Empirical approaches for non-linear site response: results for the ESG6-blind test

Abstract

As a contribution to step 3 of the ESG6 blind prediction exercise, we present an application of two different, purely empirical approaches to estimate the strong ground motion at a soft site ("KUMA") from the observed ground motion at a reference rock site ("SEVO") for the two largest shocks of the Kumamoto 2016 sequence. The two methods estimate the non-linear transfer function between a reference rock and a sedimentary site by modifying the linear transfer function derived from weak motion recordings. The modification is based either on a machine learning tool based on a wide collection of Japanese weak and strong motion recordings and the associated site metadata (method 1), or on an estimate of a site-specific parameter related to an average non-linear site response (method 2). The acceleration time series are then derived at the sedimentary site of interest using an estimation of the time delay between wave arrivals at the rock and site stations, and a minimum phase assumption for the site transfer function. These predictions were made blindly, but after the ESG6 conference they could be compared both with the actual ground motion recorded at KUMA during the two shocks, and the average and range of all other predictions preformed for this benchmark. Both of these purely empirical methods provide an honorable prediction of usual engineering ground motion parameters of the two target events. The performance of these two purely empirical approaches is at least comparable to those of the numerical simulation methods for the foreshock—if not better—and slightly worse for the (largest) mainshock. As the methods required only recordings of weak motions at the target and a referent sites and very simple description of the soil profile. The use of moderate motions to constrain the frequency shift prediction for the second method and the consideration of an alternative phase modification are possible ways to improvement.

Graphical Abstract

1 Introduction

The prediction of surface strong ground motions involves many complex contributions beginning with the seismic source and ending with the propagation of the seismic waves through the very last subsurface soil layers. Close to the surface, the seismic waves can be trapped in soft layers amplifying the surface motion compared to rock outcrop sites. These site effects differ from site-to-site and event-to-event. On a given site, the specific three-dimensional site geometry associated with the variability of the incidence of the seismic wavefield can create variability in the site response even for motions having similar amplitudes. Besides, it has been repeatedly shown that the non-linear soil behavior can significantly modify the site response depending on the incident motion intensity (Beresnev and Wen 1996; Elgamal et al. 2001; Pender 1992).

Since then, it has been shown that the non-linear soil behavior will have a different signature in earthquake data depending on the type of non-linearity involved. For instance, Bonilla et al. (2005) following the works of Archuleta et al. (2000), Iai et al. (1995), Yu et al. (1993), Zeghal and Elgamal (1994) showed that saturated dilatant soil can generate spikes in the recorded acceleration due to cyclic mobility and pore water pressure increase during the strong shaking. Others showed more classical effects of non-linear behavior, characterized by a decrease of the high-frequency amplitude and a shift of resonance frequencies to lower values due to the simultaneous shear modulus decrease and damping increase with shear strain (beginning with (Tokimatsu and Midorikawa 1981)). Some researchers also emphasize an increase of amplification at low frequency (Bonilla et al. 2011; Kawase 2006; Pavlenko and Irikura 2006; Régnier et al. 2013, 2016). The non-linear soil behavior may thus have significant effects on site response and therefore on the acceleration time histories at the surface and should be accounted for in strong ground motion predictions.

Several approaches can be used in that aim. Numerical simulation of wave propagation through the soft layers requires the definition and the calibration of a soil model including a constitutive soil model, which can be challenging to define. When earthquake recordings at the prediction site are available, the use of empirical approaches based on these recordings may be considered, such as for instance the so-called empirical Green's function (EGF) technique, which can be used either to provide some constraints on source location and mechanism after a strong event (Irikura 1986), or to predict strong ground motions for future events (e.g., Kohrs-Sansorny et al. 2005). For now however, the non-linear soil behavior is not accounted for in classical EGF approaches as only the source component is modified. When recordings of previous earthquakes are simultaneously available at the prediction site and at a near-by reference site, two purely empirical approaches that do consider the non-linear soil behavior have been proposed recently (Castro-Cruz et al. 2020; Derras et al. 2020). The purpose of this paper is to test the performance of these last two empirical approaches within the framework of the ESG6 blind prediction case (Chimoto et al. 2023; Matsushima et al. 2024; Tsuno et al. 2023). This exercise consists in predicting the (known but initially hidden) strong ground motion at an instrumented soft site "KUMA" for the two largest events of the Kumamoto 2016 sequence (a Mj6.5 foreshock and the Mj7.3 mainshock), on the basis of known recordings at a relatively close (8 km distant) reference rock site "SEVO" and known pairs of recordings at both sites for a series of small magnitude aftershocks.

After having briefly presented the principle of the two methods, the required data and correction functions are calculated based on the data provided during the blind prediction exercise. At this stage, a first comparison of the predicted correction functions of nonlinear behavior with the observation is made. Then, the results are compared not only to the real recordings at the KUMA site, but also to the results of a similar, purely empirical approach with a linear behavior assumption, and to the range of blind predictions provided by the other participants of the benchmark. This comparison is performed, for the two target events, in terms of relative transfer function between SEVO and KUMA, and surface motion at KUMA, both in the frequency (Fourier amplitude spectra FAS) and time domains (acceleration time histories), including the analysis of goodness of fit criteria as proposed by (Anderson 2004).

