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### L13_WavesAndRaysII-page2

Course: PHYSICS 384, Spring 2009
School: Stony Brook University
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Word Count: 185

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and Reflection transmission Seismic rays obey Snells Law (just like in optics) The angle of incidence equals the angle of reflection, and the angle of transmission is related to the angle of incidence through the velocity ratio. But a conversion from P to S or vice versa can also occur. Still, the angles are determined by the velocity ratios. sin iP sin RP sin rP sin RS sin rS = = = = =p VP1 VP1 VP 2 VS 1 VS 2...

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and Reflection transmission Seismic rays obey Snells Law (just like in optics) The angle of incidence equals the angle of reflection, and the angle of transmission is related to the angle of incidence through the velocity ratio. But a conversion from P to S or vice versa can also occur. Still, the angles are determined by the velocity ratios. sin iP sin RP sin rP sin RS sin rS = = = = =p VP1 VP1 VP 2 VS 1 VS 2 where p is the ray parameter and is constant along each ray. Applied Geophysics Waves and rays - II Amplitudes reflected and transmitted The amplitude of reflected, the transmitted and converted phases can be calculated as a function of the incidence angle using Zoeppritzs equations. Simple case: Normal incidence Reflection coefficient RC = AR 2V2 1V1 = Ai 2V2 + 1V1 Transmission coefficient TC = AT 2 1V1 = 1 RC = 2V2 + 1V1 Ai Reflection and transmission coefficients for a specific impedance contrast These coefficients are determined by from the product of velocity and density the impedance of the material. RC usually small typically 1% of energy is reflected. Applied Geophysics Waves and rays - II 2
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Stony Brook University - PHYSICS - 384
DeploymentImportant considerations Need good coupling to the ground spike Mini-arrays to reduce surface wave noiseOffset of geophonesSmall offsets Near-vertical incidence retains P-energy High resolution of subsurface reflectorsSeismic reflection
Stony Brook University - PHYSICS - 384
ExamplesGravityApplied geophysics IntroductionExamplesSeismic refractionApplied geophysics Introduction12
Stony Brook University - PHYSICS - 384
Definitions: Magnetic fieldstrength or intensity Biot-Savarts law definition:for a loop of wire of radius r that iscarrying a current I, H at center isgiven as:H=nI/2r [A/m]where n is a unit vector normal to theplane of the loop. The magnetic fie
Stony Brook University - PHYSICS - 384
Definitions: Magnetic fieldor flux density Definition: Vector quantity defining the magneticflux/unit area; i.e., the density of the magnetic field lines.Thus often called Flux Density Mathematical Definitions:o pp Wbr = c 2 r 2 = Tesla Air:4 r
Stony Brook University - PHYSICS - 384
Magnetization or magnetic polarization A measure of the pole strength/unit area along oneof the ends of magnetic material:J=(p/A) n [A/m]Magnetic moment Strength of a magnetic field generatorM=J V = p l [A m2]For a loop of current: M=(Ir2) n
Stony Brook University - PHYSICS - 384
Dipole nature of magnetic materialsBar MagnetN+ Although, no magnetic monopolesexist in nature, they are useful fortheory: magnetic monopoles of samesign repel, opposite signs attract.S Dipole created by two poles ofopposite sign and separated by
Stony Brook University - PHYSICS - 384
Definitions: Magnetic potential Remember that the potential is defined as thepotential to do work.o pp Wb =c W= Magnetic Potential:4 rrmwhere o=4 10-7 [H/m] is the magnetic permeability of free spaceand p [A/m] is magnetic pole strength Gravit
