INDEX to LECTURE NOTES
Sampling theorem
Linear vs. Angular Frequency
Phase shift
Fourier Theory
Fourier Analysis: Amplitude and Phase
DecibelsMECHANISMS THAT CHANGE SIGNAL STRENGTH
Geometric Spreading
Attenuation
Reflection Coefficients
Automatic Gain ControlMultiples
(1) Sea bottom multiples (long-path multiple)
(2) Internal multiples (peg-leg multiples)
(3) GhostsPROCESSING SEISMIC REFLECTION DATA
KEY WORDS:
Seismic resolution is the ability to distinguish separate features; the minimum distance between 2 features so that the two can be defined separately rather than as one. Normally we think of resolution in the vertical sense.
The seismic "measure" is a wavelength. In order for two nearby features to be seen well, they have to be about 1/4 wavelength in thickness (Rayleigh Criterion). With modeling you can improve resolution down to 1/8 wavelength.
e.g. Velocity = frequency x wavelength.
Assumption: seismic signal has one frequency and that seismic waves travel at one velocity
shallow earth: 2000 m/s, 50 Hz, l= 40 m
deep earth: 5000 m/s, 20 Hz, l= 250 m()
There is a practical limitation in generating high frequencies that can penetrate large depths.
The earth acts as a natural filter removing the higher frequencies more readily than the lower frequencies.
In effect the deeper the source of reflections, the lower the frequencies we can receive from those depths and therefore the lower resolution we appear to have from great depths such as the middle crust. Often we presume that the lower crust is more homogeneous but that can be a human perception borne by poor resolution.One could argue that we could simply increase the power of our source so that high frequencies could travel farther without being attenuated. However, larger power sources tend to produce lower frequencies.
(Figure 7.31, p. 218[ Sheriff, 1995 #1510])Vertical resolution decreases with the distance traveled (hence depth) by the ray because attenuation robs the signal of the higher frequency components more readily.
Horizontal Resolution and the First Fresnel Zone ([Yilmaz, 1988 #316] p. 470)
Lateral resolution refers to how close two reflecting points can be situated horizontally, and yet be recognized as two separate points rather than one.
If we only think of rays then we never have any problems with resolving the lateral extent of features because a ray is infinitely thin, has infinite frequencies, and can detect all changes.
(We will deal with the concept of rays versus waves later in the semester).However, when we deal with waves (reality) a reflection is not energy from just one point beneath us. A reflection is energy that bounces back at us from a region.
As waveforms are really non-planar, reflections from a surface are returned from over a region and over an interval of time. Signal that comes in at about the same time can not be separated into individual components. So, we see that reflections that can be considered as almost coincident in time at the receiver come from a region. The area that produces the reflection is known as the First Fresnel Zone: the reflecting zone in the subsurface insonified by the first quarter of a wavelength. If the wavelength is large then the zone over which the reflected returns come from is larger and the resolution is lower.
Horizontal resolution depends on the frequency and velocity.Equation: See handouts from AAPG Explorer and class handouts
KEY WORDS:
SEISMIC ACQUISITION GEOMETRY
Common-Midpoint Method
Normal moveout is the time difference between any point on a reflection hyperbola and its apex.Shotpoint Gather are the data collected in one shot and together are known as a shot-point gather.
Common Mid-Point Method (by W. Harry Mayne)
Now that we've seen the limitations involved with imaging subsurface data let's look at the geometrical techniques used in collecting field reflection data. In general, data collection always aims to collect data with the least amount of noise.
The cornerstone technique of collecting seismic reflecting data reducing the noise is known as the CMP/CDP method (common mid-point method is a better description, although CDP is most commonly used).
When we collect data we generate a seismic signal using for example, a water gun, an airgun, dynamite, a hammer etc. and collect the data with detectors.
Detectors at sea are known as hydrophones: pressure transducers: electronic equipment that converts mechanical enery (pressure) into electrical energy (voltage). Dual-sensor ocean-bottom cables combine the advantages of geophones and hydrophones.
On land the detectors are known as geophones; these convert vertical velocity ground motion into electrical energy. You can have differently oriented geophones to detect the various components of ground motion.
All data collected has some background noise such as:
cows walking across a paddock, instrumental self-generated noise, wind knocking blades of grass against geophones, rain, faulty connections, induced noise from nearby power sources. If the reflection data that you want to collect is stronger than the background noise there's no problem. But, what happens when the returning signal has traveled far, is attenuated of high frequencies and has suffered geometric spreading?
Harry Mayne thought that noise was on average random, so that if we stacked/ summed various copies of the reflections from one point, the noise would cancel out but the signal would add coherently.
We describe the ratio of wanted to unwanted data as the signal-to-noise-ratio:
If S/N =1 for x geophones,
S/N =4 times better for 16x geophones
So Mayne (1962) laid out various receivers/ geophones and shot to all of them simultaneously, then added the signal returning to all of them. The S/N ratio improves ideally as the square root of the number of geophones.
So, as we move along at fixed distances and fix shot time intervals, different receivers receive reflections from different receivers from the same position at depth. For a fixed shot all the data received and collected is known as a shotpoint gather.
Because we want information from the same position on the seafloor we re-order the traces according to where they wave source hits the seafloor. This reordering is known as a Common Mid-Point gather (of traces-) or more commonly, as a Common Depth-Point gather (of traces). With further accounting we can associate the source and receivers with the same mid-points. In the following picture, different shots with different receivers sample the same position on the seafloor. After three shots the distribution of CMP is:
Insert link to picture
In detail these curves and lines are composed of signal received from many adjacent detectors. The vertical axis is TWTT (two-way travel time (s) ) and the horizontal axis is the distance between the individual detector and the source. The data is received as a a series of sampled voltages and displayed with parts of the wiggle shaded in:
Insert link to picture
How are we going to sum the data if it's on a hyperbolic path? We can calculate a hyperbolic path and add the data along it, or we can move the traces up by an amount predicted by the hyperbola equation.
In practice, it is easier to shift the traces DTi and sum them across. We have to estimate the velocity of the medium and move the traces out until they are horizontal and we can sum or stack them:
Insert link to picture
How de we calculate this DTi ?
