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FROMDistribution authorized to U.S. Gov't.agencies and their contractors;Administrative/Operational Use; 15 APR1971. Other requests shall be referred toDefense Advanced Research Projects Agency,Arlington, VA.
AUTHORITY
USAF ltr, 31 Jul 1972
THIS PAGE IS UNCLASSIFIED
."AFTAC vroject No. VT19707 " I.- 9 C..-. AT-AC
" \ , - 70 , " " E t
RAYLEIGH-WAVE _MtLTIPATH ANALYSIS-- JSING A COMPLEX CEPSTRUMk TECHNIqUJE.-
rSPECIAL REPORT NO. 2
. x TNG-EO AA Y S-PN ROESINGOIEpORA TEDN" [ " " ertes) -up--
." Prepared by A. Frank Linville
, Aaron H. Booker, Project ScientistStanl3-ey1 'Lastei, Project Scientist -
T. W. Harley, Prograni ManagerArea Code 703, 836-,S s 8e , Ext. 6
::" [ : . -'-DA C D R S A C R ff C S A E I Y' '' '5
TEXAS-INSTRUMENT INCORPORATED"
, . '. / R A S e r c e s G .o u p 9-i" "-P-. 0. Box 621 . -Dallas Te 75Z222
as, fe,~
C; uonrtract No. F33657N69-C-1063
tof Contract: $779,8 5LLJ Beginniffg 21 Airil 1969.Ending 31 March 1971
[ tMC -Prepared for
AIR FORCE TEC'HNI.AL APPLICATIONS CENTER* ashington, D. C. 20333
[ ;**Sponsored b-ADVAN'CED RESF4ARCH PROJECTS AGEINCY
"S C 4 Niic1lear Monitoring Research. Office .V
t ARPA Order No. 624AR PA Program Code N 9F10,-
I 1~~ April 1971 C -
Acklnowledgez~~ent: -This research was sutportqd by the Advanced- Res~crc' Projects Agency, Nucle4'VMonitoring Researcb Office,
Iunder roject VELA-UNIFORM, a'nd acctnnplished under the tech-I lcal/ direction ofthe A!J Force Technical Applic-ations Center
- ,und~r Contract No. F33657-69-C-J063..
services group
a I
...
--. Statio
01
-C 3
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mn or a ted1L10JO I Usngo i * w atr om uoval UnZZ -I icles'1d1L ORIINATING £ LvIVI )rso" u. Napo"T SECURITY CLASSIFICATION
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I 1L SUPPLEMEN -ARY'NOTES 12. SPONSORING MILITARY ACTIVIT Y
Air Force Technical Applications Center
ARPA 0. rr No. 624 VEIJA Seismological Center Headquarter;
-I- . I IUSAF, Alexandria. Va.fS.- ANGTRACT
Homomorphic deconvolution is. applied to separating the components
of a convolution in which n-ultipath Rayleigh-wave propagation is modeled as
theconoluionof a signal S with a multipath7 operator nm. The technique is
applied to,a number of s~,nthetic waveforms with fairly simple multisource
and/or multipath Rayleigh-wave c~faracteristics tor demonstrate the separa-;
tion obtainable between S~and m.11 The technique is also applied to several
wrents recorded at LASA and ALPA and results of the method as applied to
actual Ra.yleigh..wave rcd sare presented.
Di Di .ov*1473 Unclas sified ____
Security Classificationl
Unclassifi~d-- -
Securty clabe.icaIus -
LINK A LINK 2 LINK C
KEY WORDS JW. VT ROLE WT MOL .!..T.
Rayleigh Wave s
Multipath -Propagation
Complex Cepstrum. Technique 4
Linear Filtering
Signal Estimate. I
Multipath Operator
I-ASA '
ALPA.
'4- tyZ1k* 1iC f~
I - . AF7.
a - .
1RAYLEIGH-WAVE MULTIPATH ANALYSIS
USING A COMPLEX CEPSTRUM TECHNIQUE
• ii SPECIAL REPORT"NO. 2" ONG-PERIOD ARRAY PROCESSING DEVELOPMENT
Prepared by A. Frank Linville
. AaronH. Booker, Project ScientistStanley J. Laster, Project Scientist'.
T_ W.. Harley, Program ManagerArea Code 703, 836-3882, Ext. 60
TEXAS, INS TRUmAENT INCORPORATED .I t- ". Services GrouP"* ~~P.O0. Box 9621 -.
o Dallas, Texas 75222,'
.. Contract No'. F33657-6.9-C-1063Amount f Contract: $779, 850
.. leginning 21 April11969Ending 31 March 1,971
U \Prepared forAIR FORCE TECHNICAL APPLICATIONS CENTER
Washingion; D.C. 20333 -
Sponsored byADVANCED. RESEARCH PROJECTS AGENCY
,.-Nucl~ar MonitorinJg'Research Office fARPA Order No. 624 .
ARPA Program Code No. 9F10
15, April 1971,
-Acknowledgement: This research' vas supported by the Advanced
Research Projects Agency, Nuclear Monitor'ing Research Office,uxfder Project VELA-UNIFORM, ahd accomplislbed under the tech-nical direction of the Air Force Technical Applicatfons Centerunder Contract No. F33657-69-C-1063. s '
b, services group
/
. ii
-TiIpg'i ntnionll-anoi"
I U
- f.
* serice griu
-j K SUMMARY2j -
Hornmorpi :deconvolution is applied to separating the corn-
* ~ ponents of a convolution in which mnultipath Rayleigh-wave propaLgation is
X modeled as the convolution of a. signal S with5ia multipath. operator tn. The
te~chnique is 'applied to a numhber of synthetic waveforms with fairly s;imple
1 ~ iultisoiirce and/or multipath Rayleigh-wave characteristics to demonstratethe separation obtainable between'.S aiid 'i. The technique is also applied to
several events- recorded at LASA Lnd ALPA and'results of. the mn~tlod as
applied, to actual Rayleigh-wave recordings are. presented.
.4'h
3'
sevie group
" !ITTABLE. OF CONTENTS
N" .
-, ;Sectioni Tifle °_Page
• , . . . I,-
SUMMARY
, - • .. -
.. I INTRODUCTION- -1 ' -. }
1 II TECNUE,-
III- THEORYAND COMPUTER APPLICATION 5
} " IV :.PROGRAM' TEST CASE' . ... 9og
SEAL. A Y OF SYNTHETIC MULTIPATH
.... ;RAYLEIGH WAVES - o12
-eci PRELTINARY CEPPage.
M ANALYSIS OF RAYL.IGH
i" "WAVES I EcORDIED AT J.KSA AND ALPA 2 8 ' i
U-MVA REFERNCES 45
" APPENDIX
IA COMPLEX CE-PSTRUM PROGRAM DOCU NTATION'- A-1
R E WAESi- \ 12 .
* I - RLMIAY E R1MAALSS.O AYEG
WAVS RCODEDAT S AN L 8I
* II RFEECE ,4
C- iv-ee;go.
APENI
C. C
Figre 'Title - - -g
-* o f Cs5 -
e I-T ILLUST ATIONS ' ""-ve "r
0 S an - w- 'SecS
o , " . " 2
:"~an 250I bv° d~rm.es litud rroa ITest ae " lnit--
"V-5 Cepstrm Result
e m RsuIti for'Synthetic Multipath" Rayleigh i
,Waves. Input is--the Sum of Two Chirp Waveforms; -
"u 500 Sec "and 500- Sec ir Length with Onsets at I Sec , ..and 250 Sec and Arplitudes ofl Unit and 1 Unit" "" V-3 "Group -yelocity vs 'Frequency__ Plots " .•-
' .
v-6 Cepstrum Results for Synthetic Multjpath Rayleigh 20-oWaves, Input isheSumof .Two Chirp Waveformps;.
600 Sec5 and -600 ec in Length with Onsets at iSecat n d , 20 c d Amplitudes of 1 Uiiit and-'Unit.
V-7 Cepstrum R'sults for Syfthetic Multipath Rayleigh ? ZLWaves, .,Input is the Sumh of Two Chirp Waveforms;
- 500 Sec ad 600Sec in Length with On.ets at 1 Sec.. -. and 250 Sec and Amplit.des of I' Unit and 1. Unit4i, . .. ,.
V-6 Ceps~trum Results for .Synthetic Multath Rayleigh 25*
Waves. Input is ;the.Sumof Three Chirp Waveforins, "
" U - 500 Sec .'5506Sec'and600 Sec'in Lenith ffsecI - - " at 1 Sec, .o201 "Sec and 40T Secand Amplitudes of " a 1.S *1Unitan IedUnit R .th F, W
V-9 Cepstrur Results for Sythetic Multipath ,Rayleigh •6 4.YWaves. ,Input is. the "Sum of Two Chirp -Waveforms; ,
400 See" andL e
6 Seand 0 ecinLength with Onsets "at 1 Secand 250 Sec and .ip'litudes. of ' Unit and 1/2. Unit
° #-8 C~epstrum Results for Stheti ltahRaegh ' 5 .., . -o. Waves..Input is the Shn. o f-Two Chirp, ~Taveforms:" .
-- . 50 Sec and,-500' Sec in- T ength with .Onsets at 1 Sec ' " ,"
T h e S e c nt C h rp W av efo m , wi h O s t a 10 S e ," .IsInertd ithl~spet o the First Gl)irp" Waveform ,- _
V-9 Cepsrmh Reslts forSynthetic Multipath ,RaleihZ'
Waves•Input is hSu ofTohipW efrs;-' .400Se nl;600 Sec in ,Length with Onsets at .I1 Sec ,. .
"andl Seceand Arnplitudes of 1 Uriit and 1 Unit '"
services group
/9 -I
* --
.i" . " Fire -- -Title 4 Page ;
-- -
1-'1 Cesrii Results for Event N3S03 Recorded a-.AL.PA. 31
mop Input-
-'" • ~Simga.: Bottom,. Sbnort-Pass Out)"" ""-
" " V1-3 Cevsfrumn Results ot- Event L01 Re~orded at ._4&A'" 35-.SGou Velocity vs Freqency for L-OI- (Top, Input-36
SSignal; -Bottom, Short-Pass Output),
VI-5 /TCepstrus Result* for E ILLeT . atS Re L-A- 57
SVI-6 Group .elocity vs Fre4ency for LL-200. (Top, hz.t .36
Si-2. . Bottom, -- Title N, Paged
" "VI Cev . strum Results for-Event LL-01'e Recode atLAA 3 - ,39
- - " I-4r Velocity vs Frequency for LL-018.- (Top Input 40 -
. .. Signal; Bottom, Short-Pass Output) ,,
YV-9 Events L 2019 and LL-2028 (fE nt t Signaclsr Cetatrm 41 -
VI-6 G~~~rou oiyv reunyfrL-0S Tp, nt 3
S VI- 0 Sroup Velocities for Input p S ga)s and ShdrtPas.
" VouItuts. Flter at 64"Se . EirentL EPX 14646, AA 39
-" "" , 1001, and EPX 146e-9 -- -
.. VI-11 Group Velocities for Input Signals and Short-Pass 4 5-6. -
"". , - . Outputs. Faiter at 64 Sec.' %In ut signals-are Single-'.C-" annel Recordings of Event 31PX 14646. -
•' -" " -'QST OF-TABLES- -A . . - . .- , .. -
" able /Title Page
Ei- . Vi-1 , Event Informati6n Catalogue J " - 30
-.. "vi" . .services group , !
