1. RECORD PROCESSING CONSIDERATIONS FOR THE EFFECTS OF TILTING AND TRANSIENTS VLADIMIR GRAIZER California Geological Survey

2. Highlights Inconsistency in the right (input) part of the differential equation of pendulum motion in European and American literature. What are we actually recording in the near-field? What can be done in case of purely translational input motion? What may happen in real life? Future: What can be done?

3. Schematic representation of an accelerograph

4. Equation of pendulum motion Longitudinal: y 1 ” + 2  1 D 1 y 1 ’ +  1 2 y 1 = -x 1 ” + gψ 2 - ψ 3 ”l 1 + x 2 ”  1 Transverse: y 2 ” + 2  2 D 2 y 2 ’ +  2 2 y 2 = -x 2 ” + gψ 1 + ψ 3 ”l 2 + x 1 ”  2 Vertical: y 3 ” + 2  3 D 3 y 3 ’ +  3 2 y 3 = -x 3 ” + gψ 1 2 /2 - ψ 1 ”l 3 +x 2 ”  3

5. List of symbols y i is recorded response of the instrument,  i is the angle of pendulum rotation, l i is the length of pendulum arm, y i =  i l i,  i and D i are respectively the natural frequency and fraction of critical damping of the ith transducer, g is acceleration due to gravity, x i ” is ground acceleration in Ith direction, ψ i is a rotation of the ground surface about x i axis.

6. Errors due to angular acceleration, tilt and cross axis sensitivity

7. “Effective” equations of pendulums Horizontal: y 1 ” + 2  1 D 1 y 1 ’ +  1 2 y 1 = -x 1 ” + gψ 2 Vertical: y 3 ” + 2  3 D 3 y 3 ’ +  3 2 y 3 = -x 3 ” Vertical sensor is much less sensitive to tilts than the horizontal sensor (for tilts < 10 deg)

8. What can be done in absence of rotations? y” + 2  Dy’ +  2 y = - Vx” T 1 T W =  [x’(t)] 2 dt +  [x’(t)] 2 dt 0 T 2 From Graizer, 1979 Baseline correction based on minimization of velocity oscillations at the beginning and the end of the record:

9. Comparison of shake-table motion with displacement calculations

11. How can tilt affect displacement calculations?

12. Acceleration and tilt

13. Tilt

14. Comparison of the true displacement with displacement contaminated by tilt

15. Comparison of the “true” displacement and displacement calculated using accelerogram contaminated by tilt

16. Acceleration and tilt

17. Tilt

18. Comparison of the true displacement with displacement contaminated by tilt

19. Comparison of the “true” displacement and displacement calculated using accelerogram contaminated by tilt

20. Conclusions Commonly used strong-motion instruments are sensitive not only to the translational motion, but also to tilts. This sensitivity can be neglected in far- field measurements, but not in the near-field studies. Numerical experiments demonstrate that ignoring tilt effects in strong-motion studies can introduce long-period error, especially for calculation of residual displacements. In contrast to horizontal sensors, vertical sensors are practically not sensitive to tilts. This makes them potentially more usable for the long-period and residual displacement calculations. Conservative methods of strong-motion data processing that involve filtering in a limited frequency band have a clear advantage, especially for routine processing, because they are getting rid of the long-period component partially introduced by tilting. It seems to be desirable to start measuring rotational component of the strong- ground motion in combination with classical translational motion measurements in the vicinity of the faults.