# Probabilistic Estimates of Ground Motion in the Los Angeles Basin from Scenario Earthquakes on the San Andreas Fault

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## Abstract

**:**

## 1. Introduction

## 2. Earthquake Source Models and Associated Probabilities

#### 2.1. Source Models

#### 2.2. Scenario Earthquake Probabilities

- (i)
- We start with “magnitude binning” of forecast earthquakes. We define as many bins as the number of distinct magnitude scenario earthquakes. The lower and upper magnitude limits of a magnitude bin are the magnitudes derived from the averages of the seismic moments of the corresponding scenario earthquake and the scenario earthquakes tied to the previous and next bins, respectively. Corresponding to the six scenario earthquake magnitudes of 6.00, 6.56, 6.92, 7.28, 7.59, and 7.89, the following six magnitude bins are defined: [5.90–6.42], (6.42–6.80], (6.80–7.15] , (7.15–7.45], (7.47–7.78], and (7.78–8.34]. The seismic moments of the ${M}_{w}$ 6.00, 6.56, 6.92, 7.28, 7.59, and 7.89 scenario earthquakes correspond to the average of the seismic moments of the upper and lower magnitude limits of the first five bins, respectively. The upper limit of the last bin is assumed higher to include all forecast earthquakes with magnitude greater than 7.89. Each of the forecast earthquakes will be assigned to one of these magnitude bins. For instance, a forecast earthquake with magnitude, say, between 6.42 and 6.80 will be assigned to the magnitude bin tied to the scenario earthquake with magnitude ${M}_{w}$ 6.56. Its probability of occurrence will be redistributed among the ten ${M}_{w}$ 6.56 scenario earthquakes (five rupture locations and two rupture directions). The dashed black lines in Figure 3 demarcate the magnitude bins.
- (ii)
- The seismic moment of forecast earthquake i, ${M}_{o}^{i}$, is multiplied by the UCERF yearly occurrence rate ${r}_{i}$ to arrive at what may be termed as the seismic moment release rate with a unit of “seismic moment/year”. Seismic moment release rates are determined for all forecast earthquakes in this manner.
- (iii)
- Within each magnitude bin, the seismic moment release rate of a forecast earthquake is distributed among the UCERF segments being ruptured by that forecast earthquake in proportion to their areas. This is based on the fact that seismic moment release rate, given by $\mu A\dot{D}$ where $\dot{D}$ is the average slip rate on the fault, $\mu $ is the shear modulus, and A is the area of rupture, scales linearly with segment area (see Equation (4.8) and Appendix G of [19]). Thus the seismic moment release rate contribution of the ${i}^{th}$ forecast earthquake to the ${j}^{th}$ UCERF segment equals ${r}_{i}{M}_{o}^{i}\frac{{A}_{j}}{{A}_{i}}$, where ${A}_{i}$ is the area of forecast earthquake i and ${A}_{j}$ is the area of the UCERF segment j.
- (iv)
- Within each magnitude bin, the contributions to fault segment j of all N forecast earthquakes in that bin are summed: $\sum _{i=1}^{N}}{r}_{i}{M}_{o}^{i}\frac{{A}_{j}}{{A}_{i}$. This represents the yearly seismic moment buildup in segment j that is expected to be released periodically by earthquakes with magnitudes lying within the bin. It may be termed as the seismic moment release rate for segment j in earthquakes from that magnitude bin.
- (v)
- Within each magnitude bin, the cumulative seismic moment release rate of segment j, determined in the previous step, is assigned to the scenario earthquake tied to that bin and whose rupture location is closest to segment j. Then the seismic moment release rate of that scenario earthquake is given by $\sum _{j=1}^{M}}{\displaystyle \sum _{i=1}^{N}}{r}_{i}{M}_{o}^{i}\frac{{A}_{j}}{{A}_{i}$, where M is the number of UCERF segments occurring within the rupture extent of that scenario earthquake. It is possible that the rupture extents of two or more scenario earthquakes may extend over the same fault segment(s). The moment release rates on such segments are evenly distributed among the overlapping scenario earthquakes.
- (vi)
- The seismic moment release rate, obtained from the last step, for scenario earthquake k is divided by its seismic moment ${M}_{o}^{S{E}_{k}}$ to obtain its yearly occurrence rate ${q}_{k}={\displaystyle \sum _{j=1}^{M}\sum _{i=1}^{N}}{r}_{i}{M}_{o}^{i}\frac{{A}_{j}}{{A}_{i}}/{M}_{o}^{S{E}_{k}}$.
- (vii)
- The probability of occurrence of scenario earthquake k over a period of $\Delta T$ years is then given by the Poisson distribution as: $P\left(S{E}_{k}\right)=1-{e}^{-{q}_{k}\Delta T}$.
- (viii)
- Steps (iii)–(vii) are repeated for all magnitude bins and the scenario earthquakes associated with them.

