Factors for Calculating

Factors for Calculating Maximum Rotated Component Watson-Lamprey Consulting

Outline Definitions of Ground Motion Maximum Rotated Component Factors Direction of Maximum Rotated Component Direction Across Multiple Periods Amplitude Across Multiple Directions Conclusions

Definitions Maximum Rotated Component: The maximum value obtained from resolving the two orthogonal components into a single direction given by a rotation angle, calculating the response spectrum, and repeating over all rotation angles.

Maximum Rotated Component: Example Landers, Lucerne, 4 Seconds 0.35 Max 03 0.3 0.25 0.2 0.15 0.1 0.05 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1

Definition: GMRotI50 GMRotI50: The median geometic mean response spectra of the two as-recorded horizontal components after a single period-independent rotation that minimizes the variation away from the median value over all usable periods

GMRotI50: Example Landers, Lucerne, 4 Seconds 0.35 Max GMRotI 03 0.3 0.25 0.2 0.15 0.1 0.05 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1 h2 geomean

Maximum Rotated Component Factors Beyer and Bommer, 2006 Watson-Lamprey and Boore, 2007 Huang et al., 2008

Max Component Factors 1.6 1.5 1.4 1.3 1.2 11 1.1 1 0.1 1 10 Period (sec) Watson-Lamprey and Boore Beyer and Bommer Huang et al. - Average Directivity

Max Component Factors Beyer and Bommer and Watson-Lamprey and Boore are very similar. Huang et al. is slightly different, steeper slope at long periods Max/GMRotI_avg not Max/GMRotI Combining effects of directivity on GMRotI and Max/GMRotI Sensitive to the input directivity parameter

Max Component Factors 1.6 1.5 1.4 1.3 1.2 11 1.1 1 0.1 1 10 Period (sec) Watson-Lamprey and Boore Beyer and Bommer Huang et al. - Average Directivity Huang et al. - 0.4

Max Component Factors If we disregard directivity, and use a slightly smaller XcosTheta as the center of the data, Huang et al. is similar in amplitude to Beyer and Bommer, 2006 and Watson- Lamprey and Boore, 2007.

Direction of Max Component The direction of the maximum rotated component of a single period is random with respect to the strike of the fault: At short periods At distances from the fault great than 3-5 km Close to the fault at long periods the direction is somewhat correlated with strike

Direction of Max Component

Direction of Max Component: Example Landers, Lucerne, 4 Seconds 0.35 SN Max GMRotI 03 0.3 0.25 0.2 0.15 0.1 0.05 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1 h2 geomean

Direction of Max Component From Huang et al., 2008 For M>6.5, Rrup<5, The direction of the maximum components all fall within 30 degrees of SN, For the period range 0-1 second, 53% of the time For the period range 1-2 seconds, 61% of the time For the period range 2-4 seconds, 72% of the time Note: The directions can still be 60 degrees apart

Direction Across Multiple Periods Very close to the fault the direction of maximum ground motion for long gperiods is associated with the strike of the fault. How strongly does the direction of the maximum rotated component for a single period correlate with other periods?

Direction Across Multiple Periods: Example Landers, Lucerne, 4 Seconds Landers, Lucerne, 2 Seconds 0.35 SN Max 0.35 SN Max 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1 Landers, Lucerne, 1 Second Landers, Lucerne, 0.01 Seconds 0.6 SN Max 0.9 SN Max 0.8 0.5 0.7 0.4 0.6 0.5 0.3 0.4 0.2 0.3 0.2 0.1 0.1 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1

Direction Across Multiple Periods: Example Landers, Lucerne 180 160 140 120 100 80 60 40 20 0 0.01 0.1 1 10 Period (sec)

Direction Across Multiple Periods

Direction Across Multiple Periods

Direction Across Multiple Periods

Direction Across Multiple Periods For M>6.5, Rrup<5, The directions of the maximum components of multiple periods fall within 30 degrees of each other, For the periods 1 and 2 seconds, 65% of the time For the periods 2 and 4 seconds, 70% of the time For the periods 1, 2 and 4 seconds, 40% of the time

Directions Across Multiple Periods

Direction Across Multiple Periods For all M and Rrup, The directions of the maximum components of multiple periods fall within 30 degrees of each other, For the periods 1 and 2 seconds, 54% of the time For the periods 2 and 4 seconds, 55% of the time For the periods 1, 2 and 4 seconds, 28% of the time

