# Twenty-Five Years of Probabilistic Fault Displacement Hazard Assessment

A. Valentini, F. Visini, P. Boncio, O. Scotti, Stephane Baize

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A. Valentini, F. Visini, P. Boncio, O. Scotti, Stephane Baize. Twenty-Five Years of Probabilistic Fault Displacement Hazard Assessment. Reviews of Geophysics, 2025, 63, pp.e2024RG000875. 10.1029/2024RG000875. hal-05137411

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# Reviews of Geophysics®

## REVIEW ARTICLE

10.1029/2024RG000875

**Key Points:**

* We review all existing Probabilistic Fault Displacement Hazard Assessment (PFDHA) models, highlighting their strengths, limitations, and applicability to different fault types

* Standardizing PFDHA terminology is essential to improve clarity, consistency, and comparability across studies

* Advancing PFDHA requires enriched databases and physics-based modeling

**Correspondence to:**

A. Valentini,
alessandro.valentini@univie.ac.at

**Citation:**

Valentini, A., Visini, F., Boncio, P., Scotti, O., & Baize, S. (2025). Twenty-five years of probabilistic fault displacement hazard assessment. *Reviews of Geophysics*, *63*, e2024RG000875. https://doi.org/10.1029/2024RG000875

Received 28 FEB 2025

Accepted 16 MAY 2025

**Author Contributions:**

**Conceptualization:** A. Valentini,
F. Visini, P. Boncio, O. Scotti, S. Baize

**Data curation:** A. Valentini

**Formal analysis:** A. Valentini

**Funding acquisition:** A. Valentini

**Investigation:** A. Valentini

**Methodology:** A. Valentini

**Project administration:** A. Valentini

**Resources:** A. Valentini, P. Boncio,
O. Scotti, S. Baize

**Software:** A. Valentini, F. Visini

**Supervision:** A. Valentini

**Validation:** A. Valentini, O. Scotti

**Visualization:** A. Valentini, P. Boncio,
S. Baize

**Writing – original draft:** A. Valentini

**Writing – review & editing:** A. Valentini,
F. Visini, P. Boncio, O. Scotti, S. Baize

<figure><img src="image_3.png" /></figure>

## Twenty-Five Years of Probabilistic Fault Displacement Hazard Assessment

A. Valentini¹ ID, F. Visini², P. Boncio³,⁴ ID, O. Scotti⁵, and S. Baize⁵ ID

¹Department of Geology, University of Vienna, Vienna, Austria, ²Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, Italy, ³Dipartimento di Ingegneria e Geologia, Università degli Studi “G. d’Annunzio”, Chieti-Pescara, Italy, ⁴UdA-TechLab, Research Center, University “G. d’Annunzio”, Chieti-Pescara, Italy, ⁵Autorité de Sureté Nucléaire et de Radioprotection, Paris, France

**Abstract** Fault displacement hazard poses significant risks to critical infrastructure such as buildings, roads, and pipelines. While some structures can tolerate a certain degree of surface displacement, others, such as buildings, are far more vulnerable. Probabilistic Fault Displacement Hazard Assessment (PFDHA) quantifies the probability of exceeding a certain level of displacement at a site due to surface-rupturing earthquakes. A PFDHA model consists primarily of two components: the surface rupture probability and the fault displacement models (FDMs). FDMs distinguish between principal fault rupture and distributed rupture. Principal fault rupture occurs along the primary fault plane, where seismic energy is primarily released during an earthquake. In contrast, distributed rupture refers to all other tectonic surface ruptures occurring away from the principal fault. This differentiation was first introduced during the pioneering PFDHA study conducted for the Yucca Mountain nuclear waste repository (Nevada, USA) in the early 2000s. Since then, the methodology has been applied to various fault types, including strike-slip and reverse faults. In recent years, the number of new models and the awareness of PFDHA has grown thanks to several international initiatives such as the worldwide and unified database of surface ruptures, the fault displacement hazard initiative, and the benchmark project led by the International Atomic Energy Agency. With growing interest and rapid advancements in this field, this article aims to provide a complete overview of all published models, discuss their applicability and limitations, standardize the terminology, and outline the necessary developments to guide future research.

**Plain Language Summary** Earthquakes can break the surface, leading to ground displacement that can damage buildings, roads, pipelines, and other critical infrastructure. Understanding and assessing how much and where the ground might break the surface is essential for reducing risks. Probabilistic Fault Displacement Hazard Assessment (PFDHA) is a method used to estimate the likelihood and extent of surface rupture due to earthquakes. Since its introduction, PFDHA has evolved with new models and data sets, but significant challenges remain. This review compiles all PFDHA models, discussing their strengths, limitations, and applicability to different fault types. We highlight the need for standardized terminology, improved databases, and the integration of advanced techniques such as remote sensing, machine learning, and physics-based modeling. Additionally, we emphasize the importance of developing a unified computational framework to ensure reproducibility and consistency in hazard assessments. By addressing these challenges, the field can progress toward more reliable predictions, ultimately improving the resilience of infrastructure and communities in earthquake-prone regions.

## 1. Introduction

Fault Displacement Hazard Assessment (FDHA), or analysis, forecasts future ground surface displacements caused by earthquake fault ruptures at a specific location or across a region. FDHA is typically used for decision-making in societal or industrial contexts, guiding applications such as infrastructure design, land-use planning, or assessing the risk for critical facilities like pipelines, bridges, dams, and nuclear power plants. Indeed, in the International Atomic Energy Agency (IAEA) Safety Standards Series No. SSR-1 (IAEA, 2019), fault displacement hazard assessment is required for the siting of nuclear installations. Moreover, the IAEA defines a capable fault as one with significant potential for producing displacement at or near the ground surface (IAEA, 2022). The IAEA recommends considering alternative sites if any capable faults exist at the candidate site. However, due to the increasing number of palaeoseismological investigations, capable faults may be found in existing sites. In such a case, the IAEA recommends evaluating the safety using probabilistic FDHA (PFDHA).

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<figure><img src="image_4.png" /><figcaption>**Figure 1.** Timeline of published studies on Probabilistic Fault Displacement Hazard Assessment (PFDHA) from 2000 to 2025. Key publications contributing to developing PFDHA models, methods, or applications are represented here. The arrow indicates the chronological progression of research, highlighting the growing interest and advancements in the field over the past 25 years.</figcaption></figure>

This approach incorporates all possible fault ruptures, accounting for their probabilities in terms of time, location, and magnitude of displacement.

PFDHA can estimate the probability of expected surface displacement and, in the last decades, is increasingly employed (and developed) for its ability to provide a more comprehensive assessment of hazards (Figure 1). Despite the uncertainties inherent in fault rupture processes, societal decisions often require estimates of potential surface displacement to inform rational decision-making, such as designing resilient infrastructure or avoiding high-risk zones. However, decision-making must balance scientific assessments with non-scientific factors, including economic feasibility, logistical constraints, and political considerations. For instance, while large fault displacements are possible in many regions, designing all infrastructure to withstand the most extreme scenarios may not be economically viable. In such cases, PFDHA can rank displacement potential, guiding land-use planning and engineering requirements.

The first example of applying a PFDHA approach was introduced in the early 2000s with the pioneering analysis conducted for the Yucca Mountain (Nevada, USA) nuclear waste repository (e.g., Stepp et al., 2001; Youngs et al., 2003). Since then, the approach has been applied to other fault types such as strike-slip faulting (e.g., Petersen et al., 2011), reverse faulting (e.g., Moss & Ross, 2011), and strike-slip and reverse faulting in Japan (e.g., Takao et al., 2013). In the last years, the number of models and the awareness of PFDHA has grown rapidly (Figure 1) thanks to several international initiatives such as the worldwide and unified database of surface ruptures (SURE database, Baize et al., 2020; Nurminen et al., 2022), the Fault Displacement Hazard Initiative (FDHI, Bozorgnia et al., 2021; Sarmiento et al., 2025), and the benchmark project led by the IAEA (Valentini et al., 2021, 2025). Due to the exponential growth in interest in this discipline and the recent evolution in the field regarding databases and methodologies, the review is timely.

Beyond its practical applications, PFDHA also holds significant scientific value, as it allows geologists and seismologists to test and refine their understanding of fault behavior, rupture processes, and the interaction of faults with the Earth's surface. This review paper aims to provide a complete overview of all published models to date, discuss their applicability and limitations, and propose a standardized terminology. A critical issue in PFDHA is the lack of standardized terminology to describe earthquake-induced surface ruptures. Terms such as “principal rupture,” “distributed rupture,” and “triggered slip” are often used interchangeably or with different meanings across the literature, leading to confusion and inconsistencies in hazard assessments. Standardizing this terminology is essential for improving communication among researchers, practitioners, and policymakers, ensuring that hazard assessments are clear, consistent, and actionable.

The article is organized into two main chapters, with a third chapter outlining the necessary developments that could guide future research. The first chapter provides an in-depth overview of the terminology and classification
of surface fault ruptures, detailing the key metrics used to describe these phenomena and compiling examples from global databases. The second chapter focuses on the methodologies and model components used in PFDHA, illustrating their application through real-world case studies involving critical infrastructures. By integrating these topics, this review paper aims to enhance our understanding of the processes that govern whether an earthquake rupture reaches the Earth's surface, the extent and distribution of displacement when it does, and the magnitude and spatial variability of such displacements along faults. Furthermore, it explores the diversity of empirical and analytical models in simulating the probability of surface rupture and predicting fault displacement, focusing on the uncertainties inherent in these assessments.

## 2. Earthquake Surface Fault Ruptures: Nomenclature, Classification, Worldwide Examples, Metrics, and Databases

### 2.1. Nomenclature and Classification of an Earthquake Surface Fault Rupture

To establish a clear framework for discussing earthquake-induced surface ruptures, it is helpful to examine the nomenclature historically adopted in the literature. A key distinction is often made between tectonic and non-tectonic deformation features, including ruptures, which are frequently categorized as primary and secondary effects. For example, McCalpin (1996) provides an overview of these terms, including definitions that differentiate tectonic faulting (primary) from secondary effects such as gravitational sliding or liquefaction. Similarly, the ESI Intensity Scale proposed by Michetti et al. (2007) also addresses these classifications and largely aligns with McCalpin's summary. Historical perspectives add further nuance, with Solonenko (1970, 1973) and Nikonov (1988, 1995a, 1995b) proposing classifications based on the interaction of seismic and gravitational forces. Solonenko (1979), for instance, describes “seismogenic residual deformations” as the result of both seismic (primary-seismotectonic) and gravitational forces, distinguishing between regional, zonal, and local seismotectonic deformations. In this framework, surface faulting is classified as a “local seismotectonic deformation.” Similar terminology has also been explored by Hagiwara (1982) and Richter (1958), discussing primary tectonic ruptures in the context of fault activity. These definitions form the foundation for understanding surface faulting and its differentiation from secondary, non-tectonic phenomena.

Earthquakes can produce various surface ruptures, but it is important to distinguish between tectonic and non-tectonic surface ruptures (Figure 2). Tectonic surface rupture, often called surface fault rupture or surface faulting in earthquake geology, is a coseismic surface effect directly caused by the movement along a fault during an earthquake. This phenomenon should not be confused with non-tectonic ruptures, which are associated with processes such as gravitational sliding, liquefaction, or other forms of non-tectonic ground deformation. In this review paper, the term “Earthquake Surface Fault Ruptures,” refers only to the phenomenon where the ground breaks and slips along a fault plane during an earthquake as shown in Figure 2. Tectonic deformation, such as tilting, subsidence, and rotation, which can occur during an earthquake or fault creep, is generally not included in fault displacement hazard assessment. Nevertheless, the scale factor is crucial, as tilting and rotation commonly accompany earthquake surface fault rupture near the fault trace. In addition, where the fault plane terminates very close to the topographic surface but does not breach it, localized tilting (i.e., warping), folding in pressure ridges, or rotation can be the only macroscopic expression of earthquake surface fault rupture. For example, one could compare the sharp surface rupture along the Kekerengu fault and the localized warping and rotation along the Papatea fault during the 2016 Kaikoura earthquake (Stirling et al. (2017) reported in their Figures 8 and 11, respectively). Both can be classified as earthquake surface fault ruptures. Therefore, other than fault creep, the primary distinction between tectonic features that are for consideration in FDHA compared to features that are not, is whether the deformation is localized onto discrete ruptures or within narrow deformation zones (i.e., from less than a few meters to tens of meters), rather than bulk deformation distributed across wide areas (i.e., for several hundreds of meters to kilometers).