VI
A -
Th xiaini-,pol~to of Rali wae thEu ln
1utlyri~ci is- wel rto.Ery*ui-s-fRv~hw eod
Rayek wav - roa- t V-- aln h-et crcepah
paat anThe gecatio frcd pth butimay of lralyhfae throu igger
vlctyl' stucue n rerto-i-o.ienamakn __c-~ekte o
1in file -tinfrtion ary -is the verageo rustal tructuri etwe the.fih
oficentrand he redn phieb veloci vrnste aurem. ~ess Tha naysi
1~ I9stantleshae deviaatn long diethe-g racl pafrm yl eighto recqfdom-the
-0- ca V
gratciie,-pt froannlcrdd stains anhtdst Inadr nedelor~ h~ircit - - - --------------
paft aloc o-e greteciycle pthdbut.myb a~al eratdtruhhg
-~~~~~~a e' eoi tutr an refractonas cif ieta margn whihrk-h'idbet~en ceaic n~cntinnta crsse a well etbi she b actiondn (19 3,us
-- ~y 194) an -rprt aofa ac1904)~ deemi hodi tisno be thli~e dted and ~ s theie aray Std, -Platn
fomyo tehnique ano racltn sps~vlciand direction o app~esroc su-.-
staniats te ~iaton f drecion of pprachof aylighwave fromei thSq P
mutp p-rbgto.-A eid whc 7or. .'
because of rnultipa proagaion
a minimum of, enrg y i s observed in the recording, the amplibde. spectia show
~largi miima and ireuarte pnterimodshes whichpa pctaocu.
These phase ir~iregularities otc albsatrIicopedhsevoii-S7
- .and svecial toechniques must be used io r~duce:the influence of beats
- 'Freque cy-wavenumber inalysis by Capdn.(1970) yield- additional-
evidence orf mrultivath propagation which is frequency jdep~rndent and conjectdres
were made conC rning the actual paths taken bFy the various- freunc 'groups [~
arriving sirnultaneously at LASA during successive'2 6 0-sec tixiie intervalg ov'et
the .RayleighA~wavei~ratibn. --
- - '-Measufrements-ihade frorih the ltecorded& Rayleigh wavefoYm to
determrine group velocity,, -phase-velocity- and, source d..e ,th are strongly infiu-7 1 11 .. I; -
* encd frjmultpathi propagation yielding resulits whch are difficult- to int pe.n
addition various discrimidnants calculated from the contafninated Rayle'gh wave
(spec 41 ratio. MS;7AA, RMS, Mchirp, etc: ),;are not representAtive 6 'the
Ryegwaeexcitation at the sour~ce._---/
-. *The analysis presented he re- then', is concerned with rec n .
the signal and the multipath, pro from the recor ded Rayteigh waveform.'
Hiomomorphic deconvolution-using a gener~alized -c oicept of lineaz filtering'~
applied, to lsepar Lting the components of a convolution in -which multipath ro-
The - is~~ - as/th convoluti~n of a F.ignai with ak mwltipat~p~tr
Tesignal would be more, repre .sentative of tiie "pure" Rayleigh way an~d the-- U3
A ~ -lipath,.qpe rator'could be i'erpreted in texrn--of differential di s ersion -
- -4---andl-/or -pe, ., - 7s-.
- . -X
*srle group,
SECIO II-
TECHNIQU
i L
sinas The a0u2p- igal. pproach- attem~pts -separati on using a"complex cepstrumV" techn i-
q/'! - que. He ivei:a- detail~d -analysis of homomorphic ytesytms forkdeconvolution
and applies the -technique lo.,a clash of signals in which one of the- members of
f he conV7o1utioni is, an impulse train. Thetechnique i s cocptal simple and'is surniiiri;;ed~a sflos:
*
o The problem here-i§ to estirm e 'asignalStt) which has been
*complicated b9 a' _ejfli (ti-i-t ut,
path qpperator); *MatheiiE'allythe problem. is X~t) =S(t)
/ - - am(t). where iq4loftvo1lti'n and X (t) is'teosvd _
~' " > 'Iwaveform. -' '*-
* ~ ,, ~hzoFouri~r tranrsform is taken andlthus, X(z) § (z) (4th e '() is tae dn I-ismd
*-~_t Timau> 1 ~ ± z g in a ry a rt- s m a d etinuous and Z) ls', log X) + log m(z).
Log X(z at t-e'ed as acomplex time aeries and a- liuhaI
-sa r
-filtei sused t \o separa~te log S(z). and log- m(z). Bascially,
7 ~~ hi 'requires. that the-spectrum of log S(i) and log m(z)
7, ot ov~erl~ap g'reatl . 'i
The exponential is 'takep to et'back .to 'an estimat of
-, - *S(.z) or m(z). There is no gimbigpity here' as exponentiation
- - - .will map any one of the multivalued" log 'function values bAck
- into the same. complex nqrmber..
* Inverse-tr-ansfo rm'.to get/an estimate of Set) j r mn(t);
3- services groujp
L
Thi -tcniu hi bee usd-csflyb'caer-n§ itn pe
daa T hecuilln-~his technique is benuewucsflyb~hher rno loeparatin logeecki
- are sep~rable fOr multipath Rayleigh Wz~-~ai - The numerical procedules-
given by Schafer were Orogrammed ibr the S/360 computer and synthetic wave-
* *forms rep-resenting fairly simple multipatj Rayegh- ae irdins were
~; -used to de-termin,- the sepa~ration ob ai!na le. The sequence of numerical calcu-,
lations involv i!;iimp entivg teter-hniqtie is described in the following
_1--ection-- --
4..
t-. -. 50h1
-. 4 /-L .. ' 4.4- 4
SEqTION II
THEORY AND !CQMAPIUTER Ai:P'LICATION PROG RAM
-~ rdon transforming a convolution
of aveor-s ito su, uing a inei lter sytmto separate the additive'-
components and transform~ing the result back to the original input space.' The
VI *numerical operations performed'are -- momarized as fo~iows:
- -<MAKE PHASE
FIT- CMI L ANDCOT EFT
FFT LOGARITHM iANTNOSREOTVE I
-COMPONENTALFT
the ~ ~ ~ ~ ~ ~ cope outputm ytsoh simt fSi o'r~)
FFT~ IF
1... ..... knVRXak Y a~ xrng NY l . .............. . . . , N -l...
NI 21
X~~~k)~ .~ N 0 1.
0 ~ n=/
se'icsgru
The signof- X(o) is checked and if negative, the sign of X(k) is changed to remove
the constant phase component. Thi -is remimbered and the sign of Y(k) is
subsequently chaniged in the inverse system. The complex logarithm is taken
* X(k) = og [X(k)] f-- log IX(k)i- +i ARG [X(k)] i i2T q k= 0, 1..., N-
and the principal value ARG [X(k)'] when q = is ob ned. where
/: -W<ARG [X(k)] < --
The complex logarithm is. multivalued and must be defined such' that there is
*no ambiguity with respedt to its imaginary pant by choosing the intger q whjich
makes the discrete phase values approximate, a continuous phase function. This
may be accomplished even for rapidly varying phase angles provided the fre-
fquency sampling is fine enough by either one of the two subroutines SCHAFR or
AARON SCHAFR-uses. a first difference and AARON uses a one-point prediction
error to detect jumps in phase. of Zr . The input signal is augmented with zeros
to give an input trace N sample points In length in order tW obtain the fine fre-
quency sampling needed. The value N is a variable input-parameter to the pro-
gram. The estimated phase curve is then rotated by removing the linear phasecor~xonent and arg [Y]is subsequently reverse rotated in the inverse system.
Next the inverse fast Fourier t'ransform (IFFT). is taken.
N-I i Z* kn /A
(n) 1 X(k)e N / n=0,1,;..,N-I
.•y.,elding--te-compppexrimeapr ily-cn c
-The pari due to S(t) in the complex cepstrum is primarily concentrat-
ed in the "short time" region while the impulses attributed to m(t) occupy 4'
the "long time- region. If these sign;at r& separated in "time", then a linearfilter system can be used to separate the signals from one another. A short-
pass or comb filter system can be used to remove the impulses due to m(t)
from the comDlex cepstrum while retaining most of the part attributed to S(t).
Similarly, a long-pass filter system can be used to remove most S(t) while
6 services group
0
. _ __ _ __ _ _ -°- ....
L;. retaining the parf due to ne(t). ',The complex cepstrum x, once calculated, 'is
saved and the program 'recycles to read -input filter -parameters, apply the
filter to the complex cepstrum and calculate the output of the inverse system .
This allows the analyst to obtain outpu ts for a suite of filters for interpretiveI ; purposes, The short-pass linear" filter-system defined as
f(n) = '1 .- n <n <n.
0 elsewhere
- - n - 0 ' ' ". "n
where n and n ar'einteger input paraineters. The long-pass system is
defined as'
f(n) , 0 I -n-I I<n <n /
•. . :. 1 . i./elsewhere \ x
A fn)
-n n
2,'- ,i,
' . where n f r integer input parameters. The comb filter system is
...... , d'efin 4 as '°/
• % Ankf-n 0 < n < nk +k = 1" 2, .':. .
1 elsewhe/re
1f(n) ~1
n 3
whei'\e nk and 'n are input parameters.
services group
7/
. " ,
The filtrstems are applied in the program by zeroing the appro-
priate _portions of the complex cepstrum. -The sharp cutoff of these filter systems
will produc of y;therefore an option is provided to
modify the filter ampltude function with a cosine tapergiving smoother transi-.,
tions between one and zero.The filtered complex ce ' then is input'to the inverse system
where the FFT is.taken, the imaginary part reyerse rotated, the- complex .
- exponential taken, and the IFFT taken to yield the filtered output y.
Preprocessing of the input signal can also be performed by the
program when necessary.. First, the mean is removed and the signal'is trun-.
cated at the nearest zero crossing at each end. A zero-phase bandpass filter
is designed and convolved with the signal to allow decimation, the cutoff'fre- Lquencies and decimation interval b ing variable input parameters. If the .
input sequence is nonminimum-phase, then exponential weighting may be ap iedin order to make the sequence more like a mphase sequence. It h
been empirically observed that this will insure that the part of the co'mpl cep-
A strun attributed to m will, occur in, the region t t where t is the spacb e t w e e n"th eo ntr t ia n d s c o d
between_ the firt and second impulses.,. The program applies a truncate, ex-
. ponential window to the input ence -"
O/t<Lt 2
(t) = d <t) O
. - 0 elsewhere . .....
wherel aI < I , L is the length of the input windo,@ and a is a ariable
inputparameter. Subsequent calculations are performed on th exponentially
weighted input sequence, and the -output of the inverse system then unweighted
using the same value of Ca.
- addition, ihe program generates CalConp plots of various
functions odned duringthe sequence of calculations to allow visual analysis of
the results.
8 "services group
(
. '
ii,-/
I, ' - , I
/ - °SECTION IV "
• PROGLAM- TEST CASE
fi In order to chec4. the programmed brocedures~a siinple example
Sgiven by Schafer was processed through the program.
... -- Input to the progr am i X whose'values are:
where th' sqen'e-S---has- values-- -,--
L sequen A, // / , ", '
t6
= . elsewhere
7'7...- ,,.
and ahend sequenceS has valuesL
4.aSt-) 80 <t-:5 880,t 0- e lsewhere
-" " " 0 ". , - "
*° an d ;he" sSa - : . ' '-
For this example, a 0.96 a 0.5.9 80, and the value N fo the L4 ast Fourrtransfori was 2048/
Calompplotedres ts of the seqeeq calculations are givent)(= (t t I fromttott tm-, aei
in Figu h5wn in the 'figure, proceedi'ng op
C- / /
(t-.":5_ (t) and their sum (t)., -Next shown a de the real and imaginary parts
of 'the. co nhplex logsaithm of theFFT of the Alt X t). The imaginary part
'
. .~ services group
,L 4 / ,
- f-I. jjfi
has beep made continuous andtthen rotated to remove -the linear component.
