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GG 450 April 3, 2008Refraction, Diffraction, Energy, Sources, and Sensors
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Critical Angle: sinq Recall: = p(Snells law) v When θ = 90° , the ray travels horizontally through the earth. The"critical" angle is a special case where the velocities are constant inthe layers. The critical angle is the angle in the upper layer wherethe ray becomes horizontal in the layer below: . v1 θ ic = sin -1 v2 In some seismic modeling, the critical angle is important, as we shall see..
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Dont confuse the refraction method with the reflection method:REFLECTION Method: Geometry: A common seismic methodinvolves the source - usually explosive, being moved along thesurface of the earth at the same speed as the receivers, so that thedistance between the source and receiver remains constant.This method is termed "profiling" and the resulting records arecalled profiles, often plotted as distance along the profile vs. timeafter the "bang". Profiles often show a close resemblance togeological profiles. A marine profile is shown as an example.
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REFRACTION METHOD: A second type of geometryhas the source remaining at one spot and the receiversspaced at increasing distances from the source. In thiscase, seismic arrivals as they change with distance areplotted. The resulting plot is an x-t plot, travel time plot,or "record section".The source MOVES WITHthe receiver in a PROFILE.The source stays fixed (usually at x=0) in anx-t plot (record section).
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Seismic ArrivalsWhen you start a seismic wave at the earths surface - aswe will with the refraction system - several waves fan outin more-or-less spherical (waves that go through the earth)and cylindrical wave fronts.Air wave: travels through the air at about 330 m/s (1,083ft/s), only seen close to the source (if at all). Velocity isconstant, so a plot of arrival time of the air wave vs.distance from the source is a line with a slope of 0.92ms/foot. This is a SLOW wave, usually mixed in withsurface wave arrivals.•How far away is lightning.
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Direct Wave: Travels at the p- Reflections: Reflections Refractions (head waves):wave velocity of the uppermost arrive from sharp changes Refractions are arrivals fromlayer of the ground, directly to in velocity (actually faster deeper layers that arrivethe receiver. Direct waves impedance, ρv) below. first at larger distances. Theycome in first close to the When plotted on a x-t are usually straight or bendingsource, but often disappear at plot, the first reflection slightly downwards withlarger distances or are lost in arrival time is asymptotic increasing distance on x-t plots.earlier, faster arrivals. The to the direct arrival. Refractions are often verydirect wave arrival time is Reflections bend upwards small amplitude arrivals, butusually a straight line or curved on x-t plots. often easy to see because theyslightly downward. come in first. t t t x x x * * * Z d i r e c t w a v e r e f l e c t i o n r e f r a c t i o n
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Surface waves: Ground roll: Ground roll are Rayleighwaves traveling at the surface of the earth. They areusually the largest signals on a seismic record, but areconsidered NOISE in most studies, because they onlyyield information about shallow layers.SLOWNESS: The slope of the arrival time vs distancecurve – or SLOWNESS - is 1/velocity of the wave at itsdeepest point. The slowness is another name for the RAYPARAMETER.
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diffractions: when a wave hits a sharp boundary along a profile,that boundary acts as a wave radiator, and a diffracted arrival isgenerated. When diffracted seismic arrivals are plotted as arrivaltime vs distance from the diffracting boundary, the arrivals arehyperbolic in shape. Diffracted arrivals come from boundaries thatare NOT directly below the source and receiver. This plot shows a Diffracted arrivals 12 PROFILE of the arrival 10 time of a diffracted arrival from point diffractors at 8 depth=1 6travel time 4 depth=4 depths of 1.5 and 4.5 km 2 below the surface for a 0 0 5 10 velocity of 1.5km/s. distance from diffractor
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seismic profile arrivals reflections diffractions geology up
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Attenuation and amplitude changes with distance.Spreading: as waves radiate away from the source, the energyspreads out. The energy of a wave is proportional to the square ofthe amplitude, so as the energy spreads out, the amplitude decreases,although the total energy remains constant.spherical spreading: For body waves (P and S), the waves spreadout in spherical shells. Since the surface area of a sphere isproportional to the square of the radius, the energy per unit area(energy density) decreases as r2 , E=E0/r2 (), and the amplitudedecreases as r, A=A0/r.cylindrical spreading: Surface waves, like ground roll, are confinedto the surface, so they spread out on a cylindrical shell. The area ofa cylindrical shell is proportional to r, so the amplitude of a surfacewave decreases only as 1/√r, and the energy density decreases as 1/r.