## 3. Ground Motion Simulation

## 4. Probabilistic Estimates of Ground Shaking

## 5. Discussion and Limitations

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Kinematic finite source model of the 2002 M

_{w}7.9 Denali earthquake mapped on the southern San Andreas fault at five locations. Blue areas represent regions of small slip and red areas represent regions of large slip (peak slip of about 12 m). The left column illustrates the five north-to-south propagating scenario earthquakes whereas the right column illustrates the south-to-north propagating earthquakes. Note that in reversing the rupture direction, the slip distribution is flipped as well. The red stars correspond to the hypocenters.

**Figure 2.**All plausible (or “forecast”) earthquakes that rupture at least one segment of the southern section of the San Andreas fault (from Parkfield in central California to Bombay Beach in southern California, shown in red). The X axis identifies the extent of rupture of a forecast earthquake and the Y axis identifies its magnitude. Line colors indicate forecast earthquake occurrence probabilities, with warmer colors (red, orange) indicative of higher probability of occurrence and colder colors (blue) indicative of lower probability of occurrence.

**Figure 3.**The 30 scenario earthquakes (shown in black) superposed on top of the forecast earthquakes from Figure 2. Two rupture directions are considered for each, bringing the total number of scenario earthquakes to 60. Dashed black lines demarcate the magnitude bins adopted in the case study. Colored lines demarcate UCERF’s plausible (“forecast”) earthquakes.

**Figure 4.**Illustrative example of the method used to derive scenario earthquake probabilities from forecast earthquake probabilities. Horizontal blue line: target fault. Dotted black lines: fault segmentation. Red unfilled rectangles: forecast earthquakes (F.E.) with seismic moment rates indicated by rectangle heights [step (ii) of method]. Yellow shaded region: seismic moment rate contributions of several forecast earthquakes to a given segment (step iii). Red shaded region: summation of yellow shaded regions (step iv). Magenta colored region: seismic moment rate of a scenario earthquake (S.E.) by summing the moment rates of all the segments (step v). Magenta lines: rupture extent of S.E.s.

**Figure 5.**(

**a**,

**b**) Median peak geometric mean horizontal displacement (m) and velocity (m/s). (

**c**,

**d**) 5%-damped spectral acceleration at 1 s and 0.2 s periods, in units of “g”, plotted as a function of earthquake magnitude from scenario earthquake simulations (blue lines) and the Campbell-Bozorgnia NGA (red lines). The vertical bars correspond to the one standard deviation spread above and below the median values.

**Figure 6.**Comparison of maps of median peak geometric mean horizontal velocities (m/s) in the Los Angeles basin from simulations of ten scenario San Andreas fault earthquakes at three magnitude levels to predictions using the CB-08 NGA relations. (

**a**,

**b**) Magnitude M

_{w}7.28, (

**c**,

**d**) Magnitude M

_{w}7.59, and (

**e**,

**f**) Magnitude M

_{w}7.89.

**Figure 7.**Basin depth (km) map for southern California. Red triangles indicate the geographical distribution of the 636 southern California sites where ground motions from the scenario earthquakes are computed. The ellipses identify the basins in southern California: Simi valley, San Fernando valley, San Gabriel valley and Los Angeles basin.

