Transcript Seismic Methods Geoph 465/565 ERB 2104 Lecture 2 – Sept 6

Lee M. Liberty
Associate Research Professor
Boise State University
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P-wave and S-wave analysis of borehole data
Dataset is now collected – I will integrate into
this week’s lab (expect a longer lab). If you
want to get a head start, the shear wave
dataset is on kelvin
/data02/lml/Borehole_shear.sgy (6 channels)
Headers contain depth (offset)
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March 17-21, 2014
A mathematical way of combining
two signals to achieve a third,
modified signal. The signal we
record is a set of time series
superimposed upon each other.
Convolution
Seismograms are the result of a convolution between the source and the
subsurface reflectivity series (and also the receiver).
source wavelet
Mathematically, this is written as:
reflectivity series output series
seismogram = source  reflectivity ,
where the operator  denotes convolution.
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The convolution calculation in the time domain is slow
Convolution is more conveniently done using Fourier
transforms, F, since F{WR} = {F(W) ∙ F(R)}
We can calculate the convolution of two series by taking
the Fourier transforms of the series, multiplying them
together and then taking the inverse transform
Since Fourier transforms are so quick to compute, this is
much faster than doing the convolution itself
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The reversal of the convolution process.
By deconvolving the source wavelet, we can obtain the
earth's reflectivity.
However, noise (unwanted signal) and other features are
also present in the recorded trace and the source
wavelet is rarely known with any accuracy.
Convolution in the time domain is represented in the
frequency domain by a multiplying the amplitude spectra
and adding the phase spectra.
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Make reflections easier to interpret - ie more like the
"real" earth
◦ improve "spikiness" of arrivals
◦ decrease "ringing"
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But without decreasing signal relative to noise.
◦ This is one of the main problems
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A statistical
measure used
to compare
two signals as
a function of
the time shift
(lag) between
them.
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A signal is compared with itself
for a range of time shifts (lags)
 useful for detecting
repeating periods within signals
in the presence of noise.
The autocorrelation function is
often normalized so that its
maximum value at zero lag is 1
(where the signal is correlated
with itself).
The autocorrelation has a zerophase spectrum.
Both auto- and crosscorrelation functions are
required in the suppression of
multiple reflections by
predictive deconvolution.
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Deconvolution removes “cyclic” noise –
anything that repeats itself on a regular
basis
2 purposes:
◦ 1) sharpen wavelet and reduce reverberations –
SPIKING Deconvolution
◦ 2) remove long-period multiples (i.e. waterbottom multiples) – PREDICTIVE Deconvolution
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Airgun bubble pulse
◦ Period depends on gun size and pressure. Use multiple guns
synchronized to initial pulse to cancel bubble pulses.
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Water multiples
◦ Effect varies with water depth.
 For shallow water, multiples are strong but reduce quickly with depth.
 For deep water, multiple is below depth of main reflectors.
 For slope depths, effect is difficult to eliminate as first (strongest)
multiple arrives at main depth of interest.
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Peg-leg multiples
◦ Due to interbed multiples which can sometimes be misinterpreted
as primaries.
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The convolutional model is the basic assumption of
deconvolution:
◦ Trace = source * reverberations (noise) *
reflections (earth)
◦ G(t) = S(t) * N(t) * R(t)
“Spiking” deconvolution shapes the source wavelet.
“Predictive” deconvolution removes reverberations
and multiples, but leaves the wavelet mostly
untouched.
Deconvolution is implemented using a “leastsquares” approach to minimize the difference
between the “desired output” and the “actual
output”.
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Shortens the embedded wavelet and attempts
to make it as close as possible to a spike. The
frequency bandwidth of the data limits the
extent to which this is possible.
This is also called whitening deconvolution,
because it attempts to achieve a flat, or
white, spectrum.
This kind of deconvolution may result in
increased noise, particularly at high
frequencies.
BUT: we can shape the frequency spectrum of the
source to equalize the frequency components, thus
making the bandwidth closer to a “boxcar” function
original
amplitude
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after decon
source bandwidth
frequency
Location 109
Time (ms)
Location 109
NO DECONVOLUTION
DECONVOLVED
• Energy of the signal wavelet W is "front loaded“
◦ peak amplitude mainly occurs at the beginning of the signal.