The IFFT of the real-and imaginary parts gives the comrplex.cepstrumn For
this example .the input X(t-) is minimum,'phase, after rotation, yieldifi a-after rotation,.yieldi"a'
complex'cepstrum thatis' zero for t,< 0.- Thle part due" to S (t) is primarily
-in the "short time" region.from t = 0 to t 80. The impulses.due to the -
echo. are in th'e "long time" iegion and occur at t = 80, t = 160, etc. The
complex cepstrum is then shbrt-pass filtere'd at sample point 74 and theFr -.
taken. 'The part shows the effect of ,removing the rapidly varying component
due. to m(t). Th'e imaginary par s reverse rotated, the complex.'exponezntial L '
* taken and the IF-I'T..t~ken to yield the short-pass output whidh is the estimate(t) miu th io -0ihc
of Si). An error trace is calculated [Sl(t) minus the short-pass ouput] Which0,. shows tAt eoutput of the short-ps, system estimates S (t)yrith negligible
7 ,error. Next, a long-pass filter at sample point 74.i. appli6d to the complex
cepstfdim and proc'erz ,-d as bdfore: "The real part shows the6ffect 9f- removing
the. slowly varying component due -o S (t). The outputof the lOng-pas! system
" *(bottom trace, --Figure IV-l) yields m,(t-with negligible er or. -.,.. , . , , --,, N * ,, , \ i. , . .
/ k, ,
/ " / ,.
.08 //
/ 2,
/ -4< N , 4
* services fjoup10.
-- 0 . ,,, . . n'4 S i
N
ISI
*. - . -
- O A
19-
--- - -- ---------
,L -. - -------- . - --------
L." .,,. *" . . " - " . . - ." " " : '
•- .,- -" . _ , /Vr
U -"- -* 0
, .1 ' " , + / -, . _ - .
L L
if.- .. . , '4 " 1 - ,I • / • '- " * +S - i ,
------ -- ---- "- f --- - ---- ----
U%0
LI J-.-----.. . - I, '-."-
" ' ' , " " " • ,- • -.
- U - - - ** * -, .- .,, .."- + : , . . . . ,-
I , ...- . . , ,,. ,,.*, - ,'.
S %M ,
, f; -'. - " "", • ' -
1... ..1 t,.lrl •
'AFigure. IV-l. -. epstrum Results ar' Program T.st Cas ..
-- -'-,I -,+ I "
L•. - * 4"
"" .. 1 2" ": services gi~up., . V•
• i, % ".. - ,' . ,.
. , , SECTION V,. .
-CEPSrRDM 4"ILYSIS OF SYNTHETIC ULTIPATH "
-YLEIG WAVES- -
-- . -" . -
Seven synthetic wav-ef6rms with smnle multisofkrce and/or multi-
path Rayleigh wave characterstics were processed using th& corp.lex-cepstrum
tecbique in order to determine the separation obtainable between 4(t) and m(t)
"and t6 serve as an aid in interpreting subsequent outputs obtained from actual
Rayleigth waye recordings. . ,
SThe waveforms used'wereproduced by adding together two or more
cosinetat:l chirp waveorms. j First, a chirp waveform is gcierated by
spcifying the jmtial frequent'iifnal frequency f , ana time duration L,' -a* [i '- - . V, C! U . .JI
where the I IVulse'resjonse is,,e '-S I,
- --P--tosinr2 t t 0
Y---- - S-- otherwise,
where
-- 0. 025 Hz and f C -- Hz
Show in igurie V-I (top) is a chirp vaveform (R(t)with a time duration of O-
'500 sec. - "
In ordeXto -make the waveform more nearly resemble an actual
-I - ~' sejsmograxn- recording of aRayleigh wave, a c 4 ine tater is a ed to R(t) where
the-output is
S(t) R(t) -. cos 0,!); t .L
This is analogous to shaping the spectrum with a recording system's responsy
. (long-period seismometer):. The cosine-.tapered chirp waveform (Figure V-1,
. middle) resem~ble5 anact al tetseismic seismogram recording of a
wave in that it exhibits "normal dispersion and its amplitude spectrum is highly
-.. *- 13.' services group
5- -
/ I• a
.. . .... --.. - - -----
M - 1
- -APR DC I R~ P 1 A~--. -S-U*
4 1 - 4 ~-.21t I SE'i
DECIATE
f4 /~~cs ru.
i . " "1 - "'" - " I' -
4'. . / *j
t , ", , _. .
* 6
-peak. - *
Application,.of the 'cepstrum tdhziique was iounii empiiicallyce.tr. .h e ao
\to work better by resampling the data at 8-sec inter-rals (Figure V-1, bottom).
Thii sample period gives a Nyquist frequency of 0. 0625 Hz which is. well out-.sidpthe fre4iefc-y-banid specified\fr th 4 .chirp waiveforms. As;.oted in Figur.e
V-iresampling f the wa;e foim yields .an irregular a;earing time function,
howeerthere is no lobs, of spectral information in th'e h~and of interet -aa
meescalculaed dire ctljrfrom the decinateA-time function such'as op
.&vel6city (wave frequency and 'arrival ti es of 17;aks and troughs) and maximum
amplitudes of given peaks and'troughs '.edeter.ifned using a three-point quad-
- tatic interf)latio'nyielding results, ess tially identical with those calculated
S 'from the -4W. efo n sampled at--1 sec.,
Tti-fi'rst multichirp wavefo analyzed consists 6f a 500 sec chirp
with onset at 1-sec plus a 500-sec. chip with onset at Z50 sec. Shown in Figdre V-
L. -are hthe two individual ciirs, their s ,-and the decimated sum which is the
input- to the compex-cepstrum a alysi r0gram. Group velocities cal- ,
cufated from ttie wav formr are prese ed in the. right ide of the figure. For.refetnce, the group velokity, curve (sol d line) -for the LASA perturbed Model
(Texas Instiurments Incorporated, 1.967 is included in the plots. - -A distance
and Qri ntime is chosen'such that the esultant group velocity pointq calculated
from S '(t), losely fit the LASA perturb d curve. Fai thin example, the di tance
was 9053 In -and, the 'origin time was&25 3 sec before tr~ace onspat. / The multi-
chirp w'vform yields a complicAted gro p velocity curve. • The smoot points at
at low frequeoncies and the smooth points a high jfreq pncies derive from the non- "
-lapping'portions of Sl(t) and S t) res ectively. A domplex intexference
pattern results,where Sl(t),'and S2 (t) overlap in time yi Wing the scattering of
ptint.s observed at-intermediate freqencies. Forgrup velocity
calculations for several earthquakes (catalogued i Table VI- 1) recorded atS LASA are preaented in F S..gs of Rayleigh waves from
N events occurring inf maPy widely separated regions yield complicated group
velocity curves with characteristics very similar tthe ultichirp examples-
presented thislsecti~n.-
s gr/up
/ ./
-S5 ' I - .
• *"- .i
, t) C
a.)."";n7. .7-.Y
- .- _. _.. .. . _ - _ _ . . . .. . 5 - --
) - V - " . . . . .. a. I
"_ '__ __ __ __/_ __,_ ___,_ ,~~
__ [K.....
) ' -nta dIU i
f .,- -
5- 5
UA219 tom
7t.
a 0.01 OC 0. 03 0 .01 0 O.Ob 1 0.3 0 0.01 0..05al OA 0.0 01.
-- - -" .C N
1.'V 1
Tigue V-. Goup elolityvs Fequncy lot
17 se-6s ru
The multichirp waveforms are not minimum phase,therefore ex-
ponential weighting is applied to the input signal and subsequent calculations
are performed on-the exponentially, weighted sequence. The valu for a used
in the weighting was 0. 98 for this exaimple, and the output of the system is then
unweighted usifig the same value foi a.
Results of the sequence of calculations using the complex-cepstrum
technique are presefite d in the remining portion of Figure V-2. Shown are the ,
real and imaginary. parts of the complex logarithm' of the, FFT of the exponen-
tially weighted input signal. The slowly varying component due to S.and th rapid,
ly varying component dueto m are clearly seen in Ithe plot of the real part oft
the log. The value N given in the figures is the number of samples for the FFT
. and is a power of two. The input signal is. augmented with zeros-before trans-
formation to give a trace N-samples jong. For the frequency-domain plots,.N
1. is the number of frequency samples and the -folding frequnecy w is a N/Z. 'or
* , " a sample period dt = 8 sec, the folding frequency is L '=-, - 0.0625 Hz. .
and for the time domain plots (complex cepstrum) o V
/ N12 Nat/2 = 819 sec
The complex ceps um is shown next. The impulses attributed to m are small Lin, amplitude relative to the puls attributed to S which occurs primarily in the
shrt time' re\gion;therefore an enlarged plot of the complex cepst um is
also shown. The large pulse about zero time is clipped i the enlarged plot
for display purposes. The impulses attributed to m oc ur at' t = 249, Lt 49 etc. and are easily discerned in the enlarged p ot.
oI tn tThe complex cepstrum was short-pass filtered at 208 sec, the
second -FF taken, the imaginary part reverse rotated, the complex exponential
taken, the second !FFT takenand the sequence unweighted to yield the short-pass LK
18 services group
i ' N
output *hich is the estimate bf S1 (t). The error trace shows that t he output
of the short-pass system estimates S (t) with little error. Group velocity calcu-
lated frqm t 'ehort-pass output Is shown to the right of the time trace.
%N _' complex\cepstrun w~asl'-as-file'red 2LQ8sec n
processed as before. ,The--real part of the second FFT shows the effect of re-
moving the slowly varying component due to'S. The output of the long-pass,
'system yields, m(t) Wvhich is. essentially 6 .(t)+ .6(t-249) for this example.
The second multichirp waveforna;analyz, ed, consists of a: 600-sec
S "- chirp with onset at 1 sec',pius a 6-sec chirp with onset at 250 sec. Results
of the sequenc6 of calc lations are shown in Figure V-4. The format of the
plots in this figure an succeeding figures in this section follows ithat of the
firstexamle"iven ()and St() ae each 100 sec longer and overlap 100
sec--more in time than for the first example. The short-pass output yields Si (t)
with practically no err r and,as seen in the figure, e long-pass, outp!! P(t)-
"is essentially identical with the first exa, ple. t
The third ntichirp waveform analyzed cons'is-tsofa-_500-
wth onset at 1 sec plus 600-sec chirp with onset at 250 sec. The analysis
results are presented,in Figure V-5. The short-pass output again yields SM(t)
with negligible error. T e long-pags output ni(t) is more complex than the first
two examples where a co stant time delay existed between S afid S2(t) for all
frequencies. For this ex mple,m(t) can be interpreted in terms lof differential
dispersion whe'rejn the ti e delay between . S (t) and S2 (t) is 249 secit 0. 025 Hz
and inc rease with increas ng frequency to 349 sec at 0. 050 Hz.
The fourth mult hirp waveform analyzed is more complex in that
it consists of three chirp waA eforins,500 sec,\ 550 sec and 600 sec in length
L with onsets at 1 sec, 201 seci and 401 sec. The analysis results are presented
in Figure V-6. The complex cep trum was short-pass filtered at 160 sec and
the output yields Sl(t) with /fit e error. The long-pass output m(t) is more
complex than the preceding ex mple in that it shows the differential dispersion
between Sl(t) and S (t) as well s between\Sl(t) and S3 (t). As nbteo, in the plot
21
services group
1/9
\ x
'a . -- i-.- N-- a
/ a
7-~
a.' -* *.,.d'I r'lU'"*Ia .