2 Methods

We are using two purely empirical approaches to predict the surface acceleration time histories and the associated Fourier spectra at KUMA site for both the foreshock and the mainshock of the Kumamoto 2016 earthquake. The two methods correct the linear transfer function for non-linear soil behavior that impacts the amplitude and frequency of the transfer function. However, the methods involve different ways of correcting the linear transfer function and also require different input parameters. The first method (Derras et al. 2020) involves a correction of the amplitude of the transfer function, while the second one involves a shift of the transfer function towards lower frequency bandwidth (Castro-Cruz et al. 2020). The first method requires the definition of site parameters based on the 1D shear wave velocity profile and linear site response, as well as the expected PGA at the site, while the second method requires the expected PGA at a reference site and the analysis of previous ground motions recorded at the site and at a reference site.

The process, for the application of two methods, can be divided into three steps: Step 1: Calculation of the linear transfer function (TF_lin) between the sedimentary (here KUMA) and the rock (SEVO) sites. Step 2: The nonlinear transfer function (TF_NL) is calculated by adjusting the linear transfer function to account for non-linear soil behavior. Step 3: Calculation of the Fourier spectrum and acceleration time history at the sedimentary site.

2.1 Step 1: Calculation of the linear transfer function (TF_lin)

The linear transfer function (TF_lin) can be obtained from weak ground motions recordings or numerical simulations and can be borehole or outcrop transfer function. In this paper focusing on purely empirical approaches, we use the available pairs of weak motions, aftershock recordings provided by the organization team at the target (KUMA) and reference (SEVO) sites: the location of the stations and epicenters of seismic events are illustrated in Fig. 1, which indicates a relative large distance between the two sites (around 8 km), comparable to the epicentral distance at KUMA from several closer aftershocks. This results in an increased variability for the instrumental TF_lin defined as the FAS_KUMA/FAS_SEVO ratio, because of the variability in geometrical spreading terms from one earthquake to another.

Fig. 1

Location of the aftershocks, foreshock and mainshock considered in the ESG6 Kumamoto blind prediction experiment, together with the locations of the SEVO and KUMA sites. Some of the events listed in Table 1 (# 47, 69 and 83) are not displayed because they are too far from the sites

2.2 Step 2: Calculation of the nonlinear transfer function TF_NL

Two methods are applied to calculate TF_NL. It is obtained by correcting TF_lin for the non-linear soil behavior.

2.2.1 Method 1: Correction for soil non-linear behavior using RSR_{NL -L}

The first method (Method 1) defines a function so-called RSR_NL-L that is to be multiplied by the TF_lin to define the modulus of the non-linear transfer function between SEVO and KUMA according to Eq. 1. The details of the calculation of RSR_NL-L are provided in the following section:

$${\text{TF}}_{\text{NL}}\left(f\right)={\text{TF}}_{\text{lin}}\left(f\right).\,{\text{RSR}}_{\text{NL-L}}(f)$$

(1)

${\text{TF}}_{\text{NL}}, {\text{TF}}_{\text{lin}}$ and RSR_NL-L are composed of real numbers.

The RSR_NL-L approach was first introduced by Régnier et al. (2013) to illustrate both the amplitude changes and frequency shift associated with the non-linear soil behavior, and later developed in a more quantitative way by Régnier et al. (2016) and Derras et al. (2020). Typically, the frequency shift due to the shear modulus reduction induced by the soil non-linearity results in a low frequency increase (RSR_NL-L > 1) for frequencies below a site-specific frequency f_NL related either to the fundamental frequency or to a higher harmonics, and a high frequency decrease associated with increased damping.

Derras et al. (2020) then used a large number of selected, large amplitude KiK-net recordings, to train an artificial neural network (ANN) in order to predict the frequency-dependent modulation function RSR_NL-L as a function of the loading level, characterized by a ground motion intensity measure (GMIM), and one or several site-condition proxies (SCPs) characterizing the soil column. More details upon the structure and the design of ANN can be found in Derras et al. (2020).

The performances of the ANN results are measured by the standard deviation of residuals between observations and model predictions (σ_res), compared to the standard deviation of the original data set (σ_obs) through the variance reduction coefficient Rc (Derras et al. 2020) as defined in the following equation:

$${R}_{c}\left(f\right)=1-\frac{{\sigma }_{res}^{2}\left(f\right)}{{\sigma }_{obs}^{2}\left(f\right)}=\left[1-\frac{\sum_{j=1}^{M}{\left({log}_{10}\left(\frac{{\text{RSR}}_{\text{NL-L,obs,}j}\left(f\right)}{{\text{RSR}}_{\text{NL-L,pred},j}\left(f\right)} \right)\right)}^{2}}{\sum_{j=1}^{M}{\left({log}_{10}\left(\frac{{\text{RSR}}_{\text{NL-L,obs},j}\left(f\right)}{mean\left({\text{RSR}}_{\text{NL-L,obs},j}\left(f\right)\right)} \right)\right)}^{2}}\right].100$$

(2)

where RSR_NL-L,obs,j(f) represents the j^th "observed" RSR_NL-L at the frequency f as derived in the data set, RSR_NL-L,pred,j(f) is the neural prediction of the j^th RSR_NL-L, and M is the size of the KiK-net data set used for developing the ANN model. This variance reduction was analyzed both in the normalized frequency domain f/f_NL as defined in (Régnier et al. 2016), and the absolute frequency domain for the final model: only absolute frequency results are used in the present application. Similarly, the ANN model chosen here to predict RSR_NL-L is only one of the various models proposed in Derras et al. (2020): it is based on six parameters: five site parameters: Vs₃₀ (mean shear Vs over the first 30 m), B₃₀ (gradient of the velocity profile over the first 30 m), V_Smin (the minimum shear wave velocity), f₀ (the fundamental resonance frequency, picked on H/V ratio), and A_0HV (the corresponding peak H/V amplitude), and on one loading parameter characterizing the intensity of ground shaking at KUMA site, here PGA.

