# The i-FSC proxy for predicting inter-event and spatial variation of topographic site effects

Aline Bou Nassif, Emeline Maufroy, Pascal Lacroix, Emmanuel Chaljub, Mathieu Causse, Pierre-Yves Bard

► To cite this version:

Aline Bou Nassif, Emeline Maufroy, Pascal Lacroix, Emmanuel Chaljub, Mathieu Causse, et al.. The i-FSC proxy for predicting inter-event and spatial variation of topographic site effects. Bulletin of Earthquake Engineering, 2025, 23 (2), pp.671 - 692. 10.1007/s10518-024-02042-4. hal-04958565

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# The i-FSC proxy for predicting inter-event and spatial variation of topographic site effects

Aline Bou Nassif<fnref n="1" /> · Emeline Maufroy<fnref n="1" /> · Pascal Lacroix<fnref n="1" /> · Emmanuel Chaljub<fnref n="1" /> · Mathieu Causse<fnref n="1" /> · Pierre-Yves Bard<fnref n="1" />

Received: 18 July 2024 / Accepted: 6 October 2024 / Published online: 7 January 2025
© The Author(s) 2024

## Abstract

Our study focuses on predicting topographic amplification of ground motion in the near-source region, where seismic rays reach the free-surface at varying incidence angles. We rely on data from previous 3D numerical simulations conducted on a topographic relief with a homogeneous medium. First, using neural networks, we identify which key parameters, describing the geometric characteristics of the relief relative to the seismic source position, control ground motion amplification. Then, we determine the functional form that relates these parameters to the simulated amplifications. Subsequently, we conduct a regression study to develop a model of topographic amplification, referred to as the i-FSC proxy (Illuminated Frequency-Scaled Curvature). Our estimator depends on the frequency-scaled (1) curvature, a parameter that accounts for the occurrence of amplifications over convex topographies and de-amplification over concave ones; (2) normalized illumination angle, a newly defined parameter that quantifies the slope exposure to the incoming wavefield, accounting for high amplification on slopes oriented opposite to the seismic source. The illumination parameter reduces the uncertainties of the proxy by a factor of 2 compared to estimators that rely solely on curvature. The proxy does not require high computational resources. It uses a digital elevation map and a seismic source position to predict amplification factors (without lithological effects) for an S-wave at any site on the surface topography. It allows exploration of variations in topographic amplification near seismic sources, representing a significant breakthrough as areas closest to the fault typically sustain the highest damages. A MATLAB script performing the i-FSC calculations is provided.

**Keywords** Topographic site effect · Site amplification · Earthquake ground motion · Ground-motion proxy

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

<footnote>Aline Bou Nassif
aline.bou-nassif@univ-grenoble-alpes.fr</footnote>

<footnote n="1">¹ Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, Univ. Gustave Eiffel, ISTerre, Grenoble 38000, France</footnote>
# 1 Introduction

Damage caused by earthquakes often exhibits significant heterogeneities in its spatial distribution near the seismic source. For example, the damage to buildings associated with the $M_W$ 6.0 Central Italy 2016 earthquake (Rossi et al. 2019; Milana et al. 2020), the landslides induced by the $M_W$ 6.6 Hokkaido 2018 earthquake (Zhang et al. 2019), or during the $M_W$ 7.8 Gorkha 2015 earthquake (Dunham et al. 2022) all demonstrate such variability.

The spatial distribution of earthquake-induced effects is frequently thought to be linked to specific geological and/or topographic factors (site effects) that amplify seismic ground motion (Trifunac and Hudson 1971; Bouchon 1973; Kawase and Aki 1990; Çelehi 1991; Bard 1995; Meunier et al. 2008). These effects are frequency-dependent (Bard 1995; Massa et al. 2014; Maufroy et al. 2015) and can coincide with the natural frequency band of man-made structures, leading to soil-structure resonance and resulting in significant damages. Although the effect of surface topography on seismic amplification has been documented since the early 70s (Trifunac and Hudson 1971; Boore 1972; Bouchon 1973), it remains much less understood than the effects of lithology (Maufroy 2010; Massa et al. 2014) and has been rarely or unsatisfactorily considered in building specification codes (European Eurocode 8, Italian NTC-08). This disparity is mainly explained by the following facts: (1) resonance due to geological site effects is usually considered a 1D phenomenon, whereas the topographic site effect is a three-dimensional one that is more difficult to study; (2) in-situ measurement of topographic effects requires placing stations in complex and often inaccessible terrain; and (3) choosing an amplification-free reference site to estimate amplification levels is more complicated in regions with pronounced topographies.

Most past studies have considered topographic site effects as a 2D problem (e.g., Boore 1972; Bouchon 1973; Sánchez-Sesma and Rosenblueth 1979; Geli et al. 1988; Pischiutta et al. 2010). However recent studies based on 3D approaches show that the 2D approximation can result in underestimated topographic amplification (Paolucci 2002; Maufroy et al. 2012; Primofiore et al. 2020). Since 3D numerical simulations are computationally expensive (Chaljub et al. 2010), especially for large earthquakes, there is a strong need to develop simplified approaches to predict topographic site effects.