Earthquake surface fault ruptures are commonly classified as principal and distributed (Coppersmith & Youngs, 2000; Youngs et al., 2003) (Figure 3). The authors defined the principal fault rupture as the slip along the main fault plane (or planes) of crustal weakness responsible for releasing seismic energy during the earthquake. In cases where the principal fault rupture reaches the Earth's surface, it can manifest as either displacement along a single, narrow line (i.e., discrete), or multiple, closely spaced (sub)parallel discrete lines (i.e., cumulative) or as a broad deformation zone (i.e., large-aperture) that can span from a few meters to several meters in width. The term large-aperture (Baize et al., 2020; Nurminen et al., 2022), also called wide-aperture (Sarmiento et al., 2024), refers
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Flowchart illustrating the classification of earthquake surface effects. Surface effects are divided into primary (tectonic origin) and secondary (non-tectonic origin) categories. Within the primary effects, coseismic surface faulting is further distinguished into principal rupture—occurring directly on the main fault plane—and distributed rupture, which includes secondary surface breaks away from the principal fault. This review paper focuses on principal and distributed ruptures, which are critical components of Probabilistic Fault Displacement Hazard Assessment.</figcaption></figure>

to the method used for measuring the displacement, with aperture being the length over which fault displacement is measured. Typically, large-aperture displacement is calculated using change detection or digital differencing methods (e.g., Milliner et al., 2015). Large-aperture measurements can capture displacement on discrete rupture (s) and inelastic or non-brittle deformation. The aperture of interest for FDHA is typically within the scale of the “narrow fault zone” previously defined. Large-aperture measurements are increasingly common, and recent databases report the aperture within which the displacement is measured and the method used for measuring it. For example, in Sarmiento et al. (2024), the measurement aperture is classified into wide (100s of m), medium (10s of m) and narrow (≤10 m). However, for several historical surface ruptures, the method used for measuring the displacement is not specified, and the reported displacement can be either discrete, cumulative or large-aperture (See Section 2.4 for more details).

Distributed fault ruptures are defined as displacement that occurs on other faults, shears, splays, or fractures near the principal rupture in response to the principal faulting (i.e., close to the principal fault). Distributed faulting includes activation of major secondary faults splaying from the principal fault. Distributed fault ruptures are discontinuous and span a region extending from several tens of meters to many kilometers (up to 10⁴ m) from the principal rupture. They can appear at locations with or without prior evidence of faulting. Nevertheless, ruptures without previous surface expression are often observed in unconsolidated sediments (e.g., Teran et al., 2015 and references therein). The IAEA adopted the distinction between principal and distributed ruptures (IAEA, 2021; IAEA, 2022; Valentini et al., 2021), which provides siting recommendations based on this classification. It is important to note that within IAEA documentation, the terms “primary” and “secondary” ruptures are consistently used to refer to what are commonly known as “principal” and “distributed” ruptures in other literature. These terms are synonyms, and special attention should be paid to this equivalence when comparing guidelines and
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Block diagram illustrating principal and distributed ruptures in a normal faulting tectonic setting. The diagram highlights the spatial distribution of surface ruptures, with the principal rupture (PF) occurring along the main fault and distributed ruptures (DR) appearing as secondary surface breaks, typically in the hanging wall and footwall areas. This figure exemplifies the distinct patterns of fault displacement considered in Probabilistic Fault Displacement Hazard Assessment.</figcaption></figure>

scientific works. To ensure clarity and consistency, in this paper, we will exclusively use the terms “principal” and “distributed” moving forward.

Several factors contribute to the variability in surface rupture patterns, such as the style of faulting, fault maturity, seismological characteristics, and local geologic structures or lithology. In some cases, triggered slip, i.e. the movement along a fault induced by the dynamic or static stress changes and permanent deformation, caused by a nearby or distant earthquake, has been considered a form of distributed displacement (e.g., Gurpinar et al., 2017). In contrast to this tectonics-based definition, Petersen et al. (2011), for strike-slip faults, defined as triggered every rupture when it occurs at more than 2 km from the principal fault and separated them from other distributed ruptures in their data set. Petersen et al. (2011) did not include these kinds of data in the empirical models used to estimate the occurrence of distributed fault ruptures.

Further subdivisions based on driving mechanisms or geological structure are also found in the literature (e.g., Nurminen et al., 2022). Nurminen et al. (2020, 2022) introduced several sub-ranking categories for distributed ruptures to account for structural complexities. The authors defined six ranks. Ranks 1 and 2 define the simplest case of principal and distributed ruptures, respectively. In this case, the distributed ruptures directly respond to the principal fault surface rupture. Instead, if distributed ruptures occur on major pre-existing secondary faults for which geological data (e.g., geologic maps, cross-sections, seismicity) indicate they are splaying from the principal fault, they are ranked as 1.5, namely “primary distributed rupture.” Typically, Rank 1.5 ruptures occur on structures that can be recognized and mapped before the earthquake, as there can be geologic and geomorphic evidence of long-term activity. Rank 3 is used for all distributed ruptures on triggered faults. Additionally, the authors defined all complex distributed ruptures related to local structural settings, such as bending-moment (Rank 21) and flexural-slip (Rank 22) distributed ruptures that could occur in reverse faulting earthquakes.
## 2.2. Documented Cases of Earthquake Surface Fault Ruptures Affecting Critical Infrastructure Worldwide

Surface ruptures caused by earthquakes can devastate critical infrastructure, such as bridges, tunnels, pipelines, and dams. The risk posed by surface fault ruptures is well-recognized in international guidelines, including the Alquist-Priolo Earthquake Fault Zoning Act in California (Bryant & Hart, 2007), the New Zealand guidelines on earthquake fault avoidance (Kerr et al., 2003), the IAEA standards (IAEA, 2019; IAEA, 2022), the Italian land use guidelines for areas with active and capable fault zones (Technical Commission on Seismic Microzonation, 2015), the U.S. Federal Guidelines for Dam Safety (FEMA, 2015), the Italian seismic safety regulations for dams (DMIT, 2014), and Japan local regulations for specific active fault investigation areas (Toda, 2015), all of which emphasize the importance of avoiding critical construction near active faults. Despite these precautions, numerous documented cases worldwide highlight the destructive effects of surface fault ruptures on infrastructure, underscoring the need for continuous evaluation of design practices and mitigation strategies. Here, we present a selection of documented cases, illustrating their impact on critical infrastructure globally (Table 1, Figure 4). These documented cases highlight the destructive potential of surface ruptures, both principal and distributed, on critical infrastructure worldwide. While international guidelines stress fault avoidance and mitigation measures, infrastructure persistence in fault zones necessitates rigorous design adaptations. Moreover, for effective hazard mitigation, it is crucial to address documentation gaps and to account for both principal and distributed fault ruptures during infrastructure planning.

One of the most well-documented success stories in fault displacement hazard mitigation is the case of the Trans-Alaska Pipeline during the moment magnitude (Mw) 7.9 2002 Denali earthquake (e.g., Honegger et al., 2004; Sorensen & Meyer, 2003). The earthquake produced a major surface rupture along the Denali and Totschunda faults, with a maximum right-lateral displacement exceeding 8 m and significant vertical displacement in some sections (e.g., Eberhart-Phillips et al., 2003; Haeussler et al., 2004). Despite crossing the Denali Fault in a region of high displacement (~4.3 m horizontal and ~0.7 m vertical displacement, which was within the design criteria; Honegger et al., 2004), the pipeline remained fully operational, avoiding a catastrophic oil spill. This resilience was primarily due to careful engineering design based on fault displacement hazard assessment. The pipeline was built with a geotechnical mitigation strategy that accounted for potential large surface displacements. Instead of being buried, it was placed on an above-ground slider system (Figure 5), allowing it to accommodate fault movement through flexibility and controlled deformation. The design included anchor points on stable ground and elevated supports that permitted lateral movement, enabling the pipeline to withstand the large displacements imposed by the fault rupture (Figure 5). The successful performance of the Trans-Alaska Pipeline highlights the effectiveness of engineering solutions in mitigating fault displacement hazards when avoidance is not an option. This case demonstrates the importance of integrating fault displacement hazard assessments into infrastructure design, ensuring that structures in fault-prone areas can accommodate potential ground deformations without loss of function.

## 2.3. Fault Displacement Metrics

The selection of design displacement inputs and hazard characterization must consider the complexity of fault geometry and displacement components because different infrastructure types are sensitive to these components differently. The displacement metrics for current models include the displacement components and the related amplitude of slip (e.g., lateral, vertical, or net), and the width, or aperture, across which is accommodated the displacement (e.g., localized on discrete faults or cumulative across multiple faults plus inelastic deformation between discrete faults). Understanding the displacement metrics used in assessing fault displacement hazard is necessary to apply the results to engineering analyses correctly, compare results between models, and appropriately use alternative models to capture epistemic uncertainty.

Earthquake surface fault ruptures produce three-dimensional ground surface displacement, and the displacement vector is the three-dimensional component of displacement (Figure 6). The full vector is the net displacement, and it consists of three orthogonal components: fault-parallel horizontal (i.e., lateral or strike), fault-perpendicular horizontal (i.e., fault-normal, heave, or shortening/lengthening), and vertical (i.e., throw). Components of this vector are measured (in data) and/or predicted (in models). The empirical data available to constrain displacement models (e.g., Nurminen et al., 2022; Sarmiento et al., 2024) do not always include or specify all the displacement components of a reported field measurement of surface fault displacement, and therefore modelers make
<table><caption>**Table 1**

*List of Documented Cases of Infrastructure Affected by Principal and Distributed Earthquake Surface Fault Ruptures*</caption><thead><tr><th>Date</th><th>Mw</th><th>Location</th><th>Infrastructure</th><th>Type of rupture</th><th>Comments</th></tr></thead><tbody><tr><td>18/04/1906</td><td>7.9</td><td>San Francisco (USA)</td><td>Alder Creek Bridge</td><td>Principal</td><td>The Alder Creek Bridge sustained severe damage during the 1906 San Francisco earthquake due to direct rupture along the principal fault, demonstrating one of the earliest documented cases of fault-induced bridge failure</td></tr><tr><td>14/10/1968</td><td>6.5</td><td>Meckering (Australia)</td><td>Water pipeline</td><td>Principal</td><td>A water pipeline sustained major damage during the earthquake, demonstrating the vulnerability of buried utilities to fault movement</td></tr><tr><td>17/08/1999</td><td>7.6</td><td>Izmit (Turkey)</td><td>Bridges</td><td>Principal</td><td>The surface fault rupture of the Izmit earthquake intersected five bridges, causing severe structural damage and having significant impacts on transportation networks. One example is shown in Figure 4</td></tr><tr><td>20/09/1999</td><td>7.7</td><td>Chi-Chi (Taiwan)</td><td>Bridges and Dam</td><td>Principal</td><td>The Pi-Feng Bridge (Figure 4) and several others experienced substantial damage during the Chi-Chi earthquake. At the Shih Kang Dam, the fault rupture caused a vertical offset of approximately 9 m, resulting in catastrophic damage to the dam structure (Figure 4)</td></tr><tr><td>12/11/1999</td><td>7.2</td><td>Düzce (Turkey)</td><td>Viaduct</td><td>Principal</td><td>The Bolu Viaduct was significantly affected by fault rupture during the Düzce earthquake, highlighting the vulnerability of large viaducts in active fault zones</td></tr><tr><td>3/11/2002</td><td>7.9</td><td>Denali (USA)</td><td>Oil Pipeline</td><td>Principal</td><td>The Trans-Alaska Pipeline was built to accommodate fault movement, and its survival during the Denali earthquake remains a prime example of effective engineering mitigation for fault rupture hazards (Figure 5)</td></tr><tr><td>12/05/2008</td><td>7.9</td><td>Wenchuan (China)</td><td>Bridge</td><td>Principal</td><td>The Xiaoyudong Bridge was heavily damaged due to fault displacement. A full reconnaissance report on the damage to bridges during this earthquake provides further insights into the mechanisms of fault rupture impact (Kawashima et al., 2009)</td></tr><tr><td>30/10/2016</td><td>6.5</td><td>Norcia (Italy)</td><td>Tunnel</td><td>Distributed</td><td>A tunnel ~7 km far from the principal fault sustained damage due to distributed faulting during the Norcia earthquake, emphasizing how secondary fault deformation can affect underground structures (Figure 4)</td></tr><tr><td>13/11/2016</td><td>7.8</td><td>Kaikoura (New Zealand)</td><td>Roads, railways, dock facilities</td><td>Principal and Distributed</td><td>Distributed deformation caused widespread damage to roadways and dock facilities. The event was extensively documented in resources such as Nature's Pic Images and earthquake reconnaissance reports</td></tr><tr><td>07/01/2022</td><td>6.7</td><td>Qinghai (China)</td><td>Railway</td><td>Principal</td><td>Faulting during this earthquake led to severe damage on the Liuhuanggou bridge that was tilted with uneven movements (Figure 4)</td></tr><tr><td>6/02/2023</td><td>7.8</td><td>Kahramanmaraş (Turkey)</td><td>Gas pipelines, Railway</td><td>Principal</td><td>Several gas pipelines were ruptured along the intersection with the principal fault, an example is given in Figure 4</td></tr><tr><td>01/01/2024</td><td>7.5</td><td>Noto (Japan)</td><td>Tunnel</td><td>Distributed</td><td>Distributed faulting in a tunnel was documented (Figure 4), though not yet published</td></tr></tbody></table>

