Correlation modules refactor #10 – Du & Ning (2021)

[raoanirudh]

Du and Ning (2021) within-event joint spatial and cross-IMT correlation model

Part of #11230.

This PR implements the recommended seven-principal-component within-event spatial cross-IMT correlation model published by Du and Ning (2021)¹. Matlab functions and coefficient arrays² related to the model were kindly shared by Wenqi Du with GEM on 25 May 2022.

The supported intensity-measure vector consists of:

• PGA
• PGV
• IA
• CAV
• RSD575 and RSD595
• 5%-damped SA from 0.01 to 10 s

Two variants are provided:

• DuNing2021 implements the publication model at its 17 tabulated SA periods, with no interpolation support.
• DuNing2021Interpolated additionally supports intermediate SA periods within 0.01–10 s.

The model combines the seven principal-component loading vectors with their nested spatial covariance functions and normalizes the reconstructed covariance to correlation. Interpolation is implemented as a separate variant because it is not specified in the publication and requires normalization to correct its non-unit off-grid diagonal; however, ordinary-period interpolation is supported in the authors' Matlab code.

PGA and SA(0.01) are distinct calibrated IMTs whose published loading rows for PC1–PC7 happen to be identical (see Table 3 of the publication), and are therefore perfectly correlated in this 7 PC model implementation.

Scientific scope and metadata

The model describes normalized within-event residuals.

The calibrated horizontal-component definitions are not uniform across the complete vector:

• PGA, PGV and SA use the RotD50 component from Campbell and Bozorgnia (2014)³.
• IA and CAV use geometric means of the two as-recorded horizontal components from Campbell and Bozorgnia (2019)⁴.
• RSD575 and RSD595 use the corresponding geometric-mean definition from Du and Wang (2017)⁵.

The implementation consequently leaves the model-wide IMC metadata unset.

Publication and authors' Matlab code

The implementation in this PR follows the equations and rounded coefficients in Tables 3 and 4 of the publication.

The author provided Matlab functions and coefficient files clarify:

• the ordering of the 23 intensity-measure loading rows
• the ordering of the nested-covariance parameters
• reconstruction of covariance from the seven principal components.

The common factor of 0.9 in Equations 22–24 cancels when Equation 12 is normalized and therefore does not affect the returned correlations.

The authors' Matlab function also bilinearly interpolates the final correlation table in ordinary period. This interpolation is not described in the publication and produces non-unit off-grid diagonal values, for example, 0.9823016 for SA(0.06), 0.9376723 for SA(0.15), and 0.9611493 for SA(0.6).

For that reason, the DuNing2021 class is restricted to the 17 published periods and does not support interpolation.

On the other hand, the DuNing2021Interpolated class is added in order to provide interpolation support explicitly. It linearly interpolates the equivalent normalized principal-component loading vectors in ordinary period and then normalizes using the two zero-distance marginal variances. This is equivalent to

$$\rho_{\mathrm{OQ}}(T_1,T_2,h) = \frac{\rho_{\mathrm{Matlab}}(T_1,T_2,h)} {\sqrt{\rho_{\mathrm{Matlab}}(T_1,T_1,0) \rho_{\mathrm{Matlab}}(T_2,T_2,0)}}.$$

The interpolated variant is exact (as expected) at all published periods, and produces symmetric, unit-diagonal, positive-semidefinite matrices (without needing to apply a subsequent matrix repair.) Log-period interpolation might be a valid option too, but is not used here because it would differ from the authors' own Matlab implementation.

Verification tables

Deterministic reference values were generated with GNU Octave 8.4.0 from the authors' Matlab functions and coefficient arrays. The Matlab files were used only to produce independent verification data; the implementation in this PR was developed separately from the publication equations. The Matlab scripts used to build the verification tables

References

Footnotes

1. Du, W., & Ning, C.-L. (2021). Modeling spatial cross-correlation of multiple ground motion intensity measures (SAs, PGA, PGV, Ia, CAV, and significant durations) based on principal component and geostatistical analyses. Earthquake Spectra, 37(1), 486–504. https://doi.org/10.1177/8755293020952442 ↩

2. Du, W. (2022). Matlab functions and coefficient arrays for Du and Ning (2021).zip ↩

3. Campbell, K. W., & Bozorgnia, Y. (2014). NGA-West2 ground motion model for the average horizontal components of PGA, PGV, and 5% damped linear acceleration response spectra. Earthquake Spectra, 30(3), 1087–1115. ↩

4. Campbell, K. W., & Bozorgnia, Y. (2019). Ground motion models for the horizontal components of Arias intensity and cumulative absolute velocity using the NGA-West2 database. Earthquake Spectra, 35(3), 1289–1310. ↩

5. Du, W., & Wang, G. (2017). Prediction equations for ground-motion significant durations using the NGA-West2 database. Bulletin of the Seismological Society of America, 107(1), 319–333. ↩