A Review of Probabilistic Approaches for Assessing the Liquefaction Hazard in Urban Areas

Abstract

Several probabilistic liquefaction triggering approaches, or liquefaction manifestation severity approaches, have been developed to consider the uncertainties related to liquefaction and its manifestations. Probabilistic approaches are essential for vulnerability and risk models that considers the consequences of liquefaction on building performance. They may be incorporated into a performance-based earthquake engineering framework through a fully probabilistic liquefaction hazard assessment. The objective is to effectively incorporate spatial interaction of two concurrent hazards, specifically earthquake-induced shaking, and liquefaction, and to develop a robust multi-hazard framework applicable to regions with limited input data. For this purpose, it is necessary to establish, according to the available probabilistic liquefaction triggering or manifestation severity assessment approaches, which set of approaches aligns optimally with vulnerability and risk models. Thus, this paper discusses the current methodologies on the ongoing probabilistic liquefaction hazard assessment approaches with the aim of defining a reliable model specific for areas with a non-liquefiable surface layer over a liquefiable layer.

1 Introduction

While compound hazards are characterized by several elements acting together above their respective damage thresholds, multiple hazards are characterized as elements of distinct natures that coincide at random or follow each other with harmful force [1]. This means that the interaction of different hazards can lead to an impact that is higher than the sum of the individual hazard effects [2]. A multi-hazard approach for the built environment takes into account the different probabilities and levels of intensity associated with each hazard and evaluates the hazards that commonly result in damage and losses [3]. It is necessary to clarify the difference between multi-hazard and multi-hazard risk. According to De Angeli et al. [3], ‘multi-hazard’ refers to a series of approaches where multiple hazards and their interrelations are analysed and modelled.

Differently, ‘multi-hazard risk’ is defined as a risk assessment that considers the impact of multiple hazards. A multi-hazard risk approach involves both multi-hazard and multi-hazard vulnerability approaches. Regarding seismic multi-hazard vulnerability approaches, research has been oriented to the simultaneous action of hazards triggered by earthquakes (cascading natural events), such as fire, floods, tsunamis, and landslides [4]. As defined by Gómez et al. [5], cascading natural events, commonly defined as a primary hazard triggering a secondary one, have induced large disasters. In the case of earthquakes, between 25 and 40% of economic losses and deaths have been reported because of secondary effects, such as tsunamis, landslides, liquefaction, fire, and others. Despite the consequences of such events, multi-hazard vulnerability and followingly risk assessment remain a partly unexplored research field with limited unified terminologies and approaches [5]. In this paper, the interactions between two hazards are considered in a multi-hazard framework: earthquakes and liquefaction-induced ground damage by earthquakes.

It is important to clarify that, there are two types of liquefaction, (i) cyclic liquefaction, which is the result of excess pore pressure and concurrent degradation of shear stiffness resulting from cyclic loading, and (ii) flow liquefaction, which is the process of strain softening of contractive, saturated cohesionless soils during undrained shear [6]. In this paper, focus is given to the first form, which is practicality hereinafter denoted only as liquefaction.

For years, civilizations have endured the effects of earthquakes in terms of human and economic losses. For instance, the 1994 Northridge and 1995 Kobe earthquakes have been among the most damaging events in history, with economic losses of approximately US$14 billion and US$150 billion, at prices when occurred, respectively [7]. Many of the impacts of earthquakes have been due to the multiple hazard actions triggered by them, e.g., fire, floods, landslides, and cyclic liquefaction. Among these multiple hazard actions, cyclic liquefaction has caused severe damage during historical events to the built environment. As reported by Arango-Serna et al. [8], notable instances include the damages reported during the 1964 Niigata earthquake and, more recently, during the earthquakes of Maule, Chile, 2010 [9, 10], Darfield, New Zealand, 2010 [11], Christchurch, New Zealand, 2011 [12], Tohoku, Japan, 2011 [13], Emilia-Romagna, Italy, 2012 [14], Sulawesi, Indonesia, 2018 [15]. For illustration purposes, Fig. 1 shows failures beneath buildings and roads that occurred around the world.

Fig. 1

Example of liquefaction consequences in the world a Niigata, Japan (1964) NOAA/NGDC, National Geophysical Data Center. b Tumaco, Colombia (1979) elpais.com.co, 12/12/2012, c Loma Prieta, USA (1989) NOAA/NGDC, Wilshire, U.S. Geological Survey, d Christchurch, New Zealand (2011) NOAA/NGDC, Steve Taylor (Ra White) (Source Herrera [16])

During liquefaction, a weakly consolidated (contractive) and saturated granular soil loses, entirety or partially, its shear strength. This phenomenon is associated with the softening that these soils exhibit when the frequency contents and amplitude of the earthquake trigger an excessive increase in porewater pressure [17]. At the point when pore pressure ratio |r_u| (the ratio of the excess pore pressure to the initial effective confining stress) equals or is close to unity, or when some threshold of cyclic shear strain amplitude is reached, liquefaction is said to occur. Thus, soil deposits with potentially liquefiable layers may present substantial variations in their seismic response, such as the change of the fundamental frequency, the amplification of frequencies below 1 Hz, and the attenuation of frequencies above 1 Hz [18,19,20].

Given that large shear strain and damage associated with liquefaction may occur, several earthquake-resistant construction regulations prohibit the construction of buildings on liquefiable soil types [21]. Conversely, regulations, such as the Colombian Norm for Earthquake-Resistant Construction (NSR10) [22], require advanced geotechnical and seismic response studies to be carried out when structures are planned on potentially liquefiable soils. It must be highlighted that before the soil reaches the liquefaction trigger, its stiffness may be dramatically reduced, as a result of the reduction of the effective confinement stress caused by the excess pore pressure. As a result, excessive soil shear strains occur, and, in general, a diverse seismic soil response that triggers differences in the structure response, so the effect of soil stiffness reduction on the structures must be considered.

Furthermore, there are soil configurations that cause a complex manifestation of surface liquefaction, and more research is needed on these effects [23]. A clear example of this is discussed here, namely, a non-liquefiable surface layer above a liquefiable sand layer. Despite the complexity involved, there are approximate approaches that suggest qualitatively and quantitatively correlating the thickness of the non-liquefiable surface layer and the liquefiable sand layer with the occurrence of damage [24,25,26].

Although some authors have developed conceptual frameworks and models for seismic risk assessment of urban centres considering the seismic hazard, physical vulnerability, exposure, and socioeconomic components of the different units of a city or country [27,28,29,30,31,32], little attention has been given to different liquefaction-induced building settlements (LIBS) approaches and multi-hazard analysis.

Due to population growth, urban centres have expanded to areas where the soils are prone to liquefaction; therefore, the region’s geological and geotechnical characteristics and seismic environment can lead to structural deformations and subsequent damage to the buildings. Neglecting the possibility of soil liquefaction triggering (or initiation) or the possibility of manifestation severity at the surface can lead to underestimating infrastructure susceptibility to damage. Undoubtedly, evaluating the vulnerability of structures to the combined action of earthquakes and liquefaction requires a characterization of both hazards. This is crucial as the simultaneous occurrence of these hazards can result in changes of the vulnerability of the exposed elements [3].

Therefore, it is important to characterize, first, both earthquakes and liquefaction hazards to accurately evaluate the vulnerability of the built environment and take appropriate measures to mitigate potential damage. This characterization, in the case of seismic hazard, is usually conducted from deterministic or probabilistic approaches, where the different sources of earthquakes must be considered, and the historical seismicity of the region must be known [27, 33, 34]. The methods have been widely used in engineering works, and, in recent years, have focused on probabilistic seismic hazard assessment (PSHA) approaches, incorporated on a performance-based earthquake engineering (PBEE) methodology, since these consider the necessity and uncertainty of seismic parameters.

The results of a PSHA are typically presented in the form of a seismic hazard curve, which graphically illustrates the relationship between the mean annual rate (${\uplambda }^{*}$) of exceeding an intensity measure (IM) versus the IM [35]. PSHA considers a range of possible seismic scenarios and generates probability distributions for earthquake intensity in a given area and are based on the method of seismic risk analysis introduced by Cornell [36]. By combining information on historical seismicity, past earthquake records, and local geology, probabilistic models can provide realistic and reliable estimates of long-term seismic hazard. On the other hand, deterministic models can be found, which are based on a single specific seismic scenario with a previously defined magnitude and location [37].

In the case of liquefaction hazard characterization, PBEE incorporates probabilistic liquefaction triggering or probabilistic liquefaction manifestation severity approaches and contributions from all hazard levels and all earthquake magnitudes through PSHA [35]. This results in a probabilistic liquefaction hazard assessment (PLHA) . The assess of this hazard in terms of liquefaction triggering or liquefaction manifestation severity assessment and seismic hazard assessment is made based on the joint interaction between the earthquake, and the geotechnical and the geological characteristics. Therefore, PLHA is a multi-hazard interaction, that is, both the earthquake and the liquefaction hazards occur interdependently.

Considering that the same phenomenon drives both hazards, it is assumed that both hazards overlap spatially and temporally. However, conducting a coupled analysis that considers both seismic shaking and liquefaction, particularly at the local or urban scale, requires extensive laboratory or field tests to determine the exact timing in which an earthquake triggers the liquefaction phenomenon. Such an approach would be impractical and resource-intensive for studies at territorial scale. Therefore, a multi-hazard analysis without the time variable will be considered. To include the spatial interaction of the two hazards, PLHA gives a better understanding of the liquefaction hazard at a location. This is achieved by combining the results of a zone liquefaction triggering assessment or manifestation severity with the results of PHSA, seismic site amplification factors (seismic site effects), and considering the geotechnical specificities (e.g., the non-liquefiable surface layer above a liquefiable sand layer).