Note. This list is not exhaustive.
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Examples of infrastructure damage caused by principal (solid red lines) and distributed (dashed red lines) fault ruptures. The images showcase different types of critical infrastructure, such as roads, pipelines, railways, and dams, impacted by earthquake surface displacement. These examples underscore the importance of considering both rupture types in fault displacement hazard assessments. Earthquakes images are from: Yang and Mavroeidis (2018) for collapsed bridges during the Chi-Chi and Izmit earthquakes; Nuss et al. (2012) for a dam collapsed during Chi-Chi earthquake; Lettis et al. (2000) for railway dislocated during Izmit earthquake; Yang et al. (2022) for Qinghai earthquake (the left panel shows the location of the principal fault relative to the damaged railways shown in the right panel); Galli et al. (2020) for Norcia earthquake; Unal et al. (2024) for Kahramanmas earthquake.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Map of Alaska showing the Denali Fault trace and the Trans-Alaska Pipeline route. In the center, two photographs show a section of the pipeline before and after the 2002 Denali earthquake, viewed from the south. Below, a schematic representation illustrates the pipeline alignment and supporting beams before and after the earthquake, highlighting the deformation at the fault crossing. The lateral shift and bending of the pipeline segment result from fault movement, which induced compression on the structure (for more details see, Honegger et al., 2004).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Block diagram illustrating the possible components of slip during an earthquake on a fault.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Example map of principal and distributed ruptures (red and blue lines) with unevenly distributed displacement measurement locations (red and blue circles) from a hypothetical earthquake. The x/L reference line indicates the nominal along-strike trajectory of the overall rupture. Displacement can be defined with respect to combined principal displacement (i.e., sum of the principal displacements in example zone) or aggregate displacement (i.e., sum of the principal and distributed displacements in example zone) (from Sarmiento et al., 2025).</figcaption></figure>

decisions regarding how to interpret historical empirical data in terms of displacement components, and what displacement component or components can be reasonably predicted by their model. Suppose displacement models provide relationships to predict one component of displacement (e.g., vertical displacement in the case of Youngs et al., 2003). In that case, practitioners should consider whether and/or how to adjust the hazard results from a single displacement component to net displacement.

The aperture of displacement refers to the general width, as measured normal to the strike of the surface rupture in map view, over which displacement is summed. Field-based displacement measurements are generally collected across discrete fault strands (e.g., offset on a single rupture plane) or fault zones (e.g., offset across a zone a few to tens of meters in width). Complex surface ruptures may contain more than one discrete rupture trace at a location, and thus, displacements measured perpendicular to the fault strike may be summed across more than one discrete fault (Figure 7). The current models (Moss et al., 2024; Petersen et al., 2011; Takao et al., 2013; Youngs et al., 2003) mainly address discrete fault displacement. While some models (e.g., Moss & Ross, 2011; Takao et al., 2013) could include data that represent cumulative fault displacement across relatively narrow apertures (e.g., the vertical displacement from scarp profiles across broad folds in reverse-faulting earthquakes), these data were treated as discrete fault displacement in the model development. As a result, the effect of surface rupture complexity on the amount of fault displacement, such as the number of fault ruptures or the amount of discrete displacement on individual ruptures, is not captured in most models (e.g., Moss et al., 2024; Petersen et al., 2011; Takao et al., 2013).

As an alternative, some recent models (e.g., Chiou et al., 2025; Lavrentiadis & Abrahamson, 2023) provide predictions for the combined discrete displacement on all ruptures by performing an additional data processing step to combine (sum) discrete fault displacement measurements on (sub)parallel ruptures across a zone as demonstrated in Figure 7. Displacement from distributed faults may or may not be included. When displacements from only principal ruptures are considered, this is referred to as the sum of principal displacement (e.g., Chiou et al., 2025); when distributed displacements are included, this is called aggregate displacement (e.g., Kuehn et al., 2024). Figure 7 shows an example of a complex surface rupture pattern with (sub)parallel principal ruptures. Principal fault displacement measurements on different ruptures are projected onto a centerline, effectively treating measurements on different faults as adjacent along a single nominal strike line.
<table><caption>**Table 2**
Databases Employed by Empirical Models</caption><thead><tr><th rowspan="2">Model</th><th rowspan="2">Probability surface rupture given a magnitude m</th><th rowspan="2">Probability surface rupture at a specific site location for distributed ruptures</th><th colspan="2">Fault displacement model</th></tr><tr><th>Principal</th><th>Distributed</th></tr></thead><tbody><tr><td>Wells and Coppersmith (1993)</td><td>276 global earthquakes</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Youngs et al. (2003)</td><td>One model with 105 extensional Cordillera earthquakes<br>One model with 32 Great Basin earthquakes<br>One model with 47 northern Basin and Range earthquakes</td><td>One model with 5 Basin and Range events from Wheeler (1989)<br>Two models with 11 global events from McCalpin and Slemmons (1998)</td><td>13 extensional Cordillera events from Pezzopane and Dawson (1996)</td><td>5 events from dePolo (1994)</td></tr><tr><td>Petersen et al. (2011)</td><td>—</td><td>13 global events from Wesnousky (2008) plus 7 global events</td><td>9 global events</td><td>8 global events</td></tr><tr><td>Moss and Ross (2011)</td><td>129 global earthquakes</td><td>—</td><td>9 events from Wesnousky (2008) and Kaneda (2008)</td><td>—</td></tr><tr><td>Moss et al. (2013)</td><td>126 reverse and 140 strike-slip global earthquakes</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Takao et al. (2013, 2014, 2016)</td><td>107 Japanese earthquakes</td><td>19 Japanese earthquakes</td><td>17 Japanese earthquakes</td><td>17 Japanese earthquakes</td></tr><tr><td>Yang et al. (2021)</td><td>11 Australian earthquakes</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Ferrario and Livio (2021)</td><td>—</td><td>21 global events</td><td>—</td><td>—</td></tr><tr><td>Pizza et al. (2023)</td><td>363 global earthquakes</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Lavrentiadis and Abrahamson (2023)</td><td>—</td><td>—</td><td>74 global events from FDHI database (Sarmiento et al., 2024)</td><td>—</td></tr><tr><td>Rodriguez Padilla and Oskin (2023)</td><td>—</td><td>5 strike-slip events from FDHI database (Sarmiento et al., 2024)</td><td>—</td><td>—</td></tr><tr><td>Chiou et al. (2025)</td><td>—</td><td>—</td><td>29 global events from FDHI database (Sarmiento et al., 2024)</td><td>—</td></tr><tr><td>Moss et al. (2024)</td><td>—</td><td>—</td><td>25 global events from FDHI database (Sarmiento et al., 2024)</td><td>—</td></tr><tr><td>Kuehn et al. (2024)</td><td>—</td><td>—</td><td>73 global events from FDHI database (Sarmiento et al., 2024)</td><td>—</td></tr><tr><td>Visini et al. (2025)</td><td>—</td><td>34 global events from SURE 2.0 database (Nurminen et al., 2022)</td><td>—</td><td>34 global events from SURE 2.0 database (Nurminen et al., 2022)</td></tr></tbody></table>

## 2.4. Existing Earthquake Surface Fault Rupture Database

The main data sets used individually by each PFDHA model (Table 2), such as the Pezzopane and Dawson (1996) data set used by Youngs et al. (2003), Wesnousky (2008) used by Petersen et al. (2011), and the Lettis et al. (1997) data set used by Moss and Ross (2011), have significant differences in their input data sets, estimated displacement metrics, modeling techniques, and treatment of uncertainty. For this reason, two international projects have been carried out in recent years to provide distinct homogeneous databases to be used for FDHA: the Surface Rupture due to Earthquakes (SURE; Baize et al., 2020, and its update, SURE 2.0; Nurminen et al., 2022) database, and the Fault Displacement Hazard Initiative (FDHI; Bozorgnia et al., 2021; Sarmiento et al., 2024) database. Those two databases, even if distinct in terms of terminology, metrics, and structure, share common cases of historical earthquakes.
The SURE project is an international effort started in 2015 and was coordinated by the Institut de Radioprotection et de Sûreté Nucléaire (IRSN), now Autorité de sûreté nucléaire et de radioprotection (ASNR). Its main aim was to constitute a worldwide and unified fault displacement database to improve further hazard estimations, to unify the existing data sets (field-based slip measurements) to incorporate recent and future cases, and to include new parameters relevant to adequately describing the rupture. The second version of the SURE database (SURE 2.0, Nurminen et al., 2022) supersedes the first version (Baize et al., 2020). It compiles data from 50 crustal earthquakes, mainly from the USA, with hypocenter depths not exceeding 25 km. The database covers three primary types of kinematics, with roughly equal representation: 16 strike-slip, 18 normal, and 16 reverse earthquakes. As of now, the database boasts a wealth of over 75,000 individual surface rupture traces and nearly 19,000 slip observations. These earthquakes span from the late 19th century to 2019, varying in Mw from 4.9 to 7.9. The SURE 2.0 database differs from its first release including new cases (such as the 2019 Mw 4.9 Le Teil earthquake) and reconsidering some cases omitted because the data were not of sufficient quality. Indeed, fault mapping precision has been significantly improved across most cases. A significant innovation introduced in this version is the application of fault ranking to all recorded events. This feature infuses geological insights into the database by categorizing surface ruptures based on their underlying geological structures. The SURE 2.0 database is structured around three principal tables: a “displacement observation point table,” a “rupture section table,” and an “earthquake table.” Each georeferenced displacement observation point is linked to the “rupture section table” and the associated “earthquake table” via a designated identifier, resulting in a one-to-many record structure. The core of the database lies in the “displacement observation table,” which contains essential data for fault displacement hazard assessment. It incorporates comprehensive observations for the earthquake set, ensuring the inclusion of important details such as latitude, longitude, and net slip for each measurement. For optimal design considerations, slip measurements are documented in horizontal and vertical components with their respective uncertainties. In SURE, the authors distinguish between principal (Rank 1) and distributed faulting (Rank 2). Moreover, they introduced four sub-ranking categories for distributed ruptures to account for structural complexities. In additions to Ranks 1 and 2, the authors defined Rank 1.5 ruptures as the ones that occur on structures that can be recognized and mapped before the earthquake, Rank 3 for all distributed ruptures on triggered faults, and Rank 21 and 22 as bending-moment and flexural-slip distributed ruptures that could occur in reverse faulting earthquakes.

The FDHI project is a multi-year and community-based research project that started in 2018, was funded by various governmental and private agencies, and was coordinated by the University of California, Los Angeles. This project's core mission is to create a state-of-the-art fault displacement database and associated models, aimed at estimating the extent and spatial distribution of both principal and distributed displacements in surface-rupturing earthquakes. The FDHI project management allowed a modeler community to work on a shared database, comparing their advancement and results based on periodic meetings. Within this initiative, a geospatially managed relational database, encompassing surface rupture maps, measurements, and related metadata, covering 75 historical earthquakes globally has been developed—the events in the database range from Mw 4.9 to 8.0 between 1872 and 2019. About 40% of the earthquakes in the data set are from Western North America, with an additional quarter of the events sourced from Japan, China, or Southeast Asia. A collection of events from Australia, a stable continental region, is also included. The earthquakes within this database represent a diversity of tectonic regimes, with roughly 45% characterized as primarily strike-slip, 20% as normal, and 35% as reverse events. Each earthquake in the database is accompanied by a detailed set of event-specific metadata, including information such as magnitude, magnitude type, hypocenter location, and faulting style. Moreover, dataset-level metadata, including citation and mapping scale, is documented. The database encompasses over 87,000 geospatially referenced point observations, comprising over 40,000 fault displacement measurements across various slip components. Another feature of this database is the classification of surficial geologic units, which includes distinctions such as bedrock, and young/old/undifferentiated alluvium. The database catalogs such classifications for over 26,000 observation sites. Furthermore, the database provides detailed surface rupture maps for each earthquake event, meticulously pinpointing rupture line vertices with precise geospatial referencing. In response to the model development teams' requirements, geological interpretation of the rupture trace and displacement measurements have been performed, including distinguishing between principal and distributed faulting, consolidating reported slip components into recommended net slip values for model development, and explicitly noting alternative measurements at the same location.
It is worth noting that both databases over-represent data from northwestern America and Japan, while the Euro-Mediterranean region remains relatively underrepresented. Additionally, there are a low number of events with magnitudes Mw 6 and below, partly because their likelihood of generating surface ruptures is lower. This is further influenced by the significant weight of historical cases in the databases, as before the 2010s, there was limited interest in documenting events around Mw 6. Another important consideration is that the SURE team modified the original data sets for processing purposes. These included creating very short rupture sections beneath “orphan” measurement points and adjusting point measurement locations when coordinates did not align with rupture sections. In contrast, the FDHI team opted not to modify the original data sets in any way. These differences in methodology and data representation are critical for understanding the structure and limitations of these databases.