2.2.2 Method 2: Correction for soil non-linear behavior using fsp

The second method (Method 2) makes use of a parameter so-called fsp (frequency shift parameter), which shifts the linear transfer function in the frequency domain according to Eq. 3. The detail of the calculation of fsp parameter is discussed in the following section:

$${\text{TF}}_{\text{NL}}(f)={\text{TF}}_{lin}\left(f.\sqrt{fsp}\right)$$

(3)

as they modify the amplitude of the Fourier transform between the reference and the target stations.

Castro-Cruz et al. (2020) utilized a subset of the KiK-net database to investigate the feasibility and effectiveness of approximating NL behavior through a simple frequency scaling method, as indicated in Eq. 4. They compared the linear Borehole transfer function (${\text{BTF}}_{\text{Lin}}$), estimated as the arithmetic average of all of the weak motion borehole transfer functions, with ${\text{BTF}}_{\text{NL}}$ derived from stronger motions recorded at the same site. This comparison aimed to define a (loading dependent) parameter that characterizes the frequency shift associated with the shear modulus reduction. A frequency scaling parameter, Ls is tuned to minimize the misfit between BTF_lin (f.Ls) and BTF_NL(f), as defined by

$$misfit=\sum_{i}|{\text{BTF}}_{\text{lin}}\left(\overline{f }.Ls\right)-{\text{BTF}}_{\text{NL}}(\overline{f })|\Delta x$$

$$\Delta x={\text{log}}_{10}\left(\frac{{f}_{i+1}}{{f}_{i}}\right)$$

$$\overline{f }=0.5({f}_{i+1}+{f}_{i})$$

(4)

with a summation performed over a frequency window going from 0.3 Hz to 30 Hz.

The final frequency shift parameter, so-called fsp, is defined as the square of Ls (fsp = Ls² when misfit is minimized), and for each site the dependence of fsp with donwhole PGA (PGA_DH) is investigated to obtain a site-specific prediction curve fsp.

2.3 Step 3: Calculation of the Fourier spectrum and acceleration time history at the sedimentary site

The Fourier Amplitude Spectrum at the surface of KUMA site (FAS_KUMA) is calculated as indicated in the following equation:

$${\text{FAS}}_{\text{KUMA}}\left(f\right)={TF}_{NL}\left(f\right).{\text{FAS}}_{\text{SEVO}}\left(f\right)$$

(5)

where FAS is the Fourier amplitude spectrum, the absolute values of the Fourier transform of the acceleration time history and TF_NL is the estimated non-linear transfer function between the reference (SEVO) and the target (KUMA) sites. FAS and TF_NL are composed of real numbers.

The acceleration time history at KUMA is obtained by inverse Fourier transform of the product of the Fourier spectrum at SEVO composed of the amplitude (FAS_SEVO) and phase (${\phi }_{SEVO}$) with the complex nonlinear transfer function. To consider the impact of the site amplification on the phase, we proceed in the same way as proposed in (Brax et al. 2018; Grendas et al. 2022; Sèbe et al. 2018). Taking advantage of the findings for vertically propagating waves in a horizontally stratified absorbing media, transfer functions are assumed to fulfill a minimum phase assumption, the corresponding phase term is called ${\phi }_{min}$. We followed the procedure described in (Pei and Lin 2006) that consists in taking the real part of the cepstrum (i.e., real part of the logarithm of the transfer function) and reconstructing a causal sequence by multiplying its inverse Fourier transform by a step function with values equal to 0 for negative times and 2 for positive times. The time domain, minimum phase finite impulse response can then be retrieved from this cepstrum approach, and its Fourier transform is the complex transfer function with the right minimum phase.

In addition, a time delay is added in the phase to consider the difference in seismic waves travel times between SEVO and KUMA, the corresponding phase term is called ${\phi }_{delay}$. For comparison purposes, we also derived in the same way (i.e., minimum phase assumption and delay) a linear complex transfer function from TF_lin to predict the Fourier spectra and acceleration time histories at the surface of KUMA when assuming a linear site response.

The flowchart illustrated in Fig. 2 synthesizes the principle of application of the two methods to any sedimentary site. In this figure, the blue annotations refer to the weak motion recordings, the red ones for the strong motion recordings and the green annotations refer to the parameters that characterize the sedimentary site.

Fig. 2

Flowchart in three steps depicting the application of two methods applied in this paper to empirically evaluated the Fourier spectrum (FAS_site in red) and the acceleration time history (A_site in red) of a strong event at a sedimentary site, from the weak motion recordings at the sedimentary and at referent rock sites (A_rock and A_site in blue), the recording of the strong event at the referent rock site (A_rock in red) and site parameters in green. The blue annotations correspond to the values for weak motions and the red ones for the strong motion. The green annotations refer to the parameters that characterize the sedimentary site

3 Application to ESG6 blind test

The linear transfer function TF_lin and the modified TF_NL are provided independently for the two horizontal components.

