An accessible approach for the site response analysis of quasihorizontal layered deposits
Abstract
This study focusses on site response analysis for sites that are neither strictly onedimensional (with flat parallel soil layers) nor clearly twodimensional (steep valleys, canyons and basins). Both these types of geometries are well studied in the literature. There is a lack of studies, however, for all those geometries that are in between these two worlds, such as sites with gently dipping layers. Theoretically, such sites should be studied with a twodimensional dynamic approach because of the formation of surface waves due to the nonhorizontal layering. In certain situations, however, the onedimensional dynamic assumption leads to minor errors and it may save a lot of effort in terms of defining a twodimensional model, computing the response and interpreting the results. As a result of these practical advantages, an accessible approach is presented here to determine when onedimensional analysis can be used for geometries consisting of quasihorizontal layers. The methodology is based on the construction of a chart, delimiting the applicability of the onedimensional approach, using simple but valid variables, such as the slope of the critical subsurface interface and the impedance contrast at this interface. Indeed, we propose our guidance on the limits of the onedimensional analysis in the form of this power law separating the onedimensional and twodimensional dynamic regimes: I_{z} = 6.95 γ^{−0.69}, where I_{z} is the impedance contrast and γ is the angle in degrees of the sloping critical subsurface interface. Site response analysis for geometries with values of I_{z} below this critical value can be computed using a standard onedimensional approach without large error whereas geometries with values of I_{z} above this threshold require twodimensional calculations.
Keywords
Seismic site response analysis Onedimensional Twodimensional Dipping layers Site effects Quasihorizontal1 Introduction
Site response analysis (SRA) is one the most powerful tools within engineering seismology as it models the influence of the nearsurface layers on earthquake ground motions. These nearsurface layers act as a filter that amplify/deamplify the seismic waves coming from the earthquake source. Based on the complexity of the nearsurface geometry and the characteristics of the layers, several SRA approaches are possible.
From all we know in the literature, we can distinguish between two macroworlds with regards subsurface geometry: flat layered sites and valleys or canyons. Each of them has its best approach for SRA. Indeed, the easiest method, onedimensional (1D) SRA, should be used whenever the stratigraphy and/or the geometry of the soil deposit is flat. This method, in fact, simplifies the reality with a single multilayered column (Kramer 1996). Whenever, on the contrary, the stratigraphy/topography requires a more complex model, two or threedimensional (2D/3D) SRA should be used. This is the case for a steep valley or canyon, where the wave path cannot be described with a 1D model. Note that in this work 3D SRA will not be discussed. Some authors have also discussed that, among geometries such as valleys, there is a critical shape ratio, which delimits the twodimensional resonance response from the onedimensional and lateral propagation (Bard and Bouchon 1980a, b). Despite this, they still focus on valleys (edges with an angle larger than 5°). This means that there is a gap in the literature of how to treat all those geometries with quasihorizontal layers (gently dipping angle). An example of this geometry is the Hinkley Point C site (located in the eastern part of the Bristol Channel basin) in LessiCheimariou et al. (2018). Most of the time, these sites are investigated by adopting the simplest and fastest method, which is 1D, but this does not mean that it is always the most correct one.
This study provides an accessible approach to identify the best option to study these particular geometries, which are neither strictly 1D nor 2D. To understand and identify a threshold between these two worlds (1D and 2D), first we need to define a model that serves as a basis for comparison. This model must present a basic geometrical irregularity, like a gentle dipping layer (slope angles of 5° or less). Indeed, we do not want to study either clearly flat layers or clearly steep valleys. For this model, we conduct parametric analyses examining the effects of the sloping angle and the stiffness of the material on the difference between 1D and 2D results. After probing these variables and collecting the results, we define a criterion to quantify these differences and finally we test it with other simulations and observations taken from the literature. The following section discusses previous studies on the limits of 1D SRA before we present our results.
2 Previous studies on the limits of 1D SRA
This approach is not valid for geometries such as steep valleys, canyons and basins. These geological formations cause a series of phenomena, related to both their geometries and also the soft material infill. Indeed, the softer the material of the alluvial basin compared to the bedrock, the higher is the effect of the waves trapped within it. These trapped waves are incident body waves that propagate through the alluvium as surface waves (Vidale and Helmberger 1988), which are responsible for stronger and longer shaking than would predicted by 1D SRA, which only considers the vertical propagation of SHwaves.