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On a sphere, you might expect a surface wave to be just aslarge at the antipode (the point directly opposite the source)as it was at the source. This doesnt happen (although thereare signs of large amplitudes on Mars opposite large impactcraters) because of ATTENUATION andSCATTERING.Scattering of waves changes the direction ofpropagation of part of the wave when it hits a rough barrieror irregular surface, or any region where the elasticconstants change over a small area. The ENERGY is still inthe waves, but the DIRECTIONS of energy movementchanges. Scattering is very important in some situations.
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Absorption: Attenuation is the result of absorption of energy. Asmall amount of energy is lost from seismic waves to heat as thewave moves through a material. Absorption takes the form: - qr where I is a measure of energy called the intensity,eI = I 0 e is the exponential constant, q is the absorbtion coefficient, and ris distance. q has units of dB/wavelength. So, at a given frequency,the energy decreases with distance at a certain number ofdB/wavelength. Note that even if q is constant, the energy in ahigh frequency wave will decrease faster than the energy in a lowfrequency wave.You often see attenuation in dB/λ (deciBells/ wavelength) for aparticular material.
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An attenuation of 0.6 dB/λ implies that a signal with a ten kmwavelength will decrease in size by a factor of 2 in 100 km.Note that if I said the material would lose one tenth of itsamplitude every wavelength, that does NOT mean that it will becompletely gone after ten wavelengths.Attenuation by a constant dB/λ implies that waves with shortwavelengths will decrease in size faster than those with longwavelengths.This means that if we want to use seismic waves to seedeep into the earth, we need to use either longwavelength waves , or very high amplitude sources.
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Energy PartitioningWhen a seismic wave hits an interface, it splits into different waves,both reflected and refracted. In the most general case an incident Pwave will split into reflected P and S waves and refracted P and Swaves, although generation of some of these waves may beforbidden by Snells Law. θπ1 θσ1 vp1, vs1 vp2, vs2 θπ2 θσ2
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Note that the ray parameter is CONSTANT for anyray -EVEN if the wave changes from a P wave to anS wave. This means that the angle of reflection of ap-wave equals the angle of incidence. In general: sin(q p1 ) sin(q p 2 ) sin(q s1 ) sin(q s2 ) p= = = = v p1 vp2 v s1 v s2 θπ1 θσ1 vp1, vs1 vp2, vs2 θπ2 θσ2
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The energy in these waves depends on the densities and velocities ofthe two materials, as given by Zoeppritz. For NORMAL incidence- that is the seismic ray hitting an interface perpendicular to thatinterface. The refracted amplitude is given by:Arfr 2Z1 = where Z i = r iVi is called the IMPEDANCE.Ai Z2 + Z1 What happens to the refracted amplitude as ρ2 approaches zero ? Does this happen? What happens to the energy of the refracted wave as ρ2 approaches zero ?
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Remember – ENERGY is conserved – not amplitude - and energychanges as the SQUARE of the amplitude. The formula forenergy (or INTENSITY) of refracted arrivals is: I rfr 4Z1Z 2 = where Z i = r iVi Ii (Z2 + Z1 )2Notice how the intensity of the refracted wave changes as ρ2approaches zero.Wave amplitudes in low-density or low-velocity materials becomelarge – which is why tsunamis get big near shore and why youshouldn’t build a house on soft fill in a place prone to earthquakes.
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Careful consideration needs to be given to the problem to beinvestigated. Some geological problems just cant be solved withseismology, while others are best attacked by looking at reflectionsfrom layers and others by looking a refractions. The deeper into the earth you need to see, the stronger your sourceenergy and lower the frequency of the source must be. This is theproblem of PENETRATION. The other side of the coin isRESOLUTION.If the feature you are trying to study has dimensions of less thanabout 1/4 of a wavelength of your seismic signal, then you wont beable to RESOLVE it in the seismic data.
