The ray paths and travel times for multiple layers can be computed using ray-tracing, as demonstrated in Lab 3.

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Todd Parrish
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1 C. Trael me cures for mulple reflecors The ray pahs ad rael mes for mulple layers ca be compued usg ray-racg, as demosraed Lab. MATLAB scrp reflec_layers_.m performs smple ray racg. (m) ref(ms) ref(ms) ref(ms)

2 The wo eamples boh show hree reflecos. Noe ha: (a) Rays are refraced a erfaces accordg o Sell s Law (b) The polary ad amplude of a refleco s due o he mpedace chage a he erface (c) Deeper reflecos are flaer (.e. hey ehb less ormal moeou) (d) The hgh amplude refleco from a shallow erface ca weake laer reflecos (from deeper srucure) (e)for a sho gaher, he ray pahs reflec (bouce) a dffere pos o he reflecor surface. (f)ths geomery, wh geophoes o each sde of he sho s called a spl-spread

I he real sho gaher oe ha: (a) The ercal scale s wo-way rael me secods. (b) Pose arrals hae he race shaded black. Ths helps he eye o correlae races whe ose s prese.

3 I he real sho gaher oe ha: (a) The ercal scale s wo-way rael me secods. (b) Pose arrals hae he race shaded black. Ths helps he eye o correlae races whe ose s prese. Beer someme helps wh hs process, bu oo much has bee kow o hae he reerse effec. Calbrao s eeded o opmze hs ask. (c) Noe he drec wae, whch s he frs arral. (d) The groud roll s log lasg ad obscures reflecos a small offse (A). (e) Slower arrals plo a a seeper agle o he sesmogram (more me eeded o rael he same dsace) (f) Reflecos ca be recogzed by her parabolc shape ad are labelled B,C,D, ad E. Noe ha hey are eer he frs arral. Ths meas ha reflecos are recorded whe he groud s sll mog from earler arrals, ad reflecos ca be obscured by ose.

4 Ieral elocy ad aerage elocy Cosder he mulplayer Earh model show aboe. The h layer has a hckess z ad a elocy. The ray raels hrough he layers o ad s refraced a each erface. I s he refleced by he h erface. The sesmc wae speds a me τ he h layer. The eral elocy s he acual elocy a specfc layer ad s defed as z / τ For he whole ray pah we ca defe a aerage elocy as V z τ τ τ Aoher way of aeragg he elocy s o use he roo-mea-square aerage. Ths s defed as: V rms, τ τ ad s eeded o compue eral eloces, as descrbed below. For he case of τ + τ V rms, τ + τ 4

5 Normal moe-ou for mulple layers ad he D equao To deerme he deph of he erfaces ad he eral eloces, a approach smlar o ha used C. s eeded. Noe ha we eed more ha jus he rael-me a ormal cdece, ad should use a plo of NMO s. offse (). Whe he sesmc sgals rael close o he ercal dreco, ca show ha he ormal moeou for he h refleco s: Δ Vrms, Ths s ery smlar o he equao C. for he ormal moeou aboe a sgle erface. The oly chage s ha he eral elocy ( ) has bee replaced by he r.m.s. elocy (.e. he aerage elocy dow o he h reflecor). I ca be show he roomea-square elocy, ges he correc aerage hs coe. Cosder he sesmogram wh hree reflecos ha s show aboe: For each refleco we ca measure he zero-offse rael mes (,, ). If ecessary a plo of ersus may be eeded (see C.) Ne he r.m.s. eloces are compued from he ormal moeou. Cosder he 4 h geophoe a dsace. The ormal moeou a hs alue of are Δ, Δ ad Δ. From hese NMO alues, he aerage (r.m.s.) eloces ca be compued. NMO Δ( ) ( ) V rms, Re-arragg hs equao ges V rms, Δ ( ) 5

6 For he oher reflecos s obous ha V rms, ( ad Δ ) V rms, Δ ( ) The e sage of he daa aalyss s o compue he eral (rue) eloces (,, ) from he aerage (r.m.s.) eloces (V rms,, V rms,, V rms,) The s acheed wh he D equao ha saes: rms, rms, Whle hs appears complcaed, s really a smple recurso equao. Ths meas ha f we kow he elocy oe layer, he equao ells us wha he elocy wll be he layer below. To sar hs process, we beg a he surface. For he frs refleco, he sesmc sgal raels oly Layer. Thus V rms, Now V rms, represes a aerage elocy of layers ad. Thus o fd we ca use he D equao wh rms, rms, Ths ca he be repeaed as may mes as ecessary. For he e layer dow we ca wre rms, rms, Eample : Look a he daa he able aboe for he hree refleco sesmogram Eample. The rael mes for he frs refleco are he same as for he eample C. The soluo o ha problem gae: 500 m s - ; z 400 m ad s Noe ha V rms, 500 m s - 6

7 Cosder he secod refleco Choose a geophoe, Zero offse rael me, Normal moeou Δ r.m.s. elocy for Layers ad V rms, Use he D equao o compue he eral elocy for layer : rms, rms, Does hs agree wh he rue model? Now compue he hckess of he secod layer. To do hs oe ha he refleco speds a me - he secod layer. I hs me, raels a dsace z. Thus we ca wre z Re-arragg he equao ges z ( ) Repea hs for he hrd refleco Choose a geophoe, Zero offse rael me, Normal moeou Δ V rms, rms, rms, z ( ) 7

8 Mulple reflecos (a) The secod refleco s a mulple bouce from he erface a 0.4 km deph. (b) The rael me of he mulple a zero-offse s eacly double ha of he sgle refleco. Ths ca help defy a mulple. (c) Noe ha whe he sesmc eergy bouces a he Earh-Ar erface, he refleco coeffce ca be large. Remember ha R ρ ρ ρ + ρ Cosder he upward raellg wae as approaches he surface. ρ s ow he desy of he ar whch s esseally zero. Ths ges R - ad almos all eergy s refleced dowwards. (d) For a eample of mulples sgle chael sesmc daa, see Kearey Fgure 4.56 ad he dscusso C. 8

(1) Cov(, ) E[( E( ))( E( ))]

(1) Cov(, ) E[( E( ))( E( ))] Impac of Auocorrelao o OLS Esmaes ECON 3033/Evas Cosder a smple bvarae me-seres model of he form: y 0 x The four key assumpos abou ε hs model are ) E(ε ) = E[ε x ]=0 ) Var(ε ) =Var(ε x ) = ) Cov(ε, ε )