DTi is known as the normal moveout: variation of reflection arrival time because of variation in the shotpoint-to-geophone distance. Normal moveout is the time difference between any point on a reflection hyperbola and its apex.
Normal moveout is equivalent to moving your receiver over to the explosion , i.e. no offset. Otherwise, you'd have to take many shots in one location, but that takes a longer period of time.
KEY WORDS Sampling theorem
The theorem determines that you must sample your data at least twice per wavelength (for the frequency being considered)
Nyquist frequency is half the sampling frequency, or the frequency above which higher frequencies are aliased (or appear to be) lower frequencies because of insufficient sampling.
Nyquist frequency is 1/2 the sampling frequency:
Nf= 1/(2DSI), where SI is the sampling interval
Phase The angle of lag or lead in a sine wave with respect to a reference
Radian When the arc=radius, the angle subtended is 1 radian
Introduction
Because of the large amount of information theory developed at MIT in the 1950's and advent of digital computers, signal recordings are now done digitally (i.e., not continuous, but through discrete sampling at fixed time intervals.) Processing is done once the signals are stored digitally.
In order to carry out seismic processing we need to review some basic concepts in signal theory:
Although the data is collected at discrete intervals of time that of the order of milliseconds, there is a basin limitation to how the sampling is done.
The rule is that you must sample at least twice per wavelength (for the frequency being considered) if you are to be able to capture the frequency during sampling (example); this requirement stems from the Sampling theorem.
e.g. 125 Hz, or 125 oscillations per second. We must sample at least 2 times per oscillation i.e. 250 times /sec, i.e. 4 ms.
(Sampling rate is the frequency at which we sample seismic signal).
So, what happens with frequencies over 125 Hz? According to the Sampling Theorem
Nf + Df will be indistinguishable from Nf - Df
That is, a 175 Hz signal will appear like a (125-50) Hz signal.
That is, signals of frequency greater that the Nf will alias as lower frequencies.
So, it's critical when you are record data that you know which frequencies you want to keep and which you cannot trust. Once you know these facts you can design the sampling interval. Just to be on the safe side, and to make sure that you don't allow any aliased frequencies to come through, the cut-off frequency is sometimes made to be half that of the Nyquist frequency. For example, if a S.I. = 4ms and Nf = 125 Hz, electronic analog filters are put in place that will start to eliminate signal at about 62.5 Hz.
The Geometrics Strataview seismograph we will use in the field automatically adjusts its input filters so that the data is not aliased.
Why are all these theorems so necessary? How do you decide what frequency you want?
Well, that will depend on how much vertical resolution you need to have of the subsurface.
Linear frequency vs. angular frequency
When we used
V = l x frequency
we were calculating how fast a wave traveled during one oscillation
Now we are considering the behavior of a wave at a fixed point in space and are seeing how it changes through time. How do we relate sine of an angle to time? Since sine waves of a fixed frequency are repetitive every 360 ° we can relate time to the degrees, expressed as radians.
We do this by working out in degrees where we are in time along a repeating oscillation. We can
Insert diagram here
A(t) = A sin(degrees)
How often does the curve cross the 0-line?
In terms of degrees we can also express this periodicity:
At sin(0), sine (180 °), sine(360 °), sine (540°) etc. Because the sine wave repeats itself regularly we can use the regular repetition of degrees in a circle to express (in radians) the same conceptSo,
A(t) = A sin [(2p/ T) t]
= A sin [(w)t]This value in the brackets (w) is the angular frequency, related to just the normal frequency we thought of before.
A(t) = A sin[ (2p f) t ]
,where f is the more familiar linear frequency =1/T, and 1/T is expressed in Hz.
We can view angular frequency as how many times in a second a single-frequency wave goes through 2 p radians;
Insert link picture of a radian
For a plot of amplitude versus time we have:Insert link to picture here
(N.B. A wave in the simplest form is described as being composed of sines and cosine waves of a single frequency added together - Fourier Analysis)
The units of angular frequency are in radians per second
The frequency is expressed in terms of a cyclical radians for greater convenience. w means that the signal takes T seconds to cycle through 2p radians.
Now with angular frequency we measure how quickly an oscillation of a fixed frequency takes to cycle completely to its beginning point.
By using radians we choose to focus on the angular cyclicity of an oscillation instead of how far it travels along the horizontal.
"How far a wave has advanced in degrees, radians distance with respect to some reference point."
Include link to Figure here
We have displaced the second wave forward in time, or back in time (remember wave oscillates for all times and angles)
As a result of moving this wave forward, the + is now a minus. The second wave is equal to the first wave. How far has the second wave moved forward? p radians (or 180 °)
In dealing with sea surface reflections: we found that waves reversed their polarity. That's the same as saying that their phase was shifted forward 180 ° or backward 180 °.
What if we shift a sine wave 90° degrees backward?
How far forward do you need to shift it to make a sine into a cosine? Sines and cosine functions are phase-shifted versions of each other.
Let's now express the phase angle mathematically:
A(t) = A sin [(2p/ T)t - phase angle]
If the phase is 45°, then
A(t) = A sin [(2p/T)t - p/4]
For example, let angular frequency = 1 and t = phase angle, then:
sin[(2p/ T)t - p/4] = 1
So, in order to move a signal forward in time, you have to make it arrive earlier, and to do this you have to -ve phase shift and vice versa.
Introduction
As I said before, discretization of the waveform allows us to analyze seismic data using the digital computer and he key aim of all processing is to enhance the geological signal over the background noise. This might mean
(1) filtering unwanted background noise, or
(2) designing the best source possible.
The digital computer also allows us to filter the data once it has been collected, but in order to achieve this we must decompose the complex signal into its simpler parts. This can be done with the help of a procedure known as Fourier Theory. With Fourier Theory we can describe waves by breaking them down into frequency and phase components.
There is a theorem (thanks to an illustrious dead French Mathematician Monsieur Fourier) that a long continuous signal (sufficiently long), no matter how complex looking, can always be decomposed into fundamental frequencies.
We achieve this transformation working on signals that have been collected as a function of time but will be expressed as a function of frequency. This transformation views complex signals as the summation of simple single-frequency signals. The Principle of Superposition assumes that these signals can be added up independently of each other.