Direction Across Multiple Periods For all M and Rrup: For the periods 1 and 2 seconds, 54% of the time For the periods 2 and 4 seconds, 55% of the time For the periods 1, 2 and 4 seconds, 28% of the time For M>6.5, Rrup<5: For the periods 1 and 2 seconds, 65% of the time For the periods 2 and 4 seconds, 70% of the time For the periods 1, 2 and 4 seconds, 40% of the time

Direction Across Multiple For M>6.5, Rrup<5: For the periods 1 and 2 seconds, 65% of the time For the periods 2 and 4 seconds, 70% of the time For the periods 1, 2 and 4 seconds, 40% of the time Periods For all M and Rrup: For the periods 1 and 2 seconds, 54% of the time For the periods 2 and 4 seconds, 55% of the time For the periods 1, 2 and 4 seconds, 28% of the time The difference in the correlations between directions of max components for different periods is not great for the close in recordings of large magnitudes. These correlations exist regardless of magnitude and distance.

Amplitude Across Multiple Directions Long period ground motion is polarized, thus a single component is close to the maximum for a range of angles.

Amplitude Across Multiple Directions: Example Landers, Lucerne, 4 Seconds 1 SN Max 0.9 0.8 0.7 0.6 0.5 0.4 0.3 02 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1

Amplitude Across Multiple Directions: Example Landers, Lucerne, 4 Seconds Landers, Lucerne, 2 Seconds 1 SN Max 1 SN Max 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 05 0.5 05 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1 Landers, Lucerne, 1 Second Landers, Lucerne, 0.01 Seconds 1 SN Max 1 SN Max 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1 0 0 20 40 60 80 100 120 140 160 180 Angle (Degrees) h1

Amplitude Across Multiple Directions For this example, the amplitude of a single component is greater than or equal to the maximum value approximately +/- 30 degrees from the direction of the maximum.

Conclusions Given the information about the direction of maximum motion across periods, and assuming that the example is correct for amplitude across multiple directions we can say: For all magnitudes and distances the amplitudes of a single component of ground motion at multiple periods are greater than 90% of their maximums: For the periods 1 and 2 seconds, 54% of the time For the periods 2 and 4 seconds, 55% of the time For the periods 1, 2 and 4 seconds, 28% of the time The direction of this occurrence is random.

Conclusions Checked this conclusion and found: For all magnitudes and distances the amplitudes of a single component of ground motion at multiple periods are greater than 90% of their maximums: For the periods 1 and 2 seconds, 97% of the time For the periods 2 and 4 seconds, 97% of the time For the periods 1, 2 and 4 seconds, 92% of the time

Conclusions Given the information about the direction of maximum motion across multiple periods, and assuming that the example is correct for amplitude across multiple directions we can say: For M>6.5, Rrup<5, the amplitudes of a single component of ground motion at multiple periods are greater than 90% of their maximums: For the periods 1 and 2 seconds, 65% of the time For the periods 2 and 4 seconds, 70% of the time For the periods 1, 2 and 4 seconds, 40% of the time

Conclusions Checked this conclusion and found: For M>6.5, Rrup<5, the amplitudes of a single component of ground motion at multiple periods are greater than 90% of their maximums: For the periods 1 and 2 seconds, 97% of the time For the periods 2 and 4 seconds, 92% of the time For the periods 1, 2 and 4 seconds, 82% of the time

Conclusions Given the information about the direction of maximum motion, and assuming that the example is correct for amplitude across multiple directions we can say: For M>6.5, Rrup<5, the amplitudes of the strike normal component of ground motion at multiple periods are greater than 90% of their maximums: For the periods 1 and 2 seconds, 61% of the time For the periods 2 and 4 seconds, 72% of the time

Conclusions For the periods 1 & 2 seconds: Estimated 0.61 for Rrup<5 Actually 0.45

Conclusions For the periods 2 & 4 seconds: Estimated 0.72 for Rrup<5 Actually 0.6

Conclusions For the periods 1, 2 & 4 seconds: No prior estimate For R<5 0.45 For R<3 0.55

Conclusions For M>6.5, Rrup<3-5, the amplitudes of the strike normal component of ground motion at long periods are greater than 90% of their maximums approximately half of the time.

Thank You

Magnitude 7, Rrup 10, XcosTheta 1 4 3.5 3 2.5 2 1.5 1 0.5 0 0.1 1 10 Period (seconds) SEA97 Huang et al. Spudich and Chiou - IDPs 0 & 4 and Beyer and Bommer

Magnitude 7, Rrup 10, Period 4s 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 XcosTheta SEA97 Huang et al. Spudich and Chiou - IDPs 0 & 4 and Beyer and Bommer