Many studies have been conducted on the effects of this kind of geometry on earthquake ground motions. Bard and Bouchon (1980a, b) extended the work of Aki and Larner (1970) to demonstrate how effective inclined interfaces are at generating surface waves, in particular Love waves, which can cause larger amplitudes in comparison with the direct incident waves. Bard and Bouchon (1980a, b) also studied the influence of a high velocity contrast between the soil deposit and the bedrock and showed that it can trap the surface waves within the basin and cause multiple reflections of them at the edge of the valley. This results in ground shaking of a longer duration in comparison with a flat site.
ChavezGarcia and Faccioli (2000), focusing on incorporating 2D site effects in seismic building codes, extended the work of Bard and Bouchon (1985). They studied a simple geometry of alluvial basins (symmetrical and homogeneous) to explore the impact of the impedance contrast and the shape ratio on site amplification. They reported their results in a similar graph to Bard and Bouchon (1985) showing the different alluvial valleys analyzed (Fig. 3).
It is important to notice that we cannot use this graph for our study because it refers to shape ratios that go from 0.1 to 0.5, which means slope angles greater than 5°. The focus of Bard and Bouchon (1985) and ChavezGarcia and Faccioli (2000) on high shape ratios is understandable because of their interest in valley/basin behavior. However, our study focuses on geometries with gentle dipping layers. All of the cases we study here are within the region entitled “1D RESONANCE + LATERAL PROPAGATION” on Fig. 3 because we focus on slopes shallower than 5°. Our study shows that even within this section of their graph there is a threshold separating geometries that clearly behave in a 2D manner and those where the 1D assumption roughly holds.
The use of an aggravation factor is also supported by Makra et al. (2012) who compared the results of different software for 2D SRA of a basin. The use of an aggravation factor is shown to be a powerful tool to quantify the additional amplification in response spectra in comparison with 1D SRA because of 2D effects. Makra et al. (2012) showed that the aggravation factors for the basins studied could be divided into three groups: a region on rock outside the basin, a region at the edge of the basin and a third region far from the edge of the basin. They concluded that the aggravation factor could be used to provide guidance on site amplification depending on the position within the basin.
Vessia et al. (2011) have reprised the problem of valley effects, stating the fact that this kind of phenomena can only be estimated on a casebycase basis through specific numerical simulations. The aim of this work was to produce a sort of “geometric coefficient” to identify the socalled “valley effects”. To do that, they propose a simple approach to predict valley effects by using 2D simple sketches of 30 m depth valleys, with a V_{s,30} characterization (according to the Italian building code), where V_{s,30} is the average shearwave velocity in the top 30 m.
Thompson et al. (2012) proposed a method to classify sites that require a complex SRA from those where the standard assumptions are sufficient. Their taxonomy is based on two criteria, the second of which is a goodnessoffit metric between the theoretical and the empirical transfer functions. For their comparison, Thompson et al. (2012) focused on the alignment of the resonances. As shown by Eq. [1], the resonance frequency depends on the geometry of the model (H), whereas the amplitude of the resonance peaks (at least for viscoelastic analysis) depends on the material damping, which is uncertain and difficult to determine. These uncertainties come from both laboratory test data and modeling issues. In a viscoelastic analysis, the amplitude depends completely on the damping value (Eq. [2]). For this reason, they have chosen to compare the theoretical and empirical transfer functions using the Pearson’s sample correlation coefficient, r, which captures how well the peaks are aligned. This correlation coefficient varies from − 1 to 1, where − 1 means completely negative correlation, 0 means no correlation and 1 means perfect positive correlation. Thompson et al. (2012) chose r = 0.6 as the threshold between poor (r < 0.6) and good (r ≥ 0.6) fits.
SanchezSesma and Velazquez (1987) derived a closedform solution for the seismic response of an elastic dipping layer using specific geometrical analysis. The exact solution is given for dipping angles of the form \(\frac{1}{2}\pi /N\), where N is an odd integer. Using this formula, they have shown the importance of modelling this kind of geometry, such as valley edges.
Furumoto et al. (2006) proposed a method to compute the transfer function of dipping layers by superposing 1D transfer functions of the upper and lower side of the slope. Then they compare their results to a 2D SRA showing that lateral site effects modify the dominant frequency.
