Bessie H. KIRKWOOD 1, Jean-Yves ROYER 2, Theodore C. CHANG 3, and Richard G. GORDON 4

1. Department of Mathematical Sciences, Sweet Briar College, Sweet Briar, VA 24595-0142, U.S.A.
2. Unité mixte de Recherche 6526 Géosciences Azur, Quai de la Darse - BP48, 06235 Villefranche-sur-mer cedex, France.

Now at:
Unité mixte de Recherche 6538 Domaines Océaniques, Institut Universitaire Européen de la Mer, Place Copernic, 29280 Plouzané - France.
3. Division of Statistics, University of Virginia, Charlottesville, VA 22903, U.S.A.
4. Department of Geology and Geophysics, Rice University MS126, Houston TX 77005, U.S.A.

SUMMARY

Plate reconstructions are a key tool for plate tectonics. Although many methods have been proposed to calculate the finite rotations that describe plate motions, only a few address the problem of the uncertainties in plate reconstructions. This information is critical for meaningful comparison of rates and directions of motion, predicted by a rotation, with geological field observations at plate boundaries. Similarly, uncertainties of a product of uncertain rotations through a plate circuit are crucial for testing plate geometry or for comparing reference frames. Here we present a method that allows one to propagate the uncertainties in the marine data (magnetic anomaly and fracture zone crossings), used to derive the finite rotations, to the uncertainties in the rotation parameters (latitude and longitude of pole, and angle of rotation). The rotation parameters are estimated by minimizing a quadratic objective function. A Taylor series approximation, centered at the true (and unknown) rotation parameters is used to approximate the non-linear least squares by a linear regression problem. From the design matrix estimated from the data, a covariance matrix describing the uncertainties in the rotation can be estimated. The covariance matrix depends upon the geometry of the plate boundary, the number of data, and the uncertainties in the data. Using a heuristic description of non-linear least squares, we show how the methodology is analogous to standard linear regression. We generalize the method for reconstructing a single plate boundary to solve for the closure of a triple junction, and we present a simple way to combine the covariance matrices of individual rotations to estimate the uncertainties in their product, and how to derive the uncertainties in the reconstructed points.

Since any statistical analysis depends upon assumptions about the errors in the data, we discuss the assumptions inherent to the proposed methodology and the limitations which result from these assumptions. One of them (referred to as the "equal kappa's" assumption) is especially troubling when two or more rotations, based on different data distributions, are combined. From similar problems arising in linear regression, we propose a method to solve the plate reconstruction problem when the equal kappa's assumption is not tenable.

Finally, we briefly review the sources of error in the magnetic anomaly and fracture zone crossings that are inverted to derive plate reconstructions, and how their uncertainties can be evaluated. A series of examples illustrate how these tools, implemented in software, can be used to solve or to test various plate geometries involving a single plate boundary, a triple junction and a combination of both.

Key words: Geostatistics, Inversion, Plate motions

Software: These statistical tools have been implemented in Fortran.  All codes are available by anonymous ftp at www.stat.virginia.edu.

See also the Appendix of:
Royer, J.-Y. & T. Chang: Evidence for relative motions between the Indian and Australian plates during the last 20 Myr from plate tectonic reconstructions. Implications for the deformation of the Indo-Australian plate.
Journal of Geophysical Research, 96: 11779-11802, 1991.