Comparison of traveltime inversions on a limestone structure

Transcription

1 Comparson of traveltme nversons on a lmestone structure Comparson of traveltme nversons on a lmestone structure Matthew D. Allen and Robert R. Stewart ABSRAC Four traveltme nverson technques were appled to two sets of sesmc data collected on a Maya pyramd run at Maax Na, Belze. In partcular, drect dvson, sngular value decomposton, damped least squares and the conjugate gradent methods were appled. he sesmc data conssts of two source-recever rngs around the crcumference of the pyramd wth one rng near the base, and the other near the top of the pyramd. All methods produced smlar velocty felds wth most veloctes n the range of m/s (characterstc of carbonate rubble). raveltme resduals (observed mnus calculated tmes) show an average of approxmately 3ms for each method. However, based on the average and standard devaton of the resduals, the most accurate technque was found to be drect dvson followed by damped least-squares, sngular value decomposton and lastly conjugate gradent. INRODUCION Sesmc tomography s startng to be appled to the feld of archaeology (Merlant and Musante 1994; Cardarell and de Nards 2001; Karastaths et al. 2001; Atken and Stewart, 2003). Archaeologcal practces generally concentrate on excavatons of a small scale as they are costly, tme consumng and nvasve. When consderng a large area such as an ancent cty t would be an mmense effort to excavate the whole area. o amelorate ths problem, geophyscal practces are beng consdered as relatvely nexpensve, rapd and non-nvasve technques. By usng geophyscs, archaeologsts mght be drected to deal locatons to excavate. In some cases, the geophyscal technques can be used to determne whether structures should be excavated or left untouched. One such area where geophyscal practces are beng used to help archaeology was n Maax Na, Belze. In the summer of 2002, sesmc magng was used on the large pyramd at Maax Na. he pyramd s approxmately 28m by 28m and stands around 15m hgh as seen n Fgure 1. he sesmc data was acqured usng sxty-one hammer sesmc sources wth sxty 1-component geophones, planted vertcally, spread around the crcumference of the pyramd. Hammer sesmc has been shown to be an effectve method for mage acquston on Maya pyramd runs (Xu and Stewart 2000; 2001). However, to further develop sesmc magng to assst archaeologcal studes the data was used to determne the best analyss technques for sesmc magng around pyramd runs. wo separate surveys were taken at Maax Na. he frst survey was performed on the lower secton of the pyramd. hs survey had a horzontal recever spacng of 2m and were spread along a contour level near the base of the pyramd. he shots were placed along the same vertcal contour space n-between the recevers. As the crcumference of the pyramd was not large enough to accommodate all of the recevers the fnal 4 recevers were placed perpendcular to the crcumference and placed gong down the CREWES Research Report Volume 19 (2007) 1

2 Allen and Stewart pyramd (Fgure 2a). he second survey was performed further up the pyramd. he horzontal recever spacng used was reduced to 1m wth the hammer-sources once agan beng placed between the recevers. In ths survey, there were not adequate recevers to cover the entre crcumference therefore a gap n the coverage was left along the eastern sde of the pyramd as seen n Fgure 2b. a) b) FIG. 1. A topographc map of the Maax Na pyramd shown n a) 3-D perspectve and b) contour plan vew. 2 CREWES Research Report Volume 19 (2007)

3 Comparson of traveltme nversons on a lmestone structure a) b) FIG. 2. he source and recever layout for the a) upper and b) lower permeter of the Maax Na pyramd. Sources are ndcated n blue and recevers are ndcated n red. DAA ANALYSIS A sesmc shot gather s consdered good, f t has clear events and consstent frst breaks. o better pck some of the more dffcult frst breaks, a 500ms AGC was appled to the trace. hs often makes the frst break clearer. A sample gather from the lower permeter showng a sngle shot wth frst break pcks s shown n Fgure 3. In both the upper and lower permeter data, there were a total of 61 shots and 60 recevers. hs amounts to a total of 3660 traveltmes per survey. Before processng the traveltmes an examnaton of the sesmc gathers s undertaken. In assessng the lower surveys sectons (See Fgure 3) channel 1 and channel 35 appear to CREWES Research Report Volume 19 (2007) 3