In our previous study (Volpini and Douglas 2017), we have already studied the effect of gently dipping layers and suggested that it could be captured by conducting 1D SRA with randomized profiles. We considered a fivelayer model using both 1D and 2D SRA. The large number of layers considered did not allow us to generalize our findings. That is why in this article we have considered just two layers, in order to understand a simpler situation.
3 Comparing 1D and 2D SRA
The purpose of this study is to investigate those geometries, which are neither strictly 1D nor 2D/3D. In reality no site is perfectly 1D and hence it is important to know when the assumptions of 1D SRA breakdown. It is clear that when possible (good knowledge of the site in terms of characteristics of material, stratigraphy and records of input motion; availability of appropriate software and skills in using this software; and time to conduct the analysis) it is worth undertaking a 2D/3D SRA for all sites significantly deviating from perfectly horizontal layering. Theoretically 2D/3D SRA should model the site amplification at such sites better than 1D SRA. From a practical viewpoint, however, 2D/3D SRA can produce erroneous and unpredictable results when there is a lack of detailed information about the site. Moreover, the more complex is the model, the higher the time taken to run the analyses, interpret the results and simplify them for engineering applications.
In the previous section, various studies on the importance of taking into account 2D effects related to the basin shape were summarized. In this study, we conduct a more general survey of stratigraphic irregularities and provide some general and simple guidance on a better method to adopt in engineering practice for sites with nearsurface geometries that are at the boundaries between the 1D and 2D worlds. The guidance is in the form of site characteristics that can be known a priori, such as sloping layers and the geomechanical characteristics of the soil, so as to avoid the need to compare the results of 1D and 2D SRA for the site.
 1.
a parametric study on the seismic response of a 2D model with different dipping layer geometries and impedance contrast ratios;
 2.
a comparison of the 2D results with a 1D analytical solution;
 3.
a numerical criterion based on the comparison between the 1D and 2D transfer functions;
 4.
definition of a boundary between the two approaches; and
 5.
verification of this guideline using other results from the literature.
3.1 Defining the tools

the influence of the dipping layer and the angle of the slope;

the influence of the impedance contrast; and

the influence of location within the model.
The length of the model is 1000 m plus two buffer zones of 1000 m each, which are fundamental to carry out the analysis in the 2D finite element software used here, Abaqus (Dassault Systèmes Simulia Corp 2013). The dimensions have been chosen following the guidance provided by Nielsen (2006, 2014) as well as the boundary conditions (a rigid base and lateral freefield boundaries).
Timedomain viscoelastic analyses are conducted. Several (four rock outcropping motion and a within motion) input accelerograms have been tested, all of them taken from the Italian ITACA database (Luzi et al. 2017). The accelerogram is input at the horizontal base of the model (Volpini et al. 2018).
We tested inputting both the horizontal and vertical accelerograms simultaneously in the model but in the final calculations we decided to input just the horizontal component because of two reasons. Firstly, making a comparison with 1D SRA is clearer in this case. Indeed, in 1D SRA the basic hypothesis is to analyze the vertical propagation of the SH wave. Inputting a vertical motion into the 2D SRA would produce P and SV waves, changing the sense of the comparison. Secondly, there is still debate over the best way of conducting vertical SRA in the site response research community (Han et al. 2017).
The ground motions at several equallyspaced control points (Fig. 5) are studied to investigate the spatial variability in the transfer functions, similarly to the approach of Makra et al. (2012). In addition, two other control points outside the main model, called C.P left and C.P right, are used to test the effect of the buffer zone (Volpini et al. 2018). The resulting transfer functions are compared to those from 1D viscouselastic SRA computed using STRATA (Kottke and Rathje 2008) and the vertical soil column below each control point.
The damping ratio chosen for both sets of analyses is 3%, which results in a smooth transfer function where the effect of noise is minimized. The choice of this damping ratio is based on the results of the Prenolin project (Regnier et al. 2016), where a series of tests were conducted to determine the most appropriate damping value for viscouselastic analysis. It is easy to fix the damping ratio in STRATA but more challenging in Abaqus because it treats damping in a different way (Volpini et al. 2018).