If we are working on combining the simpler components of a signal into an end result that looks like the signal we collect and interpret we call this a Fourier synthesis. However, if we are decomposing a signal into its simpler components we call this a Fourier analysis.
Not only is it mechanistically simpler to break down the complex parts of a signal into smaller parts but it becomes a lot faster to do the arithmetic on a computer using a very clever algorithm known as the Fast Fourier Transform or FFT. We can also filter in the time domain we would need to carry out more individual steps which include sums and multiplications whereas in the frequency domain we need only to multiply.
Applications: Filtering with Fourier
How can Fourier transforms remove unwanted data/noise?
Filtering a frequency out is a complicated procedure known mathematically as a convolution. However, in the frequency domain, filtering is accomplished by multiplication.
Let's see how we would represent a single-frequency signal in the frequency domain:
click here
If we had a more complicated signal:
click here
Now you can also see a most complicated signal, such as a seismic source:
click here
Fourier analysis would allow the removal of certain frequencies by multiplication of 0 in the
frequency domain.
Applications: Spikes and Fourier
How can Fourier transforms help us design the optimal source?
What is the ideal source that we can use to best resolve subsurface features?
It is not sufficient to just have high frequencies. The ideal source must be rich in ALL frequencies. The more frequencies that are added in the narrower the appearance of the source wavelet.A spike. But what does a spike look like in the frequency domain? What's the dominant frequency?
The best signal is one that has energy at all frequencies ("rich in frequencies"), or one that has a white spectrum:
Intuitively, one might guess that a single ultra-high frequency signal would have the greatest resolving power, because resolution depends on the wavelength and the smaller the wavelength , the greater the frequency?
However, a single-frequency signal with many sinusoids makes a very long wavelet and decreases the resolution. The best source is very short inthe time domain inorder to maximize the resolution. Note that we are not contradicting ourselves because when we explained resolution previously we spoke of dominant frequency. Dominant frequency is the nominal frequency with carrying greates energy in a wavelet. We single out dominant frequencies in simple calculations to get a practical sense of the resolving power of the wavelet. But, remember that the wavelet must be very short. While every reflector in the subsurface is unique and theoretical very thin, a reflector will produce as many returns as there are pulses in the downgoing wavelet. That is for every single interface we will create several pulses in our seismogram making interpretation ever more difficult.
Insert link to drawing here
What is the ideal source that we can use to best resolve subsurface features? How can we shape the signal to make it narrower? Add more frequencies. In the limit, an infinitely thin signal (known as a spike) is rich in all frequencies at no phase. This comes from Fourier synthesis.
If we analyze what happens when we add (superposition), we'll find that the more frequencies we add the narrower the signal.
Fourier Analysis: Amplitude and Phase
However, describing a waveform in the time domain by using the equivalent in the frequency domain is not sufficient because we are losing the phase information, or when the signal first appears.
Insert link to drawing here
The same mathematics that allows us to break the signal up into its component frequencies also allows us to break it into phases. We express the phase relationships on a phase diagram:
In Figure 1 of Yilmaz's book we see three examples of sinusoids (sine waves with different phases, i,e, also considered cosine waves with different complementary phase values). As the sinusoid advances in time the phase has a more negative value, verifying what we saw previously
Applications of Phase: Constant and linear phase shifts
Interpreters sometimes are more comfortable with data whose phase has been shifted to 0 phase. This process is known as wavelet shaping and as a result the positive-negative couplets are reduced to single peaks. This procedure requres that we apply a constant phase shift to all frequencies in the data. We can see from the following example the effect of applying different phase shifts on the shape of the wavelet:
Yilmaz has several examples in his book: click here,
Whereas a constant phase shift will create a change in the signal shape a linear shift in the phase will move the signal in time but not change its shape : click here, click here
If you want to succeed in change both the shape and arrival time of your wavelets you can combine a linear phase shift with a constant phase shift
Decibels:
db = ratio of powers: 10 log (P1/P2)
e.g. P1 =100, P2 = 10 then,P1/P2 = 10
log1010 = 1
db = 10
e.g., P1 = 4, P2 = 1 then,
P1/P2 = 4
log104 = 0.6
10 log104 = 6
octave = f, 2f, 4f, 8f
Rays as simplified wave descriptors
Advantages of Rays
I will often analyse a reflection seismology problem in terms of rays although sounds are more accurately described as travelling as waves. Rays are easier to visualise than wavefronts. Rays are ideal and have an infinite frequency content so they can image all objects regardless of how small they are. That is to say that a ray or a line of light or sound has only one dimension and therefore is always smaller than any object on which it impinges. This is a theoretical approximation only works if we have waves with a frequency content that were infinitely high.
Rays are one common way to simplify reflection seismology problems.
(Assuming flat wavefronts is another)
Limitation of RaysHowever, rays do not explain diffraction.
In an isotropic medium, I have said, seismic wave travels in a direction normal to the wavefronts and it is convenient to represent its normal as a seismic ray:
A drawing of ray-path vs. wave-path example
We know the limitations of rays in an intuitive way borrowed from optics. The term ray is adopted from optics. In optics we know that this concept of ray (or pencil of light) will travel around an object and not only travel on a straight path. When???
When the size of the object is very small in comparison to the wavelength then the object will cause diffractions
Figures of diffractions here!
So, rays are a useful concept when the size of the object we are trying to examine is several times the size of the wavelength. Rays can not predict wavefront paths if the obstacle is much smaller than the wavelength. When this situation occurs, point obstacles (a relative size problem) scatter energy in all directions (diffraction) and full wave theory is needed to explain the wavepathas and amplitudes, which are not the same in all directions.
Rays do not work well at discontinuities.
