Abstract
Epistemic uncertainty offers alternatives on decision making and various possibilities of computing the hazard integral. Generally, the logic tree approach is used while treating the epistemic uncertainty. Logic tree weight calculation is a subjective decision based on the degree of belief of the analyst on the possible contributors to the epistemic uncertainty and often leads to a different set of values by different researchers. This paper aims to develop a framework of accounting for the epistemic uncertainty in probabilistic seismic hazard analysis (PSHA) by minimizing the subjectivity involved in weight calculation. Guidelines/rules are developed for the weight calculation at each node of the logic tree. Recurrence parameters, magnitude and distance probability distributions, maximum magnitude, and selection of ground motion predictive equations (GMPEs) are considered the possible sources of epistemic uncertainty. A GMPE rule is proposed to be used with the PSHA framework to account for the propagation of epistemic uncertainty. The north-east region of India is chosen for the purpose of illustration. The study region is divided into seven seismic source zones (five in the active crustal region and two in the subduction zone). Seismic hazard is characterized in terms of the weighted mean and fractile representation of hazards using the logic tree approach. Only one sample illustration of the results are reported in terms of weighted mean and fractile representation of hazard curves and uniform hazard spectra (UHS). Further illustration of the PSHA results with possible implications from the epistemic uncertainty is reported in the companion paper.
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Data may be available from the corresponding author through making reasonable request.
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Custom code is developed in MATLAB environment and not available for sharing.
Abbreviations
- \(Vs_{30}\) :
-
Average shear wave velocity in top 30 m
- \(a\) :
-
Gutenberg-Richter recurrence relation parameter (y-intercept)
- \(b\) :
-
Gutenberg-Richter recurrence relation parameter (slope)
- \(L\) :
-
Total number of source zones
- \(q\) :
-
\(q^{{{\text{th}}}}\) Source zone
- \(a_{q}\) :
-
Number of possible candidates for GMPE in \(q^{{{\text{th}}}}\) source zone (finally selected for PSHA)
- \(b_{m}\) :
-
Number of possible rules used while defining the maximum magnitude
- \(c\) :
-
Number of possible rules used to define the probability distribution of magnitudes
- \(d\) :
-
Number of possible recurrence relations that may define the seismicity
- \(e_{r}\) :
-
Number of possible rules used to define the probability distribution of source-to-site distance
- \(N\) :
-
Number of ERFs
- \(u\) :
-
\(u^{{{\text{th}}}}\) SERF
- \(p\) :
-
\(p^{{{\text{th}}}}\) ERF
- \(m\) :
-
Earthquake magnitude (realization)
- r:
-
Source to site distance (realization)
- \(v\) :
-
\(v^{{{\text{th}}}}\) GMPE
- \(N_{{{\text{GMPE}}}}\) :
-
Total number of GMPEs considered
- \(N_{{{\text{MHC}}}}\) :
-
Total number of model hazard curves
- \(T^{ * }\) :
-
Specified or conditional time period
- \(Sa\) :
-
Spectral acceleration
- \(Sa^{ * }\) :
-
Given hazard level or specified spectral acceleration
- \(\lambda\) :
-
Mean annual rate
- \(P( \cdot )\) :
-
Probability (CDF)
- \(Rup_{k}\) :
-
\(k^{{{\text{th}}}}\) Causal rupture scenario
- \(\ln\) :
-
Natural logarithm
- \(GMPE_{j}\) :
-
\(j^{{{\text{th}}}}\) GMPE
- \(N_{q}\) :
-
Number of candidates GMPE for the \(q^{{{\text{th}}}}\) source zone screened through the criteria of Bommer et al. (2010)
- \(w_{iq}\) :
-
Associated weight for \(i^{{{\text{th}}}}\) GMPE (out of \(N_{q}\)) in the \(q^{{{\text{th}}}}\) source zone
- \(\overline{w}_{jq}\) :
-
Normalized weight for \(j^{{{\text{th}}}}\) GMPE (out of \(a_{q}\)) in the \(q^{{{\text{th}}}}\) source zone
- \(s_{q}\) :
-
Standard deviation of weights \(\left( {w_{iq} ;\,i = 1,N_{q} } \right)\) in the \(q^{{{\text{th}}}}\) source zone