The key question that arises is what is the appropriate and reliable probabilistic approach for the liquefaction triggering assessment or liquefaction manifestation severity within the context of PLHA, as well as within a vulnerability or risk model that encompasses both seismic and liquefaction hazards into a specific region? To address this query, a comprehensive literature review is conducted to gain insights into the probabilistic approaches used for evaluating the liquefaction triggering and the severity of the liquefaction manifestation severity at the surface, and to define the criteria to choose an appropriate and reliable probabilistic model. A logic tree approach is employed for this purpose, with the intent of integrating it into PLHA [38, 39]. The assessment employs the distinctive probabilistic liquefaction models tailored to a region marked by distinct geotechnical properties. Specifically, the region features a non-liquefiable surface layer overlying a stratum of liquefiable sand.

Here, it is also shown how these models are used in PLHA, which is needed within a PBEE framework. This way, it is possible to show the differences between diverse approaches, such as probabilistic liquefaction triggering assessment (PLTA) and probabilistic liquefaction manifestation severity assessment (PLMSA), and probability liquefaction hazard assessment (PLHA). The challenges inherent in these types of analyses are also discussed.

The objective of the paper is to delve into the basis for a multi-hazard framework that integrates PSHA and, PLTA or PLMSA within a comprehensive PLHA. Thus, the scope of the paper is to understand the liquefaction phenomenon and its interrelation with earthquakes from a probabilistic point of view. The ultimate objective is to incorporate these concepts into multi-hazard vulnerability assessment (MHVA). This framework will make a compared analysis of vulnerability for two different hazards (earthquake and liquefaction) easier and accounts for possible triggering effects in the structures.

This paper is organized as follows. First, considerations regarding literature review are presented in Sect. 2. A critical review and discussion of probabilistic liquefaction triggering assessment approaches and probabilistic liquefaction hazard assessment analyses is then reported (Sect. 3). This is followed by a discussion of a liquefaction probabilistic approach for a multi-hazard interaction framework to standardize the approach for use in vulnerability and risk assessment models arising from earthquake and liquefaction hazard interactions in areas with limited input data available and specific soil conditions (Sect. 4). Finally, in Sect. 5, the main conclusions and future endeavours are discussed, along with the limitations of the current study.

2 Considerations

This section consists of defining the basis for (i) a comprehensive literature review on PLTA and PLMSA used in urban scale studies focusing on semi-mechanistic “simplified stress-based” models obtained from drilling tests or in-situ sounding methods, and (ii) a comprehensive literature review on PLHA used in urban scale studies.

For the comprehensive literature review on PLTA and PLMSA approaches, it is important first to evaluate the liquefaction potential and the expected level of soil deformation. This could be obtained through determinist approaches in function of (i) a factor of safety against triggering of liquefaction (${{\text{FS}}}_{{\text{liq}}}$) or a liquefaction resistance parameter (N_req), or (ii) through an approach in function of a liquefaction manifestation severity index: liquefaction potential index (LPI), liquefaction severity number (LSN), or Ishihara-inspired liquefaction potential index (LPI_ISH). The first approach is called liquefaction triggering assessment (LTA), and the second approach is called liquefaction manifestation severity assessment (LMSA).

Several methodologies for estimating PLTA and PLMSA exist. Some of these methodologies are based on estimating the ${{\text{FS}}}_{{\text{liq}}}$ throughout the soil profile and condensing that information into an index [40]. Thus, this critical literature review seeks to define the criteria to choose a single probabilistic model for liquefaction triggering assessment or liquefaction manifestation severity for a specific zone prone to liquefaction. In this sense, is important to consider (i) how the models were built, and, if possible, (ii) establish particular geotechnical conditions for each case.

To narrow the search, the approaches based on the semi-mechanistic “simplified stress-based” models obtained from drilling tests or in-situ sounding methods were chosen. These approaches for the liquefaction evaluation are based on the simplified stress-based liquefaction triggering procedure, originally proposed by [41,42,43]. According to these authors, such approaches are more practical and reliable to use for urban-scale liquefaction risk studies.

For the comprehensive literature review on PLHA, it is important to clarify first that usually, liquefaction hazards are evaluated for a single hazard level, for example, for ground motions with a 10% probability of exceedance in 50 years, i.e., 475-years return period (RP). In some cases, two hazard levels may be considered, but different performance objectives (e.g., minimum safety factors) would typically be required for the different hazard levels [44]. For a given performance objective, only one hazard level is usually considered. In contrast, PBEE provides an approach in which contributions from all hazard levels can be incorporated into a performance evaluation to compute the mean annual rate of non-exceedance (or inverse of the RP) of a selected ${{\text{FS}}}_{{\text{liq}}}$ through PLHA [44].

The objective of the literature review on the PLHA is to review the methodologies that were carried out to combine the results of PSHA, seismic site amplification factors, and a PLTA or PLMSA approach. These studies are framed within the PBEE framework, which allows for defining the rate of occurrence of the liquefaction phenomenon and the severity of the phenomenon for a specific RP. Both are related to the conditions of the location where the phenomenon is being evaluated.

After performing the discussion of the PLTA and PLMSA approaches, in Sect. 4, a definition of a single model that is obtained from various models applying a logic tree approach is carried on. The criteria to define the unique model will be defined based on (i) the characteristics of the existing models, (ii) the aspects of feasibility and robustness in the implementation of these models in urban-level liquefaction risk studies, and (iii) the most reliable model for the soil conditions in a specific case study area.

3 State of the Art

The liquefaction potential assessment relies on two fundamental principles: (i) deterministic and (ii) probabilistic. While deterministic approaches are extensively used and offer insights into soil liquefaction behavior, they sometimes exhibit inconsistencies in predicting liquefaction potential when compared with actual data from sites where liquefaction has occurred. This inconsistency is primarily attributed to uncertainties within the liquefaction potential assessment model, encompassing uncertainties in the model itself and the associated parameters. To enhance engineering decisions and achieve more reliable measurements of liquefaction potential, these uncertainties are addressed by incorporating them into the liquefaction potential assessment models. Utilizing statistical techniques and databases from locations where the phenomenon is observed, models are developed.

A probabilistic framework for liquefaction potential assessment allows for its integration into a model for evaluating damage and losses in liquefaction-prone areas. However, this probabilistic framework needs to be coupled with the outcomes of probabilistic seismic hazard assessment (PSHA) to gain insights into multi-hazard scenarios. This integration is achieved through probabilistic liquefaction hazard assessment (PLHA).

3.1 A Primer on Deterministic Earthquake-Induced Soil Liquefaction Triggering and Manifestation Severity Approaches

The liquefaction potential assessment through a factor of safety against triggering of liquefaction (${{\text{FS}}}_{{\text{liq}}})$ or liquefaction manifestation severity assessment through liquefaction manifestation severity indices (LMSIs) at a given location is primarily influenced by (i) the susceptibility of the soil, (ii) the amplitude and duration of the ground shaking, (iii) the depth of the groundwater, and (iv) the depth of the saturated sandy soils [45].

According to Modoni et al. [46], the first step in liquefaction triggering assessment is to estimate the susceptibility of the subsoil to liquefaction. This analysis is carried out first on a large scale or local site studies based on geological studies (geological scale), estimating the tendency of the geological formation to liquefy. The most susceptible sediments are fills and alluvial, fluvial, marine, deltaic, and wind-blown deposits [47]. Then, the detailed stratigraphy is analysed at the local scale with geotechnical analyses (geotechnical scale).

According to Meslem et al. [48], the second step is estimating the tendency or potential for liquefaction to develop under a given ground shaking intensity level. This potential is generally based on two approaches (i) a qualitative approach (based on predefined hazard classification maps for liquefaction), and (ii) a quantitative approach. The last one is based on three hierarchy of liquefaction assessment approaches (i) wholly empirical, which require only geologic or geospatial data and are accessible to a broad userbase, (ii) semi-mechanistic or “simplified stress-based”, which requires a detailed in-situ soil geotechnical profile (in situ sounding tests) with specific data such as cone penetration test (CPT), standard penetration test (SPT), or small strain shear-wave velocity |V_s|. The Becker penetration test (BPT) is an advanced testing procedure that may expand the use in the future [49]. The third approaches includes (iii) wholly mechanistic constitutive models, which require laboratory tests and numerical simulations, and which use is generally limited to geoengineers trained in computational mechanics [50]. This last approach is also difficult and expensive, needing high-quality cyclic simple shear tests, as cyclic triaxial tests may inadequately represent the loading conditions of interest for most seismic problems [51]. Figure 2 shown the liquefaction triggering assessment is a schematic way.

Fig. 2

The main liquefaction triggering assessment stages

For simplicity of its use in urban-scale studies, this paper leaves aside the references related to wholly mechanistic constitutive models. They are complex due to the coupling between the seismic response of the soil and the triggering of porewater pressure. This is due to the required inputs and operator skills, but also to uncertainties about model adequacy and the interpretation of results [50]. The wholly empirical approaches will also be left out of the scope of this research, since these are used more for the liquefaction assessment on a regional and global scale, as proposed by Zhu et al. [52] and FEMA/NIBS [45] (e.g., as was presented by Yilmaz et al. [53]).