,,L ~IiI~ aI . -I
1 .. a.
s..- ~ 8 - 8 V* ~ a
.-. a.' _______________________________________________ _____
~as AZ' 4* Sb 8* Lb
.tIIIML'ti1i~.. at .81~!d!'~. * )
-~ I
- I~ ___ - K
~ I /.8 /
baj. *I - I ~I ,j-
* (I _____ -
- ~ F ~:I7 a?4 a
~1I
Aa..ona..aA
" -
* *8. a
* 8\ ---7 a
- --- ~-1-t------ -
I ------------------rT /
* ~tti****4 0 a a. a* 80 Lb ft 38
I' - ~8
i.a
'a ~ I / I iia - - a* - ' -l-,~a' U
0 - a8a~i.a
* at,*3 - 8. *~~* ~ ne~ -~rs
8R - . .8,
, basWa~a*
Cepst u Re~u1ts for ynthetic MultipathaRayleigh Waves. Input
Figure V~'*a, Is th~ uyn of Two Chir~ Waveforms; 600 Sac and 6 6 0Sec in Length
with 0 sets at 1 Sec and '250 S~c and Amp1i~udes of 1 Unit and1~' UnitVi . L
-* - - ao services group
is . 8 a
I L I-[ \ . . . - 4... / " " ."b
'\, .'., . .
- 4 3
f ........ *
.d . -.Fiu. V5 Cesru jeu\sfr *~htc 4 ltat . y ye4.-pu
(°4 , 4steSm [T Chr4 4av *'om ;"0/e and ,C"0 "Se in Legh
I \.I./ -. .-- 4'- ' / * - ' - L \with Onesa'1Sc4d,5 e n ~ptdsif1Ui n nt
I- - -- -- ------.. i , se vice-gr up
/ 1A4414"
• " r r-----, ,.-* 4
K :)I2. -,/ - -i .:
1*aa~a~aa~aa \ - a. a ~0a .I --- a--a-a~ a a a. -IL,-.
-a-a a a 0'* . a-a -a -a-a I* I-a a a0~ a-a - - '~aO ~aaaaaaa -
.- a -~ a.4I~j'ja aaaa/~
* , a g2. a: - - ~'aaa
- a.,--.- * J a *- *s £'~ -* ,-IIRt 4dta~ - a a -
* ':~-~ -
/a a ~
p9
0 0 *~1 ta~ a
a aan a.'...
t -a.- I * a' a. --
.3 - ~'7 Eaaa
*'W% aaaa - - f_
a- 9'~I - 4 - 4
ahj;a a.
ja~
?*'~~"* a-ta. as .~ Iaa
~~a9iiii~KaC~ -~*** *0, ** - -. a- - \a -.
a0 a. . C.
3
/ a a/ 4 __ -
* . a, . a *I**~a j
4-.
-a-- a aCa a a_______________ 4, a ,
a - sa a a 3
/ 'a ~ **a~a-~.~ * ~.. a -
* ~ I''*~ -- 'I, ' .0 1 i~' i'.a a
ia a ~ '~. ', a - a~
* - -a.-- .-~I ~ a
---------. ' - .!AL.-a-a-a-a- a'a
ax - ~' .a7-aa a- - - a a~ a* a - 'aaa..aa a a. . a '~~ if
a a~ ___ Aa a 4'~ a. - - ~- - -' 4 aa-, a ,,, a. .a N.
a- I , * a Ip - *a 44,
a a _____________________________________________________________________
'-a a aa.a a - ~ i a a a 4
I . a ~a aa -. aaa.a ,~ ja a a a a~
- a. - - a a\ 4
___________________ Ha Af a a a a a. a. C. aS as a.______________________ a , rala *
a 44 0
* .1 ~a~a-aa. , .~ a a*aa4n ,a - a Li
a a - ~4'a a a a
0 a.. 5 5- a-- a-
1 *~,a a Laa* a a- a a a-a a a
aaya aaaaaaaaI a a'aa.* a a.4 p a a a .. . o
a t'~ aaa 5
Laa~aa 0,4'J a I
'a-'~-~ a ~*0' \a a
a a
'~Figure V-6. Cepstrurn 'Re.~U1ts for S MultiF(ath ay1ei~h Waves. 'Input
a , Is the ~um of Three Chirp W~.veforms,; 50 Sec, '550a Sec and '&pO SecI S
in beng~h with Onsets ~ 1 S~o, 201' S~c a d 401 Sec and'A~np1itt~des a
0C1 Unit, 1 Unit 1 Unit. P a a
a a. a
* a a aLL a a
I a a, a a
'a a 4 a a a a aa -a !aa ~
a a
a - a a Ia a 0ai a ' a a j aa a k "
'a 0a a a 0~ * a
[ a*a a a a a a a a
a a- a a a aa a a a aa a 4' a a a aa a 'S
a a a a aa a a a
_N/
the ti- andl'_-(t) is, 200 sec' P021025,Hz a;d,,increa es with
me del y between S (t' Z.increasing frequeAcy to,,250 sec at 0. 050 Hz while the time delay i3el") reen
and S J*Yis -400 :kec at- 0. OZ5 Hz and increasei; wit ri'sing encyto 50
secat 6."050 Hz.,.
%The fifth mult cl irp. waveforin analyzed' is entic;j to ty4H
example with the. following exception. The waveforrr 0)'is s a ed !o'dne-#iPlAude f S (i). This.iii clearly reflectedfib th I -na*,S
hali the iL e opq gutput
fqr this e--fample (Figurer L-1) -j dre m(t),is -essentially .1 + 6.5 6 (t-Z49).
-Thq 7 sixth multichirO -o chirp wave-
Prm analyzed consists\Qf t*
forms 500 sec in-lenitif with onsets. at s .p "an4d , 101'see (RgureV- ), -.The time
separation.betiveen S- (t) and.S (t) is much lesv;.th h in pro-viotis-example's. In1 2
addition, S (t). is, scaled to-6rie-hilfthe a!qlitude of S- (t). and-the AAraVeform' in-J,
d"ith T spect to S (t),; The-corn'plex cepstrurn isl6ng-pzkss filtered at
80 secyielding *Ywhich is. e: sentiaily j -(t) 0. -5 j (t- lOOJ; complexcepstrum is also'. hort- it The7
pas: fi e-rea tit 80,:s eqj,,yieldir4 the 1! Utput 'ghown.
ealculafe d e,'r rorfor this outhut, although, still fairly small, -is largr-r-than
.'tain d from fl e ',-revious egamples. Th6 sho t-p'als W* sysfem- has ren-rove*d all'
of the contiibiftion caused-by m and part of that causea by-S.m' Itichiip wavefo m anLyz4R con wa re-
lihe se nth si su400 sec ai -1 5 a _c.; 'Tfi' analysis'forms id 9(o sec in leAgth with onsets.'at'
results are presented in Figfire. V-9. Fq f- this .-gxample the;.time d lay bretweeri
tnd S jtj 's zero t 0. Z-5-14; and increase-6 *'ith increasing.freqifency tcL-.ows tha't S and
ZOO sec-at 0. 050 inspectij n of the c6m lex cepstTum sh
m-overlap sighif cantly in the Vshp Vtj je region, cpnsequeqt-I hte portion
attributed to rn cannot remove y itho removiZ p4rt of t9h eco t -butio'n dueAll
to S anctwice versa. suite of short--I)ass filters at 14', 24;.40, 56, '72, and'
" c wSs applied to, the complex cepst.rum. Th filfer ai 56 6ec' qld-ed.the
sinallest er'fo'rtrace. the -.shqrt- - and Vong-pa s s Zered resu.Itsr for that filtL-r
ar4 shown in the figure: Simple s prt-! or pass filteririe*cafinot compleeely
separatep from m. t omb, filtering cou'ld %e used'to remove the arit. attributed
2 services group
17
10
-7 f .I
• -. . .. , - -- - -r -, L
.,f i-f
- - - -
-_ - ' -d1
- . - -..- - _ - . -
-N-
i"- ' f \ - -
- -- s rie r ui.
---. --- -
€/-.. J~m S _________________
Jj - - -t
- -
- I"
-,°_ _ _ _ • -
-- - * .. * *
~ r - -
- -es - A 5sevc s ru
- \ -~
lC-. .. . . .. . . .. . . .
I 4 .o --
-}".5 " - .... i,, . - - - -. "4j , .
II * l I I
,./ . -.
*" , " - k - '
- I
L_-a * 4 -_ - A_" .i- - -. -r
- - - - -.
*¢f - - , - : -. .. SS" "".I
* - -- - - ! : L'-" -- $. - ""
S --- , S... .-
-- L - I °
,s.... . -[
/ 4- -4 .
1 l.1
;*.....S -- -- " i
4.- " -T i
• ". - I s he____________ Simo.w ~ r aeo m; 4 Se n 60Sc-nL ngh
,..,- ,- >' LI
~~~1$ for.4- ~ -- i
to __-- -m wio.rtiigams al fte atfo Sp vi e m wsafil
5 .,
.4.
to mwhie robletaing call ofsicstliyapidWe the part froompponendt ws afi
impexcpstrain ,,he-d~idotu. Howeer,~ for this example,m repesnt dtasigna
functon silesi~al wlent with zearoy timedelayate the towes freqluencyI
adton, itHz.shu The poatibted t mi the roplex oct s bai com i~tedjjseri ss oelapt(igutraer f-9)cos -fit n the shgot Velocit ren
to~~Th whchom iultn cansnte be i succsfio appliedWhe the feasibicompoet4~-.
signficntl sith liea filftue ado myule re iov too muchfofms thMtcple cepstrm te depsreen oiutUtowe, fo thesiseale, hesignal
Ineraterfeenc frmtoo mrses.csure
estr e btanediscuring -at bter sameplocation oft sml (t)-hise cnann
ted iput sgnalanddsrioin ants noR mS, hirpsatistc t.ncluae
~~fr 0o signa es ar/il morte necrl n spoipte the tr al les
/~~ addtinetreurepone fout ta thne rourye gceypnetsbad eroh
;'sh~:pas otpuhFre disinc closl fith eifperte group velocitycuv
The e wiuth prsentdl thisseartion ds osrt hefaiiiyo
thi tehnq e eape hwta for filsipemutoucaes h and/o mmtit av eors.e Th
L ~~~ ~ 2 servicesg waesrrutigfrm
Inteferece rom wo r m~q' sismc sorce
I.j
"time', a linear filter system Can be used effectively to separate the two, 'corn-
-ponents from one another. The technique requires n apiori estimate-of*S-and&
will give an~estimate of both the signal S and the multipath operator mn.
/N
hP
- - I - ~ AI
28 serice gru
E t C
IL SECTIONVI
PRELIMINARY CEPSTRUM ANALYSIS OF RAYLEIGH- WAVES RECORDED AT LASA. AND ALPA
Several'events recorded at LASA and ALP were. processed using.
