and X(p)" by Philip B. Stark and Robert L. Parker. (Journal of Geopâ¢vsical ... measure of misfit to the r(p) data of Johnson and Lee. [1985]. Some researchers ...
JOURNALOF GEOPHYSICALRESEARCH,VOL. 93,NO. Bll, PAGES13,821-13,822, NOVEMBER10,1988
Correction•o "VelocityBoundsFrom StatisticalEstimates
of r(p) andX(p)" by PhilipB. Starkandaobm •. ?•ke• algorithm of LawsonandHanson[1974] We havediscovereda bug in the computercodethat leastsquares) generatedthe boundson core structurein the paper which was used in the original. Directly incorporat"VelocityBoundsFrom StatisticalEstimatesof r(p) ing upperboundson the modelelements(whichdeand X(p)" by Philip B. Stark and RobertL. Parker rive from the radial derivative constraint introduced by (Journalof Geop•vsical Reseat&,9•(B$), 2713-2719, Stark et al. [1986])reducesthe sizeof the matricesin 1987). The correctboundsare presented herein Fig- the problemby a factorof morethan 8. BVLS alsoalures 1 and 2. They are essentiallythe same as the lows the user to supply a starting guess,in contrast to previous(incorrect)bounds,exceptin the innercore, NNLS, whichbeginseveryproblemafresh.This "warm where the new upper boundsare reducedby about 0.1 start" feature reducesthe typical number of iterations km s-x. The notch in the incorrect bounds at a radius involvedin computinga point in the bisectionsearch of about 700km wasan artifact of the bug. The revised from several hundred to about five. As a result of these boundsdiffer very little from thoseobtainedby Stark two economies,a set of core boundswith 89 levelswas et al. [1986]usinglinearprogramming and a "strict" computedin 4.1 hourson a VAX 11/750computerand measureof misfitto the r(p) data of Johnsonand Lee 40 rain on a SUN 3/160 (with floatingpoint acceler-
[1985].Someresearchers suspected the difference betweenmodelsgeneratedunderdifferentdata-misfitcriteriamightbe large[Doman and.Jacobson, 1981]:In this problemit doesnot seemthat the choiceof misfit
ator). We estimatethe VAX 11/750run time using our originalalgorithmto be around250 hours;to perform those calculations,we used the powerful CRAY
X-MP/48, whichrequired30 rainto complete the task.
Thus the BVLS algorithmprovidesabout a 60-foldinmeasureshas a strongeffecton the results. We have increasedthe speed of the algorithm QP crease in execution speed. in this implementationusinga new algorithmcalled This work wassupportedin part by Nabounded-variable leastsquares(BVLS). We wroteBVLS Acknowledgement.
alongthe samegenerallinesasthe NNLS (nonnegativetional
Science Foundation Fellowship DMS-8705843.
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Ireloc•t31 (lcra s- •) Fig. 1. Bounds ontheseismic velocity in thecoreusingthe25uninterpolated r(p) dataof Johnson andLee[1985].Thisfigurecorresponds to Figure2 of StarkandParker.The dotted curves arethelinearprogramming bounds of Starket al. [1986].The solidcurvesaretherevised QP boundsat the 99.9%and95%confidence levels,computedwith the sameinputparameters as before. The boundsat the 99.9% and 95% confidencelevelsare again indistinguishable. Copyright 1988 by the American GeophysicalUnion. Paper number 88JB03020.
0148-0127 / 88/ 88JB-03020502.00 13,821
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geloc•tj] (krn s- •) Fig. 2. Boundson corevelocityincorporatingboth the 25 uninterpolatedr(p) data of Johnson andLee[1985]andthe five "weak"or the five "strong"X(p) data in Table1 of Stark andParker, derivedfrom the data givenby Johnsonand Lee. This figure corresponds to Figure 4 of Stark
andParker.The solidcurvesare statisticalboundsat the 95%confidence levelusingthe r(p) dataandthe weakX(p) constraints; the dashedboundsuseinsteadthe strongX(p) constraints. REFEttENCES
boundson seismicvelocity in the sphericalEarth, J. Geophys. Res., 91, 13,892-13,902, 1986.
Dotman, L.M., and R.S. Jacobson,Linear inversionof body wave data, part I, Velocity structure from traveltimes and ranges, Geophysics,$6, 138-151, 1981. Johnson, L.R., and R.C. Lee, Extremal bounds on the P velocity in the Earth's core, Bull. Seisrnol.Soc. Am., 75, 115-130, 1985.
Lawson,C.L., and R.J. Hanson,SolvingLeastSquaresProblerns,Prentice-Hall,EnglewoodCliffs, N.J., 1974. Stark, P.B., R.L. Parker, G. Masters,and J.A. Orcutt, Strict
R.L. Parker, Institute for Geophysicsand Planetary Physics, Scripps Institution of Oceanography,A-025, La Jolla, CA 92093.
P. B. Stark, Department of Statistics, University of California, Berkeley, CA 94720.
(ReceivedJuly 1, 1987)
