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Lecture 9
Todays • Critical topics: and post critical reflection
- • PhaseLayer overshifts half and space pulse distortion I. Critical Last lecture and we Post considered Critical an ReflectionSH field:
S 1
S 2
j 1
j 2
j 1 x
S 1
MEDIUM 1
MEDIUM 2
z
Adapted from Stein & Wysession^ SH wave incident on a solid-solid boundary. (2003), fig. Figure by MIT OCW.2.62, An Introductionp. 76, Blackwell to Seismology, Publishing. Earthquakes, and Earth Structure,
We know from Lecture 7 that for a post critical η 2 = iηwave:ˆ 2 (1) and for a reflected wave, / A A\ (^12) = μ μ 11 ηη 11 −+ μμ 22 ηη (^22) (2) Lecture 91
Combining the two, the reflection coefficient for a post critical wave is given by Note that equation (3) is a complex^ R^ =number^ μ^ μ^11 ηη^11 −+ divided^ iμiμ^22 ηηˆˆ^22 = by^ e iits^2 E^ complex conjugate. The complex(3) values can be visualized in the complex plane.
From this figure we can see that (^) E = tan− (^1) ( μ μ 21 η (^) ηˆ (^21) ) (4) which value The phase ofshows the shift phasethat can the shift be postcritical foundis dependent by comparing reflected on the wave (^) theangle potentials will of incidence.undergo of the a phaseincident shift. and | Rreflected |= 1, and waves: the φ^ / 1 = R φ^ \ 1 = ei^2 E^ φ
= ei^2 E^ A^ \ 1 eiω(px−η^1 z−t) =A^ \ 1 ei[2E+ω(px−η^1 z−t)] =A^ \ 1 ei[ω(^2 ωE^ +px−η^1 z−t)] Also recall that transmitted postcritical=^ A^ ^1 ewaves^ iω[px− becomeη^1 z−(t−^2 ωEevanescent:)]^ (5) φ^ \ 2 = T φ\ 1 =A^ \ 2 e−ωη^ ˆ^2 z^ eiω (px−t) (6) with Equation the e (5)−ω^ η ˆshows^2 z^ term that determining in a postcritical exponential reflected decay wave, with there depth. is a phase shift of (^2) ωE (^). Lecture 92
In A that phasethis describes case, shift the of a πsignal,vertical 2 is known slowness as a Hilbert disappears, Transform i.e. η 1 →. For0. instance, if there is a function f (t) is the Hilbert Transform of that signal.^ H[f^ ( t)]For^ = example,f^ (t)^ ⊗^ (− ifπ^ t )you^ have an impulse δ(t) with a(10) 90 degree incidence,
the graph of the Hilbert Transform for δ(t) is given by
Any incoming such phasethat wave 0 shift< E ( (^) fbetween< (t 180,)) and ψ its = Hilbert 0 and ψTransform. = π can beIn explainedother words, by aconsider linear combinationa phase shift of of the E, When f �(t) = E f (^) (=t). 0 (i = ic), sinE = f0,^ � (sot)^ =f �^ (ft^ ()t )=cosE f (t).^ + When^ H[f^ (t )]EsinE = π^ , f �(t) = −f (t). When E =(11) π 2 , Additionally, f (t) = δ(t)) results a Hilbert in − Transformf (t). on the Hilbert Transform of impulse δ (H[H[f (t)]], where Lecture 94
Also, reflection See the consider Shearer provides a textsource a π (^2) for (^) phaseimpulse more shiftinformation in the(in theearth. 2Don Ifthecase...the an Hilbert impulse 3D Transform. δ (^) caseleaves provides a source, a phase each instanceshift of π of). Now, earth. In the let’s earth,A variety consider wave of speed (^) PPthe casewaves increases where will with (^) travela ray depth. tobundle the Therefore receiver, is shot frombutan asymmetric (^) onlya source one ofon PPthem the wave surfaceis symmetric.will haveof the a shorter thus In other arriving path words, withinat theall of (^) receivera thelowvelocity source earlier energy mediumthan arrivesthe andsymmetric beforea longer thePP path actualwave within with PP wave.awhich highvelocity we are concerned. medium,
With the the PPpeak a wavefrequency of the is obvious. Hilbert (ω) approaching TransformHowever, in infinity,curve. the case Consequently, the of phase a finite shift frequency, in approaches order tothe work arrivalzero, with and of thesethethe wavearrival types is atofof waves, mistakes finite ω it is iscan to necessary calculatebe made to whatin have the the eitherwave waveforms analysis.ω → ∞ should orThe to uselookonly synthetic like.way to seismograms,properly identify otherwise waves largewith
Lecture 95
Distance
Direct waveSlope = 1/c 1
Travel time curve for rays in a layer over a halfspace.
Time
Adapted from Stein & Wysession Structure (^) ,(2003), fig^ Figure by MIT OCW. 3.22, An p.Introduction 121, Blackwell to Seismology, Publishing Earthquakes, and Earth
B. Reflected For the reflecteda reflected Wave wave wave, time, the or, travel time is given by the sum of the incident wave time and
where SC = CR = �( 1 tb^ =^ SC^ c^1 +^ CRc^1 (13) 2 x)^2 +^ h^2.^ Thus, t � b =^ c^21 h^2 + x^ x 44 (14) The for a graph reflected of t waveb is a approacheshyperbola, thewith shape (^) xlim→∞ (^) oftb (^) the= (^) c 1 travel, i.e. (^) timeas x → ∞curve, forthe a travel direct time wave: curve
Lecture 97
Distance
Reflected wave
Direct waveSlope = 1/c 1
Travel time curve for rays in a layer over a halfspace.
Time
Adapted from Stein & Wysession Structure (^) ,(2003), fig.^ Figure by MIT OCW. 3.22, An p.Introduction 121, Blackwell to Seismology, Publishing Earthquakes, and Earth Also note that when x = 0, tb = (^2) ch 1. C. Head The travel Wave time for a head wave, which occurs when i > ic, is given by
which can be written as,^ tc^ =^ SA^ c^1 +^ ABc^2 +^ BRc^1 (15) Notice that equation (16) is of the^ tc^ =form^ cx^2 t^ + =^2 c h^1 mx^ � +^1 − b,^ (thus^ c^ c^12 )^2 giving^ a straight line. The x(16) = 0 intercept (tin tercept) is then given tc(0) by, = 2 h (^1) − ( c (^1) ) (^2) (17) which is less than the value at x = 0 for t^ cb^1.^ c^2 Now, plotting the three travel time curves together gives the following: Lecture 98