Stony Brook University - PHYSICS - 384
Applications Shallow (Engineering and Environmental):contaminants, toxic waste, pipes, cables and metalinclusions Military: location of UXOs Archeology: buried walls, old fire pits Mining: iron sulfide deposits Oil and groundwater: depth to magneti
Stony Brook University - PHYSICS - 384
Derivatives Emphasizing shorter wavelength features. First vertical derivative emphasizes near surface features. It canbe measured with gradiometer, or derived from corrected data Second vertical derivative emphasizes boundaries of targetzones.Reduc
Stony Brook University - PHYSICS - 384
Removal of Regional Use formula tosubtract off IGRFvalue. Filtering processes toget the regional formula(like gravity).General Guidelines16
Stony Brook University - PHYSICS - 384
Gravity Anomalies: 2D forward calculationfor rectangular parallelepipeds with greater vertical extent than horizontalSpreadsheet: Grav2Dcolumnsee Dobrin and Savit eq 12-34Gravity anom aly2.50Define density structureProfile 10.30.30.30.30.30.3
Stony Brook University - PHYSICS - 384
SpikingdeconvolutionRecordedwaveform1-1Deconvolutionoperator101-1Output0RecoveredreflectivityseriesApplied Geophysics Seismic reflection IISpikingdeconvolutionRecordedwaveform1-1Deconvolutionoperator11100Output0-Recovere
Stony Brook University - PHYSICS - 384
Spiking deconvolutionApplied Geophysics Seismic reflection IISpikingdeconvolutionRecordedwaveformDeconvolutionoperatorOutput11-10-11RecoveredreflectivityseriesApplied Geophysics Seismic reflection II3
Stony Brook University - PHYSICS - 384
Vertical Electric Sounding When trying to probe howresistivity changes withdepth, need multiplemeasurements that each givea different depth sensitivity. This is accomplished throughresistivity sounding wheregreater electrode separationgives great
Stony Brook University - PHYSICS - 384
Geometrical FactorsArray advantages and disadvantagesArrayAdvantagesDisadvantagesWenner1. Easy to calculate a in the 1. All electrodes moved eachsoundingfield2. Sensitive to local shallow2. Less demand onvariationsinstrument sensivity3. Long
Stony Brook University - PHYSICS - 384
Current density and equipotential linesfor a current dipoledfraction total current 2z 2i f = tan1 dif=0.5 atz=d2if=0.7 at z = dWider spacing Deeper currentsApparent ResistivityPrevious expression can berearranged in terms of resistivity:=
Stony Brook University - PHYSICS - 384
Horizontal interfaceTraveltime equationsDirect wave:T=HeadwavexV1Head wave:T = TSB + TDD ' + TBDT=2h1x 2h1 tan ic+V1 cos icV2T=22x 2h1 V2 V1+V2V2V1T = ax + bslope: 1/V2intercept: gives h1Applied Geophysics Refraction IHorizontal
Stony Brook University - PHYSICS - 384
Critical incidenceWhen rP = 90 iP = iC the critical anglesin iC =VP1VP 2The critically refracted energy travelsalong the velocity interface at V2continually refracting energy back intothe upper medium at an angle iCa head waveReflection and tran
Stony Brook University - PHYSICS - 384
Classifications of magnetic materials Anti-ferromagnetic Almost identical to ferromagnetic except that themoments of neighboring sublattices are alignedopposite to each other and cancel out Thus no net magnetization is measured Example: Hematite Fe
Stony Brook University - PHYSICS - 384
Classifications of magnetic materials Ferromagnetic Materials contain unpaired electrons in incompleteelectron shells. Magnetic moment of each atom is coupled to othersin surrounding domain such they all becomeparallel. Caused by overlapping electr
Stony Brook University - PHYSICS - 384
Station spacingMust ensure thatspacing is sufficient tosample anticipatedsignalApplied geophysics IntroductionLimitations Methods require contrast in physical properties NonuniquenessDirect modeling: calculate the result of a specific structureI