Consequently, different studies have attempted to identify the link between topographic amplifications and some simple parameters describing surface geometry. Some studies used geometric parameters to classify landforms into various types, such as peaks, valleys, and slopes, and then analyzed the extent of topographic amplifications within each category (Burjánek et al. 2014). Others have examined how continuous variations in geometric parameters affect topographic amplifications. For example, Geli et al. (1988) and Kamalian et al. (2008) suggest that amplification at the top of a relief generally increases as the Shape Ratio (ratio of height to half-width of the hill) increases. Molina et al. (2019) were able to determine proxies for the Peak Ground Acceleration (PGA) and the period-dependent topographic amplification factors that depend on the shape ratio. Maufroy et al. (2015) developed a proxy (called the FSC, frequency-scaled curvature proxy) for the median topographic amplification at a given frequency based on the frequency-scaled curvature of the topography. According to this latter study: (1) most often, in the absence of lithological impedance contrast, amplification occurs at the top of a relief (characterized by positive curvatures) while de-amplification occurs at the bottom or within valleys (characterized by negative curvatures). This pattern aligns with previous findings (Bard and Méneroud 1987;
Çelebi 1987; Hartzell et al. 1994; Assimaki et al. 2005b). (2) Topographic amplification is scale-dependent as the ground-motion is amplified at wavelengths that are controlled by the characteristic length of the topography. However, despite strong correlations between curvature and amplifications in topographic highs and lows, on hillslopes where the curvature is approximately zero, their proxy consistently predicts a neutral amplification equal to 1. On the other hand, observations show significant variation on hillslopes due to azimuthal and source depth effects (Bouchon 1973; Assimaki et al. 2005a; Maufroy et al. 2015), with amplification levels being highest on slopes oriented opposite to the incident wavefield. Since the FSC proxy does not account for this property, there is a high level of uncertainty in its use near the seismic source where seismic rays reach the free surface at varying incidence angles.

The aim of this paper is to develop a new proxy for topographic amplification that considers the interaction between topographic irregularities and incident seismic waves from nearby sources. We term this proxy “illuminated Frequency-Scaled Curvature” (i-FSC). To achieve this, we use a previously available synthetic dataset of 3D numerical ground motion simulations and employ neural networks to identify the key parameters that significantly influence topographic amplification. Subsequently, we establish the functional relationship between these parameters and the simulated amplifications. Then, using regression analysis, we determine an equation for the proxy that enables the prediction of near-source topographic amplification factors for a given S-wavelength.

## 2 Dataset of the topographic amplification functions

In this study, we use the synthetic ground-motion dataset modeled by Maufroy et al. (2012) and Maufroy et al. (2015), which was previously used for the elaboration of the FSC estimator. These 3D numerical simulations of seismic wave propagation were performed using the SHAKE3D finite-differences code (Cruz-Atienza et al. 2007a, b) and applied to the mountainous area of Rustrel, France.

The considered topographic model, with a resolution of 10 m in the numerical domain, covers a wide range of topographic features, including both high and low areas (Fig. 1b). The slopes are mostly gentle, with a maximum steepness of 40°, and their azimuthal orientations vary with a preference towards the south. Two hundred double-couple point sources corresponding to $M_W = 4.5$ are considered in a wide variety of hypocenter positions to cover different azimuthal ranges of sources with respect to these slopes. However, considering that local sources are no deeper than 5 km compared to the free surface (Fig. 1a) and that there are no cliffs or extreme slopes, there is no grazing of the seismic rays: the largest incidence angle relative to the vertical can reach 60° (90° being the horizontal axis) (Fig. 1c). Hence, our study covers commonly observed topographies and not extreme ones. It should also be noted that the double-couple point sources have various focal mechanisms as the strike, dip and rake are chosen randomly.

The 3D numerical model is composed of an elastic homogeneous medium with the following rock properties: P-wave velocity $V_P = 5$ km/s, S-wave velocity $V_S = 3$ km/s and density $\rho=2.6$ g/cm³. The 200 simulations of ground motion are recorded by an array of 576 virtual stations placed within a $2.4 \times 2.4$ km² squared area, regularly spaced every 100 m (black dots in Fig. 1b). The simulations are based on a cubic mesh, which may not accu-
<figure><img src="image_4.png" /><figcaption>**Fig. 1** a Percentage distribution of the source depths compared to the station elevations. b Topographic model of the mountainous area of Rustrel, France, as represented in the dataset of 3D numerical simulations (Maufroy et al. 2012, 2015). The locations of the 576 virtual observation points are shown by the black dots on the colored topographic free surface. The sources positions corresponding to each simulation are represented by the empty blue circles in depth. The studied frames (see Sect. 3.3) are indicated by the numbers 1–9. c Percentage distribution of the seismic rays incidences to the station locations, with respect to the vertical axis</figcaption></figure>

rately represent the topography at high frequencies. To compensate for this, the free surface in the numerical mesh is represented by cubes of very small size compared to the simulated wavelengths. Maufroy et al. (2012) chose a 10 m grid spacing to simulate frequencies up to 5 Hz (the minimal *S*-wavelength is equal to 600 m, i.e., 60 grid points per minimal *S*-wavelength). Maufroy (2010) also provides a verification test for those simulations by using another simulation based on a discontinuous Galerkin finite-element method, which confirms that calculations are limited to a maximum frequency of 5 Hz.

The choice to use a homogeneous medium and a non-extended source is motivated by the following reasons. First, the purpose of our study is to understand the physics of topographic amplification alone. Assuming a simplified homogeneous medium helps isolate the topographic effect from others related to lithology. Second, for earthquakes occurring on extended faults, the topography may receive the incoming wavefield with various incidences, making our attempt to understand the effect of slope exposure to the incident

<figure><img src="image_5.png" /></figure>
wavefield more difficult. As a starting point, it is advantageous to consider point sources to capture this effect in the near-source region. This can be then expanded in future studies to include the effects of extended sources.