/ the complex cepstrum technique in order to make a preliminary evaluation of_ ._ 1 -y/ . "
the -methodas -pldRayleigh wave recordings. The events used had
S good signal-to-noise ratio and associated/information for the event (loc
date, origin time, etc.) is given in Table VI-l. The vertical comp nent of the
-. Rayleigh wave was time partitioned from each event recording for ipput to the
program. -... ,
ProCessing results for event NTS03 recorded at ALPA a presented
in Figure VIE-_l.Shown in the figure are the decimated input signal, th real and
imaginary pafs of the log, the complex-cepstrum, the short-pass filte ed out-
put, and the long-pass filtered output. The 2,utL signa as a modulated character,
Sand minima are observed in the plot of the'real part of the log. These inima
could of course result from either an interference phenomenon or low signal
excitation at the source at those frequencies. Whetheror not energy at those
frequencies was present at the source cannot be ascertained from this beaned
output alone, however an inspection of the group velocity curve calculated f1l9\m
this event (Figure VI-2, top) shows that the recorded waveform is not a simple
normally dispersed Rayleigb wave and this characteristic is indicative of inter-
fe ring waveforms._
The complex cepstrum fot this event is much more complicated than
those obtained for thes examples given in the previous section. The part
of the complex cepstrum attributed to m is not a simple impulse train. Thus it
would-be very difficult to apply a comb filter system since this requires muchi.no rnat;on aboutthe number, locations and durations of the impulses In the
complex cepstrum. An alternative is to use a short-pass system. A suite-of
short-and long-pass' filters covering the interyal from - 16 sec to - 128 sec
29 services group I.
., Co 0, m
go V- ~ ~ o cn c-c a' - ' ." if i ~' '
in C Ln a' ' .'o 3 - ' 3I
to o o Ln Ln aC-
4~~~C ?4 ' 4 ini n i
C' .4 0 r- Cn 'D A' C
~~ 4 0 mJ tn - in r3 0 - n6; .0 .- , - t!! '- c- 0 !!~
0'0 ' '3 ri 0n '3 r-. '3 '0a ' a' - a ' a
0a % 0 0 0'C')
0 go 4 s. 4A
Q - M r- -
E- Vz z z z z z z z z z zz '3 in Cn 00 in -
r~ -. C1 '3 -T M" -r -TV
00 0
V IL aV) 0 0
-Y
-T m
U) X. 0.
30 0evie 0 .u
49 ~PAR IIO
00U
5% SIC40.0
--
.- i-U- -(SOX
LA SFC An [ ic f
.k.
0 0II0 *
0 1z
31 services group
31.-
Lp
V
o . I
5" II"1..II I "* I I- J
CL.
,,~ 00
"'0/ 0.0 O .4 0' 0.6 O7 ' 'Y 0. OW /'
5- . fI
• " i u e V - . G o p V c t s. Fr q en * fo 'S 3 T p u i a
/ 3
t2
0 .10002 00 0.0 0 .0 0.7 .8
:37 ; 0 \o
Bt S O
32% 0.02i a g Iou
O.06 ~ 13.0 0.0
I - I- . ?c
0
was applied to the comlex -cepstrum and outputs obtained. An analysisof those
results was made by inspection of the plots ihowing the real part of thJ shcond .
FFT after short-and long-pass filtering. -Th short-ass filters at 16 and Z0
sec removed all of the rapidly -varying comp nent, due to m and .also removed
I , much of the part due to -S. 'The filters at 36 to 1128 sec passed alnmfost all of
the compondnt due to S and also passed inc ea ing amounts of the .rapidly
varying component due to m for increasiffg Iter times. -The filtersat'to 3&
sec passed most of the par't.due to S an re n0 e: almost all of the "4pidly vary-
ing component. ShoWn in the figutre are the re 1 part of the s'econd. FFT and theh sh r a. c, t.
output for. bth short-and long-pass filters a 28 ec.
Group velocity was calculated from th short-pass output. A conipari-'son between the g-roup velocity poinr .obtzn , -.r,,irthe -inRut wa-,gfrm-and-the'
L short-pa-s output (Figure VI-2, top and bottbn respectively) shows -that theS, i S -, • \ .
f points is largely reduced for the.output trace and~the, points are much,
- nore continuous. "In addition, the points clbs'ilfit ,the theoretical curve calcu-.
F , ted'for a continental crust. The great. circle path forhse ent: rr ./
ev a/, Oregon, Washington, parallels the Coast'Mouhtains o British Columbia,
d 1osses a portion of Alaska.;.
-- The lopg- p's, output show. a large complex impulse at about 30 sec.
Al n teroretation of the long'-pass output based on several assumptions could
of curse be made. However, additional information'would be requird to sub-
st tite that the assumed conditions ad therefore the ihterpretation were-
co ect. ;or example, two sources with identical excitation and P.tion
L occ rring with small time separation wbuld yield 'a long-pass output as shown,,'/ ,, \--.
7 fo r, examples oneand two of the previous section. However, it is conceivable' 5 - ' S/
tha an Lntirely different physicai situation tould yield an essentially identical,,,
• wav form 'ecording andc subsequent long-pass output. Therefore an analyis._ . .
of s ort-period p waves recorded at a network of statiois and subsequent hypor
s~w- of satiosa4sbeun yo
;eit r calculaticxis would be necessary in order to- confirm or deny the possibi-
lity at two sources of the sarfie type and magnitude occurre'd at the same Iccation
33 services group
S '. -
" / /..
I, - \ . °- 0 " " 0wit the time separaton observed, in the, long-pass 'outpuan the sh rt-pas fou
z "", ° sho inFigue V - The outpuina irl t.vl freo esel and o.w
ledge of the i nformation described "ob'Oe would -allow accurate hfitef paonent L
V 5 i - ' " A
4 the\complex-,cep strum technique at s yield th e ti u , signal oitep N
--.,.sn t h o ru voitshih corsodt-h oaofie ~f."ru
a-Greenlanh 8e vn at.LASA va sd
, , "- "" " " ° - " "' I. ' - • , . - ' " '
Wext , .-'' \
e* vi/n L 'Those k sults ar prese nted in Figurge VI-3 *the great circle path for s event
S SS i ' 5 ... b t
S" traverses, a portion ofthe Grr-na dSed, wrenlard and thre Canadihn S 16l.
Group velocties calc;ulated. from the input sighi and the short-pass ou l!
oum inFgure VI-4. The input signalireavlyfeofbtsad afa--S".' •/" '/ . ". ...-- A
ly.simple normally dispersedkwaveform as een in the grout velo ityion ,4g
ever,' interfer~nce is occurring late -in the waveform and a large sdatte ~is pre-
A - . , , • /. -Si",'-'-
sent at thu low gr p velocitierscwhich correspoxi to the coda ofkhe eaG o. -. rqup
Alsa'nIetr Cnd -acrpe se.so srcri en .o1en .-
S elocpt for the sho pssoutpue is sornewha smqote and as se n i Fiure VI -3,in .. signials atan e o; T. '
much of the coda of te input event is riot.. pre. nhe long-
pass output show d large complex impulse 'a about 60 secvI Th Lo. rinn
train -around 60 to 600-. sec is attributed t "h most rapidly ayirag C Pmoet
A. of abut 0. 4,03Hz
A*seeh iia the prltsof 1ge real part of the log, a ofiabout.002H
Four eventsfipe implse t curie s and atLAre f l SA were ror'et. s1 .
cayleigh wave s record'e UA9 s inm events in this region 'are h totorioisly cor-
ple, ndparameter s-.ca~kulated frbm th waveforms gieresults which are diffi- -
3 /4 -- ..
cult to-interpret as ividenced by the groui veloc~ity plots sh own in Section V, Fig-
/ i
-Alaska, and Western Canada - a co lxseries of structural niomhs
* Complete cepstrum processing resu'lts fbr LL-2OOS and LLIZ:l86 ISare 'shown in
*Figure~s VI-5 through VI-8. *Fo7; brevity, a'only'the input signals, -hort,-and long-
)pass outputs, and calculated group velocities are shown for events LL-2019 and
LL-20Z8 (Figulre VI- 9). All four events 'show beats in the input signal and
L
%,Minima are observed in the plots showing the real pai't pf the log. Ajain the
* complex cep~tra are not simple impulse trains and are-Acfficult to interpret. NA suite of short-pa'ss filte-rs was applied to the c*omplex cepstra and outputs'
* calculated as before. Presentedkjn the figi~res are,'the outputs for a filter at
*g 34 I . services group
• o - ,. .- . . - • .- - - a ...
- - -- - -- - -
- -
- • •m" " -. '', . " -Ii g S .- -
, ,/
'C- --,., "•- /-,
m ' -, I ". -
I . -s \- ---I- •aU . - . .-
-- ------------ -- ---. , -------
V - /
1IF
U- -,. , a I " "-I*}--U PI i -"
- - - - - -- - - - - - -- - - - - -
*i * -,s , s '. 4• " " '
AL - .
* , . . '.- ", _ _ _ _
, ,
', " -.,, *1 - . I - '. £. ' ""-.. , , , .. .," ,. . ,,!,* -.'.. S
LI.-Im
L-\',-.T_-7>,- -
t , o- ,
", - ,.,;~ - \ 5 __se _ies_ ru
•/ - - €
L.. .1,10
L -LU ASSArlf v I
I- I, I fi
Fiur T .-, srr eslsfrEetL-1Rcr~a A
35 sevie gru
3--
2
0 ~ ~ 17--00 -. r .4 00 00 .7 00
5-U
S COMENTA
igleIO:Gou eoiyv FeunyfrL-1 Tp nu
Siwi -3otb Shr-Ps Oupt
-3. sf.v3e ru
. .........
_ 7 --
- - -":-
N hI , I-- , --I
: ~Figure VI-5. Cepstrum Results for Event L3L-2008 Re'corded at.LASA .:
37 services group
V o
-/ - L .- '
-ii
c..
CDF
0-1 t0 .3O5 .5 00 .7 00
Sga;'otm hr-Ps Outut
38 sevcsgru
I. II
r __ __ ____ __ __
. , , - , S
,4 * . ,I I
Fi u e V -- Gp s'r e r t -o Nn L 20 S e o d d t L S
__ _ __ _ seris___ou
-!
___________________________________________________
-- -
its
D.. (ol O U M600 -
'3
Cr o 003 .4 005 .05 0.07 AO~~ 0.81 W 0 0 (it) -
UI ---
Sinl;Bttm.hotPasOupt
4
4 sevies-rm-= -I
As
/
2 - ; " 'A i t1' .
0.0 1. 00 0.03. 0 O 0.05 0 ..0 6 0.07 0.0
-" - - O C E A J IC
Figure VI-8.. Group elocity~s Freqihency for LL-Z0 18. (Top, Input "
- Signal; Bottom, Short-Pass Output)"
• -" 40 services g~o p
IZ
___ /t
II
J
-~ I ~. S - - t r,.IS -n
.~p.- as
I
- -
- - a..I -
f = - is a - a a a a -s -
I -fl-nI - -
-- -- - -
I 5-
- - .--- -r -,
4
A. --LI.
-=
4 -~ -. C- S
- -
S.-'/ >1K..
- - ~S $3 iS S.iflfl. -- '3 - 3
-. - - -J4
1 4
'I
Figure VI-9. E'~ents LLZO19and LL-2028. (Input signals, Cepstrum
- I Outputs,-an~j Group Vclocities)
41 - services group
4-
Li
0
80 sic. The sigenal estimates obtained are clearly much simplier. than the input[
wavefornms. The LL e& ent recordings obtained from Lincoln Laboratories were
band~ass' filte red at 0.025 - 0. 055 Hz during Dreprocessi ,. Since there is no
* energy above 0. 0025 Rz,the e%;ent recordings were decimated by 8 giving a
sarnole peri od of 8 Ssec *and a yuitfrecuency of 0- 662.-Hz at input to the-
compzlex cepstrum progr-am. Giroup velocit points calculated f rm te input
w-aveform sarnled at 8 sec showed variations with respect to the waveform
sampled at I sec. Tho~e variations occurred only at the bigher frequencies
-where frequency is chan-ine very slowly with time for these events. The three-
noint cuadratic interroibation used in the groupa velocity calculations to determine
:peak -.d trough times is adeduate over the frequency brand of interest for wive-
form s saninled as coarse as -sec and no smoothing wserformed on those
waveforms with .S t g 4 sec; howi-ver for the waveforms sampled at 8 sec
smoothing of the order number vs time function over about 5 points was necessary
to reduce the variation in calculated grouv velocity at the higher ffiequencies. It
should be roiidted out that smoothing over 5 points does not significantly alter
the arot=) velocity pvoints calculated from the orieinal wa-veform sampled at I sec.
