The color of the samples is proportional to ln(weight) with higher value in red and lower value in blue. The color of the samples is proportional to ln(weight) with higher value in black/red and lower value in white/blue.
Motivation, research goal and thesis plan
Decision Making—Making a decision to activate a loss mitigation measure based on system quantities predicted from EEW information. In my thesis, I try to improve each part of the EEW problem based on a Bayesian probabilistic approach and finally extend EEW to cover a wider range of applications by improving the quality of the decision-making process.
Fundamental concept of EEW
Suggest potential advanced applications of EDN (multi-measure mitigation and synergy application of EDN and structural health monitoring system) and discuss their challenges. This thesis is described as follows: Chapter 1 provides background on the recent development of EDN and its applications; Chapter 2 provides background knowledge of the theories and methodologies used in this thesis; Section 3 presents a probabilistic EEW system aimed at accurately identifying co-earthquakes with an example based on the earthquake sequence (foreshock, mainshock, and aftershocks) of the 2011 M9 Tohoku earthquake in Japan; Chapter 4 illustrates the details of the earthquake probability-based automated decision-making (ePAD) framework, which is a new framework for automated decision-making based on EEW information with a simple example of a decision to trigger an evacuation alert; Chapter 5 presents a complete study on the application of the ePAD framework for elevator control based on EEW information; Chapter 6 discusses some challenges related to two potential advanced applications of EDN - decision making for the multi-measure decision case and synergy of EDN and structural health monitoring system; Chapter seven concludes the thesis with some final remarks and suggestions for future research. The on-site system provides an alert primarily based on a single station (or sometimes a small group of two to three stations located next to each other).
Worldwide EEW development
California EEW system
However, in the current version of the ShakeAlert system, this Bayesian approach is not fully implemented yet. The current version of the ShakeAlert system does not include uncertainty in the estimation of time of origin in any of the three algorithms.
EEW applications
However, due to the limited warning time and uncertainty in EEW predictions, applications involving human decision making are not practical. As a result, the usefulness of EEW can be extended to a wider range of engineering applications.
Bayesian probability and model class selection
For example, in system identification, A may be a set of model parameters θ under a given model class Mi and B may be a set of data D obtained from empirical studies to improve the model. One can use the model class with the largest posterior probability P(Mi|D,M) for robust predictive analysis, or perform model class averaging where the final prediction is a weighted average of the predictions from each model class with the weights equal to P( Mi|D,M) (Beck, 2010).
Variations of Monte Carlo Simulation
- Monte Carlo Simulation
- Importance Sampling
- Markov Chain Monte Carlo and Subset Simulation
- Sequential Monte Carlo methods
When estimating the probabilities of rare events (often referred to as the tail of a PDF), even MCMC methods can be extremely inefficient due to the difficulty of samples converging in the small probability region. Many scientific or engineering problems involve dynamic processes that can be solved by a recursive method under the Bayesian probability framework.
Decision theory and cost-benefit analysis
Prediction phase: predict θt based on an evolution model p(θt|θt−1) for θ and the posterior PDF of θ from the previous time step t−1. Risk neutral: this represents a person who is neither of the above two cases (a linear utility function of x).
Surrogate model and relevance vector machine
The algorithm is developed based on model evidence in computing the posterior PDF of the weights. The original version of the RVM algorithm, referred to here as the top-down algorithm (Huang et al., 2014), starts from the full kernel model and recursively removes the irrelevant terms based on the recursive algorithm.
Value of information
Basic theory: Flipping coin example
The posterior of the expected profit G(π0) which depends on the possible information you can get is:. 2.19) At this stage you do not know what information you will receive. Then, a more appropriate betting strategy is to make the decision based on the posterior of x using the information you just obtained.
Engineering example: Soil contamination detection
- Problem setup and solution
- Results and discussion
After briefly reviewing the current JMA EEW system and some recent approaches to improve the system, I derive the theories embedded in the proposed algorithm, starting with the class selection theory of the basic Bayesian model. Many false alarms have been observed with the existing JMA EEW system because it does not have the ability to recognize cases of multiple events.