The outputs of the 3D numerical simulations, which are velocity seismograms, are used to compute topographic amplification factors at each frequency, up to 5 Hz, using the median reference method (MRM) (Wilson and Pavlis 2000; Poppeliers and Pavlis 2002; Maufroy et al. 2012). The MRM estimates the amplification relative to the median amplitude value in the studied zone and accounts for the statistical variability of ground motion within the array’s boundaries. Although this method is not widely used and tends to produce lower amplification coefficients compared to those obtained with the standard spectral ratio method (SSR) (Maufroy et al. 2012), it offers a significant advantage crucial for this study: unlike the SSR, which assumes a single reference site (Borcherdt 1970) and is incompatible for near-source conditions, the MRM allows for assessing amplification close to the source.

# 3 Identifying the i-FSC proxy’s key parameters and functional form

## 3.1 Methodology

To determine the parameters controlling simulated amplification levels and identify the functional form of the i-FSC proxy, we employ neural network analysis (Bishop 1994; Dreyfus 2005) implemented within the MATLAB® Neural Network Toolbox™. A neural network is a statistical learning method consisting of layers of neurons: (1) the input layer receives variables related to relief geometry with respect to the source position; (2) the output layer provides MRM amplifications; and (3) hidden layers (HL), each having a number of hidden neurons (HN), perform necessary computations. The connections between neurons in hidden layers are defined by transfer functions (TF).

We allocate 70% of the numerical simulation data for training the neural network, reserving the remaining 30% for testing and validation purposes. The selection of the neural network architecture (number of HL, HN and TF type) does not follow a specific rule. It is usually chosen based on comparable previous studies and trials and errors. The performance of the established model is measured by the coefficient of variances reduction $R_S$ (Eq. 1) (Derras et al. 2016). The closer $R_S$ is to 1, the more performant the network is.

$$R_S = 1 - \frac{\sum (O_i - P_i)^2}{\sum (O_i - \mu_o)^2} \quad (1)$$

In Eq. (1), $O_i$ are the observed output values corresponding to the input sample $i$ (with a total number of 115200 (576 virtual stations × 200 double-couple point sources) amplification factor), $P_i$ are the predicted output values for the same input sample $i$ estimated by the neural network, and $\mu_o$ is the arithmetic mean value of the output parameters $O_i$.

To identify key parameters for defining the proxy, we calculate various geometric properties of the free surface and their interaction with the seismic source position: slope inclination $S_{inc}$, slope aspect $S_{asp}$, curvature $C_S$ (Maufroy et al. 2015), topographic position index TPI (Burjánek et al. 2014), hypocentral distance $D_{hypo}$, incidence angle of the incoming wavefield relative to the vertical $Inc_{wave-vert}$, incidence angle of the incoming wavefield
relative to the free surface’s normal vector $\hat{\theta}$ (referred to as the illumination angle), azimuth orientation of the incoming wavefield $Azim_{wave}$, and incidence of the major source radiation lobes relative to the free surface’s normal vector $Inc_{rad-vert}$. Then, our methodology involves an initial run of the neural network, incorporating all computed parameters into the input layer. In a second step, we systematically exclude each tested parameter one at a time and observe changes in the network’s behavior. The impact of parameter removal is quantified by calculating the relative drop in the model’s variance reduction $R_S$. Figure 2a illustrates, as an example, the results obtained across different neural network architectures at a frequency of 2 Hz. It is observed that, regardless of the considered network architecture, the main parameters controlling the model performance, with the highest $R_S$ drop, are the hypocentral distance $D_{hypo}$, the free surface curvature $C_S$ and the incidence angle of the incoming wavefield relative to the free surface’s normal vector ($\hat{\theta}$ illumination angle). The quantitative definition of these parameters is given in the following section. Control of the curvature is of first-order significance (its removal can result in a variance reduction drop ranging between 36% and 52%), followed by the illumination angle ($R_S$ drop between 14% and 16%), and then the hypocentral distance ($R_S$ drop between 7% and 16%). The removal of the other tested parameters from the input dataset does not result in significant $R_S$ drop (not more than 5%), reflecting their limited contribution to the simulated amplifications.

After removing all the non-contributing parameters, we experiment with various neural network architectures to determine the most optimal model. This approach allows us to investigate the correlation between the identified parameters and the simulated amplifications without imposing a specific correlation form between them from the start, thus avoiding potential bias in the results. Figure 2b shows the variance reduction values obtained at 2 Hz for various model architectures. Opting for a simple neural network (1 HL, 1 HN and a linear transfer function) yields a model with an $R_S$ value of 0.17. $R_S$ is 0.19 when considering 3 HL with 5 HN and a linear TF, and not more than 0.22 with 3 HL 10 HN and a sigmoid TF (Fig. 2b). Considering that $R_S$ ranges between 0 (indicating poor model performance) and 1 (indicating the best-performing model), it can be concluded that more complex networks do not significantly enhance the model performance. Due to only mar-

<figure><img src="image_6.png" /><figcaption>**Fig. 2** a Relative drop in neural network variance reduction $R_S$ (%) when removing each tested parameter from the input parameter dataset listed in Sect. 3.1. Various neural network architectures with different numbers of hidden layers (HL) and hidden neurons (HN) are considered. b Neural network variance reduction plotted as a function of network architecture, including different numbers of hidden layers (HL), hidden neurons (HN) and layers transfer functions (TF). The results are presented for a frequency of 2 Hz</figcaption></figure>
ginal improvements brought by more complex architectures, we therefore choose to stick with the simplest model (featuring 1 HL, 1 HN and a linear TF). This decision helps mitigate the potential problem of overfitting, which often arises when the neural network becomes overly complex.