Gr6uD velocities calculated from the input signals and the short-
pass fil-tered outrputs are presented in Figures VI-6, VI-8 and VI-9. The simpli-
ciry of &he erourm velocitv curves for the outpvus is aprparent_ The curves obtain-
* ed are neither oceanic nor continentil but are-an intermiediate variant of the.
two.Grou velcitycalculations are based on the epicentral distaic ewe
event and recording site which assumes a great circle iath for the surface
waves. If' the ffirst arriving surface waves (at a given firequency) have devia-ed
- rom the -reat circle path by traversing a higher velocity structurethen the cal-
culated erouz) velocities will be lower tLhan for the actual path taken.
Tbe events were long-piss. filtered at 80 sec,yielding. a fairly.
ou~D~ t:)t;t f~jr sev.eral of the ev~ents wherein. -T sA 5> separate complex impulse
- traixns may be observed. The similarity of the outputs obtained is stronigly
-~ s.~getii~~ofSeveral fairly dlistfnct pron-agation paths from sourci! to receiver
47 services group
--for these even s. This type of analysis coupled with frequency-wavenumber
analysis (Capon,1970) could provide a powerful tool for understanding the corn-
- plex nafure of recorded Rayleigh waves.
Six additional event recordings were processed using the complex-
cepstrum technique. The results of these calculations are presented simply
16y showing the group velocities calculated from" the input signals and those
calculated from the output signals (Figures VI-1O, VI-l1). The results obtained
- are similar to previous results in that the outputs are simple normizly dispersed
wavetrains as evidenced in the group velocity plots.
*s
• 2.
- -S
-I
I
.5,
43 services group
PU
-7 .-
; 72 1 .7
- --- .o - .
-, . . -7- .
7' 7
EPX 7 144N -,"
4 . 7 777' - ' "•
- 5
, L
"I
&C. N 90 . . aj w - t I
SI-- -- "" S., -,'a" 2
Ss I . . .. .
i ' Figure VI-1'1. Gro ' Veoiisfr.nptSga n hrt-asOtus* /- a t ,
" ~ ~~~ings. of Event EPX 14646 . '
. src gru
-"5 9 9- 45/4 services -ru
I a 5 a a
-" • .
, "" SECTION VII,.-$ (. . .'
- - RE9FEAENCES
Capon, Jack, 1970, Analysis o!°Rayleigh-wave nultip;th propagation at .IASA:-.BUll. Seism, 9bc. Am.,*60, p. 1701-1731.
Evernden, 3.F.-, t953, Direction ofapprach of Rayleigh waves and relatedproblems, Parf-I: Bull. 'eisrn; Soc. Am., 43, p;. 335-374.
r Everzden, J. F., 1954. Direction of approach of Rgyleigh wa e and related. -'
( probl~iihs, Pait t1 f.Buil. Seism., Soc.- Am., 44, p. 159-184. "
Knopoff, L., S. Mueller, and W. L. Pilantj1, 966 'triicture of the crust and-upper xxtantle in'the Alps'from the phase velocity of Rayleigh waves: Bull.
! Seism. Soc. Am., 56, p. 1009-1044.
Pil;nt,.XV., L. Knopoff, 1964, Observations of multiple seisnic events: f -ull,Seism. Soc. Am.', 54, p. 19-39. / -"
Schafer-, Ronald W., 1969, Echo -removal by discrete generalized-linearfilter--ing: Te nicat Reporf-4t6, .----- c - --huvz
Research Laboratory of Electfonicsi Ciznbridge, MAss., Ph.D. .Thesis.
Texas Instruments Incorporated, 196 7, Continuatio of basic rese rchiI-crus- ses: ia nCepor. Contract AF-49 (638-1588.-
- 8 servicesroup
I -.- -- I_-- 5
a "-
-
. .
. .1?
1- "* . . 47/4 sericesgrou
! . 5.o
-A *
'- -9
I - S
- -PPE. -IX --
COMPLEX CEPSTRUM PROGRAM DOCUIJIENTATIO7 "r ,.Z -(
7! :: "I '-DESCRIPTION " *---"'IHomo lDCrOhiN deccnvolution using a genexalized cpncept of linear
fltering is applied-to separating the components o'f a -convolution. The methodfil ngi''sedo taplidto ipfLg Z
is based on traisf6rmifg a convolution of waveforms into a sum, using a linear
filter system to sepaate theadditive components and-transforming the result
back to the original input space. An extensive presentation,of theepproach is-,given by Ronald W. Schafer (199 9 ), "Echo Removal by Discrete Generalized .
,e b
Linear Filtering: Technial Repot 466, Massachusetts Institute of Technology,
' Cambridge, iss., PhDThesis. " - .
A typical run tie f6 the complex depstrun program required.
" "- _-_" " . !approi ~~y3 minutes.oh_-t. '360/65.. o .- " - _'i
"- .alComp plots-; the inputs a outputsi as well asvaxiousjunctio" obtai'ned during the. se?!uence Of calculationsare _erated "to "al"
sis of the results.
1I RES-TRICTIONS - --
;1) - The input data trace may-not exceed 1946 piints. --
" 2) The program is dimension'ed for a maximnum of Z048 complek'
- numbers in the FFT .ial~cati ras. qr
HI, rI fT " -INP U-Ca d 1. NI 0 ""..
______ NN100
Column Variable Mode De pto_ n ""
1- NKEP5" Number of events, t2 process i •' - ,. "° "t'his run. , .'
Card3 .- "' "' "1-60 NAME A." £ Al1Canunhqric title
Card4 . ,
1-10 IDIN Iput Trace 1•11-15" NSKIP \ of pjnts to skip on input16-20 NPTS. > I ' Nuriber of ppints to read oninput
,:1" A 1serices group
"" \ '
, -
1 Card 4 icontinted) -
Column- .Variable - Mode Descrintion
21_-25 T NL I - Decimation inte-xval, after band liihiting6-30 !FI'LT " I =0 sto-after calculating the cohiplex"
cepsti iM -
:" . -0 nuinber of filters to af.pl .31-35 . -IAARON I *- =1 call Aaron's 'algbrithm.
40" N-* =-2 call Schafer's algoithm
(for correcting phase)-40otal data .. nagth fdr transforms, etc.
(must bea nower of two). Zeros, - are added to give an input trace
- " "NLEN points in length. -
41 -0 DBLIM E >0 clip real part of log at DBLIMAdb below'peak. - .
. <0 no Clippkig."
51-60 .ALPH.A= a. Expinentially weight .the .inputtrace X. _W(n) an X(n).-
-61-70 DT • E - Sample period of the input data.
Card 5 " ""1-10 IDOUT i utput ID for data to:be saved op ta-
-, " . 11-20 IDIS * I Output disposition for data.t6 be. saved
on.tape.
.'Card 6 Filter ca.ds , Sftiply IFILT times ""-_ 0 ITYPE I = short-pass system.
=2 long-pass system..=3 comb filter
1 -20 INITPT . Point at -which to filter the complex, -
S.cepstrum (if ITYPE ='1 or 2).-
Number of comb filter-points. (if ITYPE=3).,
21-30 IDX I " '2IDX+I points will be zeroed at each
cornb location (suppiy if ITY'PE=3).
31-40 JTAPER I -0- c6sine taper the long-or short-
pass filter.=1 n6 taper on filter.
Card 7 Comb Filter cards, (Forriat 8110) supply if' ITYPE=3).
Supply tINOTCH(J), J=1,INITP comb centers.
A-Z services group'
*, I .
* . Colunn Variable Mode- DIescription
Card -NN 10 /
1/0 units needed - -"
- -. - - card reider -. -
"" 16 - vrinter - - -
-7 - card punch "
-Input data, tape.Output tape to save results.
- . CalComp plot tape. - " "
Ac
- S
-- -• .
- 6
oC-
P P
"- " * evceIru
* ,. A -3
4
A.D-F I=LAS -- Li AS -5-E UPFRMD~.14 -YERN
r 4c~FjS I n%-: F60 ?049 :tEaL,.DATA POINTSa, rJ I, I !'! A 17F - (I()IO) vJ a8F 115 A U 4(7) --
-C* 148) . WRK(I?749r)
S) ."A ,i.0I q, T:7C r(?049j S -
n S SS] (70)3
DATA TAPFR.948..0397..A660.7660.6424400.l2 9 7 6
xm*.rAi. NN ionin FnDMOPlI jIwS7
CALL Ptr.TS(-AQEfl,900013.0)--.-CA LL n AV; 1) aTFJ
- CEAnf-)'.7) N KFP C - ..
Q F . Pi 5.1] ) "'J-1 FPQMAT iISA741
2FODMAT jjHjIX.-,'lJAMF =.54
3 FflD-aAT.(./,IY.'O)ATF =10IIO)
.4FlP4ATUIO.)6I593F1()0.).
W1 I' 1 -6 9;)
Wo TrT'U. 9 4) I r) 1*~0,NPTS, INC .IX LT, IAARON, NLEND RLIX9 ALOHA DT
11;PODAT ( I , 7T 1). FIO-.?r.Ft*49' 1O. 2)
7 FnR,M'A*(?II)1WRITF(6.9) ffl(1UT,irTS
I FP'AT(/,.1X'TOQDjJT !.,Tr, MS 1,'15)
C ' sr-.I'rttU,%ATfC';% nATF
r T=SAMPI'F PEPr) O(DfF INPUT DATA
r NDTS=N[Jk!8FP.flF INP.UT'uINTS
- services group
-WRIFr69131 FdvE1.F2..F3 .- S
t:~ 3 OORMAT(IIix 1 'rUIOFF FREOVENC.IFS FO' PREFJILTFR',-94F 10. 41rF S 1 r114.7 Fqn HASP !RANn PASS IF TL-T EPFn-11 /iFLfAT ) *T).
On 480 1=.!4FITF r-F.T)460l.4 60,440 -
4401IF(FO-F) 450,441445'445' WnpKCR0
~4150 TF(Fl-F). *460,455#455 N- 455,-worzi ti) =f 1.+rI~s4tPf*(F--EI) /f60-E )-))/2.
-rfli -TP 490*-.
465 iwfRK(!J=1.--GOTO 480-- - --
470 TF(F3-Fl.-445947,59475475. WO'qK(T.J=(l 9(P-l*(F-E-3)/(E2-E3)IiW2.413O F7 F+FD -
r Y,*=DT*FLnlAT(LF),
STF MP (I )=WiYRK f11 7F=FD~ '' '
tin 485) J=2,,NF'- -
TEi4PIT?.=TEMP(fl1+2.*'COS(TWOPI.*F*YI*WORK(J),.485 F= F+Ffln-S -/.-
Yz-:Y+flT- -490 TEMP(TI=TF?P(U)*FD*DT
* r RFAD TRACE,TQIJNCAT6i EACH END..AT-NEARE.ST ZEFRO- CROSSI.NGC APP.L.Y COCYNF TAPFR TO FACfi.END BAND PASS FILTER.-C DECIM&TE', RFMOVE MEAN
-CALL P.CP(IDTNI4IRKNSKTPNPTS,1,AUM)AMA X=O.
