McQuarrie and Wernicke (2005) produced a series of reconstructions of the southwestern United States from 36 Ma to present, as discrete blocks displaced from one another and from the stable midcontinent. These reconstructions were defined by shapefiles incorporating polygons outlining each block. Using GeoGraphix Discovery, the polygons in the shapefiles were converted to latitude and longitude. For each block and reconstruction, finite rotations from the reconstructed age to present were calculated. For the northwestern United States, discrete reconstruction sets for two different ages are derived from maps produced by Atwater.
Method for calculating restorations
For a particular reconstruction age, the finite rotation of each block from that age to the present was derived using a four point method. Given points a1 and a2 on Block A, and corresponding points a1' and a2' on rotated Block A': determine that one of the two poles that is 90 degrees away from both points and then the angle that rotates a1' to a1 around the pole. Rotate a2' using the same pole and angle to produce a2''. Then determine the angle (positive or negative) that rotates a2'' to a2 around a1 (serving as the pole). From the two resulting sets of rotation parameters, compute the total, equivalent rotation.
Then, to allow restoration of discrete data points within the reconstruction region, form a grid comparable to or at a higher resolution than the reconstructed block spacing. Convert the reconstruction parameters to pseudovectors with magnitude equal to the rotation angle and assign them to each node of the corresponding block in present-day coordinates. Interpolate the pseudovectors onto the grid using a Gaussian distance filter and the twelve nearest block nodes to the grid node. Because the filter is exponential with distance, grid points falling within a block should produce interpolated pseudovectors equivalent to that assigned to the entire block. The result is three grids for each age corresponding with each pseudovector component and reconstruction age.
Given a discrete data point of a specific age, interpolate the pseudovectors for the two reconstruction ages that bracket the data point age using the same Gaussian distance filter method used to calculate the grid. Then linearly interpolate the pseudovector components using the ages. Convert the pseudovector to rotation parameters and apply to the data point.
Subsequent posts will provide examples of this approach.