We can use rays to simulate a diffraction if we apply Huygenís principle to a point reflector and consider it to radiate energy in all directions (a later class dealing with a demonstration of Snell's Law)
Interpreting an offset versus traveltime diagram
As we have seen during applicatin of bandpass filters there is a need to identify the type of seismic arrival in order to distinguish noise or bad data from the data which we sish to keep (signal)
Let's begin by examining a typical offset versus traveltime data set and identifying each of the key arrivals
as follows:
(N.B. that the way we disply data in class is to commonly place the increasing TWTT(s) pointing down whereas the earthquake seismological community may do the contrary. (click here for such an example)
There are several arrival geometries:
Direct wave : The direct wave (equation) is a good arrival to re-examine the concept of destructive interference because beyond a certain distance from the source the D-wave or direct wave dies out. The reason for this is that the 180 degree phase-shifted reflection from the air-water interface interferes destructively with the ray travelling directly (D-wave) from the source to each of the receivers.click here for diagram of a D-wave components as a funciton of source-receiver distance
Hyperbola
Single-layer reflections in offset-traveltime plots can be consideredto have the geometry of a hyperbola. In the parametric case shown above a and b are geophysical parameters:
The horizontal axis the source-receiver offset in meters and the vertical y axis is the TWTT(s). In this formula we will show later thatb= -1/(4h2) and a = V2/4h2,
where h is the depth to the reflector and V is the root-mean-square average (VRMS) velocity to the reflector.
Pre-critical arrivalsPrecritical reflections:
Post-critical arrivals
Refractions:
The critical distance is calculated from the following diagrm. One defintion for the simple case of a single-layer refractor is that it is the distance at which the refracted rays begin arriving before the reflected rays. This distance is important in some situation because the reflected energy near the critical distance is phase-distorted. Phase-distortion is both useful and harmful. It is useful because the new wave shape can be interperted with sophisticated methods to determine the presence of fluids below the first layer. It is harmful because if we stack the new wave shapes near the critical distance with wave shapes that are almost undisturbed at smaller source-receiver offsets then the differences may lead to destructive interference and a decrease in the signal quality. Hence it is important that we be able to calculate this critical distance at least for a simple case:
There are more sophisticated cases where more than one layer is consideredbut we will not derive them in class and I will leave them for a later homework exercise.
Head Waves
Head waves are the arrivals that appear on the data as post-critical refractions. One model of the earth can view these seismic data as rays which travel along the interface (critically refracted rays) and which generate rays returning to the surface at the same velocity. While the geomtery of these arrivals can explain the perceived velocity on the receiver offset versus traveltime displays as we said before these rays do not exist. What really exists are waves which refract near the interface and generate what are known as whispering gallery waves. These type sof waves are beyond the immediate scope of this course.
Reflection Hyperbola
Let's demonstrate that the equation for a reflection from the bottom of a single-layer is perceived as a series of seismic wavelets that line up along the branch of a theoretical hyperbola. :Let's first start by considering the following diagram:
Lets' now derive the equation
MECHANISMS THAT CHANGE SIGNAL SIZE AND SHAPE
Let's examine the reasons for which a signal changes shape and size as it travels through the earth along different paths and distances. Let's do this in a more systematic way than we have thus far. The signal as well as the time of arrival at a certain distance from the source will contain valuable subsurface information. The causes for amplitude change are several as we may have mentioned. They are, namely:
(1) Geometric Spreading
(1a) Processing techniques to compensate for Geometric Spreading (read Sect. 1.5, Yilmaz)(2) Attenuation
(3) Reflection Coefficients
(1) Geometric Spreading (Spherical divergence)
As we move away from an energy source and make recordings of the strength of the signal, the intensity diminishes. This does not only occur because individual frequencies are being filtered out by natural attenuation processes in the earth but because the energy that initially was concentrated in a very small volume around the seismic chargetravels as a wave and distributes itself over a wavefront that increases in size in all directions.
Normally, the effects of spherical divergence can be corrected during the seismic processing by taking into account the distance travelled by each part of the seismic trace. Distances can be calculated using velocity and traveltime information.
Geomteric spreading makes the amplitude of a signal falls off in proportion to the distance traveled by the ray. So that if the path of flight is doubled the amplitude will decrease by a factor of: square root of 2.
In spherical divergence energy, all the energy that leaves the source arrives at the receiver. However, with attenuation some of the energy is absorbed by the medium. Why? Should this occur in a perfectly elastic medium?
The inelastic behavior is called attenuation. In the ball and spring model, some of the bounce has been converted into heat.
Attenuation:
(1) increases with the distance (r) traveled
(2) affects higher frequency more readily (i.e. high frequency is absorbed first)
A (r) = A0 e raised to the exponent [-[rp/(lQ)]
r is the distance traveled, Q is the quality factor, and p/(lQ)is the absorption coefficient
1/Q is proportional to the decrease in amplitude a wave experiences each wavelength as it travels through a given material at a given wavelength and speed.
Q is about 1000 for low attenuation in rocks
and about 30-500 in sediments
This means that in hard sediments waves can travel three times as far before they see a decrease of their amplitude by the same amount as in soft sediments. Although Q is smaller at higher frequencies (i.e. greater attenuation at higher frequencies), for the range of frequencies we commonly experience is reflection seismology we can consider Q to be the same.
In general when a wave arrives at a surface separating two media having different elastic properties, it gives rise to reflected and refracted waves. We have seen that an incident P-wave can produce 4 other waves if the angle of incidence at the boundary is not 90 degrees. Because energy must be conserved:
EP` = ( EP`P` + EP`P´) + ( E P`S` + EP`S´)
E = 1/2 w2r A2
Through this condition, we can determine the relationship between the amplitudes of a normally incident P-wave. i.e. by looking at the conservation of this "thing" called energy.
Eventually we arrive at an estimation of how much energy comes out for the energy that goes in. In the case of normally incident P waves (reflection case) this ratio reduces to a ratio of amplitudes before and after reflection is the reflection coefficient.
Reflection coefficients in conditions of normal incidenceWe start by looking at the simplest case when the angle of incidence is 90 degrees :
EP`/EP`P´
We know that :
EP` = EP`P` + EP`P´
Illustrate this energy point with a drawing
The two key physical properties responsible for producing a reflection at normal incidence are the velocity and the density of the medium (I'm not going to prove this). These values multiplied together are known as the acoustic impedance. If a velocity does not change across a boundary there won't be refraction but there will be a reflection because the acoustic impedance has changed.