3.1 Step 1: Calculation of the linear transfer function (TF_Lin)

TF_lin is the geometrical average of spectral ratio between the smoothed Fourier spectra of weak motion recordings of the velocity at KUMA and SEVO and is obtained separately for the NS and EW horizontal components.

The available data from all aftershocks listed in Table 1 are indeed accelerometric recordings at KUMA, and velocity recordings at SEVO. Therefore, before their integration to obtain velocity time histories, the KUMA acceleration time series were pre-processed following the recommendations of (Boore and Bommer 2005). After classical mean removal and tapering, they were padded with zeros before and after the recording, and band-passed with a 2-poles Butterworth filter between 0.1 and 30 Hz. Velocity Fourier spectra were then calculated and smoothed using a Konno-Ohmachi smoothing (Konno and Ohmachi 1998) with a parameter of 60: this large value was selected to minimize the decrease of the amplitude due to the smoothing. The Fourier spectral ratio between KUMA and SEVO was then computed for each component and all seismic events, and their average TF_lin and standard deviation were then derived considering a lognormal distribution. The averaged horizontal components to vertical component spectral ratio of the aftershocks is also calculated at KUMA site and so-called HV_Lin. The results are illustrated in Fig. 3. The transfer functions from both components provide similar a first frequency peak around 0.3 Hz and then a broadband amplification from 1.3 till 7 Hz. The average H/V spectral ratio performed with the recordings at KUMA sites confirms that the fundamental resonance frequency of the site is at 0.3 Hz. and indicates a secondary peak around 0.8 Hz. One may notice the rather large event-to-event variability of the linear transfer function. Part of it is probably due to the significant (8 km) distance between reference and target sites, often comparable to the epicentral distance. As a result, the effects of the radiation pattern and crustal propagation may not be similar at both sites. This additional source of variability may lead to broader and lower amplitude peaks in the average transfer function. For instance, at the predominant frequency (1.3 Hz), the average amplitude is around 8.2 while the 16 percentiles (− 1 $\sigma$) is at 5.2 and the 84 percentiles (+ 1 $\sigma$) is at 16.6 for the east–west component.

Table 1 Characteristics of the recorded earthquakes at SEVO and KUMA

Fig. 3

Left subplot shows the spectral ratios between KUMA and SEVO velocity time histories from the aftershocks with geometrical average TF_lin and average ± 1 standard deviation and the average H/V spectral ratio at KUMA site for the NS component and right subplot for the EW component

3.2 Step2: Calculation of the nonlinear transfer function TF_NL

3.2.1 Application of method 1

The first method requires the definition of the RSR_NL-L that depend on site parameters (SCPs) and input motion parameter that is the estimated PGA at KUMA for the foreshock and the main event.

The site parameter values listed in Table 2 were taken from the preferred soil model proposed by the organization of the ESG6 benchmark (Matsushima et al. 2024). The fundamental resonance frequency f₀ (Hz) and associated amplitude A₀ were obtained from the empirical site response curves and confirmed by the horizontal to vertical spectral ratio at KUMA site.

Table 2 KUMA site-condition parameters (SCPs) used for the prediction of RSR_NL-L

Method 1 uses the PGA value at the target site as the intensity measure to tune the loading level. As such PGA values are unknown in a blind prediction for both the foreshock and mainshock, they were estimated using a linear regression between the log10 of PGA_KUMA/PGA_SEVO (for all available aftershocks) modulated by the ratio between epicentral distance at KUMA and SEVO (Depi_KUMA/Depi_SEVO), and the measured PGA values at SEVO during the foreshock and mainshock. As the mainshock recording at SEVO exhibits a large spike at 23.6 s, we measured the PGA before that spike, following the indications by Tsuno et al. (2017) that the station probably went out of power. The average PGA over the two horizontal components at SEVO are thus 1.07 and 1.96 m/s² for the foreshock and mainshock, respectively. The above-mentioned procedure resulted in estimated PGA values at KUMA of 2.36 m/s² for the foreshock and 4.85 m/s² for the mainshock. We used these values in a first iteration of prediction of the Fourier spectra at KUMA sites. For the foreshock it leads to a PGA value of 4.12 m/s² that is much higher compare to the estimation from the linear regression between the log10 of PGA_KUMA/PGA_SEVO. Therefore, we decided to update the PGA estimated at KUMA for the foreshock only and perform a second iteration of calculation using 4.12 m/s².

Comparing these blinded predicted values with the actual recordings at KUMA site, we found that the predicted PGAs are very close to the recorded ones despite a slight underestimation for the mainshock. For the foreshock the recorded PGA is 4.42 and 3.89 m/s² at the EW and NS components and 5.75 and 5.09 m/s² at the EW and NS components for the mainshock, which represents a discrepancy lower 10% in both cases. It also means that nonlinearities may have been slightly underestimated for the mainshock.

To predict the correction factor for the nonlinearity, that is the RSR_NL-L, we used the site parameters defined in Table 2: Vs₃₀, B₃₀, f₀, Vs_min and A0_H/V the estimated PGA value at KUMA that are 4.12 m/s² for the foreshock and 4.85 m/s² for the mainshock. The curves obtained for the foreshock and mainshock are illustrated in Fig. 4. Considering that the estimated PGAs are similar for both events, the RSR_NL-L curves exhibit a high degree of similarity for both occurrences.

