Study of the curvature of a triangulated surface

arnaud@sep.stanford.edu

## ABSTRACTThe depth migration of time-migrated surfaces by demigration with GOCAD is inaccurate in the case of complex surfaces, such as folded horizons, in areas where the curvature of the surface reverses its sign. I believe this problem can be avoided by cutting the original complex surface into pieces for which the depth migration is accurate. To do so, I propose an algorithm to study the sign of the curvature of the GOCAD surface in order to divide it up properly before performing the depth migration. Putting the depth-migrated pieces back together, I should then obtain a depth-migrated surface more representative of the true geology than the direct depth-migrated surface. |

INTRODUCTION

In an earlier report () I presented a method for 3-D depth migration of 3-D time-migrated maps with GOCAD. The process consists of two parts: first I use the principle of demigration; then the depth migration is performed.

The demigration of 3-D complex time-migrated surfaces produces even more complex demigrated surfaces. The result of the demigration of a 3-D syncline is a classic example of such complex surfaces: the demigrated surface presents a bow-tie, a triplication that has to be processed carefully before going on with the depth migration.

Such triplications appear when the curvature of the time-migrated surface goes to zero and reverses its sign. Consequently, examining the sign of the curvature of the time-migrated surface associated with a cut algorithm for the same GOCAD object can, I believe, enable us to migrate complex horizons in depth with greater accuracy.

In this paper, I investigate the possibility of using the curvature of a GOCAD surface to characterize its complexity. I also address some of the problems associated with analyzing the sign of the curvature. Finally, I present some results of depth migration of a synthetic complex surface.

TERMS OF THE PROBLEM

In 1992, I built a simple algorithm using GOCAD () to perform map depth migration starting with time-migrated horizons. This algorithm, described in a previous SEP report (), consists of two parts and can be summed up as follows:

- first, the time-migrated surfaces are demigrated in order to retrieve the original zero-offset time and position;
- then, the zero-offset horizons are migrated in depth, taking into account the interfaces previously migrated.

The first step requires the root mean square velocity (*V*_{rms}) that has
been used to produce the time migrated maps. In the second step, the
zero-offset horizons are migrated with a user-defined velocity model.

Interpretors feel more comfortable with time-migrated sections that show a better picture of the area under investigation than the premigrated stack sections. Time migration focuses the diffraction, and improves the general aspect of the sections, but often misplaces the horizons. Time demigration, as the inverse operation of time migration, restores these diffractions and retrieves the original position.

A syncline represents a classic example of such complex surfaces. On a stack section, the event corresponding to the syncline presents diffractions that disappear after time migration. Figure SYN25 shows a 3-D time-migrated syncline. The demigration of this syncline produces a zero-offset surface with a triplication (Figure SYNdmg25). Since GOCAD represents surfaces as a continuous triangular patch (a set of vertices connected to form contiguous triangles), the demigrated surface presents a bow-tie with two rounded edges that produce the anomalies visible on the depth-migrated surface (Figure SYNmig25): the depth-migrated surface has a M shape with two branches going upward to infinity (cut in the figure).

Figure 1

Figure 2

Figure 3

The same kind of behavior can be observed when the curvature of the initial time-migrated surface goes to zero and reverses its sign. When we look at Figure SYNmig25, or any depth-migrated complex surface, we can see that in the areas where the curvature of the initial surface has a constant sign, the result of the depth-migration process seems accurate. On the other hand, on the edges where the curvature changes its sign, the process produces strange results (mostly points going upward to infinity). This observation lead me to a new algorithm for the depth migration of complex GOCAD surfaces.

CURVATURE OF A TRIANGULATED SURFACE

The curvature of a continuous surface can be analytically calculated when the position of each point on the surface is given by a set of explicit equations or computed in the case of implicit or complex equations. In the discreet domain, where the position of a point on a surface cannot be simply expressed by equations, the calculation of the curvature requires some work.

Curvature of a discretized surface The curvature of a discontinuous set of points can be approximated by fitting a patch to this set prior to any calculation, as described in Mortenson's book . A patch is a curve-bounded collection of points whose coordinates are given by a set of continuous two-parameter single-valued functions of the form

(1) |

where the parametric variables *u* and *w* are constrained to the interval
, and is the position vector. By choosing
small subsets of the domain , we can define several particular
portions of the surface and thereby study the curvature, for example, around
different points of the surface. The chosen domain of operation is
. Any other domain
can be normalized to this unit square in the parametric space.

The curvature at a point on a surface is one of the intrinsic properties of a surface, which also include the normal and the tangent plane.

Through a point *p* on a surface *S* passes an infinity of curves *c* for
which we can compute the curvature at that point. In the general case, all
the curvature values thus calculated are different. Let us consider the
curvature of all the curves *c* that lie on the surface *S* and pass
through the point *p* with the same tangent line *t* at *p*. These curves
all have the same center of curvature, radius of curvature, curvature vector,
and principal normal curvature.