**Figure 8.**Predictions of spectral accelerations at 1 s and 3 s periods for the ten ${M}_{w}$ 7.89 scenario earthquakes (five locations and two rupture directions) by simulations and the CB-08 Next Generation Attenuation (NGA) relations. (

**a**,

**b**) Median values as a function of the Joyner-Boore source-to-site distance; (

**c**,

**d**) Median ${S}_{a}$ maps from simulations; (

**e**,

**f**) Median ${S}_{a}$ maps from CB-08 NGA relations.

**Figure 9.**Directivity effect: Comparison of simulated peak horizontal velocity from north-to-south and south-to-north ruptures of the magnitude 7.89 scenario earthquake at locations 1 (

**a**,

**b**) and 5 (

**c**,

**d**).

**Figure 10.**Geometric mean of peak horizontal ground velocities under the (

**a**) simulated ${M}_{w}$ 7.89 south-to-north propagating scenario earthquake at location 5, (

**b**) the south-to-north propagating ${M}_{w}$ 7.80 ShakeOut scenario earthquake rupturing the San Andreas fault from Bombay Beach in the south to Lake Hughes in the north, and (

**c**) the predictions by the CB-08 NGA relations.

**Figure 11.**Cumulative Distribution Functions of (

**a**,

**b**) PGV and (

**c**,

**d**) PGD at the fourteen tall building cluster locations in southern California from San Andreas fault earthquakes over the next 30 years.

**Table 1.**List of past earthquakes with fault geometry and rupture mechanisms closely matching earthquakes on the San Andreas fault whose kinematic finite source inversions are used in this study. The salient source parameters are listed as well.

Name | Date | Location | ${\mathit{M}}_{\mathit{w}}$ | Length (km) | Depth (km) | Dip (${}^{\circ}$) | Rake (${}^{\circ}$) | Reference | |
---|---|---|---|---|---|---|---|---|---|

1 | Denali | 2002 | AK, USA | 7.89 | 290.0 | 20.0 | 90.0 | 180.0 | [13] |

2 | Izmit | 1999 | Turkey | 7.59 | 155.0 | 18.0 | 90.0 | 180.0 | [14] |

3 | Landers | 1992 | CA, USA | 7.28 | 78.0 | 15.0 | 89.0 | 180.0 | [15] |

4 | Kobe | 1995 | Japan | 6.92 | 60.0 | 20.0 | 85.0 | 180.0 | [16] |

5 | Imperial Valley | 1979 | CA, USA | 6.58 | 42.0 | 10.4 | 90.0 | 180.0 | [17] |

6 | Parkfield | 2004 | CA, USA | 6.00 | 40.0 | 14.5 | 83.0 | 180.9 | [18] |

**Table 2.**UCERF3 time-independent 30-year occurrence probabilities for the 30 scenario earthquakes (six magnitudes and five rupture locations). Half of these probabilities are assigned to north-to-south propagating ruptures and the other half to south-to-north propagating ruptures.

${\mathit{M}}_{\mathit{w}}$ [Bin] | Location 1 | Location 2 | Location 3 | Location 4 | Location 5 | Total Probability |
---|---|---|---|---|---|---|