In addition to these two databases, a recent study (i.e., Milliner et al., 2024) has provided remote sensing observations of 20 historic surface-rupturing continental strike-slip earthquakes (6.1 ≤ Mw ≤ 7.9), utilizing a range of radar and optical satellite and aerial imagery processed with pixel tracking techniques, as well as interferometric synthetic aperture radar (InSAR) in selected cases. These data sets include surface displacement maps that constrain at least the horizontal component of motion (2D) and, for three events, the full 3D displacement field. Additionally, strain invariants, such as finite dilatational strain, strain magnitudes, vorticity, and maximum shear strain, have been derived for each event. The study also presents 134 manually mapped surface fault rupture traces and 2,648 objective measurements of coseismic horizontal fault-parallel surface slip, obtained through a swath profile function-fitting method. Data sets like this one can potentially enhance existing fault databases and provide valuable constraints for improving fault displacement models (FDMs) and probabilistic fault displacement hazard assessments.

# 3. Probabilistic Fault Displacement Hazard Assessment: Methods, Model Components, and Worldwide Application Examples

## 3.1. PFDHA Methodology

The methodology employed to evaluate fault displacement hazard is analogous to the well-developed formulation for probabilistic evaluation of the hazard due to earthquake-strong ground shaking (e.g., Cornell, 1968; Esteva, 1967; McGuire, 1995). The aim of PFDHA is to assess the probability of coseismic surface-fault displacement $D_0$ occurring at a site, given a specified level of exceedance probability $\lambda$, as follows:

$$ \lambda(D > D_0) = \alpha_{DE}P(D > D_0) \quad (1) $$

in which $\alpha_{DE}$ is the rate of displacement events at the site, and $P(D > D_0)$ is the probability that displacement D exceeds the threshold level $D_0$. Depending on the amount of data at or near the site of interest, two basic approaches can be followed as defined in the pioneering publication of Youngs et al. (2003): the earthquake approach and the displacement approach. The choice of the approach mainly depends on the amount of site-specific data. If there is enough data about the amount of displacement per event and age of events, to quantify the relationship between exceedance frequency and displacement amount, the displacement approach can be used to perform PFDHA. In contrast, the earthquake approach can be followed when insufficient site-specific data is available. In this case, the magnitude-frequency relationship estimates the exceedance rate for various displacement amounts at a site. Alternative approaches to PFDHA have been proposed, including methods based on stochastic source modeling (Goda, 2021; Goda & Shoaeifar, 2022). These approaches generate fault geometries and slip distributions using statistical earthquake source scaling relationships and earthquake slip synthesis, subsequently applying analytical solutions, such as the Okada equations (Okada, 1985), to estimate surface fault displacements in three dimensions. While these methods offer valuable insights, this review paper focuses on the earthquake-based approach, which has been the most widely used and extensively developed over the past 25 years (Figure 1). Its broad application and continuous refinement make it the standard framework for PFDHA.

### 3.1.1. Displacement Approach

Derived as a component of the Yucca Mountain study (Stepp et al., 2001; Youngs et al., 2003), the displacement approach relies solely on the observations and characteristics of fault displacement at a specific location to evaluate the hazard. Unlike the earthquake approach, it does not distinguish between principal and distributed
<figure><img src="image_11.png" /><figcaption>**Figure 8.** A Flowchart illustrates the key components of Probabilistic Fault Displacement Hazard Assessment (PFDHA) for the earthquake approach and highlights the essential data required for each component. PFDHA estimates the likelihood of occurrence of coseismic surface-fault displacement at a site, given a probability level of exceedance (following the equation in the upper box, where $\alpha_{DE}$ is the rate of displacement events at the site and $P(D > D_0)$ is the probability that displacement D exceeds the threshold level $D_0$).</figcaption></figure>

faulting and does not explicitly consider the causative seismic source. The rate of exceedance, denoted as v, for a given level of displacement, d, can be determined using the general formula:

$$v(d) = \lambda_{DE}P(D > d) \qquad (2)$$

In this equation, $\lambda_{DE}$ represents the rate of displacement events. $P(D > d)$ is the conditional probability that, given a slip on the fault, the single-event displacement D will exceed the value d. The rate of displacement, $\lambda_{DE}$, can be estimated based on the fault's slip rate (SR) and the average slip $D_E$ per faulting event (DE), where $\lambda_{DE} = SR/D_E$, or it can be directly calculated from the recurrence intervals ($R_{int}$) of paleoearthquake ages ($\lambda_{DE} = 1/R_{int}$). The probability term in Equation 2 can be derived from a database containing repeated slip events observed at the same location, or it can be computed using generic distributions for normalized displacement, along with an estimation of the average displacement. Examples of application of the displacement approach can be found in Youngs et al. (2003) and Angell et al. (2003).

### 3.1.2. Earthquake Approach

After Youngs et al. (2003), several groups proposed different earthquake approaches for different tectonic environments, considering regression equations updated with newly available data sets (e.g., Moss et al., 2024; Petersen et al., 2011; Takao et al., 2013; Visini et al., 2025). Generally, the input information for the earthquake approach is the seismic source characterization, equations and parameters of the surface rupture probability model and the FDM (Figure 8). In addition, depending on the seismotectonic setting and available data, additional components can be introduced, such as the uncertainty on fault location (Chen & Petersen, 2019; Ferrario et al., 2025; Petersen et al., 2011; Scott et al., 2023). Usually, the first step of PFDHA consists of identification and characterization of seismic sources to determine the rate of displacement events at the site ($\alpha_{DE}$ in Equation 1), which is computed with the overall rate of events $\alpha(m_{min})$ with magnitudes larger than a given minimum magnitude ($m_{min}$) and a probability density function $f_{m,s}$ that describes the joint probability of magnitude m earthquake occurring along the fault at distance s from the end of the fault (Figure 9), as follows:

$$\alpha_{DE} = \alpha(m_{\min}) \int_{m,s} f_{m,s}(m, s) dmds \qquad (3)$$
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Illustration showing how the rupture of length *L* for a magnitude *M* earthquake progresses or “floats” along the fault system by incrementally increasing *s*. Aleatory variability in the mapped fault location is represented with *f*<sub>R</sub>(*r*). The site can be situated (principal faulting) or not (distributed faulting) on the mapped fault.</figcaption></figure>

To implement Equation 3, all sources that are considered relevant for PFDHA must be characterized in terms of geometry (e.g., length, width, depth, dip angle), style-of-faulting (e.g., normal, reverse, strike-slip), and rate of deformation (e.g., slip rate, recurrence interval). These data are used to estimate the maximum magnitude, which is typically estimated by using source-scaling relations from either rupture length or rupture area, and the slip rate or the recurrence interval will be used to define the activity rate (i.e., the overall rate of earthquakes) of the fault (i.e., α(*M*<sub>min</sub>)). The magnitude recurrence model combines the activity rate with the magnitude-frequency distributions (MFDs), defining the relative likelihood of different magnitude earthquakes. Commonly used MFDs are the truncated exponential model (e.g., Cosentino et al., 1977; Gutenberg & Richter, 1944), the characteristic earthquake model (e.g., Schwartz & Coppersmith, 1984; Schwartz et al., 1981), and the composite magnitude model (e.g., Youngs & Coppersmith, 1985). As the characteristic and composite magnitude distribution models are more suitable for defining the seismicity on individual faults (IAEA, 2021), these models are more commonly used in PFDHA. Examples of source characterization for PFDHA are given in Nishizaka et al. (2024) and Valentini et al. (2025).

The second part of Equation 1, that is, *P*(*D* > *D*<sub>0</sub>), represents the probabilistic term, which is converted into an attenuation function for surface fault displacement using a set of conditional probabilities unique to PFDHA (Figure 8). The surface rupture probability model is defined as the probability that an earthquake with magnitude *m* produces surface rupture *sr* (*P*[*sr* ≠ 0|m|), with additionally the probability of having non-zero displacement, over a given site area *z*, for a magnitude *m* event that produces surface rupture *sr* (*P*[*d* ≠ 0|*z*, *r*, *sr* ≠ 0] if the site is located at distance *r* from the principal fault or *P*[*D* ≠ 0|*z*,*sr* ≠ 0] when the site is located on the principal fault, i.e. *r* = 0). The first term, *P*[*sr* ≠ 0|m|], is necessary because not all earthquakes result in surface rupture, and empirical data indicate a significant correlation between earthquake magnitude and the likelihood of surface rupture occurring. The second term is necessary because even in cases where an earthquake results in surface rupture, the fault rupture may not necessarily intersect/cross the site in question (e.g., Melissianos et al., 2023). Models that express the two conditional probabilities are called surface rupture probability models (Figure 8) because they are related to the probability of fault rupture, but do not predict fault displacement amount. The FDM is defined as the probability that the displacement *D* exceeds a test level *D*<sub>0</sub>, given a location parameter *l*\*, magnitude *m*, and the condition that rupture occurs at the site causing non-zero displacement, *P*[*D* > *D*<sub>0</sub>|*l*\*, *m*, *D* ≠ 0]. Models that describe the displacement amplitude (average and variability) that can express displacement exceedance are called FDMs (implicitly assuming that the displacement is at the ground surface). The available models for these terms are listed and discussed in the following sections.

It is worth noting that Takao et al. (2013) refer to the various components of PFDHA using labels such as P<sub>1p</sub>, P<sub>2p</sub>, P<sub>2d</sub>, etc. While this nomenclature is concise and convenient, it does not convey explicit information about what each term represents. Therefore, in this review paper, we have chosen to assign descriptive names to each component, aligning them with terminology used in different studies (e.g., Nurminen et al., 2020; Petersen
et al., 2011; Youngs et al., 2003) while also ensuring that the names reflect their function and purpose within the PFDHA framework.

Combining all terms and integrating over magnitudes and location distributions, Equation 1, after Petersen et al. (2011), can be written as follows:

$$ \lambda(D > D_0) = \alpha(m_{\min}) \int_{m,s} f_{m,s}(m,s) \times P[\mathrm{sr} \neq 0|m] \times \int P[\mathrm{D} \neq 0|z, r, sr \neq 0] \\ \times P[\mathrm{D} > D_0|l^*, m, D \neq 0] f_r(r) \mathrm{dmdsdr} \quad (4) $$

This general equation applies to both principal (Equation 5) and distributed faulting ruptures (Equation 6). In the first case, the location parameter $l^*$ is the along-rupture distance ratio $x/L$, where $L$ is the total surface rupture length on the fault source, $x$ is the distance from the nearest point on the principal fault to the closest end of the rupture (Figure 9), and $D$ and $D_0$ represent the displacement on the principal fault. In the second case, the location parameter is given by the distance $r$, which is the closest distance between the principal fault and the site of interest (Figure 9). Moreover, $D$ and $D_0$ are replaced by $d$ and $d_0$, which explicitly represent distributed displacement.

### Principal Fault

$$ \lambda(D > D_0) = \alpha(m_{\min}) \int_{m,s} f_{m,s}(m,s) \times P[\mathrm{sr} \neq 0|m] \times \int P[\mathrm{D} \neq 0|z, sr \neq 0] \\ \times P[\mathrm{D} > D_0|x/L, m, D \neq 0] f_r(r) \mathrm{dmdsdr} \quad (5) $$

### Distributed Rupture

$$ \lambda(d > d_0) = \alpha(m_{\min}) \int_{m,s} f_{m,s}(m,s) \times P[\mathrm{sr} \neq 0|m] \times \int P[\mathrm{d} \neq 0|z, r, sr \neq 0] \\ \times P[\mathrm{d} > d_0|r, m, d \neq 0] f_r(r) \mathrm{dmdsdr} \quad (6) $$

The last term of both integrals, the $f_r(r)$ term, accounts for the epistemic uncertainty associated with mapping and aleatory variability in the randomness of surface rupture location. This term was introduced by Petersen et al. (2011), where the authors compared the positions of observed surface ruptures with fault lines delineated on fault maps that were created before a series of eight significant surface-rupturing strike-slip earthquakes. They assessed the level of uncertainty in these positions by systematically measuring the distances between the observed surface ruptures and the corresponding fault traces as previously mapped. As recently highlighted by Scott et al. (2023), accurate prediction of coseismic rupture location before an earthquake is a critical aspect of PFDHA and, among others, it has important implications for risk management, engineering site assessments, and fault exclusion zones. Further advancements in defining the main rupture trace have been made by Thomas et al. (2023), addressing a key limitation of current approaches. Traditionally, identifying the principal rupture trace relies on subjective geological interpretations, which can vary significantly due to the often-complex network of mapped surface ruptures. This subjectivity hampers the consistency and repeatability of displacement measurements across multiple events, complicating comparative analyses and model development. To overcome these challenges, Thomas et al. (2023) propose an objective and repeatable method for defining the main rupture trace, applicable to field and remote sensing data, enhancing the reliability of rupture trace connectivity and geometry and providing a more standardized basis for developing displacement models.