Background on JMA EEW
The magnitude estimate of an event is primarily based on the maximum displacement amplitude recorded at the stations and the distance of the stations from the potential hypocenter. The idea is to maximize the fraction of solutions that can be performed analytically under a given probabilistic model in order to reduce the sample size and improve the accuracy of the estimates.
Bayesian approach for multi-events EEW
Bayesian model class selection for multi-events
- Major notation
- Probability approach of EEW
- Practical implementation to handle multi-events
Therefore, one can easily calculate the posterior parameters of each possible model class and choose the one with the maximum evidence value. This motivates the need for a suboptimal model class selection scheme that is efficient yet robust.
Estimation of earthquake parameters using RBIS
- Rao-Blackwellized Importance Sampling
- Gaussian likelihood implementation
- Analytical treatment of magnitude
The integral in equation 3.25 can only be solved analytically for some specific choices of the prior p(m), for example a Gaussian prior or a uniform prior. Now I can obtain the analytical expression for the last missing part in this problem, that is, p(Ft|Ml) in equation 3.6 or 3.8, which is simply the proof function of the earthquake parameter posterior PDF given Ml.
Multi-events EEW algorithm
- Creating, merging and canceling events
- Selective data subset
- Sample prior and update
- Algorithm summary
The spatial extent of the uniform distribution can be determined based on the Voronoi cell methodology. Shift the samples toward the direction of the expected new hypocenter location if the change in estimate from the previous time step is large.
Data description for the test
Detail for applying the algorithm to the test
- Reduction of the sampling space
- Arrived/Not-arrived picking time model
- Station selection
- Details of the suboptimal model class selection
- Proposal PDF and sample updating
After that, it is constructed based on the information from the posterior PDF of the earthquake parameters from the previous time step. The new PDF proposal allows fast convergence to the current epicenter location when it is far from the current sampling area.
Test results and discussion
Single large event
This allows a wider azimuth coverage for better appreciation of the details of the offshore event as explained in section 3.6.3. It can be observed that when the event is first fired, the values of the collection time part are relatively close to each other compared to a few seconds later.
Two overlapped events
The only problem is that the magnitude estimate of the first event is affected by the seismic waves generated by the second event. After the second event is triggered, the magnitude estimate of the first event converges to the value of the second event.
Summary of March to April events
Appendix A shows the details of all events with results of the seismic intensity estimates from the catalog, the JMA EEW system, an algorithm proposed by Tamaribuchi et al. 2014) and the proposed algorithm in this chapter. -axis shows the error for the Japanese seismic intensity calculated at: . results from Tamaribuchi et al. 2014) and the proposed algorithm in this chapter.
Conclusion
In many cases, engineers can simply set a threshold value for the expected value of IM, ignoring most of the impact of IM uncertainty. The ePAD decision-making framework presented below provides a flexible platform for further development of models used in the decision-making process.
- Decision-making by cost-benefit model
- Performance-Based Earthquake Early Warning
- Concept of decision function, decision contour, and surrogate model
- Decision function
- Decision contour
- Surrogate model and relevance vector machine
- Lead time contribution in ePAD
- Incomplete action model
- Value of information model
- ePAD framework summary
If the EEW information can be reduced to a low-dimensional space, decision contours can be. In this case, a surrogate model can be used to approximate the decision contours with another simpler functional form.
Comparisons of decision criteria
If the PDF containing EEW information can be parameterized in low-dimensional space, check whether the decision behavior represented by the resulting decision contours is consistent with the user's desired rational decision making. Also, DFeP AD can be viewed as a linear combination of sigmoid functions, which is also a sigmoid function, because E[U(DV)|IM, a] =P.
Example: Evacuation warning
Problem setup
- Structural, loss, and lead time models
- EEW model and expected value calculation
The value of r can be adjusted based on the judgment of the decision maker(s) responsible for approving the installation of the warning system in the building. A reasonable choice is r = 0.2 based on the loss analysis in Haselton et al. 2007), but the results are not found to be sensitive to small changes in the value of afr.