Among all the possible curves, some are of particular interest: the
intersection curves cut on the surface by a plane *N* passing through the
tangent line *t*, and also those containing the normal to the
surface at *p* (Figure curvature.) When the plane *N* rotates around
the normal , the curvature varies and has a maximum and a
minimum in two orthogonal directions. These extreme values, denoted as
and , are the principal normal curvatures.

Figure 4

The derivation of and , and any value of the curvature in general, is developed in Mortenson's book . The values and are useful to define two related measures of the curvature of a surface:

- the Gaussian curvature
*K*,(2) - the mean curvature
*H*,(3)

Curvature of a surface with GOCAD

In GOCAD, a triangulated surface can be approximated by a local cubic Bezier patch (). The computation of the curvature at any point on a GOCAD surface is based on this patch.

The computation of the two intrinsic properties *K* and *H* is available
in GOCAD. *K*, being a constant value for a whole surface, is not useful
for our problem. However, the mean curvature *H* characterizes well the
behavior of the dip of a surface. It has the additional advantage that it
goes to zero where inflection points are present.

Figure 5

Figure 6

Figure SYNcrvH represents the mean curvature of the syncline in Figure
SYN25. As described in the first section, I am interested in the sign
of the curvature as a means to finding out more precisely where the inflection
points are located. Figure SYNcrvHsignP is a magnified picture of a
surface whose points are at the same *x* and *y* locations as on the original
time-migrated surface (Figure SYN25), and the third coordinate is
either +1 or -1, corresponding to the sign of the curvature at the point.
The plateaux of the surface denote a positive sign; the valleys, a negative
sign.

MAP MIGRATION OF COMPLEX HORIZONS

This section explains how I modify my original depth-migration algorithm to take into account the information given by the curvature of the initial time-migrated surface. I also comment on the results obtained with the surface in Figure SYN25, and propose some improvements for future work.

Strategy

If the original surface is cut into pieces on which the sign of the curvature is constant, the problem associated with inflection points that occurs with the depth migration process should not arise. Accordingly, I want to cut the surface shown in Figure SYNcrvHsignP into plateaux and valleys. The location of the cuts will be mapped to the initial time-migrated surface. Each fragment thus created will be migrated separately.

Therefore, I propose the following algorithm to perform the map migration of complex surfaces with GOCAD:

- 1.
- Study the sign of the curvature of the surface.
- 2.
- Cut the complex surface into pieces with constant curvature.
- 3.
- Demigrate and then migrate in depth each piece separately.
- 4.
- Reconstruct the surface in depth from the fragments.

Current status of the project

This subsection presents the results I obtained with the new algorithm for depth migration, starting with the syncline surface in Figure SYN25.

Figure SYNcut represents the initial surface after the cut has been applied. The cutting algorithm uses the surface that gives the sign of the curvature (Figure SYNcrvHsignP) as a reference to indicate where to break the initial surface into pieces. The resulting surface has four cuts corresponding to the four transitions from plateau to valley or valley to plateau visible in Figure SYNcrvHsignP. The cutting algorithm detects the triangles that have to be cut and removes the connection between the three apices of the triangles.

Figure 7

Running the demigration and depth-map-migration subroutines for this surface produced the results shown in Figures SYNcutdmg and SYNcutmig. Figure SYNcutdmg represents the demigrated syncline surface. The triplication has been cut into several pieces. The left and right branches of the triplication are cut just before the round edges that caused trouble during the depth-migration process. When these edges are present in the central piece of the triplication, they create some mispositioned triangles in the depth-migrated surface.

Figure 8

Figure 9

One of the problems associated with the cutting algorithm is visible in both Figure SYNcut and Figure SYNcutmig: the syncline surface should not need to be cut on the left and right part where it is flat. The cut is made automatically; the algorithm cuts the surface using the information given by the sign of the curvature but nothing else. Consequently, the surface in Figure SYNcut is cut because these regions have been detected as areas of major changes in the curvature.

The effect of the cutting algorithm is particularly visible in Figure SYNcutmig: the depth image of the syncline itself has been improved in comparison to that shown in Figure SYNmig25, but the intervals between the syncline and the flat parts on the left and the right need some more work.

Future work

Though the cutting algorithm has proven its effectiveness in the case I considered, it needs to be improved. In its current state, only a single series of triangles is removed around a local null curvature. The results I obtained showed that the cut is not precise enough to eliminate the problems associated with the demigrated surface. Ideally, I would completely cut off the round edges visible in Figure SYNdmg25.

At this state of the project, I have not considered the problem of reconstructing the surface in depth from the different migrated pieces. One way to tackle this issue is to keep track of the triangles removed as well as the positions of the three apices constituting each of these triangles.

CONCLUSION

I have built a new algorithm for the process of demigration and map migration of complex surfaces with GOCAD. Based on a study of the curvature sign, the initial time-migrated surface is cut into simple pieces where the curvature has a constant sign, and then depth-migrated before being reconstructed.

Though some problems remain to be solved, the results show improvements in the shape of the depth-migrated surface.

[SEP,gocad9.1,geomodel]

5/11/2001