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Empirical approaches for nonlinear site response: results for the ESG6blind test
Earth, Planets and Space volume 76, Article number: 109 (2024)
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 nonlinear 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 sitespecific parameter related to an average nonlinear 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 sitetosite and eventtoevent. On a given site, the specific threedimensional 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 nonlinear 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 nonlinear soil behavior will have a different signature in earthquake data depending on the type of nonlinearity 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 nonlinear behavior, characterized by a decrease of the highfrequency 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 nonlinear 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 socalled 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., KohrsSansorny et al. 2005). For now however, the nonlinear 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 nearby reference site, two purely empirical approaches that do consider the nonlinear soil behavior have been proposed recently (CastroCruz 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 nonlinear 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 (CastroCruz 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 nonlinear 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.
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 nonlinear soil behavior.
2.2.1 Method 1: Correction for soil nonlinear behavior using RSR_{NL} _{L}
The first method (Method 1) defines a function socalled RSR_{NLL} that is to be multiplied by the TF_{lin} to define the modulus of the nonlinear transfer function between SEVO and KUMA according to Eq. 1. The details of the calculation of RSR_{NLL} are provided in the following section:
\({\text{TF}}_{\text{NL}}, {\text{TF}}_{\text{lin}}\) and RSR_{NLL} are composed of real numbers.
The RSR_{NLL} approach was first introduced by Régnier et al. (2013) to illustrate both the amplitude changes and frequency shift associated with the nonlinear 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 nonlinearity results in a low frequency increase (RSR_{NLL} > 1) for frequencies below a sitespecific 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 KiKnet recordings, to train an artificial neural network (ANN) in order to predict the frequencydependent modulation function RSR_{NLL} as a function of the loading level, characterized by a ground motion intensity measure (GMIM), and one or several sitecondition 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:
where RSR_{NLL,obs,j}(f) represents the j^{th} "observed" RSR_{NLL} at the frequency f as derived in the data set, RSR_{NLL,pred,j}(f) is the neural prediction of the j^{th} RSR_{NLL}, and M is the size of the KiKnet 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_{NLL} is only one of the various models proposed in Derras et al. (2020): it is based on six parameters: five site parameters: Vs_{30} (mean shear Vs over the first 30 m), B_{30} (gradient of the velocity profile over the first 30 m), V_{Smin} (the minimum shear wave velocity), f_{0} (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 nonlinear behavior using fsp
The second method (Method 2) makes use of a parameter socalled 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:
as they modify the amplitude of the Fourier transform between the reference and the target stations.
CastroCruz et al. (2020) utilized a subset of the KiKnet 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
with a summation performed over a frequency window going from 0.3 Hz to 30 Hz.
The final frequency shift parameter, socalled fsp, is defined as the square of Ls (fsp = Ls^{2} when misfit is minimized), and for each site the dependence of fsp with donwhole PGA (PGA_{DH}) is investigated to obtain a sitespecific 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:
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 nonlinear 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.
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 preprocessed 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 bandpassed with a 2poles Butterworth filter between 0.1 and 30 Hz. Velocity Fourier spectra were then calculated and smoothed using a KonnoOhmachi 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 socalled 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 eventtoevent 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.
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_{NLL} 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 f_{0} (Hz) and associated amplitude A_{0} were obtained from the empirical site response curves and confirmed by the horizontal to vertical spectral ratio at KUMA site.
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/s^{2} for the foreshock and mainshock, respectively. The abovementioned procedure resulted in estimated PGA values at KUMA of 2.36 m/s^{2} for the foreshock and 4.85 m/s^{2} 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/s^{2} 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/s^{2}.
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/s^{2} at the EW and NS components and 5.75 and 5.09 m/s^{2} 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_{NLL}, we used the site parameters defined in Table 2: Vs_{30}, B_{30}, f_{0}, Vs_{min} and A0_{H/V} the estimated PGA value at KUMA that are 4.12 m/s^{2} for the foreshock and 4.85 m/s^{2} 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_{NLL} curves exhibit a high degree of similarity for both occurrences.
The set of parameters (Vs30, B_{30}, f_{0} 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 deamplification 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_{0}_{HV}, and such a derivation is associated with a large uncertainty.
3.2.2 Correction from method 2
Following the procedure described in CastroCruz 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 physicsdriven 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 datadriven, 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/s^{2} for the NS component and 0.22 m/s^{2} 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 (postprediction) 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 nonlinear 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.
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 nonlinear 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 Pwave delay for the foreshock and mainshock. Figure 1, the (manually picked) delay of Pwave 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 nonlinear 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).
For the foreshock, the prediction of the second method is exactly equal to the linear prediction because no nonlinear 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 nonlinear 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:
with \({\text{FAS}}_{\text{m}}\left({f}_{i}\right)\) the predicted absolute Fourier spectra (linear, nonlinear 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.
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.
4.3 Transfer function
Figure 8 illustrates the transfer functions from the blind linear prediction and nonlinear 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).
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 nonlinear 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 nonlinear 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 nonlinear 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 nonlinear 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 nonlinear 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 lowpass 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 nonlinear 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 nonlinear 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" nonlinear 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 nonnegligible underestimation of lowfrequency 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 nonlinear 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 (bandpass 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 timewindows 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/s^{2} for the foreshock and 4.85 m/s^{2} for the mainshock, turn out to be consistent with finally obtained, as Fig. 8 shows PGA values of 3.9 and 4.9 m/s^{2} for the time series predicted by method 1.
4.5 Quantifying the differences with Anderson goodnessoffit criteria
The linear and the two nonlinear predictions can be compared to the observation in a more quantitative way using goodnessoffit 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 goodnessoffit is estimated by Eq. 7, where PGA_{1} is the PGA value predicted by the first or second method and PGA_{2} is the observed value:
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 nonlinear 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 crosscorrelation score is very low, as usual for all waveform in similar prediction exercises. This can be explained by the extreme sensitivity of crosscorrelation to the waveform details. For this event where the nonlinear soil behavior is low all three methods have the same performance.
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 nonlinear 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 nonlinear 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 crosscorrelation (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 nonlinear soil behavior either in a proxybased, average way (M1) or in a morespecific 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 nonlinear 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 nonlinear behavior with a shift of the whole linear transfer function. The two empirical methods are based on observations of effects of nonlinear behavior on site response on a very large number of sites and events. The physical mechanisms of the nonlinear 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 overestimate 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 nonlinear 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 lowfrequency 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 lowfrequency 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 crosscorrelation 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 nonlinear 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.
Availability of data and materials
All data from ESG6 Blind test step 3 (Tsuno et al 2023).
Abbreviations
 NL:

Nonlinear
 TF_{lin} :

Linear outcrop transfer function
 TF_{NL} :

NonLinear outcrop transfer function
 BTF_{lin} :

Linear borehole transfer function
 BTF_{NL} :

NonLinear borehole transfer function
 H/V:

Horizontal to vertical spectral ratio on earthquake recordings
 RSR_{NLL} :

Nonlineartolinear spectral ratio
 GMIM:

Ground motion intensity measure
 SCPs:

Sitecondition proxies
 ANN:

Artificial neural network
 Vs30:

Timeaveraged shearwave velocity in the top 30 m
 B_{30} :

Index of the velocity gradient over the top 30 m
 Vs_{min} :

Minimum shear wave velocity
 f _{0} :

Fundamental resonance frequency picked on H/V ratio
 A0_{HV} :

Amplitude of the H/V at f_{0}
 PGA:

Peak Ground Acceleration
 Fsp:

Frequency shift parameter
 GOF:

Goodnessoffit
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Acknowledgements
The authors would like to acknowledge all the ESG6 benchmark organizing team.
Funding
The author would like to thank the French Accelerometric Network (RAP/ Resif) that funded the working group GammaG on observation of nonlinear soil behavior.
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Julie Régnier is the main author of this article, she performed all the figures and all calculations of the Fourier spectra and acceleration time histories at KUMA site prediction based on the prediction of RSR_{NLL} and fsp provided by Boumédiène Derras and David CastroCruz. Boumédiene Derras performed the prediction of RSR_{NLL} and has actively participated to the prediction calculations, to all the discussions on the results and to the redaction of the paper. David CastroCruz performed the prediction of fsp and has actively participated to the prediction calculations, to all the discussions on the results and to the redaction of the paper. PierreYves Bard, has actively participated to the prediction calculations, to all the discussions on the results and to the redaction of the paper. Etienne Bertrand, has actively participated to the prediction calculations, to all the discussions on the results and to the redaction of the paper.
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Régnier, J., Bard, PY., CastroCruz, D. et al. Empirical approaches for nonlinear site response: results for the ESG6blind test. Earth Planets Space 76, 109 (2024). https://doi.org/10.1186/s4062302402048x
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DOI: https://doi.org/10.1186/s4062302402048x