4 Results
The purpose of this section is to compare the results of the 1D and 2D SRA for the 24 cases introduced above. Following the approach of Thompson et al. (2012) we make the comparison in terms of the transfer functions rather than the response spectral ordinates. Hence, timedomain results obtained with Abaqus were converted to the frequency domain. It is well known that the frequency content of the input motion becomes very important in SRA, especially in nonlinear analyses, whenever it is linked to a certain kind of soil deposit. Assimaki and Li (2012) have defined a frequency index that is a crosscorrelation between the transfer function and the input motion’s amplitude spectrum. The higher this value is the more similar are these two functions, implying resonance.
5 Investigating numerically the boundary between 1D/2D SRA for quasihorizontal layers
In the previous section, a qualitative comparison of the transfer functions was shown. For the chosen examples, it was clear which graph represented a good and poor match. Indeed, they have been selected with that aim. It is important to quantify the match, especially for those situations that are at the boundaries between visually good and poor matches. Indeed, although the transfer function plot immediately indicates the match between 1D and 2D SRA, it does not measure it. Hence, following the approach of Thompson et al. (2012), the Pearson’s sample correlation coefficient (r), is used to measure the goodness of fit between the two transfer functions. As discussed above r can vary from − 1 to 1. In this context r measures the alignment of the resonance frequencies.
From Fig. 11 it can be seen that at the control point of 750 m most r values are below the threshold of 0.6. To be conservative this location is chosen as the basis for the guidance derived below. Other analyses were conducted for control points 300 m, 600 m, 700 m, 800 m and 900 m to check whether 750 m is indeed the most critical location. These analyses demonstrated that the worst match between 1D and 2D SRA occurs at the farthest distances from the origin. We have decided to base the guidance on the results for 750 m because the results are more consistent here than at 800 m and 900 m.
Pearson’s sample correlation coefficient for geometries with different widths
Location (m)  125  250  375  500  750  1000  1250  1500  1750  2000 

Two degrees  
ORIGINAL (1000)  ×  0.8  ×  0.71  0.62  0.77  ×  ×  ×  × 
2000  ×  0.88  ×  0.88  0.86  0.73  0.68  0.73  0.74  0.77 
500  0.92  0.6  0.5  0.71  ×  ×  ×  ×  ×  × 
Four degrees  
ORIGINAL (1000)  ×  0.49  ×  0.68  0.52  0.22  ×  ×  ×  × 
2000  ×  0.81  ×  0.67  0.85  0.68  0.6  0.62  0.64  0.7 
500  0.68  0.7  0.47  0.6  ×  ×  ×  ×  ×  × 
We also made calculations using the shearwave velocity profile from an invasive test (crosshole, from Fugro) performed in Mirandola (Italy) for the Interpacific project (Garofalo et al. 2016a, b). Results from these calculations are also considered when checking the guidance (see below). In addition, results from our previous study (Volpini and Douglas 2017) for extreme cases are also considered below. It should be noted that following publication of that study in the conference proceedings we found errors in our calculations, which have been corrected for consideration here.
Pearson’s sample correlation coefficient for geometries a, b and c
Location (m)  0  250  500  750  1000 

r_{c}  0.93  0.70  0.56  0.41  0.71 
r_{a}  0.91  0.71  0.53  0.53  0.70 
r_{b}  0.94  0.57  0.45  0.43  0.70 
Pearson’s sample correlation coefficient for geometries b, c, Vs,1 = 200 m/s and 3°
Location (m)  0  250  500  750  1000 

r_{c}  0.91  0.68  0.31  0.35  0.64 
r_{b}  0.92  0.58  0.26  0.54  0.64 
6 Development of the chart
As discussed above, the 750 m control point (Fig. 11) is the worst location in terms of values of r and, therefore, to be conservative (i.e. to recommend 2D SRA when there is a doubt) results for this location are used in this section to develop the guideline. The purpose of this guideline is to choose the best analysis method (1D or 2D SRA), a priori, based on the slope angle and the impedance contrast.
To determine this guideline, in this section we: firstly define a relation from the 750 m location point graph separating the regions when 1D SRA gives acceptable results from those regions when it does not; and secondly verify this relation with additional calculations taken from the literature as well as computed here for more realistic shearwave velocity profiles.