## 3.2. Surface Rupture Probability Models

### 3.2.1. Probability of Surface Rupture Given a Magnitude $m$

The conditional probability of surface rupture for an earthquake given a magnitude $m$ is expressed by the term $P[\mathrm{sr} \neq 0|m]$ in Equation 4. All the models rely on logistic regression, as follows:
<table><caption>**Table 3**
Number of Published Models for Each PFDHA Component</caption><thead><tr><th rowspan="2"></th><th colspan="4">Probability surface rupture given a magnitude m<sup>a</sup></th><th colspan="4">Probability surface rupture at a specific site location for distributed ruptures</th><th colspan="8">Fault displacement model</th></tr><tr><th>N</th><th>R</th><th>SS</th><th>All</th><th>N</th><th>R</th><th>SS</th><th>N</th><th>R</th><th>SS</th><th>N</th><th>R</th><th>SS</th><th>N</th><th>R</th><th>SS</th></tr></thead><tbody><tr><th>Fault style</th><td>N</td><td>R</td><td>SS</td><td>All</td><td>N</td><td>R</td><td>SS</td><td>N</td><td>R</td><td>SS</td><td>N</td><td>R</td><td>SS</td><td>N</td><td>R</td><td>SS</td></tr><tr><th>Number of models<sup>b</sup></th><td>5</td><td>7</td><td>5</td><td>2</td><td>5</td><td>3</td><td>3</td><td>4</td><td>5</td><td>5</td><td>2</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td></tr></tbody></table>

$$ \frac{e^{f(x)}}{1 + e^{f(x)}} \qquad (7) $$

$$ \text{with } f(x) = a + bm \qquad (8) $$

with earthquake magnitude *m* as the predictor variable, *a* and *b* are coefficients inferred from the data. To date, 16 empirical models have been published for the conditional probability of surface rupture given a magnitude (Table 3), with some of these models applying to multiple faulting styles, allowing for a broader range of applications in FDHA. In particular, four models for normal faulting (three models by Youngs et al., 2003 and one by Pizza et al., 2023), four models for strike-slip faulting (one model by Takao et al., 2013; Pizza et al., 2023, and two models by Moss et al., 2013), six models for reverse faulting (one model by Moss & Ross, 2011; Takao et al., 2013; Yang et al., 2021; Pizza et al., 2023, and two models by Moss et al., 2013), and two models for all kinematics (Pizza et al., 2023; Wells & Coppersmith, 1993). The distinction among these models lies in the specific empirical data subsets used in the logistic regression (see Table 2), which are developed for various factors, such as faulting style, geographic region, inferred near-surface soil condition, or a combination of these elements. For instance, the empirical model by Wells and Coppersmith (1993) was established using data from 276 earthquakes worldwide, encompassing all faulting styles. Youngs et al. (2003) introduced the earliest models dedicated to normal faulting, derived from three distinct data subsets compiled by Pezzopane and Dawson (1996). Models designed for reverse faulting were formulated by Moss and Ross (2011), based on 129 global seismic events. Later, Moss et al. (2013) extended their work to create separate models for reverse and strike-slip faulting, considering the style of faulting and inferred near-surface soil stiffness. On the other hand, Takao et al. (2013) employed a database of 107 Japanese earthquakes to develop a model applicable to reverse and strike-slip kinematics. Recent contributions include a model developed by Yang et al. (2021), based on surface ruptures from 11 historical reverse earthquakes in Australia, spanning 1968 to 2018 and ranging from Mw 4.7 to 6.6. Pizza et al. (2023) conducted a comprehensive update of the initial catalog by Wells and Coppersmith (1993), using a global data set of 363 earthquakes representing all faulting styles, and developed models both for the entire data set and for data subsets specific to normal, strike-slip, and reverse earthquakes. It is worth noting that the assessment of these models underscores the significance of factors like faulting style, regional location (regionalization), and generalized soil conditions when selecting models for the conditional probability of rupture. It is also important to note that numerical methods have been applied to compute this conditional probability (e.g., Mammarella et al., 2024). Numerical simulations become particularly important when empirical models—often developed from region-specific data sets (e.g., limited to the Basin and Range Province)—are applied to different tectonic settings, such as Central Italy. In such cases, numerical approaches can help account for geological and structural differences (e.g., seismogenic depth) not captured by the original empirical data sets.

Figure 10 compares the probability of surface rupture for the empirical models listed and described in Table 4 and the numerical model proposed by Mammarella et al. (2024). The plots display models suitable for faulting styles, as well as the models encompassing all faulting styles. The probability of surface rupture increases with magnitude for all models, but the rate and absolute probability at any given magnitude level are highly contingent on regionalization and faulting style. For example, the four alternative models for normal faulting yield probabilities ranging from 48% to 80% for Mw 6.5, emphasizing the relevance of regionalization. Compared to normal

<footnote><sup>a</sup>Including both empirical and numerical models. <sup>b</sup>Different models derived from the same database are not considered. For example, Petersen et al. (2011) developed three models (bilinear, elliptical, and quadratic) using the same data set, but in this table, they are counted as a single model.</footnote>
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Probability of surface rupture given *m* for different earthquake mechanisms and different databases (see Tables 2 and 4 for more details). Probability curves for Mammarella et al. (2024) are obtained with a dip angle of 50°, 30°, and 90°, for normal, reverse and strike-slip, respectively, and the seismogenic zone thickness is equal to 9 km in all cases.</figcaption></figure>

and strike-slip faulting, reverse faulting typically exhibits a lower probability of surface rupture. For instance, the Moss and Ross (2011) model, the Moss et al. (2013) model for soft soil conditions, and the Pizza et al. (2023) model indicate probabilities of less than 50% for Mw 7.0, while for other faulting styles, the probability at this magnitude exceeds 70%. However, when the model is strictly derived from a regional database (e.g., Takao et al., 2013; Yang et al., 2021), the probability of surface rupture tends to be higher. Evaluation of the probability of surface rupture models suggests that style-of-faulting, geographic location, and generalized soil conditions may be important factors to consider in selecting models. Other parameters, such as down-dip width, seismogenic zone thickness, and hypocentral depth, strongly affect the probability of surface rupture in numerical simulations (Mammarella et al., 2024).

### 3.2.2. Conditional Probability of Surface Rupture at a Specific Site Location Along the Principal Fault

A limited number of models are available for the conditional probability of surface rupture (i.e., of non-zero displacement) at a specific site located along the principal fault. Generally, the probability of fault rupture on the fault is considered one, and the lack of surface rupture directly along the fault is not considered.

Takao et al. (2013) provide a model for the conditional probability of non-zero principal displacement for strike-slip and reverse faults based on data from earthquakes in Japan (the $P_{2\mu}$ term). The $P_{2\mu}$ term considers that the surface rupture length is statistically shorter than the source fault length for small-to-moderate magnitude events. The authors calibrated the ratio of the length of the surface earthquake fault to the length of the earthquake source fault through a literature survey.

Lavrentiadis and Abrahamson (2023) provide separate models for the conditional probability that no principal ruptures occur at a location *x*/L and the conditional probability that *x*/L is in a gap where no surface ruptures (principal) occur. The probability of the site being in a gap is the chance that the site will be located between two
<table><caption>**Table 4**
Synoptic View of Published Models</caption><thead><tr><th rowspan="3">Reference model</th><th rowspan="3">Style of faulting</th><th rowspan="3">Applicable magnitude range</th><th rowspan="3">Notes</th><th rowspan="3">Probability surface rupture given a magnitude m</th><th colspan="6">Fault displacement model</th><th rowspan="3">Notes</th></tr><tr><th rowspan="2">Function type</th><th rowspan="2">Cell size (m)</th><th colspan="2">Principal</th><th colspan="2">Distributed</th><th rowspan="2">Slip component</th><th rowspan="2">Classification</th></tr><tr><th>Predicted variable</th><th>Statistical distribution</th><th>Predicted variable</th><th>Statistical distribution</th></tr></thead><tbody><tr><td>Wells and Coppersmith (1993)</td><td>All</td><td>5.0-8.2</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Youngs et al. (2003)</td><td>N</td><td>1) 4.5-7.6<br>2) 4.9-7.2<br>3) 4.9-7.4</td><td>Three different models for three databases:<br>1) Cordillera<br>2) Great Basin<br>3) northern Basin and Range</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Petersen et al. (2011)</td><td>SS</td><td></td><td></td><td>Logistic<br>500 x 500</td><td>1) D/MD<br>2) D/MD<br>3) D/AD</td><td>1) Beta<br>2) Beta<br>3) Gamma</td><td>d/MD</td><td>Gamma</td><td>Vertical</td><td>Single Principal and Distributed</td><td>For distributed rupture:<br>5.5 ≤ Mw ≤ 7.4<br>0 ≤ r ≤ 15 km</td><td>Three different FDMs for principal for three database (see Table 1)</td></tr><tr><td>Moss and Ross (2011)</td><td>R</td><td>5.5-8.0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Surface rupture probability model for distributed with and without random effect<br>Different FDMs for HW and FW</td></tr><tr><td>Moss et al. (2013)</td><td>R and SS</td><td>4.2-8.7</td><td>Models for stiff and soft soil</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Takao et al. (2013, 2014, 2016).</td><td>R and SS</td><td>5.5-7.4</td><td>Published in Takao et al., 2013</td><td>Logistic<br>50 x 50<br>100 x 100<br>250 x 250<br>500 x 500</td><td>1) D/MD<br>2) D/AD<br>3) D/AD</td><td>1) Beta<br>2) Gamma<br>3) Weibull</td><td>d/AD</td><td>Gamma</td><td>Lateral</td><td>Single Principal and Distributed</td><td>For distributed rupture:<br>6.5 ≤ Mw ≤ 7.5<br>0 ≤ r ≤ 2.5 km</td><td>Bilinear, quadratic, or elliptical profile<br>Mw range of applicability for FDM for principal fault is 6.0-8.0</td></tr><tr><td>Yang et al. (2021)</td><td>R</td><td>4.7-6.6</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Ferrario and Livio (2021)</td><td>N</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Pizza et al. (2023)</td><td>All</td><td>1) 5.5-7.9<br>2) 5.5-7.1<br>3) 5.5-7.9<br>4) 5.5-7.8</td><td>Four different models depending on the kinematic:<br>1) All<br>2) Normal<br>3) Reverse<br>4) Strike-Slip</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Different models for HW and FW (HW)<br>Regular and conservative model</td></tr></tbody></table>
<table><caption>**Table 4**
*Continued*</caption><thead><tr><th rowspan="3">Reference model</th><th rowspan="3">Style of failling</th><th rowspan="3">Applicable magnitude range</th><th rowspan="3">Notes</th><th colspan="2">Probability surface rupture at a specific site location for distributed ruptures</th><th colspan="4">Fault displacement model</th><th rowspan="3">Slip component</th><th rowspan="3">Classification</th><th rowspan="3">Applicable ranges</th><th rowspan="3">Notes</th></tr><tr><th rowspan="2">Function type</th><th rowspan="2">Cell size (m)</th><th colspan="2">Principal</th><th colspan="2">Distributed</th></tr><tr><th>Predicted variable</th><th>Statistical distribution</th><th>Predicted variable</th><th>Statistical distribution</th></tr><tr><th colspan="2">Rodriguez, Padilla and Oskin (2023)*</th><th colspan="2"></th><th colspan="2">Unit area</th><th colspan="2"></th><th colspan="2"></th><th colspan="2"></th><th rowspan="2">0 ≤ r ≤ 3 km</th><th rowspan="2">The authors recommend the application in near-field region of immature strike-slip faults (&lt;25 km of cumulative displ.)</th></tr><tr><th colspan="2">Lavrentiadis and Abrahamsson (2023)*</th><th colspan="2">All</th><th colspan="2">D<sup>0.3</sup></th><th colspan="2">Normal</th><th colspan="2">Net</th><th colspan="2">Aggregate: Sum of Principal</th><th>5 ≤ Mw ≤ 8.5</th></tr><tr><th colspan="2">Mass et al. (2024)*</th><th colspan="2">R</th><th colspan="2">1) DAMD<br>2) D/AD</th><th colspan="2">1) Gamma<br>2) Gamma</th><th colspan="2">Vertical</th><th colspan="2">Single Principal</th><th>4.7 ≤ Mw ≤ 8.0</th></tr><tr><th colspan="2">Kuehn et al. (2024)*</th><th colspan="2">All</th><th colspan="2">D/λ<sup>1</sup></th><th colspan="2">Normal</th><th colspan="2">Net</th><th colspan="2">Aggregate</th><th>6.0 ≤ Mw ≤ 8.0 (SS, N)<br>5.0 ≤ Mw ≤ 8.0 (R)</th></tr><tr><th colspan="2">Mamumella et al. (2024)*</th><th colspan="2">All</th><th colspan="2">5.0-8.0</th><th colspan="2">Three different numerical models:<br>1) Normal<br>2) Reverse<br>3) Strike-Slip</th><th colspan="2">In(D)</th><th colspan="2">Negative Exponentially Modified Gaussian</th><th colspan="2">In(d)</th><th colspan="2">Sum of Principal</th><th>Probabilities are estimated as a function of magnitude from:<br>1) the position of the earthquake hypocenter along the fault's width,<br>2) the total rupture width, and<br>3) the up-dip and down-dip extents of fault rupture from the hypocenter</th><th>Supersede the FDM in Petersen et al. (2011)</th></tr><tr><th colspan="2">Chiou et al. (2025)*</th><th colspan="2">SS</th><th colspan="2">Logistic</th><th colspan="2">10 × ∞</th><th colspan="2">In(D)</th><th colspan="2">Negative Exponentially Modified Gaussian</th><th colspan="2">In(d)</th><th colspan="2">Vertical</th><th>5.5 ≤ Mw ≤ 7.9 (N)<br>4.9 ≤ Mw ≤ 7.9 (R)<br>0 ≤ r ≤ 8 km (FW)<br>0 ≤ r ≤ 10 km (HW)</th><th>Different models for HW and FW and for type of distributed ruptures<br>It supersedes the model published in Nunnem et al. (2020)</th></tr><tr><th colspan="2">Visini et al. (2025)*</th><th colspan="2">N and R</th><th colspan="2"></th><th colspan="2"></th><th colspan="2"></th><th colspan="2"></th><th colspan="2"></th><th colspan="2"></th><th colspan="2"></th></tr></thead></table>

*Note.* Models marked with * are available for download. For the code download link, please refer to the associated publication.
segments and have zero aggregate displacement. The probability of being in a gap scales the conditional probability of exceedance, which scales the hazard. However, their conditional probability of surface rupture models at a specific site location were developed for use with their own PFDHA model. They are not recommended for use with other FDMs, unless the modeling framework is consistent.