Obtaining high-quality undisturbed samples of saturated sandy soils is complicated and expensive. They can only be obtained for large projects for which the consequences of liquefaction may result in large costs. Therefore, simple, economic procedures are needed to estimate the cyclic resistance of sandy soils, particularly for low-risk projects and the initial screening stages of high-risk projects [54]. The most popular direct simple methods for estimating the cyclic resistance of sandy soils make use of (i) penetration resistance from the SPT (the oldest, and most widely used [51]) and CPT tests, and (ii) small shear strains like |Vs| procedure.

Concerning the qualitative approaches, they represent the levels of liquefaction susceptibility, i.e., they are used to classify the liquefaction hazard through pre-computed liquefaction hazard maps. Qualitative approaches classify the hazard from very low to very high. Differently, the quantitative approaches (Fig. 2) consist of the number of analyses to be performed in sequence, the first one includes liquefaction triggering analysis to estimate the tendency for liquefaction to develop under a given seismic input. The analysis is based on the calculation of (i) the ${{\text{FS}}}_{{\text{liq}}}$ for drilling tests—semi-mechanistic approaches—or (ii) the soil shear strains (γ) or the variation with time of the porewater pressure for a numerical model adjusted to a case study—wholly mechanistic constitutive model. Subsequently, the second analysis evaluates the liquefaction-induced surface manifestation severity to assess ground-level impacts, where LMSIs are used to quantify liquefaction severity [48].

In this study, given the practicality in urban scale studies, the second set of quantitative or semi-mechanistic approaches is focused on drilling tests or in-situ sounding methods. These approaches are known in the literature as stress-based simplified liquefaction triggering approaches or “simplified stress-based”, or simply, Seed–Idriss simplified method as it was proposed by Whitman [41] and Seed and Idriss [42]. The triggering approaches are based mainly on obtaining the ${{\text{FS}}}_{{\text{liq}}}$ at various depths (defined as the capacity of the soil to resist liquefaction divided by the seismic demand). These approaches have been subject to continuous modification since 1971, but their basic framework is unchanged. Furthermore, the approaches remain the most commonly used to evaluate liquefaction triggering in practice today [55]. In the “simplified stress-based” approaches, the ${{\text{FS}}}_{{\text{liq}}}$ is defined as:

$${{\text{FS}}}_{{\text{liq}}}=\frac{{{\text{CRR}}}_{7.5}}{{{\text{CSR}}}_{7.5}}\times {{\text{K}}}_{\upsigma }\times {{\text{K}}}_{\propto }$$

(1)

where ${{\text{CRR}}}_{7.5}$ is the cyclic resistance ratio $({\text{CRR}})$ of the soil for an M_w 7.5 earthquake. The ${{\text{CRR}}}_{7.5}$ represents the capacity of the soil to resist liquefaction during a magnitude of 7.5 earthquakes for level ground conditions and an effective overburden stress of 100 kPa [54]. This parameter may be obtained from empirical correlations between the data of the CPT test, SPT test, |V_s| profile, and liquefaction case histories or laboratory tests procedures such as [47, 56, 57]. The terms ${{\text{K}}}_{\upsigma }$ and ${{\text{K}}}_{\propto }$ are adjustment factors to incorporate the effect of effective vertical stress (overburden stress) and the initial shear stress on the horizontal plane on liquefaction resistance, respectively [57]. ${{\text{CSR}}}_{7.5}$ is the earthquake-induced cyclic (shear) stress ratio (seismic demand) for an M_w 7.5 earthquake. ${{\text{CSR}}}_{7.5}$ is measured as a function of seismic intensity and depth (liquefaction susceptible layer), and is obtained for a stress-based simplified liquefaction assessment triggering model as follows:

$${{\text{CSR}}}_{7.5}=0.65\frac{{{\text{a}}}_{{\text{max}}}}{{\text{g}}}\frac{{\upsigma }_{{\text{vo}}}}{{\upsigma }_{{\text{vo}}}^{\mathrm{^{\prime}}}}\frac{{{\text{r}}}_{{\text{d}}}}{{\text{MSF}}}$$

(2)

where ${{\text{a}}}_{{\text{max}}}$ is the peak ground acceleration (PGA) on the ground surface in units of g for a specific earthquake scenario (e.g., 475-year of RP), ${\upsigma }_{{\text{vo}}}$ is the initial total vertical stress at a depth of interest, ${\upsigma }_{{\text{vo}}}^{\mathrm{^{\prime}}}$ is the initial effective vertical stress at the depth of interest, and ${{\text{r}}}_{{\text{d}}}$ is an empirical depth-stress reduction coefficient that accounts for the non-rigid response of the soil profile [58]. ${\text{MSF}}$ is a magnitude scaling factor that accounts for the effects of the shaking duration on liquefaction triggering [58], and is given by:

$${\text{MSF}} = \left( {{\raise0.7ex\hbox{${{\text{M}}_{{\text{w}}} }$} \!\mathord{\left/ {\vphantom {{{\text{M}}_{{\text{w}}} } {7.5}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${7.5}$}}} \right)^{ - 2.56}$$

(3)

According to [58], the studies proposed in Idriss and Boulanger [47] and Boulanger and Idriss [56] pose variants concerning the stress-based simplified liquefaction evaluation procedure using an ${{\text{r}}}_{{\text{d}}}$ relationship that was developed in Idriss [59]. This simplification for obtaining ${\text{CSR}}$ is widely used in liquefaction risk studies, as recommended in the technical guide developed by Lai et al. [60]. The work developed by Lai et al. [60] serves as technical guidelines for updating the standards and the operational criteria currently used in different countries worldwide to construct seismic microzonation maps of liquefaction hazards. Also, the authors recommend working with the simplified method to calculate ${\text{CSR}}$ for evaluating the liquefaction triggering or liquefaction potential in an area. Some modifications have been done to calculate the ${{\text{FS}}}_{{\text{liq}}}$, in terms of the ${{\text{r}}}_{{\text{d}}}$ and ${\text{MSF}}$ parameters, as described in Green et al. [58].

Regarding the ${\text{CRR}}$ parameter, commonly, the CPT test estimated the ${\text{CRR}}$ as a function of the normalized overburden corrected cone tip resistance $\left({{\text{q}}}_{{\text{C}}1{\text{N}}}\right)$ or the equivalent clean sand of the $({{\text{q}}}_{{\text{C}}1{\text{N}}})$, denoted as ${\left({{\text{q}}}_{{\text{C}}1{\text{N}}}\right)}_{{\text{CS}}}$. The SPT test is based on a connection with the SPT blow count normalised to an over-burden pressure of approximately 100 kPa and a hammer energy ratio or hammer efficiency of 60% ${({{\text{N}}}_{1})}_{60}$, or the equivalent clean sand of the ${({{\text{N}}}_{1})}_{60}$, denoted as ${\left({{\text{N}}}_{1}\right)}_{60,{\text{cs}}}$. Finally, |V_s| test estimates the resistance as a function of the shear wave velocity corrected to an effective over-burden pressure of approximately 100 kPa, $\left({{\text{V}}}_{{\text{s}}1}\right)$ or the equivalent clean sand of the ${{\text{V}}}_{{\text{s}}1},$ denoted as ${\left({{\text{V}}}_{{\text{s}}1}\right)}_{{\text{cs}}}$. In this paper, it refers to any of these parameters as liquefaction resistance parameters (N_req).

When dealing with a continuous test, the CPT is the most popular because of its higher repeatability and the continuous nature of its profile [54]. Besides, due to the inherent difficulties and poor repeatability associated with the SPT, several correlations have been proposed to estimate ${\text{CRR}}$ for clean sands and silty sands using a corrected CPT penetration resistance [54]. However, the cost of the CPT and SPT tests stands out, which sometimes makes it impractical in urban-scale studies and where, due to lack of financial resources and execution time, other methods are required.

Because of this, the results of the |V_s| test are recommended to obtaining the N_req. The emphasis lies in highlighting the continuous nature of data acquisition in this test, along with the capacity to quantitatively assess a fundamental soil property [61]. However, some studies, such as NZGS [62], have reported that the |V_s| test is not as robust as the CPT test to obtain the ${\text{CRR}}$ of the soil. The test must be chosen according to the project or research conditions and simplifications to obtain reliable, not excessively costly, and fast results. In this sense, the work developed by [63], compiled the advantages and disadvantages of various in-situ tests for assessment of the ${\text{CRR}}$.

There are several models for obtaining ${\text{CRR}}$ at various depths using data derive from comprehensive in situ sounding methods. Works that have made important contributions regarding ${\text{CRR}}$ models are shown in Table 1.

Table 1 Liquefaction triggering assessment models to obtain the CRR at each depth of a soil profile from CPT, SPT or |V_s| data

A common way of the deterministic approach is to use charts, as suggested by Andrus and Stokoe II [65], as is shown in Fig. 3. The horizontal axis could correspond to any of the N_req, and the vertical axis measures the intensity of ground motion through the cyclic stress ratio (${\text{CSR}}$). Points are drawn to represent cases of liquefaction (filled marks) and non-liquefaction (unfilled marks). The deterministic boundary curve or liquefaction limit state, is subjectively drawn to separate events of the two types (derived from empirical data) [66]. The line is then used as a deterministic classification criterion and represents the ${\text{CRR}}$ obtained from some models shown in Table 1.