7 Probing the guidelines
To check the guidance shown in Fig. 13 it is useful to consider simulations or observations from the literature. We do not consider valleys with slopes larger than about 15° and “sine” shape because it is commonly agreed that beyond a certain level (h/l = 0.25, narrow valley) (Silva 1988) 1D SRA will always give incorrect results. It is important to include both sites that are clearly 1D and clearly 2D but also cases between these two worlds. Each site will be classified by two parameters: slope and impedance contrast.
 1.
The paper must present the geology of the site as well as geomechanical characteristics.
 2.
A study is excluded from consideration if a 1D SRA is presented without giving a rough estimation of the subsurface stratigraphy/geology because we cannot estimate the slope for this situation.
 3.
In case of an irregular shape, a simplified shape will be taken into consideration (e.g. Fig. 12).
 4.
As previously mentioned, the sloping angle must be < 15°. Our focus is on studies for slopes between 2° and 8° as this is the critical zone between 1D and 2D response.
 5.
The results of Bard and Bouchon (1980a, b, 1985) and ChavezGarcia and Faccioli (2000) are also considered.
Summary of the real cases used to test the guideline
Site  References  Description/Info  Slope (°)  Impedance contrast ratio, Cv  1D  2D 

[–]  Bard and Bouchon (1980a)  Type 1  4  2.2  ×  
Type 1  4  8.2  ×  
Type 2  12  2.2  ×  
Type 2  12  8.2  ×  
[–]  ChavezGarcia and Faccioli (2000)  HC  9  5.2  ×  
1.8  ×  
Same angle  13  2.6  ×  
3.5  ×  
5.2  ×  
Thessaloniki, Greece  Raptakis et al. (2004)  LEP  7  3.5  ×  
Val di Sole, Trento, Italy  Faccioli et al. (2002)  []  17  3.0  ×  
Kirovakan Valley, Armenia  Bielak et al. (1999)  Zone 3  6  1.7  ×  
Zone 2  > 20  2.3  ×  
Grenoble, France  Bonilla et al. (2006)  [–]  12  5.2  ×  
[–]  Prenolin project  Alpha < 1  3  2.4  ×  
Alpha < 1  4  2.4  ×  
Alpha = 2.5  5  4.5  ×  
Alpha = 1  5  3.3  ×  
Mirandola, Italy  Interpacific projectMirandola  Fugrocrosshole  2  4.4  ×  
5  3.3  ×  
Nice basin, France  Semblat et al. (2002)  [–]  3  4.6  ×  
Caracas basin, Venezuela  10  5.5  ×  
Mississippi enbayment, U.S.  Park and Hashash (2004)  [–]  0  2.5  ×  
[–]  Volpini and Douglas (2017)  [–]  1  2.4  ×  
5  3.4  × 
8 Conclusions
In this article, a comparison between transfer functions from 1D and 2D site response analysis was presented. 1D analyses are easy to understand, they are rapid and uncertainties in the geomechnical properties of the soil layers can be easily incorporated. When the subsurface geometry/stratigraphy does not present marked derivation from the assumption of flat layers 1D analysis can provide accurate results. In contrast, 2D analyses are more complex and require much more detailed information about the site. In addition, they require more computational resources and time, especially if uncertainties in the site properties are considered. For these reasons, most of the time 2D analyses are not used in engineering practice unless strictly necessary, e.g. a steep valley. The result of this study was guidance in the form of a power law, based on the subsurface slope of the soil deposit and the impedance contrast, was proposed to decide on when 1D analysis provides acceptable results or in contrast when 2D analysis is required. Linear viscoelastic analyses were performed, where the main geomechanical characteristic is the material stiffness (expressed through the shearwave velocity). The model proposed presents a simple geometry, defined by two layers, where the shallowest one is inclined. This configuration can be seen as the edge of a valley.
This guidance was the result of a parametric analysis, which was then checked using results from the literature. In future it will be interesting to add nonlinearity to this parametric study, which could bring more realistic results.
Notes
Acknowledgements
The first author of this article is undertaking a Ph.D. funded by a University of Strathclyde “Engineering The Future” studentship, for which we are grateful. We thank: Stella Pytharouli; CH2M Hill (now Jacobs), in particular, Iain Tromans, Guillermo Aldama Bustos, Manuela Davi and Angeliki Lessi Cheimariou; Andreas Nielsen; and Alessandro Tarantino for their help with various aspects of this study. Finally we thank an anonymous reviewer for their detailed comments on a previous version of this study.
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