It is worth noting that the $R_s$ values for this simple model, considering the three identified parameters (i.e., hypocentral distance, curvature and illumination angle), range between 0.11 and 0.18 at different frequencies, indicating relatively low performance (closer to 0 than 1). However, in section 3.3., we proceed to enhance the model’s performance to increase $R_s$ values before deriving the i-FSC proxy. This enhancement involves removing irrelevant features in the dataset used to train the neural network, which are unrelated to topographic site effects and not explained by the identified parameters.

## 3.2 Parameters of the proxy

The contribution of hypocentral distance arises from propagation effects observed in the simulated amplification maps. Since this does not stem from topographic site effects, we will mitigate its influence through a series of path-effect normalizations in Sect. 3.3. Hereafter, we will focus solely on curvature and illumination angle as the two key parameters controlling topographic amplification.

### 3.2.1 Topographic curvature

Curvature is the second derivative of the elevation map and is used to measure land convexity (Zevenbergen and Thorne 1987; Moore et al. 1991). A positive curvature value represents convex features (such as ridges, edges, or local peaks), whereas a negative curvature value represents concave features (like valleys). Curvature is scale-dependent; for instance, at a specific location, a positive value may indicate a local peak at small scales, but at larger scales, that peak may be located within a large valley associated with negative curvatures.

When seismic waves encounter curved topography, they focus or defocus depending on the shape of the free surface. This behavior is analogous to the laws of optics. Wave focusing occurs at convex features, leading to amplified ground motion amplitudes. Conversely, wave defocusing occurs at concave features, resulting in reduced ground motion amplitudes. According to Maufroy et al. (2015), amplification at a given S-wavelength $\lambda_s$ is correlated to the frequency-scaled curvature, smoothed over a kernel size of $L_s = \frac{\lambda_s}{2}$. Curvatures can be calculated from a digital elevation map (DEM) using Eq. (1) to (4) in Maufroy et al. (2015). Equation (6) in their paper quantifies the relationship between curvature and median topographic amplification as a function of the S-wavelength. Figure 3 displays curvature $C_s$ maps derived for various values of characteristic length $L_s$ in the mountainous area of Rustrel, France.

### 3.2.2 Seismic illumination angle

In addition to curvature, topographic amplification is also influenced by the seismic source position relative to the relief. Topographic irregularities can focus seismic waves and amplify ground motion, but this phenomenon only occurs when the waves reach the surface. The concept of “illumination angle” refers to spatial variations in slope exposure to incom-
<figure><img src="image_7.png" /><figcaption>Fig. 3 Maps of smoothed topographic curvature (C_S) computed based on Zevenbergen and Thorne (1987) and Maufroy et al. (2015) for three different characteristic smoothing length L_S = λ_S/2 a 1500 m, b 750 m and c 500 m</figcaption></figure>

ing wavefield. Similar to how an object is illuminated by a light source, we use “illuminated slopes” to describe slopes that directly receive seismic waves, while those less exposed are in the “seismic shadow”. The amplification maps in Fig. 4a, b and c, obtained from the 3D synthetic dataset of ground motion on the same relief (Maufroy et al. 2012, 2015), illustrate this. When the seismic source is directly beneath the recording array (Fig. 4a), the incident wavefield is predominantly vertical, and amplification is primarily controlled by curvature (see Fig. 4a and 3b): areas with positive curvature are amplified, while those with negative curvature are de-amplified. However, as the incidence angle of the seismic rays varies along the slopes of the region (Fig. 4b and c), a systematic behavior appears: slopes facing away from the incident wavefield (as denoted by the circles — slopes facing South in Fig. 4b and South-East in Fig. 4c) experience greater amplification.

Although these observations have long been qualitatively identified (Bouchon 1973; Assimaki et al. 2005a; Maufroy et al. 2015), no study has proposed a quantitative measure of this effect. In our study, we consider the angle formed between the incident seismic rays and the normal to the free surface topography to measure the exposure of the slope to the incoming wavefield. We refer to this as the “seismic illumination angle”. An angle greater than 90° (θ₁ in Fig. 5) indicates that the slope is pointing in the opposite direction of the source and is expected to have higher exposure to incoming seismic rays (ground motion will consequently be amplified). Conversely, an angle smaller than 90° (θ₂ in Fig. 5) means that the slope obscures the arrival of the seismic waves (ground motion will consequently be de-amplified).

To determine the illumination angle, it is necessary to first compute the surface normal vector at the topographic site. This computation involves selecting neighboring points within a specified distance from the site, which we call the mid-span extent M_S. We tested several values of M_S and found that the optimal one, which maximizes the correlation between the illumination angle and the target amplification factors, varies according to the S-wavelength as described by Eq. (2).

$$M_S(m) = \frac{\lambda_S}{5} \qquad (2)$$
<figure><img src="image_8.png" /><figcaption>**Fig. 4** Amplification maps obtained from 3D numerical simulation of seismic waves propagation at a frequency of 2 Hz ($\lambda_s=1500$ m) for: **a**, **b** and **c** different seismic source positions (indicated by the red stars); **c** and **d** for similar seismic source position with different focal mechanisms (for **c**: Strike=184° Dip=67° Rake=76°; for **d**: Strike=240° Dip=49° Rake=126°). Amplified circled regions indicate slopes opposing seismic wavefield direction and receiving the highest degrees of illumination</figcaption></figure>

Based on these findings, the calculation of the illumination angle is frequency-scaled. It should be noted that the mid-span for a given S-wavelength may not necessarily be an integer multiple of the DEM resolution. In such cases, the closest integer multiple can be selected, as we observe no significant variation when slightly adjusting this parameter. While Eq. (2) sets a minimum DEM resolution requirement for a specific S-wavelength, the most restrictive criterion remains the one specified by Maufroy et al. (2015): a minimum of 12 DEM points per minimum S-wavelength is required.