### 3.2.3. Conditional Probability of Surface Rupture at a Specific Site Location for Distributed Ruptures

The conditional probability of surface rupture at a specific site for distributed ruptures relies on the condition that surface-fault rupture happens during the earthquake ($sr \neq 0$). It predicts the probability of a distributed rupture occurring within a designated site area, typically represented as a square with an area of $z$, at a distance $r$ from the principal rupture. The surface rupture probability models currently accessible to describe this conditional probability are limited by empirical data (Tables 2 and 4). A prevalent method for developing models that estimate the probability of distributed rupture is to superimpose a grid with individual cells of area $z$ onto a rupture map (i.e., the “gridding” approach). This approach calculates the frequency of cells (or pixels) that contain one or more distributed ruptures situated at a distance $r$ from the principal rupture (Figure 11). Notably, this methodology has been employed in previous models, such as Youngs et al. (2003), Petersen et al. (2011), Takao et al. (2013), Ferrario and Livio (2021), and Rodriguez Padilla and Oskin (2023). Recently, Nurminen et al. (2020) and Visini et al. (2025) utilized fault-parallel slices as an alternative to square cells to assess the frequency of distributed ruptures (Figure 11) to address the issue of likely incompleteness of the distributed rupture mapping. The “slicing” approach generalizes the probability of distributed rupture occurrence along the strike of the principal fault by not assuming the completeness of the database. Furthermore, $P[d \neq 0|z, r, sr \neq 0]$ is replaced by two terms: $P[d \neq 0|r, m]$ and $P[d \neq 0|r, dim, F]$. The former represents the probability of having a distributed rupture occurrence in a slice, being a function of the distance $r$, with no regard for the site’s dimensions of interest. The latter expresses the conditional probability of having a distributed rupture within the site, given the occurrence of a distributed rupture at the given distance. This term is a function of the site’s dimensions ($dim$) and fraction ($F$) of the total distributed rupture length with respect to the length of the principal fault.

To date, 11 empirical models have been published for the conditional probability of surface rupture at a specific site location for distributed ruptures (Table 3), with some of these models applied to multiple faulting styles. In particular, five models for normal faulting (two by Youngs et al., 2003; Ferrario & Livio, 2021, and one by Visini et al., 2025), three models for strike-slip faulting (Petersen et al., 2011; Takao et al., 2013; Takao et al., 2014), and three models for reverse faulting (Takao et al., 2013, 2014; Visini et al., 2025). See Table 4 for more details. All models primarily consider the influence of the distance $r$ from the principal rupture. A common trend across all models is a decline in the probability as the distance from the principal fault increases (Figure 12). While some models (e.g., Visini et al., 2025; Youngs et al., 2003) incorporate earthquake magnitude as a predictive factor, linking higher probabilities to larger magnitudes, some model developers observed that earthquake size did not strongly affect the results (e.g., Ferrario & Livio, 2021; Petersen et al., 2011; Takao et al., 2014). Moreover, all models are crafted from empirical data (Table 2) related to a specific style of faulting, which inherently incorporates the effects of fault kinematics. For dip-slip faults (i.e., normal and reverse), some model developers offer distinct attenuation relationships for the hanging wall and footwall. Petersen et al. (2011) and Takao et al. (2014), who employ the “gridding” approach, have devised a set of models tailored to a broad spectrum of cell areas, with the probabilities that decrease with the decrease in cell area (Figure 12).

Despite differences in databases and methodologies (Tables 2 and 4), such as the inclusion of magnitude-dependency and variations in grid size, all models consistently exhibit a similar trend. The probability of a distributed rupture is highest in the immediate proximity of the fault and decreases rapidly as one moves away from the seismic source. It is also noteworthy that in Figure 12, for comparison purposes, some models are extrapolated out of the model’s range of applicability (see Table 4 for more details).

In addition to the models described above, Rodriguez Padilla and Oskin (2023) proposed a mathematical approach to estimate the probability per unit area of finding a rupture at a distance $r$ away from the principal rupture, resulting from an event of a given magnitude $m$, and with slip greater than a threshold $S_0$, as follows:

$$P(S > S_0|x, m) = P(\text{rupture}|r) P(S > S_0|r, \text{rupture}, m) \quad (9)$$
<figure><img src="image_14.png" /><figcaption>**Figure 11.** This figure illustrates the two primary methods for analyzing the occurrence of distributed ruptures (DR, blue lines) about the principal fault (PF, red line): the Gridding and Slicing approaches. In the Gridding approach, the study area is divided into a raster grid of cells with a predefined size (e.g., 500 × 500 m in Youngs et al., 2003; Ferrario & Livio, 2021, or a range of cell sizes from 25 × 25 m to 250 × 250 m in Petersen et al., 2011). For each earthquake, the number of grid cells containing DRs is counted and then divided by the total number of cells within the faulting area, providing a spatial rate of DR occurrence. The Slicing approach divides the study area into slices parallel to the PF. The probability of DR occurrence in each slice is calculated as a function of the distance *r* from the fault, given a response of occurrence/non-occurrence (gray areas indicate slices with mapped ruptures, while white areas indicate slices without mapped ruptures). A Monte Carlo simulation is then used to account for the site's position and dimension relative to the PF, which, with its length, influences the F ratio and consequently the expected number of DRs at the site.</figcaption></figure>

Computing this probability requires knowledge of the spatial distribution of ruptures and the displacements that these ruptures could accommodate. The authors address the former by analyzing the distribution of rupture density and the latter by examining the distribution of surface displacements measured for each selected event. The authors recommend applying the model for a site in the near-field region (<3 km) of immature strike-slip faults (<25 km of cumulative displacement) where widespread, distributed fault ruptures are expected, such as in the Eastern California Shear Zone or the Walker Lake Belt of the western United States. A similar approach was followed by Hu and Ren (2024) to predict displacement hazard for distributed ruptures from surface rupturing strike-slip faults with an earthquake potential of Mw 6.6–7.5 of the Tibetan Plateau.

## 3.3. Fault Displacement Models

### 3.3.1. Conditional Probability of Exceedance on the Principal Fault

The conditional probability of exceedance on the principal fault hinges on the likelihood that the principal displacement will exceed $D_0$ at a specified location (i.e., $x/L$) along the surface rupture during an earthquake of magnitude *m*. This probability is computed using principal FDMs, which provide a probability density function for the amount of principal displacement. It is essential to note that these models are typically tailored to empirical data (Table 2) and employ various statistical distributions like lognormal, gamma, or beta, which can vary between models.

To date, 14 empirical models have been published for the conditional probability of exceedance on the principal fault, with some of these models applied to multiple faulting styles (Table 4). In particular, four models for normal faulting (two by Youngs et al., 2003, and one by Lavrentiadis & Abrahamson, 2023; Kuehn et al., 2024), five models for strike-slip faulting (Chiou et al., 2025; Kuehn et al., 2024; Lavrentiadis & Abrahamson, 2023; Petersen et al., 2011; Takao et al., 2013), and five models for reverse faulting (Kuehn et al., 2024; Lavrentiadis &
<figure><img src="image_15.png" /><figcaption>**Figure 12.** Conditional probability of slip for distributed faulting for different kinematics. All curves for normal faulting are for a 500 m × 500 m grid size, and the curves of Youngs et al. (2003) and Visini et al. (2025) are computed for Mw 6.5. All curves from Visini et al. (2025) are only for distributed ruptures ranked as 2 (see Visini et al. (2025) for more details). For reverse faulting, the models of Takao et al. (2013, 2014) are the same for footwall and hanging wall. For strike-slip kinematics, the probabilities are assumed to be equal in both the hanging wall and footwall.</figcaption></figure>

Abrahamson, 2023; Moss et al., 2024; Moss & Ross, 2011; Takao et al., 2013). Recently, thanks to the FDHI Program (Bozorgnia et al., 2021), a set of four new FDMs was introduced, namely Lavrentiadis and Abrahamson (2023), Kuehn et al. (2024), Moss et al. (2024), and Chiou et al. (2025). They diverge from the old models (e.g., Petersen et al., 2011; Youngs et al., 2003) regarding database size, statistical modeling techniques, and the predicted displacement metric. Additionally, these FDHI models, to some extent, replace earlier models. e.g., the new model by Moss et al. (2024) is an update to Moss and Ross (2011), while the new model by Chiou et al. (2025) updates the FDM for principal fault as presented in Petersen et al. (2011).

All models primarily account for earthquake size and location along the rupture (i.e., *x/L*) in calculating the conditional probability of exceedance and are based on empirical data for a specific faulting style, thus inherently encompassing the influence of fault kinematics as captured in the data. Two approaches are generally employed to capture the effects of earthquake size. The first approach utilizes normalized displacement as the predicted variable in the model formulation. These models forecast displacement as a ratio normalized by the maximum (MD) or average displacement (AD). To calculate *D*, a separate scaling relation estimates MD or AD as a function of earthquake magnitude based on empirical relations (e.g., Wells & Coppersmith, 1994). In the second approach, *D* (or ln(*D*)) is directly predicted, and earthquake magnitude is used as an input variable in the model. An essential distinction between these approaches lies in the treatment of aleatory variability. In the first approach, earthquake size is not explicitly parametrized in the model formulation, and therefore, the statistical distribution does not encompass the data dispersion related to earthquake size. Instead, the aleatory variability due to earthquake size is introduced in the scaling relationship utilized to estimate AD or MD. To calculate the overall aleatory variability in the second approach, the statistical distribution for the FDM must be combined or convolved with the statistical distribution for the scaling relationship to ensure a comprehensive estimation of aleatory variability. In the second approach, earthquake size is parametrized directly within the model formulation, so the statistical distribution within the model includes variability linked to earthquake size.

Figure 13 illustrates displacement profiles for the normalized displacement models in Table 4. These profiles illustrate how the predicted median displacement varies with respect to the normalized position along the rupture length (*x/L*). All models in Figure 13 employ functional forms that exhibit symmetry around the rupture midpoint (*x/L* = 0.5). Generally, displacement amplitude decreases near the rupture endpoints (i.e., closer to *x/L* = 0) and peaks at the midpoint. Exceptions to this pattern are the location-independent models (e.g., Takao et al., 2013 for fault lengths less than 10 km). Disparities in these slip profiles result from differences in the data sets used to develop the models (see Table 2 for more details) and varying modeling decisions concerning the shape or functional form of the displacement profile (e.g., beta, gamma, Weibull). Figure 14 provides similar plots for
<figure><img src="image_16.png" /><figcaption>**Figure 13.** Median (50th percentile) principal displacement profiles for normalized displacement models (D/MD, top row; and D/AD, bottom row) in Table 4. Note the Takao et al. (2013) models are the same for reverse and strike-slip styles of faulting.</figcaption></figure>

models that predict the non-normalized amount of displacement. These profiles are generated for an Mw 6.5 earthquake. In most models, the profile shape remains consistent across earthquake sizes. One exception is Laventriadis & Abrahamson (2023), which employs surface rupture length as a predictive variable for both the shape and amplitude of the slip profile.

### 3.3.2. Conditional Probability of Exceedance on the Distributed Ruptures

The conditional probability of distributed faulting exceeding a certain level $d_0$ is analogous to principal faulting. This probability calculates the likelihood of the distributed displacement surpassing $d_0$ at a distance $r$ from the principal surface rupture during an earthquake of magnitude $m$.