- \(w_{q}\) :
-
Weight assigned to the \(q^{{{\text{th}}}}\) source zone
- \(w_{{{\text{SERF}}}}\) :
-
Weight associated with any typical SERF
- \(w_{{{\text{ERF}}}}\) :
-
Weight of the defining ERF
- \(w_{{{\text{MHC}}}}\) :
-
Weight associated with any typical MHC
- \(\lambda_{k}\) :
-
\(k^{{{\text{th}}}}\) Realization of the rate
- \(z\) :
-
\(z^{{{\text{th}}}}\) Fractile
- \(R_{jb}\) :
-
Joyner-Boore distance (source to site)
- \(R_{epi}\) :
-
Epicentral distance (source to site)
- \(R_{hypo}\) :
-
Hypocentral distance (source to site)
- \(R_{rup}\) :
-
Rupture distance (source to site)
- \(X\) :
-
Observed sample dataset
- \(x_{k}\) :
-
\(k^{{{\text{th}}}}\) Realization of the observed sample data set \(\left( {k = 1,n} \right)\)
- \(M_{k}\) :
-
Magnitude associated with \(k^{{{\text{th}}}}\) realization of the observed sample data set
- \(R_{k}\) :
-
Source to site distance associated with \(k^{{{\text{th}}}}\) realization of the observed sample data set
- \(T_{k}\) :
-
Time period associated with \(k^{{{\text{th}}}}\) realization of the observed sample data set
- \(\overline{Y}_{k}\) :
-
Median of the intensity measure from GMPE for \(k^{{{\text{th}}}}\) triplet of (M-R-T)
- \(\sigma_{k}\) :
-
Logarithmic dispersion of the intensity measure from GMPE for \(k^{{{\text{th}}}}\) triplet of (M-R-T)
- \(p( \cdot )\) :
-
Probability mass function
- \(f( \cdot )\) :
-
Probability density function
- \(\pi\) :
-
Pi
- \(e\) :
-
Exponential
- \(dx\) :
-
Small interval of \(x\)
- \(B\) :
-
Base of the logarithm
- \(\log_{B}\) :
-
Logarithm with base \(B\)
- \({\text{LLH}}_{iq}\) :
-
Log-likelihood of \(i^{{{\text{th}}}}\) GMPE (out of \(N_{q}\)) in the \(q^{{{\text{th}}}}\) source zone
- \(\lambda_{m}\) :
-
Mean annual rate of exceedance of magnitude \(m\)
- \(T_{c}\) :
-
Total period of a catalogue
- \(s\) :
-
Total number of magnitude intervals or sub-catalogues
- \(T_{m}^{s}\) :
-
Completion period in years from recent associated with \(s^{{{\text{th}}}}\) magnitude interval
- \(T_{s}\) :
-
Duration of \(s^{{{\text{th}}}}\) sub-catalogue
- \(m_{\min }^{s}\) :
-
Minimum magnitude of completeness of \(s^{{{\text{th}}}}\) sub-catalogue
- \(n_{s}\) :
-
Total number of events in \(s^{{{\text{th}}}}\) sub-catalogue
- \(m_{{n_{i} }}^{i}\) :
-
Magnitude of \(n^{{{\text{th}}}}\) event in \(i^{{{\text{th}}}}\) sub-catalogue
- \(\beta\) :
-
Gutenberg-Richter recurrence relation parameter \(\left[ {\beta = b\ln \left( {10} \right)} \right]\)
- \(L( \cdot )\) :
-
Likelihood function
- \(\hat{\beta }\) :
-
Estimated Gutenberg-Richter recurrence relation parameter
- \(\overline{m}^{i}\) :
-
Mean magnitude of \(i^{{{\text{th}}}}\) sub-catalogue
- \(r_{i}\) :
-
Ratio of number of events in \(i^{{{\text{th}}}}\) sub-catalogue to the total number of events in the catalogue \(\left( {{{n_{i} } \mathord{\left/ {\vphantom {{n_{i} } {N_{c} }}} \right. \kern-\nulldelimiterspace} {N_{c} }}} \right)\)
- \(\hat{N}_{i}\) :
-
Estimated total number of events in \(i^{{{\text{th}}}}\) sub-catalogue between \(m_{\min }^{i}\) to \(m_{\max }\)
- \(N_{c}\) :
-
Total number of events in the catalogue
- \(\hat{N}\) :
-
Estimated total number of events in the catalogue
- \(m_{\min }\) :
-
Minimum magnitude of the catalogue
- \(m_{\max }\) :
-
Maximum magnitude of the catalogue
- \(m_{\max }^{obs}\) :
-
Maximum observed magnitude of the catalogue
- \(\hat{\lambda }_{{m_{\min } }}\) :
-
Estimated mean annual rate of exceedance of minimum magnitude
- \(N_{m}^{k}\) :
-
Number of events in \(k^{{{\text{th}}}}\) magnitude interval, within completion period \(\left( {T_{m}^{k} } \right)\)
- \(\lambda^{{{\text{obs}}}}\) :
-
Observed mean annual rate of exceedance
- \(E_{SL}^{q}\) :
-
Mean squared error in the estimation of mean annual rate of exceedance by straight line fitting method in \(q^{{{\text{th}}}}\) source zone
- \(\lambda^{{k,{\text{obs}}}}\) :
-