At normal incidence, whatever does not get reflected gets transmitted:
Reflection coefficient
AP`P´ / AP´ = difference in the acoustic impedance
sum of the acoustic impedanceAP`P´ / AP´ = (r2V2 - r1V1) / (r2V2 +r1V1)
(ratio of reflected to incident amplitude)
Transmission coefficient:
AP`P` / AP` = 2 (r1V1)/ (r2V2 +r1V1)
(ratio of transmitted to incident amplitude at normal incidence)
These equations are derived from energy relations that reduce to amplitude relations by the square root of the energy. They are very useful even up to angles of reflection of about 20 degrees.
Link to image of reflection coefficients
(and Table 3.1 Sheriff )
Example of sea water/air interface and
calculate reflection coefficients.
Let's calculate reflection coefficients so that they are negative in a solid rock example
Summary of all the above effects
If we combine all of the above effects we can semi-quantitatively express the total control of these factors on the amplitude:Final amplitude at receiver = attenuation (r, frequency) x Geometric Spreading (r) x Reflection Coefficient
For the range of frequencies we will encounter, losses by attenuation are insignificant by comparison to the effects of geometric spreading.
Reflection Coefficients when the angle of incidence is not a right angle, i.e., as a function of offset (AVO)
I'd like to show you what happens to the signal energy if you are not in conditions of normal incidence.
Figure 3.2(b) from Sheriff, P. 78
Other types of waves, known as shear waves start to appear. So from an incident P wave, you can produce reflecting P and S waves as well as refracting P and S waves. Please refer to the section on Snell's law where we discuss this in more detail.
Reflection coefficients are far more complicated and depend on the Vs, density and VP. Why? Remember that at non-normal incidence you will begin to create S waves.I'd like to show you what happens to the signal energy if you are not at normal incidence conditions. Reflection coefficients are far more complicated and depend on the Vs, density and VP. Why? Remember that at non-normal incidence you will begin to create S waves.
In basic physics classes you probably envountered the term critical reflection. At the critical angle of reflection all the enrgy that hits a boundary is reflected backA rule of thumb used in the early days of estimating fluid content according to the patterns in the Amplitude versus offset for a given reflector was that increasing reflection coefficients values with offset revealed a higher Poissonís ratio in the layer below the origin of the reflection and that this higher Poissonís ratio implied gas. The first part of the argument can be true but Poissonís ratio is sensitive to even a small amount of gas so that the amount of gas cannot be readily ascertained.
Fig. 3.4 from Sheriff, p. 80
Signals lose strength by ..... transmissivity losses, geometric spreading and attenuation. They could be counteracted exactly if we knew the velocity of the medium and its non-elastic properties. However, we don't always know that (We could guess, we have for many other problems!)
The Automatic Gain Control technique is cosmetic, visual, improves the look, and tries to counteract the effects of amplitude decay in a rather artificial way as well as improve the structural continuity and stabilize calculations.
(If there are large numbers in the data it tries to suppress them and obtain a smoother continuous set of amplitudes-- Some calculations cannot handle very small or very large numbers, it's a mathematical limitation)
There are many ways of controlling the amplitude: Two of the most common are: fixed time gate and sliding time gate
1. Fixed Time Gate
1. A signal is divided into fixed intervals, or windows or gates. They can be of any size.
2. Within each gate, we square the amplitudes, average the square and find the square root:
known as the ARMS. Then we
3. The we say that the RMS has to be a certain value: e.g. 2000, and find the factor
by which the ARMS has to be multiplied, in order to obtain 2000.
4. The value is applied to the center of the gate and interpolated between them:
2. Sliding Time Gate
In a sliding time gate a similar process is carried out, but the value is assigned to any sample, and the calculation is done by sliding the window over one sample at a time. If the window you use is small it has the effect of boosting small amplitudes and deteriorating the signal character
NOISE
Multiples
Seismic energy which has been reflected more than once is known as a multiple. An unknown amount of energy received at geophones and hydrophones may actually be multiple energy. Recall that every time a ray crosses a reflecting boundary on the way up and on the way down there will be a reflection generating in principle a myriad of possible reflections.
Multiple noise only really becomes a problem when it arrives coincident with other important information.
Let's look at different types of multiples:
(1) Sea bottom multiples (long-path multiple)We can examine sea-bottom multiples in CDP/CMP/SP gathers as well as the stacked sections
Include hand-outsMultiples in seismic gathers
Sea-bottom multiples can also be seen as deeper reflections, but which travel at lower velocities than a true reflection from those depths
Handout of seafloor multiples
A deeper reflection will produce a flatter hyperbola. The hyperbola will be asymptotic to the D-wave.
However, how would a true reflection from the same depth compare? Remember that if we assume that velocity generally increases with depth, the real hyperbola will cut the extension of the D-wave
Multiples in stacked seismic sections
Multiples are best distinguished when there is a slight dip to the seafloor. The dip of the multiples increases with the order of the multiple.
(2) Internal multiples (peg-leg multiples)
-- When both long and short-path multiples reflect repeatedly within a layer beneath the sea surface or land surfaceInternal multiples can be determined in a seismic normal-incident profile (staked seismic section) by looking for pairs of strong reflectors and then trying to predict later arrivals separated by the time it takes to bounce between these two reflectors.
(3) Ghosts
In marine seismic data, ghost energy forms part of the signal return from each reflector. The ghosts are a follow-on sea surface reflection. They can not be eliminated; only minimized. If the depth to both the receivers and shots are known, then you can filter out the signal. The total signal comprises four components:Include ODP drawing here.
Example: Direct wave at sea is destroyed by a ghost reflection
The D-wave dies at sea at a certain distance of a few hundred meters.
Q. At what distance is the difference in arrival time smaller than the sampling interval ("window of observation") of about 4 ms? Assume that the hydrophone and a point source are both at 3 meters below the surface of the water.
PROCESSING SEISMIC REFLECTION DATA
Definitions
There are three velocities that we will see in reflection seismology processing:
VINT Velocity of a subsurface layer determined from travel time through the layer of known thickness but in the context of today's class it is derived using the Vrms and TWTT(s) to the top and bottom of the layer using the Dix equationBut, for real cases we tend to use a type of average velocity that fulfills the equation for a hyperbola VRMS (another type of average velocity)g
VRMS is the root mean square velocity. It is a type of weighted average velocity. Weighted means that the thicker the layer in which the ray travels, the greater the contribution to the final estimated VRMS
VSTACK Velocity used for stacking data calculated from the best-fit hyperbola to gather data through any of the techniques that follow.
