# Prospective Fault Displacement Hazard Assessment for Leech River Valley Fault Using Stochastic Source Modeling and Okada Fault Displacement Equations

^{1}

^{2}

^{*}

## Abstract

**:**

^{−4}. For critical infrastructures, such as bridges and pipelines, quantifying the uncertainty of fault displacement hazard is essential to manage potential damage and loss effectively.

## 1. Introduction

## 2. Probabilistic Fault Displacement Hazard Analysis Using Stochastic Source Models and Okada Equations

#### 2.1. Methodology

_{min}or larger, while f

_{M}is the conditional probability distribution of earthquake magnitude M above M

_{min}. f

_{SM}

_{|M}is the probability density function of a source model SM given M. P (D ≥ d|sm,m) is the probability of exceeding a specified fault displacement given an earthquake source model SM and magnitude M.

_{M}). To account for epistemic uncertainty of earthquake occurrence and magnitude models, a logic tree can be implemented (see Section 2.3). The second step is to specify a fault plane geometry (which may consist of multiple segments) and generate source parameter samples (e.g., fault length, fault width, mean slip, and maximum slip) from statistical scaling relationships [18]. Then, heterogenous earthquake slip distributions [19,20] are synthesized to generate various stochastic source models (see Section 2.4). A stochastic source model is a collection of earthquake slip values on the fault-rupture plane. In the third step, for each stochastic source model, three orthogonal displacement fields can be calculated using Okada equations (see Section 2.5). Finally, a fault displacement hazard curve ν

_{D}(D ≥ d) for a single site or a differential fault displacement hazard curve for paired sites can be obtained as a product of the assessment. The aforementioned procedure is summarized in Figure 1.

#### 2.2. Target Fault Source: Leech River Valley Fault

#### 2.2.1. Geological Setting

_{w}6.9.

#### 2.2.2. Fault Plane Model

#### 2.3. Earthquake Magnitude Model

_{r}of a fault zone with an area A

_{f}and with the shear modulus μ (i.e., seismic moment release = µ S

_{r}A

_{f}). The probability density function of the characteristic magnitude model is given by [29,32]

_{min}and M

_{max}are the minimum magnitude and maximum magnitude for the fault source, respectively; ∆m

_{1}and ∆m

_{2}are the magnitude intervals used to specify the probability density values for the exponential distribution part (magnitude range between M

_{min}and M

_{max}– ∆m

_{2}) and the uniform distribution part (magnitude range between M

_{max}– ∆m

_{2}and M

_{max}). The truncated exponential magnitude model can be achieved by setting ∆m

_{2}= 0 (i.e., C = 0) in the above equations.

_{r}), b-value (equivalent to β value), and M

_{max}—are varied by considering three possible values with different weights. The parameter values and weights are listed in Table 1. According to sensitivity analyses conducted by [27], one of the most influential parameters is the slip rate. The range of the variability of S

_{r}considered in Table 1 (i.e., 0.15–0.35 mm/year) is consistent with [25] (i.e., 0.2–0.3 mm/year). Other relevant parameters are set as follows: μ = 35 GPa; A

_{f}is calculated based on the fault geometry shown in Figure 2b; M

_{min}= 6.0; ∆m

_{1}= 1.0; ∆m

_{2}= 0.5. These values are consistent with those reported in [26]; details can be found in [27].

_{M}) were calculated.

#### 2.4. Stochastic Source Model

_{a}, D

_{m}, and λ

_{BC}determine the marginal distribution of the earthquake slip on the fault-rupture plane. In particular, the Box–Cox transformation with λ

_{BC}is used to achieve a desirable right-skewed feature of the slip marginal distribution [20]. To characterize the spatial complexity of earthquake slip, spectral synthesis of random slip fields adopts an anisotropic 2D von Kármán wavenumber spectrum, with its amplitude spectrum being parametrized by along-strike correlation length C

_{L}, along-dip correlation length C

_{W}, and Hurst number H, and its phase being randomly distributed between 0 and 2π [19]. Once a random earthquake slip is generated for a given fault geometry, the slip distribution is converted using the Box–Cox transformation and its associated parameter (λ

_{BC}) to achieve a realistic heavier right-tail distribution. The obtained distribution is adjusted by the mean and maximum slips (D

_{a}and D

_{m}), and the corresponding seismic moment is calculated for the generated source model with consistent values of L, W, and D

_{a}. The candidate stochastic source model is accepted if it produces the seismic moment within the target range (e.g., M

_{w}6.5 and M

_{w}6.6); otherwise, the procedure is repeated [18].