Fig. 3

Example of the deterministic method for liquefaction assessment derived from empirical data (Source Andrus and Stokoe II [65])

As for the second set of quantitative approaches, called liquefaction manifestation severity approaches, they use information from the first group in terms of the ${{\text{FS}}}_{{\text{liq}}}$ at depth within a profile. These approaches are based on stress-based simplified liquefaction triggering, and give an indication of the severity of the phenomenon on the surface by integrating the estimated effects of liquefaction in the first 20 m depth [67]. The most widespread LMSIs to evaluate the liquefaction severity or damage potential, given a measure of the liquefaction-induced surficial manifestations evidence or liquefaction-induced ground damage (in free conditions, i.e., without structures) are (i) the liquefaction potential index (LPI), (ii) the Ishihara-inspired liquefaction potential index (LPI_ISH), and (iii) the liquefaction severity number (LSN) [67].

In this sense, the LPI defined by Iwasaki et al. [68] (after Iwasaki et al. [69, 70]) or its modified version defined by Ishihara [24] is commonly used to quantify the manifestation severity of liquefaction in the free field. This parameter represents the cumulative response of a soil deposit [71] besides the influence of the soil properties, the ground motion IM (e.g., PGA), the ${{\text{FS}}}_{{\text{liq}}}$, and the thickness of the liquefiable layer (< 20 m in depth) at the site [72]. The effect of liquefaction at depths greater than 20 m is assumed to be negligible since no surface effects from liquefaction at such depths have been reported [73]. The ${\text{LPI}}$ is computed as:

$${\text{LPI}}={\int }_{0}^{20\mathrm{ m}}{\text{F}}\cdot {\text{w}}\left({\text{z}}\right){\text{dz}}$$

(4)

where w(z) is a weighting function given by ${\text{w}}\left({\text{z}}\right)=10-0.5{\text{z}}$, in which ${\text{z}}$ represents the depth evaluated in meters, and ${\text{F}}=1-{{\text{FS}}}_{{\text{liq}}}$, with ${{\text{FS}}}_{{\text{liq}}}$ calculated at each depth for the first 20 m of depth. Note that liquefaction occurs when ${{\text{FS}}}_{{\text{liq}}}$ ≤ 1. For ${{\text{FS}}}_{{\text{liq}}}$ > 1, F = 0, which means no liquefaction. Moreover, according to the research given by [69], significant liquefaction is anticipated in areas where LPI > 15. LPI has been used to evaluate liquefaction danger across the world based on this standard [23].

Inspired by Ishihara [24], a modified LPI was proposed by Maurer et al. [74] using limit-state curves or charts for predicting manifestations as a function of the non-liquefiable surface layer thickness (H₁), among other factors (these curves will be explained at the final of this subsection). Given its provenance, the result was termed Ishihara-inspired LPI (LPI_ISH) and is defined as:

$${\text{LPI}}_{{{\text{ISH}}}} = \int_{{{\text{H}}_{1} }}^{{20{\text{~m}}}} {{\text{F}}_{{{\text{LPI}}_{{{\text{ISH}}}} }} \left( {{\text{FS}}_{{{\text{liq}}}} } \right)\frac{{25.56}}{{\text{z}}}{\text{dz}}}$$

(5)

where

$$\begin{gathered} {\text{F}}_{{{\text{LPI}}_{{{\text{ISH}}}} }} \left( {{\text{FS}}_{{{\text{liq}}}} } \right) = 1 - {\text{FS}}_{{{\text{liq}}}} {\text{ if FS}}_{{{\text{liq}}}} \le 1{\text{ and H}}_{1} \cdot {\text{m}}\left( {{\text{FS}}_{{{\text{liq}}}} } \right) \le 3 \hfill \\ {\text{F}}_{{{\text{LPI}}_{{{\text{ISH}}}} }} \left( {{\text{FS}}_{{{\text{liq}}}} } \right) = 1 - {\text{FS}}_{{{\text{liq}}}} {\text{ if FS}}_{{{\text{liq}}}} \le 1{\text{ and H}}_{1} \cdot {\text{m}}\left( {{\text{FS}}_{{{\text{liq}}}} } \right) \le 3 \hfill \\ \end{gathered}$$

(6)

and

$$\begin{gathered} {\text{m}}\left( {{\text{FS}}_{{{\text{liq}}}} } \right) = {\text{exp}}\left\{ {\frac{5}{{25.56 \cdot \left( {1 - {\text{FS}}_{{{\text{liq}}}} } \right)}}} \right\} - 1\quad {\text{if FS}}_{{{\text{liq}}}} \le 0.95 \hfill \\ {\text{m}}\left( {{\text{FS}}_{{{\text{liq}}}} } \right) = 100\quad {\text{if FS}}_{{{\text{liq}}}} > 0.95 \hfill \\ \end{gathered}$$

(7)

where H₁ is the thickness of the non-liquefiable surface layer, taken as the depth from the ground surface to the first instance where ${{\text{FS}}}_{{\text{liq}}}$ < 1.

According to Green et al. [75], the most notable differences between the original LPI and LPI_ISH frameworks are that the latter better accounts for the influence of the non-liquefiable surface layer on the severity of surficial liquefaction manifestations and more appropriately weights the contribution of shallower liquefied layers to surficial manifestations [76].

van Ballegooy et al. [76] proposed an alternative metric for the effects of liquefaction, the LSN. The authors stated that for the range of seismic demands imposed by the Canterbury earthquake sequence on the residential areas, LSN provides a more consistent correlation with the datasets of observed damage than LPI and thickness of the non-liquefiable surface layer [76]. The LSN is defined by:

$${\text{LSN}}=1000\int \frac{{\upvarepsilon }_{{\text{v}}}}{{\text{z}}}{\text{dz}}$$

(8)

where ε_v is the calculated post-liquefaction volumetric reconsolidation strain based on the Zhang et al. [77] strain equations. These equations are a function of the ${{\text{FS}}}_{{\text{liq}}}$ calculated from a stress-based liquefaction triggering assessment approach and the relative density of the soil determined from the CPT tip resistance and z is the depth below the ground surface in meters for depths greater than 0.0. LSN is calculated as the sum of the post-liquefaction volumetric reconsolidation strains calculated for each soil layer, divided by the depth to the midpoint of that layer [76].

A modified LSN was proposed by Upadhyaya et al. [78]. The new LMSI was denoted as LSN_ISH. According to Upadhyaya et al. [78], the LSN_ISH index merges the positive attributes of the LPI_ISH and LSN models. The derivation of the LSN_ISH index follows a procedure similar to the derivation of the LPI_ISH index and is defined as:

$${\text{LSN}}_{{{\text{ISH}}}} = \int_{{{\text{H}}_{1} }}^{{20{\text{~m}}}} {{\text{F}}_{{{\text{LSN}}_{{{\text{ISH}}}} }} \left( {\upepsilon _{{\text{v}}} } \right)\frac{{36.929}}{{\text{z}}}{\text{dz}}}$$

(9)

where

$$\begin{gathered} {\text{F}}_{{{\text{LSN}}_{{{\text{ISH}}}} }} \left( {{\upvarepsilon }_{{\text{v}}} } \right) = { }\frac{{{\upvarepsilon }_{{\text{v}}} }}{{{\text{z}}5.5}}\quad {\text{if FS}}_{{{\text{liq}}}} \le 2{\text{ and H}}_{1} \cdot {\text{m}}\left( {{\upvarepsilon }_{{\text{v}}} } \right) \le 3 \hfill \\ {\text{F}}_{{{\text{LSN}}_{{{\text{ISH}}}} }} \left( {{\upvarepsilon }_{{\text{v}}} } \right) = 0\quad {\text{if FS}}_{{{\text{liq}}}} > 2{\text{ or H}}_{1} \cdot {\text{m}}\left( {{\upvarepsilon }_{{\text{v}}} } \right) > 3 \hfill \\ \end{gathered}$$

(10)

where

$$\begin{gathered} {\text{m}}\left( {{\upvarepsilon }_{{\text{v}}} } \right) = {\text{exp}}\left\{ {\frac{0.7447}{{{\upvarepsilon }_{{\text{v}}} }}} \right\} - 1 \hfill \\ {\text{m}}\left( {{\upvarepsilon }_{{\text{v}}} < 0.16} \right) = 100 \hfill \\ \end{gathered}$$

(11)

In literature, some categories of manifestation severity of the phenomenon have been defined according to the probability of liquefaction manifestation severity, which depends on the chosen liquefaction manifestation severity index (LMSI). Table 2 defines categories or ranges for the three previously defined LMSI. However, it must be clarified that other authors have defined different ranges depending on the particular conditions of the study (depending on the liquefaction triggering model or the correlations between superficial damages and events that occurred with different magnitudes [79]).

Table 2 Definition of the categories for the liquefaction manifestation severity indices

As mentioned above, another way of defining the severity of liquefaction is from charts, which are based on the results of the manifestation on the surface after the earthquake has generated liquefaction. In this sense, the study developed by Ishihara [24] is one of the first that explore the impact of liquefaction when it is evidenced at the surface and is a basis for recent studies on liquefaction manifestation severity. Since the correlations provided are for settlements in free-field or very light structures, and do not consider the physical properties of the building and foundation, the study involves the effect on the soil’s surface rather than the damage produced in the building [81].

Ishihara [24] demonstrated that one of the factors influencing the liquefaction manifestation severity is the presence of a non-liquefiable surface layer above the liquefiable sand layer (H₁ and H₂ in Fig. 4, respectively) in the soil deposit. His findings revealed that if the non-liquefiable surface layer is thick, the uplift force caused by excess pore pressure is not strong enough to cause a breach in the superficial layer, and, there will not be manifestation of liquefaction. Thus, guidelines were proposed based on literature surveys in areas with field case histories and where liquefaction affectation had been evidenced. After the 1976 Tangshan Earthquake, with an acceleration close to 0.2 g, and the 1983 Nihonkai-Chubu earthquake, with an acceleration between 0.4 and 0.5 g, the author examined the conditions of occurrence or non-occurrence of ground damage due to liquefaction. The field data collected from liquefaction-affected areas revealed boundary curves for site identification of liquefaction-induced ground damage for both earthquakes, as shown in Fig. 4. Finally, the third curve with an acceleration of 0.3 g was incorporated for an intermediate level of shaking intensity.

