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UNCLASSIFIEDAD NUMBER
LIMITATION CHANGESTO:
FROM:
AUTHORITY
THIS PAGE IS UNCLASSIFIED
AD480406
Approved for public release; distribution isunlimited. Document partially illegible.
Distribution authorized to U.S. Gov't. agenciesand their contractors; Critical Technology; MAY1965. Other requests shall be referred to AirForce Technical Applications Center,Washington, DC. Document partially illegible.This document contains export-controlledtechnical data.
usaf ltr 25 jan 1972
B - M
"> KJ M sS^.^: uS raff ~ o SHR--.-i^;. II : ' HH
iS :-
INTERPOLATED ,IEFFR=YS AND PULLEN SEISMOLOGICAL TABLES
D D C 1W APR2 01S56 jij
THIS nrcu^E'lT iS SUBJECT TO SPECiAl -i c-rcLs AMO vm TRftHSMtim ToKblGHGCVERNMEMTSORFOREl^ HAT10HL MAY BE MAOE CNIY WiTH PRi^H APPROVAL OF CHtEF^rWS.
THE BEDTSCHNICAI CORPORATiaN a..iLANP, Tt:x.^^Il
-_ _. .,-,v..~~^i^-Aiitv-^J--;;^
BEST AVAILABLE COPY
TECHNICAL REPORT NO. 65-35
INTERPOLATED JEFFREYS AND BULLEN SEISMOLOGICAL TABLES
by
H. S, Travis
AFTAC Project No: ARPA Order No: AR PA Code No: Contractor:
Contract No;
VT/40.,P 104-6 8100 The Gv;otechnical Corporation, Garland, Texas AF 33(657)-i2145
THE GEOTECHNICAL CORPORATION 3401 Skiloh Road T: ? n^^sST irSJlCT TO SPECIAL Garland, Texas I . ' , C!'::;T-GlS AF EACH TRANSf,";"Al
TC FGKEoH GGVErfaOTS OR Fi-tiSN mmm. mt DE MIDI: ONLY mn PKIOR AH^GVAL CF CHiff^FB!.
May 1965
CONTENTS
ABSTRACT
1. INTRODUCTION
2. DESCRIPTION OF METHOD
3. NUMERICAL TECHNIQUES 3. 1 Lagraugian interpolation formula 3.2 Linear interpolation formula
4.
5.
6.
REMARKS
INDEX OF PHASES
INTERPOLATED JEFFREYS AND BULLEN SEJSMOLUGICAL TABLES
Page
1
2
$ 3 4
4
5
TR 65-35
ABSTRACT
The travel times for all body phases tabulated in the Jeffreys and Bullen SeismoJogical Tables (1958) have been calculated by interpolation for distance intervals of 1 degree at each of the 14 focal depths established by Jeffreys and Bullen. Travel times have been similarly derived for the phases "diffracted P" (P at distances beyond 105 degrees) and "PKPPKPPKP, " based on data extrac- ted from the Gutenberg and Richter travel-time charts (1934-1939), The cal- culations were effected by means of a digital computer program principally emtloying Lagrange's four-point interpolation formula. This report discusses the techniques involved and presents the fully interpolated tables in a convenient form.
TR 65.35
INTERPOLATED JEFFREYS AND BULLEN SEISIviOLOGICAL TABLES
I. IN'IROBUCTION
This report prestnts a fully interpolated version of the Jeffreys and Bullen (J-B) Seismological Tables and discusses the manner in which it was gen- erated. The travel times for all teleseismic body phases contained in the J-B tables are given for diotance intervals of 1 degree at each of the 14 standard J-B focal depths. The times are tabulated in a convenient reference form very similar to that employed for the "P1' and "S:' phases in the existing J-B tables. Tra/el times are similarly presented for the phases "diffracted PH {P at distances beyond 105 degrees) and "PKPPKPPKF. " based on data extracted from the Gutenberg and Richter travel-time charts.
