Class 11: Continuation of
Reflection
UMass Lowell
Applied Geophysics
Fall 2016
Midterms
• Solutions are generally written on the exam if you’ve missed points
somewhere
• Mean score = 60.34
• All questions had at least one full-credit answer, some questions had
all correct answers
• Final exam questions are likely to be conceptual and will cover all
topics discussed
Lab Follow-up
• Unlike previous labs, you have an extra week to complete this one
(Happy Thanksgiving!)
• Field sheets will be posted very soon…
• PDF versions and .wcl versions of the files will be posted to the
website over the next couple of days
• A set of questions will be given to allow you to make some qualitative
interpretations of the results
Presentations
• Three presentations scheduled today
• 10 minutes to talk, 10 minutes for questions
• Everyone must ask two questions over the course of the next three
weeks!
Reflection (Continued): RECALL…
• Travel-time equation for reflection:
𝑡2
𝑥 2 + 4ℎ12
1 2 4ℎ12
=
= 2𝑥 + 2
𝑉12
𝑉1
𝑉1
which plots as a straight line on an 𝑥 2 − 𝑡 2 with a slope equal to
1
𝑉1 2
• Projecting this line back to zero gives 𝑡0 2
• Using the value of velocity V1 and t0, we can set x = 0 in the 𝑡 2 equation to get
2
4ℎ
1
𝑡0 2 =
𝑉1 2
so
𝑥 2 + 4ℎ12
1 2 4ℎ12
1 2
2
𝑡 =
= 2 𝑥 + 2 = 2 𝑥 + 𝑡0 2
2
𝑉1
𝑉1
𝑉1
𝑉1
• Taking the square root of both sides and rearranging to solve for h1 gives
ℎ1 =
𝑡0 𝑉1
2
Relationship between velocity, depth,
and curvature of the reflection hyperbola
• Holding the velocity constant and
changing the depth from shallow to
deeper, it is apparent that the
curvature is greatest for shallow
reflectors and least for deep reflectors
• Holding the depth constant and
increasing velocity from low to high, it
is apparent that the curvature is
greatest for slower velocities and least
for high velocities
• Curvature is dependent on travel path!
Normal Move-Out (NMO)
• NMO is defined as the
difference in reflection travel
times from a horizontal
reflecting surface due to the
variations in source-geophone
distance
• NMO behavior: NMO decreases
with increasing depth or velocity
and increases with increasing
source-geophone distance
Multiple Horizontal Interfaces
• To reach deeper interfaces, incident seismic energy must first refract
through shallower layers with bends toward and away from the normal
according to Snell’s Law
• In practice, we don’t have enough information to calculate this ray path
from a seismic reflection experiment
• Options to deal with this:
• Green Method – assume that source-receiver separation is small enough that you
can pretend the wave isn’t refracting from layers (angle of incidence = normal) and
create an 𝑥 2 − 𝑡 2 plot for each reflection that can be seen on a field record – each
line will give a velocity value, a thickness value, and a 𝑡0 value so we can obtain the
approximate model
• Dix equation – use the root-mean-squared velocity, 𝑉𝑟𝑚𝑠 , to relate travel times to
actual paths!
Dix Equation
• For the case where there are n horizontal beds and ∆𝑡𝑖 is ONE WAY vertical travel
time through the ith bed, the Dix equation is
𝑛
2
𝑉
𝑖=1 𝑖 ∆𝑡𝑖
2
𝑉𝑟𝑚𝑠 ≈
𝑛
𝑖=1 ∆𝑡𝑖
• Expand the equation for however many layers of interest and taking the square
root of both sides to get 𝑉𝑟𝑚𝑠
• This is still an approximation and won’t work for the case where source-receiver
distances are >> distances to reflecting interfaces
• Substituting in the 𝑉𝑟𝑚𝑠 for the velocity terms in the reflection travel time
equation gives
1 2
2
𝑡 = 2 𝑥 + 𝑡02
𝑉𝑟𝑚𝑠
2
so the slope of the 𝑥 2 − 𝑡 2 graph is now = 1/𝑉𝑟𝑚𝑠
Straight Line Path Assumption for Dix Eq.
Far from the source, the straight
line approximation (dashed lines)
falls apart – the angle of incidence
on V2-V3 interface is large enough
that the bend away from the
normal creates a path with a
significant departure from the
straight line approximation.