Fig. 4

Proposed boundary curves for site identification of liquefaction-induced ground damage (Source Ishihara [24])

Later, following the 1990 Luzon earthquake, Ishihara [25] compiled a series of SPT data and Swedish cone penetration test data in the city of Dagupan (Philippines) to clarify the relationship between the thicknesses H₁ and H₂. The proposed relationship by the authors might or might not result in the liquefaction manifestation severity at the surface, which would be accompanied by disastrous damage to the ground surface. The results of each test are shown in Fig. 5, where the continuous line separates the areas prone to soil damage (filled dots) and those without damage (unfilled dots).

Fig. 5

Surface manifestation of liquefaction through different in situ sounding methods: (left) Effects of the non-liquefiable surface layer on the damage on the ground surface (by SPT), (right) Effects of the non-liquefiable surface layer on the damage on the ground surface (by Swedish cone) (Source Ishihara [25])

Finally, Towhata [26] using data from residential areas impacted by the 2011 M_w 9.0 in Tohoku, Japan, earthquake, developed a procedure that combines the influence of the thickness of the non-liquefiable surface layer (H₁) with LPI in the chart format, shown in Fig. 6. The chart consists of five zones of surficial liquefaction manifestations from unlikely to highly probable, as listed in Table 3.

Fig. 6

Qualification chart in terms of LPI and H1 (Source Towhata [26])

Table 3 Judged liquefaction quality of subsoil conditions in residential areas (Source Towhata [26])

3.2 Probabilistic Liquefaction Triggering Assessment (PLTA) and Probabilistic Liquefaction Manifestation Severity Assessment (PLMSA) Approaches

Approaches to assess the probability of liquefaction triggering or the probability of liquefaction manifestation severity are used as an alternative or complement to deterministic assessment. On many occasions, the results of both probabilistic approaches may lead to better engineering decisions, and those are more consistent measures of liquefaction potential [82]. The available approaches differ in their basis on either empirical or numerical data, and, if they predict the occurrence of liquefaction for a complete soil profile or individual layers. The approaches also differ in how they distinguish between liquefaction and non-liquefaction. Nevertheless, none of these methods is yet able to account for all types of site-specific effects, including soil profile and site-specific ground motion hazard [40].

Probabilistic liquefaction triggering assessment or probabilistic liquefaction manifestation severity approaches in the form of a probabilistic equation, expressing a parameter of intensity versus a probability of liquefaction triggering or probability of liquefaction manifestation severity, respectively. In their basis, these approaches considered the model and parameters uncertainties, giving a more realistic assessment of the liquefaction hazard. Due to these uncertainties, there is a probability that liquefaction will occur even if ${{\text{FS}}}_{{\text{liq}}}$ > 1 and, also, there is probability that liquefaction will not happen even if ${{\text{FS}}}_{{\text{liq}}}\le$ 1 [83].

The probabilistic approaches are essential for vulnerability and risk models which consider the consequence of liquefaction in the performance of buildings [47] and have the advantage that the user can set an acceptable level of risk [64]. Furthermore, they may be used within a PBEE framework, which includes contributions from different return periods, not only the return periods but by various codes or regulations [35]. Estimating the probability of liquefaction at a given site is a critical first step in calculating the seismic risk on lifelines and structures, such as lateral spreading and ground settlement [40].

The estimation of the probability of either liquefaction triggering or the probability of liquefaction manifestation severity (denoted as ${{\text{P}}}_{{\text{L}}}$), or the conditional probability of an engineering demand parameter (EDP) in the presence of liquefaction given a certain intensity measure (IM), often referred to as P(EDP|IM) in the context of the PBEE framework, is an essential requirement for the inclusion of this condition in Eq. 12. As proposed by Bird et al. [84], in Eq. 12, three components are considered, (i) the probability of liquefaction, P(EDP|IM₁), which corresponds to a given level of an IM₁, (ii) a probabilistic term to predict a consequence of liquefaction on buildings to a given level of an IM₂ and a damage state level DS_i, assuming liquefaction has occurred, P[DS_i|Liquefaction induced by shaking], and (iii) a probabilistic term to predict a consequence of seismic on buildings to a given level of an IM₃ and a damage state level DS_i, assuming liquefaction has not occurred, P[DS_i|Shaking only]. The IM₁, IM₂, and IM₃ could be the same. In this sense, working with a probability liquefaction triggering or probability liquefaction manifestation severity approach gives the possibility of the comparison of the liquefaction-induced ground damage with other hazards. Moreover, it enables the exploration of potential concurrent effects arising from both hazards as part of a multi-hazard vulnerability assessment (MHVA) framework. This approach allows for an evaluation of the relative likelihood of foundation damage resulting from soil liquefaction in comparison to the likelihood of structural damage induced by earthquakes [85], as exemplified below:

$${\text{P}}\left( {{\text{DS}}_{{\text{i}}} } \right) = {\text{P}}\left( {{\text{EDP|IM}}_{1} } \right) \times {\text{PDS}}_{{\text{i}}} |{\text{Liquefaction}}\,{\text{induced}}\,{\text{by}}\,{\text{shaking}} + \left( {1 - {\text{P}}\left( {{\text{EDP}}|{\text{IM}}_{1} } \right)} \right) \times {\text{ P}}\left[ {{\text{DS}}_{{\text{i}}} |{\text{Shaking}}\,{\text{only}}} \right]$$

(12)

In the literature, there are several approaches that define the ${{\text{P}}}_{{\text{L}}}$ or the P(EDP|IM₁) term. The EDP may be the ${{\text{FS}}}_{{\text{liq}}}$, which is smaller than the ${{\text{FS}}}_{{\text{liq}}}$* value. This may also be the liquefaction resistance parameter (N_req), obtained from an SPT, CPT or |V_s| test, which is larger than the N_req* value, when it is referring to PLTA approaches. The EDP may also be a LMSI (e.g., LPI, LPI_ISH, or LSN), which is larger than a LMSI_i value or damage state (e.g., i: 1 = slight damage, i: 2 = moderate damage, and i: 3 = severe damage), when is referring to PLMSA approaches. In both cases the IM is the peak ground acceleration (PGA) on the ground surface ${({\text{a}}}_{{\text{max}}})$.

These approaches have been obtained from (i) real field cases where the manifestation of liquefaction on the surface was evidenced after an earthquake (wholly empirical approaches), (ii) numerical simulations and validated from real field cases or experimental tests (wholly mechanistic constitutive approaches), and (iii) in situ sounding methods like SPT, CPT or |V_s| combine with empirical formulations (semi-mechanistic approaches).

There is a large number of available probabilistic approaches, which differ in terms of their basis on either empirical data [40], i.e., the existing studies have been obtained mainly with a predominance of Japanese, California, and New Zealand data. Thus, the guiding question for urban-scale research is, which is/are the most appropriate approach/es for a specific case study area where no liquefaction cases have been reported and limited input data are available?

Due to the simplicity, easy and practical use for urban-scale, semi-mechanistic approaches obtained from drilling tests or in-situ sounding methods (i.e., require subsurface information) are discussed here, which include (i) CPT and SPT-based probabilistic methods, (ii) shear wave velocity-based probabilistic methods. Both develop relationships between the CPT, SPT or Vs data and the simplified procedures for soil liquefaction assessment (mainly in the ${\text{CRR}}$ value, see Table 1), and then, use probabilistic methods to estimate the probability of liquefaction triggering as a function of the ${{\text{FS}}}_{{\text{liq}}}$ or as a function of a N_req. Finally, the third one, (iii) LMSI-based probabilistic methods which develop relationships between the LPI, LSN or LPI_ISH data and the liquefaction triggering assessment, and then use probabilistic methods to estimate the probability of liquefaction manifestation severity as a function of a LMSI.

From an engineering point of view and as evidenced in Eq. 12, probabilistic liquefaction triggering assessment or probabilistic liquefaction manifestation severity approaches are useful in allowing the comparison of liquefaction-induced structural damage with seismic-induced structural damage. The probabilistic liquefaction approaches can be used as an alternative to deterministic methods, which provide a better engineering decision-making [86]. For example, the engineer would like to know how the vulnerability of foundation failure induced by liquefaction compares to the risk of seismically induced structural damage [86]. Some authors who have worked on probabilistic methods for PLTA or PLMSA approaches based on the sounding methods or semi-mechanistic approaches are listed in Table 4, which was built according to the criteria defined in Sect. 2.

Table 4 Some relevant works about PLTA and PLMSA approaches obtained from semi-mechanistic approaches

Table 4 summarises contributions of most investigations in this subsection with a brief description of the formulation, including the statistical technique used to evaluate the probability of liquefaction triggering or the probability of liquefaction manifestation severity (${{\text{P}}}_{{\text{L}}})$, as well as the type of in-situ sounding method or in-situ test to obtain the liquefaction resistance parameter (N_req). Furthermore, it is also reported whether the research verified geotechnical profiles with a non-liquefiable surface layer or not, considering that the final step in this article is to define a probabilistic model in places with this condition.