These interpolated J-B tables have been calculated as a convenience for seis- mologists who must have frequent access to accurate, closely resolved (in distance and depth) travel times for ail phases. The only phases in the exist- ing J-B tables listed for distance intervals of 1 degree at all depths are "P" and "S. " It is anticipated that the tables will save m^ny hours of needless, repetitious manual interpolation on the part of analysts trying to identify and associate phases, Particular application is seen for compiling seismological bulletins, such as those assembled under the Long-Range Seismic Measure- ments Program, which routinely involve a vast amount of table lookup for accurate association of phases.
Section 2 of this report discusses ia detail the interpolation and computational methods used to create the interpolated tables, and section 3 specifies mathematically the interpolation formulas used. Section 5 provides an index of the phasei available, thair various branches, and the ranges of distance
Jeffreys, Harold, and Bullen, K. E, , 1958, Seismological tables: British Association for the Advancement of Science Publication
^Gutenberg and Richter, 1934-1939, Seisnnic wave papers, I-IV: Gerlands Beitrge Zur Geophysik
TR 65-35 -1-
Jeffreys and Bullen, op. cit.
TR 65-35 -2-
i over which they are tabulated. Depth phases such as "pP" and "sS" are not included because of the ease with which their times can be calculated from the "P" and "S" tables.
2. DESCRIPTION OF METHOD
A program for the Control Data 160-A digita) computer was written to inter- polate the existing Jeffreys and Bullen seismological travel-time tables and tabulate the results. Primarily, the program employed the Lagrangian four- point interpolation polynomial to interpolate between the times; however, when four points could not be obtained, a linear interpolation method was used. For each phase where it was necessary, the program first calculated by inter- polation the depth allowances at all standard J-B depths for distance intervals of 1 degree. Depth allowances in the Jeffreys and Bullen tables are generally given at distance intervals of only 10 degrees. Subsequently, the interpolated depth allowances were subtracted from the surface times of the phase to obtain the corrected travel times at depth. In some cases, the times for surface focus were given at 5-degree intervals and also had to be interpolated for each degree.
For core phases other than "PKP, " "SKS, " "PcP, " and "ScS, " the allowances for depth had to be obtained by first determining the dt/dA at 1-degree intervals for each phase's surface travel-time curve. Then the allowance 5^r depth were taken as equal to those given (or interpolated) for the above core phases at distances where dt/dA were the same. For example, to obtain the desired depth allowance for the phase "SKKS" at 150 degrees for depth ?. 06R, it is determined that dt/dA at that distance is 5.5 sec/degree, which is matched by "SKS" at 94 degrees. Therefore, the depth allowance for "SKKS" at 150 degrees for depth 0. 06R is equal to that for "SKS" at 94 degrees and depth 0, 06R. 3 For those phases whose depth allowances were not defined by the J-B tablea, such as "PP" at 10 degrees and depth 0,06R, the times were left blank.
Travel-time tables were tabulated for those phases which produce multiple com- binations, such as "PcSPKP. " Tables for the phases "ScPPKP" and "PKPScP" (or PKPPcS) were tabulated by obtaining the proper depth allowances (for ScP and PKP, respectively) by th3 atandard procedure and subtracting these times i
from :.he surface times of "PcSPKP. " The travel times for iffrac^ed nP" and ^FKPFKFPKP11 were obtained from the Gutenberg and Richter travel- time charts at distance intervals of approximately 0, o degree and at depth intervals of 100 kilometers. From these, travel times were determined for distance intervals of 1 degree at thi standard J-B focal depths.
Generally, the times for each phase given in the standard Jeffreys and Bullen travel-time tables represent average times with standard errors of under 1 second. Inevitably, the interpolated tables presented erein have the same error; however, it should be pointed out that possibilities of small disagree- ments with the standard tables arise from the smoothing technique introduced. In the majority of cases, the interpolated times agree to 0. 1 second with the standard tables. However, the derivatives (dt/di) of the travel-time curves for several of the more exotic body phases (e.g. , PKPPKS) were so erratic the interpolation technique employed smoothed over these points. Conse- quently, the times differ occasionally from a linearly interpolated time. The magnitude of the disagreement is generally less than half a second.