Once the surface normal vector has been computed at the topographic site, the illumination angle $\hat{\theta}$ can be determined for a given source position using Eq. (3). In this equation, $\vec{n}$ is the surface normal vector at the topographic site computed for the mid-span closest to $M_s$ and $\vec{SR}$ is the vector connecting the site to the seismic source.
<figure><img src="image_9.png" /><figcaption>Fig. 5 2D representation of the seismic illumination of a topographic hill with two slopes: (1) the first slope on the right is pointing in the opposite direction of the incident wave field and is thus illuminated; The seismic illumination angle $\hat{\theta}_1$, which is the angle formed by the surface normal vector $\vec{n}_1$ and the site-source vector, exceeds 90°. (2) The second slope on the left is facing the source direction and is thus shadowed, receiving less seismic energy; Its illumination angle $\hat{\theta}_2$ is less than 90°. The red star indicates the location of the seismic source</figcaption></figure>

$$ \cos \hat{\theta}(M_S) = \frac{\vec{S} \cdot \vec{n}(M_S)}{\|\vec{S} \cdot \vec{n}(M_S)\|} \qquad (3) $$

Figure 6 shows the illumination angle values calculated according to Eq. (3) at 2 Hz ($\lambda_S=1500$ m), for the three source positions shown in Fig. 4a, b and c. It can be seen that the illuminated slopes within the circled regions in Fig. 6b and c correspond to the amplified slopes within the circled regions in Fig. 4b and c respectively.

### 3.3 Normalization of path and source effects to isolate topographic site effect

The variance reduction values $R_S$ obtained through the trained neural network range between 0.11 and 0.18, depending on the studied frequency. These low $R_S$ values indicate that the variability of ground motion in the synthetic dataset remains too high. This issue primarily arises from the presence of noisy features related to path and source effects. For instance, variations in simulated amplification due to the source’s proximity, influenced by parameters such as the focal mechanism, create variability that is not captured by the identified site
<figure><img src="image_10.png" /><figcaption>**Fig. 6** The illumination angle $\hat{\theta}$ maps in °, calculated based on Eq. (3) at 2 Hz ($\lambda_s$=1500 m), for three different hypocenter positions (indicated by the red stars). Circled regions indicate slopes opposing seismic wavefield direction and receiving the highest degrees of illumination</figcaption></figure>

parameters. These features, which are unrelated to topographic site effects, will be removed
to derive the proxy based on the two key parameters identified in the previous section.

**3.3.1 Removing the focal mechanism effect from the amplification maps**

Besides hypocentral distance, curvature and illumination angle, the focal mechanism sig-
nificantly contributes to the variability of near-source ground motion. For example, ampli-
fication maps in Fig. 4c and d), obtained using the same topographic model for two sources
located in the same area but with different focal mechanisms, exhibit clear differences. To
mitigate this effect while keeping the influence of the seismic illumination, we adopt the
following approach: We divide our region into nine frames (labeled 1 to 9, as shown in
Fig. 1b). For a fixed frequency, we compute the median amplification maps generated by
all sources located within each of these frames. Each frame contains at least 12 sources that
have various focal mechanisms but illuminate the station array in a similar way. Conse-
quently, our analysis will be focused on these averaged amplifications, rather than those for
individual sources.

**3.3.2 Removing the distance effect from the amplification maps**

In near-source conditions, the effect of distance on ground motion (i.e., geometrical attenu-
ation) can influence amplification estimations. This issue is similar to the previous one con-
cerning focal mechanisms: it is not a topographic site effect. Therefore, in order to remove
it we normalize the amplification maps obtained from the 3D numerical simulations by
the corresponding 1D amplification maps (i.e., maps that only include the distance effect,
without the topographic one). These 1D amplification maps were obtained from a new set
of 1D numerical simulations that use the same model properties (same source positions,
source focal mechanisms, medium properties and frequencies), but assume a flat free sur-
face instead of a topographic one. A total of 576×200 1D simulations were conducted, each
representing a unique combination of a seismic source and an individual station. In each
simulation, the flat surface was positioned at the elevation of the individual station. These
simulations were performed using the “AXITRA” code developed by Cotton and Coutant
(1997), which employs the discrete wavenumber method to simulate ground motion dis-
placements. Subsequently, the normalized amplifications, where source and path effects are removed, are obtained by subtracting the 1D amplification factors from the 3D ones and adding 1 to maintain a reference amplification threshold of 1 for flat areas with no topography.

### 3.3.3 Removing the distance effect from the illumination angle

The illumination angle described in Sect. 3.2.2 includes a distance effect. In a homogeneous medium with a flat surface, the illumination angle is 180° just above the seismic source, where seismic rays reach the free surface vertically. It gradually decreases to 90° as we move away from the epicenter, where the rays become grazing (Fig. 7). Since the goal is to develop an estimator that predicts distance-independent amplifications, it is crucial that the parameters of the proxy are also distance-independent. To remove the distance effect from the illumination parameter, we define a normalized version of this angle $\hat{\theta}_n$. This is obtained by subtracting the 1D illumination angle (calculated using Eq. (3)) considering a flat free surface at the receiver position) from the 3D illumination angle calculated with the actual 3D topography, considering the same seismic source position. In this case, the reference threshold for a neutral zone corresponds to a normalized illumination angle $\hat{\theta}_n$ equal to 0. A negative normalized angle indicates a shadowed slope less exposed to the incoming wavefield, while a positive angle indicates an illuminated slope with greater exposure. According to this definition, the effect of illumination is not observed when there is no topography; rather, it describes a modification of illumination caused by free surface irregularities. Finally, the median value of the normalized illumination angles computed for each frame, is removed to ensure that they are centered on 0, closely aligning with the defined reference (the free surface without topography).

