Estimating the Relative Rotation of Two Tectonic Plates from Boundary CrossingsAuthor(s): Ted ChangSource: Journal of the American Statistical Association, Vol. 83, No. 404 (Dec., 1988), pp. 1178-1183Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2290152 .

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Estimating the Relative Rotation of Two Tectonic Plates From Boundary Crossings

TED CHANG*

Let .7, . . ., a, be unknown vectors on the sphere and Ao be an unknown rotation. Suppose that uii are estimates of points lying on the great circle normal to Ii and Vik are estimates of points lying on the great circle normal to Ao11i. This article discusses a method to construct a confidence region for A,. This problem arises in the reconstruction of the relative motion of two tectonic plates on opposing sides of a rift. The boundary on each side is represented by a collection of great circle segments, and Ao is the rotation that takes one boundary into the other. The data consist of measured crossing points uij and Vik of the various segments on the opposing boundaries. The analysis completes an analysis of Hellinger (1981). The errors in tectonic data are quite concentrated, and the problem reduces to linear regression. Once this is realized, many interesting problems such as triple junctions or multiple time periods can be examined. The analysis is aided substantially by a parameterization of the rotation group, which does not destroy the inherent symmetry in the problem because it satisfies a group model (in a statistical sense). Consistency of the estimator is shown in an Appendix. For the case when the unknown parameter is estimated by minimizing some continuous function, a consistency lemma is given that requires only compactness in the parameter space, a "unique minimum condition," and a law of large numbers for the sampling distribution. KEY WORDS: Estimated rotations on spheres; Reconstruction of tectonic plate motion.

1. INTRODUCTION Chang (1986, in press), Rivest (in press), and Bingham

and Chang (1988) discuss the question of estimating a rotation A, from pairs of points (ui, v,) on the sphere that would have satisfied vi = A0ui in the absence of error. This problem arises, among other places, in estimating the relative motion of diverging tectonic plates. A summary of the results of these papers appears in Chang (1987).

The geometric picture (see Fig. 1) shows that the present shape of the plates has (not necessarily perpendicular) staircase-shaped boundaries and that the two staircase boundaries once meshed together. In this picture the ui represent measurements of the pivot points on one bound- ary and the vi represent measurements on the other. On the boundary the straight segments in one direction are called magnetic anomaly lineations and those in the other direction are called fracture zones. Thus the pivot points represent intersections of magnetic anomaly lineation and fracture zones on the two plates.

The staircase boundaries are under water; hence the ui and vi are not measured directly. The chief complaint of geophysicists is that the points (ui, vi) are not data. As ships crisscross the boundary they measure the points at which they cross a magnetic-anomaly-lineation segment or a fracture-zone segment. There is no correspondence be- tween the crossings on one boundary and the crossings on the other boundary. All that is known is that the segments on one boundary can be rotated into the segments on the other boundary. Because the fracture zones continue across the rift, it is possible to determine the pairing between the segments on opposite sides of the rift. The crossings are then mapped and the intersection points (ui and vi) are "interpreted." This article discusses the construction of

* Ted Chang is Associate Professor, Department of Mathematics, Uni- versity of Virginia, Charlottesville, VA 22903. The author acknowledges support from the Kansas Geological Survey at the University of Kansas. He also thanks Peter Molnar for geophysical advice and suggesting the problem explained in this article.

confidence regions for the unknown rotation using the raw data-that is, the ship crossings.

Hellinger (1979, 1981) proposed an estimation tech- nique for the unknown rotation Ao. Hellinger's approach was to model each segment of the staircase-shaped bound- ary by a segment of a great circle. Although the segments are not great, each segment is short and the error in ap- proximating a segment by a great circle is negligible com- pared with the errors in the data. If 11, . . . , q, denote the normals to the great circle segments on one boundary, then AOq1,. . . , A0is are the normals to the corresponding segments on the opposing boundary. The data consist of estimated crossings uij on one side and Vik on the other; the first index denotes the crossed segment. Hellinger used a least squares technique to simultaneously estimate Ao and q1, * * . , as. He then constructed "uncertainty re- gions" for Ao.

