# Conditioning Ground Shaking#

Given an earthquake rupture model, a site model, and a set of ground shaking intensity models (GSIMs), the OpenQuake scenario calculator simulates a set of ground motion fields (GMFs) at the target sites for the requested set of intensity measure types (IMTs). This set of GMFs can then be used in the Scenario Damage Assessment and Scenario Risk Assessment calculators to estimate the distribution of potential damage, economic losses, fatalities, and other consequences. The scenario calculators are useful for simulating both historical and hypothetical earthquakes.

For historical events, ground motion recordings and macroseismic intensity data (“ground truth”) can provide additional constraints on the ground shaking estimates in the region affected by the earthquake. The U.S. Geological Survey ShakeMap system (Worden et al. 2020; Wald et al. 2022) provides near-real-time estimates of ground shaking for earthquakes conditioned on station and macroseismic data where available. Support to read USGS ShakeMaps and simulate GMFs based on the mean and standard deviation values estimated by the ShakeMap — for peak ground acceleration (PGA) and spectral acceleration (SA) at 0.3s, 1.0s, and 3.0s — was added to the OpenQuake-engine in 2018, and a link to the scenario_damage and scenario_risk calculators was added based on the methodology described by Silva & Horspool (2019). Further support for reading ShakeMaps from sources other than the USGS, support for MMI, and performance improvements were introduced to the engine in 2021.

Therefore, starting from OpenQuake engine v3.16, it is possible to condition the ground shaking to observations, such as ground motion recordings and macroseismic intensity observations. The implementation is tightly coupled with the existing scenario calculator.

## Methodology#

The main source of uncertainty in the evaluation of earthquake scenarios is the ground shaking component. The estimation of the ground shaking, considering past events or hypothetical future earthquakes, is characterized by two types of uncertainty: epistemic and aleatory. The former is related to our lack of knowledge or data regarding the expected ground shaking in the region, and it is usually incorporated in earthquake scenarios by considering multiple ground motion models or GSIMs. Alternatively, a ground motion model (backbone model) that depends on different parameters (that are region-dependent) can be used. In theory, with additional data, this source of uncertainty could be minimized. The second type of uncertainty is related to the randomness of the ground shaking process and it cannot be completely reduced. A ground shaking intensity measure (IM) at a given site is typically estimated using the following equation:

Where IM stands for an intensity measure such as peak ground acceleration (PGA), peak ground velocity (PGV), spectral acceleration (SA), amongst others. μ(MRθ) represents the mean logarithm of the estimated ground shaking as a function of the magnitude (M), some measure of distance (R), and other parameters (θ) such as local site conditions or faulting mechanism. ζ represents the total residual of log(IM). By treating the total residual as a linear mixed-effects model (Abrahanson and Youngs 1992; Engler et al. 2022), equation above can be re-written as follows:

Where B stands for the between-event residual (i.e., representing the variability in the ground shaking amongst events with the same characteristics) and W refers to the within-event residual (i.e., representing the variability within the same event for sites at the same distance and soil conditions). These residuals typically follow a normal distribution and for the assessment of earthquake damage or losses they can be randomly sampled or numerically integrated using the respective standard deviations associated with each ground motion model.

## OpenQuake implementation#

The implementation of the conditioning of ground motion fields in the OpenQuake-engine was performed following closely the procedure proposed by Engler et al. (2022). For additional details, readers are referred to the Appendix A and B of Engler et al. (2022). The module in the OpenQuake-engine is similar to the well-established scenario damage or risk calculator in which users have to provide an earthquake rupture, a site model, the list of locations where ground shaking will be calculated, and a single GSIM or a GSIM logic tree. However, for the constraining of the ground shaking, it is necessary to provide additional input files and adjustments to the configuration file as presented in the OpenQuake input files section.

The generation of the conditioned cross-spatially correlated ground motion fields was performed in the following steps described below. All variables associated with the target sites (i.e., locations where the ground shaking will be computed) are identified with the subscript “T” while all variables related to the observations at the seismic station are identified with subscript “O”.

The median ground shaking is calculated at the location of the seismic stations considering the set of GMMs selected by the user (and using the library of GMMs of the OpenQuake-engine).

The residuals (\(\zeta_{IMO}\)) at every station location are computed through the difference between the recorded IM and the median IM from the GMMs.

A covariance matrix (\(\Sigma_{WOWO}\)) is computed for all IMs and station locations using the within-event sigma (W). Since each GMM has its own uncertainty, this step is repeated for each GMM.

The mean (\(\mu_{BIM}\)) and variance (\(\sigma^2_{BIM}\)) of the between-event residual conditioned on the observations from the seismic stations for each IM (\(BIM\)) is computed using the formulae below (Worden et al. 2018). This parameter is used to adjust the ground shaking (per IM) for the entire affected region, thus allowing the possibility of not sampling the between-event residual.

\[\mu_{BIM} = \frac{Z^T \Sigma_{WOWO}^{-1} \zeta_{IMO}}{\tau_{IM}^{-2} + Z^T \Sigma_{WOWO}^{-1} Z}\]\[\sigma^2_{BIM} = \frac{1}{\tau_{IM}^{-2} + Z^T \Sigma_{WOWO}^{-1} Z}\]Where Z is a column vector of ones (Jayaram and Baker 2010).

