We present a new method for obtaining relative horizontal motions on the surface of a sphere from strain rate data. Strain rates can be obtained from the summation of earthquake moment tensors or from estimates of Quaternary rates of deformation on major faults. The method is particularly useful for determining the kinematics within zones of distributed continental deformation, or any region where there is distributed strain. All relative motions, including rotation rates about the vertical axis, are uniquely determined when the three rates of horizontal strain &egr;˙&thgr;&thgr;, &egr;˙ϕϕ, and &egr;˙&thgr;ϕ are everywhere defined within the region of interest. The forward problem is set up such that all relative velocities u on the surface of the sphere are defined by rW(xˆ)¿xˆ, where W(xˆ) is the three-dimensional rotation vector that describes the velocities u&thgr; and uϕ at all points on the surface of Earth of radius r with position unit radial vectors xˆ.

The three-dimensional rotation vector W(xˆ) is expressed as an infinite power series expansion, truncated at finite order N-1. Coefficients of this polynomial are sought in a damped least squares inversion such that the strain rates &egr;˙&thgr;&thgr;, &egr;˙ϕϕ, and &egr;˙&thgr;ϕ, which define all of the coefficients in the power series expansion of W(xˆ), are optimally matched by the smooth polynomial function. Formal uncertainties are introduced that take into account observational error as well as the inability of the polynomial function to accommodate the more rapid spatial variations of the rate-of-strain field. We demonstrate the method on a deforming part of Asia. Strain rates for the region were obtained from the summation of moment tensor elements of moderate and large-sized earthquakes in this century. Solutions, both velocity fields and rotation rates, are investigated as a function of polynomial smoothing and polynomial order. We demonstrate that the velocity field, obtained by the polynomial fitting of the regions where strain rates were averaged, is by nature extremely robust and is almost independent of the amount of polynomial smoothing. The rotation rate field on the other hand shows the same order of smoothing as the polynomials used in the fitting procedure. ¿ American Geophysical Union 1993