.1-406 f1l.0PTS
406 SUm=<(JM*WfRKUT)SI)mf S M / AJTS Innf 407' T=1.NPTUr
407 WORK(1)=WORK(T)-S(JMINIlEX~l 0 1
DO 1'1 J=1,NPTSJF(1fPK(J).E.O..AND.WORK(J+1).LE.O.) GO 'TO 816.1F(wnRKUJ).LF.0..ANl.wfRK(J+1).GE.0.) GO TO'8t6
GO-5 Tn ioXTolE4-P16 TF(AriS,(wfR((NlEx)).GE.ABS(WRK(INEX+1)))IN0EX:;INDFX+I
NPTS=NQTS-f-nflE Y+1 I'
4 PnP A17 J=hNPTS -
servicesi groupA-5
ni7 wnPK (J)=AMR~'f I ~N(EX-lI+J).I f~nF)C I Sc ID+ In orx-1 IW01.TF(6#4193) INOEX
4Q1 fQ4T/,YTrLNME OF POINTS SKIPPED ON INPUT ,10
P .1iF Y = T .- - -
rF&n s' riEY).LF.Q..'&Nf.WORI('t'NDX-1).GE.9).1; Go 'Tf 8190AIR n rfl I N r)FI -:y
* r~r- . Tn
I =10Tc -7!
rDK(..4-APFR( I)*HflQk(jj.4 -WhQK(Ki=TAPFQ I)*WnRK(.K)
A?(; K'=(- KN HA F=N PF /9nn* 4?75 r1 INHA F
47'; TRArF(T)zfl.--00-. 8 ?'i ~=, NP-T S
A2)TAF(NHAF~j)=WeqK(Ji)
* 100( wrOKlJI=TQACE(NHAF+J)--CALL WRr(Pf.Tfl'IT,WJRKVTS.!,.I DIS, IDAT.ENAME, AJ'4)
lr1)d)JT=!OTD1IiT+ I-K=NHA1+ I'JO.TSj
L.=K+NHAF-tnn 430 r=?e.1-
410 TRArFf 1)=O. . .
-+' 1T F R INPOT TPE R /lFQU1=TFMfl(N4AF+l)
n0n43-1-1 =jNP*-
K=NPF+r .
DO1 426 zhl.NHAF426 SI~t4SIJm+TFM(j)*(TRACE(J4L),+TRACE(k-J,))431 WPQ K (TI zUm 47SP.0OTRACE(NHAF+1) V
r DFmATjF tHF B~AND 1, 1'4ITFTr) TRACE*.
nDn 405 1jNPTi ,TNC
40 5 TRACF~I )=WCqK(I)
riNnit TflF 11AN") 1TITFf)t DFfMATED 114PUT TRACE, FDq.mATM5El.7)L
o C'ALL WJ'Cf( T0PUJT,TRArF,NP,.,1,90, IfATF,NAME,AtUm)
LPTS;Nt'Fr FXOANn TrV4F SCAL F nN 114P.JT DAAA6OUPTDT
I Fxfl~n=l4.
A-6 services group
~A T 1-T* I.NCIdRiTF(6*4011 AT
4(31 F O"AT II..,(SAMPLE PFIDO.ADLIMITED ANODF flECITF9) OATA',-
fP)O 70JTiiPA3NI)<P ijff( 6
CII L ETTER i-b. . 2, NAM4E, 90..60)-
xC1RTG=Y5;TZF, I1CALL ORIGjN(X4Rl G ... fCAttlLETTEPO.?5.0.,O.15'f.NPUJ TRACE 990.,.v111CALL PLOTA(tRAcr-.KPTS.'AELX'YSIZE,XLEN'*YMAX,'YNININ,Igok;fN., OrCALL :WPCP(IoUT.TR.ACE.NPT S.1,1,ID)IS,IDATE,NA .E, A4).IDflUT=mfotJl41 * I- -- -C.
Tf4AX=YP4AXTMlN=*Mffd.1 DO. 7SO J= I aPTS
750'WOP.K (Jj=TRACE(JI-*(ALPHA**AJl00 7-70 J=TPTS,PIPTS,.-4
770 W0RK#Jl=A0AXC, . (STnPF- REAL TNP.UT INTO COMPLEX ARRAY X
On 8 J=1.NPTS.71. X(J)=tMPLX(Wf)RK '(Ji0.0)
c TAKF.F OF',urAR~RAY XCALL .NLOGN(XNP-TS,--1.0)
C TAKF lOG. OF Xf-iPNf=NPTS/,?.1
L* .FALP=RFAL-(X( 1))TF(RFAI.P.LT.0. ) ISW=2Gf) TO (1IO,120).ISw
t 70 nnO 130 j=1TFNnRFALP=-REAL((J))-Al=-2-AT'Ar,(X(J))
130 X(J)=CMPLX-(RFhLP,AI)WRITF(6,1401
1,4n F0QMATilXq,'CfNSTA.NT PHASE.COMPONENT REMOVED@)11O CONTINUJE I
R.FALP=qFALX( IENO))
TNflEXP=].:On 12 j _irEpn)
4REALP=REAL IX(J) IWfltK(.l)=RFALP
* TF('PEAl-P.LT.Q10AX) kqTfl1?
/A-7 services group-
* RAY=QrALP -
I NrFXP=i. -
- 1:? ~NT-TNJF .*.P- T O FAL nAT~F LrG". * qr-1
YS T7EF=?. ,
- * - * CALL ORIGTN' fXORfG9..'AMINTO,!COJ'r A it I ETTFQ(O.26',O.#0.15,0REAL PARY OF JO,. 1-00o. 171"
:r.ALI PV)A(Q~ 4,D~,YIE LN MXt. N N BX N TO MAX=YPAX V
flMIN=Yt4TN* .~w~fTF(6*'j4) NnEXP9RNAX
14 FORMAT(fI.1%.-*!NlFX OF PFAK V'f 10 9IOX MAX. VALUE, REAL PART OF L* - 1f3G. I .PF15. 7) . .
* . FTf NnD7FC -AWll -THE PEAK *FOR WHICH-'VALUES R"FCEJ ARE ABOVE',r- JHF CL ,IP I FVEL <
TNOjrFxn=T ENl* ~INflFXA=l
* FIORLITMLT.0 ' GO TO 2?0*rLTP=f I./(lO~fR1/2.JCLIP=XLIJG(C(IP)
' I TP=RS,tIP)
IFflNDEP16E0;1') GO TO 17NTIMFS' =!NnEXP-i - '*
nn If, J=19NT TUES '' LX= *KNOF XP-JT1NDrXA=K ' '-
jFjpmAX-WlRK(K)).GE.ACLTP) ~QTO 17:16 r ONT FINUE17 CONTINUIF -.-..
TF(fND)EXP EO. I.F Nr) GO-TO 20NlTMFS~TN0FXPf I IOn, 19 J=NIT!4ES.HEnD
!NnFXRB=J,.-.~
iq crINTINUF ' i 7L?W0Qi(J)=ATMAn(X(J)) ~cri-ro (610.-$?01,!AAQO.N'
mocfNT!UF o
CALL AARON(XWPRK.CC1R,TN6FE&,#INDEX8,CLIP,NPTS,!ENDPITWOPI)L* GO TO'630-
A CAL LSC H AFPX ,WOR K ,cnR , INbE X A 'INDE X 0,C L IP, NPT I[E N D 9P-19TWO P I ,T F 4P)6-io r nNrIl TNOI ,Uw (IF
WPITF(6-,1) TDX*N)X21 FORMAT(//.1K,'TNOfXA =',T1OIOX,'.TNDEXB =49,1l0).
r Rn1TATF THF PHASF* T1F('INDFx8.FQ.TFND) INfFXB=INnEXB-l
I~ ~Jl'T= TNOFX9-,IND)FXA+I
services groupA-S8 '
K= 1. ' .
nO- 601 .=f-JDFXA.1NlE'SAVF(JJ=SLIPF*T-EMP(KII -- TFMP(J)=bIORKtJ)-SAVE(j).0
601 K-K-elonfl 697 J=INDEXA.IN DEXiR~RFALP=stEAf (YX(J) I ICORA(J(.=CRl3J +-SAVE (j) -
607 X'(.I=CMPLX(AFALP.TFMPfj))
i fCzNPTS/-, . -
CflR(NPTS-IP)=--rOR(J)X(NPTS-TP)=CMPLX(RFEiP,Ari
TF(fAARhN.Fd.2) Ga TO 835oPLOT REAL. PART OF LOG. LiLPPEb
YST.7E=2..
YTN=MT
CAOR I G !1. 1 -
rAL LETtR(J50' 0 5.RA PART OFLG.C.LIPPED9 .9425) i.4,
nPo0 4 J=, PTS54'VW6RK WXJ))
PIT REALS PART. OF: LOG.T (OMPEX N CFPSTRU~ n
L evce ruYc0,7A-9
1N=0
xflQTG=YS1f 75;+1. .~~1. - T0 CALL' OPT(U'IfXORG,0.) -
CAtL LETTFqr 1:O.,~5, ORFAL PART OF IFFT(CMEXEPTYI
Pl. PLT C EP S TQ1J*. FROki -w'-Y'f +-w - .4 -
wnQ<(J+4HALF-I
NHALF=JHALF-I)FLXf)*
nn .-31, j=1NiqALF *'
wn k (j)=Rr si i(fTojn+ j))31 ST'I(J)=wnRK'(J)Z *
(A11PfOAIWORKV TSD)FLXYS1ZEXLEN.YM4AX,YMU"I,fXN.T-rAlI WRCPC f[IT ,RKITS,1,I19'dS,TATE,NAME,AUMJT nnurT= inrDOI)T +.I
201 Ekii.rsPtXfPEALP..0Q.)
1F~f RP(J)) 494,403,495 L* 94 1 F (STM4 tJ.L T.SVMN) STOR(J)=StJMN
GO.rfl 493 9
495 TF(ST()R(J).GT.StJMP) 4STr)Rtjf~sump *-4
493 roNT iNUr-
I'N=-
XORTG=Y' IZFtI.* CALL 'ORi!CJNtKORiG.0.)
r~l.1 PL1ITAIST"',.NPT 5,ELX,-YSlEXLEN,YMAX,YMTN4IN,1P0X,!NC,OTISUM"ThS Jmp / -0.
CALL. NLOGN (X.NPTq,+.O)r PLOT RF.AL PART OF IFFT OF REAL PART
YS17F=?. KCALL. flRTiN(xRI IGO;)
nfl 203 J.=lNPTS 9* S4
TFPDO J, or = OE A (.1 P1FTRP
SJOR (J )=T9:4P J) --
CALL PLQTAtWORK,NPTSOELXYSIZEXLFNtyMAXYMIN, !N,t8Ox.14cDr?-
- s erviceimgroup 4
A-1
no04i=.PI
qf- F(STP6 (J).LT.5;lJAN) S fRCJI=SUMN
905 tFSfR Y TSiIMV) SfOROj)=SUMP.90.4 rPNi!JW mi
1,4,
XfR G=YSTZE+ I
CALL PL]Tk( JflRK.vUTSDFL YS!E,XLE'J,-YHAAKVY't*NN,i80X TNC,.DT)-
DO70 1J=.l.NPTS
20 COJ091 jI= oLM .9"A-
* YS'17=YS?.
'CALL 4.GN
X RI , ~
DO 70 J;-INPT-q
3 n K- ) R A * ((. S -)
- CAL I cTASK
~Flo fILTF flrE4;-S
- ~n ORR KrlLT~I.IFIl!