4 Allen and Stewart be dead traces. hese recevers were klled. Durng acquston four source postons had to be abandoned due to an angry wasps nest. Fnally, snce we are usng 2D traveltme nverson further excluson of sources and recevers must be made. In order to perform an accurate nverson all locatons must be on the same vertcal contour. However, snce there were extra sources and recevers some were placed further down the pyramd then the others. hese shots can be re-ncluded f 3D nverson technques are appled. After the excluson of these sources and recevers there wll be a total of nne shots and sx recevers beng excluded. akng ths nformaton nto account the lower permeter s left wth 54 recevers and 52 shots resultng n 2808 traveltmes nstead of the orgnal FIG. 3. A sample shot from the lower permeter wth a 500ms AGC. Frst break pcks are shown n blue. Recevers 1 and 35 are dead. he upper permeter experenced dffcultes as well. Due to a looter s trench that was dug out of the pyramd through the open gap n the survey (Fgure 4), many rays had to be rejected. Snce many rays passed through the empty trench they would result n many transmsson effects that wll alter the correct velocty structure of the pyramd. In order to compensate for ths trench all rays passng through the trench were gnored. hs brngs the ray total from 3660 down to a total of CREWES Research Report Volume 19 (2007)

5 Comparson of traveltme nversons on a lmestone structure FIG. 4. he source recever layout of the upper permeter lne wth the approxmate locaton of the looter s trench n green. he hammer-sesmc source produced on average a relatvely broadband sgnal up to 200Hz. However, some shots produced sgnal up to 300 Hz. hese varatons can be assocated wth dfferences n hammer speed and power. Fgure 5 shows the FK spectrum of the shot from poston 1. he remanng shots dsplay smlar results. FIG. 5. he FK spectrum (left) and the ampltude spectrum for the average data trace of shot A mathematcal expresson for the measure of error n the measurement of arrval tme of the frst moton s gven by Stewart (1984) as Δt = f m 1 log 1 S 2 + N 2 2, (1) where f m s the frequency of the arrval and S/N s the sgnal-to-nose rato. As seen n Fgure 5 the peak frequency s 40 Hz. hs frequency was used as the frequency of the arrval. An estmated sgnal-to-nose rato of 5.26 was found by dvdng the RMS of the CREWES Research Report Volume 19 (2007) 5

6 Allen and Stewart peak arrval ampltude by the RMS of the peak nose ampltude. Solvng equaton (1) usng these numbers a traveltme error of 4.8ms was found. INVERSION he four methods of nverson appled to the data were drect dvson, damped leastsquares, sngular value decomposton and conjugate gradent. All four methods are based on ray tracng technques. Assumng straght ray paths the traveltme nverson s cast as a seres of lnear equatons = A P (2) j where s the total traveltme of th shot-recever par, A j s the dstance of the th ray through the j th pxel and P j s the slowness of the j th pxel. Expressed n matrx form = AP (3) hs s the equaton used for drect dvson. However, the goal s to fnd the slowness value so to get the fnal drect dvson soluton equaton (3) must be modfed to gve P = (4) A he damped least-squares soluton s seen as an mprovement on drect dvson n the case of nadequate or nosy data. he soluton for damped least-squares s gven by j j P A A 2 1 = ( + λ I) A (5) where λ s the dampng factor. he dampng factor used for the pyramd run data s 1e- 06. Sngular value decomposton (SVD) was also developed as an mprovement on drect dvson. Sngular value decomposton breaks up the dstance matrx to gve 1 P = VL U (6) where the columns of U are equvalent to the egenvectors of AA and the columns of V are the egenvectors of A A and L contans the non sngular values of A. he last technque used was the conjugate gradent method. he conjugate gradent method s a recursve method that works by mnmzng a resdual vector. After each teraton a more accurate velocty parameter s produced. A constrant of 30 teratons was placed for work on the pyramd data. Further descrpton of the nverson technques can be found n the appendx. RAY RACING For the ray tracng a grd system of 2m by 2m pxels was found to gve a hgh fold, see Fgure 6, whle remanng small enough to gve a good pcture of the velocty structure. For the lower crcumference survey a grd system of 19 by 19 pxels was used. For the 6 CREWES Research Report Volume 19 (2007)

7 Comparson of traveltme nversons on a lmestone structure upper survey a layout of 12 by 12 pxels were used. hs results n an A matrx of 2808x361 for the lower and 3054x144 for the upper survey. a) b) FIG. 6. otal coverage of rays per pxel for a) lower and b) upper survey. Perspectve and plane vews of the coverage are shown. he x and y axes are the horzontal dmenson of the survey and are gven n meters. RESULS Four dfferent technques were used to fnd the velocty structure of the pyramd ncludng drect dvson, damped least-squares, sngular value decomposton and conjugate gradent. All four technques produced smlar results for the lower survey wth veloctes n the range of m/s. Fgure 7 shows the velocty structure of the lower survey found usng SVD. CREWES Research Report Volume 19 (2007) 7