The conditioned between-event residual can now be used to adjust the total residuals calculated in Step 2 as presented below. Since the between-event component has now been removed from \(\zeta_{IMO}\), we are left with the residuals due to the within-event component (\(w_O\)).

\[w_O = \zeta_{IMO} - \mu_{BIM}\]With the calculation of \(w_O\), it is now necessary to compute the covariance matrix between the target IMs (in general, considering the intensity measures used by the vulnerability or fragility functions) and the target locations (\(\Sigma_{WTWT}\)) as well as the covariance matrix considering both the target/observed IMs and target/observed locations (\(\Sigma_{WTWO}\)) and vice-versa (\(\Sigma_{WOWT}\)).

Finally, the mean (\(\mu_{TIM|O}\)) and covariance (\(\Sigma_{WTWT|O}\)) of the ground shaking for the target IMs and locations conditioned on the observations can be computed using the following formulae:

\[\mu_{TIM|O} = \mu_{TIM} + \mu_{BIM} + \Sigma_{WTWO} \Sigma_{WOWO}^{-1} w_O\]\[\Sigma_{WTWT|O} = \Sigma_{WTWT} - \Sigma_{WTWO} \Sigma_{WOWO}^{-1} \Sigma_{WOWT}\]

We can assume that the ground shaking for the various IMs are multivariate random variables, and these two parameters can be used to randomly sample cross-spatially correlated ground motion fields conditioned on the observations.

In addition to the generation of cross-spatially correlated ground motion fields conditioned on data from seismic stations, it is also possible to constrain the resulting ground shaking based on observational data usually characterized using macroseismic intensity. This option can be advantageous for regions with poor seismic networks or when considering older events for which strong motion data might also be limited. This process requires the conversion of macroseismic intensity into ground motion, which will inevitably introduce additional uncertainty. As a result, the ground shaking converted from macroseismic intensity will be represented not only by a single value (as done when a seismic station is used) but by a mean and standard deviation. This variability can be taken from the chosen conversion equation. The incorporation of macroseismic intensity in this procedure has also been implemented within the OpenQuake-engine. Additional details about the mathematical formulation of this procedure can be found in Worden et al. (2018).

## Future improvements#

Additional improvements and testing are required for a more robust tool. Some of the capabilities foreseen in the coming years include:

The conditioning process requires the specification of a spatial correlation model of the within-event residuals between two points for the intensity measures involved in the calculation, a model for the cross-measure correlation of the within-event residuals, and a model for the cross-measure correlation of the between-event residuals. Assuming a conditional independence of the cross-measure correlation and the spatial correlation of a given intensity measure, the spatial cross-correlation of two different IMs at two different sites can be obtained as the product of the cross-correlation of two IMs at the same location and the spatial correlation due to the distance between sites. Given the limited set of correlation models available in the engine currently, the choice of the three aforementioned correlation models could be hardcoded in the initial implementation of the conditioned GMF calculator using JB2009CorrelationModel as the spatial correlation model of the within-event residuals, BakerJayaram2008 as the cross-measure correlation model for the within-event residuals, and GodaAtkinson2009 as the cross-measure correlation model for the between-event residuals.

Ideally, the choice of the different correlation models should be exposed to the user through parameters in the job file. Direct spatial cross-correlation models for the within-event residuals (Loth and Baker 2013 and Du and Ning 2021) could also be used instead of separate models for the spatial correlation and cross-measure correlation. Supporting these choices would entail a substantial refactoring of the existing engine and is thus left for a future implementation.

## References#

Worden, C. B., Thompson, E. M., Hearne, M. G., & Wald, D. J. (2020). ShakeMap Manual Online: technical manual, user’s guide, and software guide. U.S. Geological Survey. URL: http://usgs.github.io/shakemap/. DOI: https://doi.org/10.5066/F7D21VPQ.

Wald, D. J., Worden, C. B., Thompson, E. M., & Hearne, M. G. (2022). ShakeMap operations, policies, and procedures. Earthquake Spectra, 38(1), 756–777. DOI: https://doi.org/10.1177/87552930211030298.

Silva, V., & Horspool, N. (2019). Combining USGS ShakeMaps and the OpenQuake-engine for damage and loss assessment. Earthquake Engineering and Structural Dynamics, 48(6), 634–652. DOI: https://doi.org/10.1002/eqe.3154.

Engler, D. T., Worden, C. B., Thompson, E. M., & Jaiswal, K. S. (2022). Partitioning Ground Motion Uncertainty When Conditioned on Station Data. Bulletin of the Seismological Society of America, 112(2), 1060–1079. DOI: https://doi.org/10.1785/0120210177.