This article proposed adopting the Hellinger approach to estimate Ao. For a variety of reasons, I feel that Hel- linger's uncertainty regions are not a satisfactory solution to the confidence-region problem, so I propose a different solution that is more consonant with standard statistical practice.

This article is asymptotic in the sense that the variance of the data error distribution goes to 0. The errors in tectonic data are miniscule (relative to the circumference of the earth), and it seems plausible that concentrated error asymptotics are more relevant than large-sample asymptotics. My previous work with Bingham on spheri- cal-regression (ui, vi) models found concentrated error ap- proximations to be highly satisfactory. In addition, for this problem concentrated error distribution theorems are eas- ier to prove and the results are simpler to express than for large-sample theorems.

In Section 2 the construction of a confidence region for

? 1988 American Statistical Associatlon Joumal of the American Statistical Association

December 1988, Vol. 83, No. 404, Theory and Methods

1178

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Chang: Estimating Rotations From Boundary Crossings 1179

o(11

2 0(2

0(

0(4

d 4 2 &41Q

\

Figure 1. Geometric Representation of Two Opposing Tectonic Plate Boundaries. Dark lines represent plate boundaries; light lines represent ship tracks. The data for this study are measurements u,j of a,i and Vik

Of ,ik. The data for spherical regression studies are measurements u, of a and v, of &,.

A, is discussed. Section 3 discusses Hellinger's uncertainty regions and other relevant papers from the geophysical literature. Section 4 discusses some two-sample tests.

2. THE MAIN RESULTS

The uij (j = 1, . . . , mi) are assumed to be measure- ments of unknown points aij on the ith great circle seg- ment; that is, a('qi = 0. Similarly, Vik (k = 1, . . . , ni) is a measurement of the unknown point &ik on the great circle perpendicular to Ao0qi. Choose unit vectors /fJj and flik SO that aij, /,ij, 1i and &ik, fki Ao0i form a right-rule-oriented orthonormal basis of Euclidean 3-space. Let N = (mi + ni).

Write uij = (u,jaij)aij + xij and Vik = (Vik&ik)&ik + Yik Thus xij and Yik live in the tangent planes to the sphere at aij and &ik, respectively. Finally, assume the existence of constants Kij, kik, Aij, and 2ik SO that as K -X 00, K12Xij and K"/Yik converge in probability to independent bivariate normal distributions with mean 0 and covariance matrices Ki q( i?1' + j1fi ijjfi'and k-'A0?1iq(iAt + fik fi,k, respec- tively. The proposed methods require known Kij and kik.

Application of these methods does not, however, require knowledge of the values of Aij and Aik.

One way to accomplish these probability assumptions is for uij to have the Fisher-Bingham distribution c(,lij, vij)1 exp{uij a,uij + Vij((-(Uij)2

_ (fi,(uij)2)}, with/ui = 2K(Kij + Aij) and vij = 'K(2ij - Kij). Similar assumptions are placed on the Vik. The properties of the Fisher-Bingham distribution were discussed by Kent (1982).

Following Hellinger, choose estimates A and ?1 to min- imize r(A, vj) = 1j'j Kij((U5?17)2 + Yi,k Kik(v vkA 11)2 where q represents the s tuple ('*l, . . , ns). Since r(A, ) =

zi ?15(i + A'E1A)q1j, where >i = Ej KijUijU(j and >Z = Ek kikVikVik, it follows that for any given A, r is minimized by choosing ̂i(A) as the eigenvector corresponding to the smallest eigenvalue of 1i + A'EiA.

It is well known that A can be expressed as A = A0(I + B + 132/2! + *o+ Bk/k! + *)= A0exp B, where B is 3 >x 3 skew symmetric. Similarly, write ?1b = (1?i? + Xri-

Theorem 1. Suppose that ni + mi ? 2 for all i and either (a) mi n 2 and ni 2 1 for at least four values of i or (b) mi ' 1 and ni ' 2 for at least four values of i. Then for almost all configurations (Ao, ij aij, aik), as K -> Co, (A, rl) > (AO, q). The proof is in the Appendix.