Steps to Using the Dix Equation
• Pick arrivals for the reflector(s)
• Plot travel time and distance on 𝑥 2 − 𝑡 2 graph
2 =0 axis to obtain a value for 𝑡 2 for the
• Project the resultant straight line(s) back
to
the
𝑥
0
reflector(s) – the square root of the 𝑡02 value(s) is the two-way vertical travel time for the
reflector(s)
• Calculate 𝑉𝑟𝑚𝑠 = 1/𝑠𝑙𝑜𝑝𝑒 of the line(s)
• Note that 𝑉𝑟𝑚𝑠 for reflector 1 = 𝑉1 because this path is actually a straight line
• Calculate 𝑉2 and 𝑉3 (for example) by starting with the Dix equation and expanding the
summation to write an expression for 𝑉𝑟𝑚𝑠 for the (𝑛 − 1)th reflector (see board)
• Substitute this back into the equation and rearrange to obtain an expression for 𝑉𝑛
• Recall that ∆𝑡𝑖 is one-way vertical travel time through bed 𝑖 and 𝑡0𝑛 is two-way vertical
travel time from reflector n
2
2
• Arrive
at
an
equation
for
𝑉
that
you
can
solve
from
observable
quantities
on
the
𝑥
−
𝑛
2
𝑡 graph: you can do this for any layer where you have reflections from the top and
bottom of the layer  𝑉𝑛 is called INTERVAL VELOCITY
Determining Thickness from Interval Velocity
• Once you’ve calculated interval velocity, 𝑉𝑛 , you can easily calculate
the layer thickness by using V = d/t where d = travel path distance and
t = travel time: rearranging,
ℎ𝑛 = 𝑉𝑛
𝑡0𝑛 −𝑡0𝑛−1
2
Assumption and a Rule of Thumb for Dix
Method
• The Dix method still relies on the assumption that data points will lie
on a straight line in an 𝑥 2 − 𝑡 2 graph
• This requires the assumption that the travel paths from the (n-1)th
layer to the nth layer are essentially identical except for the additional
travel in the layer between the two reflectors
• Actual data shows a deviation from straight line behavior that
becomes more apparent with increased distance from source to
receiver
• Rule of thumb: keep source-receiver offsets small relative to the
targeted depth of exploration!
Departure of Actual Arrival Times from Straight Travel-path
Assumption at Far Offsets
Calculated arrival times assuming straight travel path
Observed arrival times from field seismogram
Offset = ½ Depth
Travel path is close to a straight line
Offset = Depth
Travel times begin to deviate
toward end of array, but are fairly close
to straight travel-path assumption
Offset = 2 x Depth
Significant deviation from
straight path assumption
Approach to Deriving a Travel-time Equation
for Reflections from a Dipping Interface
• All discussions up until this point have focused on horizontal layers
• Just like with refraction, we now want to understand what happens to
the appearance of reflected arrivals when you’re dealing with a
dipping interface
• Luckily, we all understand image proofs now (right…?) so there’s an
easy approach that uses the Law of Cosines (as opposed to
Pythagorean Theorem):
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 (cos 𝐴)
• The end result for the travel-time curve is still a hyperbola in x-t
space, but it’s shifted off of the center!
Some test values…
Noteworthy Observations
• For the case of a horizontal layer, the axis of symmetry for the reflection hyperbola was parallel to the time
axis and the minimum was at the point x = 0
• This implied that the shortest travel path was the vertical path located beneath the energy source
• For the case of a dipping interface, the shortest path is the path is located somewhere up-dip of the shot
point and the minimum point on the hyperbola is offset up-dip from x = 0
• The opposite would be true for a negative dip!
• Note also that the arrival time is earlier for the minimum point on the hyperbola for the positive dip angle!
Determining the values of β and j from observation of the
reflected arrivals
• Differentiate the travel time equation with respect to x and set the results equal to zero to find the value of x
for which the time term is a minimum:
𝑥𝑚𝑖𝑛 = 2𝑗 sin 𝛽
2
• Plug this value into the travel time equation to find 𝑡𝑚𝑖𝑛
and take square root of both sides to get:
2𝑗 cos 𝛽
𝑡𝑚𝑖𝑛 =
𝑉
• Then find an expression for 𝑡0 by setting x = 0 in travel time equation to find that:
2𝑗
𝑡0 =
𝑉
• The quantities 𝑡𝑚𝑖𝑛 and 𝑡0 should be observable from the travel time curve
• Take the ratios of these quantities to solve for 𝛽:
2𝑗 cos 𝛽
𝑡𝑚𝑖𝑛
𝑡𝑚𝑖𝑛
𝑉
−1
=
= cos 𝛽 → 𝛽 = cos
2𝑗
𝑡0
𝑡0
𝑉
• Obtaining 𝑥𝑚𝑖𝑛 from observation, we can rearrange to calculate the value for j:
𝑥𝑚𝑖𝑛
𝑗=
2 sin 𝛽
• Finally, solve for h from the diagram and triangle geometry:
𝑗
ℎ=
cos 𝛽
Next time:
• Another way to determine dip, velocity, and thickness
• Survey setup and data collection
• Data processing/corrections
Final class:
• Data review
• Data processing
• Final exam questions/end of semester questions