8 Allen and Stewart FIG. 7. he fnal velocty (m/s) map of the Maax Na pyramd found usng sngular value decomposton. Solvng for the velocty of the upper crcumference multple negatve veloctes were derved. hese are clearly not physcal therefore were set equal to zero for dsplay purposes. All the negatve values appeared n areas of low fold makng them unrelable. Whle most of the technques produced very smlar results for the upper permeter several hgh unphyscal values were derved from the conjugate gradent method. hese values can be seen n Fgure 8. Once agan these hgh values are n areas of lower fold resultng n values that are unrelable. a) b) FIG. 8. he fnal conjugate gradent velocty (m/s) map of the Maax Na pyramd for the upper survey wth a) no flterng and b) all veloctes greater then 2000 set equal to hese conjugate gradent results can be compared to the smlar velocty structures gven by the other three methods. Fgure 9 shows the velocty map derved from the SVD method that s very smlar to the damped least squares and the drect dvson maps. 8 CREWES Research Report Volume 19 (2007)

9 Comparson of traveltme nversons on a lmestone structure FIG. 9. he fnal velocty (m/s) map of the upper permeter survey as derved from sngular value decomposton. A vew of both the upper and lower velocty maps n relaton to one another can be seen n Fgure 10. FIG. 10. he velocty (m/s) map of both the upper and lower surveys derved by SVD. he upper survey has been fltered so all veloctes over 700 m/s are equal to 700 m/s. COMPARISON Usng the velocty structures derved by the four dfferent technques and the dstance matrx A a set of traveltmes were found. hese traveltmes were compared to the orgnal traveltmes measured from the raw sesmc. Lower Permeter he dfferences between the measured and calculated were graphed and can be seen n Fgure 11. CREWES Research Report Volume 19 (2007) 9

10 Allen and Stewart a) b) c) 10 CREWES Research Report Volume 19 (2007)

11 Comparson of traveltme nversons on a lmestone structure d) FIG. 11. he dfference between measured and calculated lower permeter traveltmes (ms) for a) drect dvson, b) damped least-squares, c) sngular value decomposton and d) conjugate gradent. he dfferences for each method are very smlar. Most of the dfferences appear to be wthn 5ms of the measured traveltmes. However there are a few cases where the dfferences are around the 15ms range. hese traveltmes are typcally from pxels of low fold and therefore the velocty structure wll not be as accurate resultng n erroneous traveltmes. o determne the most accurate technque an average of the absolute value of the dfferences as well as the standard devaton of the dfferences was taken for each method as seen n able 1. able 1. he average of the absolute value of the dfferences and the standard devaton values for all technques on the lower permeter. Drect Dvson Damped Leastsquares Sngular Value Conjugate Decomposton Gradent Average Standard Devaton Upper Permeter he dfferences between the measured and calculated traveltmes for the upper permeter survey can be seen n Fgure 12. CREWES Research Report Volume 19 (2007) 11

12 Allen and Stewart a) b) 12 CREWES Research Report Volume 19 (2007)

13 Comparson of traveltme nversons on a lmestone structure c) d) FIG. 12. he dfference between measured and calculated upper permeter traveltmes (ms) for a) drect dvson, b) damped least-squares, c) sngular value decomposton and d) conjugate gradent. Smlar to the lower survey the dfferent technques are resultng n very smlar dfferences. For all four technques the majorty of the dfferences are wthn the 5ms range ndcatng a farly accurate velocty map. However there are larger dfferences up to approxmately 14ms. Upon nspecton of the large dfference values the majorty are found to be n these areas of low fold meanng the velocty structure of the regon s not as accurate resultng n large traveltme dfferences. Snce the dfferences between the four methods are so smlar t s hard to determne whch method s the most accurate. o get a better ndcaton the average and standard devaton of the dfferences were once agan taken to help get a determnaton. hese values can be seen n able 2. CREWES Research Report Volume 19 (2007) 13

