Evidences of thickening and thinning

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Problem 10.10a

Figure 10.10a shows a schematic 1:1 geologic section with three parallel beds of equal thicknesses dipping Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 18^{\circ}} . Assume that the sediment velocity is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V=1.5+0.5z\ {\rm km/s}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z} is the depth below the sea floor in kilometers, density changes providing the impedance contrasts. The water bottom slopes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 14^{\circ}} between 6 and 12 km so that the increase of velocity does not begin until the sea floor. Show how this section would appear on an unmigrated CMP seismic section. Assume coincident source and geophone.

Figure 10.9a.  Migrated map.
Figure 10.9b.  Map of unmigrated seismic data showing lines of control by diagonal dashes (courtesy of Prakla-Seismos).
Figure 10.10a.  Assumed geologic section.

Solution

Because source and geophone are coincident, reflection raypaths are incident on the beds at right angles; such raypaths are shown by short dashes on Figure 10.10a. We let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \xi = {\rm dip}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x= {\rm CMP}} location, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x'=h\sin \xi =} location of reflecting point, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h=} slant depth at CMP = raypath length, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z=h\cos \xi =} depth of reflecting point, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=} depth of seafloor above reflecting point, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V=} velocity at reflecting point, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bar{V}=(V+1.5)/2} is the average velocity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t={\rm two}} -way reflection time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =2z/(\bar{V}\cos \xi)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t=} apparent thickness in time.

In Table 10.10a we have calculated (neglecting raypath curvature) the arrival times for the four interfaces at locations 0, 6, and 10 km. Figure 10.10b shows the time section. Note that the beds, which are of equal thickness, apparently thin with depth and also downdip. Also note that an overlying water layer distorts structure and changes the apparent dips and thicknesses.

Problem 10.10b

Figure 10.10c shows a geologic section with constant velocity layers. Assume constant density and no out-of-the-plane effects; draw the unmigrated zero-offset seismic section. Scale ratio is 1:1.

Table 10.10a. Calculation of apparent thicknesses.
Refln. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x'} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z/\cos \xi} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bar{V}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t\dagger} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t_{i}}
A 4.00 0.0 1.24 3.60 3.79 3.30 2.40 3.158
B 5.00 0.0 1.55 4.55 4.78 3.78 2.64 3.621 0.463
C 6.00 0.0 1.80 5.50 5.78 4.25 2.88 4.014 0.393
D 7.00 0.0 2.17 6.45 6.78 4.72 3.11 4.360 0.346
A 2.05 6.0 6.64 1.85 1.95 0.16 2.42 1.96 1.990
B 3.05 6.0 6.95 2.76 2.90 0.24 2.88 2.19 2.648 0.658
C 4.05 6.0 7.31 3.66 3.85 0.33 3.33 2.42 3.182 0.534
D 5.05 6.0 7.62 4.57 4.80 0.40 3.78 2.64 3.636 0.454
* 10.0 1.00* 0.97* 1.50 1.294
** 10.0 1.14 1.50 1.520
B 1.75 10.0 10.54 1.66 1.84 1.13 1.81 1.66 0.976† 2.496
C 2.75 10.0 10.85 2.62 2.89 1.21 2.27 1.89 1.973† 3.493 0.997
D 3.75 10.0 11.16 3.57 3.94 1.28 2.74 2.12 2.745† 4.265 0.772
*Seafloor reflection
**Portion of reflection travel paths in water
†For travel paths in sediments
Figure 10.10b.  Seismic time section.
Figure 10.10c.  Assumed geologic section.
Figure 10.10d.  Unmigrated time section of Figure 10.10c.

Solution

Figure 10.10d shows reflections with solid lines, diffractions with dashed lines, and multiple reflections with short-dashed lines. Note that

  1. reflections are tangent to the diffractions and this would make determining the ends of the reflections difficult;
  2. the horizontal terminations of reflections are at the same horizontal locations as the terminations of the reflectors;
  3. there is no unconformity reflection where there is no change in properties even though the lithology may be different;
  4. the angle of the high-velocity wedge is changed and features below it are distorted;
  5. the top of the basement layer has a kink in it at C, and there would be a small bow-tie effect [see (7) below], because of the overlying high-velocity wedge;
  6. there is no fault-plane reflection on this part of the display, the fault can be located by connecting the crests of the diffractions;
  7. the syncline reflection becomes a buried-focus bow-tie “anticline.”

Ordinary migration would not remedy the effects of velocity changes in the horizontal direction, i.e., it would continue the erroneous appearance of the high-velocity wedge, and it would not remove the kink in the basement surface. However, with the correct migration velocity, diffractions would collapse, reflections would terminate correctly, and the syncline would appear correctly. The fault could then be located by the reflection terminations.

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