Stony Brook University - PHYSICS - 384
Analysis and interpretationOnce we have made our gravity observations,corrected for surface effects,we attempt to deduce sub-surface structureConsiderations: Anomaly profile (2D structure) or map (3D structure)?If anomaly length &gt; twice the width a
Stony Brook University - PHYSICS - 384
Bouguer anomalyApply all the corrections:g B = g obs + C + C F C B + CTwatch the signs!Smaller scale engineering/environmental surveys: Not tied to absolute gravity Use corrections with accuracy necessarydetermined by the size of the target signal
Stony Brook University - PHYSICS - 384
Gravity and potentialsg is a vector field:g=GM EREGravitational potential:U=2r1where r1 is the unit vector pointingtoward the center of the EarthGmrU is a scalar field which makes iteasier to work withDefinition: The gravitational potential
Stony Brook University - PHYSICS - 384
Remote sensingConstraining the Earths sub-surface withobservations at the surfaceGeophysical techniques measure physical phenomena: Gravity Magnetism Elastic waves Electricity Electromagnetic wavesWhich are sensitive to sub-surface physical prope
Stony Brook University - PHYSICS - 384
P and S-velocitiesP-velocity+43VP =S-velocityVS =change of shape and volumechange of shape onlyFor liquids and gases = 0, thereforeVS = 0 and VP is reduced in liquids and gasesHighly fractured or porous rocks have significantly reduced VPThe b
Stony Brook University - PHYSICS - 384
Body wavesP and S-wavesApplied Geophysics Waves and rays - IElastic moduliYoungsmodulusShear modulus, Force per unit area tochange the shape ofthe materialdescribe the physical properties of the rockand determine the seismic velocityPoissons
Stony Brook University - PHYSICS - 384
ExampleRockhead determinationfor waste disposal siteRefractor velocity determination Velocity is 1/slope ofoptimal TV plot Decided on threerefractor velocities asdata fitted well withthree straight linesegmentsApplied Geophysics Refraction III
Stony Brook University - PHYSICS - 384
ExampleRockhead determinationfor waste disposal sitePhantomingIncreased velocityor reduced depth? Used to generate arefraction arrival timemuch longer that it ispossible to collectNear-constantvelocity refractor Could be used todetermine inte
Stony Brook University - PHYSICS - 384
Gravity:Analysis and examplesReading:Today: p39-64Next Lecture: p65-75Applied Geophysics Analysis and examplesGravity Anomalies: 2D forward calculationfor rectangular parallelepipeds with greater vertical extent than horizontalSpreadsheet: Grav2Dc
Stony Brook University - PHYSICS - 384
Concept of hysteresis Complex relationship betweenB and H that occurs inferromagnetic materials. B flattens off with increasing Hat saturation When H is decreased, B does notfollow same curve Will have remanent B value atzero H
Stony Brook University - PHYSICS - 384
Magnetic properties
Stony Brook University - PHYSICS - 384
AttenuationThe amplitude of an arrival decreases with distance from the source1. Geometric spreadingEnergy spread over a sphere: 4r2Amplitude 1/r2. Intrinsic attenuationRocks are not perfectly elastic.Some energy is lost as heat due tofrictional d
Stony Brook University - PHYSICS - 384
Velocity sensitivityThe amplitude of wave motiondecreases with depthRelated to depth/wavelengthLonger wavelengths sample deeper(This fig isfor water-waves)Seismic velocity generally increaseswith depth.Surface waves are dispersive,which means th
Stony Brook University - PHYSICS - 384
