This page presents the code found on the spatial correlation repository: https://github.com/ucgmsim/stochastic_event_set. The repository has all the references cited here, which can be consulted when something is not clear.
There are two main usages of the spatial correlation. The first one is related to Ground Motion Prediction Equations (GMPE) and the second is relevant for physics-based simulations (like CS).
Spatial correlation can used in two ways for GMPE. In the first case, it will allow us to generate consistent IMs for a set of close geolocations. The second usage is to use spatial correlation between sites with observed IMs, coming from Strong Motion Stations (SMS) to obtain GMPE median and sigma values that are more in accordance to what has been observed.
The idea is to allow the study the impact over several close locations at the same time, in opposition to the more common approach of generating IM observations for a single location.
For this purpose, we have implemented the equations as in Jayaram, 2018 and Loth and Baker, 2013. Basically, the algorithm is as follows:
The main script that calls all the needed functions is generate_realisations.py.
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In this part, we use SMS observations to influence mean and sigma values from GMPEs. The way to do that is to use the spatial correlation between the sites where we have data and all the other sites.
The implementation follows both Bradley, 2013 and Stafford, 2009:
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For the case of the simulated correlations, we have explored two types of prototypes. The first one to analyse one simulation and the GMPE to compare the simulated spatial correlation with the Jayaram et al. model. The second type of computation is applied to a set of simulations for a given rupture, to obtain an estimation of the correlation. In the following, both will be explained, including what needs to be done going further.
This is the case presented in the NZ correlation study paper by Chen et al. 2019. This paper uses the process to compute the within-event correlations for observed IMs. In this case, we have adapted the calculations to take a single simulation.
The workflow for a site j and a rupture i:
On the paper this is compared to the stationary approach for the spatial correlation. We still need to decide what to do forward, as this could be used for CS.
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This part emulates what is being done in the PSHA textbook for a set of simulations performed by SCEC. The idea here is to have a large number of simulations for the same fault. The calculations can be found on Brendon's PSHA book, 2020.
The only trick here is to store the IM values in a table where rows are station data and columns with the lnIM for each simulation. With this trick, quantities like the median across all simulations (average of all columns for a given row) or the total residual on the equations are easily calculated.
Once we have the within-event values, we calculate the Pearson correlation to obtain the covariance matrix for the within-event part.
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The stochastic event set is a subset of a large number of simulations which is representative. This is useful as the subset is easier to analyze.
So far, we implemented a simple Monte Carlo method to select random simulations and we have tested that.
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Bradley, B.A. (2014) Site-specific and spatially-distributed ground-motion intensity estimation in the 2010–2011 Canterbury earthquakes. Soil Dynamics and Earthquake Eng. Volumes 61–62, June–July 2014, Pages 83-91.
Loth, C. and Baker, J.W. (2013), A spatial cross‐correlation model of spectral accelerations at multiple periods. Earthquake Engng Struct. Dyn., 42: 397-417. https://doi.org/10.1002/eqe.2212
Jayaram, N. (2010) Probabilistic seismic lifeline risk assessment using efficient sampling and data reduction techniques. Stanford University.
Stafford PJ. (2012) Evaluation of structural performance in the immediate aftermath of an earthquake: a case study of the 2011Christchurche Earthquake. Int J Foren Eng;1(1):58–77.