Click here for a picture of NMO and stacking and here for tutorials on velocity analysis: by former students and those used for former labsThe equation for a hyperbola for the two layer case uses VRMS and the hyperbolic approximation is a good model to deal with reflections, at least initially.
If we know the VRMS down to the bottom of a layer and down to bottom of the previous layer, we can estimate the individual interval velocity of the last layer:
(Place the Dix formula here)
Remember, that in order for this equation to be true, you must have conditions of near-normal reflections.
When stacking data we use a velocity which is statistically derived. We assume that the best stacking velocity gives us the highest amplitude signal in the data, and we can derive it in several ways:
1. Constant Velocity AnalysisThrough a trial a error we plot move out the traces at different times with a constant velocity (a Vrms)
Click here to see the effects of using different stacking velocities (too much and too little)
2. Semblance Analysis
Measure the similarity of signals across a CDP gather and is expressed as:
the sum of energy across traces over an interval of time Ö normalized to Ö
the sum of energies in each trace
As the semblance increases across a gather, the stacking improves, and the output signal preserves all the summed energy. Think that an ideal signal will simply amplify the input:
This is done on a trial and error basis too.
click here for drawing of Semblance Analysis
V stack is a cosmetic (looks good) best-fit velocity and it's derived assuming signals do not change shape with offset
V stack is distorted by processes that change the signal:
e.g. NMO stretch and the need for NMO stretch mute (click here for and exa,mple)
We know that the deeper a reflection the flatter the hyperbola, and the faster the velocity the flatter the hyperbola. Therefore if we try to correct for a shallow reflector, we'll over correct for the deeper reflector. We therefore have to apply a different correction at different depths. However, because we only have a finite number of samples, where many hyperbola cross the data the signal will become artificially stretched, essentially filtered to let only low-frequency signals. Remember that deep signals will have lower frequencies and therefore we may have lower resolution.
Let's calculate this difference and see what sort of error, using the equation for a hyperbola.e.g., near-critical phase changes
e.g. reflected signals can overlap
e.g., there's also random and coherent noise
e.g. we assume that the signal to noise ratio is the same in each trace and down each trace
We've seen how this is done for single-layer cases. How is it done for multiple-layer cases?In case you haven't noticed, the equation that we have obtained for a hyperbola has always been the case for a single layer. If you were to develop the exact form of a reflection for a double layer, it would be a hideous exercise, because you'd have to take into account 2 or more layers with a different thickness and a different velocity.
Multiples will stack at lower velocities than expected, and on average the signal/noise ration will improve.
Improving the signal-to-noise ration means adding signal from the same point.
Process by which dipping reflections are plotted in their true spatial positions rather than directly beneath the half-way point between source and receiver.
Why do we use migration?Because the assumption of horizontal layers has been violated. Unmigrated sections give a distorted picture of the subsurface, the distortion increasing with the amount of dip and structural complexity.
When do we use it ?1. When we want to remove diffractions:
See figure on diffractions
2. When we want to restore the true dip and position of reflectors in order to make stacked seismic sections more interpretable.
See figure before and after migration
What are the effects of migration?
1. Migration steepens reflectors .
The dip angle of the reflector in a unmigrated section is greater than in the migrated section. Migration moves reflectors in the updip direction.
2. migration shortens reflectors. The length of the reflector is shorter in the migrated section than in the unmigrated section.
Semi-circle Superposition Method of Migration
One way of examining these consequences is to consider the very simple case of a constant-velocity medium. This method is known as the semi-circle superposition method of migration. The true position of the reflector will be along a line that is tangent to all the semicircles. These semicircles represent the wavefront that was generated by an incident plane wave. We swing arcs whose radius is equal to the traveltime
LimitationWhen using migration one important point to remember is that toward the edge of your profile your migration cannot be complete. When you restore reflections to heir proper location you need to restore enough of them so that interfere constructively to produce the true reflector. If, however, you only have a partial set of reflections to migrate you can produce artificial
Insert figure
Application of migration to very complex geology
If your geology is not too complex and you want your results in time you can use the migration technique in the time domain. (time migration)
If your geology is complex and there are lateral variations in velocity (not a horizontally layered case) then you'll want a depth-migration technique. Time domain based algorithms are simply not accurate enough. You'll have to shoot rays through your model to do depth migration.
If we migrate before stacking it's because so far we've always considered that reflectors and receivers are coincident. This effect was achieved during NMO which implicitly assumed no geological dip (POSSIBLY A BAD ASSUMPTION). So, instead of accumulating errors why don't we just migrate before the stacking? (pre-stack depth migration)
Spatial Aliasingreflection seismology attempts to remove noise. A common noise type is streamer noise which comes in producing lines across seismic sections. Or, refracated energy? How do we remove this? All we have is that the noise is linear, has a certain slope. How canwe use this information to remove the noise? When we study this problem we'll also find that if the processing and acquisistion has not been done correctly misleading stratigraphy may be introduced into the seismic section.
Just as frequencies are aliased or mistaken if the sampling in time is not fine enough, dipping events can aslo be aliased if we do not have enough samples of a particular geometry.
Insert figure
How can we avoid ambiguity of dipping events. Add more traces.
What's the maximum shift (dip) that can take place between traces before you produce an ambiguous interpretation? 1/2 cycle per trace. How can we remove this ambiguity? Use a lower frequency. Add more traces.
This problem, known as spatial aliasing leads in to a type of frequency we have not looked at before. We know of the temporal frequency (oscillations per second).
Spatial frequency is trhe number of oscillations per second, or wavenumber
1/l= k (wavenumber)
Now we want to know the number of oscillations over a given distance. (spatial frequency) or cycles/km or wavenumber. That is first you convert the field into frequencies (frequency spectra)
let's look at the example of spatial aliasing in the overhead:
Insert overhead
for a 36 hz wavelet, if you have more that 14 ms shift per trace you will produce aliasing.