<figure><img src="image_11.png" /><figcaption>**Fig. 7** a 2D cross-section of the decreasing illumination angle with the increasing hypocentral distance for a flat free surface b the illumination angle ($\hat{\theta}$) map in °, calculated based on Eq. (3) for a flat free surface (the seismic source position is indicated by the red star)</figcaption></figure>
## 4 The final model for the topographic site effect proxy

With the series of normalizations carried out in Sect. 3.3, we significantly increase the neural network performance. The model now achieves an $R_S$ value of 0.65 at the lowest frequency (1 Hz) and decreases with increasing frequency to an $R_S$ value of 0.44 at 5 Hz. The optimal neural network is a simple network consisting of 1 hidden layer with 1 hidden neuron and a linear transfer function. Consequently, the proxy takes a linear form that relates the identified controlling parameters (curvature $C_s$ and normalized illumination angle $\hat{\theta}_n$) to the amplification factors (AF) at a given S-wavelength $\lambda_S$ (Eq. 4).

$$AF(\lambda_S) = \alpha(\lambda_S) C_s(L_S) + \beta(\lambda_S) \hat{\theta}_n(M_S) + \gamma \quad (4)$$

$\alpha$, $\beta$ and $\gamma$, are determined using wavelength-by-wavelength regression analyses. The optimal values shown in Eq. (5) are the one maximizing the correlation between the equation parameters ($C_s$ ($L_S$) and $\hat{\theta}_n$ ($M_S$)) and the simulated amplification factors (AF ($\lambda_S$)) at a given S-wavelength. $\gamma$ is set to unity so that for a flat free-surface with zero curvature and zero normalized illumination angle, the resulting amplification is equal to 1. This implies that such a site is neutral and does not exhibit any topographic site effect. It should be noted that the $\alpha$ and $\beta$ are expressed as a function of $\lambda_S$, but if the S-wave velocity of the medium is known, they can be expressed as a function of frequency by simply replacing $\lambda_S$ with $\frac{V_S}{f}$.

$$ \begin{aligned} \alpha &= 0.00022\lambda_S^{1.2} \quad (\text{with } \sigma = 0.0254) \\ \beta &= 0.0042\lambda_S^{0.13} \quad (\text{with } \sigma = 0.00086) \\ \gamma &= 1 \quad (\text{with } \sigma = 0.0065) \end{aligned} \quad (5) $$

Compared to the normalized simulated amplifications, the new linear model exhibits a variance reduction ranging from 0.41 to 0.64, depending on the considered frequencies. In contrast, Maufroy et al. (2015)'s FSC proxy, which only accounts for the effect of surface curvature, shows a variance reduction ranging from 0.22 to 0.43. Consequently, by incorporating the illumination parameter into the new proxy, we reduce uncertainties of amplification prediction by a factor of up to 2.

To apply the i-FSC proxy and generate the topographic amplification map at a specified S-wavelength, a digital elevation model and a seismic source position are required. The following steps should be followed:

(1) Use Eq. (4) from Maufroy et al. (2015) to calculate the smoothed curvatures $C_S(L_S)$ considering a characteristic length $L_S = \frac{\lambda_S}{2}$.

(2) Derive the surface normal vectors using the integer mid-span closest to $M_S = \frac{\lambda_S}{5}$ (Eq. 2).

(Note that steps 1 and 2 require a minimum DEM resolution of 12 points per S-wavelength).

(3) Using the DEM along with the seismic source position, apply Eq. (3) to determine the 3D illumination angles.

(4) Compute the 1D illumination angles using Eq. (3) for the same seismic source position, but considering a flat free-surface (vertical normal vectors).

(5) Calculate the normalized illumination angles $\hat{\theta}_n(M_S)$ by subtracting the 1D illumination angles from the 3D illumination angles.

(6) Apply Eqs. (4) and (5) to calculate the topographic amplification map.
This procedure is easy and quick to apply, providing a user-friendly tool to estimate the topographic amplification factors at any topographic site. A MATLAB script is provided to perform all the steps mentioned above and generate the amplification maps. The user only needs to provide a digital elevation model, the position of the seismic source and the S-wavelength value (see Sect. 7).