To date, 10 empirical models have been published for the conditional probability of exceedance on distributed ruptures, with some of these models applied to multiple faulting styles. In particular, two models for normal faulting (Visini et al., 2025; Youngs et al., 2003), four models for strike-slip faulting (Petersen et al., 2011; Takao et al., 2013, 2014, 2016), and four models for reverse faulting (Takao et al., 2013, 2014, 2016; Visini et al., 2025).

<figure><img src="image_17.png" /><figcaption>**Figure 14.** Median (50th percentile) principal displacement profiles for displacement models in Table 4 using Mw 6.5.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Figure 15.** 95th percentile of distance attenuation of *d*/MD (left) and 50th percentile of *d*/AD (right). The models of Petersen et al. (2011) and Takao et al. (2013, 2014, 2016) do not distinguish between hanging wall and footwall. Takao et al. (2016), *a* and *b*, represent the model published by Takao et al. (2016) applied to the data set of Takao et al. (2013, 2014), respectively.</figcaption></figure>

These models are empirical, offer a probability density function for the amplitude of distributed displacement, and are developed for specific faulting styles, capturing the effects of fault kinematics. Like principal FDMs, they employ various statistical distributions (e.g., lognormal, gamma, beta) to characterize variability and necessitate a separate scaling relation to estimate MD or AD, and the statistical distribution for MD or AD prediction should be convolved with the distribution for *d*/MD or *d*/AD prediction. Some model developers have provided simplified formulations for high percentiles as a function of distance *r* (e.g., 95th percentile in Youngs et al., 2003, 90th percentile in Takao et al., 2013, 2014, 2016).

Several existing distributed FDMs consider the influence of earthquake size and the distance from the site to the principal rupture, with the *r* term as a predictor (input) variable in most models. The impact of earthquake size is integrated into the normalization variable (MD or AD) for the normalized models. Recently, Nurminen et al. (2020), which was superseded by Visini et al. (2025), went a step further by considering the predicted principal displacement amplitude *D* at the distance *r*. The model proposed by Visini et al. (2025) is a function of *m*, *r*, and *D*, with amplitude decreasing with distance *r* and contingent on the type of distributed ruptures (i.e., Rank, see Nurminen et al., 2020; Visini et al., 2025 for more details).

Figures 15 and 16 show how the distributed displacement amplitude (95th percentile for *d*/MD and 50th percentile for *d*/AD) attenuates with distance from the principal rupture for the displacement models in Table 4. In all models, the distance factor is significant but exhibits variations contingent on the faulting style. Youngs et al. (2003) and Visini et al. (2025) provide distinct attenuation relationships for the hanging wall and footwall for dip-slip faults. The difference is primarily lead by fault kinematics and dip. For normal and reverse faults, the hanging wall curve tends to be more strained compared to the footwall, and the spatial distribution of distributed earthquake surface ruptures is concentrated mainly in the hanging wall compared to the footwall (Boncio et al., 2012, 2018; Visini et al., 2025). It is also noteworthy that in Figures 15 and 16, for comparison purposes, some models are extrapolated out of the model's range of applicability (see Table 4 for more details).

### 3.4. Example of PFDHA Applications Worldwide

This section presents three case studies that illustrate the application of PFDHA for critical infrastructure, offering valuable insights for stakeholders involved in seismic hazard assessment and risk management. Each case highlights different geological settings, fault characteristics, and methodological approaches, demonstrating how
<figure><img src="image_19.png" /><figcaption>**Figure 16.** Median distributed displacement profiles for Visini et al. (2025). All curves are for Mw 6.5 and only for distributed ruptures ranked as 2 (see Visini et al., 2025 for more details).</figcaption></figure>

PFDHA can be tailored to specific site conditions and regulatory requirements. The selected examples cover a range of infrastructure types, including nuclear power plants and strategic water reservoirs, emphasizing the role of FDHA in ensuring the safety and resilience of essential facilities. While this section focuses on the Yucca Mountain repository in the United States, the Diablo Canyon Power Plant (DCPP) in California, and the Montedoglio Dam in Italy, other notable applications of PFDHA include the Krsko Nuclear Power Plant in Slovenia (Quittmeyer et al., 2017) and the Ikata Nuclear Power Plant in Japan (Nishizaka et al., 2024), further underscoring the global relevance of this methodology in seismic risk mitigation for critical infrastructure.

### 3.4.1. Yucca Mountain (Nevada, USA)

In the quest for a long-term solution to manage spent nuclear fuel and high-level radioactive waste, the U.S. began evaluating a potential underground geological repository near Yucca Mountain, approximately 160 km northwest of Las Vegas, Nevada, in the 1970s. The seismic hazard investigations at Yucca Mountain represented the first-ever Level 4 study by the Senior Seismic Hazard Advisory Committee. The analyses described here, as detailed in Stepp et al. (2001), highlight the groundbreaking methodologies developed for Yucca Mountain that paved the way to the evolution of modern PFDHA (Youngs et al., 2003). Among the key contributions of the Yucca Mountain study was the classification of fault displacement hazards into principal and distributed faulting categories and the introduction of two distinct modeling approaches: the displacement-based approach and the earthquake-based approach.

Yucca Mountain is situated in the southern portion of the Basin and Range Province in the western U.S., a region characterized by tectonic extension at a low strain rate, with predominant late-Quaternary normal faulting and moderate historical seismicity. Paleoseismic studies have documented earthquakes of magnitude 7 and above along several Quaternary faults within 100 km of the site. The Solitario Canyon fault bounds the proposed repository to the west and the Bow Ridge fault to the east. Extensive geological, geophysical, and paleoseismic research focused on faults within a 10–20 km range from the repository site, revealing multiple surface-rupturing events during the Quaternary period. The mapped faults are 27 km or shorter, with preferred slip rates ranging from 0.001 to 0.05 mm/year. Paleoseismic trenching suggests that some subparallel faults may have ruptured simultaneously during the Quaternary period. In addition to fault length and slip rates, expert teams assessed key fault source parameters, including fault dip, slip sense, maximum potential magnitude, and rupture behavior.
Using the earthquake-based approach, expert teams estimated the rate of surface fault ruptures on principal faults by analyzing earthquake frequencies identified in ground motion studies, combined with the conditional probability of surface slip. This conditional probability encompassed the likelihood of the earthquake rupturing the ground surface and the probability of rupture occurring specifically at the site of interest, referred to as the along-strike intersection probability. Empirical relationships derived from historical ruptures in the Basin and Range Province were used for distributed faulting. The conditional probability of distributed slip was found to correlate with both the earthquake magnitude and the distance from the principal fault rupture. Logistic regression models were developed to predict the probability of distributed slip within a 0.5 km × 0.5 km site, accounting for different tectonic settings, such as the hanging wall and footwall. The teams also explored how the angular relationship between principal and secondary faults influenced the conditional probability of distributed slip. Similar methodologies were applied for principal and distributed faulting in the displacement-based approach. The preferred method for estimating the rate of surface-rupturing events involved dividing the fault slip rate by the average displacement per event. This average displacement was determined through direct paleoseismic observations or empirical scaling relationships linking displacement to fault length or cumulative fault offset. These empirical relationships were calibrated explicitly for the Yucca Mountain region using local and sub-regional data.

Overall, expert teams slightly preferred the displacement-based approach over the earthquake-based approach when characterizing fault displacement hazards at Yucca Mountain, as reflected in the logic-tree weighting. The most significant uncertainty was associated with the characterization of distributed faulting under the earthquake approach. Variations in methodologies were most pronounced when estimating displacement per event for distributed ruptures. The PFDHA was calculated for nine demonstration sites within the proposed repository area. For a return period of 100,000 years (corresponding to an annual exceedance probability of 10⁻⁵), fault displacements were generally less than 0.1 cm. However, significant uncertainties persisted in the fault displacement hazard assessments. These uncertainties stemmed from factors such as limited data on fault parameters, variations in the expertise and background of the expert teams, and the absence of established standards for conducting PFDHA. Following the completion of the study, the findings were incorporated into the safety assessments of the proposed repository, considering both pre-closure and post-closure scenarios.

### 3.4.2. Diablo Canyon (California, USA)

The DCPP is located on California's central coast, roughly equidistant between Los Angeles and San Francisco. In 2008, several small seismic events revealed the Shoreline fault, a strike-slip fault less than one km offshore from DCPP. To evaluate the potential seismic hazard and risk posed by this fault, a PFDHA was carried out, focusing on the plant's auxiliary saltwater intake pipes, which are critical safety components vulnerable to minor permanent ground shifts. This case study outlines the PFDHA conducted at DCPP, applying the methodology developed by Petersen et al. (2011) for strike-slip fault environments. It illustrates how PFDHA can be used to assess distributed rupture displacement hazard that affect specific, safety-related infrastructure at nuclear facilities that are not situated directly on a principal fault.

DCPP is located within a tectonically active transpressional zone west of the San Andreas Fault. The most prominent fault near DCPP is the offshore strike-slip Hosgri fault, approximately 5 km from the plant at its nearest point. Additionally, the plant lies within the hanging wall of the Los Osos and San Luis Bay faults. The region has experienced significant seismic events, such as the 1927 offshore Lompoc earthquake (M<sub>S</sub> 7.0) and the 2003 San Simeon earthquake (Mw 6.5). In 2008, seismic monitoring identified a pattern of small earthquakes (magnitudes <1 to 3.5) offshore between the coastline and the Hosgri fault, occurring from 1987 to 2007 (Hardebeck, 2010). Follow-up investigations by Pacific Gas and Electric (PG&E) confirmed the Shoreline fault, a previously undocumented strike-slip fault (PG & E, 2011). Initial evaluations estimated the Shoreline fault's length at about 20 km, with its closest points being 600 m from DCPP's power block and 300 m from the cooling water intake structures.

Since the auxiliary saltwater intake pipes were the only safety-related structures at risk from minor displacements near the Shoreline fault, PG&E conducted a PFDHA focusing on potential secondary fault deformation at the dresser coupling sections. The analysis followed methods from a study funded by the Pacific Earthquake Engineering Research Institute, as detailed in Petersen et al. (2004) and later refined in Petersen et al. (2011). PG&E assumed that any rupture of the Shoreline fault would occur along the fault plane's section closest to DCPP. The analysis used two fixed distances between the fault trace and the plant: 600 m (nearest to the power block) and
300 m (nearest to the intake structure). The PFDHA used a simplified fault source characterization to develop the magnitude-recurrence distribution based on a preliminary evaluation of the Shoreline fault. Epistemic uncertainties in the magnitude-recurrence relationship were addressed by considering two characteristic earthquake magnitudes (Mw 6.0 and Mw 6.25) and two end-member fault slip rates (0.01 and 0.3 mm/year).

The PFDHA was conducted for two displacement thresholds, 1 and 2 cm, reflecting the estimated tolerance of the dresser couplings on the saltwater intake pipes. Annual exceedance frequencies were calculated for 28 scenarios, encompassing the two displacement thresholds, two characteristic magnitudes, and two fault slip rates. Rather than combining uncertainties into a single model, PG&E analyzed each scenario separately to understand better how different parameters influenced hazard levels. The annual exceedance frequency for a 1 cm displacement ranged from approximately $2.0 \times 10^{-7}$ to $2.6 \times 10^{-10}$ per year, with the highest hazard linked to the largest magnitude, highest slip rate, and shortest distance. Increasing the displacement threshold to 2 cm reduced the exceedance frequency by less than a factor of two. The U.S. Nuclear Regulatory Commission (NRC) reviewed PG&E's findings and conducted an independent confirmatory analysis (USNRC, 2012). The NRC's results were generally higher: for the Mw 6.0 case, their calculated hazard was about six times greater than PG&E's estimates and about four times greater for the Mw 6.25 case. When the NRC analyzed a hypothetical Mw 6.7 event, the hazard increased by less than a factor of 1.5 compared to the Mw 6.25 scenario. Differences between PG&E and NRC findings were primarily due to variations in the probability models for distributed surface rupture and displacement exceedance. These methodological differences resulted in four to six times in hazard discrepancies between the two studies, which is more significant than the variability caused by different magnitude assumptions (typically less than a factor of two). The uncertainty in slip rate, however, results in a factor of 30 difference, which is a direct result of the factor of 30 difference between the low (0.01 mm/year) and high (0.3 mm/year) slip rate ranges. These findings highlight the critical role of sensitivity analyses in probabilistic hazard assessments, as they help identify which epistemic uncertainties most significantly impact the overall hazard. For DCPP, comparing the annual exceedance probabilities of 1 and 2 cm displacements to the plant's annual core damage frequency helped assess the potential impact of surface-fault rupture hazards. Both PG&E and the NRC concluded that the risk from surface-fault rupture contributed less than 1% to the annual core damage frequency, two orders of magnitude lower, and was therefore considered a negligible risk factor for DCPP's overall safety profile.

