Seismic reflection at a single interface

For a seismic ray to travel through a single layer, the seismic velocity at the  below (underlying) layer of Material must be lower than the above (overlying) layer. This is important in seismic reflection. Today I want to describe to you through this post on single layered seismic reflection, However before proceeding let's start with some terms that may be used in this description,

Terminologies

two way travel time (twt): Is the measure of time taken for a seismic wave to travel from a source into the ground and reflected back to the surface and detected by a receiver. See figure 1, below 

offset (x): Is the distance between seismic shot and receiver at which seismic travel.

zero - off set: It occurs when ray incident on a particular reflector at 90°, and causes it to become reflected back along the same travel path towards the receiver. Such that x = 0

TWT time along the normal incident path.

t = 2 x depth/ Velocity.

where t - Zero offset time / Normal incident TWT.

N.B: Practically, receiver and shot are always at the same distance (offset) from one another.

reflection time - distance plots (T - X)

Consider the figure 1, below where A is the source point (shot point). Geophones are spread out along the x - axis on either side of the shot point.

Figure 1: seismic Ray path from source to receiver.

Ray path from A - C or A - E, calculated using Pythagorus theorem as shown in ∆ABC in figure 1, 

AC = 2√(h2 + (x/2)2).................(1)

Travel time (t), is the Raypath (AC) divided by the velocity.

t = AC/V

 t = 2√(h2 + (x/2)2)/V.............(2)

After squaring both side the equation 2 can be re-written as :

V2t2 = x2 + 4h2................(3)

When we divide by 4h2 throughout the hyperbolic symmetric equation about the t - axis, is

V2t2/4h2 - x2/4h2 = 1...........(4)

Figure 2: Time (t) versus offset (x) hyperbolic curve.

Determine the velocity and thickness (h) from a plot of  t2 against x2.

As from equation 3.

t2 = x2/V2 + 4h2/V2..........(5)

Slope (m) of the line = 1/V2 .Velocity (V) = √(1/m). 

The y-intercept (c) gives depth (h) as 

y-intercept (c) = 4h2/V2

Then, depth (h) = √(cV2/4), since c = t02

Depth (h) = t0V/2

Also we can obtain h = √(c/4m), since 1/V2 = m

Objective: To determine the magnitude of correction required to flatten the hyperbolic events (Travel time curve) in seismic reflection.

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