Analytic Signal Combination of derivatives:2 F(x , y, z ) + x2A(x , y ) = F(x, y, z ) +z F(x, y, z ) z2 Shape is independent ofinclination/ declination ofinduced field.Other analysis or filters Fourier transforms Wavelets Upward and downw
Stony Brook University - PHYSICS - 384
Derivatives Emphasizing shorter wavelength features First vertical derivative emphasizes near surface features. It canbe measured with gradiometer, or derived from corrected data Second vertical derivative emphasizes the plan view boundariesof target
Stony Brook University - PHYSICS - 384
Velocity and densityNafe-DrakecurveVPigneous andmetamorphic rockssediments andsedimentary rocksVSThis curve has beenapproximated usingthe expression = aVP14(a is a constant: 1670 when in km/m3 and VP in km/s)Applied Geophysics Waves and r
Stony Brook University - PHYSICS - 384
Head waveYou can see:a head wave, trapped surface wave, diving body waveApplied Geophysics Waves and rays - IFactors affecting velocityVP =Density velocity typically increases with density+43VS =( and are dependant on and increase more rapidly t
Stony Brook University - PHYSICS - 384
ExamplesArcheological investigationsP-wave refraction lines Sledge hammer source 2m geophone spacing 50m lines reversedS-wave refraction lines Rubber maul hammerstriking horizontally on avertical plate 2m horizontal geophonespacingFrom Sharma:
Stony Brook University - PHYSICS - 384
ExampleRockhead determinationfor waste disposal siteRefractor depth profile Using TG, the refractorand overburden velocity,we can calculate thedepth of the refractorbeneath each geophoneGround surface Note it is the distance ofthe refractor fro
Stony Brook University - PHYSICS - 384
Methodology of interpretationInverse modelingForward modeling:Make a skilled guess of the structure (the model)Calculate the anomaly this would produceCompare to the observations (the data)Adjust the model and recalculate etcInverse modeling essent
Stony Brook University - PHYSICS - 384
Gravity Anomalies: 2D forward calculationfor rectangular parallelepipeds with greater vertical extent than horizontalSpreadsheet: Grav2Dcolumnsee Dobrin and Savit eq 12-34Gravity anom aly0.000Define density structure24681012141618-0.3-0.
Stony Brook University - PHYSICS - 384
Seismic methods:Seismic reflection - IVReflection reading:Sharma p130-158; (Reynolds p343-379)Applied Geophysics Seismic reflection IVSeismic reflection processingFlow overviewThese are the mainsteps in processingThe order in whichthey are appli
Stony Brook University - PHYSICS - 384
Stacking velocityIn order to stack the waveforms weneed to know the velocity. We find thevelocity by trial and error:TNMO =x22T0V12 For each velocity we calculate the hyperbolae and stack the waveforms The correct velocity will stack the reflectio
Stony Brook University - PHYSICS - 384
ExamplesElectrical resistivityApplied geophysics IntroductionExamplesElectrical resistivitypolluted sandclean sandpolluted sandApplied geophysics Introduction13
Stony Brook University - PHYSICS - 384
Multiple-layered modelsFor multiple layered models wecan apply the same process todetermine layer thickness andvelocity sequentially from the toplayer to the bottomHead wave from top of layer 2:T=22x 2h1 V2 V1+V2V2V1Head wave from top of lay
Stony Brook University - MATH - 300
100LECTURE 10. THE MODULAR EQUATIONExample 10.1. Take N = 2 and f = ( ) M(1)6 . We see that (2 )/( )belongs to the space M(X0 (2). Observe that q = e2i changes to q 2 when wereplace with 2 . SoQYq 2 =1 (1 q 2m )24mQ(2 )/( ) ==q(1 + q m )24 = 2
Stony Brook University - MATH - 300
101show that each matrix A from AN is contained in the left-hand-side. This follows fromthe well known fact that each integral matrix can be transformed by integral row and` 0column transformations to the unique matrix of the form n n , where n|n . Th