3.3 A Probabilistic Liquefaction Hazard Assessment (PLHA)

The previously mentioned deterministic/probabilistic methods are applicable for a specific performance level or earthquake scenario (e.g., 475-year of RP); however, they do not address overall liquefaction potential by taking all possible earthquake scenarios into account, i.e., possible combinations of earthquake magnitude and source-to-site distance [123]. The actual probability of liquefaction would be the probability of liquefaction occurring during an earthquake when considered together with the probability of an earthquake of such magnitude occurring [98].

PLHA is related to a performance-based liquefaction potential evaluation and allows us to answer the question: what is the probability that liquefaction will occur at a given site during a given time period, given the subsurface conditions and seismic hazard at the site? [40]. According to Marrone et al. [124], a modified version of the advanced seismic hazard/liquefaction evaluation (ASHLE) method, referred to as ASHLES, was the pioneering development in formulating a PSHA analysis. ASHLES innovatively integrated emerging relationships on the conditional probability of liquefaction into the computational framework of probabilistic seismic hazard assessment. This integration greatly facilitated the simultaneous computation of probabilistic ground motions and the probability of liquefaction.

PLHA involves developing a probabilistic seismic hazard model, that considers the variability and uncertainty in the seismic hazard and the soil properties. Then a PLTA or PLMSA approach is used to estimate the liquefaction RP in a region, as shown in Kramer and Mayfield [35]. PLHA is framed into a PBEE framework to incorporate the occurrence of the ground motion (PGA and M_w pairs) obtained from PSHA into the calculation of a liquefaction triggering or manifestation approach [125]. The PBEE framework allows to incorporate contributions from all hazard levels into a performance evaluation to compute an RP for liquefaction-triggering [44].

Overall, probabilistic liquefaction hazard assessment approaches can provide a more comprehensive understanding of the potential for liquefaction in a given area, considering all possible scenarios, rather than the conditional one (i.e., liquefaction potential evaluation carried out for a specific earthquake scenario or ground motion level). However, these methods require accurate and reliable data, and expert knowledge in statistics and geotechnical engineering to properly apply and interpret the results. Therefore, PLHA is not a popular liquefaction evaluation procedure in the practice [125].

According to Kramer and Mayfield [35], in assessing liquefaction hazards at a potentially liquefiable site, the initial responsibility of the geotechnical engineer primarily revolves around the evaluation of ${{\text{P}}}_{{\text{L}}}$ or P(EDP|IM). Subsequently, these findings are integrated with the results obtained from a PSHA approach, which includes considerations of seismic site amplifications. Finally, this comprehensive approach facilitates the calculation of the mean annual rate of no exceedance of a selected ${{\text{FS}}}_{{\text{liq}}}$ [35] or the mean annual rate of exceedance of a liquefaction manifestation severity index [67]. The reciprocal of the mean annual rate of no exceedance of liquefaction triggering and the reciprocal of the mean annual rate of exceedance of liquefaction manifestation is the RP of the liquefaction triggering and the liquefaction manifestation severity at a site of interest, respectively.

According to Kramer and Mayfield [35] PLHA could be computed for probabilistic liquefaction triggering as:

$${\uplambda }_{{{\text{FS}}_{{{\text{liq}}^{*} }} }} = \mathop \sum \limits_{{{\text{j}} = 1}}^{{{\text{N}}_{{{\text{M}}_{{\text{w}}} }} }} \mathop \sum \limits_{{{\text{i}} = 1}}^{{{\text{N}}_{{{\text{PGA}}}} }} {\text{P}}\left[ {{\text{FS}}_{{{\text{liq}}}} \left\langle {{\text{FS}}_{{{\text{liq}}^{*} }} } \right|{\text{pga}} = {\text{pga}}_{{\text{i}}} ,{\text{m}} = {\text{m}}_{{\text{j}}} } \right]\Delta {\uplambda }_{{{\text{pga}}_{{\text{i}}} ,{\text{m}}_{{\text{j}}} }}$$

(13)

where ${N}_{{M}_{w}}$ and ${N}_{PGA}$ are the number of magnitude and peak acceleration increments into which hazard space is subdivided, $\Delta {\uplambda }_{{pga}_{i},{m}_{j}}$ is the incremental mean annual rate of exceedance for intensity measure, ${{\text{pga}}}_{{\text{i}}}$, and magnitude, ${{\text{m}}}_{{\text{j}}}$, ${\text{P}}\left[{{\text{FS}}}_{{\text{liq}}}<{{\text{FS}}}_{{{\text{liq}}}^{*}}|{\text{pga}}={{\text{pga}}}_{{\text{i}}},{\text{m}}={{\text{m}}}_{{\text{j}}}\right]$ is the conditional probability of liquefaction triggering that ${{\text{FS}}}_{{\text{liq}}}$ no exceeds some threshold value, ${{\text{FS}}}_{{{\text{liq}}}^{*}}$, conditioned on PGA and M_w pairs, and can be calculated using a PLTA approach (e.g., some of the Table 4). The RP of liquefaction corresponds to the reciprocal of the mean annual rate of non-exceedance ${\uplambda }_{{FS}_{{liq}^{*}}}$, i.e., RP = 1/${\uplambda }_{{FS}_{{liq}^{*}}}$. According to Geyin and Maurer [67], PLHA could be computed for probabilistic liquefaction manifestation severity as:

$${\uplambda }_{{\text{LMSL}}}=\frac{1}{{\text{RP}}}={\int }_{{\text{MSI}}=0}^{\infty }{{\text{F}}}_{{\text{LMSL}}}\left({\text{LMSI}}\right)\times \left|\frac{{\mathrm{d\lambda }}_{{\text{LMSI}}}}{{\text{dLMSI}}}\right|\times {\text{dLMSI}}$$

(14)

where ${{\text{F}}}_{{\text{LMSL}}}\left({\text{LMSI}}\right)$ is the fragility function or PLMSA approach for a particular liquefaction manifestation severity level (LMSL) (e.g., minor manifestation, moderate manifestation, or severe manifestation), ${\uplambda }_{{\text{LMSL}}}$ is the mean annual exceedance frequency of the LMSL, and $\left|\frac{{\mathrm{d\lambda }}_{{\text{LMSI}}}}{{\text{dLMSI}}}\right|$ is the absolute value of the derivative of the liquefaction manifestation severity index hazard curve. This curve can be computed as follows:

$${\uplambda }_{{{\text{LMSI}}}} = \mathop \sum \limits_{{{\text{j}} = 1}}^{{{\text{N}}_{{{\text{M}}_{{\text{w}}} }} }} \mathop \sum \limits_{{{\text{i}} = 1}}^{{{\text{N}}_{{{\text{PGA}}}} }} {\text{P}}\left[ {{\text{LMSI}} > {\text{LMSI}}_{{\text{i}}} { }|{\text{pga}} = {\text{pga}}_{{\text{i}}} ,{\text{m}} = {\text{m}}_{{\text{j}}} } \right]\Delta {\uplambda }_{{{\text{pga}}_{{\text{i}}} ,{\text{m}}_{{\text{j}}} }}$$

(15)

The data obtained from a comprehensive PLHA enables the establishment of concurrent seismic and liquefaction scenarios or damage scenarios for these hazards, typically for a defined return period (RP). This RP is often chosen to correspond to a specific level of liquefaction severity at the surface. This information is then employed to conduct numerical analyses and understand how the soil deposit responds when considering its influence on structural behaviour. Utilising the Poisson model, it becomes feasible to compute the probability of either liquefaction triggering or the probability of liquefaction manifestation severity within a specified time frame (${\lambda }_{t })$ as:

$${\lambda }_{t }=1-{e}^{-{\lambda }_{LIQ }t}$$

(16)

where ${\lambda }_{LIQ}$ is the annual probability or rate of no exceedance or exceedance, according to the approach used.

Authors who have engaged in developing probabilistic methodologies for comprehensive probabilistic liquefaction hazard assessment (PLHA), using stress-based simplified procedures for liquefaction triggering or manifestation, are presented in Table 5. The compilation of Table 5 adhered to the criteria delineated in Sect. 2. This table provides a concise overview of the primary innovative contributions from most of the studies within PLHA. It briefly describes the formulation approach and specifies whether a probabilistic or deterministic methodology was employed to assess liquefaction hazard.

Table 5 Some relevant works on PLHA

4 Definition of a Probabilistic Liquefaction Approach to Do a Fully PLHA in Sites with a Non-liquefiable Surface Layer Over a Liquefiable Layer

The above-introduced studies provide a whole range of probabilistic approaches to assess different aspects of soil liquefaction. The investigations include the characterization with probabilities ranging from the deterministic boundary curves to liquefaction limit state curves. These curves define a boundary that separates liquefaction from no-liquefaction occurrence, obtained from SPT, |V_s|, CPT-based evaluation methods (e.g., Fig. 3), from a probabilistic estimation of soil liquefaction using logistic regression, artificial neural networks, and maximum likelihood estimation (MLE) method [135]. Approaches based on the optimisation algorithms, decision trees, and Bayesian networks are also mentioned. Juang et al. [135] conducted a systematic literature review of only PLTA approaches, discussing the formulations, key assumptions, advantages and limitations, and their applications for liquefaction assessment. Challenges and the need for further research were also addressed.