# 5 Discussion

## 5.1 Impact of seismic illumination on topographic amplification variations

In order to illustrate how different topographic features behave in terms of amplification with respect to the seismic illumination, we chose some specific sites located on slopes (sites a, b, c and d as shown in Fig. 3b), summits (site e in Fig. 3b) and valleys (site f in Fig. 3b), and analyzed their frequency-dependent amplification functions (Fig. 8). The non-normalized amplifications, corresponding to the 200 simulated sources, are represented at each frequency. The values of the normalized illumination angles are indicated by the color scale (red color when the site is illuminated by the source, blue color when the slope is shadowed, and white color for a neutral zone). The frequency-dependence of the slope curvature is also represented by the purple line corresponding to the right Y axis. For almost all the sites (except for site d), a clear correlation is seen between the amplification levels and the illumination angle. The non-normalized amplification functions (Fig. 8a, b, c, e and f) clearly show that the highest amplification levels are associated with high positive illumination angles (red color in the upper part), whereas lower amplifications and de-amplifications are associated with negative illumination angles (blue color in the lower part). For some sites, this general trend may hold for all studied frequencies (e.g. site b), but for others, variations may occur at certain frequencies (e.g. variations at 3.5 Hz for site a, 1 Hz for site c). For site d, facing the site c but on the opposite slope, we observe significant variabilities (Fig. 8d): The correlation between amplification and illumination angle is no longer visible and it can be slightly identified for high frequencies (>3 Hz only) and only for large amplification values indicated by the circle (Amplification>1.6). The observed variabilities may be associated with other controlling factors masking the illumination effect. For sites e and f located respectively on a summit and within a valley (Fig. 8e and f), the effect of illumination is weak at low frequencies (<2 Hz), but it becomes more noticeable as the frequency increases. It is worth noting that when we are at the valley’s bottom area, where the negative curvature values are large, curvature is the main controlling factor and illumination has no effect. In fact, according to Maufroy et al. (2015), valleys have generally stable behavior with little variability.

## 5.2 Performance and sources of errors of the i-FSC proxy

Compared to the amplifications calculated from the numerical simulations, the i-FSC proxy predicts the amplifications with a root mean square percentage error ranging from 35 to 56%, increasing with frequency. The errors can be attributed to two main factors. First, the residual effects of the focal mechanism which contributed to the remaining variability. Our attempts to mitigate these effects through normalization and averaging were constrained by
<figure><img src="image_12.png" /><figcaption>**Fig. 8** Non-normalized amplification variation, at each of the sites marked in Fig. 3b, as a function of frequency, for all of the 200 simulated sources. The color scale shows the normalized illumination angle values at the chosen sites for each simulated source. The purple line corresponding to the right y-axis represents the curvature variation with frequency</figcaption></figure>
the available sources included in the dataset, which may not cover all possible strike, dip and rake combinations. Consequently, the focal mechanism’s impact was reduced but not entirely eliminated. Second, in our study, we only considered direct waves as sources of amplification variability. However, the generated surface and diffracted waves also play a role and can interfere with the direct incident waves (Pedersen et al. 1994; Maufroy 2010) to contribute to the overall amplification dynamics. These wave interferences introduce additional variability that cannot be solely explained by the local properties of the free surface, such as the localized parameters of curvature and illumination angle, as they will also depend on the topographic and lithologic irregularities encountered by the waves along their path.

Despite the margin of error, the i-FSC proxy significantly improves amplification estimation results compared to the previous FSC proxy (Maufroy et al. 2015). In Fig. 9a, b and c, we show the amplifications predicted by the i-FSC for the same seismic source positions that were presented in Fig. 4a, b and c, at a frequency of 2 Hz. Compared to the amplifications predicted by the FSC proxy, the illumination parameter provides further insight into hill response dynamics: while the previous FSC proxy always predict an amplification close to 1 (neutral zone) at slope sites with approximately zero curvature (Fig. 9e), the new i-FSC proxy can well capture the amplification variations (as indicated by the circles in Fig. 9b

<figure><img src="image_13.png" /><figcaption>**Fig. 9** Amplification maps at a frequency of 2 Hz ($\lambda_s$=1500 m) predicted by: **a–c** the i-FSC proxy for different seismic source positions indicated by the red stars (amplified circled regions indicate slopes opposing seismic wavefield direction and receiving the highest degrees of illumination); **d** the i-FSC proxy for a vertical incidence and **e** the FSC proxy (Maufroy et al. 2015)</figcaption></figure>
and c) observed through numerical simulations (as indicated by the circles in Fig. 4b and c) at these sites. On the other hand, the i-FSC proxy can also account for summits and valleys variations as it enables predicting de-amplification at the top of the relief and amplification or neutral zone at the bottom (which was also not possible using the FSC proxy).

One should be aware that the effect of illumination is a near-source effect in the sense that incident seismic rays can be inclined and illuminate the surface topography with various seismic illumination angles. In the far-source region, the rays, having a curved path in the Earth’s crust, illuminate the whole region vertically. Note that here the notions of near-source regions differ from the classical near-field, which refer to source-site distances respectively restricted within a range of distances from the fault rupture (Stewart et al. 2002; Li and Xie 2007). Figure 9d shows the amplification map provided by the i-FSC for a distant source (vertical incidence), which agrees with the FSC map (Fig. 9e). The small differences come from the term preceding the curvature in the i-FSC proxy equation, which is no longer linear compared to the FSC proxy, thereby giving slightly better fitting results.

## 5.3 Limitations of the i-FSC proxy

Despite large improvements compared to our previous works on the estimation of the topographic amplification, the i-FSC proxy is elaborated under relatively simplified conditions. Further studies should be considered to validate its applicability for more complex scenarios and with datasets obtained on other topographic and source configurations. Therefore, there are several key points that should be highlighted.

First, our study was carried out using a topographic model with slopes below 40°, and the considered sources are mostly located at depths that do not exceed 5 km compared to the free surface, with a maximum simulated frequency of 5 Hz. For more extreme topographies (e.g. cliffs), the curvature at high frequencies may exceed the range for which the i-FSC proxy was developed, and the illumination may be grazing (the maximum normalized illumination angle within our dataset is ±35°, resulting in a maximum gain on amplification of ±0.4). There are currently few 3D simulations on cliffs, primarily because discretizing, properly representing, and simulation on these sites are quite delicate. Some studies have considered 2D simulations on cliffs to understand the behavior of these pronounced topographies (Paolucci 2002; Pagliaroli et al. 2007), but 3D simulations remain essential in order to avoid underestimating the amplifications. According to Paolucci et al. (2002), 3D simulation results on a cliff in Italy show a significant reduction in amplitude in the “shadow zone” and an increase on the other side; hence, further exploration of this behavior on cliffs would be valuable.