### 3.4.3. Montedoglio Dam (Italy)

The Montedoglio Dam, an earth-fill structure standing 60 m high and stretching 570 m long, is situated in the Northern Apennines of Italy, impounding the Tiber River, Italy's third-longest river. The reservoir supplies 102 million cubic meters of potable water annually and is classified as a strategic infrastructure. According to the ITaly HAzard from CApable faults (ITHACA) database, the dam was believed to be intersected by the Montedoglio fault, an oblique-slip capable fault striking southwest to northeast (ITHACA Working Group, 2019). Additionally, the dam is located within the hanging wall and approximately 1.5 km from the northeast-dipping Anghiari normal fault, which paleoseismic studies have confirmed as capable (Testa et al., 2023). Recent updates in Italian regulations and guidelines governing the seismic safety of large dams mandated a comprehensive seismotectonic assessment for Montedoglio Dam, including an FDHA (Basili et al., 2017; DMIT, 2014). The analysis of the Montedoglio Dam evaluated fault displacement hazards originating from principal and distributed faulting. At the same time, the initial phase of the study ruled out hazards from the principal Montedoglio fault; subsequent efforts concentrated on assessing distributed faulting hazards. The findings of this pioneering PFDHA for critical infrastructure in Italy were published by Testa et al. (2024).

Montedoglio Dam is located at the northwestern margin of the Sansepolcro Basin, a Quaternary basin bordered by normal faults dipping to the northeast and southwest. The primary fault, the Anghiari-Città di Castello fault, dips northeast and delineates the basin's western boundary, while an antithetic fault defines its eastern limit. The northern edge of the basin, where the dam is located, is shaped by the Montedoglio fault's linear geometry. The region is currently experiencing active extensional tectonics at a rate of approximately 2 mm/year, with the extension direction oriented southwest-northeast, nearly perpendicular to the bounding faults. Historically, the basin has witnessed seven damaging earthquakes over the past 700 years, with magnitudes ranging from 5.0 to 6.3. The dam is within its hanging wall about 1.5 km east of the Anghiari fault. According to Testa et al. (2023),
the Anghiari fault, a segment of the Anghiari-Città di Castello fault system, extends for 11 km and is recognized as active and capable. The fault is divided into four segments, with paleoseismological investigations revealing up to seven surface-rupturing events over the last 25,000 years. The most recent rupture correlates with the 1458 Sansepolcro earthquake (Mw 5.8). The fault is attributed to a slip rate of 0.14–0.35 mm/year and a potential to generate earthquakes with magnitudes up to 6.2, or 6.7 if the entire Anghiari-Città di Castello fault system ruptures.

The study's initial phase focused on verifying the Montedoglio fault's location and capability through geological, geophysical, and paleoseismological methods. This phase concluded that the Montedoglio fault does not pose a principal fault hazard. The subsequent phase concentrated on a PFDHA for distributed faulting linked to the Anghiari-Città di Castello fault system. The PFDHA employed the earthquake-based approach using the methodology by Youngs et al. (2003), complemented with empirical regressions for distributed ruptures from Ferrario and Livio (2021). Both the standard and conservative scenarios proposed by Ferrario and Livio, alongside Youngs et al.'s original regressions, were applied to analyze distributed faulting hazards. For principal faulting scenarios, the analysis considered various rupture patterns based on the segmentation of the Anghiari-Città di Castello fault. This yielded four scenarios, each characterized by different magnitudes, rupture lengths, maximum and average displacements, and distances from the dam. The earthquake magnitudes ranged from 6.3 to 6.6, with rupture distances to the dam spanning from 1.5 to 2.3 km.

The first part of the study determined that the Montedoglio fault is neither active nor capable of generating surface-rupturing earthquakes. Characterized as a network of short, discontinuous, and ancient bedrock faults, the Montedoglio fault has been inactive for over 56,000 years and lacks the potential to trigger significant seismic events. However, the potential for distributed faulting associated with the Anghiari fault, including the possible reactivation of existing discontinuities in the Montedoglio fault system, could not be dismissed. Consequently, the PFDHA assessed the distributed fault displacement hazard at the dam site. The results indicated a very low hazard level. The mean hazard curve consistently remained below the annual exceedance frequency of $1 \times 10^{-5}$, corresponding to displacement values of less than 1 cm for a return period of 100,000 years. These findings led the Italian Ministry responsible for dam safety to deem the risk level acceptable, allowing the dam to continue its operations. The publication by Testa et al. (2024) emphasized several areas for improving PFDHA methodologies. Key recommendations included: (a) developing tailored regressions for distributed faulting that consider the influence of pre-existing secondary structures on faulting probabilities; (b) enhancing empirical constraints on distributed displacement attenuation models, necessitating updated and comprehensive data sets, (c) refining displacement attenuation models for distributed faulting to incorporate parameters that more accurately represent the variability of displacements along the principal rupture, beyond solely using maximum or average displacement metrics.

## 4. Future Perspectives

In this section, we outline key improvements needed for surface rupture probability assessment and FDMs. By addressing these challenges, PFDHA can continue to evolve as a vital tool for mitigating risks associated with surface fault displacement hazards, ensuring the safety and resilience of critical infrastructure worldwide. Future advancements should focus on refining probabilistic frameworks, integrating high-resolution geologic and geophysical data, and reducing epistemic uncertainties in rupture modeling. Additionally, interdisciplinary approaches combining geology, geodesy, seismology, and numerical simulations will be essential for improving model accuracy and applicability. Strengthening empirical data sets, incorporating physics-based modeling techniques, and developing more reliable segmentation and rupture propagation models will further enhance PFDHA's capabilities.

The number of PFDHA models published to date is limited. For example, the number of FDMs (Table 3) is minimal compared to the number of ground motion models, which have exceeded 500 publications since 1964 (Douglas, 2022). While it is unrealistic to expect FDMs to reach similar numbers within the next decade, it is crucial to make a concerted effort to increase the availability of published models. In our view, this goal can be achieved through three main directions: first, by enriching existing databases with new high-quality observations and measurements; second, by adopting physics-based models to bridge gaps in empirical data and improve predictive capabilities; and third, by refining existing modeling approaches. This order reflects our perspective on the key priorities necessary to foster robust and sustainable growth in the field.
## 4.1. Database Enrichment

The current data sets on principal and distributed fault ruptures are limited, in scope and resolution. Expanding these databases with detailed fault rupture histories, including large and moderate events, will provide a more comprehensive basis for probabilistic assessments. Increasing the number of documented cases will improve the statistical robustness of these models, capturing the complexity and variability of principal and distributed faulting.

Besides the classical post-earthquake field survey, the enrichment of databases can be achieved through multiple approaches, including geodetic and remote sensing techniques, which provide high-resolution measurements of surface deformation, as well as machine learning methods, which can help identify patterns and extract information from large data sets. High-resolution remote sensing techniques, such as LiDAR and InSAR, offer valuable insights into surface deformations and fault geometries. Incorporating these data into rupture models can improve the detection and characterization of principal fault ruptures. This can also enhance the understanding of distributed rupture patterns and their spatial distribution. To improve databases, machine learning methods can be employed to recognize and classify earthquake surface ruptures. For instance, a convolutional neural network can be used to analyze remote sensing images and synthetic digital elevation models to identify, classify, and categorize earthquake surface ruptures. This will allow us to have a more spatially complete surface rupture database, which can improve existing and future collected data.

Once databases have been enriched with extensive and high-quality data, we can explore epistemic uncertainties in the models by considering the following aspects:

* *Control of near-surface lithology:* near-surface lithological variations play a significant role in influencing fault displacement patterns. Incorporating detailed lithological data can refine displacement models, reducing uncertainties in hazard assessments.

* *Inclusion of site-specific factors:* factors such as style-of-faulting, geographic location, and generalized soil conditions significantly affect the probability of principal faulting and distributed ruptures. Models should integrate these parameters to enhance their predictive accuracy.

* *Structural complexity considerations:* recognizing the role of structural complexities, such as fault stepovers and bends, in partitioning slip between principal and distributed ruptures is crucial. Denser distributed ruptures often occur in these areas, and models should account for these variations.

* *Refinement of empirical relationships:* empirical models need continuous refinement based on new data and case studies. Validating these models against well-documented seismic events will improve reliability and reduce uncertainties.

## 4.2. Physics-Based Modeling

A significant challenge in PFDHA is the variability in surface displacement, ranging from localized slip along a single fault trace to distributed deformation across a broad zone. This variability is influenced by factors such as the accumulated slip on a fault, slip magnitude, fault geometry, and the presence of secondary structures. The magnitude and spatial extent of surface displacement depend on the earthquake's size, the fault's mechanical properties, and the local geologic setting. Displacement can range from centimeters to several meters and may extend for kilometers along the fault. Empirical models, such as those developed by Youngs et al. (2003) and Petersen et al. (2011), provide valuable insights but are often limited by sparse data and local-to-regional scale specific conditions. Regionalizing the models will account for the non-ergodic nature of earthquake processes, capturing differences in rupture behavior based on geographic location, fault type (e.g., strike-slip, reverse, normal), and site-specific conditions. For example, Liou and Abrahamson (2025) developed a generalized non-ergodic framework for FDM, enabling robust estimation of epistemic uncertainty even in the absence of displacement metadata. The authors provide a practical methodology to incorporate site- and fault-specific variability for more realistic hazard assessments, also in data-limited contexts.

Next-generation models that overcome these limitations by integrating physics-based approaches are essential. Earthquake cycle simulators (e.g., Zielke & Ma, 2023) may offer a promising framework for improving FDHA, particularly by allowing explicit consideration of fault geometries, models of fault strength and long-term accumulation. These elements could play a key role in controlling coseismic surface ruptures and should be further explored in future studies. Discrete element method (DEM), dynamic, and physical/numerical modeling require
detailed input parameters that are not always available. Improving the accuracy of these models will require better constraints on fault behavior and more comprehensive data sets. For this reason, the enrichment of the databases has the priority. These models can simulate complex rupture dynamics, offering a more detailed understanding of potential fault behaviors. DEM models are well suited to this investigation because they can effectively model the geologic process of faulting at depth in cohesive rocks, as well as the granular mechanics of soil and sediment deformation in the shallow subsurface (e.g., Chiama et al., 2023; Garcia & Bray, 2022). Dynamic rupture simulations have been employed successfully to validate fault displacements observed during past earthquakes (e.g., Wang & Goulet, 2021). Moreover, physical and numerical models have been used to explore the physics of fault displacement through near-surface soils. For example, Moss et al. (2018), using physical fault-box testing and analytical approximations, argued that the 30-m shear-wave velocity estimated from slope is a useful proxy variable for assessing the likelihood of surface fault displacement in reverse-faulting events. Existing models for distributed displacement attenuation are poorly constrained by empirical data. Developing updated models that reflect the variability in displacement through physics-based approaches will improve hazard assessments.

### 4.3. Refining Existing Models

Regarding the refinement of existing models, there are numerous possible improvements, most of which depend on the enrichment of databases, as discussed above. Here, we want to highlight three fundamental aspects that have been little to not at all explored:

*   **Advanced rupture modeling:** incorporating the possibility of multi-fault ruptures (such as Kaikoura earthquake) and the concept of floating ruptures, i.e., floating the earthquakes along the length of the fault, as is commonly done in probabilistic seismic hazard assessment (e.g., Pagani et al., 2014). This approach treats epistemic uncertainty by considering various rupture scenarios, including simultaneous segment ruptures leading to higher potential magnitudes.

*   **Displacement definitions and component consistency:** a key limitation in comparing and integrating different FDMs lies in the inconsistent definitions of displacement and rupture participation. Methods for adjusting model outputs based on different rupture assumptions (such as aggregate, sum-of-principal, or single-principal ruptures) are currently unavailable, complicating harmonization across models. Additionally, definitions of displacement vary (e.g., net vs. principal components), and systematic methods to decompose or convert between these are lacking. The incompleteness of component-level displacement data further complicates this issue. For example, as shown by Sarmiento et al. (2025) in their evaluation of the FDHI Database, net displacement data for reverse-faulting events often omit the fault-normal (dip-slip) component, making the recorded values more representative of displacement on a hypothetical vertical fault. Consequently, attempts to scale or reconcile models (e.g., via trigonometric corrections for dip) are unreliable without consistent data and standard definitions.

*   **Hazard code and verification:** currently, there is no unified computational platform for conducting PFDHA, with different researchers and institutions relying on their in-house codes. This lack of standardization introduces inconsistencies in how models are implemented, and results are compared. A dedicated open-source platform, similar to OpenQuake for probabilistic seismic hazard analysis (Pagani et al., 2014), would ensure that calculations, model integrations, and uncertainty quantifications are performed transparently, reproducibly, and verifiably.

### Acknowledgments

We would like to express our gratitude to the Editor-in-Chief for inviting us to contribute with this paper. We also extend our appreciation to the members of the Fault2SHA community, and in particular, the Fault Displacement Hazard Laboratory, for the ongoing discussions that have greatly enriched our understanding of this topic. We are especially grateful to the reviewers, A. M. Rodriguez Padilla and R. E. S. Moss, for their constructive comments and valuable suggestions, which helped us to significantly improve the quality of the manuscript. Open access funding provided by Universitat Wien/KEMÖ.

### Data Availability Statement

Data were not used, nor created for this research.

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