Then, using standard Taylor series arguments, B and Xi are O(K- l/2). Since Aiqi = (1 - At)1/2, iti, Wuaija, and V~ik&ik are all 1 - O(K 1). Thus

r(A, i1) = > Ki1[(a(i + x,1 + O(K1))'Q1, + 4, + O(K1))]2 1,)

+ E k1k(&,k + Y,k + O(K 1))t *, k

x A#(I + B + O(K 1))(qi + 4, + O(KQ1))]2

- E Kjaj + x,tqi + O(K 1)]2 .,

+ E Kik[a,tAo,j + &IkA,B?j + YtkA^ +O(K1)]2 ik

=EKij(X(tqi + atiX')

+ > K,k(Y1tkAo?7 + (Ato&k)tQU, + Bi ))2 + O(K '). ik

Recalling that K112Xqj"/ and K112Y (kAO71i approach in prob- ability independent normal variates with variances K171 and

ik I respectively, we see that r(A, ) is, up to o(K-1), the error sum of squares of a weighted linear regression of x,'jj and y('Aoqi, with unknown parameters Xi and B and a design matrix constructed from Ao, aiaij, and alk.

Theorem 2. As K ?-> (assume that N > 2s + 3 and the consistency of A and A), (a) K(r(Ao, i1(Ao)) - r(A,

-*(A))) x2(3), (b) Kr(A, (A)) - 2(N - 2s - 3), and

r(Ac ) i(A0)) - r(A, 11(A)) r(A, Aij(A))

3 -2 F(3, N - 2s - 3).

N -2s- 3

Now consider Hellinger's (1981) anomaly 13 reconstruc- tion. This data set is reproduced in Table 1. Hellinger assumed that xij and Yik have covariance matrices of the form acrjI or &o2I, respectively, where aij and &ik are given in the table. The given estimates of aij and &ik were derived from the navigational uncertainty of the ship that made the observation and the estimated width of the magnetic anomaly or fracture-zone profile that the ith section rep- resents. Since the estimated width of the profile only con- tributes to error in the directions 1i and Ao0i, the assump- tion of circularly symmetric error is too restrictive and the preceding analysis demonstrates that it is unnecessary. After converting the aij and 0ik to radian measure, set Kij = ai 2andKik = ik A

Hellinger (1981) found A by calculating r(A, ij(A)) on a grid of trial rotations A. A smaller grid was constructed around the best A, and the procedure was iterated until A was found to an acceptable precision. The ZXMWD routine of the International Mathematical and Statistical

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1180 Journal of the American Statistical Association, December 1988

Table 1. Anomaly 13 Data From Hellinger (1981)

East of rise (v,,) West of rise (u,1)

SD SD Section Coordinates error Coordinates error

1 -55.59, -100.38 20 -51.19, -133.64 20 -56.39, -100.97 15

2 -57.61, -119.37 30 -51.67, -150.00 15 -58.84, -121.16 15

3 -59.28, -121.98 17.5 -51.90, -150.59 15 -60.18, -123.74 1 5 -52.57, -151.89 15

4 -59.95, -130.56 12.5 - 53.23, -159.05 15.4 -54.37, -160.64 12.5

5 -56.97, -104.60 15 -50.10, -137.00 4.1 -50.95, -134.70 6.1

6 -60.24, -125.60 3.2 -53.77, -151.82 9.6 -59.76, -127.50 3.2

7 -59.08, -104.88 8.1 -52.48, -137.00 13.1 -58.49, -107.23 10.5

8 -57.89, -109.35 8.6 -51.45, -140.10 12.7 -58.49, -107.23 10.5

9 -57.71, -115.66 7.1 -50.30, -146.85 10.8 -50.90, -145.00 6.1

10 -59.01, -110.17 12.7 -52.42, -140.17 7.5 -51.70, -142.83 10.3

NOTE: Coordinates are degrees of latitude and longitude. SD represents standard deviation.

Libraries is sufficient for finding A. The algorithm for ZXMWD involves calculating r(A, i^(A)) for a preset number of systematically placed candidates A for A. A quasi-Newton method (in which the gradient and the Hes- sian are numerically approximated) is used with a small number of iterations to refine each candidate A. The five best candidates are iterated to convergence, and the best is selected as A.

