Using a velocity function linear with depth
When the velocity increases linearly with depth according to the relation
being constant, show that
where velocity, velocity at depth , source-geophone distance, raypath parameter, arrival time, and angle of incidence.
When the velocity is a function of depth only, as in equation (4.17a), expressions for the offset (see problem 4.1) and traveltime can be found by dividing the medium into horizontal layers, each of infinitesimal thickness (Figure 4.17a) and then integrating. We have
Also, using equation (3.1a), we have
In the limit we get the following integrals for and :
When the velocity increases monotonically with depth, a ray must eventually return to the surface (see Figure 4.20a). For horizontal velocity layering the raypaths are symmetrical about the deepest point.
In equation (4.17d) we substitute , from equation (4.17a), also . Thus, we get
Show that the angle of incidence and the depth can be written
From equation (4.17c) we get
Solving equation (4.17a) for gives
Given the velocity function km/s, find the depth and offset of the point of reflection and the reflector dip when s and s/km. What interpretation would you give the result?
First, we note that in the preceding equations is one-way time, so we take s, km/s, . To get , we find the angle of approach [see equation (4.2d)] for s/km:
The dip of the reflector equals , 50.9 in this case, because, in order for the ray to return to the source, it must have been incident on the reflector at right angles.
Because of the unusually steep dip, this event might be from a fault plane or it might represent a steeply dipping bed in an area of highly folded sediments.
If the ray continued without reflection, when and where would it emerge? What moveout would be observed at the recording point? Calculate the maximum depth of penetration.
[Equations (4.20a,b,c) could be used here instead of the following.]
Because the path is symmetrical about the midpoint, the angle of approach at the receiver equals .
From equations (4.17b,c) we have
Because of symmetry, the moveout on emergence would be s/km. To find the maximum depth of penetration , we use the fact that the ray is traveling horizontally at the depth , that is, . From part (c) we have , so . Equation (4.17f) now gives
|Weathering corrections and dip/depth calculations
|Head waves (refractions) and effect of hidden layer
|Partitioning at an interface
Also in this chapter
- Accuracy of normal-moveout calculations
- Dip, cross-dip, and angle of approach
- Relationship for a dipping bed
- Reflector dip in terms of traveltimes squared
- Second approximation for dip moveout
- Calculation of reflector depths and dips
- Plotting raypaths for primary and multiple reflections
- Effect of migration on plotted reflector locations
- Resolution of cross-dip
- Variation of reflection point with offset
- Functional fits for velocity-depth data
- Relation between average and rms velocities
- Vertical depth calculations using velocity functions
- Depth and dip calculations using velocity functions
- Weathering corrections and dip/depth calculations
- Using a velocity function linear with depth
- Head waves (refractions) and effect of hidden layer
- Interpretation of sonobuoy data
- Diving waves
- Linear increase in velocity above a refractor
- Time-distance curves for various situations
- Locating the bottom of a borehole
- Two-layer refraction problem