# Depth structure maps

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

We now have a set of time horizon maps (Figure 9.4-4) and a corresponding set of interval velocity maps (Figure 9.4-7). Recall from introduction to earth imaging in depth that, in principle, a time-migrated image can be converted to a depth section by mapping the amplitudes along image rays. This notion also can be employed to convert time horizons into depth horizons. The process is done layer by layer starting with the shallowest horizon.

A comprehensive mathematical discussion on image-ray tracing is given by Hubral and Krey ^{[1]}. Refer to Figure 9.4-8 for a brief description of image-ray tracing. Consider an image ray that departs the *n*th surface at point *S _{n}* with coordinates (

*x*) and emerges at the right angle at the earth’s surface at point

_{n}, y_{n}, z_{n}*S*

_{0}with coordinates (

*x*

_{0},

*y*

_{0},

*z*= 0). Our goal is to determine the coordinates of the output point (

*x*) on the depth map from image-ray depth conversion of the input point (

_{n}, y_{n}, z_{n}*x*

_{0},

*y*

_{0},

*t*) on the time map. To achieve this goal, we want to trace the image ray from the point of emergence (

_{n}*x*

_{0},

*y*

_{0}, 0) back to its point of departure (

*x*).

_{n}, y_{n}, z_{n}**Figure 9.4-7**Interval velocity maps derived from the horizon-consistent rms velocity maps shown in Figure 9.4-6.

Suppose that the first *n* − 1 horizons have already been converted to depth, and that next we want to convert the *n*th horizon to depth. Since the earth model is known for the first *n* − 1 layers, then we know the coordinates of the intersection point *S*_{1} of the image ray with the first layer, (*x*_{1}, *y*_{1}, *z*_{1}), where by definition of the image ray, *x*_{1} = *x*_{0} and *y*_{1} = *y*_{0}. By using Snell’s law, we can determine the direction of the ray as it departs point *S*_{1} reaching point *S*_{2} on the next surface. As the image ray moves from one surface to the next, we add up the time it takes to travel. When it reaches the (*n* − 1)st layer, the elapsed two-way time *t*_{n−1} is

**(**)

where *v _{k}* is the interval velocity of the

*k*th layer, and Δ

*s*is the distance between the intersection points of the image ray,

_{k}*S*

_{k−1}and

*S*, on the (

_{k}*k*− 1)st and

*k*th surfaces given by

**(**)

Now, we examine the situation when the image ray departs the (*n* − 1)st layer at point *S*_{n−1} on the way to the *n*th surface. Again, by Snell’s law we know the direction of the ray. We also know the elapsed time *t _{n} − t*

_{n−1}from the (

*n*− 1)st surface to the

*n*th surface since we know the total elapsed time

*t*

_{n−1}from equation (

**2**) and the total elapsed time

*t*from the input time horizon read at point (

_{n}*x*

_{0},

*y*

_{0},

*z*= 0). Finally, we know the interval velocity

*v*of the

_{n}*n*th layer from the interval velocity map. Therefore, we can calculate the elapsed distance Δ

*s*along the raypath as it departs the point

_{n}*S*

_{k−1}on the (

*k*− 1)st surface in the direction dictated by Snell’s law. The quantity Δ

*s*is given by

_{n}

**(**)

Finally, the coordinates of the point *S _{n}* that we need to know to perform the time-to-depth conversion are given by

**(**)

**(**)

**(**)

where *α, β,* and *γ* are the directional cosines of the ray at point *S*_{n−1}. The directional cosines are known by the application of Snell’s law at point *S*_{n−1} with known coordinates (*x*_{n−1}, *y*_{n−1}, *z*_{n−1}).

To summarize, given the depth and interval velocity maps for the first *n* − 1 horizons, and the time and interval velocity maps for the *n*th horizon, we can trace an image ray associated with the time *t _{n}*(

*x*

_{0},

*y*

_{0}) on the time map and derive the depth value

*z*(

_{n}*x*) on the depth map.

_{n}, y_{n}Figure 9.4-9 shows the depth maps derived from image-ray depth conversion of the time maps shown in Figure 9.4-4 using the interval velocity maps shown in Figure 9.4-7. As for the time maps, the usual way to post the depth maps is to contour them (Figure 9.4-10).

The depth maps are compatible with the time maps; nevertheless, there can be subtle differences because of velocity variations that would give rise to the departure of image rays from the vertical. To quantify the differences between vertical-ray and image-ray trajectories, and thus to quantify the differences between the time maps and depth maps, we can calculate the modulus Δ*d _{n}* of the lateral displacement vector between the points

*S*

_{0}and

*S*as

_{n}

**(**)

and create the displacement modulus maps shown in Figure 9.4-11. Note that the most significant displacement between the vertical rays and image rays is at the fault zones.

The displacement vector also has a directional azimuth *ϕ _{n}* which is given by

**(**)

as measured from the inline *x* direction. The displacement azimuth maps are shown in Figure 9.4-12. Again, note that the most significant azimuthal variations are along the fault zones.

**Figure 9.4-9**Depth horizons derived from time-to-depth conversion of the time horizons shown in Figure 9.4-4 using the interval velocity maps shown in Figure 9.4-7.

## References

- ↑ Hubral and Krey (1980), Hubral, P. and Krey, T., 1980, Interval velocities from seismic reflection time measurements: Soc. Expl. Geophys.

## See also

- Model building
- Time-to-depth conversion
- Time structure maps
- Interval velocity maps
- Calibration to well tops
- Layer-by-layer inversion
- Structure-independent inversion