For the anomaly 13 data of Table 1, r is minimized by choosing A to be a rotation of 27.860 around the axis 74.600N, 57.92?W, and r = 3.06. Using Theorem 2(b), a 95% confidence interval for K iS .71 < K< 5.74; hence we derive no contradiction to the assumption that K = 1. In other words, the values of ui, and Aik in Table 1 can be assumed correct instead of just relatively correct. (Theo- rem 2 is asymptotic as K - oo. Since the Kij and kik are quite large, we are implicitly dividing them by a large constant c and multiplying K by c before using Theorem 2.)

In principle, Theorem 2(a) (for K known) or Theorem 2(c) (for K unknown) can be used to produce a confidence region for A,. Such a procedure is rather cumbersome; I prefer working with an estimated covariance matrix of the skew symmetric matrix B defined by A = A0exp B.

For the 3 x 1 vector t = [t1 t2 t3]t, define

-0 -t3 t2_

M(t) = t3 t t2 t, ?j

Note that M(t)q = M(i)'t = t x i. For i 1, . ..s, let 0, be a 3 x 2 matrix whose columns form a basis of the vectors perpendicular to ai. Letting B = M(t) and Xi = 01s, (where s, is 2 x 1), from the argument of Theorem 2, up to order o(K'1), r(A, i) is equal to the error sum

of squares of the regression

-VKiXj = (V'ia1Oj,)Si + cil

- ky IA ,,i (k?tk&A ,Oi)si

+ (\/ k,kA0M( j71))t + C1k,

i= 1,... ,s, j=1,. .. ,mi, k= 1,... ,n,.

Let X be the design matrix of this regression. Then

-X [Hll H12] H21 H22

where i= Yj Kijaij a(., Zi = Sk 1kaikakk Hll =

M(jj)Ao1jA,M(qj)t (3 x 3), H12 = [M(11)AI 2IA0OO * M(s)AO sA00s] (3 x 2s), H21 = Ht12, and H22 = block diagonal[O1(E1 + Ao1ZAo)O1 *... O(ls + At!sAO)OS] (2s x 2s). Accordingly, we arrive at Theorem 3.

Theorem 3. Let H1l.2 = Hll - H12H-1H2j. Then, as K-~ 00, (a) r(A0, ri(A0)) - r(A, ai(A)) = ttH11.2t + o(K-1);(b) the covariance matrix of K112t is asymptotically Hi1j; and (c) if H1I.2 is estimated using the estimates A for Ao0, i for t7j, Ij KijUijU, for Si, and zk KikVikVik for li,then C = {A exp M(t) I ttH11-2t < x2(3)IK} (K known) or C = {A exp M(t) I ttH11.2t < 3r(A, i)F0(3, N - 2s - 3)1(N - 2s - 3)} (K unknown) is an asymptotic 100(1 - a)% confidence region for AO. [Here x2(3) and Fa(3, N - 2s - 3) refer to the appropriate critical points of the indicated distributions.]

For the data set of Table 1,

1.420 -.106 -.281 H11.2 = -.106 .265 -.097 x 106.

L- .281 -.097 .113

In principle, the confidence region C can be written as (axis latitude, axis longitude) EE a and g(latitude, longi- tude) < angle of rotation < f(latitude, longitude) for an appropriate region a in the plane. Many points on the boundary of C were calculated, and a contouring program was used to produce contour maps of the graphs of f and g (the upper and lower surfaces, respectively). The results appear in Figure 2. Thus, for example, all rotations around 750N, 56?W and with an angle of rotation between 27.80 and 28.70 are in the confidence region of Figure 2. A complete description of the algorithm used to produce these figures appears in Hanna and Chang (1987).

3. RELATIONSHIPS TO PREVIOUS WORK IN THE GEOPHYSICAL LITERATURE

Hellinger (1981) constructed uncertainty regions for AO by including in that region all rotations A such that u'jA (A)l <2ij and IV( A ii(A) I< 2&ik. By considering the analogous problem on the circle, it is easily seen that this does not constitute a confidence region in the statistical sense. For some reason (which I do not understand) the uncertainty region that he constructed in this manner is strikingly sim- ilar to Figure 2.