9nA 3-10, J=j,'J*TSTFEP(J3=OALSAVE(J)
VolV) R dl I =~A I Ar(SAVE (j)* - £ FAPl-,.V)?3 F11PF. P111TOT, IDX.JTAPER -
- - l4jTr-.-71 I TYPEF*POJTPT* tozJTAPEER, A321V FflQ"AT(1X*411'1)
I F IFNJfT=YPS- ?J I P7,2
- : Tn (339,110.315R.JTYPE
VX'-flP 304 j=Ij~jrT,!FlV-PTUOQK(J)=O.
306 TF"P(J)=0.IF(JTAPEQ0,FM)I GO TP 31-6
-. *ITAPFP=INfTPT-4-Dr-G=0.
nflf .J:-'TAPF-P*1N1TPTARG=DFG57.79518
UflqK(J )=Wfl"K(J)*AQGXFiP(J1=TF!'D(JJ*hRG
lTAiPF:lzIF!NDT*4 -
DOr'667 J=1FIN0T1rAPER'
ARG=DEr,/57.29513
* - -. ' APC,(S1.4-CfS(AQG)7
TF"P LI)= TFWP (J I AQ
66 .0ts-2 r.4-
(0f TO) 1 J-
r ILiNG DASS SYSTF M --
1 f)') 311, J=!.IkJJ;P1 --
WflDK(J)'=0. '-
A.1 TF-AP(J).='n. -* -fl 312 J=IFT*4PT,NPTSwflRK(J)=f). -
*F.l 31? TOP~i 316
ITAPr0=lA1TPT+4
nnO 666',J rTITTITAPFR S
ARC-fl5G/57.2958 -
* ARr,=(!..+C('S(A0G))/-2.* WnflQ(J)=WflPK(.JVtARG
-A-12 'servces group
A- -( - -A G 2
WOR (J I WI I*QG
-3.15 - 1TI -
-RF~)(59 1.R)(-j~TCH(),J-.I~lPI-
3tR F~lAATRTIC
* DEG=O. -.
-- DOn 6V7 J=!,TSRI!P- P=~t0Ji-0KJ
ARG'(J=flFI57P95 X-EL,-r TARG=(FIFTOE A'f;)112
318 FOMT(X(J01 +OR(-
PrOTRFA-PA= ONTHJ-Fl-CN - -
- ' T M ()O --- I
n . 1 5 L '' -
- CA'I. LOGNE(.PTS,.O)..,RA AT FSCN T -' -3
.. DO f-. J=l.NPTS-
- RFAIY=-REi(X(J))'* ~ ~~ AT=.- A , (,x(j)+CRJ .
17 )= MtTF.PART OF SACOP AFF
* I~~AlIA-13
r4.
1~) C(,%TTIkNIJF-Zr TA'F IFFT F: ARRAY X ---
* CiAL. P4[OG(X,NPTS*I.o)P1 O! RFAL PART OF 5ECOND TFFT- (OUTPUT I
* - Y.S7F=2.
S ~ -GY ZF_
* CACL ORIGIN (XORTG,O.Jr. n TO (791,791,19-3J,ITYPE
71 CL1i) . TF f02.0 0;1 RA PART OF SECOND) !FFT (SHORT PASS OUT
* GO- TO 794. ~I 792 CALt LETTER(.25..1.'- *REAL. PART OF SECOND IFFT (LN AS UP
I UT199.943) \* *.;-
qn- GfTO 94Ig 3 C.ALL F~TTFR k0.25,...15, #REAL' PART OF 'SECOND 1FF! (COMB OUTPUT)@,
794 CONTYNLJF *
* DO06 62..J=1,NPTS *-I
-WrlQK (J)=PF ~A X, p)')- .
* ~~~ IFTYOE.EQ.2Y GO TO .095 --. y
*flfl 760' J=1;,LPTS -. * * --
760 WORK(J)=WORK(fJi-(ALPHA**AJI -*-C
;68'5 IN=0
;DO 6 8-7 J=i,LPTS GOT68-IYTF(.WORKf(4hGT-D.5) r 1 8
* ,6~C.OW4TNUE.WF *WTF( 6.690) K .*
6PO' FORMAT (/?,lX;jIN.ITIAL' PON1T: FOR UNWEIGHTY*N6 LONG PASS OUTPUT='I F10
L=LPTS*+K'-l -
689 WORK(J)WOR ((j)*jALlPHA**AJ)
CA-WCI0T/OQP~1 Ij A tE 9NAME- ,A) . .
A-14TTnlJ~ .//ice Iru
l~i TYPFF0.V' f0 -T 88
YP41N=T4T NXORl" 1VTF+I.tALL ORTGIN (XflRIGQ,.)*CALL CFTTFR(0.25.O.,04I5-**INPUT SIGNAL MINUS OUTPUTI,90.",5100 81 J=1.NPTS.'
81. rEMP(J)=TiACE(J)-4R(JiPL1ThA(TEMP,KPTS,AELX#'YSIZE - YA*MN9~IOqNT
I DOUT=OOTi888 Cn41IIE
I' CALL TTASK(T2)WRTTE(6,8889) IFILT*T2
999 CO INUF1C . (R!GIN(4.90.)
C LL SLOkNO.1000) NT!NIJE1009 ,0 NT INtIF
-CKL- FNnPLT -
* ALL NN100'STOP
S.U8ROUT4 NE PLOTA(DATA9NPTS,D)EL,SIZEXLEN,YMAX,YMININ,180*),TNC,
01?4FNSTON DATA(1CALL RLORNOXLIY=J(NPT5-I)*F-X*INClWDTIF(iN*FO.I )'GO TO 10
0' YMTN=DATA(l)
no 1 J=?.*NDTS* IF(fATA(,q.GT.VMAX) YP9AX=DATA(J),
I TF(nATA(J).LT.-YMIN) YMINd4DATA(-J!10 SF=YSiLI E MAX-!:IA~rqN
TF(1!iObX*FQ.ljCAft SQUARE(-:YSIZE,(V.90.9XLEN)N-;6
lF(ARSfYMIN*10**N).LT.840060b0*)GO TO 5' 4 N=Nl
5 CALfI.R W4-6-i0. .061 *YMyN99o.,N)'I V(YM1TN.GT.0..OR. MAX.LT.O.) GO TO 2
ZO=YMTN*SFCA L RIGHTJ(ZFR0,o.90rhgo.iCAtIL PL6T(ZFRO.,3)
CALDASH(-ZfRO.XLE Nt-0.*1,.1).
?Ft'A$S(YMAX*1O**N).LT.84ooooo.,) GO TO 8
7 N N-
I'
1, CALL RIGHTJ(-VSIZE,O.,,O.1,YMAX,90.,NI-~9 SF =-SF
AY-0.
- A-15 *-services group
4, CALL PLflT(AX.AY.3)
00n 3 Jz2,lJPTS* AY(J-ti*nELX*INC*DT
AX(OlATA(J)-YMJ1NI*SFA3 CALL PLOT(AXAY,21
-RFTURN.
ENr)
C SIORROtJTINO LG CMUE THE DISCRETE FOURIER TRANSFORM .8YTHE=C N LOG N METHCr) OR- THE VA~ST FOURI-ERTRANS'FORM M.THQD-
rTg'F\ARrUWENTS ARE-
r N W HFRE 2**N IS 'THE NMBER OF: TERM§ IN THE X" ARRAYC x THF- ARPAY OF COMPLEX*NUMBSERS FOR-BOTH INPUT AND OUTOUT.C SIGN EfTHER -j.0 OR- 41.0
-< CIx COMPOTED AS 2**Nr NMiAX=I ARGEcT VALUE OF N TO BE'PROCESSED -
C NONOUMiIY 01#ENSION MINMAX?~c n FORirAiPLFvIF NMAX-25 THEN
nit4ENSION.XtI)CO'4PLEYX X.,WK*HOLD , -
* FLX=LX'ILKa!00 33SQ 1-1925,'
- IF (LX.,LE.!LKI GO TO 3381 -
14380 ftK=ILK* 2 *2 -
3381 CONTINUE
I MI11rn2** IN-InO 10L-,I*NNRLOCK=2**(L-1)
- LBLflCK=LX/N8LOCKLHALF=L KLOCK 12 ,
no It 'ISLOCK-I.NBLOCKFK*K
I GN*6. 28fI" 2cvc 1 F K, f
WK=CMPLX(COS(Vh*SIN(V))I START=1BLOCK*(IBL0CK-1)-no 2 IzI.LMHALF
SJwTSTART41 --ilH=J4LRBHALF
QXJH)*WK~-X(Jjl-X(Ji40
'Ij 'V I V
DO3 T(=-(T
A-16 sevio grVI@9o~up
!F (K .. JGo TO 5
X(Jix (K+1VYX(K+ll)=HOLo
-150O6 1=19N
MFK.LT.M(I) GO TO 7-6 K=K-M( -7 K= 'K+MUIft) -
JF(STGN.LT.0.0) RETURN
83 X(!.)=X(f)IFLXRFTtJQNENO
- SUARMIT!NE AARON(X,WORK,COR.INnEXAINDEXBCLIPNPTSFENOPT,
CO)4PLE X XISIJR=NPTS/2*IF([NDEXA.EO.1) GO' TO 40WORK(1)=fl.0 . -
* Xfl)=CMPLX(CLIP*WORKflll
X(2)=CMPLX(CLIP,WORK(2l).Gb J-0~ 42*
40- CONTYNIJE4? eCnTINUF
VORt 1) =0.
JDO 25 LL=34,!SIMR!F(Ll..LE.LjNrEXA) GO TO 30TF(LL.GT.T.N0EXq)-G0 TO 30> -
MFL.LF.INDiEXA+2). GO TO-410YHAT=,2*WORK(t-Il-WnRK (LL-2)X MOn=AMOD.(Y HAT ,TWOP 1,lIF(Ym.OD.G'T.PI) XMOD=XMOD-TWOP.IIF(-XM0DA.E.--PJ) xmnn=xMOO+TWdPIPtHA=VnfRK (L InflF r-=AEts ( x 'f1D- PHA)
iHPITFF.LT.P!) WORK(LL)#-YHAT+PHA -XMOO-!F(DTFF.GT.PI) u4RK(LtA=YHAT+PHA -XMOD-(TWO]PI*SIGN)CORfIL)=PHA -wnRK(Lt'RX =R EA I- (I )X(LL)=CM~tX(RXtWORK( L))K 0-Gn TO-75
servcesgroupA-17
X( LL).=rMPLXRX9WRKtLL))
30 wnRKfLLh=O.0CCR(LL,)zO.
* X(LL )=CMPLX(CL-[PWrRK(LL-J75 (7ONTINUE * -
* fFjPnFlx9.LE.ISUB) X(,IEND)=CMPLX(CLIPWD0RK(IEND)))
RrTIJRNENDSIBROUT INE.eHA7R X, WORK CR 9:1N9EXAtI.NbOExBCL IP,NPT.S, IENDIP!,
(p'FfNSI;,ON -X(1 ).,WORK(1) ,COR(l)I., TEMP(IM
~* ISUR4EPTS/2
FP=1 .*-EP=TW6PI-FP'
INDEXA=I.- lNfFXTENn
* C TNLS ROUTTNF ALLOWS NO CLIPPI.NG
CORMi)0.
. C0i(Jh=C0R(J-l)
IF()I*F..GT.EP ') COR(JIuCOR(J-1I-TW3PTL* IF(DTFF*LT.-EP) COR(J)-rCORIJ-1)+TWOP!
I S5 TEMP(JI=WORK(J)tC0R(J)00 20 A=1,ISUB
PFALO-REAL(X(j))
WORK(J.=T~iIFAI TM~II)-
RETURN-ENO -
-A 1