In general, until 2002, the most common approaches were the logistic regression model and the Bayesian mapping model. According to Andrus et al. [90], the last one is considered to be an improvement over the logistic regression models and is suggested for engineering risk-based design, as mentioned in [93]. Furthermore, according to [106] the studies before than the developed by Cetin et al. [51] were based on earlier and less comprehensive databases. Bayesian models relate the ${{\text{FS}}}_{{\text{liq}}}$ obtained from a deterministic method to the probability of liquefaction triggering (${{\text{P}}}_{{\text{L}}}$) obtained by Bayesian interpretation [82]. After 2010, the MLE method was used, which is a special case of the Bayesian method when a distinct prior distribution is used [135]. The study by Cetin et al. [49] states that the MLE essentially serves as an analogous function to multidimensional probabilistic “regression” analyses. However, it offers distinct advantages by (i) enabling the independent handling of diverse origins of aleatory uncertainty and (ii) streamlining the treatment of a wider array of descriptive variables (model parameters). Simultaneously, it facilitates monitoring interactions and covariances among these model parameters.

These studies highlight the importance of considering the uncertainty that is associated with the liquefaction limit state curve or the ${\text{CRR}}$ model and the uncertainties in the seismic load (${\text{CSR}}$) and the liquefaction resistance parameters (N_req) in the assessment of soil liquefaction. In this context, the utilization of Bayesian regression analysis and the first-order reliability method (FORM) proves advantageous as it allows for the incorporation of both parametric uncertainties related to input variables and uncertainties linked to the chosen CRR model [132].

Overall, these studies seek to improve the understanding and prediction ability of soil liquefaction potential using probabilistic approaches that consider the uncertainty and variability inherent to this geotechnical phenomenon to complement the deterministic liquefaction assessment [106]. When the uncertainty in the input parameters is much larger than that in the assumed database, a more advanced analysis should be performed, using, for example, reliability theory, which explicitly considers the uncertainty in each input parameter. Finally, it is also highlighted that most of the approaches correspond to the PLTA, and few ones focus on PLMSA.

The variability in liquefaction probability estimates along the liquefaction limit state curves within probabilistic liquefaction triggering assessment (PLTA) approaches is attributable to the dependence on specific liquefaction triggering assessment models. This dependency extends to the calculation of the cyclic resistance ratio (${\text{CRR}}$) and encompasses the inherent uncertainties associated with both the models themselves and the pertinent parameters, encompassing seismic and geotechnical factors. For example in the study of Juang et al. [82], which employed the SPT-based simplified model developed by Seed et al. [136], the boundary curve was characterised by an average probability of approximately 30%. However, in a subsequent work by Juang et al. [83], the same boundary curve was associated with a probability of 24%. The PLMSA approaches are seen in the form of PLMSA functions [67, 80, 121, 122], and are a good option when there is no clear boundary of the LMSIs to delineate the liquefied from the non-liquefied sites [118]. The PLMSA approaches can effectively account for the uncertainty in the LMSIs predictions [118].

Deterministic triggering and manifestation severity approaches have been used worldwide, considering deterministic classification thresholds. Although they have been verified in real scenarios where liquefaction has occurred, in some cases, they are not reliable, as evidenced in the work of Lee et al. [137] and Maurer et al. [71]. As mentioned by Geyin and Maurer [67], liquefaction manifestation severity indices based on deterministic classification thresholds are often used, unaware that such thresholds have inherent limitations. Namely, these approaches (i) are sensitive to the liquefaction triggering assessment analyses used to calculate them, (ii) are tied to the method used to select them and consequently to the relative consequences of the misprediction assumed therein, and (iii) inherently hide the probabilities of possible outcomes. In this sense, motivated by these uncertainties and limitations, researchers have formulated PLMSA functions that probabilistically predict the occurrence and severity of liquefaction manifestations on free-field level ground, as is the case of Geyin and Maurer [67], Upadhyaya et al. [80, 122] and Ge et al. [121] who proposed PLMSA functions at the free field level, conditioned on the LMSI: LPI, LPI_ISH, LSN, and the soil indicator ${{\text{I}}}_{{\text{c}}10}$. The approaches focused mainly on the occurrence and severity of liquefaction ejecta and not any other land damage metric, e.g., as shown in Fig. 7 for Geyin and Maurer [67].

Fig. 7

Fragility functions based on LPI for Canterbury (left) and Global (right) databases of historical cases of liquefaction-induced ground damage. Note The CPT data were computed using the liquefaction triggering assessment model defined in Boulanger and Idriss [56] (Source Geyin and Maurer [67])

Given that the probabilistic models, whether PLTA or PLMSA, may be used to evaluate damage and losses in places exposed to the multi-hazard of earthquakes and liquefaction, mainly where shallow foundation systems and surficial or near-surface infrastructure exist, it is advisable to employ PLMSA approaches. The LPI, LPI_ISH and LSN indices have been shown to correlate well with the potential of liquefaction-induced ground damage for infrastructures (e.g., shallow foundations) [67]. Furthermore, these indices are related to the contribution of the non-liquefiable surface layer of soil in the manifestation of liquefaction on the surface or liquefaction-induced ground damage [55], as evidenced in the work of Green et al. [75] among others. Additionally, the use of liquefaction manifestation severity indices is useful in engineering practice for prioritising sites for more detailed investigations and regional hazard mapping [55]. The LSN_IHS index is not included in the discussion since it is a recently developed index that the academic community has not widely discussed.

Without a doubt, PLMSA approaches are convenient for working with damage models (vulnerability and risk) since LMSIs can predict different levels of liquefaction-induced ground damage or a certain LMSL, and its possibility of occurrence according to the earthquake intensity level. The opposite occurs with PLTA since they are not directly associated with the levels of liquefaction-induced ground damage, and what they reflect is the probability of liquefaction triggering occurring in the susceptible layer (i.e., yes/no liquefaction), which due to soil conditions may not be reflected on the surface (liquefaction-induced ground damage) [128].

This paper also aims to provide criteria for choosing a unique probabilistic approach after applying a logic tree approach for liquefaction triggering or manifestation severity assessment. This unique approach aims to be integrated within a multi-hazard interaction assessment seen in a PLHA. According to Geyin and Maurer [67], this ensemble approach or logic tree, in which different models are weighted in proportion to their predictive capabilities, is conventional in seismic hazard analyses and has the advantage of avoiding large swings as a result of changing a single adopted model [38, 39]. This approach has not been widely used to date for liquefaction hazard assessment but could be used to incorporate the results from the different probabilistic liquefaction triggering or manifestation relationships to account for the epistemic uncertainty [55]. Indeed, the unique approach will facilitate a comparability analysis of vulnerability or risk for two different hazards (earthquake and liquefaction) and consider possible triggering effects.

Moss et al. [100] highlighted a potentially significant concern associated with instances where liquefaction might have occurred at subsurface depths but did not manifest at the surface. This phenomenon is attributed to substantial and uninterrupted layers of non-liquefiable surface layers above the liquefied zone. Also, it should be noted that the PLTA approaches detailed herein are assessed only for a single layer, not for the potential of the entire soil profile. Meanwhile, PLMSA approaches are based on LMSI and consider the contribution of the soil profile at depths extending up to 20 m.

In this sense, in this paper it works with the probability curves obtained by Geyin and Maurer [67], Upadhyaya et al. [122] and Ge et al. [121] since they meet the condition of models based on PLMSA and consider different levels of intensity (different earthquakes). Additionally, these studies used the information on sites with non-liquefiable layers above liquefiable layers, how is the cases of Canterbury earthquakes between 2010 and 2016 in New Zealand. The work of Geyin and Maurer [67] defined different PLMSA approaches with three liquefaction manifestation severity indices (LMSIs) in addition to six liquefaction triggering evaluation procedures based on CPT analysis, and two CPT processing possibilities (CPT data as measure and CPT data with baseline inversion). Furthermore, the work of Upadhyaya et al. [122] defined different PLMSA approaches as a function of (i) the LMSI and the soil behaviour type index averaged over the upper 10 m of a profile (${{\text{I}}}_{{\text{c}}10}$) to quantify the prevalence of high fines-content (${\text{FC}}$), high-plasticity strata in a profile, and (ii) only the three LMSI (LPI, LPI_ISH, LSN), i.e., without considering the index ${{\text{I}}}_{{\text{c}}10}$. It was shown that the predictive capabilities of the models of the first group were better than those of the models of the second group. Finally, the work of Ge et al. [121] defined the PLMSA approach as a function of the index LPI only.

Thus, it is necessary to establish the appropriate model for specific soil conditions. It is worth nothing that all studies used the information of the most notable earthquakes in terms of the well-documented spatial extent and the severity of liquefaction-induced ground damage in the Canterbury region of New Zealand [122] (i) the 2010 M_w 7.1, in Darfield, earthquake, (ii) the 2011 M_w 6.2, in Christchurch, earthquake, and (iii) the 2016 M_w 5.7, Valentine’s Day earthquake.

Finally, the work of Geyin and Maurer [67] is the only one of the three PLMSA approaches to establish three different levels of damage or LMSL (i.e., minor, moderate, and severe) for a Canterbury database. However, the work of Upadhyaya et al. [122] proposed for each of the four (including any manifestation) LMSI approaches specific threshold probabilities of liquefaction manifestation for different classes of LMSL (Table 6). Despite its high implementation cost, the three selected works use the CPT test for the liquefaction triggering assessment procedure to obtain ${{\text{FS}}}_{{\text{liq}}}$, so the condition for obtaining this term will not be considered as a criterion for the definition of the PLMSA model. Furthermore, van Ballegooy et al. [138] suggested that the choice of liquefaction-triggering assessment model slightly impacts the correlation between the liquefaction manifestation severity index and the liquefaction-induced ground damage.