(Parkfield) | (Bombay Beach) | (All Locations) | ||||

6.00 [5.90–6.42] | 0.6449 | 0.0459 | 0.1910 | 0.2485 | 0.0685 | 0.8081 |

6.58 (6.42–6.80] | 0.0051 | 0.0100 | 0.0854 | 0.1280 | 0.0183 | 0.2288 |

6.92 (6.80–7.15] | 0.0180 | 0.0171 | 0.0060 | 0.0764 | 0.0271 | 0.1380 |

7.28 (7.15–7.45] | 0.0211 | 0.0182 | 0.0059 | 0.0153 | 0.0365 | 0.0935 |

7.59 (7.47–7.78] | 0.0124 | 0.0121 | 0.0061 | 0.0082 | 0.0192 | 0.0568 |

7.89 (7.78–8.34] | 0.0339 | 0.0281 | 0.0236 | 0.0225 | 0.0215 | 0.1231 |

Total Probability | ||||||

${M}_{w}$ [5.90–8.34] | 0.6760 | 0.1249 | 0.2904 | 0.4221 | 0.1773 | 0.8553 |

**Table 3.**Comparison of ground motion intensities from the ten (five locations and two rupture directions) simulated ${M}_{w}$ 7.89 scenario earthquakes against CB-08 NGA predictions at fourteen locations in southern California where a significant number of tall buildings exist.

Site Location | Latitude | Longitude | Simulated | CB-08 | Soil Type | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PGV (m/s) | PGD (m) | ${\mathit{S}}_{\mathit{a}}^{1\mathit{s}}$ (g) | ${\mathit{S}}_{\mathit{a}}^{3\mathit{s}}$ (g) | PGV (m/s) | PGD (m) | ${\mathit{S}}_{\mathit{a}}^{1\mathit{s}}$ (g) | ${\mathit{S}}_{\mathit{a}}^{3\mathit{s}}$ (g) | UBC | UBC | |||||||

$\mathit{Md}$ | $\mathit{\sigma}$ | $\mathit{Md}$ | $\mathit{\sigma}$ | $\mathit{Md}$ | $\mathit{\sigma}$ | $\mathit{Md}$ | $\mathit{\sigma}$ | $\mathit{Md}$ | $\mathit{Md}$ | $\mathit{Md}$ | $\mathit{Md}$ | 94 | 97 | |||

Irvine | 33.67 | 117.80 | 0.46 | 0.35 | 0.56 | 0.29 | 0.21 | 0.17 | 0.15 | 0.18 | 0.21 | 1.18 | 0.17 | 0.08 | ${S}_{3}$ | ${S}_{d}$ |

Encino | 34.16 | 118.50 | 0.30 | 0.46 | 0.43 | 0.24 | 0.12 | 0.19 | 0.13 | 0.27 | 0.19 | 0.89 | 0.14 | 0.06 | ${S}_{2}$ | ${S}_{c}$ |

Downtown LA | 34.05 | 118.25 | 0.79 | 0.75 | 0.79 | 0.60 | 0.26 | 0.15 | 0.38 | 0.43 | 0.28 | 1.70 | 0.23 | 0.10 | ${S}_{3}$ | ${S}_{d}$ |

Canoga Park | 34.20 | 118.60 | 0.94 | 0.53 | 0.70 | 0.41 | 0.38 | 0.24 | 0.30 | 0.37 | 0.18 | 0.87 | 0.14 | 0.06 | ${S}_{2}$ | ${S}_{c}$ |

Pasadena | 34.16 | 118.13 | 0.13 | 0.09 | 0.20 | 0.13 | 0.04 | 0.10 | 0.01 | 0.03 | 0.15 | 0.77 | 0.12 | 0.05 | ${S}_{3}$ | ${S}_{d}$ |

Anaheim | 33.84 | 117.89 | 0.73 | 0.61 | 0.70 | 0.48 | 0.26 | 0.18 | 0.42 | 0.41 | 0.22 | 1.16 | 0.17 | 0.07 | ${S}_{2}$ | ${S}_{c}$ |

Long Beach | 33.77 | 118.19 | 0.26 | 0.21 | 0.33 | 0.27 | 0.14 | 0.10 | 0.08 | 0.09 | 0.23 | 1.38 | 0.19 | 0.09 | ${S}_{3}$ | ${S}_{d}$ |

Glendale | 34.17 | 118.25 | 0.26 | 0.33 | 0.40 | 0.30 | 0.15 | 0.09 | 0.08 | 0.15 | 0.20 | 0.93 | 0.15 | 0.06 | ${S}_{2}$ | ${S}_{c}$ |