_{a}), and maximum slip (D

_{max}). The magnitude range between 6 and 7.5 is considered and divided into 15 bins with 0.1 widths. The maximum magnitude of M

_{w}7.5 corresponds to the upper limit of the moment magnitude for the fault plane shown in Figure 2b and is selected based on the fault area scaling relationship [33] by taking a mean plus 1.5 times the standard deviation. For each magnitude bin, 1000 stochastic source models are generated. The discrete characteristics of the fault length and width shown in Figure 4a,b, respectively, are because the fault dimensions can only take the odd multiples of subfault sizes (= 2 km), i.e., 2, 6, 10, etc. This requirement of the odd multiples of the subfault size stems from the generation of wavenumber vectors specified at the lower and upper boundary wavenumbers and at zero wavenumbers in the earthquake slip synthesis method proposed by [19]. In the simulated samples of the fault dimensions, the effects of saturation of the fault area can be observed (i.e., the maximum dimensions are limited to 70 km and 30 km for the fault length and width, respectively). Consequently, to satisfy the target seismic moment for large earthquake scenarios (e.g., M

_{w}greater than 7.1), the mean and maximum slips start to deviate upward from the mean scaling relationships. On the other hand, for small to moderate earthquake scenarios (e.g., M

_{w}smaller than 7.1), simulated stochastic source models have representative characteristics similar to the empirical scaling relationships.

_{w}6.9–7.0. The figure shows that the simulated source models can have smaller geometry than the defined source zone; hence, they can be floated within the fault zone. Additionally, the slip distributions vary according to the scaling relationships and realizations of the earthquake slip synthesis. Some models have large slip values that are concentrated near the center of the fault-rupture plane, which is not densely populated (Figure 5a), while others have large slip concentrations that are relatively close to populated areas near Victoria (Figure 5b,c). In short, by using the 1000 stochastic source models, a wide range of possible earthquake ruptures for the specified earthquake magnitude can be captured.

#### 2.5. Fault Displacement Calculation

## 3. Probabilistic Fault Displacement Hazard Analysis of the Leech River Valley Fault

#### 3.1. Simulation Setup and Site Selection

_{w}6+ events; Figure 3).

#### 3.2. Fault Displacement Hazard Curves

^{−4}), some large displacements exceeding 0.1 m (up to 0.5 m vertical uplift at hanging wall sites) are likely to occur. Relatively low fault displacement hazards in Langford are mainly attributed to a low frequency of major earthquake ruptures of the LRVF.

^{−4}, for instance, the differential vertical displacement of 0.5 m or greater is possible for Sites 2 and 3. The differential displacements tend to become large when the two sites experience different relative fault movements. For the reverse faulting case, vertical differential displacements are more critical than horizontal differential displacements. However, different situations would arise for different faulting mechanisms (e.g., for strike-slip events, differential horizontal displacements become dominant for two sites across the fault strike). Importantly, the results from the differential fault displacement hazard curves can be used for further analyses of engineered structures, such as bridges and pipelines, to assess their structural integrity and seismic performance under extreme loading conditions.

_{w}7.0+ events for both differential displacement cases. With the increase in the displacement threshold, larger earthquake events tend to contribute more to the differential displacement hazards, which is expected.

#### 3.3. Fault Displacement Hazard Maps for Critical Scenarios

^{−5}and 3.8 × 10

^{−5}annual frequency of exceedance. Figure 12a,b illustrate the regional views of the slip distribution for two selected source models. Figure 12c,d show the horizontal fault displacement hazard maps, whereas Figure 12e,f show the vertical displacement hazard maps. Figure 13 shows the local views of Figure 12 near Sites 2 and 3. As shown in Figure 12c (Figure 13c), there is a larger horizontal displacement on the footwall side of the fault plane (Site 2) than on the hanging wall side of the fault plane (Site 3). This is reversed in the case of the vertical displacement and larger surface displacement appears on the hanging wall side of the fault plane, as shown in Figure 12e (Figure 13e). With the increase in the fault displacement hazard threshold, the extent of fault displacements tends to be more extensive (as expected). Although the frequencies of the hazard for 0.5 m and 1 m fault displacements in the LRVF are low, such probable rupture scenarios are relevant and should be considered because of the potential consequences that can be caused by the failures of critical infrastructure in Victoria and the surrounding region. These critical fault displacement hazard maps are useful for emergency management purposes.

## 4. Conclusions

#### 4.1. Summary and Key Findings

- The results for the LRVF region indicate that a surface vertical displacement of 0.5 m is possible at a low annual frequency of exceedance (annual frequency of 10
^{−4}). The low chance of a major fault displacement hazard is attributed to the low seismic activity of the LRVF (i.e., mean recurrence periods of moderate-size earthquakes are around 1000 years). Although the probability is low, the consideration of the surface fault displacement of the LRVF is important as it may affect local critical infrastructures widely and simultaneously. - The differential fault displacement hazards for two sites that are located across the fault strike (i.e., one site is on the footwall side of the fault, while the other site is on the hanging wall side of the fault) are significantly greater than other paired sites that are located on the same side of the fault. For critical linear infrastructure crossing the fault trace, assessing the structural integrity against possible differential fault displacement hazards is important.

