over this section and identify the faults.

Take a few moments to interpret this section

How did you do? Note the familiar pattern of stretched anticlines, collapsed synclines with reverse branches.

At any single surface point, the coincident source-receiver record views normal incidence reflections off to the side from dipping structure.

Thus reflections from a continuous reflector will be displaced relative to their actual surface location and may even appear discontinuous.

anticline syncline vertical limb Left limb of anticline

Make a quick interpretation of this section.