Hellinger (1979) also proved that for each i, if

ri= f,i(A0)t( KijUijUI; + Ao 2 KikVikVikA0) Vi?(A0)

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Chang: Estimating Rotations From Boundary Crossings 1181

-0 -66 -62 -58 -54 -50 -46

76 3O 76

75- 2 9 a -75

74- (--2 ~74

72 26 ~ 272

71 -70 -66 -62 -58 -54 -50 -46 71 Upper Surface

-70 -66 -62 -58 -54 -50 -46

76 76

750 -66 -62 -58 -54 -50 -46 Lower Surface

Figure 2. 95% Confidence Region for Anomaly 13 Reconstruction (Table 1) With K = 1.

then as K ?? o, Kri iS asymptotically X2(mi + n, - 2). He then proposed constructing a size-ax confidence region for Ao by including in it all rotations A such that Kr, <x~ (m, + n7 - 2), where 27 = a. Although correct, by analogy with linear regression, one expects this procedure to be inefficient relative to the one used in this article.

Pilger (1978) proposed a different scheme for estimating the rotation Ao from data such as those considered here. Given a point u11, if n1 ' 2, let dij(A) be the distance between ua 1 and the great circle through two closest points of the form Aovsk. Similarly, given a point Vik, if mo 2 let dAk(A) be the distance between Aivok and the great circle through the two closest points of the form uhi. Pilger chose A to minimize 4>7(A) = p dif(A)2 + siek dik(A)tn For example, in the data set of Table 1, cn = de + dh2 + d41 + d51 + d1l + d2o01 + dhl + d2, + d31 + d32 + d + dGv + dpi. He did not discuss the errors in such an esti- mate.

Using heuristic arguments similar to the construction of uncertainty regions in Hellinger (1981), Stock and Molnar (1983) arrived at "partial uncertainty regions," which have certain qualitative features that can be verified here. Sup- pose for simplicity that Kij = Kik = 1 for all i, the Mia are evenly dispersed over the ith segment, and the gr ik are evenly dispersed over the ith rotated segment. Then N, -~ mi(c1aeice + (1 - ci)flBI,B) and S, -~ niA0(cicaiae + (1 -

ct)hrio)Au , where at is the center of the ith great circle segment, /, A = )i x a1, and c= is a dispersion constant determined by the length of the ith segment. For short segments c+ would be close to 1. If we let the columns of O, be heu andist, reasonably straightforward calculations

then show that

H11.2 -E nimi (cflAif/ + (1 - ci)aia().

i ni + mi Since /3'ai = 0 and ci is close to 1, if the ai tend to cluster near a0o, then a' H11.2ao will be relatively small. In other words, the eigenvector corresponding to the smallest ei- genvalue of H11.2 should be close to the center of the uij. Physically this means that the small rotation AI A is least constrained if its axis is close to the center of the data. Stock and Molnar called this the "skewed fit" axis.

For the data set of Table 1, the eigenvalues of H11.2 are 2.644 x 10, 3.113 x 10, and 1.483 x 106. The eigen- vector corresponding to the eigenvalue 2,644 is at 62.6?S, 116.70W.

It is often necessary to reconstruct the relative positions of two plates by working through a third plate [see Molnar and Stock (1985) for a brief discussion of the reasons]. In other words, to estimate A = A1A2 we first calculate Al and A2 and then set A - A1A2. Setting A = A exp M(t) and Ai = Aiexp M(ti) we get

exp M(t) = (exp M(Alt1))(exp M(t2)).

Thus M(t) =M(A2t1) + M(t2) + o(K-112) and t = A2t1 + t2 + o(K- 12). Then the covariance matrices of t, tl, and t2 satisfy cov(t) =A2cov(t1)A2 + cov(t2) + o(K-1), assuming the errors in the data used along the two rifts are independent. Theorem 3(b) can then be used to con- struct confidence regions for A. This approach for deter- mining the uncertainties in a composite reconstruction was used by Jurdy and Stefanick (1987), but with a different parameterization of the rotation group. For roughly the same reason that the uncertainty regions of Hellinger (1981) are not confidence regions, the matrices that Jurdy and Stefanick used are not covariance matrices in a statistical sense, but rather a reformulation in matrix form of Stock and Molnar's partial uncertainties.