#### 4.2. Perspectives

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Flowchart of the fault displacement hazard analysis based on the stochastic source modeling method.

**Figure 2.**(

**a**) Topographical characteristics of the southern Vancouver Island and (

**b**) fault plane model of the Leech River Valley Fault. The fault scarp is based on [25].

**Figure 3.**Magnitude–recurrence relationships for the Leech River Valley Fault based on the characteristic model (

**a**) and the truncated exponential model (

**b**). For each magnitude model, 27 variations were considered by assigning discrete values to slip rate, b value, and M

_{max}.

**Figure 4.**Earthquake scaling relationships and simulated parameters for the stochastic earthquake source models for the fault length (

**a**), fault width (

**b**), mean slip (

**c**), and maximum slip (

**d**).

**Figure 6.**Average earthquake slip distributions of the 1000 stochastic source models: (

**a**) M

_{w}6.6–6.7 scenario, (

**b**) M

_{w}6.9–7.0 scenario, and (

**c**) M

_{w}7.2–7.3 scenario.

**Figure 7.**Spatial distributions (

**a**–

**c**) and cross-sectional profiles (

**d**–

**f**) of fault displacements in east–west (

**a**,

**d**), north–south (

**b**,

**e**), and up–down (

**c**,

**f**) directions due to a fault rupture with unit slip in a reverse faulting mechanism.

**Figure 8.**Locations of Sites 1 to 5 for the fault displacement hazard analysis of the Leech River Valley Fault.

**Figure 9.**Fault displacement hazard curves for Sites 1 to 5: (

**a**) horizontal displacement and (

**b**) vertical displacement.

**Figure 10.**Differential fault displacement hazard curves for four pairs of Sites 1 to 5: (

**a**) horizontal displacement and (

**b**) vertical displacement.

**Figure 11.**Magnitude distributions of fault-rupture events that produce total differential displacements for Sites 2 and 3, exceeding the threshold values of 0.5 m (

**a**) and 1.0 m (

**b**).

**Figure 12.**Regional views of earthquake slip distribution (

**a**,

**b**), horizontal displacement (

**c**,

**d**), and vertical displacement (

**e**,

**f**) due to stochastic source models that produce total differential displacements of 0.5 m (

**a**,

**c**,

**e**) and 1.0 m (

**b**,

**d**,

**f**) for Sites 2 and 3.

**Figure 13.**Local views of earthquake slip distribution (

**a**,

**b**), horizontal displacement (

**c**,

**d**), and vertical displacement (

**e**,

**f**) due to stochastic source models that produce total differential displacements of 0.5 m (

**a**,

**c**,

**e**) and 1.0 m (

**b**,

**d**,

**f**) for Sites 2 and 3.

**Table 1.**Logic-tree parameter values and associated weights of the magnitude model for the Leech River Valley Fault. Parameter values and weights were adopted from [26].

Parameter | Values | Weights |
---|---|---|

Slip rate (mm/year) | 0.25, 0.15, 0.35 | 0.68, 0.16, 0.16 |

b-value | 0.796, 0.730, 0.862 | 0.68, 0.16, 0.16 |

M_{max} | 7.37 ^{1}, 7.22, 7.52 | 0.6, 0.3, 0.1 |

^{1}The best estimate of M

_{max}is determined using the fault area–magnitude scaling relationship [33].

Stochastic Source Parameters | Functions |
---|---|

Fault length (L) | Define a fault geometry that can be smaller than the fault plane and can be floated within the source zone. |

Fault width (W) | |

Mean slip (D_{a}) | Define a marginal distribution of earthquake slip on the fault plane |

Maximum slip (D_{m}) | |

Box–Cox parameter (λ_{BC}) | |

Correlation length along strike (C_{L}) | Define a heterogeneous slip distribution based on the von Kármán wavenumber spectrum [19] within the source zone. |

Correlation length along dip (C_{W}) | |

Hurst number (H) |

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**MDPI and ACS Style**

Goda, K.; Shoaeifar, P.
Prospective Fault Displacement Hazard Assessment for Leech River Valley Fault Using Stochastic Source Modeling and Okada Fault Displacement Equations. *GeoHazards* **2022**, *3*, 277-293.
https://doi.org/10.3390/geohazards3020015

**AMA Style**

Goda K, Shoaeifar P.
Prospective Fault Displacement Hazard Assessment for Leech River Valley Fault Using Stochastic Source Modeling and Okada Fault Displacement Equations. *GeoHazards*. 2022; 3(2):277-293.
https://doi.org/10.3390/geohazards3020015

**Chicago/Turabian Style**

Goda, Katsuichiro, and Parva Shoaeifar.
2022. "Prospective Fault Displacement Hazard Assessment for Leech River Valley Fault Using Stochastic Source Modeling and Okada Fault Displacement Equations" *GeoHazards* 3, no. 2: 277-293.
https://doi.org/10.3390/geohazards3020015