Second, in order to determine the proxy, we used a dataset that considers (1) a homogeneous medium to isolate the topographic site effects, excluding the effects of lithology, and (2) point sources to facilitate the identification of the key parameter governing the illumination phenomena. However, in common earthquake events, there is often, if not always, a superposition between the lithological and the topographic effects (Davis and West 1973; Nechtschein et al. 1995; Assimaki et al. 2005a), and the illumination becomes more complex in the case of an extended source.

Regarding the first issue, in line with Massa et al. (2014), it would be more appropriate to refer to scenarios involving the coexistence of lithology and topography as complex seismic amplification effects, rather than solely topographic site effects. Further studies, such as
the investigation conducted by Wang et al. (2018) on the FSC proxy, should be considered to assess the variations of the i-FSC proxy in heterogeneous topographic media. It is also important to note that in a heterogeneous medium, as the shear velocity increases with depth, the incident wavefield ray path may experience multiple refractions when passing through contrasting impedance layers. In highly heterogeneous soil characterized by significant velocity contrasts between layers, seismic rays will bend towards the vertical, thereby reducing the illumination effect. In this case, it is advisable to calculate the potential changes in incidence angles following Snell’s law, taking into account the shear wave velocity model of the medium before applying the i-FSC proxy. However, it is important to note that thin heterogeneous layers, even with high velocity contrasts, cannot have a significant impact on the illumination angle for considerably larger wavelengths.

Regarding the second issue (the case of an extended seismic source), it involves a geometrical problem that can be addressed by testing different source configurations. This can be achieved by varying the location of the source along the ruptured fault or considering multiple source positions to calculate the range of illuminations instead of relying on a single value. In such cases, it is possible to estimate the average illumination or consider the most critical scenario, which corresponds to the highest degree of slope exposure to the incoming wavefield.

Finally, the FSC and the i-FSC proxies were elaborated by considering the average amplification of the two horizontal components. On the other hand, several researchers have studied the directionality of ground motion (Hong and Goda 2010; Shahi and Baker 2014) and found that a greater amplification can be observed according to a preferential orientation. Studies show that the ground motion over topography is generally polarized in the direction perpendicular to the elongation of the topographic feature (Wood and Cox 2016; Stolte et al. 2017). By averaging both horizontal components, one cannot determine whether one component is more amplified than the other, which results in significantly lower amplification values. However, by following the same approach and assuming the same hypotheses as for the mean component, we were able to determine an equation that predicts the values of the amplification factors for the maximized component, without specifying which component it is (see code in Sect. 7).

# 6 Conclusion

In this study, we developed a comprehensive methodology to investigate the origins of near-source ground motion variability due to topographic site effects. Using neural networks and previously available 3D numerical simulations performed on a topographic relief with a homogeneous medium, we determined the key parameters (related to surface irregularities and relative source position) that control ground motion amplification.

We identified two frequency-scaled parameters: surface curvature (Maufroy et al. 2015) and a newly defined parameter called the normalized illumination angle. This angle helps quantify the slope’s exposure to the incoming wavefield. A site with a positive normalized illumination angle will be more exposed to the incoming wavefield, resulting in higher ground motion amplification. Conversely, a negative value refers to a “shadowed slope” that obstructs the arrival of seismic waves, resulting in de-amplification.
Next, through wavelength-by-wavelength regression analysis, we determined a quantitative estimator of topographic site effects that we call the “i-FSC proxy,” which stands for illuminated Frequency-Scaled Curvature proxy. This proxy is a linear equation that predicts the amplification factors at any given topographic site for an *S*-wavelength, using only a digital elevation map and a seismic source position.

Results show that near-source topographic amplification is highly sensitive to the direction of the incoming wavefield, particularly on slopes. This sensitivity can largely be explained by the i-FSC proxy. Therefore, this estimator allows investigation of amplification variations caused by a nearby seismic source, marking a significant breakthrough. This is crucial because areas closest to faults typically sustain the most damage during earthquakes, often lacking seismological records.

The i-FSC proxy can serve as a valuable tool for investigating the influence of variable topographic amplification on the spatial distribution of earthquake-induced effects in mountainous regions, such as structural damage or co-seismic landslides. Several events have shown that after earthquakes, the spatial distribution of landslides exhibits specific patterns, characterized by significant concentrations along mountain ridges and slopes oriented opposite to the seismic sources (e.g., Kawase and Aki 1990; Meunier et al. 2008; Lacroix et al. 2013, 2016; Xu et al. 2014; Rault et al. 2019, 2020). This suggests the potential contribution of topographic site effects in landslide trigerring, and the i-FSC proxy can provide insights into these observed patterns.

# 7 Code availability

A MATLAB script is provided to perform the i-FSC proxy’s calculations. It is shared publicly on the following link: https://gricad gitlab.univ-grenoble-alpes.fr/i-fsc-proxy-for-topographic-amplification/i-fsc

**Acknowledgments** The authors would like to thank Prof. Domniki Assimaki and Dr. Paola Traversa for their constructive feedback, which significantly enhanced the quality of the manuscript.

**Funding** This work was supported by a doctoral funding from the French Ministry of Research.

# Declarations

**Competing interests** The authors have no relevant financial or non-financial interests to disclose.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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