Hollywood | 34.10 | 119.33 | 0.31 | 0.41 | 0.49 | 0.27 | 0.16 | 0.12 | 0.19 | 0.29 | 0.18 | 0.85 | 0.14 | 0.06 | ${S}_{2}$ | ${S}_{c}$ |

El Segundo | 33.92 | 118.41 | 0.63 | 0.39 | 0.60 | 0.29 | 0.18 | 0.13 | 0.22 | 0.19 | 0.20 | 1.09 | 0.16 | 0.07 | ${S}_{3}$ | ${S}_{d}$ |

Santa Monica | 34.02 | 118.48 | 0.66 | 0.32 | 0.68 | 0.30 | 0.17 | 0.10 | 0.16 | 0.12 | 0.19 | 0.94 | 0.14 | 0.06 | ${S}_{2}$ | ${S}_{c}$ |

Century City | 34.08 | 118.42 | 0.66 | 0.41 | 0.68 | 0.45 | 0.16 | 0.10 | 0.21 | 0.16 | 0.20 | 0.99 | 0.15 | 0.06 | ${S}_{2}$ | ${S}_{c}$ |

Universal City | 34.14 | 118.35 | 0.27 | 0.13 | 0.38 | 0.19 | 0.06 | 0.05 | 0.06 | 0.06 | 0.14 | 0.54 | 0.10 | 0.04 | ${S}_{2}$ | ${S}_{c}$ |

Park La Brea | 34.06 | 118.35 | 0.30 | 0.46 | 0.43 | 0.24 | 0.12 | 0.19 | 0.13 | 0.27 | 0.19 | 0.89 | 0.14 | 0.06 | ${S}_{2}$ | ${S}_{c}$ |

**Table 4.**Probabilistic estimates of PGV and peak ground displacements (PGD) at 14 stations in southern California with probabilities of exceedance of 10% and 2% in 30 years from earthquakes on the San Andreas fault.

Location | PGV (m/s) | PGD (m) | ||
---|---|---|---|---|

10% | 2% | 10% | 2% | |

Irvine | 0.16 | 0.70 | 0.16 | 0.89 |

Encino | 0.09 | 0.46 | 0.11 | 0.71 |

Downtown LA | 0.22 | 0.82 | 0.19 | 1.02 |

Canoga Park | 0.16 | 1.63 | 0.19 | 1.22 |

Pasadena | 0.04 | 0.20 | 0.09 | 0.44 |

Anaheim | 0.40 | 1.38 | 0.34 | 1.23 |

Long Beach | 0.08 | 0.40 | 0.12 | 0.63 |

Glendale | 0.05 | 0.52 | 0.09 | 0.74 |

Hollywood | 0.06 | 0.84 | 0.10 | 0.91 |

El Segundo | 0.20 | 0.89 | 0.21 | 0.85 |

Santa Monica | 0.16 | 0.83 | 0.20 | 0.95 |

Century City | 0.18 | 0.94 | 0.23 | 1.05 |

Universal City | 0.06 | 0.48 | 0.11 | 0.74 |

Park La Brea | 0.09 | 0.46 | 0.11 | 0.71 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mourhatch, R.; Krishnan, S. Probabilistic Estimates of Ground Motion in the Los Angeles Basin from Scenario Earthquakes on the San Andreas Fault. *Geosciences* **2018**, *8*, 126.
https://doi.org/10.3390/geosciences8040126

**AMA Style**

Mourhatch R, Krishnan S. Probabilistic Estimates of Ground Motion in the Los Angeles Basin from Scenario Earthquakes on the San Andreas Fault. *Geosciences*. 2018; 8(4):126.
https://doi.org/10.3390/geosciences8040126

**Chicago/Turabian Style**

Mourhatch, Ramses, and Swaminathan Krishnan. 2018. "Probabilistic Estimates of Ground Motion in the Los Angeles Basin from Scenario Earthquakes on the San Andreas Fault" *Geosciences* 8, no. 4: 126.
https://doi.org/10.3390/geosciences8040126