14 Allen and Stewart able 2. he average of the absolute value of the dfferences and the standard devaton values for all technques on the upper permeter. Drect Dvson Damped Leastsquares Sngular Value Conjugate Decomposton Gradent Average Standard Devaton CONCLUSIONS Four dfferent traveltme nverson technques were used on two dfferent surveys from the Maax Na pyramd n Belze. All four methods produced velocty maps smlar to those seen n Fgure 7 for the lower survey and Fgure 9 for the upper survey. Upon comparson wth the frst break traveltmes measured from the raw data we see that all four methods are very smlar but there are dfferences. he most accurate method appears to be drect dvson followed by damped least-squares, sngular value decomposton and fnally conjugate gradent. Drect dvson s perhaps the most accurate due to the scale and coverage of the surveys. Snce we are dealng n a small scale (28m by 28m) there would be lmted nose n the data. Also due to the almost 360 degree coverage we have good fold over most pxels there s adequate coverage for drect dvson to be accurate. Whle the velocty structure appears good there s stll an average of approxmately 3ms dfference between our calculated and measured traveltmes. hs average s nsde the estmated 4.8ms error for frst break traveltmes. In order to mprove on these results a curved ray as opposed to a straght ray tracer should be employed. FUURE WORK In the future, we would lke to solve ths data set usng dfferent methods ncludng fnte dfference traveltmes and 2d curved ray tracng. hs data set can also be expanded nto 3D. Along wth the 2 permeter surveys dscussed n ths paper multple lnes were shot vertcally up the pyramd. Usng all the avalable sesmc lnes a comparson of dfferent technques can be undertaken n 3D. REFERENCES Atken, J. and Stewart, R.R., 2003, Ground-Penetratng Radar (GPR) magng of near- surface structure n a carbonate envronment: CREWES Research Report, 15. Cardarell, E., de Nards, R., 2001, Sesmc refracton, sotropc ansotropc sesmc tomography on an ancent monument (Antonno and Faustna temple AD 141): Geophyscal Prospectng, 49, Karastaths, V.K., Papamarnopoulos, S., and Jones, R.E., 2001, 2-D velocty structure of the bured ancent canal of Xerxes: an applcaton of sesmc methods n archaeology: Journal of Appled Geophyscs, 47, Lnes, L.R., and retel, S., 1984, utoral a revew of least-squares nverson and ts applcaton to geophyscal problems: Geophyscal Prospectng, 32, CREWES Research Report Volume 19 (2007)

15 Comparson of traveltme nversons on a lmestone structure Mahmoudan, F., 2006, Lnear AVO nverson of mult-component surface sesmc and VSP data: M.Sc. hess, Unversty of Calgary. Merlant, F., and Musante, B., 1994, Sesmc tomography: applcaton for archaeologcal purposes: Geoarchaeology, 9, Mchelena, R. J., 1993, Sngular value decomposton for cross-well tomography: Geophyscs, 58, Stewart, R. R., 1984, Vertcal-sesmc-profle (VSP) nterval veloctes from traveltme nverson: Geophyscal Prospectng, 32, Stewart, R.R., 1991, Exploraton Sesmc omography: Fundamentals, SEG Course Notes Seres, 3. Xu, C. and Stewart R.R., 2000, Sesmc tomography of a Maya pyramd: Chan Chch, Belze: CREWES Research Report, 12, Xu, C. and Stewart R.R., 2001, Sesmc tomography of a Maya pyramd run, Chan Chch, Belze: CREWES Research Report, 13, Ylmaz, O., 2001, Sesmc data analyss: processng, nverson and nterpretaton of sesmc data. Vol 2. Socety of Exploraton Geophyscsts, APPENDIX Drect Dvson o fnd the velocty structure nsde the temple we must use an nverson technque. In ths case we use a seres expanson method. hs method nvolves dvdng the area of nterest, n ths case the pyramd, nto pxels. From here we can use ray tracng through the pxels to fnd a sum of the pxel values (Stewart, 1991). he pxels can be defned to have a length l and a heght h. he dstance that the ray travels through a pxel can be gven by d where D gves the total length of the ray as seen n Fgure 5. Fnally we set the slowness to be equal to p. FIG. A-1. A ray propagatng through a sngle pxel where d j s the dstance that ray travels n pxel j and p j s the slowness of pxel j and the traveltme s gven by t=dp, (Stewart, 1991) Snce we know the travel tmes of the rays we want a formula that ncorporates both the travel tme, whch we know, and the velocty or slowness, whch we want to fnd. We can fnd the travel tme of a ray travelng through a sngle pxel to be equal to t = dp. (A-1) Usng the equaton for the travel tme through a sngle pxel we can fnd the equaton for the travel tme through all the pxels. hs can be expressed as for the frst gven ray as, CREWES Research Report Volume 19 (2007) 15

