Bounded Exponential Magnitude Model
Class Name
- RBoundedExponentialMagnitudeModel
Location in Objects Pane
- Models > Model > Hazard > Earthquake > Magnitude > Bounded Exponential Magnitude
Model Description
Model Form
- The basis for this model is the following PDF:
$$f(m) = {{\beta \cdot \exp \left[ { - \beta \cdot (m - {M_{\min }})} \right]} \over {1 - \exp \left[ { - \beta \cdot ({M_{\max }} - {M_{\min }})} \right]}} \text{ for }{M_{\min }} \le m \le {M_{\max }}$$
where \({\beta}\) is the model parameter that depends on the relative frequency of different magnitudes.
- In order to obtain the outcome of \(m\), a standard normal random variable, \({\theta}\), is given to the model, and transformed according to the probability-preserving transformation \(F(m)={\Phi}({\theta})\), where \(F(m)\) is the CDF corresponding to the given PDF.
- Hence, the model reads
$$m=-\frac{1}{\beta}\ln \left[ 1-\Phi (\theta )\cdot \left( 1-\exp \left( -{\beta}\left( {{M}_{\max }}-{{M}_{\min }} \right) \right) \right) \right]+{{M}_{\min }}$$
- For further information on this model, refer to Rahimi and Mahsuli (2019) and Mahsuli et al. (2019).
DDM Sensitivities
Properties
Object Name
- Name of the object in Rt
- Allowable characters are upper-case and lower-case letters, numbers, and underscore (“_”).
- The name is unique and case-sensitive.
Display Output
- Determines whether the model is allowed to print messages to the Output Pane.
Minimum Magnitude
- \({M _{\min}}\) = Magnitude Lower bound
Maximum Magnitude
- \({M _ {\max}}\) = Magnitude upper bound
Beta
- \({\beta}\) = Model parameter, usually in the range of 1 to 2
Theta
- \({\theta}\) = A standard-normal random variable
Output
- \(m\) = Magnitude
- The output is an automatically generated generic response object, which takes the object name of the model plus “Response”.
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References
- Rahimi, H., Mahsuli, M. (2019) “Structural reliability approach to analysis of probabilistic seismic hazard and its sensitivities,” Bulletin of Earthquake Engineering, 17(3): 1331-1359 (DOI)
- Mahsuli, M., Rahimi, H., Bakhshi, A. (2019) “Probabilistic seismic hazard analysis of Iran using reliability methods,” Bulletin of Earthquake Engineering, 17(3): 1117-1143 (DOI)