◦ This results in a Fourier transform of the wavelet which has a
minimum phase
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If W is not minimum phase, then we cannot find the
operator D (ie W-1) to convert the signal W into a spike
(d) (Fig. 2.3-2)
Minimum phase thus becomes one of the basic
assumptions of seismic processing
Convolution Model
• S = W  R + N (noise)
• Five Main Assumptions
– #1: R is composed of horizontal layers of
constant velocity
– #2: W is composed of a compressional
plane wave at normal incidence which does
not change as it travels, ie is stationary
– #3: noise N = 0
– #4: R is random. There is no "pattern" to the
set of reflectors R
– #5: W is minimum phase
• Generally #3 is NOT valid
– ie. there will always be some noise on our
seismic records
– We will need to investigate what happens
when N ≠ 0
• We generally do not know W
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The filter attempts to shape the input seismic
trace x(t) into the desired output r(t) by
minimizing the mean-squared error between
the desired output and the actual filter output
y(t).
The actual output is simply the input x(t)
convolved with the filter f(t).
The least-squares error is
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Predict and eliminating multiple reflections
How does it work?
Design a filter that recognizes and eliminates
repetitions in the signal
Uses the autocorrelation to remove the
multiples.
Predictability means that the arrival of an
event can be predicted from knowledge of
earlier events.
INPUT
OUTPUT
*
FILTER
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In the convolutional model, one assumption is
that the reflectivity sequence (reflection
coefficients) are random. This means that the
autocorrelation of the seismic trace is the
same as the autocorrelation of the input
wavelet, scaled by the amplitude of the
reflectivity sequence.
A plot showing 100 random numbers with a "hidden"
sine function, and an autocorrelation of the series on
the bottom.
A measure of how well a signal matches a timeshifted version of itself, as a function of the amount
of time shift.
Short-period
reverberations
can also be
caused by bubble
oscillations in
airgun sources,
shallow water
layers, or thin
reflective layers
near the source
or receiver.
“pegleg” multiple
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Regularly-spaced cycles
“predictable” – given a model, we can
predict the times of the noise.
We can add the “predictable” noise
(reverberations, multiples) to our
convolutional model by convolving our
original source wavelet with a noise model
The convolutional model
Source
wavelet
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We can deconvolve the reverberations, as
long as we do not touch the original source
wavelet.
We can use “predictive”
deconvolution to remove the
minimum phase reverberations –
we are “predicting” the times and
amplitudes of the reverberations.
This is called “predictive error
filtering” when using least-squares
error method to implement it.
“Predictive”
deconvolution
Remove the
spike train
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The process by which seismic events (changes in
energy) are geometrically re-located in either
space or time to the location the event occurred
in the subsurface rather than the location that it
was recorded at the surface, thereby creating an
accurate image of the subsurface.
-Migration is dependent on
seismic velocity
-seismic profile (or gather)
length must include the
reflected energy
Point source
in subsurface
Diffraction on
zero offset
seismic
section with
velocity of
overlying
medium
t2 = 02 +4x2/V2RMS
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Migration steepens reflectors
Migration shortens reflectors
Migration moves reflectors updip
Danbom Geophy sics
Schematic that shows the imaging problem for a vertical fault.
In an un-migrated time section reflectors do not represent the true
subsurface geometry.
•See examples below…
(C)
(A)
From Kearey et al., 2002
Seafloor
Dipping
reflectors
Time section
(A) a syncline on the seafloor is
imaged as a “bow-time section
Bow-tie effect
(B)
Time section
Geological
Crosssection
(B) The addition of diffractions from
the end of reflectors results in a very
complex time section
(B) A dipping reflector is shallower in
a time section
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Unmigrated
image with
bowtie
reflectors
Migrated
image
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Original image
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Unmigrated image
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Wave-equation migration (computationally
expensive
◦ Spread every data point along the wavefront
◦ (Huygens Principal)
◦ Or downward continue the wavefront
 (usually finite Finite-difference)
 the most exact pursuit of migration – assuming the
use the ‘exact’ subsurface velocity field
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Kirchoff migration (intermediate speed)
◦ Sum diffraction curves
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F-k (Stolt) migration (fastest)
◦ Remapping in frequency-wavenumber domain
◦ assume constant velocity (or simple gradient)