Or the same signal will be confused as a negative and a positive slope!
To eliminate aliasisng you can do two things:
(1) use more traces or decrease trace separation.
(2) lower frequency (but you lose resolution)
Nk= 1/2. trace interval
Nwavenumber + Dwavenumber = -(Nwavenumber + Dwavenumber)
Analogous but not exactly, when you see old motionpictures of stage-coach pursuits, the wheels seem to spin backward, because there are not enough frames per second!
e.g. for a 36 Hz wavelet, the T = 28 ms. Therefore shift must not exceed 14 ms/trace.
It's a convenient way to look at this data because we can represent it in a way where one
dip becomes a point in a two-dimensional plot. If we could delete the dips of no interest and reverse the process we should be able to remove unwanted dipping noise.
Let's now look in a little more detail at how we can remove the dips:
In a frequency wavenumber plot the more to the right we go, the greater the dip, until we reach a point where the dip is so great we actually begin to see a negative dip (negative k).
Insert figure
let's see how we could use this to eliminate real noise:
Insert figure
For a fixed frequencyThe greatest energy would lie where several arrivals have the same slopes. In f-k space, these slopes would appear as :
Insert figure
Differences between Reflection and Earthquake seismology
Letís examine the scope of each field in order to understand the differences and similarities a little better.
Seismology is literally the study of earthquakes (i.e. seismos comes from the Greek word meaning earthquake and logo means science, also derived from the Greek). So, what is seismology and whatís the difference between earthquake seismology and reflection seismology?
1) Causes of waves/propagating vibrations generated by Earthquake & Reflection Seismology
In Earthquake Seismology :
The uppermost part of the earth is divided into a rigid shell about 100 km thick that comprises both rigid mantle and rigid crust. This shell (known as the lithosphere) is broken into say a dozen small plates, which do not coincide with the continents.
As these plates move with respect to each other, you can imagine large stresses build up at the edges where these plates meet. Plates collide at a velocity of 2-10 cm/year.
image of plates in motion and plate boundary earthquakes
It is obvious in this simple description that there can be a significant build up of stress as a result of plate movement. Rocks can accumulate stress to a small degree but when the stresses exceed the rock strength the rock fractures along a plane of weakness (fault ).
When the rocks rebound back to their original position, or close to it, the mapping motion shakes creating a vibration that propagates in all directions.
In reflection seismology we tend to use artificially induced vibrations created by a variety of seismic sources:
Images of
(2) bobcat-mounted auger and shotgun (San Juan Basin) and
(2) vibrating trucks
Thereís a tradeoff between energy released by a seismic source and the dominant frequency. High frequency sources require less energy to produce than low frequency sources. But, high frequency sources are more easily attenuated with depth.
2) Types of Waves studied by Earthquake Seismology vs. Reflection Seismology
In Earthquake seismology we want to record all the different types of waves created by earthquakes.
Essentially there are two types of seismic waves, namely body waves that propagate through the earthís interior and surface waves that propagate along discontinuities e.g., surface of the earth, bottom of the ocean, between geological layers.
The vibrations that are set up naturally will use either shearing strain or dilation or compressions. They are the two types of deformation rods experience.
If, for example, the medium can not maintain the type of deformation, e.g. water canít maintain shearing, then certain vibrations will not be set up.
That is, if we had a homogeneous earth with no sharp changes anywhere at any scale, we could only detect body waves.
Surface waves are usually the primary cause of destruction and carry the greatest amount of energy from shallow sources.Body waves move more quickly than surface waves. There are two types of body waves. (This was postulated last century by S. D. Poisson, in 1829.) There are P waves and S waves.
In P waves, the vibrations in the rods are in the direction of propagation of the wave. Note that the rod moves back and forth, the vibration is carried out.
P-stands for primary waves. P waves are about 1.7 times faster than S waves.
S waves, on the other hand, incur a vibration which is at right angles to the direction in which the wave is propagated. S-stands for secondary or shear.
The second group of waves depends on the existence of geological layers. This is mathematically demonstrable complex and beyond the realm of this course.
I will limit myself here by saying that as far as surface waves are concerned we have Love waves and Rayleigh waves. Love and Rayleigh waves are noise for reflection seismology. Love waves are shearing horizontal waves. Rayleigh waves, named after the British mathematician, produce retrograde elliptical motion.
In Reflection Seismology, we are concerned with only P waves ( S waves are not generated for conditions of normal incidence because none of the motion across an interface can be converted into shearing motion.
VP = 1.7 VS
VR = 0.92 VS
VS1
VS1 < VL < VS2 --------------------
VS2
e.g., P waves 1,500 m/s in sea water at 0°C
P waves 6,500 m/s in granite
P waves 8100 m/s in upper mantle (peridotite)
P waves ;4,500 m/s
S waves 0 m/s in water
3) Differences in scale between Earthquake and Reflection Seismology
In general there is a difference in scale between Reflection and Earthquake Seismology.
In Earthquake Seismology, we have very large energy sources and we can study the earth at the scale of 10,000km.In Reflection Seismology, we tend to look at reflections from the Moho (about 30 km near sea level under continents).
4) Mathematical representations of wave propagation
And finally in Earthquake Seismology we tend to use more complex mathematical representations because at the scale of work, the earth is round whereas in Reflection Seismology it is approximately flat.
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Key Words
Amplitude, Frequency and Wavelength are very important DESCRIPTORS of the behavior of ground motion. We see that they can help characterize the energy of the motion. Waves: perturbation/vibration propagated through a medium Amplitude--how far the ground oscillates up and down, about the center or line of reference. (units: m, volts)
Frequency (f) Wavelength ( l ) "lambda" velocity V= l * f Stress: force/unit area Elasticity Strain Stress |
WAVESWaves are a type of kinetic energy or energy of motion.What is energy ? In physics today we have no knowledge of what energy is, but we know how to calculate its conservation (Feynman Lectures, Vol. 1, 1963). We can calculate energy, have laws for it (Conservation of Energy) but it doesnít tell us why things happen.
e.g. imagine you experienced an earthquake. The ground moved under your feet - up and down and you could record the elevation of the floor at each moment ( in m ). A moment of greater amplitude conveys a sense of greater motion--greater energy. We can derive that the energy associated with the motion of the medium at any point depends on
w2 (faster motion; greater energy)
r (heavier material, requires more energy)
A2(greater height, greater energy)
K.E. = 1/2 w2r A2
The 1 / 2 and (2) come from translating our intuition into the conventional units for energy.