16 Allen and Stewart t = d1 p1 + d 2 p2 + d3 p3 + d M p M, (A-2) = M d 1 j p j j= (A-3) where M s the number of pxels n the object, j s the pxel number, and s the ray number. (Stewart, 1991) Each shot recever par makes up one equaton gvng us a system of lnear equatons. Expressng ths equaton n matrx form we get = AP (A-4) where s the traveltme vector, A s the raypath geometry matrx and P s the slowness vector. Usng ths equaton we can fnd the traveltmes of the rays gven the slowness values and the ray path. However, snce we have the traveltmes and are lookng for the slowness values we must solve the nverse problem. he exact soluton for ths nverse problem s gven by 1 P = A. (A-5) In order to solve for the slowness usng matrx values equaton (A-5) was modfed to the drect dvson equaton gven by Damped Least Squares P =. (A-6) A In the presence of nose or nadequate observatons the least squares soluton s seen as an mprovement on drect dvson. he least-squares soluton (Lnes and retel, 1984) s gven by 1 P = ( A A) A. (A-7) In order to obtan more stablty n the soluton a dampng term may be added. hs results n the damped least squares equaton P A A 2 1 = ( + λ I) A, (A-8) 2 where λ s the dampng parameter and I s the dentty matrx. Sngular Value Decomposton he beneft of sngular value decomposton (SVD) s that t provdes a precse way of analyzng a matrx, whch results n a stable and approxmate nverse (Mahmoudan, 2006). he SVD method allows the matrx to be expressed as the product of three matrces, 16 CREWES Research Report Volume 19 (2007)

17 Comparson of traveltme nversons on a lmestone structure A = ULV, (A-9) where U s an m x m orthonormal matrx of egenvectors that span the data space, V s an n x n orthonormal matrx that spans the model space and L s the m x n matrx that contans the non zero sngular values of A n t s dagonal elements (Mchelena, 1993). he columns of U are equvalent to the egenvectors of AA and the columns of V are the egenvectors of A A. Insertng the SVD of matrx A nto equaton A-5 the equaton for the slowness s gven by 1 P = VL U, (A-10) here are a few advantages of the SVD method over the least squares method. he man advantage comes when dealng wth a matrx that s sngular or near sngular. he SVD wll stll work wth these matrces and can produce a numercal answer that s stable (Mahmoudan, 2006). Conjugate Gradent he conjugate gradent method starts wth the least squares soluton of equaton (A-7). hs equaton can be arranged to by the normal equaton gven by A AP = A. (A-11) Followng the method descrbed n Ylmaz (2001) D s defned by and B s defned as resultng n D = A A, (A-12) B = A, (A-13) DP = B. (A-14) In order to recursvely fnd an estmated value for P we start wth an ntal estmate X o. hs ntal estmate can be defned as a null vector and s termnated when the resdual vector R defned by R = B (A-15) DX becomes a null vector. Where X s the parameter vector estmate after the th teraton and f the matrx D has dmensons n x n the conjugate gradent method wll yeld the soluton after m < n teratons. he ntal values used n the conjugate gradent method are gven by and c = 1, 1 (A-16) G = 0, 1 (A-17) CREWES Research Report Volume 19 (2007) 17

18 Allen and Stewart R0 = B DX 0. (A-18) o start the recurson, frst compute the ntal resdual error c 0 usng c = R R, = 0,1,2, m, (A-19), where c s the resdual energy after the th teraton. Input the derved value of c 0 and the value of c -1 gven n equaton 16 to the formula c b, (A-20) = 1 c 1 usng the derved value of b -1 and the values of G -1 (equaton A-16) and R 0 (equaton A- 17) n the equaton a value of G 0 can be obtaned. G (A-21) = R + b 1G 1 Applyng the equatons: Q = DG, m n, (A-22) d = G Q, (A-23) and c a =, (A-24) d X = X + a G +1 (A-25) a value for the parameter vector estmate X 1 can be derved. Fnally the new resdual vector R 1 s gven by R = R a Q +1 (A-26) Plug the value of R 1 back nto equaton A-19 and repeat the calculatons of equaton A- 18 through equaton A-26. hs could be done for any specfed number of teratons where m < n. 18 CREWES Research Report Volume 19 (2007)