Table 6 Optimum threshold probabilities for different LMSL [122]

From the above considerations, it becomes essential to identify an appropriate approach for a PLMSA according to the existing literature. This need arises primarily due to two key factors. Firstly, there are areas where no surface evidence of liquefaction exists and without documented cases of liquefaction manifestations during historical earthquakes. Such occurrences have not been reported in the available literature. The lack of available data poses a challenge in formulating a tailored model for assessing the severity of probabilistic liquefaction manifestations (referred to as the liquefaction manifestation severity model) within a specific case study area. Secondly, it’s important to recognize that PLMSA curves tend to exhibit variations from one region to another, as illustrated by studies such as Ge et al. [121]. Given these conditions, there is a significant challenge in validating any of the models.

Furthermore, the concern regarding model uncertainty in the context of an urban-scale analysis justifies using a straightforward probabilistic liquefaction prediction model documented in existing literature. It is preferable to choose a model which has been previously applied in settings with environmental and soil configurations like those in the specific study area case instead of embarking on a site-specific numerical model derived from site response analysis.

In this context, it is imperative to establish criteria for selecting a robust yet straightforward probabilistic liquefaction manifestation severity assessment (PLMSA) approach by applying a logical tree framework. For a particular soil configuration, the criteria should be delineated based on the principal model characteristics as defined by Geyin and Maurer [67], Upadhyaya et al. [122], and Ge et al. [121]. These criteria should be contingent upon the liquefaction manifestation severity indices or soil indicators that most suit the specific soil conditions. This consideration aligns with findings from a separate study, particularly focusing on soils susceptible to liquefaction featuring a non-liquefiable surface layer overlying a liquefiable sand layer. These indicators have been extensively discussed in the literature [26, 71, 74, 75, 139] and exhibit sound correlations with post-liquefaction-induced ground damage reports.

In alignment with the geotechnical characteristics intrinsic to a particular study area, the weighting factors for each model related to probabilistic liquefaction manifestation severity are determined. This is carried out within the framework of a logic tree methodology with the explicit aim of establishing a singular model suitable for its application in large-scale urban liquefaction risk assessments. As shown in Table 7, this model is devised with specific regard to its suitability for the unique soil conditions prevalent in the region.

Table 7 Weights to each probabilistic liquefaction manifestation severity assessment approach for places with a non-liquefiable surface layer over a liquefiable layer

When historical liquefaction case data are absent for the development of area-specific thresholds for liquefaction manifestation severity, it is advised to conservatively adopt the thresholds proposed by Upadhyaya et al. [122] (as presented in Table 6) for forthcoming investigations. These threshold values are contingent upon the selected PLMSA approach.

In general, more emphasis is placed on the Upadhyaya et al. [122] model, primarily due to the incorporation of the soil behaviour type index averaged over the upper 10 m of a profile (${{\text{I}}}_{{\text{c}}10}$) to quantify the prevalence of high fine-content (${\text{FC}}$), and high-plasticity layers in a profile compared to the Geyin and Maurer [67] model. Additionally, efforts were made to ensure a balanced consideration of LMSI in both models within the logical tree approach. Concerning the criteria or LMSI, when dealing with models featuring more than one LMSI, greater significance is attributed to the LPI_ISH. This decision is guided by the more robust formulation of the LMSI, particularly in terms of the effects of the non-liquefiable surface layer on the liquefaction-induced ground damage, as was presented in Upadhyaya et al. [78] and Green et al. [75] and cited by Green et al. [133].

However, further research is needed to verify the proposed model. The verification could be done in a real multi-hazard scenario with a PLHA. From the results of the PLHA, it is possible to adjust the weights of the logic tree approach. Nevertheless, this logic tree approach provides a structured framework that can be easily applied to gain valuable insights in performance-based earthquake engineering (PBEE). In particular, it excels in characterizing liquefaction hazards, which is not possible with conventional deterministic methods.

Finally, it is important to emphasise that the ideal scenario for such soil configurations involves the availability of a model to predict the liquefaction triggering along with the PLTA or PLMS model. However, it is essential to note that the suitability of such a model may be questionable. As Green et al. [133] emphasise, the simplified models are naturally semi-empirical. The empirical components were primarily designed for active regions with shallow-crustal tectonic regions (e.g., California, Japan, and New Zealand). Furthermore, the majority of PLTA and PLMS models do not account for the presence of a non-liquefiable surface layer.

5 Remarks and Challenges

This paper provides a comprehensive and detailed literature review of probabilistic approaches concerning liquefaction, encompassing probabilistic liquefaction triggering assessment (PLTA), probabilistic liquefaction manifestation severity (PLMSA), and probabilistic liquefaction hazard assessment (PLHA). These approaches are primarily founded on semi-mechanistic “simplified stress-based” models derived from in-situ drilling tests, intended for application in urban environments to address issues of vulnerability and risk.

This review places particular emphasis on probabilistic studies conducted post-2004, as it seeks to delve into advancements made after that period [51] findings were based on earlier and less comprehensive databases [106]. The discussion within this research is chronologically organised into four principal categories (i) a primer on deterministic earthquake-induced soil liquefaction triggering and manifestation severity approaches (research done before 2014), (ii) PLTA approaches, (iii) PLMSA approaches, and (iv) PLHA analyses.

This paper aims to highlight the distinctions between various methodologies and analyses, specifically (i) probabilistic liquefaction triggering assessment (PLTA) and probabilistic liquefaction manifestation severity assessment (PLMSA) approaches, and (ii) a comprehensive approach known as fully probabilistic liquefaction hazard assessment (PLHA). The main objective is to elucidate the primary challenges associated with these forms of analysis.

In essence, the PLTA and PLMSA methodologies are utilized to determine the likelihood of encountering specific acceleration levels. Moreover, these approaches facilitate meaningful comparisons between the effects of liquefaction on different infrastructure and the seismic events’ impact on structural integrity, all within a defined timeframe. In contrast, the PLHA is employed to generate a comprehensive map of liquefaction hazard within a specific site, considering potential seismic scenarios (pairs PGA and M_w) within the region. Furthermore, this analysis integrates seamlessly into the performance-based earthquake engineering (PBEE) framework, providing insights into the combined effects of both hazards for various return periods. This combination explicitly includes uncertainties in regional seismicity parameters and the conditional probability of liquefaction [124].

The literature review proposes a probabilistic liquefaction approach to be incorporated into a multi-hazard vulnerability assessment (MHVA) approach with an easier comparability analysis of vulnerability for two different hazards (earthquake and liquefaction). In this sense, this study is relevant because it allows us to define, among several probabilistic liquefaction models, a single model and apply it in areas of high seismicity and prone to liquefaction where the seismic and geotechnical information is available. Furthermore, the approach may be employed in areas where the cases of liquefaction are limited or absent and with specific characteristics of the soil profile (non-liquefiable surface layer on a layer of liquefiable sand).

Undoubtedly, considering the variation of the PLMSA approach from one region to another, it is optimal to tailor its development based on specific data inherent to the respective region. However, there are places without information or evidence of the manifestation of liquefaction on the surface, or cases of liquefaction manifestations that occurred in historical earthquakes, but without information reported in the literature. In such cases, it is advisable to consider the use of an existing model, with the selection closely aligned with the geotechnical and seismic conditions of the region in question. The key is to ensure that both aspects closely resemble those under which the model was initially developed.

Therefore, the recommendation here is to define a single PLMSA model from several PLMSA approaches reported in the literature applying a logic tree approach, where the weight of each model is defined according to the liquefaction manifestation severity index or soil indicator more reliable in soils with a non-liquefiable surface layer. Although these parameters are the most related to a specific geotechnical configuration, it must be noted that more detailed studies should be carried out on the mechanism when the generation of excess pore pressure occurs in places with a non-liquefiable surface layer.

The use and verification of this probabilistic liquefaction model will be undertaken in a separate study concerning a fully PLHA for a real multi-hazard scenario in Aguablanca, Cali, Colombia. This scenario consists of (i) soil profile with a non-liquefiable surface layer [clayey soil with high fines-content (${{{\text{FC}}}}$)] above a liquefiable sand layer, and (ii) subduction earthquakes that may generate liquefaction as stipulated in the seismic microzonation study of the city [140].

Some challenges in these types of analysis are (i) the use of artificial intelligence (AI) for developing a region-specific liquefaction-induced ground damage PLMSA function for a region with only a limited amount of data as was proposed by Ge et al. [121], (ii) modify the LMSIs approaches as a function of the probability of liquefaction triggering to remove the concerns of model and parameter uncertainty in the calculation of the ${{\text{FS}}}_{{\text{liq}}}$. The work of Li et al. [73] produces a reasonable estimate of the probability of liquefaction manifestation severity for a given LPI, however, the method was demonstrated only for a one liquefaction case, and (iii) as mentioned by Upadhyaya et al. [122] an improved liquefaction manifestation severity index is needed that better accounts for the effects of high-${\text{FC}}$, high-plasticity soils more directly, because according to the authors the efficacy of the existing ones decreases at sites with high Ic₁₀. The inclusion of a new PLMSA based on a new LMSI like LSN_ISH as proposed by Upadhyaya et al. [80] could be included within our logical tree. However, it has not been included in this study considering the novelty of the new LMSI, which was proposed by Upadhyaya et al. [78]. Further case studies should be compared to ascertain the merits of this new index in the assessment of liquefaction manifestation severity.

Finally, while it is a component of fully mechanistic constitutive models, it becomes important to formulate a probabilistic liquefaction model that encompasses the impact of the non-liquefiable surface layer on excess pore pressure. The study of Bullock, et al. [141] defined a probabilistic model for predicting the cumulative absolute velocity (CAV) needed to generate a threshold in the pore pressure ratio |r_u| at a specified depth in a layer of liquefiable soil, but without considering the non-liquefiable layer.