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SEISMIC SOURCESWe need to put enough energy into the seismic waves to besure we can see the necessary signals at our receivers.• Earthquakes: Signals from earthquakes have largeamplitudes, but the source can be complicated by energycoming from many places and poorly known origin time.• Underground Nuclear explosions. Great seismicsources, but wildly unpopular.• Conventional explosives: Can be made as large asdesired but somewhat unpredictable amplitudes, expensive,and dangerous. Permitting and drilling required. Explosivesare used for refraction work and used to be used forreflection profiling.
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• Conventionalexplosives: Can bemade as large asdesired but havesomewhat unpredictableamplitudes, expensive,and dangerous.Permitting and drillingrequired. Explosivesare used for refractionwork and used to beused for reflectionprofiling.
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Airguns fire a pulse of air into the water as a marine seismic source. A problem is that the bubble of air oscillates generating a complex “bubble pulse.” Many guns are used to kill the bubble pulse and add more energy. QuickTimeª and aTIFF (Uncompressed) decompressor are needed to see this picture.
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• Shotgun source. Thesesources use 12 gaugeshotgun shells to send asignal into the earth.
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• Vibroseis: vibroseis is themost commonly used seismicsource on land. Rather thansend an impulse into theground, these trucks send a“chirp” into the ground thamust be processed to see theseismogram.The big advantages areexcellent control, no loudnoise, minimal permitting, andno drilling.
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• hammer: We will use ahammer.The hammer source isgood for small-scaleshallow studies.
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SEISMIC INSTRUMENTSWhat is a seismometer?What is meant by "motion of the ground"?What about tides and gravity?What does a seismometer measure? Displacement? Velocity? Acceleration? Stress? Strain? Propagation velocity?
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A TRANSDUCER: changes one type of energy into another[motors, generators, etc.]. In seismometers we change groundshaking into electrical signals.There are several ways to do this. If we use a magnet surrounded bya coil of wire to generate an electric signal when the magnet and coilare moved with respect to each other, then the VELOCITY of thecoil relative to the magnet gives us the signal.This gives a measurement of the velocity of the ground motion IFthe frequency is high - what if it isnt? NOTE: this is NOT thepropagation velocity - its the PARTICLE velocity!Nearly all land seismic sensors used for exploration have velocitytransducers.
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To measure the motion of the earth, we need to be able to measurethe motion of some point connected to the earth RELATIVE tosome point that is NOT moving with the earth.The simplest way we know of to do this is with mechanicaloscillators, of which there are two basic types: masses attached tosprings (used for detection of VERTICAL motion of the earth), andpendulums, used to measure HORIZONTAL motion of the earth.
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While these two instruments look pretty much alike, one measureshorizontal motion in-and-out-of the page, and one measuresvertical motion.The SIGNAL is the relative motion between the frame fixed to theearth and the mass. The frame moves with the ground at allfrequencies. At high frequencies, the mass is stationary, or inertial, so therelative position of the mass relative to the frame is a measure ofthe displacement of the ground.At very low frequencies, the mass is no longer stationary. Does itstill move relative to the frame? What would cause a change inthe location of the mass relative to the frame at very lowfrequencies?
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A horizontal seismometer like that shown on the previouspages can be centered easily in the same way that youwould adjust a swinging gate to always remain closed.How do you adjust a gate to always swing to the “closed”position?Which seismometer measures vertical motion and whichhorizontal motion?A gravity meter uses the same design as the verticalseismometer. In the case of the gravity meter, thedisplacement of the mass (stretch of the spring) is ameasure of what parameter?If the seismometer frame is tilted, what is the effect on theseismometers?
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This seismometer is called a GEOPHONE. The motion ofthe ground is detected by a coil of wire moving through amagnet attached to the frame generating a current. If thecoil is not moving relative to the frame, no signal isgenerated. Geophones sense the VELOCITY of the groundat high frequency.
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What are desirable characteristics of a seismometer? Fidelity: A seismometer should yield the motion of the ground with high "fidelity", where fidelity is a measure of accuracy. It should be possible to reconstruct the motion of the ground from the recorded signal. Any distortions should be linear, or at worst, well known. Noise: A seismometer should have the lowest possible noise level Bandwidth: It should have a large frequency band across which it has a low noise level. Dynamic Range: Large dynamic range to record both very large and very small signals without distortion.
