Rotation about an inclined axis: Three dimensional matrices for reconstructing paleomagnetic and structural data

Abstract

We derive matrices for calculating the three-dimensional finite rotation that restores a rigid block to its pre-displacement attitude. To use this technique an initial horizontal bedding is assumed, and the following orientations must be known: the expected field orientation of magnetization, the situ orientation of the paleomagnetic vector and the bedding normal. The restoring rotation is decomposed into two convenient steps, represented by rotation matrices, such that they restore the in situ paleomagnetic vector together with the tilted bedding normal to their original orientation. These auxiliary rotations combine algebraically to a single finite rotation from which an inclined axis and amount of rotation arc obtained. We demonstrate this method by reconstructing a distinct section of the Mount Sedom salt diapir (Dead Sea Rift Valley, Israel). Here, the conventional tilt correction that brings the bedding to horizontal fails to position the paleomagnetic vector in its expected orientation. Only a rotation of 125 ° about an axis which plunges 50 ° southward can restore both bedding and magnetization to their expected orientations.