Fig. 4

Displays the so-obtained RSR_NL-L curves for the foreshock and the mainshock. Both curves are very close since the predicted PGA at KUMA are very close for both events. Below 1.2 Hz, the non-linear soil behavior results in a slight increase of the amplification (around 10%), and above this frequency, in a significant decrease that exceeds 40% between 10 and 20 Hz. As expected, the impact of non-linear soil behavior is slightly more important for the mainshock (dashed line)

The set of parameters (Vs30, B₃₀, f₀ and PGA) is not the one that performs the best in terms of variance reduction, but it is a good compromise between ease of use (knowing the velocity profile) and model performance: the corresponding standard deviation reduction (Rc) is 18%.

The use of the additional parameter f_NL, that is a frequency from which we observe a de-amplification in the site response, would have increased Rc to 25%. However, in this case, considering that no other large events were recorded at KUMA and SEVO sites, it would have been necessary to infer a value for f_NL from f_0HV, and such a derivation is associated with a large uncertainty.

3.2.2 Correction from method 2

Following the procedure described in Castro-Cruz et al. (2020), the frequency shift parameter was derived for each TF with respect TF_lin. The values of fsp for each horizontal component are summarized in Table 3, for all the aftershocks, the foreshock and the mainshock. The fsp is then plotted as a function of ground motion intensity, here characterized by the PGA at SEVO (see Fig. 5), and fsp is predicted for the two large events were blindly predicted by using a physics-driven correlation (hyperbolic model) between fsp and PGA at SEVO. In Fig. 5, the gray markers represent the fsp and PGA of the aftershocks while the two red markers represent the two target events. The blind fsp prediction is based on the weak aftershock data only and is illustrated by the blue lines (solid for the mean and dashed for the associated uncertainty). The uncertainty is very large because the fit is only data-driven, and in the present case, the training data set consists only of the weak aftershocks provided by the organizing team, which are characterized by low PGA values (below 0.15 m/s² for the NS component and 0.22 m/s² for the EW component): extrapolating beyond these values is highly uncertain. For the NS component, no trend can be derived, and the predicted fsp curve is equal to one, corresponding to a fully linear soil behavior. For the EW component, although the uncertainties are very high, the hyperbolic fit predicts an average fsp value of 0.95 for the foreshock and 0.91 for the mainshock. The (post-prediction) measured fsp values are in the range 0.6–0.86 for the foreshock, and 0.54–0.65 for the mainshock (NS and EW, respectively). Given the set of initially available recordings, this method significantly underpredicts the non-linear shift of the transfer function for both events. For comparison purposes, we also add to this figure the prediction of fsp curves that would be made with the two target events (red curves). We can observe that the prediction is very different from the blind prediction and that the uncertainties are much smaller, indicating that a future prediction of even greater seismic motion could now be made at the KUMA site with greater accuracy.

Table 3 Values of frequency shift parameter (fsp) for all events relative to the linear transfer function (TF_lin)

Fig. 5

fsp curves for the two horizontals directions, NS, EW. The blue curves represent blind predictions generated from the aftershock recordings used in the forecast. The red curves are computed using the foreshock and mainshock fsp values calculated once the surface acceleration data were provided. These red curves are presented for comparison purposes only and were not utilized in the prediction

3.3 Step 3: Calculation of the Fourier spectrum and acceleration time history at the sedimentary site

Using the NL correction of the linear transfer function, the Fourier spectra of the two events can be predicted at KUMA site by multiplying the Fourier spectra of the SEVO recordings by the predicted transfer functions. For each event, we obtain three predictions: a "reference", linear one using the linear transfer function for comparison purposes, and two non-linear using the two NL corrections provided by methods 1 and 2.

The next step is to predict the acceleration time histories. In that aim, a phase spectrum needs to be associated with the modulus of the so obtained Fourier spectra. This phase spectrum has three components: the first one is the phase spectrum corresponding to the observed motion at SEVO during the target event, the second one corresponds to the phase shift associated with the difference in travel time, and the third is the phase assigned to the estimated transfer function TF_NL through the minimum phase assumption.

To account for the arrival time differences between KUMA and SEVO, we use the provided aftershocks to estimate the P-wave delay for the foreshock and mainshock. Figure 1, the (manually picked) delay of P-wave arrival, is illustrated with the grey color scale. The delays depend on the location of the epicenters with respect to the location of KUMA and SEVO stations. The difference of epicentral distances (and consequently the delay) decreases when the epicenters are further from the KUMA/SEVO alignment. The closest aftershocks to the target events are events #48 for the foreshock, and # 71 for the mainshock, and the corresponding delay values are applied, i.e., − 0.98 s for the foreshock and − 0.87 s for the mainshock, as listed in Table 1.

We multiply the obtain complex transfer function by the Fourier spectra recorded at SEVO for both events and use an inverse transfer function to recover the predicted acceleration time histories at KUMA. Then, in time domain, we corrected for the arrival time delay. As for the Fourier spectra, three predictions are obtained: one linear and two non-linear using method 1 and method 2.