4. TWO-SAMPLE TESTS

Since r(Ancn) is asymptotically the error sum of squares of a linear regression, and since Rivest (in press) notes that the regression test of a linear hypothesis is asymp- totically valid for any submanifold hypothesis as the error variance goes to 0, many hypotheses of interest can be tested using an extra error sum of squares principle.

Hellinger (1981) also included data from anomaly 18 from the same rift system in the South Pacific. Using the dating of LaBrecque, Kent, and Cande (1977), anomaly 13 is about 35.56 million years before present and anomaly 18 is 42.14 million years. One possible question of interest is whether the axis of rotation is different for the two reconstructions and, if it is not, if the angular speed is the same. Geophysicists generally believe that the axis and angular speed do change with time. Nevertheless, when fitting rotations, they often assume constant axis and an- gular velocity over certain time spans, so it is of interest to be able to assess this assumption.

Under the assumption that the shape of the rift did not change from anomaly 18 to anomaly 13 time, we can hy-

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1182 Journal of the American Statistical Association, December 1988

pothesize the existence of rotations A2 from 18 west to 13 west, A3 from 18 west to 13 east, and A4 from 18 west to 18 east. To estimate A2, A3, and A4 and to perform hy- potheses tests involving them, we need to identify the corresponding sections of Hellinger's anomaly 13 (Table 1) and 18 data sets.

In producing the data, Hellinger (1981) divided several physical features into two sections because he detected significant curvature in the feature: "This was detected by careful inspection of the data as it was nearly impossible to detect by eye" (pp. 9313-9314). In Table 1, sections 2 and 3, 7 and 8, and 9 and 10 represent parts of the same physical features. To identify the corresponding sections of Hellinger's anomaly 13 and 18 data, I found it neces- sary first to rejoin the sections of those data correspond- ing to the same physical features. The result is shown in Table 2.

One can check the necessity of using 10 sections in Table 1 as opposed to 7 sections by an extra error sum of squares test. If in Table 1 we assume that q2 = q3' m7 = p18, and ?9= ?l0, one gets r = 4.20 with 14 df. Otherwise one has r = 3.06 with 8 df. Thus the F statistic is F = .49 with (6, 8) df. One concludes that seven sections in Table 1 are in fact adequate.

Letting AI = I, we estimate Ai and 'i to minimize r = ak= Y ij Kkij(UkijAkkT)2, where Al = I and Ukij are given in Table 2. The arguments of Section 2 apply.

Geophysicists generally assume symmetric spreading from the center rift, although asymmetric spreading does occur in rare cases. Assuming symmetry, let B1 and B2 represent the rotations from the unknown center to 13E and 18E, respectively, so that Bft and B-1 are the rotations from

the center to 13W and 18W. Then A2 = Bf1B2, A3 = B1B2, and A4 = B2. Then A2 = A1'2A A-'112.

The total number of data points is N = 58. The number of sections is s = 9 and hence N - 2s = 40. Fitting asymmetric spreading (i.e., arbitrary A2, A3, and A4), one gets r = 31.41 with 40 - 9 = 31 df. Fitting symmetric spreading (i.e., arbitrary A3 and A4, with A2 set to

1/2 A-'A 1/2), one gets r = 37.53 with 40 - 6 = 34 df. Fitting a symmetric spreading with a common axis and separate angular velocities, one gets r = 75.60 with 36 df. Fitting symmetric spreading with a constant angular ve- locity, one gets r = 144.54 with 37 df. Thus F = 2.01 with (3, 31) df for symmetric spreading and F = 18.79 with (2, 31) df for a common axis assuming symmetric spreading. We thus conclude that the spreading is symmetric with a changing axis.

Alternatively, if one is not willing to assume that the shape of the rift did not change between anomaly 18 and anomaly 13 time, one can treat the data as two indepen- dent samples. The details are left to the reader's imagi- nation.

APPENDIX: CONSISTENCY PROOFS

The proofs of Section 1 can easily be made rigorous once consistency of (A, f(A)) and fij(A0) are proven. To prove this, we rely on the following proposition.

Lemma 1. Suppose that f is a continuous function on bX x '.9 with -,X compact. Suppose for a particular y0 e 9 that f(-, y0) has a unique minimum x0. Suppose that Y, are random variables taking values in 'l such that Y, -> y0 in distribution. Furthermore, suppose that X, is a choice of minimum for f (, Y,). Then X, xo in distribution.

