Depth structure maps

Seismic Data Analysis

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 . Refer to Figure 9.4-8 for a brief description of image-ray tracing. Consider an image ray that departs the nth surface at point S_n with coordinates (x_n, y_n, z_n) and emerges at the right angle at the earth’s surface at point S₀ with coordinates (x₀, y₀, z = 0). Our goal is to determine the coordinates of the output point (x_n, y_n, z_n) on the depth map from image-ray depth conversion of the input point (x₀, y₀, t_n) on the time map. To achieve this goal, we want to trace the image ray from the point of emergence (x₀, y₀, 0) back to its point of departure (x_n, y_n, z_n).

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  Figure 9.4-4  Smoothed versions of the gridded surfaces shown in Figure 9.4-3.

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  Figure 9.4-7  Interval velocity maps derived from the horizon-consistent rms velocity maps shown in Figure 9.4-6.

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  Figure 9.4-8  Principles of image-ray depth conversion. See text for details.

Suppose that the first n − 1 horizons have already been converted to depth, and that next we want to convert the nth horizon to depth. Since the earth model is known for the first n − 1 layers, then we know the coordinates of the intersection point S₁ of the image ray with the first layer, (x₁, y₁, z₁), where by definition of the image ray, x₁ = x₀ and y₁ = y₀. By using Snell’s law, we can determine the direction of the ray as it departs point S₁ reaching point S₂ 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

$t_{n-1}=2\sum _{k}^{n-1}{\frac {\Delta s_{k}}{v_{k}}},$

(2)

where v_k is the interval velocity of the kth layer, and Δs_k is the distance between the intersection points of the image ray, S_k−1 and S_k, on the (k − 1)st and kth surfaces given by

$\Delta s_{k}=\left[(x_{k}-x_{k-1})^{2}+(y_{k}-y_{k-1})^{2}+(z_{k}-z_{k-1})^{2}\right]^{1/2}.$

(3)

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 nth 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 nth surface since we know the total elapsed time t_n−1 from equation (2) and the total elapsed time t_n from the input time horizon read at point (x₀, y₀, z = 0). Finally, we know the interval velocity v_n of the nth layer from the interval velocity map. Therefore, we can calculate the elapsed distance Δs_n along the raypath as it departs the point S_k−1 on the (k − 1)st surface in the direction dictated by Snell’s law. The quantity Δs_n is given by

$\Delta s_{n}={\frac {v_{n}}{2}}(t_{n}-t_{n-1}).$

(4)

Finally, the coordinates of the point S_n that we need to know to perform the time-to-depth conversion are given by

$x_{n}=x_{n-1}+\Delta s_{n}\cos \ \alpha ,$

(5a)

$y_{n}=y_{n-1}+\Delta s_{n}\cos \ \beta ,$

(5b)

$z_{n}=z_{n-1}+\Delta s_{n}\cos \ \gamma ,$

(5c)

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 nth horizon, we can trace an image ray associated with the time t_n(x₀, y₀) on the time map and derive the depth value z_n(x_n, y_n) on the depth map.

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 Δd_n of the lateral displacement vector between the points S₀ and S_n as

$\Delta d_{n}={\sqrt {(x_{n}-x_{0})^{2}+(y_{n}-y_{0})^{2}}},$

(6a)

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

$\phi _{n}=\tan ^{-1}{\frac {y_{n}-y_{0}}{x_{n}-x_{0}}}$

(6b)

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.

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  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.

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  Figure 9.4-10  Contour maps of the depth horizons shown in Figure 9.4-9.

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  Figure 9.4-11  Modulus maps for the displacement vectors that describe the lateral departure between vertical rays and image rays. See text for details.

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  Figure 9.4-12  Azimuth maps for the displacement vectors that describe the directional difference between vertical rays and image rays. See text for details.

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

External links

find literature about Depth structure maps