4 Results

4.1 Fourier spectra

Figure 6 displays a comparison of our predictions to the observed Fourier spectra at KUMA for both target events, together with the range of all predictions from the ESG6 blind test participants (average ± 1 standard deviation (Tsuno et al 2023).

Fig. 6

Fourier spectra at the surface of the observed event (Obs event) for foreshock (Ev1) and for the mainshock (Ev2) and for both horizontal components compared with the prediction without correction of NL behavior (Lin), with corrections of NL behavior by the two methods (NL-M1 and NL-M2) and with the overall predictions coming from the step 3 of the ESG6 benchmark (All Pred $\pm 1\upsigma )$

For the foreshock, the prediction of the second method is exactly equal to the linear prediction because no non-linear soil behavior is predicted for both components (fsp equal to one, see Fig. 5). On the other hand, the correction associated with the first method leads to slightly larger FAS at low frequencies (f < 1 Hz) and reduced one at high frequencies (more than 30% reduction for f ≥ 5 Hz). The observed Fourier spectrum is significantly different from our predictions only in a limited frequency band (0.4–0.5 Hz), but is comparable for all other frequencies, especially for method 1 predictions. Regarding the whole set of predictions, they generally underestimate the actual motion, since observed FAS are close to the average + 1 standard deviation of all predictions. Similar results hold also for the mainshock, with slight differences in the low frequency domain: unpredicted large amplitudes occur in the frequency band 0.5–0.6 Hz, while our methods overpredict the observations between 0.15 and 0.35 Hz.

To quantify this comparison, we calculated the mean absolute error (in percentage) (Eq. 6), between the observations and our three predictions (linear and non-linear using methods 1 and 2), as well as the average of all predictions. It is the area between the observation and the prediction curves (in log scale) divided by the area below the observed curve multiplied by 100. It has been calculated over the frequency band 0.1–10 Hz:

$$\text{Err}\left(\%\right)=100\frac{\sum_{1=1}^{N}\left|{\text{log}}_{10}\left(\frac{{\text{FAS}}_{\text{m}}\left({f}_{i}\right)}{{\text{FAS}}_{\text{obs}}\left({f}_{i}\right)}\right){\text{log}}_{10}\left(\frac{{f}_{i}}{{f}_{i+1}}\right)\right|.}{\sum_{i=1}^{N}{\text{log}}_{10}{(\text{FAS}}_{\text{obs}}\left({f}_{i}\right)){\text{log}}_{10}\left(\frac{{f}_{i}}{{f}_{i+1}}\right)}$$

(6)

with ${\text{FAS}}_{\text{m}}\left({f}_{i}\right)$ the predicted absolute Fourier spectra (linear, non-linear methods 1 and 2, or the average of all predictions) at a given frequency ${f}_{\text{i}}$ and ${\text{FAS}}_{\text{obs}}$ the observed absolute Fourier spectra for the foreshock or the mainshock. This error is the same on the transfer function and on the absolute Fourier spectra FAS, because the latter is the product of the predicted transfer function by the same quantity (FAS_SEVO).

The results are synthetized in Table 4. For the foreshock, the method 1 predicts the Fourier spectra with the lowest errors (13.9 and 8.2% for NS and EW components) whereas the method 2 and the linear predictions provide similar results with an error of 15.2 and 9.1% for NS and EW components and the average of all predictions has 15.8 and 13.1% of errors for NS and EW components. For the mainshock, the first method still provides lower percentage of errors (11.7% and 12 for NS and EW components). Then comes the second method with 12.7 and 12.4% for NS and EW components. The average of all predictions gets closer to the observations with 12.9 and 12.5% of error. Finally, the linear prediction with the errors of 13% for both components is the less accurate.

Table 4 Average errors between predictions and observations in percentage for the absolute Fourier spectra (FAS) (Err %)

4.2 Response spectra

We also compared the pseudo response spectra in acceleration in Fig. 7. We can observe clearly the discrepancies between the observation and the predictions at low periods. For the foreshock, the first method provides very good results for both components; the decrease of the amplification at high frequency improve the prediction compared to the linear one. For the mainshock, the NS component is not well captured by any predictions while the EW component is well reproduced between 0.2 s till 0.45 s.

Fig. 7

Pseudoresponse spectra at the surface of the observed event (Obs event) for foreshock (Ev1) and for the mainshock (Ev2) and for both horizontal components compared with the prediction without correction of NL behavior (Lin), with corrections of NL behavior by the two methods (NL-M1 and NL-M2) and with the overall predictions coming from the step 3 of the ESG6 benchmark (All Pred $\pm 1\upsigma )$

4.3 Transfer function

Figure 8 illustrates the transfer functions from the blind linear prediction and non-linear predictions with the two methods (method 1 in blue, method 2 in red) along with the observed transfer function of the foreshock and mainshock (black lines) and the range of all predictions (average ± 1 standard deviation) from the ESG6 blind test participants (Tsuno et al. 2023). Compared to the weak motion transfer function, the first method modifies mainly the amplitude while the second method applies a shift of the resonance frequencies (only on the mainshock).