Analysis and results
- Results with ePAD for single warning
- Results with ePAD for multiple-warning
If the EEW information lies above the incomplete action decision contour, the original action is triggered; otherwise if it is in between. As the expected lead time µT falls below the minimum time needed for the action to be completed (20-25 seconds), the decision contour converges to that in Figure 4.6.
Conclusion and ongoing work
The expected peak floor acceleration is calculated based on EEW information, a ground motion prediction equation, and a lumped mass building model. A surrogate model is trained based on Importance Vector Machine (RVM) from machine learning to mimic the complex cost-benefit model involving value of information (VoI).
Applying ePAD
- Basic model (No lead time contribution)
- Structural model
- Incomplete action model
- Value of information model
In this chapter, I examine the contributions of the uncertainty in the EEW information and the warning lead time to decision making, which were not included in the previous studies. Also, let the benefit of action be to prevent injury or death to passengers trapped in the elevator after an earthquake.
Analyses and Discussion
- Closed form solution for basic model
- Closed form solution for incomplete action model
- Numerical solution for VoI model using surrogate model
- Example
However, compared to Figure 5.7, the largerσIM is, the larger the difference in the decision contours. The solid black line in Figure 5.10 shows the causative fault segment and the star indicates the location of the building.
Conclusion
Traffic control of bridge network
Each entry to the highway is equipped with traffic lights to control traffic flows on the network. EEW information can be used to predict sections of the network at risk from seismic shaking, where there is a high probability of major damage or bridge collapse.
Case study for 1989 Loma Prieta earthquake
The resulting ROC curve of ePAD in this case is only slightly better than a random guess decision policy. This indicates that the proposed approach for a multiple-action decision problem is not effective (next section explains the reasons).
Discussion
If a reasonable result is obtained from this method, this simplification can be used to approximate the multi-action decision problem. Another reason why one cannot effectively reduce the multi-action decision problem to many single-action decision problems is that spatial correlation of the ground motions can play an important role in accurately predicting the local shaking intensity (Jayaram and Baker, 2009).
Potential advanced application 2: Synergy with SHM
Background on Bayesian SHM approach
The simplest method is to minimize the difference between the identified modal parameters from the response data and those calculated from a predefined structural model. Another problem is how to choose the appropriate weights α and β in Equation 6.1, since their values can significantly affect the optimal model parameter estimates.
Proposed synergistic framework
An example of near real-time loss estimation is given in Porter et al. 2006) based on the PBEE method, which relies on real-time structural monitoring system and structural models. This may also allow the decision support for near-real-time loss assessment mentioned in Porter et al.
Methodology for synergy of EEW and SHM data
- Method 1: In-series BN model (define θ in terms of DM )
- Method 2: θ-centered BN model (define DM in terms of θ)
Again, p(DS|θ) is known from the SHM framework, p(DS|DE) is a normalizing constant, and p(DM|θ) is a probabilistic relationship for DM in terms of θ to be fitted. Also, p(IM|DE) can be determined from EEW data and a ground motion prediction equation for the IM measure of shaking intensity.
Method to evaluate the benefits of the proposed synergy
Discussion
A New Triggering Measure for Improved Real-Time Performance of In-Situ Earthquake Early Warning in Southern California. Overview of the 2011 Pacific Coast Tohoku Earthquake (Mw 9.0)—Earthquake Early Warning and Observed Seismic Intensity.
Probability model of PBEEW assuming EEW provides M and R prediction
Example of a decision map in R 2
Fragility curves for structural model (C: global collapse, LC: local collapse)
Lead time model β 1 for the action a 1
Decision contours for the single warning case (with an incomplete action model) with
Decision contours of multiple warning case (include incomplete action model and value
Decision contours of multiple warning case (include incomplete action model and value
Mean of floor acceleration demand (based on Taghavi-Ardakan (2006)). z is the height
Standard deviation of floor acceleration demand (based on Taghavi-Ardakan (2006))
Decision contours with fixed µ ST and σ ST values but varying P 0 values. Region above
Decision contours with incomplete action model
Detail of the Tohoku foreshock on March 9, 2011
Detail of the two overlapped concurrent events on March 19, 2011
Choice of parameter values for fragility function models and loss models
Values for the parameters for reduced expected life loss, the fragility curves p(DM i |SA),