3. NUMERICAL TECHNIQUES
3. 1 LAGRANGIAN IKTERPOLATION FORMULA
Numerical four-point interpolation (Hildebrand, F. B. , 1956) can be obtained from the Lagrangian interpolation polynomial of degree 3 expressed in the following general form:
y(x)= Mx)f'xk; k=0
where:
y{x) is the Lagrangian interpolation polynomial of degree 3 that approximates the assumed continuous function f{x).
f(xjt) is a series of discrete ordinate points (corresponding to abcissa values XQ. X|, X2. and X3) representing the function to be approximated.
4Ibid
TR 65-35 -3-
and the desired Lagrangian coefficients are given by:
^k(x) = (x-x0) . . . (x-xj^.,!) {x-xk+l) . . . (X-X3)
{xk-x0). . . (xk-Xk-iMxk-xjcf i). -. {xk-X3^
where vhe factors (x-xk) and {xk-xk) are to be omitted in the products. Thus, a vaJv i of y may be calculated for any value of x by means of the Lagrangian polynomial. To obtain the highest degree of accuracy, the value of x should be centered between the two innermost data pcintp (xj and X2).
3.2 LINEAR INTERPOLATION FORMULA
Linear numerical interpolation (Hildebrand, F, B. . 1956) between two tabular points can be obtained from the following formula:
f(x)f(x0) + (X'X0) (X^XQ)
fl.X,) . f{Xn)
where:
f(x) is a function assumed to be linear between the two tabular points, XQ and xj.
4, REMARKS
The interpolated Jeffreys and Bullen seismological tables are also available on punched cards.
TR 65-35 -4.
5. INDEX OF PHASES
PHASE BRANCH
1. P 2. P diff i. S 4. PP 5. PS 6. SS 7. s? 8. PPP 9. PPS
10. PSS li SSS 12. SSP 13. SPP 14. PcP 15. PcS )i. ScS 17. ScP IS. PKP
DISTANCE RANGES PAGE
19. PKS
20. SKS
21. SKP
22. PKKP
AB BC DEF
AB BC DEF
AC DEF
AB BC DEF
AB BC DF
0o-lC5o 9 105-150 12
o'-io/ 14 10-180 17 50-140 22 10-180 25 4^-140 30 10-220 33 60-180 39
100-220 42 10-220 45 90-220 51 45-180 55 0- 80 59 0- 70 61 0- 80 63 0'- 80 65
143-180 67 143"-147 68 110-180 68
13 >0-14&0 70 130-140" 70 l-.-o-i80o 70
70-133^ 73 99-180 74
ISO'-HS"' 77 130P-140 7? ),r,4o-180 77
104-126 80 93-126 80 r -162 81
TR 65-35 .c.
PHASE BRANCH
23. PKKS AB BC DF
24. SKKS AC DF
25. SKKP AB BC DF
26. SKKKS ABC
27. PcPPKP AB DF
28. PcSPKP AC DF
DISTANCE GANGES PAGE
2v. ScSP
30. JcSPKP ABC D
31. ScPPKP AC DF
32. PKPPcP AB DF
33. PKPPcS or PKPScP AC
DF 34. PKPScS
ABC D
35. PKPPKP DF
36. PKPPKS or PKPSKP B
DF
134'-146' 86 100-146 86
0*-168' 87
85*-243 92 0o-173o 96
134,,-x460 101 100 "146*' 101
0'-i68o 102
90-370
82-187 132-180
110o-1949 1278-180,'
s o 112 -140 11Z0-1300
l60o-Z0b' IZV-ISO0
110o^194,' 12'o-180o
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1600-2060 i2r-180o
0e-i4no
o 75 - 84 0e-145o
107
114 116
li8 120
122 122
124 125
127 129
131 133
135 137
139
142
1 6 146
TP 65-35 -6-
PHi3 iSE BRANCH
37. PKSPKP B
DF 38. SKPPKP
B DF
DISTANCE RANGES PAGE
39. SKSP 115"-205 158 40. PKPPKPPKP 446-104o 161
TR 65-35
f
75- 84 150 0o-i45o 150
75 - 84 154 0o-145o 154
6. INTERPOLATED JEFFREYS AND BULLEN SEISMOLOGICAL TABLES
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