Fig. 8

Transfer function at the surface of the observed event (Obs event) for foreshock (Ev1) and for the mainshock (Ev2) and for both horizontal components compared with the prediction without correction of NL behavior (Lin), with corrections of NL behavior by the two methods (NL-M1 and NL-M2) and with the overall predictions coming from the step 3 of the ESG6 benchmark (All Pred $\pm 1\upsigma )$

The observed foreshock transfer function shows three consistent peaks on the two components at 0.5, 0.9 and 1.5 Hz, while the weak motion transfer functions shows a main peak at 1.3 Hz and smaller peaks at 0.3 and 0.65 Hz. The main effects of non-linear soil behavior, which are a shift to lower frequency band and a decrease in amplification cannot be clearly observed for this event. The linear and non-linear methods provide a very good estimation of the observed transfer function especially for the EW component. The observed transfer function of the foreshock indicates a high amplification between 1 and 2 Hz for both components, which is well reproduced by both methods, while it corresponds to the upper 84 percentile of all ESG6 predictions—in this frequency band. This may indicate that most predictions slightly overestimated the non-linear behavior of the soil.

The mainshock transfer function is characterized by two high peaks at 0.7 and 1 Hz and by a reduced amplification above 7 Hz. The large amplitude of the first observed peak (compared to the average linear transfer function) is not a classical effect of non-linear soil behavior; however, the peak at 1 Hz could be related to the shift of the main peak of the weak motion transfer function (at 1.3 Hz) and the decrease in amplification above 7Hz could also be attributed to non-linear soil behavior. None of our methods (1 and 2) could reproduce the large low frequency peak. Nevertheless, the agreement is quite satisfactory for method 1 for frequencies between 1.5 and 20 Hz. The agreement definitely drops at high frequencies (beyond 20 Hz), but it may also be due to some data processing issues (such as a low-pass filtering of the accelerograms recordings). For the second method, the predicted shifts (0.95 and 0.91 for the foreshock and mainshock) are too low compared to the observations and consequently overestimate the amplification above 2 Hz.

Both predictions are based on the linear transfer function modified to account for the non-linear soil behavior. Therefore, the discrepancy observed for the mainshock, especially for the EW component and for the large peaks at 0.7 and 1 Hz, can be attributed either to an underestimation of the non-linear soil behavior, or to an underestimation of the linear transfer function, or both. The first method slightly underestimates the observed PGA and the second method underestimates the frequency shift parameter. As the first method is based on a regression over a large number of sites and events, its predictions correspond to "average" non-linear soil characteristics for rather stiff soils of Japan and it might be that the NL behavior of the rather soft soils at KUMA site is significantly departing form such an average. The inability of our methods to reproduce the high amplitude of the two peaks observed at 0.7 and 1 Hz, might also be related to the limited duration of the reference signal provided at SEVO, as the relevant signal is before the spike, i.e., before 20 s, which could lead to a non-negligible underestimation of low-frequency input (some other participants actually used other recordings as input than the SEVO mainshock, truncated recording). As outlined before, the estimate of the linear transfer function is affected by a large uncertainty. Consequently, the average curve may also underestimate the amplitude of the site response at some frequencies and for some events.

In summary, the blind predictions reproduced well the transfer function of the foreshock but the prediction of the mainshock is less accurate especially for the second method. The first method still provides the lowest percentage of error. Both methods seem to underestimate the non-linear soil behavior effects, while the other ESG6 predictions seem to overestimate these effects for the foreshock event. Besides, the average linear transfer function calculated with a large uncertainty may underestimate the amplitude of the site response.

4.4 Acceleration time histories at KUMA site

The time domain predictions (band-pass filtered between 0.1 and 15 Hz with a Butterworth filter of order 2) are displayed, for the EW component only in Fig. 9. We also display the recording at SEVO for both target events. For sake of clarity, we selected two time-windows that show the beginning of the recordings and the main phase (13 to 14.5 s and 15 to 18 s for the foreshock event and 13 to 14.5 s and 17 to 20 s for the mainshock). The signal envelopes are well reproduced by both methods for the two events except for the spurious peak at 22.87 s of the mainshock. Looking at the first selected time window, we can observe that the first arrival of the foreshock is well reproduced, while the predicted arrival is 0.28 s is too early for the mainshock. Inversely, for the strong phase, a delay of 0.44 s appears very clearly on the low frequency wavelet for the foreshock. The comparison of the largest amplitudes indicates that the maximum amplitudes are well reproduced by the first method and are overestimated by the second method. The roughly estimated PGA values used for the method 1 were 4.12 m/s² for the foreshock and 4.85 m/s² for the mainshock, turn out to be consistent with finally obtained, as Fig. 8 shows PGA values of 3.9 and 4.9 m/s² for the time series predicted by method 1.

Fig. 9

Acceleration time histories at the surface of KUMA site and for the EW component by method 1 (blue curve), method 2 (red curve) and using a linear transfer function (dotted black line) and recorded (black line) for the forshock (left subplots) and the mainshock (right subplots) and looking at the whole signal (upper subplots), a zoom at the first arrival middle plots and a zoom at the main amplitude lower plots

4.5 Quantifying the differences with Anderson goodness-of-fit criteria

The linear and the two non-linear predictions can be compared to the observation in a more quantitative way using goodness-of-fit measures (GOF). For this purpose, some of the 10 criteria proposed by Anderson (2004) are considered here and additional parameters are also proposed. For instance, for the PGA parameter, the Anderson goodness-of-fit is estimated by Eq. 7, where PGA₁ is the PGA value predicted by the first or second method and PGA₂ is the observed value:

$$GOF\left({\text{PGA}}_{1},{\text{PGA}}_{2}\right)=10.\text{exp}\left[-{\left(\frac{{\text{PGA}}_{1}-{\text{PGA}}_{2}}{\text{min}\left({\text{PGA}}_{1},{\text{PGA}}_{2}\right)}\right)}^{2}\right]$$

(7)

Given this formula, a GOF score from 8 to 10 is considered an excellent fit, from 6 to 8 a good fit, from 4 to 6 a medium fit and below 4 a poor fit.