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Another type of seismic transducer consists of two plates, oneconnected to the seismic mass and one to the frame.As the distance between the plates changes the capacitancechanges, and we get a measure of the DISPLACEMENT of themass relative to the frame of the instrument.How does this relate to the particle motion?
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REFRACTION SEISMOLOGY METHODSThe DIRECT WAVE travels straight from the seismic sourceto the seismometer. The x-t plot for the direct arrival lookslike: slope= ray parameter time =slowness =1/propagation velocity =1/v0 distance from shot up v 0
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As soon as we let velocity change with depth in a flat model, thex-t graph will no longer be a straight line, as the ray path betweenany two points will no longer be a straight line, in general.If the earth is made up of constant-velocity layers, the x-t plot willbe made up of a sequence of straight lines, one for each layer IFthe velocity always increases with depth.When we have a single horizontal interface separating two layersthat have constant velocities, its relatively simple to describe theresulting refracted arrival.The ray that will arrive at the geophones along the critical path(horizontal in the lower layer) hitting the lower layer at the criticalangle, thus:
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slope= ray parameter =slowness time =1/propagation velocity =1/v1 distance from shot up v 0 v 1The critical angle is important here. It allows us to determinethe travel time of the refracted arrival, and from there tocalculate the depth to the 2nd layer.
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The travel time from source to receiver for a refraction through aflat 2-layer model is: 2 2 2h1 v - v 2 1x t rfr = + . v 2v1 v2The trfr equation is much simpler than it looks, since x only appearsin the 2nd term, it is a straight line with slope equal to the rayparameter and a y-intercept equal to the first term. Since we can measure the y-intercept of the refraction (called theintercept time), and the two velocities can be measured from theslopes of the direct and refracted arrival, we can solve the aboveequation for h1, and obtain the depth to the layer:
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ti v2 v1hi = 1/ 2 , where ti is the y intercept time of the refraction arrival. 2 (v - v ) 2 2 2 1Evaluation of refraction data using these formulae, and theirexpansion to multiple layers, has been used extensively - so muchthat many people have been given the impression that the earth ismade up of constant velocity layers! While the models often fit thedata quite well, so do models with gradients and low velocityzones.Great care must be taken in over interpreting model results.
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A refracted arrival has a slope of 5 ms/10 m. The direct layer arrival has a slope of 2 ms/m. The intercept time of the refracted arrival is 20 ms.• What is the velocity of the upper layer?• What is the velocity of the lower layer?• What is the depth to the lower layer?
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HOMEWORK FOR Tuesday, Apr 8, 2008:Find the appropriate formulas in text books or on the Web todetermine the critical distance, crossover distance, and intercepttime for a single-layer model with zero dip where the upperlayer has a velocity of 500 m/s and the depth to the lower layer is5 m. Construct a graph of critical and crossover distance andintercept time vs. velocity of the lower layer between 510 m/sand 2000 m/s. What do these results imply in terms of geophonespacing and detection of the two layers?Hint: Find the formulas and enter them into a spread sheet orMatlab, then use the formulas to make the graphs. Come see me ifyou have problems.
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reflection critical distance crossover distance distance ξ χ ριτ δ ε πτη θχ v1 v2The critical distance is the distance from the source where the refraction canfirst be observed. Notice that at the critical distance the reflection from the layerand the refraction have the same travel time AND the ray parameter (slope) ofboth the reflection and the refraction are the same. The cross-over distance isthe distance where the direct arrival and the refracted arrival come in at the sametime.
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IN CLASS PROBLEM: You observe the following x-t .refraction plot: 50 40 30 t rav el t im e, m s 20 10 0 0 5 10 15 20 dist ance f rom shot , m
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The closest geophone is 5 m from the shot point. Note that when you extrapolate the first arrival time back to zero distance it doesnt go through the origin.3) What is the velocity of the layer observed?5) Is this the velocity of the surface layer?3) What limits can you place on the thickness and velocity of the surface layer?4) How could you prevent this problem and get the surface layer model parameters “exactly”?
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