Extensions of the Galperin Transformation Matrices for Triaxial Seismometers

Modern triaxial seismometers often employ a symmetric configuration that includes three identical sensors, each with a proof mass suspended on a boom pointing at the same “Galperin” tilt angle (tan^(-1)⁡〖1/√2〗 or 35.26°) with respect to vertical. A well-known transformation matrix is used to convert measured displacement of the suspended mass orthogonal to the three booms into ground displacement in the cardinal horizontal/vertical directions. In this paper, we extended the Galperin transformation matrix to cover arbitrary values of the tilt angle. We also discovered a gap in our understanding of the Galperin triaxial seismometer. The transformation matrix was derived with the assumption of a point mass on a massless boom. In practice, the masses are distributed along the boom. It is not clear where, along the boom, one should make measurements of the displacements, so that one can get the correct ground displacement using the transformation matrix. We found that this location should be at a distance of D_n from the pivot, where D_n=J/(mD_g ), and J, m, D_g are, respectively, the moment of inertia, the mass, and the center of gravity of the boom with all the components mounted on it. Another limitation of the Galperin transformation matrix is that it only works at high frequencies. We use D_n to extend the transformation to cover signals at all frequencies. We also extended the equation of motion for a rotational mass-spring oscillator to include ground accelerations. With this, we address how torque feedback signals are transformed into ground accelerations in the x, y and z directions. We apply our results to understand how correlated and uncorrelated noise propagates from the sensor to the seismometer’s output. In the Appendices, we also extended the transformation matrices to cover non-identical component seismometers that are not oriented symmetrically.