In addition to the comparison of the PGA, we also compare the average discrepancy of the acceleration response spectra for 3 ranges of periods. The first range, between 0.1 and 0.5 s reflect the difference between the short periods (at high frequency), the second range between 0.5 and 1.5 s refers to the period close to the predominant resonance frequency (1.3 Hz) and the last range between 2 and 4 s refers to the long period (low frequency below 0.5 Hz, where the fundamental resonance frequency of site occurs). Then, as proposed by Anderson we compare the duration of the strong motion part of the signal (Trifunac and Brady 1975) and the maximal cross correlation and the Cumulative Absolute Velocity (CAV).

The values of the criteria for the linear approach and both non-linear methods (M1 and M2) and for both target events (foreshock (left plots) and the mainshock (right plots)) are displayed in Fig. 10 for the NS and EW components. All spectral parameters (PGA, SA) and the CAV of the North–South and East–West components of the foreshock exhibit very high scores for all methods, while the cross-correlation score is very low, as usual for all waveform in similar prediction exercises. This can be explained by the extreme sensitivity of cross-correlation to the waveform details. For this event where the non-linear soil behavior is low all three methods have the same performance.

Fig. 10

Criteria of goodness of fit between the surface acceleration predicted by the method M1 and the method M2 and the observed acceleration at KUMA Site (A) for the North–South component and (B) for the East–West component

The scores for the prediction of the mainshock are lower except for the PGA and duration parameter (we recall that the very high amplitude at 23.6s was not considered to pick the PGA) but the spectral parameter predictions are still higher than 6 meaning a good fit. For the mainshock, where the non-linear soil behavior is greater, the first method provides slightly higher scores for the spectral acceleration for low period bandwidth (0.03–0.5 s). The better performance of the approaches involving correction of non-linear behavior (M1 and M2) compared to the linear prediction is not obvious in these plots, but overall, the GOF scores are always very good to excellent except for the cross-correlation (as for all other predictions in this blind prediction Tsuno et al., (2023) and all other similar benchmarking exercises, even in the linear domain (e.g., (Maufroy et al., 2015a; Régnier et al., 2018).

5 Conclusions

These two purely empirical approaches provide acceleration time histories predictions accounting for the non-linear soil behavior either in a proxy-based, average way (M1) or in a more-specific way (M2). The first (machine learning) method involves a simple modulation for the linear amplification curve, combining a slight increase at low frequencies, and a significant decrease at high frequencies, reproducing the two main observations of non-linear behavior on site response, i.e., the shift of the resonance frequency towards lower values, and the amplification decrease associated with increase of soil damping with larger strain. The second method emphasizes the first aspect of soil non-linear behavior with a shift of the whole linear transfer function. The two empirical methods are based on observations of effects of non-linear behavior on site response on a very large number of sites and events. The physical mechanisms of the non-linear behavior are complex and highly variable depending on the type of soil, the presence or not of water and also on the incident wave field. Therefore, the results of any type of statistical analysis will result in a median behavior that, for a specific site, may under- or over-estimate the effects when only the average result is considered.

Recordings of the target events (Foreskock and Mainshock) are compared in frequency and time domains with the predictions from these two non-linear empirical approaches, and also with a similar, purely linear approach, and also with the average of all predictions provided by benchmark participants, whenever possible (i.e., only in the frequency domain).

The comparison of the transfer functions indicate that these purely empirical methods perform almost as well as generally more sophisticated, numerical simulation techniques. Their predictions are better for the foreshock, but worse for the mainshock (especially M2), mainly because they miss a large low-frequency amplification. While this might be due to an underestimation of the NL effects, there might be also explanations: the truncation of the reference SEVO mainshock waveform (instrument malfunction) which may have prevented from accounting for the right low-frequency input, or the too large distance between the reference and target sites, which significantly impacts the reliability of these empirical approaches (as observed spectral ratios also include some source and path components).

The comparison in time domain and the quantification of the discrepancy between the observation and the prediction using quantitative parameters indicates that the cross-correlation between the observation and the predictions is low. However, this is the case for all other predictions in this particular benchmark, and for all other benchmarking exercises even in the linear domain (Maufroy et al., 2015b; Régnier et al. 2018). Thus, it should not, be considered as a failure, especially as most of the spectral parameters (PGA, spectral acceleration) and CAV are very well predicted by the first method for the foreshock and well reproduced for the mainshock.

Considering that these purely empirical methods require only recordings of weak motions at the target and reference sites, and a very simple description of the soil profile, their prediction performance can be estimated as fairly good in both the frequency and time domains. However, additional recordings with at least moderate accelerations would undoubtedly improve the prediction of the second method by better constraining the non-linear domain. It is also expected that considering a much closer reference site would improve the prediction by allowing to avoid spurious path and source contaminations.