Table 2. Anomaly 13 and 18 Data From Hellinger (1981)

18E 13E 13W 18W Section (i) coordinates (u411) coordinates (U311) coordinates (u2i,) coordinates (u,,,)

1 -55.52, -98.60 (20) -46.52, -133.28 (10) - 47.33, -133.92 (15.4)

2 -55.59, -100.38 (20) -51.19, -133.64 (20) -56.39, -100.97 (15)

3 -59.28, -118.68 (17.5) -57.61, -119.37 (30) -51.67, -150.00 (15) -51.52, -152.15 (30) -60.50, -120.90 (15) -58.84, -121.16 (15) -51.90, -150.59 (15) -51.58, -152.29 (15)

-59.28, -121.98 (17.5) -52.57, -151.89 (15) -60.18, -123.74 (15)

4 -60.05, -127.82 (15) -59.95, -130.56 (12.5) -53.23, -159.05 (15,4) -52.64, -160.28 (21.9) -54.37, -160.64 (12.5) -53.90, -161.86 (14)

5 -66.04, -140.84 (15) -58.01, -168.34 (15) -58.84, -172.95 (41) -59.08, -175.01 (20)

6 -56.97, -104.60 (15) -50.10, -137.00 (4.1) -50.95, -134.70 (6.1)

7 -60.24, -125.60 (3.2) -53.77, -151.82 (9.6) -59.76, -127.50 (3.2)

8 -59.95, -101.75 (7.6) -59.08, -104.88 (8.1) -52.48, -137.00 (13.1) -52.48, -137.00 (12.7) -59.08, -104.88 (8.6) -58.49, -107.23 (10.5) -51.45, -140.10 (12.7) -51.45, -140.10 (12.7) -58.49, -107.23 (10.5) -57.89, -109.35 (8.6) -50.44, -143.00 (12.4) -57.89, -109.35 (7.6)

9 -59.94, -107.95 (6.1) -57.71, -115.66 (7.1) -50.30, -146.85 (10.8) -51.70, -142.83 (10.3) -59.01, -110.17 (12.4) -59.01, -110.17 (12.7) -50.90, -145.00 (6.1) -50.90, -145.00 (6.1) -57.71, -115.66 (10) -52.42, -140.17 (7.5) -48.42, -151.00 (10.3)

-51.70, -142.83 (10.3) -50.30, -146.85 (10.8)

NOTE: Sections-1, lineation between fracture zones Menard and IV; 2, lineation between fracture zones IV and V; 3, lineation between fracture zones Tharp and Udintsev; 4, lineation between fracture zones Udintsev and VIII; 5, lineation between fracture zones IX and X; 6, fracture zone V; 7, Udintsev fracture zone; 8, Heezen fracture zone; 9, Tharp fracture zone. Standard errors are in parentheses.

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Chang: Estimating Rotations From Boundary Crossings 1183

Proof. Let U C )X be open with x0 E U. We want to prove that lim,1 Pr(X, ? U) = 0. We can assume without loss of generality that f(xo, yo) = 0. By the compactness of tXV - U, there is an e > 0 such that f (x, yo) > E for all x ? U. By continuity, there are open sets xo E U, C .X and yo E VI C .'- such that f(x, y) < E for all (x, y) E U, x V,. Furthermore, there is an open yo E V2 C -b such that if y E V2 and x ? U, f (x, y) > e. Otherwise there would be y,, -n yo and x,, ? U such that f(xn yYn) < E. By the compactness of tX - U we can assume xn -_ x* with x* ? U. Then f(x*, yo) ' E, which is a contradiction. Now if Y, E V1 n V2, we have f(xo, Y,) < e. Therefore, f(X,, Y,) < E and hence X, E U. In other words, Pr(X, E U) 2 Pr(Y, C V1l nV2) and the proposition follows.

Remark. If Y, -- yo with probability 1, then X, -* xo with probability 1.

Jennrich (1969) and Wu (1981) discussed conditions for the consistency of a nonlinear least squares estimator that seem re- lated to those of Lemma 1. Even allowing for the different sense of asymptotics (K -? oo instead of n -- oo) and the sphericity of the data (which is essentially unimportant as K -? oc), I was unable to fit the situation at hand into the framework of Jennrich and Wu.