Observed mean annual rate of exceedance in \(k^{{{\text{th}}}}\) magnitude interval
- \(\hat{\lambda }_{m}^{k,SL}\) :
-
Estimated mean annual rate of exceedance in \(k^{{{\text{th}}}}\) magnitude interval by straight line fitting method
- \(E_{LLh}^{q}\) :
-
Mean squared error in the estimation of mean annual rate of exceedance by Log-likelihood method in \(q^{{{\text{th}}}}\) source zone
- \(\hat{\lambda }_{m}^{k,LLH}\) :
-
Estimated mean annual rate of exceedance in \(k^{{{\text{th}}}}\) magnitude interval by log-likelihood method
- \(E_{LLH}\) :
-
Mean squared error in the estimation of mean annual rate of exceedance by log-likelihood method over all the source zones
- \(E_{SL}\) :
-
Mean squared error in the estimation of mean annual rate of exceedance by straight-line fitting method over all the source zones
- \(\theta\) :
-
Slope
- \(W_{SL}\) :
-
Associated weight of straight-line fitting method
- \(W_{LLH}\) :
-
Associated weight of log-likelihood method
- \(Mw\) :
-
Moment magnitude
- \(R^{2}\) :
-
Goodness-of-fit measure
- \(M\) :
-
Earthquake magnitude as random variable
- \(dm_{i}\) :
-
Small interval (delta) of earthquake magnitude
- \(H( \cdot )\) :
-
Heaviside step function
- \(\delta ( \cdot )\) :
-
Dirac delta function
- \(n\) :
-
Sample data set of events
- \(W_{{g_{j} }}\) :
-
Associated weight of \(j^{{{\text{th}}}}\) recurrence relation model
- \(\hat{m}_{\max }\) :
-
Estimated maximum magnitude
- \(\hat{m}_{\max }^{2}\) :
-
Estimated maximum magnitude by second method
- \(L_{f}\) :
-
Fault length
- \(L_{0}\) :
-
Extended fault length up to perpendicular line to the fault from the site
- \(X\left( m \right)\) :
-
Fault rupture length as a function of the earthquake magnitude
- \(D\) :
-
Perpendicular distance between source (fault line) and the site
- \(R_{\min }\) :
-
Minimum source-to-site distance
- \(R_{\max }\) :
-
Maximum source-to-site distance
- \(R\) :
-
Source to site distance as random variable
- \(N_{sc}\) :
-
Total sample count
- \(W_{f\left( r \right)}\) :
-
Weight associated with distance probability distribution
- \(C_{\beta }\) :
-
Normalizing coefficient
- \(p_{G}\) :
-
Gamma distribution parameter
- \(q_{G}\) :
-
Gamma distribution parameter
- \(\sigma_{{\hat{\beta }}}\) :
-
Standard deviation of estimated Gutenberg-Richter recurrence relation parameter
- \(L_{f}^{j}\) :
-
Length of \(j^{{{\text{th}}}}\) fault
- \(m_{\max }^{j}\) :
-
Expected maximum magnitude associated with \(j^{{{\text{th}}}}\) fault
- \(m_{0}\) :
-
Factored maximum magnitude
- \(L_{f}^{\max }\) :
-
Maximum fault length in a source zone
- \(L_{f}^{{{\text{median}}}}\) :
-
Median length of all the faults in a source zone
- \(W_{{\hat{m}_{\max }^{l} }}\) :
-
Weight associated with \(l^{{{\text{th}}}}\) method of estimating maximum magnitude
- \(N_{S}\) :
-
Number of potential earthquake sources in a region around the site
- \(N_{M}\) :
-
Number of discrete magnitude segments considered
- \(N_{R}\) :
-
Number of discrete distance segments considered
- \(\nu_{i}\) :
-
Mean annual rate of exceedance of events greater than certain magnitude in \(i^{{{\text{th}}}}\) source
- \(\Delta m\) :
-
Size of a discrete magnitude segment
- \(g\) :
-
Gravitational acceleration (9.81 \({m \mathord{\left/ {\vphantom {m {\sec^{2} }}} \right. \kern-\nulldelimiterspace} {\sec^{2} }}\))
- \(\Delta r\) :
-
Size of a discrete distance segment
- \(\lambda_{IM}\) :
-
Mean annual rate of exceedance associated with IM- Hazard curve
- \(\lambda_{{{\text{Rate}}}}\) :
-
Mean annual rate of exceedance associated with Rate- Hazard curve
- \(Sa_{IM}\) :
-
Hazard level (Spectral acceleration) associated with IM- Hazard curve
- \(Sa_{{{\text{Rate}}}}\) :
-
Hazard level (Spectral acceleration) associated with Rate- Hazard curve
- PSHA:
-
Probabilistic seismic hazard analysis
- GMPE:
-
Ground motion predictive equation
- UHS:
-
Uniform hazard spectra
- 1D:
-
One dimensional
- MECE:
-
Mutually exclusive and collectively exhaustive
- LLH:
-
Log-likelihood
- SSHAC:
-
Senior Seismic Hazard Analysis Committee
- USNRC:
-
United States Nuclear Regulatory Commission
- CBR:
-
Center, body, and range
- TDI:
-
Technically defensible interpretations
- MCE:
-
Maximum considered earthquake
- DBE:
-
Design basis earthquake
- NEHRP:
-
National earthquake hazard reduction program
- ERF:
-
Earthquake rupture forecast
- SERF:
-
Sub-earthquake rupture forecast
- MHC:
-
Model hazard curve
- IM, im:
-
Intensity measure
- ACR:
-
Active crustal region
- SZ:
-
Subduction zone
- SL:
-
Straight line
- GR:
-
Gutenberg-Richter
- MB:
-
Main and Burton
- EDP:
-
Engineering demand parameter
- CDF:
-
Cumulative distribution function
- z-UHS:
-
Z-fractile representation of uniform hazard spectrum
- 10PE50:
-
10% Probability of exceedance in 50 years
- 2PE50:
-
2% Probability of exceedance in 50 years
- GMRotI50:
-
50th percentile (median) orientation or rotation independent geometric mean intensity measure
- RotD50:
-
50th percentile (median) orientation or rotation dependent intensity measure
- km:
-
Kilometer
- CFVB15:
-
GMPE by Cauzzi and Faccioli (2015)
- SDBK09:
-
GMPE by Sharma et al. (2009)
- KNMF06:
-
GMPE by Kanno et al. (2006)
- ASB14:
-
GMPE by Akkar et al. (2014)
- OZBE04:
-
GMPE by Özbey et al. (2004)
- ZHAO16c:
-
GMPE by Zhao et al. (2016)
- ZHAO06:
-
GMPE by Zhao et al. (2006)
- ASK14:
-
GMPE by Abrahamson et al. (2014)
- IDRI14:
-
GMPE by Idriss (2014)
- BSSA14:
-
GMPE by Boore et al. (2014)
- CB14:
-
GMPE by Campbell and Bozorgnia (2014)
- CY14:
-
GMPE by Chiou and Youngs (2014)
- MZAS06:
-
GMPE by McVerry et al. (2006)
- NATH12:
-
GMPE by Nath et al. (2012)
- ANBA13:
-
GMPE by Anbazhagan et al. (2013)
- GUPTA10:
-
GMPE by Gupta (2010)
- YOUNG97:
-
GMPE by Youngs et al. (1997)
- BCHY16:
-
GMPE by Abrahamson et al. (2016)
- ZHAO16:
-
GMPE by Zhao et al. (2016)
- ATBO03:
-
GMPE by Atkinson and Boore (2003)
- LNLE08:
-
GMPE by Lin and Lee (2008)
- ARRO10:
-
GMPE by Arroyo et al. (2010)
- NDMA10:
-
GMPE from National disaster management authority report (2010)
- SZ2:
-
Second source zone in subduction zone
- ACR5:
-
5th source zone in active crustal region
- PDF:
-
Probability density function
- PGA:
-
Peak ground acceleration
- CS:
-
Conditional spectra
- GCR:
-
Generalized causal rupture
- GCIM:
-
Generalized conditional intensity measure
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Acknowledgements
This research is funded by the Ministry of Earth Sciences, Seismology Division, Government of India, under the Grant No. MoES/P.O.(Seismo)/1(370)/2019 and the financial support is acknowledged. We thank Prof. Andrzej Kijko for his insightful help in understanding the methods of calculating maximum magnitude and providing the required MATLAB codes. The authors greatly acknowledge the Geological Survey of India for providing Seismotectonic Atlas of North-east India used in this study. The authors also acknowledge the sources of seismic catalogues used in this study: International Seismological Centre, UK.; US Geological Survey; National Center for Seismology, India; and IRIS Data management centre, New York, USA.
Funding
This research is funded by the Ministry of Earth Sciences, Seismology Division, Government of India, under the Grant No. MoES/P.O.(Seismo)/1(370)/2019.
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DB proposed the idea, performed conceptual design, interpreted the results, prepared the revised draft, and managed the overall research. NG performed the analysis, developed the coding, processed the data, generated and interpreted results, written the original draft.
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Gurjar, N., Basu, D. Epistemic Uncertainty in PSHA and Seismic Hazard Characterization Using the Logic Tree Approach: Part I, Developing the Framework. Pure Appl. Geophys. 179, 3647–3676 (2022). https://doi.org/10.1007/s00024-022-03143-4
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DOI: https://doi.org/10.1007/s00024-022-03143-4