Stony Brook University - MATH - 300
106LECTURE 10. THE MODULAR EQUATIONObserve also that F 2 = 1 so that Fr2 = identity. It is called the Fricke involution.By taking the inverse transform of functions, the Fricke involution acts on modularfunctions of weight k byFr (f )( ) = f (1/N ) =
Stony Brook University - MATH - 300
7Assume = 0. Then the Moebius transformation dened by M 1 is thetranslation z z + and hence takes z out of the domain 1 Re z 122unless = 0 or = 1 and Re z = 1 . In the rst case M = I and f = f . In21 1the second case M = , f = ax2 axy + cy 2 and
Stony Brook University - MATH - 300
105Remark 10.2. Notice that Q( d) if and only if the lattice has complexmultiplication (see Lecture 2). By Exercise 2.6 this is equivalent to that E hasendomorphism ring larger than Z. An elliptic curve with this property is called anelliptic curve wi
Stony Brook University - MATH - 300
115Corollary 11.3. Keep the notation from the previous lemma. Assume f is normalizedso that c1 = 1. Thencm cn = cmn if (m, n) = 1,cp cpn = cpn+1 + p2k1 cpn1where p is prime and n 1.Proof. The coecient cn is equal to the eigenvalue of T (n) on Mk (1)
Stony Brook University - MATH - 300
114LECTURE 11. HECKE OPERATORSFrom now on we shall identify M(1)k with Fk . So we have linear operatorsT (n) in each space M(1)k which also leave the subspace M(1)0 invariant.kTo avoid denominators in the formulas one redenes the action of operators
Stony Brook University - MATH - 300
113( + )2k f (Z + Z) = ( + )2k f ( ).By property (iii) of Lemma 1, we obtain that T (n) leave the set of functions f on Hsatisfying (11.6) invariant.Let Fk be the space of functions on L of the form f where f M(1)k .Theorem 11.1. For any positive int
Stony Brook University - MATH - 300
112LECTURE 11. HECKE OPERATORSXpT (pn1 ) Rp () = pT (pn1 )(p) = pb ,[: ]=pn+1where(b =10if p.if p.(11.7)Comparing the coecients at we have to show that(a) a = 1 if p;(b) a = p + 1 if p.Recall that a counts the number of of index p in whic
Stony Brook University - MATH - 300
959.2 In view of this theorem the cross-ratio R can be thought as a functionR : H/(2) C.The next theorem shows that this function extends to a meromorphic function onX (2) = H /(2):Theorem 9.3. The cross-ratio function R extends to a meromorphic func
Stony Brook University - MATH - 300
94LECTURE 9. ABSOLUTE INVARIANT AND CROSS-RATIOTheorem 9.1. The cross-ratio R denes a bijective mapR : X/GL(2, C) C \ cfw_0, 1.Proof. Let (x1 , x2 , x3 , x4 ) X . Solving a system of three linear equations with 4unknowns a, b, c, d we nd a transforma
Stony Brook University - MATH - 300
Lecture 9Absolute Invariant andCross-Ratio9.1 Letx1 = (a1 , b1 ),x2 = (a2 , b2 ),x3 = (a3 , b3 ),be four distinct points on P1 (C). The expressiona1 b1 a3a2 b2 a4R=a1 b1 a2a3 b3 a4x4 = (a4 , b4 )b3 b4 b2 b4 (9.1)is called the cross-rati
Stony Brook University - MATH - 300
918.4 Find all normal subgroups of (1) for which the genus of the modular curve X ()is equal to 0. [Hint: Use Theorem 10.4 and prove that r2 = /2, r3 = /3, r | ].8.5 Generalize the Hurwitz formula to any non-constant holomorphic map f : X Yof compact
Stony Brook University - MATH - 300
90LECTURE 8. THE MODULAR CURVEProof. We know this already when = (1). So we may assume that &gt; 1. ByTheorem 8.5 we can identify the space M6 () with L(D), wheredeg D = 6 deg KX + 6r + 3r2 + 4r3 = 12g 12 + 6r + 3r2 + 4r3 .I claim that deg D &gt; 2g + 1. I
Stony Brook University - MATH - 300
89Let us compute (). We know that8&gt;2&lt; (j j ( ) = 3&gt;:1if i (1) ,if e2i/3 (1) ,otherwise.This immediately implies that8&gt;1&lt; (j ) = 2&gt;:0if i (1) ,if e2i/3 (1) ,otherwise.Thusx () = k(ex (j ) 1)/e(x).Now, let x = ci be a cusp represent