(2) Use of wave descriptors to describe velocityIt's important to make a note here of the use of frequency and wavelength together
Velocity of a wave = distance travelled by perturbation / unit of time (s) or, said differently, how many wavelengths (distance/Amplitude space) pass a given point in one second i.e. How long does a wavelength take to pass a given fixed reference point in space?
Include sketch of oscillations here
The answer is by definition, the period of a wave (measured in time-Amplitude space)
Velocity = l/ T
But, 1/T = 1/T oscillations per second, ore frequency.
So, Velocity = wavelength x frequency
(3) Controls on velocity
Introduction
Wave propagation is the propagation of a deformation through the rock. This strain (or internal deformation) is not permanent, is very small, and is completely reversible (elastic strain). In ideal elastic conditions, the strain expereienced by a rock is in direct proportion to the amount of stress it experiences. The constant of proportionality are the elastic constants or moduli.
Fig. 2.4
Wave propagations In geological situations inside rock bodies we always have three principal stress at right angles to each other. One is vertical (overburden) and the other two horizontal.
Normally stress is a mathematical entity known as a second-order tensor, that describes stress in all directions in all planes (shear stress) and at right angles to all planes (normal stresses). The analysis of these entities can not de described by simple vectorial decomposition. We can describe general stress in terms of three principal stress tensors that are know to have a predictable empirical relationship to faults.
Fig. 2.2
When we say that a rock or sediment has a velocity we do not mean that the rock travels at a certain speed but that the deformation, be it a shearing propagation as in the case of shear waves or a dilatational perturbation ,as in the case of P waves, travels through the medium at a certain pace. For example Vp in granite is about 6.5 km/s and VS is about 60% that of VP and VR is about 92% of VS.
VP=
sqrt (m/r)
m a general variable related to two elastic
constants
and
VS= sqrt (m/r)
Not that in both cases the velocity is inversely proportional to the density of the medium:
VP
m includes both the dilatational properties as well as the shearing properties. Why?
Well, for a when the dilatation propagates in the direction of wave travel, the material is also deformed ina direction at right angles to the direction of this propagation.
m is therefore = k + 4/3 m
Vs
Passage of a shear wave , on the other hand only requires that we know , the shear modulus.
In water, m=0 and therefore m = k which implies that
Vp = sqrt (k /r)
VS = sqrt (m /r) = 0
The velocity of waves through rock is associated with the density and elastic properties of the mediumLook at Fig. 5.19 from Stacey and Fig. 4.20 from Telford
Where would you predict velocity to be the greatest? By how much? Calculate!
What's the density of the mantle? And of the crust?
Now look at Fig. 5.17 from StaceyWhy do our predictions not agree?
The cause of this disparity is that the elastic moduli increase faster than does density and it is the elastic moduli that dominate.
SNELLíS LAW
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Definitions
Refraction A change in the direction of a seismic ray upon passing into a medium with a different velocity Ray parameter The quantity p = sin q1 /Vi which is constant everywhere along a raypath in the case of horizontal layering. Reflections Waves that have returned from a geological interface because of an impedance contrast and do not enter into the underlying medium. Diving rays Are refracted rays which have been bent back toward the surface because of zone with a strong velocity gradient. Head waves Refracted wave which travels along an interface because the incident wave impinged on the boundary at a critical angle
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Reflection is a specific form of wave behavior. In the general case we have energy being transmitted into the lower medium at a different angle to the incident angle. This behavior is known as refraction as is sufficiently pervasive in reflection seismology data that we should become acquainted with its behavior.
Refraction and reflection describe the paths rays take and are based on Snellís Law.
Geometric Confirmation of Snell's Law of Refraction and Reflection (for an isotropic medium)Consider two successive wavefronts in a medium A of velocity V1 and that propagate into another medium of velocity V2. The 2 successive wavefronts are separated by an interval of time t and distance V1 x t. in the first ,medium and V2 x t in the second medium.
Let's assume that the boundary between the two media is stuck together (close to reality) so that if we pull the first medium up, the second will follow and not separate at the seam and if you shear the first medium, the second medium will shear in continuity. This condition implies that all the energy hitting the boundary will go into propagating the wave, either back into the first medium or through into the second medium.
Since the second medium is faster, for the same t, the wavefront will travel a greater distance, although it will travel the same distance in that unit of time along the interface. (a units)
;If V1 = V2, the wavefronts in both media are parallel to each other:
Include mathematical development of proof
Application of Snell's Law to non-normal incident (refraction)at an interface
According to Snell's Law we must maintain the horizontal rate of travel of the wavefront at the interface (i.e. a). We can also see Snell's Las as a law of constancy of the apparent speed of the horizontal wavefront. Let's add another layer still.Include another drawing
t/a is also known as the ray parameter. This one value characterizes all the travel path and does not change. The only thing that does change is the angle and instantaneous velocity of the ray. As you can see the wavefront is continually changing direction.
What happens when q2 becomes a right angle?
Include mathematical development here
The angle of incidence when the angel of refraction is 90 ° is known as the critical angle of refraction., and it's that angle at which the wavefront begins to travel horizontally.As we make V2 greater, keeping V1 the same then the critical angle becomes smaller. That is if we go from a low velocity medium, such as unconsolidated clays (2-3 km/s) into a deeper medium, say through an angular unconformity, into granite then the large velocity jump will bring us closer to critical conditions ,i.e., the rays will be forced to move horizontally and will not be able to penetrate more deeply.
Snell's Law applied to converted phasesWe've said before that unless you impinge on an interface between materials that have different velocities at exactly 90 °, you can set up shear waves for incident P waves.
Conservation of the ray parameter still applies to these cases and so does Snell's Law.
However, we know that if shear waves are produced at an interface they will travel more slowly.
If there velocity is lower, what does that imply about the angle of refraction?Include graphical explanation here