As K -> 00, u,, Oa,, and V,k l a,k. Thus using Lemma 1, consistency of (A, r,(A)) and ',(AO) as K oo iS implied by unique minima for the functions r(A, 4, . . ., s) = , ( + A'S,A)(, and r,(() = '(1, + Aol,AO)(. This is not guaranteed. Roughly speaking, if fl = {(Ao, ts,, at, a,k) I a, = 0, (tkAotQ, = O}, then except for a set of measure 0 in fl, consistency will hold. [Here fl is a manifold of dimension 3 + 2s + N, which is naturally embedded in Euclidean (9 + 4s + 3N)-dimensional space, and hence we can use a surface measure on fl.] Nature is surely not so sadistic as to choose the rare configuration that leads to an inconsistent estimator. In addition, since ii is defined only up to sign, we should think of ii as a point in the projective plane, not the sphere.

Proposition 1. If ni + m, > 2, then r, has a unique minimum except on the set of measure 0 in fl consisting of the points (Ao, th, ai,, ?tk) for which a, and A

Proof. Since ri(() = E Kat(a')2 + 2k Kik(ikAoQ), r, is always nonnegative and r,(i,) = 0. If r,() = 0, then at( = 0 and (Ato?&k)t( = 0. Thus if the a,, and Ao &k do not all lie on a line, C = +? , and hence represents the same point in the pro- jective plane.

Lemma 2. Let 'q, .. , 11 be fixed unit vectors so that no three lie in a plane. Suppose that a1, . . ., a, satisfy a,'1i = 0 [the set ilt of possible (a,, . . . , a,) is a Cartesian product of r circles]. For r > 4, except for a set of measure 0 (in UlL), if qtA'ai = 0 for all i and some rotation A then A = I.

Proof. It suffices to prove the lemma for r = 4. LetT C 't be the set of (a,, a2, a3, a4) so that no three of the vectors q,, X a,,, t12 X a2, and ,13 x a3 are dependent. V is an open four- dimensional subset of ,t with complement of measure 0. Let SO(3) be the collection of rotation matrices, and let f: (SO(3) - I) x -t R4be the map f (A, a1, ... , a4) = (q A'la1, i4tA'ta4). I claim that M = f-1(0, 0, 0, 0) is a submanifold of (SO(3) - I) x 729 of dimension 3. It is sufficient to show that if tAta, = 0 (all i), then the 4 x 7 matrix

J [diag(qtA't#,) 11tA'M(t,)aJ,]

has rank 4, where t1, t2, and t3 are the usual basis of R3 and /,B = q, x ai (see Spivak 1979, p. 65). Note that the left 4 x 4 submatrix of J will have rank 4 unless q'A'fl, = 0 for at least one i. Since 'A'la, = 0, if ,Atf/3, = 0, then i, is an eigenvalue of A, Ar, = +?,, and it follows that at most three of the q'A'fl, are 0. Now if A, = +?, then I'A'M(t)a = +?'(t x a) = ?(a x q)'t. Since no three of the a, x q, are dependent, it easily follows that J has rank 4. Now let ir: (SO(3) - I) x -lI -* -It be the projection. Since M has dimension 3 and tl has dimension 4, it follows from Sard's theorem (see Spivak 1979, p. 55) that ir(M) has measure 0. This proves the lemma.

Proof of Theorem 1. Let (AO, 'v) be fixed. Suppose that (Ao, 'j, aq , &ik) E fl Note that r(A , I . ., ) = 1, j K,(t)2 +

Eik Kik(tlakA (i)2 and hence, at any minimum, at, = 0 and atkA(, - 0. Suppose that m, 2 2 and n, - 1 for i = 1, 2, 3, 4. Unless a,l . . t ?lare colinear, q, = i,. But by Lemma 2, 0 = &t A1, - (Ato&,k)'At A1, (i = 1, ., 4) implies that A = AO except on a set of measure 0 in &?k. Thus for each (AO, 'v), the collection of aq, and &,k for which r has a nonunique minimum has measure 0. The theorem follows by Fubini's theorem.

[Received September 1987. Revised February 1988.]

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